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[<img[images/png/1-form.png]]A ''1-form'', or ''//cotangent vector//'', $\sf{f}$, is a geometric object that acts on a [[tangent vector]] at a point, $p$, to give a real number. It may be written in terms of the [[coordinate basis 1-forms]] as\n\s[ \sf{f} = f_i \sf{dx^i} \sin T_{p}^{*}M \s]\nIt is a linear operator, and so may be written as a function of a vector or more simply as a [[vector-form contraction|vector-form algebra]] (product),\n\s[ \sf{f}(\sve{v}) = {\sbf i}_{\sve{v}} \sf{f} = \sve{v} \sf{f} = v^j f_i \sve{\spa_j} \sf{dx^i} = v^j f_i \sde_j^i= v^i f_i \sin \sRe \s]\nThe vector space of [[1-form]]s at each point, $p$, of a [[manifold]], $M$, is the ''cotangent space'', $T_{p}^* M$, and is spanned by the $\sf{dx^i}$.\n
A ''2-sphere'' embedded in flat 3-space is a two dimensional [[manifold]], $M$, defined by the equation $x^e x^f \sde_{ef} = r^2$ -- it is the surface of constant $r=r$ in [[spherical coordinates]]. The angular spherical coordinates, $(\sth,\sph)$, cover patches of the manifold (other patches are needed for the poles). The [[metric]] induced on the 2-sphere is $g_{ij} = {\srm diag}(r^2, r^2 \ssin^2(\sth))$. The simplest [[frame]] compatible with this metric is\n$$\n\sf{e} = \sf{d \sth} \s, r \ssi_1 + \sf{d \sph} \s, r \ssin(\sth) \ssi_2\n$$\nin which $\ssi_{1/2}$ are the [[Clifford basis vectors]] for [[Cl(2,0)|Clifford matrix representation]]. The coframe is\n$$\n\sve{e} = \ssi^1 \sfr{1}{r} \sve{\spa_\sth} + \ssi^2 \sfr{1}{r \ssin(\sth)} \sve{\spa_\sph}\n$$\nThe [[torsion]]less [[spin connection]], found by solving [[Cartan's equation]], $0=\sf{d} \sf{e} + \sf{\som} \stimes \sf{e}$, is\n$$\n\sf{\som} = - \sve{e} \stimes \sf{d} \sf{e} + \sfr{1}{4} \slp \sve{e} \stimes \sve{e} \srp \slp \sf{e} \scdot \sf{d} \sf{e} \srp = - \sf{d \sph} \scos(\sth) \ssi_{12}\n$$\nThe [[Clifford vector bundle]] curvature is\n$$\n\sff{F} = \sf{d} \sf{\som} + \sha \sf{\som} \sf{\som} = \sf{d \sth} \sf{d \sph} \ssin(\sth) \ssi_{12}\n$$\nThe [[Clifford-Ricci curvature]] is\n$$\n\sf{R} = \sve{e} \stimes \sff{F} = \sf{d \sph} \sfr{1}{r} \ssin(\sth) \ssi_2 + \sf{d \sth} \sfr{1}{r} \ssi_1 = \sfr{1}{r^2} \sf{e}\n$$\nshowing that the 2-sphere is an [[Einstein space|Einstein's equation]] with cosmological constant $\sLa = 0$ (as do all two dimensional spaces). The [[Clifford curvature scalar]] is $R = \sve{e} \scdot \sf{R} = \sfr{2}{r^2}$.
A ''3-sphere'' embedded in flat 4-space of positive signature is a three dimensional [[manifold]], $M$, defined by the equation $x^w x^x \sde_{wx} = r^2$ -- it is the surface of constant $r=r$ in $4d$ [[hyperspherical coordinates]],\n\sbegin{eqnarray}\nx^1 &=& r \scos(a^1) \s\s\nx^2 &=& r \ssin(a^1) \scos(a^2) \s\s\nx^3 &=& r \ssin(a^1) \ssin(a^2) \scos(a^3) \s\s\nx^4 &=& r \ssin(a^1) \ssin(a^2) \ssin(a^3)\n\send{eqnarray}\nThe angular hyperspherical coordinates, $(a^1,a^2,a^3)$, cover patches of the manifold (other patches are needed for the poles). The [[metric]] induced on the 3-sphere is $g_{ij} = {\srm diag}(r^2, r^2 \ssin^2(a^1),r^2 \ssin^2(a^1) \ssin^2(a^2))$. The simplest [[frame]] compatible with this metric is\n$$\n\sf{e} = \sf{d a^1} \s, r \ssi_1 + \sf{d a^2} \s, r \ssin(a^1) \ssi_2 + \sf{d a^3} \s, r \ssin(a^1) \ssin(a^2) \ssi_3\n$$\nin which $\ssi_{1/2/3}$ are the [[Clifford basis vectors]] for [[Cl(3,0)|Cl(3)]]. The coframe is\n$$\n\sve{e} = \ssi^1 \sfr{1}{r} \sve{\spa_1} + \ssi^2 \sfr{1}{r \ssin(a^1)} \sve{\spa_2} + \ssi^3 \sfr{1}{r \ssin(a^1) \ssin(a^2)} \sve{\spa_3}\n$$\nThe [[torsion]]less [[spin connection]], found by solving [[Cartan's equation]], $0=\sf{d} \sf{e} + \sf{\som} \stimes \sf{e}$, is\n\sbegin{eqnarray}\n\sf{\som} &=& - \sve{e} \stimes \sf{d} \sf{e} + \sfr{1}{4} \slp \sve{e} \stimes \sve{e} \srp \slp \sf{e} \scdot \sf{d} \sf{e} \srp \s\s\n&=& - \sf{d a^2} \scos(a^1) \ssi_{12} - \sf{d a^3} \scos(a^1) \ssin(a^2) \ssi_{13} - \sf{d a^3} \scos(a^2) \ssi_{23}\n\send{eqnarray}\nThe [[Clifford vector bundle]] curvature is\n\sbegin{eqnarray}\n\sff{F} &=& \sf{d} \sf{\som} + \sha \sf{\som} \sf{\som} \s\s\n&=& \sf{d a^1} \sf{d a^2} \ssin(a^1) \ssi_{12} + \sf{d a^1} \sf{d a^3} \ssin(a^1) \ssin(a^2) \ssi_{13} + \sf{d a^2} \sf{d a^3} \ssin^2(a^1) \ssin(a^2) \ssi_{23}\n\send{eqnarray}\nThe [[Clifford-Ricci curvature]] is\n\sbegin{eqnarray}\n\sf{R} &=& \sve{e} \stimes \sff{F} \s\s\n&=& \sf{d a^1} \sfr{2}{r} \ssi_1 + \sf{d a^2} \sfr{2}{r} \ssin(a^1) \ssi_2 + \sf{d a^3} \sfr{2}{r} \ssin(a^1) \ssin(a^2) \ssi_3 \s\s\n&=& \sfr{2}{r^2} \sf{e}\n\send{eqnarray}\nshowing that the 3-sphere is an [[Einstein space|Einstein's equation]] with cosmological constant $\sLa = \sfr{1}{2 r^2}$. The [[Clifford curvature scalar]] is $R = \sve{e} \scdot \sf{R} = \sfr{8}{r^2}$.
<<note HideTags>>$$\n\sbegin{array}{rclclc}\n\sf{\som} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^k} \sha \som_k^{\sp{k}\smu\snu} \sga_{\smu\snu} \s!&\s!\s! \sin \s!\s!&\s! \sf{Cl}^2(3,1)\n&\n\squad\n\sf{e} = \sf{dx^k} (e_k)^\smu \sga_\smu \s, \sin \s, \sf{Cl}^1(3,1) \svp{|_{(}} \s\s\n\n\sf{W} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^k} W_k^{\sp{i}\spi} \sfr{i}{2} \ssi_\spi \s!&\s!\s! \sin \s!\s!&\s! \sf{su}(2)\n&\n\squad\n\slb \smatrix{\n\sph_+ \s\s \sph_0\n} \srb\n\sqquad\n\slb \smatrix{\n\snu_{eL} \s\s e_L\n} \srb\n\s\s\n\n\sf{B} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^k} B_k i \s!&\s!\s! \sin \s!\s!&\s! \sf{u}(1)\n&\n\squad\nY \s\s\n\n\sf{g} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^k} g_k^{\sp{k}A} \sfr{i}{2} \sla_A \s!&\s!\s! \sin \s!\s!&\s! \sf{su}(3)\n&\n\squad\n\slb u^r, u^g, u^b \srb \svp{|^{(^(}_{(}}\n\send{array}\n\sbegin{array}{c}\n\squad\n\slb \smatrix{\ne_L^\swedge \s\s e_L^\svee \s\s e_R^\swedge \s\s e_R^\svee\n} \srb\n\s; \s\s\n\s; \s\s\n\send{array}\n$$\n$$\n\supdownarrow \svp{{\shuge(}_{\sbig(}}\n$$\n$$\n\sbegin{array}{rcl}\n\sudf{A} \s!\s!&\s!\s!=\s!\s!&\s!\s! {\ssmall \sfrac{1}{2}} \sf{\som} + {\ssmall \sfrac{1}{4}} \sf{e} \sph + \sf{W} + \sf{B} + \sf{g} + ( \sud{\snu}{}_e + \sud{e} + \sud{u} + \sud{d} ) \s\s\n&& + \s, (\sud{\snu}{}_\smu + \sud{\smu} + \sud{c} + \sud{s}) + (\sud{\snu}{}_\sta + \sud{\sta} + \sud{t} + \sud{b}) \svp{|_{\sBig(}}\n\send{array}\n$$\n$$\n\sudff{F} = \sf{d} \sudf{A} + {\sscriptsize \sfrac{1}{2}} \sbig[ \sudf{A}, \sudf{A} \sbig]\n$$
!!How did you come up with the title, "Deferential Geometry"?\nMy favorite interpretation is that it's about geometry in the service of physics. There is a lot of bad theoretical physics out there without math, and a lot of good math without physics &mdash; good physics uses math, and this site is about using only the math needed by physics, differential geometry in particular. There shouldn't be any mathematical tangents here without physics ideas motivating them, the geometry is deferential to the physics &mdash; a beautiful idea, and a bad pun.
Modified BF action, using $\sff{\sod{B}} = \sff{B} + \sfff{\sod{B}} \s,$:\n\n\sbegin{eqnarray}\nS &=& \sint \sbig< \sff{\sod{B}} \sudff{F} + \snf{\sPhi} ( \sf{H}{}_1, \sf{H}{}_2, \sff{B} ) \sbig> \s\s\n&=& \sint \sbig< \sfff{\sod{B}} \sf{D} \sud{\sPs}\n+ \sff{B} \sff{F} + {\sscriptsize \sfrac{\spi G}{4}} \sff{B}{}_G \sff{B}{}_G \sga + \sff{B'} \sff{*B'} \sbig> \s\s\n&=& \sint \sbig< \sfff{\sod{B}} \sf{D} \sud{\sPs}\n+ \snf{e} \sfr{1}{16 \spi G} \sph^2 \sbig( R - \sfr{3}{2} \sph^2 \sbig) + \sfr{1}{4} \sff{F'} \sff{*F'} \sbig>\n\send{eqnarray}\n\nCosmological constant from the Higgs VEV: $\squad \sLa = \sfr{3}{4} \sph^2$\n\nImplies frame VEV is de Sitter: $\squad \sff{R} = \sfr{\sLa}{6} \sf{e} \sf{e} \sqquad R = 4 \sLa$\n\nVacuum expectation value of the curvature vanishes: $\squad \sudff{F} = 0$\n<<note HideTags>>
<<note HideTags>>$$\n\sudff{F} = \sf{d} \sudf{A} + \sudf{A} \sudf{A} = \sbig( \sf{d} \sf{H} + \sf{H} \sf{H} \sbig) + \sbig( \sf{d} \sf{G} + \sf{G} \sf{G} \sbig) + \sbig( \sf{d} \sud{\sps} + \sf{H} \sud{\sps} + \sud{\sps} \sf{G} \sbig)\n$$\nModified BF action for everything, using $\sff{\sod{B}} = \sff{B} + \sfff{\sod{B}} \s,$:\n\sbegin{eqnarray}\nS &=& \sint \sbig< \sff{\sod{B}} \sudff{F} + \snf{\sPhi} ( \sf{H}, \sf{G}, \sff{B} ) \sbig> \s\s\n&=& \sint \sbig< \sfff{\sod{B}} \sbig( \sf{d} \sud{\sps} + \sf{H} \sud{\sps} + \sud{\sps} \sf{G} \sbig)\n+ \sff{B} \sff{F} - {\sscriptsize \sfrac{1}{4}} \sff{B_s} \sff{B_s} \sga + \sff{B_{m,h,G}} \sff{*B_{m,h,G}} \sbig>\n\send{eqnarray}\nFermionic part, using [[anti-ghost|BRST technique]] [[Grassmann|Grassmann number]] 3-form, $\sfff{\sod{B}} = \snf{e} \sod{\sps} \sve{e} \s,$:\n\sbegin{eqnarray}\nS_f &=& \sint \sbig< \sfff{\sod{B}} \sbig( \sf{d} \sud{\sps} + \sf{H} \sud{\sps} + \sud{\sps} \sf{G} \sbig) \sbig> \s\s\n&=& \sint \sbig< \snf{e} \sod{\sps} \sve{e} \sbig( \sf{d} \sud{\sps} + {\sscriptsize \sfrac{1}{2}} \sf{\som} \sud{\sps} + {\sscriptsize \sfrac{1}{4}} \sf{e} \sph \sud{\sps} + \sf{B} \sud{\sps} + \sf{W} \sud{\sps} + \sud{\sps} \sf{G} \sbig) \sbig> \s\s\n&=& \sint \snf{d^4 x} |e| \s, \sbig< \sod{\sps} \sga^\smu (e_\smu)^i \sbig( \spa_i \sud{\sps} + {\sscriptsize \sfrac{1}{4}} \som_i^{\sp{i} \smu \snu} \sga_{\smu \snu} \sud{\sps} + B_i \sud{\sps} + W_i \sud{\sps} - \sud{\sps} G_i \sbig) + \sod{\sps} \s, \sph \s, \sud{\sps} \sbig>\n\send{eqnarray}
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author: [[Garrett Lisi]]\narxiv: http://arxiv.org/abs/0711.0770\nlocally: [[AESToE|papers/AESToE.pdf]]\nabstract:\nAll fields of the [[standard model]] and [[gravity|modified BF gravity]] are unified as an [[E8]] [[principal bundle]] [[connection]]. A non-compact real form of the [[E8 Lie algebra|e8]] has [[G2]] and [[F4]] subalgebras which break down to strong [[su(3)]], electroweak [[su(2)]] x u(1), gravitational [[so(3,1)|spacetime]], the [[frame]]-Higgs, and three generations of fermions related by [[triality]]. The interactions and dynamics of these [[1-form]] and [[Grassmann|Grassmann number]] valued parts of an E8 [[superconnection]] are described by the [[curvature]] and action over a four dimensional base [[manifold]].\n\nA [[talk for ILQGS 07]] on 11/13/07 was presented on this paper.\n\nInternet discussion of the paper, in chronological order: \n*[[Backreaction|http://backreaction.blogspot.com/2007/11/theoretically-simple-exception-of.html]]\n*[[Physics Forums|http://www.physicsforums.com/showthread.php?t=196498]]\n*[[The Reference Frame|http://motls.blogspot.com/2007/11/exceptionally-simple-theory-of.html]]\n*[[Hidden Variables|http://blog.domenicdenicola.com/post/2007/11/Criteria-for-a-Theory-of-Everything.aspx]]\n*[[Not Even Wrong|http://www.math.columbia.edu/~woit/wordpress/?p=617]]\n*[[Arcadian Functor|https://www.blogger.com/comment.g?blogID=28857369&postID=5548882952979522971]]\n*[[Freedom of Science|http://globalpioneering.com/wp02/an-exceptionally-simple-theory-of-everything/]]\n*[[Theoreman Egregium|http://egregium.wordpress.com/2007/11/10/physics-needs-independent-thinkers/]]\n*[[Science Forums|http://www.scienceforums.net/forum/showthread.php?t=29522]]\n\nAnd previously:\n*[[This Week's Finds 253]]
<<note HideTags>>Start with $E8$ principal bundle connection and its curvature,\n$$\n\sf{A} = \sf{H} + \sf{\sPs} \sqquad \squad\n\sff{F} = (\sf{d} \sf{H} + \sf{H} \sf{H} + \sf{\sPs} \sf{\sPs})\n+ (\sf{d} \sf{\sPs} + \sf{H} \sf{\sPs} + \sf{\sPs} \sf{H})\n$$\nAction such that $\sf{\sPs}$ part is pure gauge,\n$$\nS = \sint \sbig< \sff{B} \sff{F}\n+ {\sscriptsize \sfrac{\spi G}{4}} \sff{B}{}_G \sff{B}{}_G \sga + \sff{B'} \sff{*B'} \sbig>\n$$\nBRST: Replace $\sf{\sPs}$ part with ghosts, $\sud{\sPs}$, in extended connection, \n$$\n\sudf{A} = \sf{H} + \sud{\sPs} \sqquad \squad\n\sudff{F} = \sbig( \sf{d} \sf{H} + \sf{H} \sf{H} \sbig) + \sbig( \sf{d} \sud{\sPs} + [ \sf{H}, \sud{\sPs} ] \sbig)\n= \sff{F}{}_H + \sf{D} \sud{\sPs}\n$$\nEffective action for gauge fields, ghosts, and anti-ghosts:\n\sbegin{eqnarray}\nS &=& \sint \sbig< \sff{\sod{B}} \sudff{F}\n+ {\sscriptsize \sfrac{\spi G}{4}} \sff{B}{}_G \sff{B}{}_G \sga + \sff{B'} \sff{*B'} \sbig> \s\s\n&=& \sint \sbig< \sfff{\sod{B}} \sf{D} \sud{\sPs}\n+ \snf{e} \sfr{1}{16 \spi G} \sph^2 \sbig( R - \sfr{3}{2} \sph^2 \sbig) + \sfr{1}{4} \sff{F'} \sff{*F'} \sbig>\n\send{eqnarray}
<<note HideTags>>$\sde \snf{L} = 0$ under [[gauge transformation]]: &nbsp;&nbsp; $\sde \sf{A} = - \sf{\sna} C = -\sf{d} C - \sbig[ \sf{A}, C \sbig]$\nAccount for gauge part of $\sf{A}$ by introducing [[Grassmann|Grassmann number]] valued ''ghosts'', $\sud{C} \sin \sud{\srm Lie}(G)_g$, ''anti-ghosts'', $\snf{\sod{B}}$, ''partners'', $\snf{\sla}$, and [[BRST transformation|BRST technique]]:\n$$\n\sbegin{array}{rclcrcl}\n\sud{\sde} \sf{A} &=& - \sf{\sna} \sud{C} & \s;\s;\s;\s;\s;\s;\s;\s;\s; & \sud{\sde} \sud{C} &=& - \sha \sbig[ \sud{C}, \sud{C} \sbig] \s\s\n\sud{\sde} \sff{B} &=& \sbig[ \sff{B}, \sud{C} \sbig] & \s;\s;\s;\s;\s;\s;\s;\s;\s; & \sud{\sde} \snf{\sod{B}} &=& \snf{\sla} \s\s\n\sud{\sde} \snf{\sla} &=& 0 & \s;\s;\s;\s;\s;\s;\s;\s;\s; & & &\n\send{array}\n$$\nThis satisfies $\sud{\sde} \snf{L} = 0_{\sphantom{\sbig(}}$and $\sud{\sde} \sud{\sde} = 0$.\nChoose a ''BRST potential'', $\snf{\sod{\sPs}} = \sbig< \snf{\sod{B}} \sf{A} \sbig>$, to get new Lagrangian:\n$$\n\snf{L'} = \snf{L} + \sud{\sde} \snf{\sod{\sPs}} = \snf{L} + \sbig< \snf{\sla} \sf{A_g} \sbig> + \sbig< \snf{\sod{B}} \sf{\sna} \sud{C} \sbig>\n$$\nBRST partners act as Lagrange multipliers; ''effective Lagrangian'':\n$$\n\snf{L^{\srm eff}} = \snf{L}[\sff{B'},\sf{A'}] + \sbig< \snf{\sod{B}} \sf{\sna'} \sud{C} \sbig>\n$$
//This is a speculative description of the [[BRST technique]] based on conversations with [[Michael Edwards]]//\n\nStart with a [[connection]] 1-form, $\sf{\som}$, defined over the entire space of a [[fiber bundle]], and some fiber bundle section, $\ssi$. The connection field over the base manifold is the pullback of the connection along the section,\n$$\n\sf{A} = \ssi^* \sf{\som}\n$$\nA BRST transformation may be a way of describing how $\sf{A}$ changes under a change of section. (Though I think this is just a gauge transformation, with a funny pair of Grassmann valued parameters.) Consider a vector field,\n$$\n\sve{\sva} = \sva^A(x) \sve{\sxi_A}(p)\n$$\non the entire space, with $\sve{\sxi_A}$ the flow fields corresponding to the group generators, $T_A$. The gauge transformation parameters can be written in terms of a Grassmann valued parameter and Grassmann valued ghost fields as $\sva^A(x)= \sva C^A(x)$. The BRST transformation then is\n$$\n\sf{\sde A} = - \ssi^* L_{\sve{\sva}} \sf{\som} = \sva s \sf{A} = - \sva (\sf{d} C + \sf{A} \stimes C)\n$$\nin which $C=C^A T_A$.\n//Hmm, this seems to give the change in A from flowing $\som$ under $\ssi$...// \n\n\nAnother idea, from Picken. Instead of pulling the [[Ehresmann connection]] back along a section, use the surface [[vector projection]] on the E conn to project to the gauge 1-form on the surface and a ''ghost'' -- a 1-form off of the surface. This ghost is the projection of the [[Maurer-Cartan form]], and its value determines the shape of the section. May be able to connect this with other descriptions, like the one above.\n\n\nNah, none of this is going to work right. Have to work in the space of connections.\n\nvariational bicomplex\n\nRefs:\n*M. Ghiotti\n**[[Gauge fixing and BRST formalism in non-Abelian gauge theories|papers/Ghiotti - Gauge fixing and BRST formalism in non-Abelian gauge theories.pdf]]\n***Excellent new thesis.\n*G. Catren and J. Devoto\n**[[Extended Connection in Yang-Mills Theory|http://arxiv.org/abs/0710.5698]]\n*Bonora and Cotta-Ramusino\n**[[Some Remarks on BRS Transformations, Anomalies and the Cohomology of the Lie Algebra of the Group of Gauge Transformations|papers/1103922136.pdf]]\n***This is one of the first, and probably the best, description of ghosts as 1-forms in the space of connections.\n***${\scal G}$ is group of vertical [[automorphism]]s of $E$, equals group of [[gauge transformation]]s.\n*Stora and Kastler\n**[[A Differential Geometric Setting for BRS Transformations and Anomalies|papers/A Differential Geometric Setting for BRS Transformations and Anomalies.pdf]]\n***detailed exposition.\n***gauge transformation bundle\n***old (hard to read scanned text) but good. lengthy.\n***same gauge trasf as Viallet (below)\n*Viallet\n**[[The Geometry of the Space of Fields in Yang-Mills theory|papers/The Geometry of the Space of Fields in Yang-Mills theory.pdf]]\n***space of fields as bundle, physical fields as base\n***strange definition for group automorphism, which disagrees with mine and Wikipedia's.\n***gauge transf are ''equivariant'' automorphisms, $f(p)=p\sph(p)$, satisfying $f(ph)=f(p)h$ and hence $\sph(ph)=h^- \sph(p) h$.\n*J.W. van Holten\n**[[Aspects of BRST Quantization|papers/JHolten_BRST.pdf]]\n***excellent elementary practical intro\n***ghost <--> M-C form mapping not an identification\n**[[The BRST Complex and the Cohomology of Compact Lie Algebras|papers/The BRST Complex and the Cohomology of Compact Lie Algebras.pdf]]\n***BRST analysis analogous to [[Hodge decomposition]]\n***this paper's content is included in the paper above\n*http://en.wikipedia.org/wiki/BRST_Quantization\n*[[Principal Bundles, Connections and BRST Cohomology|papers/9408003.pdf]]\n**(//read this now//)\n**BRST cohomology in the space of connections\n**mathematically dense, but I'm hacking it so far\n**[[lots of geometric brst papers from spires|http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+c+cmpha,87,589&SKIP=0]]\n*Jeffrey A. Harvey\n**[[TASI 2004 Lectures on Anomalies|papers/0509097.pdf]]\n***for particle physicists\n***ghosts are 1-forms in the space of gauge transformations\n***BRST operator is exterior derivative in this space\n*Barnich, Brandt, and Henneaux\n**[[Local BRST cohomology in gauge theories|papers/0002245.pdf]]\n***these guys are the big shots in the field, but I don't like this antifield approach yet.\n*[[Four-Dimensional Yang-Mills Theory as a Deformation of Topological BF Theory|papers/9705123.pdf]]\n**pretty good and succinct intros, plus treats BF\n**agrees with Viallet's def of equivariant automorphism.\n*Moritsch, Sorella, et al\n**[[Algebraic characterization of gauge anamolies on a nontrivial bundle|papers/9611168.pdf]]\n***nice algebraic treatment, with generalized connection\n**[[Algebraic characterization of the Wess-Zumino consistency conditions in gauge theory|papers/9302136.pdf]]\n***Sorrella's paper introducing $\sde$. Seems to work with jets without calling them that.\n**[[Algebraic structure of gravity in Ashtekar variables|papers/9409046.pdf]]\n***Blaga, using $\sde$.\n*Jim Stasheff\n**[[The (secret?) homological algebra of the Batalin-Vilkovisky approach|papers/9712157.pdf]]\n***abstract mathematical overview (jets) of relations between physics and math structures\n***ghost = Chevelley-Eilenberg generator\n***anti-ghost = Tate generator\n***anti-field = Koszul generator\n*Yang, Lee\n**[[Lie algebra cohomology and group structure of gauge theories|papers/9503204.pdf]]\n***maybe or maybe not useful\n*Kelnhofer\n**[[On the Geometrical Structure of Covariant Anomalies in Yang-Mills Theory|paper/9302012.pdf]]\n***damn this is a (unavoidable) mess\n***universal bundle\n*Thomas Schucker\n**see his book on Amazon for introductory material:\n***[[Differential Geometry, Gauge Theories and Gravity|http://www.amazon.com/Differential-Geometry-Cambridge-Monographs-Mathematical/dp/0521378214/ref=si3_rdr_bb_product/104-9709999-3726336]]\n**[[The Cohomological Construction of Stora's Solutions|papers/The Cohomological Construction of Stora's Solutions.pdf]]\n*Picken: http://www.iop.org/EJ/abstract/0305-4470/19/5/001\n**//(scan this)//\n*J. P. Zwart\n**[[BRST Reduction and Quantization of Constrained Hamiltonian Systems|papers/zwart98brst.pdf]]\n*Witten\n**[[Topological Quantum Field Theory|papers/1104161738.pdf]]\n**differential forms on the space of connections\n*Laurent Baulieu\n**[[On the Cohomological Structure of Gauge Theories|papers/On the Cohomological Structure of Gauge Theories.pdf]]\n***adds two Grassmann coordinates to spacetime\n*Rudolf Schmid\n**[[Local Cohomology in Gauge Theories BRST Tansformations and Anomalies|papers/localbrst.pdf]]\n***mathematically abstract, but geometric\n***ack, [[jet]]s. \n***connects ghosts to [[Maurer-Cartan form]]\n***pretty tough going, maybe delete this ref\n**[[A Few BRST Bicomplexes|papers/nankai.pdf]]\n*discussion with [[Michael Edwards]] on\n**[[Not Even Wrong|http://www.math.columbia.edu/~woit/wordpress/?p=436]]\n*Ian Anderson\n**[[The Variational Bicomplex|papers/The Variational Bicomplex.pdf]]
The [[BRST technique]] fixes and accounts for [[gauge symmetries|gauge transformation]] by introducing new fields with [[Grassmann valued|Grassmann number]] coefficients having dynamics and interactions with existing fields that breaks the original local gauge symmetry but includes a new global (super) symmetry -- the BRST transformation -- that's a "rotation" between old and new fields. This method of gauge fixing is an indispensable tool in the application of path integral methods in the quantum field theory of non-abelian gauge fields ([[principal bundle]] connections), and has a natural extension to describe the existence and dynamics of fermionic [[spinor]] fields.\n\nA restricted BF Lagrangian,\n$$\n\snf{L} = \sli \snf{B} \sff{F} + \snf{\sPhi}(\sf{A},\snf{B}) \sri\n$$\ninvariant, $\sdelta_{G} \snf{L} = 0$, under some subset of the gauge transformation, $G \sin {\srm Lie}(H) \ssubset {\srm Lie}(G)$, is amenable to the BRST technique. A ''ghost field'', $\sud{C} = \sud{C^A} T_A \sin \sud{\srm Lie}(H)$, is introduced with Grassmann coefficients multiplying [[Lie algebra]] elements, along with an anti-Grassmann valued $(n-1)$-form ''antighost field'', $\snf{\sod{B}} = \snf{\sod{B}{}^A} T_A$, and a real valued $(n-1)$-form ''BRST partner field'', $\snf{\slambda} = \snf{\slambda^A} T_A$. This new system is equipped with a global ''BRST transformation'' -- a ''supersymmetry rotation'' between real and Grassmann valued variables,\n$$\n\sbegin{array}{rclcrcl}\n\sud{\sde} \sf{A} &=& -\sf{\snabla} \sud{C} & \s;\s;\s; & \sud{\sde} \sud{C} &=& -\sha \slb \sud{C}, \sud{C} \srb \s\s\n\sud{\sde} \snf{B} &=& \slb \snf{B}, \sud{C} \srb & \s;\s;\s; & \sud{\sde} \snf{\sod{B}} &=& \snf{\sla} \s\s\n\sud{\sde} \snf{\sla} &=& 0 & & & &\n\send{array}\n$$\nthat is nilpotent, $\sud{\sde} \sud{\sde} = 0$, and leaves the Lagrangian invariant (''BRST [[closed]]''), $\sud{\sde} \snf{L} = 0$. Physical observables are in the [[cohomology]] of this ''BRST operator'', $\sud{\sde}$. Dynamics are introduced for the ghosts by adding a ''BRST [[exact]]'' term to get a ''BRST extended Lagrangian'',\n$$\n\snf{L'} = \snf{L} + \sud{\sde} \snf{\sod{\sPsi}}\n$$\nwith some ''BRST potential'', $\snf{\sod{\sPsi}}$, chosen. For example, choosing\n$$\n\snf{\sod{\sPsi}} = \sli \snf{\sod{B}} \sf{A} \sri\n$$\ngives\n$$\n\sud{\sde} \snf{\sod{\sPsi}} = \sli \snf{\sla} \sf{A} \sri + \sli \snf{\sod{B}} \sf{\snabla} \sud{C} \sri\n$$\nThe BRST partner field, $\snf{\sla}$, acts as a Lagrange multiplier constraining the gauge freedom of the connection, so the ''gauge fixed connection'' is $\sf{A} = \sf{A'}$, with $\snf{\sla} \sf{A'} = 0$. The resulting ''effective Lagrangian'' is\n$$\n\snf{L^{\srm eff}} = \sli \snf{B'} \sff{F'} + \snf{\sPhi}(\sf{A'},\snf{B'}) \sri \n+ \sli \snf{\sod{B}} \sf{\snabla'} \sud{C} \sri\n$$\n\nThis form of the Lagrangian suggests the introduction of a ''BRST extended connection'',\n$$\n\sudf{A} = \sf{A'} + \sud{C}\n$$\nwith ''BRST extended curvature'',\n$$\n\sudff{F} = \sf{d} \sudf{A} + \sha \slb \sudf{A} , \sudf{A} \srb = \sff{F'} + \sf{\snabla'} \sud{C} + \sha \slb \sud{C} , \sud{C} \srb\n$$\nallowing the effective Larangian to be written as\n$$\n\snf{L^{\srm eff}} = \sli \snf{\sod{B'}} \sudff{F} + \snf{\sPhi}(\sf{A'},\snf{B'}) \sri\n$$\nwith $\snf{\sod{B'}} = \snf{B'} + \snf{\sod{B}}$.\n\nRef:\n*J.W. van Holten\n**[[Aspects of BRST Quantization|papers/JHolten_BRST.pdf]]
<<note HideTags>>$$\sbegin{array}{rcl}\n\sf{H} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sha \sf{\som} + \sfr{1}{4}\sf{e}\sph + \sf{B} + \sf{W} =\n{\sscriptsize\n\slb \sbegin{array}{cccc} \n\sha \sf{\som_L} \s!+\s! i \sf{W^3} & i \sf{W^1} \s!+\s! \sf{W^2} & - \sfr{1}{4} \sf{e_R} \sph_0^* & \sfr{1}{4} \sf{e_R} \sph_+ \s\s\ni \sf{W^1} \s!-\s! \sf{W^2} & \sha \sf{\som_L} \s!-\s! i \sf{W^3} & \sp{-} \sfr{1}{4} \sf{e_R} \sph_+^* & \sfr{1}{4} \sf{e_R} \sph_0 \s\s\n- \sfr{1}{4} \sf{e_L} \sph_0 & \sfr{1}{4} \sf{e_L} \sph_+ & \sha \sf{\som_R} \s!+\s! i \sf{B} & \s\s\n\sp{-} \sfr{1}{4} \sf{e_L} \sph_+^* & \sfr{1}{4} \sf{e_L} \sph_0^* & & \sha \sf{\som_R} \s!-\s! i \sf{B}\n\send{array} \srb_{\sp{(}}\n} \s\s\n\s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^a} \sha h_a^{\sp{a} \sal\sbe} \sga_{\sal\sbe} \s;\s; \sin \s;\s; \sf{so}(1,7) = \sf{Cl}^2(1,7) \ssubset \sf{\smathbb{C}}(8\stimes8)\n\send{array}$$\n@@display:block;text-align:center;[[Clifford bivector|Clifford algebra]] parts:@@$$\n\sbegin{array}{rcl}\n\sf{\som} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^a} \sha \som_a^{\sp{a} \smu \snu} \sga_{\smu \snu}\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\n\sleftarrow \stext{spin connection} \s\s\n\sf{e} \sph \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^a} \slp e_a \srp^\smu \sph^\sph \sga_{\smu \sph}\n\s, \sleft\s{\n\sbegin{array}{rcl}\n\sf{e} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^a} \slp e_a \srp^\smu \sga_\smu\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\n\sleftarrow \stext{frame (vierbein)} \s\s\n\sph \s!\s!&\s!\s!=\s!\s!&\s!\s! \sph^\sph \sga_\sph\n\s, \sleft\s{\n\sbegin{array}{rcl}\n\sph_+ \s!\s!&\s!\s!=\s!\s!&\s!\s! (-\sph^5 \s!+\s! i \sph^6) \s\s\n\sph_0 \s!\s!&\s!\s!=\s!\s!&\s!\s! (\sph^7 \s!+\s! i \sph^8)\n\send{array}\n\sright\s}\n\sbegin{array}{c}\n\s;\s;\s;\s;\n\sleftarrow \stext{Higgs} \s\s\n\sph \sph = -M^2\n\send{array}\n\send{array}\n\srd\n\send{array}\n$$\n$$\n\sbegin{array}{rcl}\n\sf{B} \s!\s!&\s!\s!=\s!\s!&\s!\s! - \s! \sf{dx^a} \sha B_a \sbig( \sga_{56} - \sga_{78} \sbig) \n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s,\n\sleftarrow \s; \sdownarrow \stext{electroweak gauge fields} \s\s\n\sf{W} \s!\s!&\s!\s!=\s!\s!&\s!\s! - \s! \sha \sf{W^1} \sbig( \sga_{67} + \sga_{58} \sbig)\n- \sha \sf{W^2} \sbig(-\sga_{57} + \sga_{68} \sbig)\n- \sha \sf{W^3} \sbig( \sga_{56} + \sga_{78} \sbig) \n\s;\s;\s;\s;\s;\s;\s;\s, \s\s\n\send{array}\n$$\n@@display:block;text-align:center;[[indices]]: $\s;\s;\s;\s; 0 \sle a,b \sle 3 \s;\s;\s;\s;\s;\s; 0 \sle \smu,\snu \sle 3 \s;\s;\s;\s;\s;\s; 5 \sle \sph,\sps \sle 8$@@
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/TED08/images/Soft Coral_620.jpg" width="827" height="620"></embed>\n</center></html>@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/TED08/images/Soft Coral_620.jpg" width="827" height="620"></embed>\n</center></html>@@\n
Creating bulleted lists is simple.\n* Just add an asterisk\n* at the beginning of a line.\n** If you want to create sub-bullets\n** start the line with two asterisks\n*** And if you want yet another level\n*** use three asterisks\n* You can also do [[Numbered Lists]]\n{{{\nCreating bulleted lists is simple.\n* Just add an asterisk\n* at the beginning of a line.\n** If you want to create sub-bullets\n** start the line with two asterisks\n*** And if you want yet another level\n*** use three asterisks\n* You can also do [[Numbered Lists]]\n}}}
\n\nRef:\n*M. Berg, C. DeWitt-Morette, S. Gwo, E. Kramer, [[The Pin Groups in Physics: C, P, and T|papers/The Pin Groups in Physics- C, P, and T.pdf]]
[[arxiv|http://arxiv.org/abs/0706.0217]]\nS. Raby, A. Wingerter\nAbstract: We investigate whether the hypercharge assignments in the Standard Model can be interpreted as a hint at Grand Unification in the context of heterotic string theory. To this end, we introduce a general method to calculate U(1)_Y for any heterotic orbifold and compare our findings to the cases where hypercharge arises from a GUT. Surprisingly, in the overwhelming majority of 3-2 Standard Models, a non-anomalous hypercharge direction can be defined, for which the spectrum is vector-like. For these models, we calculate sin^2 theta to see how well it agrees with the standard GUT value. We find that 12% have sin^2 theta = 3/8, while all others have values which are less. Finally, 89% of the models with sin^2 theta = 3/8 have U(1)_Y in SU(5). \n\n*computation to find hypercharge directions in E8xE8 root system
[>img[images/person/Carlo Rovelli.jpg]]Homepage: http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html\n*Location: Marseille\n*CV: http://www.cpt.univ-mrs.fr/%7Erovelli/vita.pdf\n*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Rovelli_C/0/1/0/all/0/1\n\nSelected work:\n*[[Quantum Gravity|http://www.cpt.univ-mrs.fr/%7Erovelli/book.pdf]]\n*[[Graviton propagator in loop quantum gravity|http://arxiv.org/abs/gr-qc/0604044]]\n**nice treatment. includes basic example of canonical and path integral QM, field theory, then does LQG via GFT.
A ''Cartan H-bundle'', with total space $E_H$, is a [[principal bundle]] with $n_H$ dimensional [[Lie group]], $H$, as the typical fiber (and structure group) and $n_M$ dimensional base, $M$. This bundle is not [[associated]] to the [[Ehresmann Cartan geometry]], $E_G$, since the structure group of $E_H$ is $H \ssubset G$; however, $E_H$ does serve as a base space under $E_G$, and the [[Ehresmann Cartan connection]] over $E_G$ does pull back to give a "connection" over $E_H$.\n\nThe Cartan H-bundle is mapped into a section (a [[submanifold]]), $E'_H$, of the Ehressmann Cartan geometry, $E_G$, by the reference section of the [[Cartan homogeneous space bundle]],\n\sbegin{eqnarray}\n\ssi'^S &:& E_H \sto E_G \s\s\n\ssi'^S(x,y) &=& (x,x_{s\ssi}(x),y)\n\send{eqnarray}\nThe [[Ehresmann Cartan connection]] form [[pulls back|pullback]] along this map to give the ''Ehresmann Cartan H-connection form'' over $E_H$,\n\sbegin{eqnarray}\n\sf{{\scal C}_H}(x,y) &=& \ssi'^{S*} \sf{\scal C} = \slp \sf{C^J}(x) L^I{}_J(x_{s\ssi}(x),y) + \ssi'^{S*} \sf{\sxi_R^I}(x_{s\ssi}(x),y) \srp T_I \s\s\n&=& g^-(x_{s\ssi}(x),y) \s, \sf{C}(x) \s, g(x_{s\ssi}(x),y) + g^-(x_{s\ssi}(x),y) \s, \sf{d} \s, g(x_{s\ssi}(x),y) \s\s\n&=& h^-(y) \sBig( r^-(x_{s\ssi}(x)) \s, \sf{C}(x) \s, r(x_{s\ssi}(x)) + r^-(x_{s\ssi}(x)) \s, \sf{d} \s, r(x_{s\ssi}(x)) \sBig) h(y) + h^-(y) \s, \sf{d} h(y) \n\send{eqnarray}\nin which the [[coset representative section|homogeneous space]], $r:G/H \sto G$, is used to write $g(x_s,y)=r(x_s) \s, h(y)$. Choosing the homogeneous space bundle zero reference section, $r(x_{s\ssi_0}(x)) = r(0) = 1$, this gives\n$$\n\sf{{\scal C}_H}(x,y) = h^-(y) \s, \sf{C}(x) \s, h(y) + h^-(y) \s, \sf{d} \s, h(y)\n$$\nPulling this back along the [[canonical reference section|Ehresmann principal bundle connection]] gives the [[Cartan connection|Cartan geometry]], $\ssi_0^{H*} \sf{{\scal C}_H} = \sf{C}$, over $M$.\n\nIf $H$ is [[reductive]] in $G$ (as is usually assumed) the Ehresmann Cartan H-connection form splits into the ''Ehresmann Cartan H-connection frame form'' and '' Ehresmann H-connection form'',\n\sbegin{eqnarray}\n\sf{{\scal C}_H} &=& \sf{{\scal E}_H} + \sf{{\scal A}_H} \s\s \n\sf{{\scal E}_H} &=& h^-(y) \s, \sf{e}(x) \s, h(y) \sin \sf{\srm Lie}(G/H) \s\s\n\sf{{\scal A}_H} &=& h^-(y) \s, \sf{A}(x) \s, h(y) + h^-(y) \s, \sf{d} \s, h(y) \sin \sf{\srm Lie}(H)\n\send{eqnarray}\nThe Ehresmann H-connection form, $\sf{{\scal A}_H}(x,y)$, over $E_H$ is an [[Ehresmann principal bundle connection]] form for the bundle.\n\nWhen the Ehresmann Cartan H-connection form equals the [[Maurer-Cartan form]], $\sf{{\scal C}_H} = \sf{\scal I}$, the Cartan H-bundle is an [[Ehresmann homogeneous space geometry]], $E_H = G$. In this way, the Cartan H-bundle may be considered to be a [[reductive Lie group geometry]], $G$, that has gone wavy along $G/H$ -- with $\sf{{\scal C}_H}$ deviating from $\sf{\scal I}$ to give the new [[frame]] 1-forms, $\sf{{\scal C}_H^J} = \sf{E^J}$, of the [[Cartan tangent bundle geometry]] over what was $G$.
A ''Cartan geometry'' is a [[Lie group geometry]], $G$, that's allowed to go wavy while maintaining some of its symmetry, represented by a subgroup, $H \ssubset G$, usually assumed to be [[reductive]] in $G$. The wavy ''Cartan geometry base manifold'', $M$, is ''modeled'' on the [[homogeneous space]], $M \ssim S=G/H$, and has the same dimension, $n_S = (n_G - n_H)$. The ''Cartan connection'' over $M$,\n$$\n\sf{C}(x) = \sf{e} + \sf{A} \sin \sf{\srm Lie}(G)\n$$\nis a [[Lieform]] modeled on the [[Maurer-Cartan frame|homogeneous space]], $\sf{C} \ssim \sf{I} = r^- \sf{d} r(x)$, and splits (for $H$ reductive in $G$) into the ''Cartan frame'', $\sf{e}(x) = \sf{e^A} K_A \sin \sf{\srm Lie}(G/H)$, and ''Cartan H-connection'', $\sf{A}(x) = \sf{A^P} H_P \sin \sf{\srm Lie}(H)$, which (unlike their homogeneous space counterparts) may vary freely. \n\nThe ''Cartan [[curvature]]'' of the connection is\n\sbegin{eqnarray}\n\sff{F}(x) &=& \sf{d} \sf{C} + \sha \slb \sf{C}, \sf{C} \srb \s\s\n&=& \sf{d} \sf{e} + \sf{d} \sf{A} + \sha \slb \sf{e}, \sf{e} \srb + \slb \sf{A}, \sf{e} \srb + \sha \slb \sf{A}, \sf{A} \srb \s\s\n&=& \sff{F^A} K_A + \sff{F^P} H_P \n\send{eqnarray}\nwhich (like the [[homogeneous space curvature|homogeneous space tangent bundle geometry]]) splits into\n\sbegin{eqnarray}\n\sff{F^A} &=& \sf{d} \sf{e^A} + \sf{A^P} \sf{e^B} C_{PB}{}^A + \sha \sf{e^C} \sf{e^B} C_{CB}{}^A \s\s \n\sff{F^P} &=& \sff{F_H^P} + \sha \sf{e^C} \sf{e^D} C_{CD}{}^P\n\send{eqnarray}\nwith the ''curvature of the Cartan H-connection'' defined by:\n$$\n\sff{F_H^P} = \sf{d} \sf{A^P} + \sha \sf{A^Q} \sf{A^R} C_{QR}{}^P\n$$\nNote that $\sha \sf{e^C} \sf{e^B} C_{CB}{}^A = 0$ if $G/H$ is a [[symmetric space]].\n\nThere are many relationships between a Cartan geometry and other structures. A [[natural]] [[Ehresmann Cartan geometry]] is a description of a Cartan geometry as an [[Ehresmann principal bundle connection]] for a [[G-bundle|principal bundle]] -- and this description splits via $G/H$ into the [[Cartan H-bundle]] and [[Cartan homogeneous space bundle]]. These two bundles relate to the way the [[Lie group tangent bundle geometry]] of $G$ turns wavy, described by the [[Cartan tangent bundle geometry]].\n\nRefs:\n*http://en.wikipedia.org/wiki/Cartan_connection\n*[[Differential Geometry of Cartan Connections|papers/9412232.pdf]]\n**by [[Peter Michor]] and Alekseevsky (note $G \ssubset H$)\n*[[The Works of Charles Ehresmann on Connections: From Cartan Connections to Connections on Fibre Bundles|papers/CMMarle.pdf]]\n**Ehresmann version of Cartan, nicely explained. See p9 for main def.\n**http://www.math.jussieu.fr/~marle/\n*[[MacDowell-Mansouri Gravity and Cartan Geometry|papers/0611154.pdf]]\n**a new paper by Derek Wise\n*[[Natural Operations on the Bundle of Cartan Connections|papers/Natural Operations on the Bundle of Cartan Connections.pdf]]\n*[[The Existance of Cartan Connections and Geometrizable Principal Bundles|papers/0206136.pdf]]\n**a very concise and interesting mathematical treatment.\n
A ''Cartan homogeneous space bundle'', with total space $E_S$, is a [[fiber bundle]] with $n_S$ dimensional [[homogeneous space]], $F=S=G/H$, as the typical fiber and $n_M = n_S$ dimensional base, $M$. This bundle may be visualized as the set of homogeneous spaces tangent to the base space. (If $n_M \sneq n_S$ this is a ''generalized Cartan homogeneous space bundle''.) The structure group, $G$, of the bundle is the subset of [[homogeneous space geometry symmetries]] corresponding to the [[left action|group]] of $G$ on the space.\n\nThe Cartan homogeneous space bundle, $E_S$, is [[associated]] to the [[Ehresmann Cartan geometry]], $E_G$, and $E_S$ also serves as a base space under $E_G$. The $n_M$ coordinates, $x^a$, cover patches of the base manifold, $M$, and the $n_S$ homogeneous space coordinates, $x_s^a$, cover patches of $S$ -- the combined coordinates, $(x,x_s)$, cover patches of $E_S$. The [[reference section|Ehresmann gauge transformation]], $\ssi^S : M \sto E_S$, of the Cartan homogeneous space bundle determines the ''points of tangency'' -- the points, $x_{s\ssi}(x)$, of the $S_x$ in contact with $x$. Since a homogeneous space has a natural zero point, we often use the ''zero point reference section'', $\ssi_0^S(x) = (x,0)$.\n\nThe Cartan homogeneous space bundle, $E_S$, is mapped into a section (a [[submanifold]]), $E'_S$, of the Ehressmann Cartan geometry, $E_G$, by the reference section of the [[Cartan H-bundle]],\n$$\n\ssi'^H(x,x_s) = (x,x_s,y_\ssi(x))\n$$\nThe [[Ehresmann Cartan connection]] form [[pulls back|pullback]] along this map to give the ''Cartan homogeneous space connection form'' over $E_S$,\n\sbegin{eqnarray}\n\sf{{\scal C}_S}(x,x_s) &=& \ssi'^{H*} \sf{\scal C} = \slp \sf{C^J}(x) L^I{}_J(x_s,y_\ssi(x)) + \ssi'^{H*} \sf{\sxi_R^I}(x_s,y_\ssi(x)) \srp T_I \s\s\n&=& g^-(x_s,y_\ssi(x)) \s, \sf{C}(x) \s, g(x_s,y_\ssi(x)) + g^-(x_s,y_\ssi(x)) \s, \sf{d} \s, g(x_s,y_\ssi(x)) \s\s\n&=& h^-(y_\ssi(x)) \sBig( r^-(x_s) \s, \sf{C}(x) \s, r(x_s) + r^-(x_s) \s, \sf{d} \s, r(x_s) \sBig) h(y_\ssi(x)) + h^-(y_\ssi(x)) \s, \sf{d} h(y_\ssi(x)) \n\send{eqnarray}\nin which the [[coset representative section|homogeneous space]], $r:S \sto G$, is used to write $g(x_s,y)=r(x_s) \s, h(y)$. Choosing the canonical H-bundle reference section, $h(y_{\ssi_0}(x)) = h(0) = 1$, this gives\n$$\n\sf{{\scal C}_S}(x,x_s) = r^-(x_s) \s, \sf{C}(x) \s, r(x_s) + r^-(x_s) \s, \sf{d} \s, r(x_s)\n$$\nPulling this back along the zero point reference section gives the [[Cartan connection|Cartan geometry]], $\ssi_0^{S*} \sf{{\scal C}_S} = \sf{C}$, over $M$.
<<note HideTags>>Mutually [[commuting|commutator]] set of $r$ [[Lie algebra]] generators:\n$$\n\sleft\s{ T_1, T_2, ..., T_r \sright\s} \s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s; [ T_i, T_j ] = 0\n$$\n[[Cartan subalgebra|Lie algebra structure]]: $\s;\s;\s; C=c^i T_i \s;\s; \sin {\srm Lie(G)} \sp{{}_{(}}$\n[[Eigenvalues|eigen]], $\sal^a$, and [[eigenvectors|eigen]], $V_a \sin {\srm Lie(G)}$, using the Lie bracket:\n$$\n[ C , V_a ] = \sal^a V_a = \ssum_i c^i \sal_i^a V_a\n$$\nUnique eigenvalue for each of the $(n-r)$ eigenvectors, corresponding to $(n-r)$ ''roots'', $\sal_i^a$, in $r$ dimensional vector space.\n\nCartan subalgebra of the standard model and gravity:\n$$\nC = {\sscriptsize \sfrac{1}{2}} \som^{01} \sga_{01} + {\sscriptsize \sfrac{1}{2}} \som^{12} \sga_{12} + W^3 i \sSi_3 + B i Y + G^3 i \sla_3 + G^8 i \sla_8 \n$$\nEigenvectors are elementary particles, roots are their charges:\n$$\n\sal(e_L) = ( \spm {\sscriptsize \sfrac{1}{2}}, \smp {\sscriptsize \sfrac{1}{2}}, -1, -1, 0, 0 ) \sp{{}_{\sBig(}}\n$$
The [[Riemann curvature]] for the [[Cartan tangent bundle geometry]] is calculated from the [[Cartan tangent bundle spin connection]],\n$$\n\sff{R}^J{}_I = \sf{d} \sf{W}^J{}_I + \sf{W}^J{}_K \sf{W}^K{}_I\n$$\nWe'll tackle this in pieces. Using a [[left-right rotator]] identity,\n$$\n\sf{d} \slp L^h \srp^J{}_I = \sf{e_H^P} C_P{}^J{}_K \slp L^h \srp^K{}_I\n$$\nthe [[exterior derivative]]s are:\n\sbegin{eqnarray}\n\sf{d} \sf{W}^B{}_A &=& \sf{d} \slp \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \slp L^h \srp_A{}^F - \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_A{}^E - \sf{e_H^P} C_P{}^B{}_A \srp \s\s\n&=& \slp \sf{d} \sf{\snu}^E{}_F \srp \slp L^h \srp^B{}_E \slp L^h \srp_A{}^F - \sf{\snu}^E{}_F \sf{e_H^P} C_P{}^B{}_D \slp L^h \srp^D{}_E \slp L^h \srp_A{}^F - \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \sf{e_H^P} C_P{}^F{}_C \slp L^h \srp^C{}_A \s\s\n&-& \sha \slp \sf{d} \sf{A^Q} + \slp \sf{d} \sf{e_H^P} \srp \slp L^h \srp_P{}^Q - \sf{e_H^P} \sf{e_H^R} C_{RP}{}^T \slp L^h \srp_T{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_A{}^E \s\s\n&+& \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp\n\slp \slp \sf{d} F^H_{DEQ} \srp \slp L^h \srp^{BD} \slp L^h \srp_A{}^E\n+ F^H_{DEQ} \sf{e_H^R} C_R{}^B{}_C \slp L^h \srp^{CD} \slp L^h \srp_A{}^E\n+ F^H_{DEQ} \slp L^h \srp^{BD} \sf{e_H^R} C_R{}_{AC} \slp L^h \srp^{CE} \srp \s\s\n&-& \sf{d} \sf{e_H^P} C_P{}^B{}_A \s\s\n\s\s\n\sf{d} \sf{W}^B{}_R &=& \sf{d} \slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \slp L^h \srp_R{}^Q \srp \s\s\n&=& \sha \slp \sf{d} F^H_{DEQ} \srp \slp L^h \srp^{BE} \slp L^h \srp_R{}^Q \n- \sha \sf{e^D} F^H_{DEQ} \sf{e_H^P} C_P{}^B{}_C \slp L^h \srp^{CE} \slp L^h \srp_R{}^Q \n- \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \sf{e_H^P} C_{PRS} \slp L^h \srp^{SQ} \s\s\n\s\s\n\sf{d} \sf{W}^Q{}_R &=& - \sha \sf{d} \slp \sf{A^S} \s, \slp L^h \srp^P{}_S + \sf{e_H^P} \srp C_P{}^Q{}_R \s\s\n&=& - \sha \slp \slp \sf{d} \sf{A^S} \srp \slp L^h \srp^P{}_S\n- \sf{A^S} \sf{e_H^U} C_U{}^P{}_T \slp L^h \srp^T{}_S\n+ \sf{d} \sf{e_H^P} \srp C_P{}^Q{}_R \n\send{eqnarray}\nThe pieces quadratic in the spin connection are:\n\sbegin{eqnarray}\n\sf{W}^B{}_K \sf{W}^K{}_A &=& \sf{W}^B{}_C \sf{W}^C{}_A + \sf{W}^B{}_P \sf{W}^P{}_A \s\s\n&=&\n\slp \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \slp L^h \srp_C{}^F - \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_C{}^E - \sf{e_H^P} C_P{}^B{}_C \srp \s\s\n&\stimes&\n\slp \sf{\snu}^G{}_H \slp L^h \srp^C{}_G \slp L^h \srp_A{}^H - \sha \slp \sf{A^R} + \sf{e_H^S} \slp L^h \srp_S{}^R \srp F^H_{HGR} \slp L^h \srp^{CH} \slp L^h \srp_A{}^G - \sf{e_H^Q} C_Q{}^C{}_A \srp \s\s\n&-& \slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \slp L^h \srp_P{}^Q \srp\n\slp \sha \sf{e^C} F^H_{CFR} \slp L^h \srp_A{}^F \slp L^h \srp^{PR} \srp \s\s\n\s\s\n\sf{W}^B{}_K \sf{W}^K{}_R &=& \sf{W}^B{}_C \sf{W}^C{}_R + \sf{W}^B{}_P \sf{W}^P{}_R \s\s\n&=& \slp \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \slp L^h \srp_C{}^F - \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_C{}^E - \sf{e_H^P} C_P{}^B{}_C \srp\n\slp \sha \sf{e^G} F^H_{GHS} \slp L^h \srp^{CH} \slp L^h \srp_R{}^S \srp \s\s\n&-& \slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \slp L^h \srp_P{}^Q \srp\n \sha \slp \sf{A^S} \s, \slp L^h \srp^T{}_S + \sf{e_H^T} \srp C_T{}^P{}_R \s\s\n\s\s\n\sf{W}^Q{}_K \sf{W}^K{}_R &=& \sf{W}^Q{}_C \sf{W}^C{}_R + \sf{W}^Q{}_P \sf{W}^P{}_R \s\s\n&=& - \slp \sha \sf{e^A} F^H_{AFR} \slp L^h \srp_C{}^F \slp L^h \srp^{QR} \srp\n\slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{CE} \slp L^h \srp_R{}^Q \srp \s\s\n&+&\n\sha \slp \sf{A^S} \s, \slp L^h \srp^T{}_S + \sf{e_H^T} \srp C_T{}^Q{}_P\n\sha \slp \sf{A^U} \s, \slp L^h \srp^V{}_U + \sf{e_H^V} \srp C_V{}^P{}_R\n\send{eqnarray}\nCombining these gives the curvature,\n\sbegin{eqnarray}\n\sff{R}^B{}_A &=& \slp \sf{d} \sf{\snu}^E{}_F \srp \slp L^h \srp^B{}_E \slp L^h \srp_A{}^F - \sf{\snu}^E{}_F \sf{e_H^P} C_P{}^B{}_D \slp L^h \srp^D{}_E \slp L^h \srp_A{}^F - \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \sf{e_H^P} C_P{}^F{}_C \slp L^h \srp^C{}_A \s\s\n&-& \sha \slp \sf{d} \sf{A^Q} + \slp \sf{d} \sf{e_H^P} \srp \slp L^h \srp_P{}^Q - \sf{e_H^P} \sf{e_H^R} C_{RP}{}^T \slp L^h \srp_T{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_A{}^E \s\s\n&+& \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp\n\slp \slp \sf{d} F^H_{DEQ} \srp \slp L^h \srp^{BD} \slp L^h \srp_A{}^E\n+ F^H_{DEQ} \sf{e_H^R} C_R{}^B{}_C \slp L^h \srp^{CD} \slp L^h \srp_A{}^E\n+ F^H_{DEQ} \slp L^h \srp^{BD} \sf{e_H^R} C_R{}_{AC} \slp L^h \srp^{CE} \srp \s\s\n&-& \sf{d} \sf{e_H^P} C_P{}^B{}_A \s\s\n&+& \slp \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \slp L^h \srp_C{}^F - \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_C{}^E - \sf{e_H^P} C_P{}^B{}_C \srp \s\s\n&\stimes&\n\slp \sf{\snu}^G{}_H \slp L^h \srp^C{}_G \slp L^h \srp_A{}^H - \sha \slp \sf{A^R} + \sf{e_H^S} \slp L^h \srp_S{}^R \srp F^H_{HGR} \slp L^h \srp^{CH} \slp L^h \srp_A{}^G - \sf{e_H^Q} C_Q{}^C{}_A \srp \s\s\n&-& \slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \slp L^h \srp_P{}^Q \srp\n\slp \sha \sf{e^C} F^H_{CFR} \slp L^h \srp_A{}^F \slp L^h \srp^{PR} \srp \s\s\n\s\s\n\sff{R}^B{}_R &=& \sha \slp \sf{d} F^H_{DEQ} \srp \slp L^h \srp^{BE} \slp L^h \srp_R{}^Q \n- \sha \sf{e^D} F^H_{DEQ} \sf{e_H^P} C_P{}^B{}_C \slp L^h \srp^{CE} \slp L^h \srp_R{}^Q \n- \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \sf{e_H^P} C_{PRS} \slp L^h \srp^{SQ} \s\s\n&+& \slp \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \slp L^h \srp_C{}^F - \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_C{}^E - \sf{e_H^P} C_P{}^B{}_C \srp\n\slp \sha \sf{e^G} F^H_{GHS} \slp L^h \srp^{CH} \slp L^h \srp_R{}^S \srp \s\s\n&-& \slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \slp L^h \srp_P{}^Q \srp\n \sha \slp \sf{A^S} \s, \slp L^h \srp^T{}_S + \sf{e_H^T} \srp C_T{}^P{}_R \s\s\n\s\s\n\sff{R}^Q{}_R &=& - \sha \slp \slp \sf{d} \sf{A^S} \srp \slp L^h \srp^P{}_S\n- \sf{A^S} \sf{e_H^U} C_U{}^P{}_T \slp L^h \srp^T{}_S\n+ \sf{d} \sf{e_H^P} \srp C_P{}^Q{}_R \s\s\n&+&\n\sha \slp \sf{A^S} \s, \slp L^h \srp^T{}_S + \sf{e_H^T} \srp C_T{}^Q{}_P\n\sha \slp \sf{A^U} \s, \slp L^h \srp^V{}_U + \sf{e_H^V} \srp C_V{}^P{}_R\n\send{eqnarray}\n\nFrom this ugly mess, the [[Ricci curvature]], $\sf{R}{}_I = \sve{E_J} \sff{R}^J{}_I$, is\n\sbegin{eqnarray}\n\sf{R}{}_B &=& \sve{E_A} \sff{R}^A{}_B + \sve{E_R} \sff{R}^R{}_B \s\s\n\sf{R}{}_R &=& \sve{E_B} \sff{R}^B{}_R + \sve{E_Q} \sff{R}^Q{}_R \s\s\n\send{eqnarray}\n\n//ack, I've got to work on something else for awhile.//\n\n[[curvature scalar]]\n\nCheck that it matches [[reductive Lie group tangent bundle geometry]] as special case.\n
A ''Cartan tangent bundle geometry'' is a [[reductive Lie group tangent bundle geometry]] that has gone a little wavy. The [[frame]] 1-forms, $\sf{E^J}$, over what was the Lie group manifold, $E_H \ssim G$, split in adapted coordinates as\n\sbegin{eqnarray}\n\sf{E^A}(x,y) &=& \sf{e^B}(x) \s, \slp L^h\srp^A{}_B(y) \s\s\n\sf{E^P}(x,y) &=& \sf{A^Q}(x) \s, \slp L^h \srp^P{}_Q(y) + \sf{e_H^P}(y)\n\send{eqnarray}\nin which $\sf{e^B}$ and $\sf{A^Q}$ are the [[Cartan frame|Cartan geometry]] forms and [[Cartan H-connection|Cartan geometry]] forms, $\slp L^h \srp^A{}_B = \slp H^A, h^- H_B h(y) \srp$ is the [[left-right rotator]] over $H$, and $\sf{e_H^P}$ are the frame 1-forms over $H$. These frame 1-forms are components of the Ehresmann Cartan H-connection form, $\sf{E^J} = \sf{{\scal C}_H^J}$, over the [[Cartan H-bundle]], $E_H$, with $\sf{E^A} = \sf{{\scal E}_H^A}$ and $\sf{E^P} = \sf{{\scal A}_H^P}$. The Cartan tangent bundle, $TE_H$ IS the bundle of tangent vectors over the Cartan H-bundle, $E^H$, but from the point of view of treating the Ehresmann Cartan H-connection forms as a frame. Holding this point of view, we need to figure out what the [[Cartan tangent bundle spin connection]], $\sf{W}{}^J{}_K$, is from this frame, and its curvature.
The [[tangent bundle spin connection|tangent bundle connection]] for a [[Cartan tangent bundle geometry]] is determined by insisting the [[torsion]] vanishes over $E_H$, giving [[Cartan's equation]],\n$$\n\sff{T^J} = 0 = \sf{d} \sf{E^J} + \sf{W}{}^J{}_K \sf{E^K}\n$$\nwhich may be solved for the ''Cartan tangent bundle spin connection'', $\sf{W}{}^J{}_K$. To construct the solution, we first compute the [[exterior derivative]] of the [[frame]] 1-forms,\n\sbegin{eqnarray}\n\sf{d} \sf{E^A} &=& \slp \sf{d} \sf{e^B} \srp \slp L^h\srp^A{}_B - \sf{e^B} \sf{d} \slp L^h\srp^A{}_B \s\s\n&=& \slp \sf{d} \sf{e^B} \srp \slp L^h\srp^A{}_B - \sf{e^B} \sf{e_H^P} \slp L^h \srp^D{}_B C_{DP}{}^A \s\s\n\sf{d} \sf{E^P} &=& \slp \sf{d} \sf{A^Q} \srp \slp L^h \srp^P{}_Q - \sf{A^Q} \sf{d} \slp L^h \srp^P{}_Q + \sf{d} \sf{e_H^P} \s\s\n&=& \slp \sf{d} \sf{A^Q} \srp \slp L^h \srp^P{}_Q - \sf{A^Q} \sf{e_H^R} \slp L^h \srp^T{}_Q C_{TR}{}^P - \sha \sf{e_H^Q} \sf{e_H^R} C_{QR}{}^P\n\send{eqnarray}\nusing the [[left-right rotator]], $\slp L^h\srp^I{}_J = \slp T^I, h^- T_J h \srp$, and the [[Maurer-Cartan equation|Maurer-Cartan form]] over $H$. Next we write down the [[orthonormal basis vectors|frame]] (satisfying $\sve{E_J} \sf{E^K} = \sde_J^K$),\n\sbegin{eqnarray}\n\sve{E_A}(x,y) &=& \slp L^h \srp_A{}^B \s, \sve{e_B}(x) - \slp L^h \srp_A{}^C \slp \sve{e_C} \sf{A^Q} \srp \slp L^h \srp^P{}_Q \s, \sve{e^H_P} \s\s\n\sve{E_P}(x,y) &=& \sve{e^H_P}(y)\n\send{eqnarray}\nand compute the [[anholonomy|Cartan's equation]] coefficients, $f_{IJ}{}^K = \sve{E_J} \sve{E_I} \slp \sf{d} \sf{E^K} \srp$, getting\n\sbegin{eqnarray}\nf_{AB}{}^C &=& \sve{e_E} \sve{e_D} \slp \sf{d} \sf{e^F} \srp \slp L^h \srp_B{}^E \slp L^h \srp_A{}^D \slp L^h \srp^C{}_F \n+ 2 \slp - \slp L^h \srp_{\slb B \srd}{}^E \s, \sve{e_E} \slp L^h \srp_{\sld A \srb}{}^G \slp \sve{e_G} \sf{A^Q} \srp \slp L^h \srp^R{}_Q \s, \sve{e^H_R} \srp \slp - \sf{e^F} \sf{e_H^P} \slp L^h \srp^D{}_F C_{DP}{}^C \srp \s\s\n&=& f^M_{DE}{}^F \slp L^h \srp_A{}^D \slp L^h \srp_B{}^E \slp L^h \srp^C{}_F \n- 2 \slp \sve{e_E} \sf{A^Q} \srp \slp L^h \srp^R{}_Q \slp L^h \srp_{\slb A \srd}{}^E \s, C_{\sld B \srb R}{}^C \s\s\nf_{AQ}{}^C &=& - C_{AQ}{}^C \s\s\nf_{AB}{}^R &=& \sve{e_E} \sve{e_D} \slp \sf{d} \sf{A^Q} \srp \slp L^h \srp_B{}^E \slp L^h \srp_A{}^D \slp L^h \srp^R{}_Q\n- \slp L^h \srp_A{}^E \slp \sve{e_E} \sf{A^Q} \srp \slp L^h \srp^P{}_Q \slp L^h \srp_B{}^D \slp \sve{e_D} \sf{A^S} \srp \slp L^h \srp^T{}_S C_{TP}{}^R \s\s\n&=& \sve{e_E} \sve{e_D} \slp \sff{F_H^Q} \srp \slp L^h \srp_B{}^E \slp L^h \srp_A{}^D \slp L^h \srp^R{}_Q \s\s\n&=& F^H_{DE}{}^Q \slp L^h \srp_A{}^D \slp L^h \srp_B{}^E \slp L^h \srp^R{}_Q \s\s\nf_{AQ}{}^R &=& 0 \s\s\nf_{PQ}{}^C &=& 0 \s\s\nf_{PQ}{}^R &=& - C_{PQ}{}^R\n\send{eqnarray}\nin which $f^M_{DE}{}^F(x)$ is the anholonomy for $\sf{e^A}$ and $\sff{F_H^Q}(x) = \sf{d} \sf{A^Q} + \sha \sf{A^P} \sf{A^R} C_{PR}{}^Q$ is the [[curvature]] for $\sf{A^Q}$. Using these, the solution to Cartan's equation, $W_{IJK} = \sha \slp f_{IJK} - f_{JKI} + f_{KIJ} \srp$, gives the Cartan tangent bundle spin connection coefficients,\n\sbegin{eqnarray}\nW_{ABC} &=& \snu_{DEF} \slp L^h \srp_A{}^D \slp L^h \srp_B{}^E \slp L^h \srp_C{}^F - \slp L^h \srp_A{}^E \slp \sve{e_E} \sf{A^Q} \srp \slp L^h \srp^R{}_Q C_{B R C} \s\s\nW_{ABR} &=& \sha f_{ABR} = \sha F^H_{DEQ} \slp L^h \srp_A{}^D \slp L^h \srp_B{}^E \slp L^h \srp_R{}^Q \s\s\nW_{AQR} &=& 0 \s\s\nW_{PBC} &=& - C_{PBC} - \sha f_{BCP} = - C_{PBC} - \sha F^H_{DEQ} \slp L^h \srp_B{}^D \slp L^h \srp_C{}^E \slp L^h \srp_P{}^Q \s\s\nW_{PQC} &=& 0 \s\s\nW_{PQR} &=& - \sha C_{PQR}\n\send{eqnarray}\nwith $\snu_{DEF}(x)$ the coefficients of the torsionless spin connection for $\sf{e^A}$. From these, the ''Cartan tangent bundle spin connection'', $\sf{W}{}_{JK} = \sf{E^I} W_{IJK}$, is\n\sbegin{eqnarray}\n\sf{W}{}_{BC} &=& \sf{E^A} W_{ABC} + \sf{E^P} W_{PBC} \s\s\n&=& \sf{e^D} \slp \snu_{DEF} \slp L^h \srp_B{}^E \slp L^h \srp_C{}^F - \slp \sve{e_D} \sf{A^Q} \srp \slp L^h \srp^R{}_Q C_{B R C} \srp \n+ \slp \sf{A^R} \s, \slp L^h \srp^P{}_R + \sf{e_H^P} \srp \slp - C_{PBC} - \sha F^H_{DEQ} \slp L^h \srp_B{}^D \slp L^h \srp_C{}^E \slp L^h \srp_P{}^Q \srp \s\s\n&=& \sf{\snu}{}_{EF} \slp L^h \srp_B{}^E \slp L^h \srp_C{}^F \n- \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp_B{}^D \slp L^h \srp_C{}^E - \sf{e_H^P} C_{PBC} \s\s\n\sf{W}{}_{BR} &=& \sf{E^A} W_{ABR} = \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp_B{}^E \slp L^h \srp_R{}^Q \s\s\n\sf{W}{}_{QR} &=& \sf{E^P} W_{PQR} = - \sha \slp \sf{A^S} \s, \slp L^h \srp^P{}_S + \sf{e_H^P} \srp C_{PQR}\n\send{eqnarray}\nThis may be used to calculate the [[Cartan tangent bundle curvature]].
''Cartan's equation'' relates the [[spin connection]] to the [[exterior derivative]] of the [[frame]] by asserting that the [[torsion]] is zero,\n$$\n0 = \sf{d} \sf{e} + \sf{\som} \stimes \sf{e} \n$$\nor, equivalently,\n$$\n0 = \sf{d} \sf{e^\sal} + \sf{\som}^\sal{}_\sbe \sf{e^\sbe} \n$$\nThis equation may be solved in closed form for the spin or [[tangent bundle connection]] coefficients. To generalize, lets solve\n$$\n\sf{\som} \stimes \sf{e} = -\sff{f}\n$$\nfor $\sf{\som}$ in terms of the frame and an arbitrary Clifford vector valued 2-form, $\sff{f}$. In components, using the [[index bracket]], this is\n$$\n\som_{\slb i \srd}{}^{\sal \sbe} \slp e_{\sld j \srb} \srp_\sbe = -\sha f_{ij}{}^\sal\n$$\nUsing the frame to change from coordinate to Clifford indices, and using antisymmetry of the last two spin connection coefficient indices, this may be expressed simply as\n$$\n\som_{\slb \sbe \sga \srb \sal} = \sha f_{\sbe \sga \sal}\n$$\nBy once again juggling spin connection indices, we see from this that\n$$\n\som_{\slp \sbe \sga \srp \sal} = \som_{\slb \sal \sbe \srb \sga} + \som_{\slb \sal \sga \srb \sbe} = f_{\sal \slp \sbe \sga \srp}\n$$\nAdding these last two expressions gives the explicit solution for the spin connection coefficients:\n$$\n\som_{\sal \sbe \sga} = \som_{\slb \sal \sbe \srb \sga} + \som_{\slp \sal \sbe \srp \sga} = \sha \slp f_{\sal \sbe \sga} - f_{\sbe \sga \sal} + f_{\sga \sal \sbe} \srp\n$$\nPutting these indices in their more familiar positions is done using the frame and [[Minkowski metric]]: $\som_{i}{}^{\sde \sep} = \slp e_i \srp^\sal \set^{\sde \sbe} \set^{\sep \sga} \som_{\sal \sbe \sga}$. Cartan's equation is solved by simply plugging $\sff{f}=\sf{d} \sf{e}$ into the above equation -- in coefficients,\n$$\nf_{\sal \sbe \sga} = \slp e_\sal \srp^i \slp e_\sbe \srp^j \set_{\sga \sde} \slp \spa_i \slp e_j \srp^\sde - \spa_j \slp e_i \srp^\sde \srp\n= \sve{e_\sbe} \sve{e_\sal} \slp \sf{d} \sf{e}{}_\sga \srp\n$$\n\nNote that this last tensor, the ''anholonomy'', may also be expressed using the [[Lie bracket|Lie derivative]] of the orthonormal basis vectors,\n$$\n\slb \sve{e_\sal} , \sve{e_\sbe} \srb_L = 2 \slb \slp e_{\slb \sal \srd} \srp^j \spa_j \slp e_{\sld \sbe \srb} \srp^i \srb \sve{\spa_i}\n= - 2 \slb \slp e_{\slb \sal \srd} \srp^j \slp e_{\sld \sbe \srb} \srp^k \slp e_\sga \srp^i \spa_j \slp e_k \srp^\sga \srb \sve{\spa_i}\n= - f_{\sal \sbe}{}^\sga \sve{e_\sga} \n$$\n\nIt is possible to express the solution to Cartan's equation in a particularly pretty, index free way using [[Clifform algebra]]:\n$$\n\sf{\som} = - \sve{e} \stimes \sff{f} + \sfr{1}{4} \slp \sve{e} \stimes \sve{e} \srp \slp \sf{e} \scdot \sff{f} \srp\n$$\n(A cute, if not particularly useful expression.)
[[Lie algebra]]\nhttp://en.wikipedia.org/wiki/Casimir_invariant\n\nrelated to [[Laplacian]]\n
For a $2 \stimes 2$ square matrix or $2$ or $3$ dimensional [[Clifford algebra]] element, $A$, using the [[trace]] and products gives\n$$\n0 = A^2 - \sli A \sri A + \sha \slp \sli A \sri^2 - \sli A^2 \sri \srp\n$$\nFor a $3 \stimes 3$ square matrix, $A$,\n$$\n0 = A^3 - \sli A \sri A^2 + \sha \slp \sli A \sri^2 - \sli A^2 \sri \srp A - \sfr{1}{6} \slp \sli A \sri^3 - 3 \sli A^2 \sri \sli A \sri + 2 \sli A^3 \sri \srp\n$$\nThis generalizes to formula for larger matrices,\nhttp://arxiv.org/hep-th/0701116
A ''Chern-Simons form'', $\snf{\som_p}$, is a grade $p$ [[differential form]] defined (for odd $p$) to satisfy\n$$\n\sf{d} \snf{\som_p} = Tr\slp \sff{F}^{\sfr{p+1}{2}} \srp\n$$\nin which $\sff{F}=\sf{d} \sf{A} + \sf{A} \sf{A}$ is the [[curvature]] for some [[principal bundle]] [[connection]], $\sf{A}$. The first few Chern-Simons forms are\n\sbegin{eqnarray}\n\sf{\som_1} &=& Tr\slp \sf{A} \srp \s\s\n\snf{\som_3} &=& Tr\slp \sff{F} \sf{A} - \sfr{1}{3} \sf{A} \sf{A} \sf{A} \srp \s\s\n\snf{\som_5} &=& Tr\slp \sff{F} \sff{F} \sf{A} - \sfr{1}{2} \sff{F} \sf{A} \sf{A} \sf{A} + \sfr{1}{10} \sf{A} \sf{A} \sf{A} \sf{A} \sf{A} \srp \s\s\n\snf{\som_7} &=& Tr\slp \sff{F} \sff{F} \sff{F} \sf{A} + ? \sff{F} \sff{F} \sf{A} \sf{A} \sf{A} + ? \sff{F} \sf{A} \sf{A} \sf{A} \sf{A} \sf{A} + ? \sf{A} \sf{A} \sf{A} \sf{A} \sf{A} \sf{A} \sf{A} \srp\n\send{eqnarray}\n\nThe [[integral|integration]] of a Chern-Simons p-form over a $p$ dimensional [[manifold]] is a homotopy invariant called the ''Chern number'',\n$$\nc_p = \sint \snf{\som_p}\n$$\ncorresponding to the topology of the manifold. For a $(p+1)$ dimensional manifold with a boundary,\n$$\n\sint Tr\slp \sff{F}^{\sfr{p+1}{2}} \srp = \sint \sf{d} \snf{\som_p} = \sint_\spa \snf{\som_p} = c_p \n$$\n\nAlso of potential interest is the relationship to the ''Pfaffian'',\n$$\n\sff{F}^{\sfr{p+1}{2}} = Pf\slp F \srp \snf{d^{p+1}x}\n$$\nwhere $Pf(F) = \ssqrt{\sll F \srl}$\nref:\nhttp://en.wikipedia.org/wiki/Chern-Simons_form
Using the alternative notation for the [[covariant derivative]] employing the [[tangent bundle connection]] and [[cotangent bundle connection]], the covariant derivative of a suitably indexed tensor is written as\n$$\nD_{i}T^{k}{}_{j}=\spartial_{i}T^{k}{}_{j}+\sGamma^{k}{}_{im}T^{m}{}_{j}-\sGamma^{m}{}_{ij}T^{k}{}_{m}\n$$\nwith the ''Christoffel symbols'', $\sGa^k{}_{ij}$, defined as tangent bundle connection coefficients, \n$$\n\sna_i \sve{\spa_j} = \sGa^k{}_{ij} \sve{\spa_k} \n$$\nThe Christoffel symbols are determined from the assumptions that the [[torsion]] vanishes,\n$$\n\sGa^k{}_{\slb ij \srb} = 0\n$$\nand that the covariant derivative is ''[[metric]] compatible'',\n$$\n0 = D_i g_{jk} = \spa_i g_{jk} - \sGamma^{m}{}_{ij} g_{mk} - \sGamma^{m}{}_{ik} g_{jm}\n$$\nIt is then computed explicitly in terms of the metric, metric inverse, and its partial derivatives as\n$$\n\sGa^k{}_{ij} = \sha g^{km} \slp \spa_j g_{mi} + \spa_i g_{jm} - \spa_m g_{ij} \srp\n$$\nComputing the Christoffel symbols from vanishing torsion and metric compatibility is equivalent to calculating the [[spin connection]] from [[Cartan's equation]].\n\nThe Christoffel symbols (with torsion) may alternatively be computed from the [[tangent bundle spin connection|tangent bundle connection]], using the expression for the covariant derivative of the [[orthonormal basis vectors|frame]],\n$$\n\sna_i \sve{e_\sal} = \slp \spa_i \slp e_\sal \srp^j + \slp e_\sal \srp^k \sGa^j{}_{ik} \srp \sve{\spa_j} = w_{i}{}^\sbe{}_\sal \sve{e_\sbe}\n= w_{i}{}^\sbe{}_\sal \slp e_\sbe \srp^j \sve{\spa_j}\n$$\nto get\n$$\n\sGa^j{}_{ik} = \slp e_k \srp^\sal \slp w_{i}{}^\sbe{}_\sal \slp e_\sbe \srp^j - \spa_i \slp e_\sal \srp^j \srp = \slp e_\sbe \srp^j \slp w_{i}{}^\sbe{}_\sal \slp e_k \srp^\sal + \spa_i \slp e_k \srp^\sbe \srp\n$$\nThis last expression may be used to easily determine how the Christoffel symbols, which do not constitute a tensor, transform under a [[coordinate change]] to\n\sbegin{eqnarray}\n\sGa^n{}_{ml} &=& \sfr{\spa x^k}{\spa y^l} \slp e_k \srp^\sal \sfr{\spa x^i}{\spa y^m} \slp w_{i}{}^\sbe{}_\sal \slp e_\sbe \srp^j \sfr{\spa y^n}{\spa x^j} - \spa_i \slp \slp e_\sal \srp^j \sfr{\spa y^n}{\spa x^j} \srp \srp \s\s\n&=& \sfr{\spa y^n}{\spa x^j} \sfr{\spa x^i}{\spa y^m} \sfr{\spa x^k}{\spa y^l} \sGa^j{}_{ik} - \sfr{\spa x^i}{\spa y^m} \sfr{\spa x^j}{\spa y^l} \sfr{\spa^2 y^n}{\spa x^i \spa x^j} \n\send{eqnarray}\nFrom the last term we see that it's possible to choose a set of coordinates in which the Christoffel symbols vanish if and only if the torsion vanishes, $\sGa^k{}_{\slb ij \srb} = 0$. It is sometimes argued, along the lines of the [[equivalence principle|frame]], that such a choice should be possible and hence torsion should vanish. \n\nUsing the Christoffel symbols is quite old fashioned, but sometimes practical. Things may be fancied up a bit by defining the ''Christoffel 1-form''s, $\sf{\sGa^k{}_j} = \sf{dx^i} \sGa^k{}_{ij}$, and using the [[vector-form algebra]] and [[partial derivative]] to get $\sf{\sna} \sve{\spa_j} = \sf{\sGa^k{}_j} \sve{\spa_k}$ and\n$$\n\sf{\sna} \sve{e_\sal} = \sf{\sna} \slp e_\sal \srp^k \sve{\spa_k} = \sf{\spa} \sve{e_\sal} + \slp e_\sal \srp^k \sf{\sGa^j{}_k} \sve{\spa_j} = \sf{\som^\sbe{}_\sal} \sve{e_\sbe}\n$$\n
The $n=4$ dimensional ''[[spacetime]] [[Clifford algebra]]'', Cl(1,3), is built from 4 anti-commuting, [[Clifford basis vectors]], $\sga_\smu$, with positive time [[Minkowski norm|Minkowski metric]],\n$$\n\sga_\smu \scdot \sga_\snu = \sha \slp \sga_\smu \sga_\snu + \sga_\snu \sga_\smu \srp = \set_{\smu \snu}\n$$\nThe full algebra has $2^4 = 16$ [[Clifford basis elements]],\n| !Element(s) | !Grade | !Multiplicity |!Names |\n| $1$ | $0$ | $1$ |scalar |\n| $\sga_\smu$ | $1$ | $4$ |vector |\n| $\sga_{\smu \snu}$ | $2$ | $6$ |bivector |\n| $\sga_{\smu \snu \ska } = \sfr{1}{3!} \sep_{\smu \snu \ska \sla} \sga^\sla \sga$ | $3$ | $4$ |trivector |\n| $\sga_{\smu \snu \ska \sla} = \sep_{\smu \snu \ska \sla} \sga$ | 4 | $1$ |4-vector, pseudoscalar |\nThe ''spacetime [[pseudoscalar]]'', $\sga = \sga_0 \sga_1 \sga_2 \sga_3$, satisfies $\sga \sga = -1$ and anti-commutes with odd-graded elements. This algebra has several nice [[Clifford matrix representation]]s in real or complex $4\stimes4$ matrices -- the [[Dirac matrices]].
The 6 bivector [[Clifford basis elements]] of the spacetime Clifford algebra, [[Cl(1,3)]] may be represented by multiplying the 4 basis vectors, in the [[Weyl representation|Dirac matrices]], to get:\n\sbegin{eqnarray}\n\sga_{01} &=& - \ssi^P_3 \sotimes \ssi^P_1 \s\s\n\sga_{02} &=& - \ssi^P_3 \sotimes \ssi^P_2 \s\s\n\sga_{03} &=& - \ssi^P_3 \sotimes \ssi^P_3 \s\s\n\sga_{12} &=& -i 1 \sotimes \ssi^P_3 \s\s\n\sga_{13} &=& +i 1 \sotimes \ssi^P_2 \s\s\n\sga_{23} &=& -i 1 \sotimes \ssi^P_1\n\send{eqnarray}\nAny ''Cl(1,3) bivector'' may thus be represented as\n\sbegin{eqnarray}\nB &=& \sha B^{\smu \snu} \sga_{\smu \snu} =\n\slb \sbegin{array}{cc}\nB_L & 0 \s\s\n0 & B_R\n\send{array} \srb\n=\n\slb \sbegin{array}{cc}\n- B^{0 \sva} \ssi^P_\sva - i \sha B^{\sva \sze} \sep_{\sva \sze \sta} \ssi^P_\sta & 0 \s\s\n0 & B^{0 \sva} \ssi^P_\sva - i \sha B^{\sva \sze} \sep_{\sva \sze \sta} \ssi^P_\sta\n\send{array} \srb \s\s\n&=&\n\slb \sbegin{array}{cccc}\n-B^{03}- i B^{12} & -B^{01}+B^{13}+i B^{02}- i B^{23} & 0 & 0 \s\s\n-B^{01}- B^{13}-i B^{02}-i B^{23} & B^{03}+i B^{12} & 0 & 0 \s\s\n0 & 0 & B^{03}- i B^{12} & B^{01}+B^{13}-i B^{02}- i B^{23} \s\s\n0 & 0 & B^{01}- B^{13}+i B^{02}-i B^{23} & B^{03}+i B^{12}\n\send{array} \srb\n\send{eqnarray}\nin which $B_{L/R}$ are the ''left and right [[chiral]] bivector parts'', projected out by the [[left/right chirality projector]], and $\sep_{\sva \sze \sta}$ is the three dimensional [[permutation symbol]]. These $2\stimes2$ matrices satisfy $B_L^\sdagger = - B_R$, using Hermitian conjugation. Note that a bivector is completely determined by one of its chiral parts. The bivectors of [[Cl(3,1)]] have the same expression, with signs reversed since the expressions of all vectors pick up $i$'s.
The $n=16$ dimensional [[Clifford algebra]], $Cl(16,0)$, is built from 16 anti-commuting, positive norm [[Clifford basis vectors]], $\sga^{\slp16\srp}_\sal$. The full algebra has $2^{16} = 65,536$ [[Clifford basis elements]]. This algebra has many [[Clifford matrix representation]]s in real or complex $256\stimes256$ matrices. One particularly nice representation, built using the [[Kronecker product]] of [[Cl(8)]] elements, is\n\sbegin{eqnarray}\n\sga^{\slp16\srp}_\sal &=& \sGa_\sal \sotimes 1 \s\s\n\sga^{\slp16\srp}_{\slp\sal+8\srp} &=& \sGa \sotimes \sGa_\sal\n\send{eqnarray}\nwith $1 \sle \sal \sle 8$. (Note that this rep is not [[chiral]].) These $16$ ''Cl(16) basis vectors'' may be multiplied to get the $120$ ''Cl(16) basis bivectors'',\n\sbegin{eqnarray}\n\sga^{\slp16\srp}_{\sal \sbe} &=& \sGa_{\sal \sbe} \sotimes 1 \s\s\n\sga^{\slp16\srp}_{\slp\sal+8\srp \slp\sbe+8\srp} &=& 1 \sotimes \sGa_{\sal \sbe} \s\s\n\sga^{\slp16\srp}_{\sal \slp\sbe+8\srp} = -\sga^{\slp16\srp}_{\slp\sbe+8\srp \sal} &=& \slp \sGa_\sal \sGa \srp \sotimes \sGa_{\sbe}\n\send{eqnarray}\nThe [[pseudoscalar]] in this rep, $\sga^{\slp16\srp} = \sGa \sotimes \sGa$, satisfies $\sga^{\slp16\srp} \sga^{\slp16\srp} = 1$ and anti-commutes with odd-graded elements.\n\nA chiral representation for $Cl(16)$ may be built by starting with a chiral [[Cl(8)]] rep and picking out one of the vectors, such as $\sGa_8$, and using it to build the ''chiral Cl(16) basis vectors'':\n\sbegin{eqnarray}\n\sga^{\slp16\srp}_\sal &=& \sGa_\sal \sotimes 1 \s\s\n\sga^{\slp16\srp}_8 &=& \sGa_8 \sotimes \sGa \s\s\n\sga^{\slp16\srp}_{\slp\sal+8\srp} &=& \sGa_8 \sotimes \sGa_\sal \s\s\n\sga^{\slp16\srp}_{(16)} &=& \sGa_8 \sotimes \sGa_8\n\send{eqnarray}\nwith $1 \sle \sal \sle 7$. The pseudoscalar in this rep is\n$$\n\sga^{\slp16\srp} = ( \sGa \sotimes \sGa ) ( 1 \sotimes \sGa ) = \sGa \sotimes 1\n$$\nThe (1st level) chirality projector is $P_{\spm} = \sha \slp 1 \smp \sga^{\slp16\srp} \srp = \sha \slp 1 \smp \sGa \srp \sotimes 1$. The basis vectors may be used to build the ''chiral Cl(16) basis bivectors'',\n\sbegin{eqnarray}\n\sga^{\slp16\srp}_{\sal \sbe} &=& \sGa_{\sal \sbe} \sotimes 1 \s\s\n\sga^{\slp16\srp}_{\sal 8} &=& \sGa_{\sal 8} \sotimes \sGa \s\s\n\sga^{\slp16\srp}_{\sal \slp\sbe+8\srp} &=& \sGa_{\sal 8} \sotimes \sGa_\sbe \s\s\n\sga^{\slp16\srp}_{\sal (16)} &=& \sGa_{\sal 8} \sotimes \sGa_8 \s\s\n\sga^{\slp16\srp}_{\slp\sal+8\srp \slp\sbe+8\srp} &=& 1 \sotimes \sGa_{\sal \sbe} \s\s\n\sga^{\slp16\srp}_{\slp\sal+8\srp (16)} &=& 1 \sotimes \sGa_{\sal 8} \s\s\n\sga^{\slp16\srp}_{8 \slp\sbe+8\srp} &=& 1 \sotimes \sGa \sGa_\sbe \s\s\n\sga^{\slp16\srp}_{8 (16)} &=& 1 \sotimes \sGa \sGa_8\n\send{eqnarray}\n
The three dimensional [[Clifford algebra]], Cl(3,0), is generated by three [[Clifford basis vectors]], $\ssi_\sio$. These basis vectors have a matrix representation as the three [[Pauli matrices]], $\ssi_\sio=\ssi_\sio^P$. The eight [[Clifford basis elements]] are formed by all possible products of these Clifford basis vectors. The complete multiplication table for the algebra, calculated from the general [[Clifford basis product identities]], is (row header times column header equals entry):\n| | !$1$ | !$\ssi_1$ | !$\ssi_2$ | !$\ssi_3$ | !$\ssi_{12}$ | !$\ssi_{13}$ | !$\ssi_{23}$ | !$\ssi$ |\n| !$1$ | $1$ | $\ssi_1$ | $\ssi_2$ | $\ssi_3$ | $\ssi_{12}$ | $\ssi_{13}$ | $\ssi_{23}$ | $\ssi$ |\n| !$\ssi_1$ | $\ssi_1$ | $1$ |bgcolor(#a0ffa0): $\ssi_{12}$ |bgcolor(#a0ffa0): $\ssi_{13}$ | $\ssi_2$ | $\ssi_3$ | $\ssi$ |bgcolor(#a0ffa0): $\ssi_{23}$ |\n| !$\ssi_2$ | $\ssi_2$ |bgcolor(#a0ffa0): $-\ssi_{12}$ | $1$ |bgcolor(#a0ffa0): $\ssi_{23}$ | $-\ssi_1$ | $-\ssi$ | $\ssi_3$ |bgcolor(#a0ffa0): $-\ssi_{13}$ |\n| !$\ssi_3$ | $\ssi_3$ |bgcolor(#a0ffa0): $-\ssi_{13}$ |bgcolor(#a0ffa0): $-\ssi_{23}$ | $1$ | $\ssi$ | $-\ssi_1$ | $-\ssi_2$ |bgcolor(#a0ffa0): $\ssi_{12}$ |\n| !$\ssi_{12}$ | $\ssi_{12}$ | $-\ssi_2$ | $\ssi_1$ | $\ssi$ |bgcolor(#88ccff): $-1$ |bgcolor(#88ccff): $-\ssi_{23}$ |bgcolor(#88ccff): $\ssi_{13}$ | $-\ssi_3$ |\n| !$\ssi_{13}$ | $\ssi_{13}$ | $-\ssi_3$ | $-\ssi$ | $\ssi_1$ |bgcolor(#88ccff): $\ssi_{23}$ |bgcolor(#88ccff): $-1$ |bgcolor(#88ccff): $-\ssi_{12}$ | $\ssi_2$ |\n| !$\ssi_{23}$ | $\ssi_{23}$ | $\ssi$ | $-\ssi_3$ | $\ssi_2$ |bgcolor(#88ccff): $-\ssi_{13}$ |bgcolor(#88ccff): $\ssi_{12}$ |bgcolor(#88ccff): $-1$ | $-\ssi_1$ |\n| !$\ssi$ | $\ssi$ |bgcolor(#a0ffa0): $\ssi_{23}$ |bgcolor(#a0ffa0): $-\ssi_{13}$ |bgcolor(#a0ffa0): $\ssi_{12}$ | $-\ssi_3$ | $\ssi_2$ | $-\ssi_1$ | $-1$ |\nThe blue square shows the bivector subalgebra. This bivector subalgebra is the [[three dimensional special unitary group Lie algebra|su(2)]], with the identification $T_A = \ssi \ssi_A = \sepsilon_{ABC} \ssi_{BC} = i \ssi_A^P$ giving the three $su(2)$ generators,\n$$\n\sbegin{array}{ccc}\nT_1 = i \ssigma_{1}^{P} = \ssi_{23}\n&\nT_2 = i \ssigma_{2}^{P} = -\ssi_{13}\n&\nT_3 = i \ssigma_{3}^{P} = \ssi_{12}\n\send{array}\n$$\nwhich form a closed subalgebra under the [[commutator]]. The green entries illustrate the two ways the bivector basis elements can be represented -- as the product of vectors, or as the product of vector and pseudoscalar. The [[pseudoscalar]], $\ssi=\ssi_1 \ssi_2 \ssi_3$, squares to $-1$ and has the matrix representation $\ssi = i$.\n\nThe ''three dimensional Clifford algebra of negative signature'', $Cl(0,3)$, is obtained by using $\ssi'_\sio = i \ssi_\sio$ as the basis vectors.
The $n=4$ dimensional ''[[spacetime]] [[Clifford algebra]]'', Cl(3,1), is built from 4 anti-commuting, [[Clifford basis vectors]], $\sga_\smu$, with negative time [[Minkowski norm|Minkowski metric]],\n$$\n\sga_\smu \scdot \sga_\snu = \sha \slp \sga_\smu \sga_\snu + \sga_\snu \sga_\smu \srp = \set_{\smu \snu}\n$$\nThe full algebra has $2^4 = 16$ [[Clifford basis elements]],\n| !Element(s) | !Grade | !Multiplicity |!Names |\n| $1$ | $0$ | $1$ |scalar |\n| $\sga_\smu$ | $1$ | $4$ |vector |\n| $\sga_{\smu \snu}$ | $2$ | $6$ |bivector |\n| $\sga_{\smu \snu \ska } = \sfr{1}{3!} \sep_{\smu \snu \ska \sla} \sga^\sla \sga$ | $3$ | $4$ |trivector |\n| $\sga_{\smu \snu \ska \sla} = \sep_{\smu \snu \ska \sla} \sga$ | 4 | $1$ |4-vector, pseudoscalar |\nThe ''spacetime [[pseudoscalar]]'', $\sga = \sga_0 \sga_1 \sga_2 \sga_3$, satisfies $\sga \sga = -1$ and anti-commutes with odd-graded elements. This algebra has several nice [[Clifford matrix representation]]s in real or complex $4\stimes4$ matrices -- the [[Dirac matrices]].
The $n=8$ dimensional [[Clifford algebra]], ''Cl(5,3)'', is built from 5 positive norm and 3 negative norm [[Clifford basis vectors]], $\sGa_\sal$. It is the same as [[Cl(8)]] except for the signature.\n\nThis algebra has many [[Clifford matrix representation]]s in real or complex $16\stimes16$ matrices. One particularly nice, [[chiral]], complex representation, built using the [[Kronecker product]] of [[Pauli matrices]], is\n\sbegin{eqnarray}\n\sGa_1 &=& i \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \s\s\n\sGa_2 &=& i \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_3 &=& i \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \s\s\n\sGa_0 = \sGa_4 &=& \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \s\s\n\sGa_5 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \s\s\n\sGa_6 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \s\s\n\sGa_7 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_8 &=& \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \sotimes \ssi^P_0\n\send{eqnarray}\ngiving $\sGa = - i \s, \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0$.\n\nStandard model using this Cl(5,3) rep:\nHas correct Higgs.\nAh, no good -- would give negative [[cosmological constant]].\n$$\n{\sscriptsize\n\slb \sbegin{array}{cccccccc}\n\sha \sf{\som_L} \s!+\s! i \sf{W^3} & i \sf{W^1} \s!+\s! \sf{W^2} & - \sfr{1}{4} \sf{e_R} \sph_0^* & \sfr{1}{4} \sf{e_R} \sph_+ &\n\sud{\snu_L} & \sud{u_L^r} & \sud{u_L^b} & \sud{u_L^g} \s\s\n\ni \sf{W^1} \s!-\s! \sf{W^2} & \sha \sf{\som_L} \s!-\s! i \sf{W^3} & \sfr{1}{4} \sf{e_R} \sph_+^* & \sfr{1}{4} \sf{e_R} \sph_0 &\n\sud{e_L} & \sud{d_L^r} & \sud{d_L^b} & \sud{d_L^g} \s\s\n\n\sfr{1}{4} \sf{e_L} \sph_0 & -\sfr{1}{4} \sf{e_L} \sph_+ & \sha \sf{\som_R} \s!+\s! i \sf{B} & &\n\sud{\snu_R} & \sud{u_R^r} & \sud{u_R^b} & \sud{u_R^g} \s\s\n\n-\sfr{1}{4} \sf{e_L} \sph_+^* & -\sfr{1}{4} \sf{e_L} \sph_0^* & & \sha \sf{\som_R} \s!-\s! i \sf{B} &\n\sud{e_R} & \sud{d_R^r} & \sud{d_R^b} & \sud{d_R^g} \s\s\n\n& & & & i \sf{B} & & & \s\s\n& & & & & \sfr{-i}{3} \sf{B} \s!+\s! i \sf{G^{3+8}} & i\sf{G^1} \s!-\s! \sf{G^2} & i\sf{G^4} \s!-\s! \sf{G^5} \s\s\n& & & & & i\sf{G^1} \s!+\s! \sf{G^2} & \sfr{-i}{3} \sf{B} \s!-\s! i \sf{G^{3+8}} & i\sf{G^6} \s!-\s! \sf{G^7} \s\s\n& & & & & i\sf{G^4} \s!+\s! \sf{G^5} & i\sf{G^6} \s!+\s! \sf{G^7} & \sfr{-i}{3} \sf{B} \s!-\s!\s! \sfr{2i}{\ssqrt{3}}\sf{G^8}\n\send{array} \srb\n}\n$$
The $n=8$ dimensional [[Clifford algebra]], $Cl(8,0)$, is built from 8 anti-commuting, positive norm [[Clifford basis vectors]], $\sGa_\sal$. The full algebra has $2^8 = 256$ [[Clifford basis elements]],\n| !Element(s) | !Grade | !Multiplicity |!Names |\n| $1$ | $0$ | $1$ |scalar |\n| $\sGa_\sal$ | $1$ | $8$ |vector |\n| $\sGa_{\sal \sbe}$ | $2$ | $28$ |bivector |\n| $\sGa_{\sal \sbe \sga}$ | $3$ | $56$ |trivector |\n| $\sGa_{\sal \sbe \sga \sde}$ | $4$ | $70$ |4-vector |\n| $\sGa_{\sal \sbe \sga \sde \sep}$ | $5$ | $56$ |5-vector |\n| $\sGa_{\sal \sdots \sbe}$ | $6$ | $28$ |6-vector |\n| $\sGa_{\sal \sdots \sbe}$ | $7$ | $8$ |7-vector |\n| $\sGa_{\sal \sdots \sbe} = \sep_{\sal \sdots \sbe} \sGa$ | $8$ | $1$ |8-vector, psuedoscalar |\nThe [[pseudoscalar]], $\sGa$, satisfies $\sGa \sGa = 1$ and anti-commutes with odd-graded elements. This algebra has many [[Clifford matrix representation]]s in real or complex $16\stimes16$ matrices. One particularly nice, [[chiral]], complex representation, built using the [[Kronecker product]] of [[Pauli matrices]], is\n\sbegin{eqnarray}\n\sGa_1 &=& \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \s\s\n\sGa_2 &=& \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_3 &=& \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \s\s\n\sGa_4 &=& \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \s\s\n\sGa_5 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \s\s\n\sGa_6 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \s\s\n\sGa_7 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_0 = \sGa_8 &=& \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0\n\send{eqnarray}\ngiving $\sGa = \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0$. These may all be expressed in a $16\stimes16$ matrix (using $2\stimes2$ sub-matrices) as\n\sbegin{eqnarray}\n& & \sGa_\spi + z \sGa_4 + a \sGa_5 + b \sGa_6 + c \sGa_7 + \sGa_8 = \s\s\n& & \slb\n\sbegin{array}{cccccccc}\n& & & & 1-i\ssi^p_\spi & & -z-ic & -b-ia \s\s\n& & & & & 1-i\ssi^p_\spi & b-ia & -z+ic \s\s\n& & & & z-ic & -b-ia & 1+i\ssi^p_\spi & \s\s\n& & & & b-ia & z+ic & & 1+i\ssi^p_\spi \s\s\n1+i\ssi^p_\spi & & z+ic & b+ia & & & & \s\s\n& 1+i\ssi^p_\spi & -b+ia & z-ic & & & & \s\s\n-z+ic & b+ia & 1-i\ssi^p_\spi & & & & & \s\s\n-b+ia & -z-ic & & 1-i\ssi^p_\spi & & & &\n\send{array}\n\srb\n\send{eqnarray}\n\nA nice chiral, real representation of $Cl(8,0)$ is\n\sbegin{eqnarray}\n\sGa_1 &=& \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_2 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_3 &=& \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_4 &=& \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \s\s\n\sGa_5 &=& \ssi^P_2 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \s\s\n\sGa_6 &=& \ssi^P_2 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \s\s\n\sGa_7 &=& \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \s\s\n\sGa_8 &=& \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0\n\send{eqnarray}\n\nThe ''chirality operator for Cl(8)'' is $P^{\slp8\srp}_\spm = \sha \slp 1 \spm \sGa \srp$.
The 28 bivector [[Clifford basis elements]] of [[Cl(8,0)|Cl(8)]] may be represented by multiplying the 8 basis vectors, in the complex rep, to get:\n\sbegin{eqnarray}\n\sGa_{01} &=& i \ssi^P_3 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \s\s\n\sGa_{02} &=& i \ssi^P_3 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_{03} &=& i \ssi^P_3 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \s\s\n\sGa_{12} &=& i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \s\s\n\sGa_{13} &=& -i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_{23} &=& i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \s\s\n&\s,& \s\s\n\sGa_{04} &=& i \ssi^P_3 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \s\s\n\sGa_{05} &=& i \ssi^P_3 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \s\s\n\sGa_{06} &=& i \ssi^P_3 \sotimes \ssi^P_1 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \s\s\n\sGa_{07} &=& i \ssi^P_3 \sotimes \ssi^P_1 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_{14} &=& -i \ssi^P_0 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \s\s\n\sGa_{15} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \s\s\n\sGa_{16} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \s\s\n\sGa_{17} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_1 \s\s\n\sGa_{24} &=& -i \ssi^P_0 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \sotimes \ssi^P_2 \s\s\n\sGa_{25} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_2 \s\s\n\sGa_{26} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \s\s\n\sGa_{27} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_2 \s\s\n\sGa_{34} &=& -i \ssi^P_0 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \s\s\n\sGa_{35} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_3 \s\s\n\sGa_{36} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \s\s\n\sGa_{37} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_3 \s\s\n&\s,& \s\s\n\sGa_{45} &=& -i \ssi^P_0 \sotimes \ssi^P_3 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \s\s\n\sGa_{46} &=& -i \ssi^P_0 \sotimes \ssi^P_3 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \s\s\n\sGa_{47} &=& -i \ssi^P_0 \sotimes \ssi^P_3 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_{56} &=& i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_{57} &=& -i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \s\s\n\sGa_{67} &=& i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \sotimes \ssi^P_0\n\send{eqnarray}\n
The ''Clifford adjoint'' transformation of a [[Clifford element]], $A$, by a [[Clifford group]] element, $U$, is\n\s[ A' = U A U^- \s]\nThe ''Clifford inner [[automorphism]]'', a.k.a. //''similarity transformation''//, is the Clifford adjoint transformation of the [[Clifford basis vectors]], \n\s[ \sga'_\sal = U \sga_\sal U^- \s]\nThis subsequently produces the Clifford adjoint transformation of all Clifford elements built from these basis vectors, since\n$$\n\sga'_{\sal \sdots \sbe} = \sga'_\sal \sdots \sga'_\sbe = U \sga_\sal U^- \sdots U \sga_\sbe U^- = U \sga_{\sal \sdots \sbe} U^-\n$$\nIt is an automorphism because it is a map, specified by $U \sin Cl^*$, from the [[Clifford algebra]] into itself.\n\nThe adjoint transformation does not necessarily preserve the [[grade|Clifford grade]] of elements. It does, however, preserve scalars, $\sli A' B' \sri = \sli UAU^- UBU^- \sri = \sli A B \sri$. The [[fundamental Clifford identity|Clifford basis vectors]], $\sga_\sal \scdot \sga_\sbe = \set_{\sal \sbe}$, is preserved by the Clifford automorphism, $\sga'_\sal \scdot \sga'_\sbe = \set_{\sal \sbe}$, preserving the structure of the Clifford algebra and the decomposition of [[Clifford element]]s even though the transformed basis "vectors", $\sga'_\sal$, may no longer be grade 1 with respect to the old basis.\n\nFor Clifford group elements near the identity, $U \ssimeq 1 + \sha C$, the Clifford adjoint is approximately\n\s[ A' = U A U^- \ssimeq \slp 1 + \sha C \srp A \slp 1 - \sha C \srp \ssimeq A + C \stimes A \s]\nwith the "small" Clifford element, $C$, acting via the [[cross product|antisymmetric bracket]].
An "$n$ dimensional" ''Clifford algebra'', $Cl(p,q)$, is a $2^n$ dimensional [[Lie algebra]] of [[Clifford element]]s consisting of coefficients multiplying [[Clifford basis elements]] constructed from $p$ positive norm and $q$ negative norm ($n=p+q$) [[Clifford basis vectors]], $\sga_\sal$. The Clifford algebra product of any two Clifford elements, equivalent to the [[matrix product in a suitable representation|Clifford matrix representation]], is non-commutative and decomposes into ''symmetric product'' (''//dot product//'') and [[antisymmetric product|antisymmetric bracket]] (''//cross product//'') parts,\n\sbegin{eqnarray}\nAB &=& A \scdot B + A \stimes B\s\s\nA \scdot B &=& \sha \slp AB+BA \srp\s\s\nA \stimes B &=& \sha \slp AB-BA \srp\n\send{eqnarray}\nThe product is associative and distributive,\n\sbegin{eqnarray}\nA \slp B C \srp = \slp A B \srp C\s\s\nA \slp B + C \srp = A B + A C\n\send{eqnarray}\nAnd, just as for matrices, [[almost all elements|Clifford group]] have an [[inverse]], $AA^-=1$.\n\nA Clifford algebra is a graded "geometric algebra" in that the elements of [[Clifford grade]] $0,1,2,3,\sdots$ may be considered as scalars, vectors, areas, volumes, ... and the Clifford product as operations between them. For example, the product of two vectors is a scalar (their dot product) plus an area (their cross product). The antisymmetric product of three vectors is a 3-vector, or volume. The product and its decomposition are completely described by the [[Clifford basis product identities]].
Any two [[Clifford basis elements]] are orthogonal under the [[scalar part|Clifford grade]] operator. Taking the scalar part of two multiplied basis elements of grade $r$ gives the orthogonality relation,\n\s[ \sli \sga_{\sal \sdots \sbe} \sga^{\sga \sdots \sde} \sri = r! \sde^\sga_{\slb \sbe \srd} \sdots \sde^\sde_{\sld \sal \srb} \s]\nin which the basis element indices have been raised with the [[Minkowski metric]]. The scalar part of any two multiplied basis elements of unequal grade is 0.\n\nThe orthogonality relation may be used to determine the ''scalar product'' of any two multivectors. For example, between a multivector, $A$, and bivector, $B$, the scalar product is\n\s[ \sli A B \sri = \sfr{1}{4} A^{\sal \sbe} B_{\sga \sde} \sli \sga_{\sal \sbe} \sga^{\sga \sde} \sri = \sfr{1}{4} A^{\sal \sbe} B_{\sga \sde} 2 \sde^\sga_{\slb \sbe \srd} \sde^\sde_{\sld \sal \srb} = \sha A^{\sal \sbe} B_{\sbe \sal} \s]\n\nThe scalar part operator, and the orthogonality relations, is equivalent to the matrix [[trace]] and [[Lie algebra]] generator orthogonality through the [[Killing form]].
The $2^n$ ''Clifford basis elements'' are formed by all possible products of the $n$ [[Clifford basis vectors]]. Because of the [[fundamental Clifford identity|Clifford basis vectors]], basis elements are antisymmetric under the exchange of indices, like [[coordinate basis forms]], and may be written via the [[antisymmetric bracket]]. Each basis element has a [[grade|Clifford grade]], $q$, corresponding to the number of constituent basis vectors, and a multiplicity, ${n \schoose q} = \sfr{n!}{q!(n-q)!}$, equal to the number of their ordered combinations,\n| !Element(s) | !Grade | !Multiplicity |!Names |\n| $1$ | $0$ | ${n \schoose 0} = 1$ |scalar, real number |\n| $\sga_\sal$ | $1$ | ${n \schoose 1} = n$ |vector, [[Clifford basis vectors]]|\n| $\sga_{\sal \sbe} = \sga_{\slb \sal \sbe \srb} = \sga_\sal \sga_\sbe = \slb \sga_\sal,\sga_\sbe \srb_A$ | 2 | ${n \schoose 2} = \sha n (n-1)$ |bivector, 2-vector |\n| $\sga_{\sal \sbe \sga} = \sga_{\slb \sal \sbe \sga \srb} = \sga_\sal \sga_\sbe \sga_\sga = \slb \sga_\sal,\sga_\sbe,\sga_\sga \srb_A$ | 3 | ${n \schoose 3} = \sfr{1}{3!} n (n-1)(n-2)$ |trivector, 3-vector |\n| $\svdots$ | $\svdots$ | $\svdots$ |$\svdots$ |\n| $\sga_{\sal \sdots \sbe} = \sep_{\sal \sdots \sbe} \sga$ | $n$ | ${n \schoose n} = 1$ |n-vector |\nEach of these $\ssum_{k=0}^n {n \schoose k} =2^n$ Clifford basis elements is a [[Lie algebra]] generator, with structure coefficients corresponding to [[Clifford basis product identities]]. The Clifford basis elements also satisfy [[Clifford basis element orthogonality]].\n\nUsing [[Clifford dual]]ity, it is often convenient to express high grade basis elements in terms of the Clifford [[pseudoscalar]],\n$$\sga = \sga_0 \sga_1 \sdots \sga_{n-1}$$\nand the [[permutation symbol]]. In this way, the basis r-vectors can be written as\n$$\sga_{\sal \sdots \sbe} = \sfr{1}{\slp n-r \srp!} \sep_{\sal \sdots \sbe \sga \sdots \sde} \sga^{\sga \sdots \sde} \sga$$\nFor example, the $n$ basis (n-1)-vectors are\n$$\sga_{\sal \sdots \sbe} = \sfr{1}{\slp n-1 \srp!} \sep_{\sal \sdots \sbe \sga} \sga^\sga \sga$$\nThis reduces the number of indices necessary to represent high grade [[Clifford element]]s.\n\nThe $2^n$ basis elements can also be written via a generalized index as $\sga_A$, with $A$ enumerating\neach different antisymmetric combination of the usual Clifford indices.
The ''Clifford basis product identities'' are derived from the [[fundamental Clifford identity|Clifford basis vectors]] by splitting the product of two basis vectors into a scalar (a [[Minkowski metric]] component) plus a bivector,\n\s[ \sga_\sal \sga_\sbe = \sga_\sal \scdot \sga_\sbe + \sga_\sal \stimes \sga_\sbe = \set_{\sal \sbe} + \sga_{\sal \sbe} \s]\nor going in reverse &mdash; rewriting a bivector as a scalar plus the product of two basis vectors. By selectively applying this rule, all [[Clifford element]]s can be written as sums of coefficients times [[Clifford basis elements]]. The structure coefficients characterizing the [[Clifford algebra]] as a [[Lie algebra]] can be read off the [[cross product|antisymmetric bracket]] identities,\n\sbegin{eqnarray}\n\sga_\sal \stimes \sga_\sbe &=& \sga_{\sal \sbe}\s\s\n\sga_\sal \stimes \sga_{\sbe \sga} &=& \set_{\sal \sbe} \sga_{\sga} - \set_{\sal \sga} \sga_{\sbe}\s\s\n\sga_{\sal \sbe} \stimes \sga_{\sga \sde} &=& - \set_{\sal \sga} \sga_{\sbe \sde} + \set_{\sal \sde} \sga_{\sbe \sga} + \set_{\sbe \sga} \sga_{\sal \sde} - \set_{\sbe \sde} \sga_{\sal \sga}\s\s\n\sga_\sal \stimes \sga_{\sbe \sga \sde} &=& \sga_{\sal \sbe \sga \sde} \s\s\n&\svdots& \n\send{eqnarray}\nEqually useful identities arise for the symmetric product,\n\sbegin{eqnarray}\n\sga_\sal \scdot \sga_\sbe &=& \set_{\sal \sbe}\s\s\n\sga_\sal \scdot \sga_{\sbe \sga} &=& \sga_{\sal \sbe \sga}\s\s\n\sga_{\sal \sbe} \scdot \sga_{\sga \sde} &=& \slp \set_{\sal \sde} \set_{\sbe \sga} - \set_{\sal \sga} \set_{\sbe \sde} \srp + \sga_{\sal \sbe \sga \sde}\s\s\n\sga_\sal \scdot \sga_{\sbe \sga \sde} &=& \set_{\sal\sbe} \sga_{\sga\sde} - \set_{\sal\sga} \sga_{\sbe\sde} + \set_{\sal\sde} \sga_{\sbe\sga} \s\s\n &\svdots& \n\send{eqnarray}\nContinuing the series, the product of two basis elements of [[grade|Clifford grade]]s $p$ and $q$, such as\n\sbegin{eqnarray}\n\sga_{\sal \sbe} \sga_{\sga \sde} &=& \sga_{\sal \sbe} \scdot \sga_{\sga \sde} + \sga_{\sal \sbe} \stimes \sga_{\sga \sde}\s\s\n &=& \slp \set_{\sal \sde} \set_{\sbe \sga} - \set_{\sal \sga} \set_{\sbe \sde} \srp + \slp \set_{\sal \sga} \sga_{\sbe \sde} + \set_{\sal \sde} \sga_{\sbe \sga} + \set_{\sbe \sga} \sga_{\sal \sde} + \set_{\sbe \sde} \sga_{\sal \sga} \srp + \sga_{\sal \sbe \sga \sde}\n\send{eqnarray}\ngives a result of mixed grades $|p-q|$ through $p+q \sle n$ in steps of $2$. For example, a bivector times a 3-vector typically gives a vector plus a 3-vector plus a 5-vector if $n$ is at least 5, otherwise just a vector plus a 3-vector.\n\nThe products of even or odd graded elements are\n| !grade of $A$ | !grade of $B$ | !grade of $AB$ |\n| even | even | even |\n| odd | odd | even |\n| odd | even | odd |\nThe cross product of anything with a bivector is grade preserving.
A rest [[frame]] exists at each point in a curved [[spacetime]]. A sufficiently small surrounding region is described locally by a diagonal [[Minkowski metric]], $\seta_{\sal \sbe}$, with $p$ positive and $q$ negative unit entries. This may be visualized by considering a set of $n$ ''orthonormal'' (orthogonal and unit length) geometric "vector" elements, the ''Clifford basis vectors'', or ''//Clifford algebra generators//'', $\sga_\sal$. These Clifford basis vectors provide a means for invariantly describing local geometric objects.\n\nLike [[coordinate basis 1-forms]], two unequal Clifford basis vectors anti-commute, and their product represents a geometric area, or ''bivector'', element such as $\sga_1 \sga_2 = - \sga_2 \sga_1$, representing a unit area element spanned by $\sga_1$ and $\sga_2$. The orthonormality of Clifford basis vectors is expressed by the ''fundamental Clifford identity'',\n\s[ \sga_\sal \scdot \sga_\sbe = \sha \slp \sga_\sal \sga_\sbe + \sga_\sbe \sga_\sal \srp = \set_{\sal \sbe} \s]\nwhich gives a Clifford scalar (real number) as a result of the symmetric product of two Clifford vectors. The antisymmetric product of every combination of two unequal Clifford vectors gives the $\sha n (n-1)$ bivector [[Clifford basis elements]],\n\s[ \sga_\sal \stimes \sga_\sbe = \sha \slb \sga_\sal, \sga_\sbe \srb = \sha \slp \sga_\sal \sga_\sbe - \sga_\sbe \sga_\sal \srp = \sga_\sal \sga_\sbe = \sga_{\slb \sal \sbe \srb} = \sga_{\sal \sbe} \s]
The ''Clifford bundle'', $Cl M$, with base [[manifold]] $M$ is an [[automorphism bundle]] and a [[vector bundle]] with $2^n$ fiber basis elements equal to the [[Clifford basis elements]], $\sga_{\sal \sdots \sbe}$. The fiber at each base manifold point, $p$, is the space of [[Clifford element]]s. The transition functions for the basis elements over overlapping patches, $U_1$ and $U_2$, are given by [[Clifford adjoint]]s,\n$$\n\sga_{\sal \sdots \sbe}^2 = U_{21} \sga_{\sal \sdots \sbe}^1 U_{21}^-\n$$\nwhich don't necessarily preserve [[Clifford grade]]. The structure group, $Aut(Cl)=Cl^*$, the automorphism group, is the [[Clifford group]] with adjoint action on the fiber. //(Is that true?)//\n\nFor a section, $C(x)$, transforming under the adjoint action [[gauge transformation]], $C \smapsto C'=U C U^-$, the [[covariant derivative]] is\n$$\n\sf{\sna} C = \sf{d} C + \sha \sf{A} C - \sha C \sf{A} = \sf{d} C + \sf{A} \stimes C\n$$\n(defined with a $\sha$ in it to keep things pretty) with the [[Clifford connection]], $\sf{A}$, applied using the [[cross product|Clifford algebra]].\n\nAny fiber element, $C$, at $t=0$ may be [[parallel transport]]ed to $C(t)=U(t)CU^-$ along a path on the base by a parameter dependent Clifford element, the path holonomy, $U(t) = Pe^{- \sha \sint_0^t \sf{A}}$, satisfying the [[path holonomy]] equation,\n$$\n\sfr{d}{dt} U(t) = - \sha \sve{v} \sf{A} U\n$$\n\nApplying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),\n\sbegin{eqnarray}\n\sf{\sna} \sf{\sna} C &=& \sf{d} \slp \sf{d} C + \sha \sf{A} C - \sha C \sf{A} \srp + \sha \sf{A} \slp \sf{d} C + \sha \sf{A} C - \sha C \sf{A} \srp + \sha \slp \sf{d} C + \sha \sf{A} C - \sha C \sf{A} \srp \sf{A} \s\s\n&=& \sha \slp \sf{d} \sf{A} \srp C - \sha \sf{A} \sf{d} C - \sha \slp \sf{d} C \srp \sf{A} - \sha C \sf{d} \sf{A} \n + \sha \sf{A} \slp \sf{d} C + \sha \sf{A} C - \sha C \sf{A} \srp + \sha \slp \sf{d} C + \sha \sf{A} C - \sha C \sf{A} \srp \sf{A} \s\s\n&=& \sff{F} \stimes C\n\send{eqnarray}\ngives the [[Clifford curvature|Clifford-Riemann curvature]],\n$$\n\sff{F} = \sf{d} \sf{A} + \sha \sf{A} \stimes \sf{A}\n$$\na Clifford valued 2-form. This expression for the curvature may alternatively be obtained from the [[holonomy]] (minding the new factor of $\sha$ in the path holonomy equation).\n\nUnder a gauge transformation, $C(x) \smapsto C'(x) = U(x) C(x) U^-(x)$, the covariant derivative changes to\n\sbegin{eqnarray}\n\sf{\sna'} C' &=& U \slp \sf{\sna} C \srp U^-\s\s\n\sf{d} \slp U C U^- \srp + \sha \sf{A'} U C U^- - \sha U C U^- \sf{A'} &=& U \slp \sf{d} C \srp U^- + \sha U \sf{A} C U^- - \sha U C \sf{A} U^-\n\send{eqnarray}\ngiving the transformation law for the connection,\n$$\n\sf{A'} = U \sf{A} U^- - 2 \slp \sf{d} U \srp U^- = U \sf{A} U^- + 2 U \slp \sf{d} U^- \srp \n$$\nAn infinitesimal transformation, $U \ssimeq 1 + \sha C$, changes the connection to\n$$\n\sf{A'} \ssimeq \sf{A} - \sf{d} C - \sha \sf{A} C + \sha C \sf{A} = \sf{A} - \sf{\sna} C\n$$\nThe curvature consequently transforms under a gauge transformation to\n$$\n\sff{F'} = \sf{d} \sf{A'} + \sha \sf{A'} \stimes \sf{A'} = U \sff{F} U^- \ssimeq \sff{F} + C \stimes \sff{F}\n$$\n\nThe covariant derivative acting on a [[Clifform]] such as the curvature, transforming under the adjoint action, $\sff{F'} = U \sff{F} U^-$, is still \n$$\n\sf{\sna} \sff{F} = \sf{d} \sff{F} + \sf{A} \stimes \sff{F} \n$$\n\nThe [[graded Clifford bundle|Clifford vector bundle]] has the same fiber as the [[Clifford bundle]], but the transition functions (which for the graded Clifford bundle are grade preserving) are [[Clifford rotation]]s.
One may carry out several unary operations on [[Clifford element]]s.\n\nThe [[inverse]] of a Clifford element, $A^-$, is most generally computed by working in a [[Clifford matrix representation]]. However, some cases may be handled easily, such as the inverse of a Clifford vector, $v^- = \sfr{v}{v \scdot v}$.\n\nThe [[Clifford dual]] of an element, $A \sga^-$, is often a useful object.\n\nThe ''involution'' operator inverts the signs of all vectors in an element, producing a [[grade|Clifford grade]] dependent sign change for the parts of an element, $\shat{A^r} = \slp -1 \srp^r A^r$, also expressible as $\shat{A} = A^e - A^o$.\n\nThe ''reversion operator'', a.k.a. ''//reverse//'', reverses the order of all vectors multiplied in an element, producing $\stilde{A^r} = \slp -1 \srp^{\sha r(r-1)} A^r$.\n\n''Clifford conjugation'' combines these last two, $\sbar{A} = \stilde{\shat{A^r}} = \slp -1 \srp^{\sha r(r+1)} A^r$.\n\nFor a set of [[Dirac matrices]] in which $\sga_0$ is represented by a Hermitian matrix and all spatial [[Clifford basis vectors]] are represented by anti-Hermitian matrices, the ''Hermitian conjugate'' of a Clifford element is $A^\sdagger = \sga_0 \stilde{A} \sga^0$. When written as a matrix, this gives the transpose of the complex conjugate, $A^\sdagger = A^{*T}$.\n\nOne often encounters the ''Dirac conjugate'', $\soverline{A} = A^\sdagger \sga_0 = \sga_0 \stilde{A}$, which shouldn't be confused with the Clifford conjugate.
The [[vector bundle connection]] for the [[Clifford bundle]] is defined through the operation of the suitable [[vector bundle covariant derivative|vector bundle connection]] on the [[Clifford basis vectors]] for the $Cl$ fiber. The structure group for the bundle is the [[Clifford group]], with group elements acting on the fiber through the [[Clifford adjoint]]. The covariant derivative may therefore be represented using a ''Clifford connection'', $\sf{A} \sin \sf{Cl}$, acting on basis elements via the [[cross product|antisymmetric bracket]],\n$$\n\sf{\sna} \sga_\sal = \sf{A} \stimes \sga_\sal \n$$\nwhich gives the ''Clifford covariant derivative'' acting on any Clifford valued field (Clifford bundle section),\n$$\n\sf{\sna} C = \sf{d} C + \sf{A} \stimes C \n$$\n \nNote that the covariant derivative for the Clifford bundle does not necessarily preserve [[Clifford grade]].
The ''Clifford curvature scalar'' is obtained by taking the [[scalar part|Clifford grade]] of the [[frame]] contracted twice with the [[Clifford bundle]] curvature,\n$$\nR = \sli \sve{e} \sve{e} \sff{F} \sri\n$$\nIf, specifically, we are working with the [[Clifford vector bundle]], the Clifford curvature scalar is then the result of the [[dot product|Clifford algebra]] of the frame with the [[Clifford-Ricci curvature]],\n$$\nR = \sli \sve{e} \sve{e} \sff{R} \sri = \sve{e} \scdot \sf{R} = \slp \sve{e} \stimes \sve{e} \srp \scdot \sff{R}\n$$\nand equals the [[curvature scalar]] written in terms of the [[spin connection]] and frame.\n\nThe Clifford curvature scalar also comes from the expression:\n\sbegin{eqnarray}\n\sfr{2}{\slp n-2 \srp!} \sli \slp \sf{e} \srp^{n-2} \sff{R} \sga^- \sri\n&=& \sfr{2}{\slp n-2 \srp!} \sf{e}^\sal \sdots \sf{e}^\sbe \sf{e}^\smu \sf{e}^\snu \sfr{1}{4} R_{\smu\snu}{}^{\srh\ssi} \sli \sga_{\sal \sdots \sbe} \sga_{\srh \ssi} \sga^- \sri \s\s\n&=& \sfr{2}{\slp n-2 \srp!} \snf{e} \sep^{\sal \sdots \sbe \smu \snu} \sfr{1}{4} R_{\smu\snu}{}^{\srh\ssi} \sep_{\sal \sdots \sbe \srh \ssi} \s\s\n&=& \snf{e} \sde_{\slb \srh \ssi \srb}^{\smu\snu} R_{\smu\snu}{}^{\srh \ssi}=\snf{e} R\n\send{eqnarray}\nusing the [[volume form]] and [[permutation identities]].
The ''Clifford dual'' of any [[Clifford element]], $A$, is obtained by right multiplying it by the inverse [[pseudoscalar]], $A \sga^-$. For a [[Clifford grade]] $r$ element, this gives a grade $(n-r)$ element,\n\s[ A^r \sga^- = \sfr{1}{r!} A^{\sal \sdots \sbe} \sga_{\sal \sdots \sbe} \sga^- = \sfr{1}{r! \slp n - r \srp !} A^{\sal \sdots \sbe} \sli \sga_{\sal \sdots \sbe \sga \sdots \sde} \sga^- \sri \sga^{\sga \sdots \sde} = \sfr{1}{r! \slp n - r \srp !} A^{\sal \sdots \sbe} \sep_{\sal \sdots \sbe \sga \sdots \sde} \sga^{\sga \sdots \sde} \s]\nin which $\sep_{\sal \sdots \sbe \sga \sdots \sde}$ is the [[permutation symbol]], and [[indices]] are raised with the [[Minkowski metric]].\n\nThe Clifford dual transformation is analogous to the [[Hodge dual]].
All [[Clifford algebra]] elements may be written as a sum of $2^n$ real coefficients multiplying [[Clifford basis elements]], with multiplicative factors included to account for the redundant sums over [[indices]],\n\sbegin{eqnarray}\nA &=& A^s + A^\sal \sga_\sal + \sha A^{\sal \sbe} \sga_{\sal \sbe} + \sfr{1}{3!} A^{\sal \sbe \sga} \sga_{\sal \sbe \sga} + \sdots + A^p \sga\s\s\n&=& A^0 + A^1 + A^2 + A^3 + \sdots + A^n\n\send{eqnarray}\n(Some people choose to limit the sums so they don't run over all index values &mdash; but this isn't done here.) Like the coefficients of [[differential form]]s, the Clifford element coefficients are [[antisymmetric|index bracket]] in their indices, $A^{\sal \sdots \sbe}=A^{\slb \sal \sdots \sbe \srb}$. Unlike differential forms, Clifford elements may be of mixed [[grade|Clifford grade]].\n\nA clifford element has a geometric interpretation as a collection of variously sized scalar, vector, oriented area set, ..., and n-volume objects.\n\nClifford elements have a faithful [[matrix representation|Clifford matrix representation]].\n\nThe high grade terms of Clifford elements may be written with fewer indices by using the [[pseudoscalar]],\n$$A^r = \sfr{1}{r!} A^{\sal \sdots \sbe} \sga_{\sal \sdots \sbe} = \sfr{1}{r!} A^{\sal \sdots \sbe} \sfr{1}{\slp n-r \srp!} \sep_{\sal \sdots \sbe \sga \sdots \sde} \sga^{\sga \sdots \sde} \sga\n= \sfr{1}{\slp n-r \srp!} \slp \sfr{1}{r!} A^{\sal \sdots \sbe} \sep_{\sal \sdots \sbe \sga \sdots \sde} \srp \sga^{\sga \sdots \sde} \sga\n= \sfr{1}{\slp n-r \srp!} A^r_{\sga \sdots \sde} \sga^{\sga \sdots \sde} \sga$$\nSo, for example, the pseudoscalar (n-vector, grade $n$) part is\n$$A^n = \sfr{1}{n!} A^{\sal \sdots \sbe} \sga_{\sal \sdots \sbe} = A^p \sga$$\nand the (n-1)-vector part is\n$$A^{n-1} = \sfr{1}{\slp n-1 \srp!} A^{\sal \sdots \sbe} \sga_{\sal \sdots \sbe} = A^{n-1}_\sal \sga^\sal \sga$$
A ''Clifford [[gauge transformation|vector bundle gauge transformation]]'' is a change of the fiber basis elements for a [[Clifford bundle]], [[Clifford vector bundle]], or any graded Clifford bundle. The change may be induced by the action of an arbitrary, position dependent element of the fiber bundle's structure group -- a subgroup of the [[Clifford group]] acting on the [[Clifford basis elements]] via the [[Clifford adjoint]],\n$$\n\sga'_\sal = U \sga_\sal U^-\n$$\nThis gauge transformation is an active transformation of bundle elements, and transforms any Clifford valued field (section), $\sPh$, to\n$$\n\sPh' = U \sPh U^-\n$$\nBy definition, the [[Clifford covariant derivative|Clifford connection]] of any Clifford valued field transforms under a gauge transformation such that,\n$$\n\sf{\sna'} \sPhi' = \slp \sf{\sna} \sPhi \srp'\n$$\nWriting out the covariant derivative operators in this equation using the [[Clifford connection]],\n\sbegin{eqnarray}\n\sf{\sna'} \slp U \sPhi U^- \srp &=& U \slp \sf{\sna} \sPhi \srp U^- \s\s\n\slp \sf{d} U \srp \sPhi U^- + U \slp \sf{d} \sPhi \srp U^- + U \sPhi \slp \sf{d} U^- \srp + \sf{A'} \stimes \slp U \sPhi U^-\srp &=& U \slp \sf{d} \sPhi + \sf{A} \stimes \sPhi \srp U^- \s\s\nU^- \slp \sf{d} U \srp \sPhi + \sPhi \slp \sf{d} U^- \srp U + \sha U^- \sf{A'} U \sPhi - \sha \sPhi U^- \sf{A'} U &=& \sha \sf{A} \sPhi - \sha \sPhi \sf{A}\n\send{eqnarray}\ngives the transformation law for the connection under a gauge transformation:\n$$\n\sf{A'} = U \sf{A} U^- - 2 \slp \sf{d} U \srp U^- \n$$\nFor an infinitesimal gauge transformation, $U \ssimeq 1 + \sha C$, the connection changes to\n$$\n\sf{A'} \ssimeq \sf{A} - \sf{d} C - \sha \sf{A} C + \sha C \sf{A} = \sf{A} - \sf{\sna} C\n$$\ngiving the change $\sde \sf{A} = - \sf{\sna} C$.
The ''grade'' of a [[Clifford element]] corresponds to the number of [[Clifford basis vectors]] used in the [[Clifford basis elements]] needed to represent it. An element may be a single grade, $q$, in which case it is called a ''q-vector'', or it may be of mixed grade, and called a ''multivector''. For example,\n\s[ t = \sfr{1}{3!} t^{\sal \sbe \sga} \sga_{\sal \sbe \sga} \s]\nis a 3-vector, or ''trivector'', while\n\s[ w = w^s + \sha w^{\sal \sbe} \sga_{\sal \sbe} \s]\nis a multivector of grades 0 and 2.\n\nThe ''grade operator'', $\sli A \sri_q = A^q$, acts as a filter, passing only the grade $q$ parts of $A$. For example, the bivector part of $w$ is\n\s[ \sli w \sri_2 = \sha w^{\sal \sbe} \sga_{\sal \sbe} \sin \sli Cl \sri_2 = Cl^2 \s] \nThe grade operator may also be used to filter the even or odd graded parts of an element, such as $\sli w\sri_e = \sli w\sri_2$ and $\sli w\sri_o = 0$. Of special interest is the grade 0 operator, $\sli A\sri = \sli A\sri_0 = A^0 = A^s$, or //''scalar part''// operator which gives the scalar part of $A$. This operator is proportional to the [[trace]] of an element in a [[Clifford matrix representation]]. It is useful since the grade 0 (scalar) part of a Clifford element is a real number.
Combining [[Clifford basis product identities]] with the [[grade|Clifford grade]] operator gives a ''Clifford graded [[commutation|commutator]]'' relationship for two Clifford elements of grades $r$ and $s$,\n\s[ \sli A^r B^s \sri_q = \slp -1 \srp^\sep \sli B^s A^r \sri_q \s]\nwith\n\s[ \sep = \sha \slp q^2 + r^2 + s^2 - q - r - s \srp \s]\nThis relation implies that any two Clifford elements commute inside the scalar part operator, $\sli AB \sri = \sli BA \sri$.\n\nThe Clifford product of two elements of grades $r$ and $s$ can produce elements of various grades,\n\s[ A^r B^s = \sli A^r B^s \sri_{\sll r - s \srl} + \sli A^r B^s \sri_{\sll r - s \srl + 2} + \sdots + \sli A^r B^s \sri_{r + s} \s]
The ''Clifford group'' consists of [[Clifford algebra]] elements having an inverse,\n\s[ Cl^* = \sleft\s{ U \sin Cl \smid \sexists \s; U^- \sni U U^- = 1 \sright\s} \s]\nIt is the [[Lie group]] corresponding to [[exponentiation]] of the [[Clifford basis elements]],\n$$U = e^{B^A \sga_A}$$\nThe [[Lie algebra]] corresponding to the Clifford group is the Clifford algebra.
Each [[Clifford algebra]] has a faithful representation in the complex matrices, $GL(2^{[n/2]},\smathbb{C})$, with the Clifford product isomorphic to matrix multiplication. This corresponds to the traditional definition of [[Pauli matrices]] and [[Dirac matrices]] as the $\sgamma_{\salpha}$ for the purpose of using matrix algebra to do Clifford Algebra calculations, or simply for writing Clifford elements as matrices. Various unary operations on Clifford elements, the [[Clifford conjugate]]s, are equivalent to various matrix conjugates.\n\nA Clifford algebra is built by starting with the basis vectors and creating all possible multiples. For a seed example, we can build a representation for ''Cl(2,0)'' by starting with two Pauli matrices as the two [[Clifford basis vectors]],\n$$\n\sbegin{array}{cc}\n\ssi_1 = \ssigma_{1}^{P} =\n\sleft[\sbegin{array}{cc}\n0 & 1\s\s\n1 & 0\n\send{array}\sright]\n&\n\ssi_2 = \ssigma_{2}^{P}=\sleft[\sbegin{array}{cc}\n0 & -i\s\s\ni & 0\send{array}\sright]\n\send{array}\n$$\nand multiplying to get the scalar and bivector,\n$$\n\sbegin{array}{cc}\n1 = \ssi_1 \ssi_1 =\n\sleft[\sbegin{array}{cc}\n1 & 0\s\s\n0 & 1\send{array}\sright]\n&\n\ssi_{12} = \ssi_1 \ssi_2 =\n\sleft[\sbegin{array}{cc}\ni & 0\s\s\n0 & -i\n\send{array}\sright]\n= i \ssi_3^P\n\send{array}\n$$\ncompleting the list of $Cl(2,0)$ [[Clifford basis elements]] represented as $2 \stimes 2$ complex matrices. To build larger Clifford algebras we can use the [[Kronecker product]] of any smaller Clifford algebras. For example, $Cl(2,2) = Cl(2,0) \sotimes Cl(2,0)$. The tricky part is finding a set of orthogonal, anticommuting, ''Clifford basis vector matrix representatives'' after doing the product, such as picking out:\n$$\n\sbegin{array}{cc}\n\sga_1 = \ssi_1 \sotimes 1 =\n\sleft[\sbegin{array}{cccc}\n0 & 0 & 1 & 0\s\s\n0 & 0 & 0 & 1\s\s\n1 & 0 & 0 & 0\s\s\n0 & 1 & 0 & 0\n\send{array}\sright]\n&\n\sga_2 = \ssi_2 \sotimes \ssi_2 =\n\sleft[\sbegin{array}{cccc}\n0 & 0 & 0 & -1\s\s\n0 & 0 & 1 & 0\s\s\n0 & 1 & 0 & 0\s\s\n-1 & 0 & 0 & 0\n\send{array}\sright]\n\s\s\n\sga_3 = \ssi_2 \sotimes \ssi_1 =\n\sleft[\sbegin{array}{cccc}\n0 & 0 & 0 & -i\s\s\n0 & 0 & -i & 0\s\s\n0 & i & 0 & 0\s\s\ni & 0 & 0 & 0\n\send{array}\sright]\n&\n\sga_4 = \ssi_2 \sotimes \ssi_{12} =\n\sleft[\sbegin{array}{cccc}\n0 & 0 & 1 & 0\s\s\n0 & 0 & 0 & -1\s\s\n-1 & 0 & 0 & 0\s\s\n0 & 1 & 0 & 0\n\send{array}\sright]\n\send{array}\n$$\nA matrix representation, such as above, allows any $Cl(2,2)$ element to be represented by a $4 \stimes 4$ complex matrix. For example,\n$$\na \s, \sga_{12} + b \s, \sga_{34} = a \s, \ssi_{12} \sotimes \ssi_2 + b \s, 1 \sotimes \ssi_2 =\n\sleft[\sbegin{array}{cccc}\n0 & a - i b & 0 & 0\s\s\n-a + i b & 0 & 0 & 0\s\s\n0 & 0 & 0 & -a - i b\s\s\n0 & 0 & a + i b & 0\n\send{array}\sright]\n$$\nTo get a different signature we can multiply any basis vector representaive by $i$, such as multiplying $\sga_3$ above by $i$ to get a ''real representation'' of $Cl(3,1)$ -- in which all basis vectors, and hence all elements, are represented by real matrices. And to represent a Clifford algebra of one less dimension we can discard a vector.\n\nEverything done with Clifford algebra can be identified with the corresponding matrix manipulation; however, it will almost always be more geometrically revealing to deal with the Clifford algebra elements directly.\n\nRefs:\n*http://en.wikipedia.org/wiki/Representations_of_Clifford_algebras\n*Andrzej Trautman\n**[[Clifford Algebras and their Representations|papers/Clifford Algebras and their Representations.pdf]]\n***p20 describes construction of reps for arbitrarily high dimension
Iteration of the [[cross product|antisymmetric bracket]] produces the ''Clifford Jacobi identity'',\n\s[ A \stimes \slp B \stimes C \srp + B \stimes \slp C \stimes A \srp + C \stimes \slp A \stimes B \srp = 0 \s]\nand the ''cross product distributive rule'',\n\s[ A \stimes \slp B C \srp = \slp A \stimes B \srp C + B \slp A \stimes C \srp \s]\nA combination of [[Clifford algebra]] dot and cross products is\n\s[ A \scdot \slp B \stimes C \srp + A \stimes \slp B \scdot C \srp = \sha \slp ABC - CBA \srp = \slp A \scdot B \srp \stimes C + \slp A \stimes B \srp \scdot C \s]\n\nA string of cross products without parenthesis, $A \stimes B \stimes C$, is not well defined because $A \stimes \slp B \stimes C \srp \sne \slp A \stimes B \srp \stimes C$; but a string of dot products, $A \scdot B \scdot C$, or a string of Clifford products, $ABC$, is well defined. In general, parenthesis should always be used to group multiple operations when the cross product is employed.\n\nTo calculate Clifford products, it is best to use the [[Clifford basis product identities]]
A bivector crossed with a vector gives a vector orthogonal to the original, in the plane (or planes) of the bivector. Using the [[Clifford basis product identities]] and antisymmetry of bivector indices,\n\sbegin{eqnarray}\nB \stimes v &=& \sha B^{\sal \sbe} v^\sga \sga_{\sal \sbe} \stimes \sga_\sga = B^{\sal \sbe} v^\sga \sga_{\slb \sal \srd} \set_{\sld \sbe \srb \sga} = B^{\sal \sbe} v_\sbe \sga_\sal \s\s\nv \scdot \slp B \stimes v \srp &=& v^\sde B^{\sal \sbe} v_\sbe \sga_\sde \scdot \sga_\sal = B^{\sal \sbe} v_\sal v_\sbe = 0\n \send{eqnarray}\nA small rotational transformation in the plane (or planes) of a bivector may be carried out by\n\s[ v' = v + \sfr{1}{N}B \stimes v \ssimeq \slp 1 + \sfr{1}{2N} B \srp v \slp 1 - \sfr{1}{2N} B \srp \s]\nfor a large parameter, $N$. A finite rotation comes from [[exponentiating|exponentiation]] the bivector,\n\s[ v' = \slim_{N \sto \sinfty} \slp 1 + \sfr{1}{2N} B \srp^N v \slp 1 - \sfr{1}{2N} B \srp^N = e^{\sha B} v e^{- \sha B} = U v U^- \s]\nFor example, rotating $v$ by an angle of $\sth$ in the $\sga_{12}$ plane gives\n\s[ v' = e^{\sha \sth \sga_{12}} v e^{- \sha \sth \sga_{12}} = \slp \scos \sha \sth + \sga_{12} \ssin \sha \sth \srp v \slp \scos \sha \sth - \sga_{12} \ssin \sha \sth \srp \s]\nSince $U^- = e^{- \sha B} = \stilde{U}$ is the [[reverse|Clifford conjugate]] and the [[inverse]] of $U = e^{\sha B}$, Clifford rotation is a special case of the [[Clifford adjoint]]. Such a $U$ is sometimes called a ''rotor''. Any [[Clifford element]] may be rotated by, $A' = U A U^-$ -- which preserves the [[Clifford grade]] of the element.\n\nA Clifford rotation may be readily translated into the standard matrix coefficient notation for a [[Lorentz rotation]] via\n\s[ \sga'_\sal = \sga_\sbe L^\sbe{}_\sal = U \sga_\sal U^- \s]\n\nThe group of Clifford rotations, $\smbox{Spin}{}^+$, in any [[spacetime]] is a double cover of the [[special orthochronous Lorentz group|Lorentz group]]. Boosts along any [[spatial|indices]] direction, $\snu = \snu^\spi \sga_\spi$, are Clifford rotations in the $\sga_0 \snu$ plane,\n\s[ v' = e^{\sha \sga_{0 \spi} \snu^\spi} v e^{- \sha \sga_{0 \srh} \snu^\srh} \s]\nFor example, a boost of $\snu$ along $\sga_3$ gives\n\s[ v' = e^{\sha \sga_{03} \snu} v e^{- \sha \sga_{03} \snu} = \slp \scosh \sha \snu + \sga_{03} \ssinh \sha \snu \srp v \slp \scosh \sha \snu - \sga_{03} \ssinh \sha \snu \srp \s]\n\nA simple rotor is defined as a rotor that can be written as the product of two vectors, $U_{s}=ab=e^{\sfrac{1}{2}i_{2}\stheta}$, in which $i_{2}$ is a unit bivector of a rotation plane and $\stheta$ is a rotation angle. A rotor may always be factored into a product of $\sleq\sfrac{n}{2}$ simple rotors. A standard decomposition uses the choice of a non-singular vector, $v$, to factor a rotor into $U=\spm U'U_{s}$, in which $U'$ is a rotor that leaves $v$ invariant, $U'v=vU'$, and $U_{s}$ is a simple rotor that rotates $v$. As an example, in a four dimensional Lorentzian spacetime, a rotor can be factored using the time-like frame vector, $\sgamma_{0}$, into the spatial rotation and Lorentz boost,\s[\nU=e^{\sfrac{1}{2}\sgamma\sgamma_{0}\sgamma_{\spi}\stheta^{\spi}}\s: e^{\sfrac{1}{2}\sgamma_{0}\sgamma_{\srh}\snu^{\srh}}\s]\nin which $\sth = \sgamma_{\spi}\stheta^{\spi}$ is the spatial rotation vector and $\snu = \sgamma_{\srh}\snu^{\srh}$ is the spatial boost vector used to construct the [[Cl(1,3) bivector]] corresponding to the Clifford rotation.
The ''Clifford vector bundle'', $Cl^1 M$, with base [[manifold]] $M$ is a [[vector bundle]] with $n$ fiber basis elements equal to the [[Clifford basis vectors]], $\sga_\sal$. The fiber at each base manifold point, $p$, is the space of grade 1 Clifford elements, $Cl^1 = \sli Cl \sri_1$. The transition functions for the basis elements over overlapping patches, $U_1$ and $U_2$, are given by [[Clifford rotation]]s,\n$$\n\sga_\sal^2 = U_{12} \sga_\sal^1 U_{12}^- = \slp t^{12} \srp_\sal{}^\sbe \sga_\sbe^1 = \slp L^{12} \srp^\sbe{}_\sal \sga_\sbe^1\n$$\nThrough equating the transition functions, $L^\sbe{}_\sal$, and using the [[frame]], $\sve{e_\sal} \sf{e} = \sga_\sal$, the Clifford vector bundle may be [[associated]], $\sga_\sal \sleftrightarrow \sve{e_\sal}$, to the [[tangent bundle]], with a corresponding equivalence between all their geometric structures. The structure group of the Clifford vector bundle, $\smbox{Spin}{}^+$, is a double cover of the [[special orthochronous Lorentz group|Lorentz group]]. A Clifford vector field, $v = v(x) = v^\sal(x) \sga_\sal$, over the manifold is a section of the bundle, and gives a Clifford vector at each manifold point.\n\n[[Clifford grade]] $p$ fields are sections of the ''Clifford p-vector bundle'', $Cl^p M$, which has the $\sfrac{n!}{\sleft(n-p\sright)!p!}$ grade $p$ [[Clifford basis elements]], $\sga_{\sla \sdots \sbe}$, as basis. The combined collection of these Clifford vector product bundles is the ''graded Clifford bundle'', $Cl^g M = \sbigoplus_{p=0}^{n} Cl^p M$, having dimension $2^{n}$. The transition functions for the graded Clifford bundle are also [[Clifford rotation]]s,\n$$\sga_{\sal \sdots \sbe}^2 = U_{12} \sga_{\sal \sdots \sbe}^1 U_{12}^-$$\nwhich preserve the grade of the basis elements. The graded Clifford bundle fiber, $Cl$, is the same as for the [[Clifford bundle]] &mdash; but the transition functions (which for the graded Clifford bundle are grade preserving) are in different groups for the two bundles -- the Clifford vector bundle is a Clifford bundle with a [[reduction of the structure group]].\n\nFor a section, $C(x)$, transforming under the Clifford rotation [[gauge transformation]], $C \smapsto C'=U C U^-$, the [[covariant derivative]] is\n$$\n\sf{\sna} C = \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} = \sf{d} C + \sf{\som} \stimes C\n$$\n(defined with a $\sha$ in it to keep things pretty) with the [[spin connection]], $\sf{\som}$, applied using the [[cross product|Clifford algebra]].\n\nAny fiber element, $C$, at $t=0$ may be [[parallel transport]]ed to $C(t)=U(t)CU^-$ along a path on the base by a parameter dependent Clifford element, the path holonomy, $U(t) = Pe^{- \sha \sint_0^t \sf{\som}}$, satisfying the [[path holonomy]] equation,\n$$\n\sfr{d}{dt} U(t) = - \sha \sve{v} \sf{\som} U\n$$\n\nApplying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),\n\sbegin{eqnarray}\n\sf{\sna} \sf{\sna} C &=& \sf{d} \slp \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} \srp + \sha \sf{\som} \slp \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} \srp + \sha \slp \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} \srp \sf{\som} \s\s\n&=& \sha \slp \sf{d} \sf{\som} \srp C - \sha \sf{\som} \sf{d} C - \sha \slp \sf{d} C \srp \sf{\som} - \sha C \sf{d} \sf{\som} \n + \sha \sf{\som} \slp \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} \srp + \sha \slp \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} \srp \sf{\som} \s\s\n&=& \sff{R} \stimes C\n\send{eqnarray}\ngives (after using one of the [[Clifford basis product identities]]) the [[Clifford-Riemann curvature]],\n$$\n\sff{R} = \sf{d} \sf{\som} + \sha \sf{\som} \stimes \sf{\som}\n$$\nThis expression for the curvature may alternatively be obtained from the [[holonomy]] (minding the new factor of $\sha$ in the path holonomy equation).\n\nUnder a gauge transformation, $C(x) \smapsto C'(x) = U(x) C(x) U^-(x)$, the covariant derivative changes to\n\sbegin{eqnarray}\n\sf{\sna'} C' &=& U \slp \sf{\sna} C \srp U^-\s\s\n\sf{d} \slp U C U^- \srp + \sha \sf{\som'} U C U^- - \sha U C U^- \sf{\som'} &=& U \slp \sf{d} C \srp U^- + \sha U \sf{\som} C U^- - \sha U C \sf{\som} U^-\n\send{eqnarray}\ngiving the transformation law for the spin connection,\n$$\n\sf{\som'} = U \sf{\som} U^- - 2 \slp \sf{d} U \srp U^- = U \sf{\som} U^- + 2 U \slp \sf{d} U^- \srp \n$$\nAn infinitesimal transformation, $U \ssimeq 1 + \sha B$, in which $B$ is a Clifford bivector, changes the spin connection to\n$$\n\sf{\som'} \ssimeq \sf{\som} - \sf{d} B - \sha \sf{\som} B + \sha B \sf{\som} = \sf{\som} - \sf{\sna} B\n$$\nThe curvature consequently transforms under a gauge transformation to\n$$\n\sff{R'} = \sf{d} \sf{\som'} + \sha \sf{\som'} \stimes \sf{\som'} = U \sff{R} U^- \ssimeq \sff{R} + B \stimes \sff{R}\n$$\nThese expressions equate to those for a [[tangent bundle gauge transformation]].\n\nThe covariant derivative acting on a [[Clifform]] such as the curvature, transforming under a Clifford rotation, $\sff{F'} = U \sff{F} U^-$, is still \n$$\n\sf{\sna} \sff{F} = \sf{d} \sff{F} + \sf{\som} \stimes \sff{F} \n$$\n\nClifford vector bundles or graded Clifford bundles may alternatively be defined as [[automorphism bundle]]s -- for which outer automorphisms may prove interesting.
The ''Clifford-Ricci curvature'' is a [[Clifform]] obtained by taking the [[cross product|Clifford algebra]] of the [[frame]] with the [[Clifford bundle]] curvature,\n$$\n\sf{R} = \sve{e} \stimes \sff{F}\n$$\nIf, specifically, we are working with the [[Clifford vector bundle]], the Clifford-Ricci curvature is then a Clifford vector valued 1-form,\n$$\n\sf{R} = \sf{dx^i} R_i{}^\sal \sga_\sal = \sve{e} \stimes \sff{R} = \sve{e} \stimes \slp \sf{d} \sf{\som} + \sha \sf{\som} \stimes \sf{\som} \srp\n$$\nwith coefficients equal to those of the [[Ricci curvature]], $R_i{}^\sal = \slp e_\sbe \srp^j R_{ji}{}^{\sbe \sal} = \set^{\sal \sbe} R_{i \sbe}$.
The ''Clifford curvature'' is a [[Clifform]] describing the [[curvature]] of a [[Clifford bundle]],\n$$\n\sff{F} = \sf{d} \sf{A} + \sha \sf{A} \stimes \sf{A}\n$$\nIf, specifically, we are working with the [[Clifford vector bundle]], the Clifford curvature is then the ''Clifford-Riemann curvature'', a Clifford bivector valued 2-form calculated from the [[spin connection]],\n$$\n\sff{R} = \sf{d} \sf{\som} + \sha \sf{\som} \stimes \sf{\som} = \sf{dx^i} \sf{dx^j} \sfr{1}{4} R_{ij}{}^{\sal \sbe} \sga_{\sal \sbe}\n$$\n$$\nR_{ij}{}^{\sal \sbe} = 2 \spa_{\slb i \srd} \som_{\sld j \srb}{}^{\sal \sbe} + 2 \som_{\slb i \srd}{}^\sal{}_\sga \som_{\sld j \srb}{}^{\sga \sbe}\n$$\nwith coefficients equal to those of the [[Riemann curvature]], $R_{ij}{}^{\sal\sbe}$, when the [[tangent bundle connection]] and spin connection coefficients are identified, $\sf{w^{\sal\sbe}}=\sf{\som^{\sal\sbe}}$.
A ''Clifform'' is a [[Clifford algebra]] valued [[differential form]], or, conversely, a [[Clifford element]] with form valued coefficients. A Clifform has a single form grade, $p$, but may consist of pieces with different Clifford grades. In terms of [[coordinate basis forms]] and [[Clifford basis elements]], an arbitrary Clifform may be written as\n$$\n\snf{A} = \sf{dx^i} \sdots \sf{dx^k} \sfr{1}{p!} \slp A_{i \sdots k}{}^s + A_{i \sdots k}{}^\sal \sga_\sal + \sha A_{i \sdots k}{}^{\sal \sbe} \sga_{\sal \sbe} + \sfr{1}{3!} A_{i \sdots k}{}^{\sal \sbe \sga} \sga_{\sal \sbe \sga} + \sdots + A_{i \sdots k}{}^p \sga \srp\n$$\nFor example, a bivector 2-form is written (using the coordinate or [[frame]] basis forms) as\n$$\n\sff{R} = \sff{R^2} = \sf{dx^i} \sf{dx^j} \sfr{1}{4} R_{ij}{}^{\sal \sbe} \sga_{\sal \sbe}\n= \sf{e^\sga} \sf{e^\sde} \sfr{1}{4} R_{\sga \sde}{}^{\sal \sbe} \sga_{\sal \sbe}\n$$\nThe form elements and Clifford elements act in different algebras. All scalar valued form elements commute with all Clifford basis elements. By convention, the form basis elements will be collected on the left and the Clifford basis elements on the right.\n\nThe product of Clifforms may be computed using [[Clifform algebra]]. A Clifform is a [[Lieform]] in which the [[Lie algebra]] generators are Clifford basis elements.
The algebra of [[Clifform]]s is the disjoint union of [[vector-form algebra]] and [[Clifford algebra]]. When performing calculations, it is best to move all [[coordinate basis 1-forms]] to the left of the expression (without commuting them) and all [[Clifford basis elements]] (and the operations between them) to the right. Then the basis contractions and products play out in their independent algebraic sandboxes. Clifford algebra operators like $\scdot$, $\stimes$, $[,]$, and $<>_q$ do not act on the forms, only on the Clifford basis elements. As an example, the dot product of a bivector (-2)-form and a bivector 2-form is a scalar plus a 4-vector,\n\sbegin{eqnarray}\n\svv{L} \scdot \sff{R} &=& \slp \sve{\spa_i} \sve{\spa_j} \sfr{1}{4} L^{i j \sal \sbe} \sga_{\sal \sbe} \srp \scdot \slp \sf{dx^k} \sf{dx^m} \sfr{1}{4} R_{km}{}^{\sga \sde} \sga_{\sga \sde} \srp\s\s\n&=& \slp \sve{\spa_i} \sve{\spa_j} \slp \sf{dx^k} \sf{dx^m} \srp \srp \sfr{1}{16} L^{i j \sal \sbe} R_{km}{}^{\sga \sde} \slp \sga_{\sal \sbe} \scdot \sga_{\sga \sde} \srp\s\s\n&=& \slp - 2 \sde_i^{\slb k \srd} \sdelta_j^{\sld m \srb} \srp \sfr{1}{16} L^{i j \sal \sbe} R_{km}{}^{\sga \sde} \slp \slp \set_{\sal \sde} \set_{\sbe \sga} - \set_{\sal \sga} \set_{\sbe \sde} \srp + \sga_{\sal \sbe \sga \sde} \srp\s\s\n&=& - \sfr{1}{8} L^{i j \sal \sbe} R_{ij}{}^{\sga \sde} \slp \slp \set_{\sal \sde} \set_{\sbe \sga} - \set_{\sal \sga} \set_{\sbe \sde} \srp + \sga_{\sal \sbe \sga \sde} \srp\s\s\n&=& \sfr{1}{4} L^{i j \sal \sbe} R_{ij \sal \sbe} - \sfr{1}{8} L^{i j \sal \sbe} R_{ij}{}^{\sga \sde} \sga_{\sal \sbe \sga \sde}\s\s\n\send{eqnarray}\nusing vector-form algebra and [[Clifford basis product identities]].\n\nClifform product identities can be inferred from the identities of the two respective algebras. For example, since 1-forms anti-commute,\n\s[ \sf{A} \sti \sf{B} = \sha \slp \sf{A} \sf{B} + \sf{B} \sf{A} \srp = \sf{B} \sti \sf{A} \s]\nSome useful identities can be computed using the [[frame]]. For example, for any Clifford vector valued 2-form, $\sff{f}$,\n\sbegin{eqnarray}\n\sve{e} \sti \sff{f} & = & -\slp \sve{e} \sti \sve{e} \srp \slp \sf{e} \scdot \sff{f} \srp + \sve{e} \sti \slp \slp \sve{e} \sti \sff{f} \srp \sti \sf{e} \srp\s\s\n\sff{f} & = & \slp \sve{e} \sti \sff{f} \srp \sti \sf{e} - \sve{e} \scdot \slp \sf{e} \scdot \sff{f} \srp\s\s\n\slp n-2 \srp \sff{f} & = & \sve{e} \sti \slp \sf{e} \sti \sff{f} \srp - \sf{e} \scdot \slp \sve{e} \scdot \sff{f} \srp\n\send{eqnarray} \n(//add identities as needed//)
The [[Coleman-Mandula theorem|http://prola.aps.org/abstract/PR/v159/i5/p1251_1]] states:\n<<<\nLet G be a connected symmetry group of the S matrix, and let the following five conditions hold: (1) G contains a subgroup locally isomorphic to the Poincaré group. (2) For any M>0, there are only a finite number of one-particle states with mass less than M. (3) Elastic scattering amplitudes are analytic functions of s and t, in some neighborhood of the physical region. (4) The S matrix is nontrivial in the sense that any two one-particle momentum eigenstates scatter (into something), except perhaps at isolated values of s. (5) The generators of G, written as integral operators in momentum space, have distributions for their kernels. Then, we show that G is necessarily locally isomorphic to the direct product of an internal symmetry group and the Poincaré group.\n<<<\n\nThe E8 theory proposed in [[An Exceptionally Simple Theory of Everything]] avoids condition (1) of this theorem because $G = E8$ does not containing a subgroup locally isomorphic to the Poincaré group. The expected vacuum spacetime of E8 theory is [[de Sitter spacetime]], which has $SO(4,1)$ as symmetry group, which is nearly, but not, the Poincaré group. At low energies the deviation from the Poincaré group is infinitesimally small, and the Coleman-Mandula theorem applies to a good approximation, with gravity separate from the other symmetries.\n\nRef:\n*K. Cahill\n**[[On the unification of the gravitational and electronuclear forces|papers/Cahill - On the unification of the gravitational and electronuclear forces.pdf]]\n*** Phys. Rev. D 26, 1916 - 1922 (1982).\n*T. Love\n**The Geometry of Grand Unification\n***Int. J. Th. Phys., 801 (1984).\n*F. Nesti and R. Percacci\n**[[Gravi-Weak Unification|http://arxiv.org/abs/0706.3307]]\n*S. Alexander\n**[[Isogravity: Toward an Electroweak and Gravitational Unification|http://arxiv.org/abs/0706.4481]]
/***\nAuthors: Eric Shulman & Bradley Meck\nversion: 2007.30.03\nsource: http://www.tiddlytools.com/\n***/\n/*{{{*/\nconfig.commands.collapseNote = {\ntext: "-",\ntooltip: "Collapse this note",\nhandler: function(event,src,title)\n{\nvar e = story.findContainingNote(src);\nif(e.getAttribute("template") != config.noteTemplates[DEFAULT_EDIT_TEMPLATE]){\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nif(e.getAttribute("template") != t ){\ne.setAttribute("oldTemplate",e.getAttribute("template"));\nstory.displayNote(null,title,t);\n}\n}\n}\n}\n\nconfig.commands.expandNote = {\ntext: " | ",\ntooltip: "Expand this note",\nhandler: function(event,src,title)\n{\nvar e = story.findContainingNote(src);\nstory.displayNote(null,title,e.getAttribute("oldTemplate"));\n}\n}\n\nconfig.macros.collapseAll = {\nhandler: function(place,macroName,params,wikifier,paramString,note){\ncreateTiddlyButton(place,"-","Collapse all notes",function(){\nstory.forEachNote(function(title,note){\nif(note.getAttribute("template") != config.noteTemplates[DEFAULT_EDIT_TEMPLATE])\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nstory.displayNote(null,title,t);\n})})\n}\n}\n\nconfig.macros.expandAll = {\nhandler: function(place,macroName,params,wikifier,paramString,note){\ncreateTiddlyButton(place,"expand all","Expand all notes",function(){\nstory.forEachNote(function(title,note){\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nif(note.getAttribute("template") == t) story.displayNote(null,title,note.getAttribute("oldTemplate"));\n})})\n}\n}\n\nconfig.commands.collapseOthers = {\ntext: "Ø",\ntooltip: "Expand this note and collapse all others",\nhandler: function(event,src,title)\n{\nvar e = story.findContainingNote(src);\nstory.forEachNote(function(title,note){\nif(note.getAttribute("template") != config.noteTemplates[DEFAULT_EDIT_TEMPLATE]){\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nif (e==note) t=e.getAttribute("oldTemplate");\n//////////\n// ELS 2006.02.22 - removed this line. if t==null, then the *current* view template, not the default "ViewTemplate", will be used.\n// if (!t||!t.length) t=!readOnly?"ViewTemplate":"WebViewTemplate";\n//////////\nstory.displayNote(null,title,t);\n}\n})\n}\n}\n/*}}}*/
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http://arxiv.org/abs/gr-qc/0603062\n*Concise treatment of Hamiltonian formulation of GR with a conformal factor.\n*uses metric instead of frame
[[Consequences of Propagating Torsion in Connection-Dynamic Theories of Gravity|papers/9403058.pdf]]\nAuthors: Sean M. Carroll, George B. Field\n\nWe discuss the possibility of constraining theories of gravity in which the connection is a fundamental variable by searching for observational consequences of the torsion degrees of freedom. In a wide class of models, the only modes of the torsion tensor which interact with matter are either a massive scalar or a massive spin-1 boson. Focusing on the scalar version, we study constraints on the two-dimensional parameter space characterizing the theory. For reasonable choices of these parameters the torsion decays quickly into matter fields, and no long-range fields are generated which could be discovered by ground-based or astrophysical experiments. \n
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<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/TED08/images/Coral_reef_s620.JPG" width="827" height="620"></embed></center></html>@@\n
<<note HideTags>>$$\n\sbegin{array}{rcll}\n\sff{F} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{d} \sf{H} + \sf{H} \sf{H}\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s; \sf{H} = \sha \sf{\som} + \sfr{1}{4}\sf{e}\sph + \sf{B} + \sf{W}\n\s\s\n\n\s!\s!&\s!\s!=\s!\s!&\s!\s! \sBig( \sha ( \sf{d} \sf{\som} + \sha \sf{\som} \sf{\som} ) + \sfr{1}{16} M^2 \sf{e} \sf{e} \sBig)_{\sp{(}}\n\s!&\s!\s! \sleftarrow \stext{spacetime} \s; \sga_{\smu\snu} \s\s\n\n&&\s!\s!\s! + \sBig( \sfr{1}{4} \sbig( \sf{d} \sf{e} + \sha [ \sf{\som}, \sf{e} ] \sbig) \sph - \sfr{1}{4} \sf{e} \sbig( \sf{d} \sph + [ \sf{B} \s!+\s! \sf{W}, \sph ] \sbig) \sBig)_{\sp{(}}\n\s!&\s!\s! \sleftarrow \stext{mixed} \s; \sga_{\smu\sph} \s\s\n\n&&\s!\s!\s! + \sBig( \sf{d} \sf{B} + \sf{d} \sf{W} + \sf{W} \sf{W} \sBig)_{\sp{\sbig(}}\n\s!&\s!\s! \sleftarrow \stext{higher} \s; \sga_{\sph\sps} \s\s\n\n\s!\s!&\s!\s!=\s!\s!&\s!\s! \sha \sbig( \sff{R} + \sfr{1}{8} M^2 \sf{e} \sf{e} \sbig)\n+ \sfr{1}{4} \sbig( \sff{T} \sph - \sf{e} \sf{D} \sph \sbig)\n+ \sbig( \sff{F_B} + \sff{F_W} \sbig) \s\s\n\n\s!\s!&\s!\s!=\s!\s!&\s!\s! \sff{F_s} + \sff{F_m} + \sff{F_h}\n\send{array}\n$$\nModified BF action over 4D base [[manifold]]:\n\sbegin{eqnarray}\nS &=& \sint \sbig< \sff{B} \s, \sff{F} + \sPh(\sf{H},\sff{B}) \sbig>\n= \sint \sbig< \sff{B} \s, \sff{F} - {\sscriptsize \sfrac{1}{4}} \sff{B_s} \sff{B_s} \sga + \sff{B_m} \sff{*B_m} + \sff{B_h} \sff{*B_h} \sbig> \s\s\n&=& \sint \sbig< \sff{F_s} \s, \sff{F_s} \sga^- + {\sscriptsize \sfrac{1}{4}} \sff{F_m} \sff{*F_m} + {\sscriptsize \sfrac{1}{4}} \sff{F_h} \sff{*F_h} \sbig>\n\send{eqnarray}
\nNew paper. How to go from a higher dimensional gauge theory, with Chern Simons or Born Infeld action, to Einstein gravity in 4D:\n*[[D=4 Einstein gravity from higher D CS and BI gravity and an alternative to dimensional reduction|papers/0703034.pdf]]
Welcome
The kinetic [[Lagrangian]] term for a [[Weyl spinor]] field in curved [[spacetime]] is\n$$\nL = \sPsi^\sdagger \sga_0 \sve{e} \slp \sf{\spa} + \sha \sf{\som} \srp \sPsi \n$$\n//(that's not necessarily real...but maybe it is, up to a divergence term?)// Using the [[chiral]] representation for the [[Cl(1,3)]] [[Dirac matrices]], the [[spacetime frame]] and [[spacetime spin connection]] break up to give\n\sbegin{eqnarray}\nL &=& \slb \sPsi_L^\sdagger \s;\s; \sPsi_R^\sdagger \srb\n\slb \sbegin{array}{cc}\n0 & 1 \s\s\n1 & 0\n\send{array} \srb\n\slb \sbegin{array}{cc}\n0 & \sve{e}_L \s\s\n\sve{e}_R & 0\n\send{array} \srb\n\slp \sf{\spa} + \n\sha\n\slb \sbegin{array}{cc}\n\sf{\som}{}_L & 0 \s\s\n0 & \sf{\som}{}_R\n\send{array} \srb\n\srp\n\slb \sbegin{array}{c}\n\sPsi_L \s\s\n\sPsi_R\n\send{array} \srb\n \s\s\n&=& \sPsi_L^\sdagger \sve{e}_R \slp \sf{\spa} + \sha \sf{\som}{}_L \srp \sPsi_L + \sPsi_R^\sdagger \sve{e}_L \slp \sf{\spa} + \sha \sf{\som}{}_R \srp \sPsi_R\n\send{eqnarray}
The ''Dirac matrices'' provide a $4\stimes4$ [[Clifford matrix representation]] of [[Cl(1,3)]] or [[Cl(3,1)]]. There are several standard choices, built from the [[Kronecker product]] of [[Pauli matrices]]:\n\nThe ([[chiral]]) ''Weyl representation'' comes from:\n\sbegin{eqnarray}\n\sga_0 &=& \ssi^P_1 \sotimes 1 \s\s\n\sga_1 &=& i \ssi^P_2 \sotimes \ssi^P_1 \s\s\n\sga_2 &=& i \ssi^P_2 \sotimes \ssi^P_2 \s\s\n\sga_3 &=& i \ssi^P_2 \sotimes \ssi^P_3\n\send{eqnarray}\ngiving a complex rep for ''Cl(1,3) vectors'',\n\sbegin{eqnarray}\nv &=& v^\smu \sga_\smu =\n\slb \sbegin{array}{cc}\n0 & v_R \s\s\nv_L & 0\n\send{array} \srb\n=\n\slb \sbegin{array}{cc}\n0 & v^0 + v^\sva \ssi^P_\sva \s\s\nv^0 - v^\sva \ssi^P_\sva & 0\n\send{array} \srb\n\s\s\n&=& \n\slb \sbegin{array}{cccc}\n0 & 0 & v^0+v^3 & v^1-iv^2 \s\s\n0 & 0 & v^1+iv^2 & v^0-v^3 \s\s\nv^0-v^3 & -v^1+iv^2 & 0 & 0 \s\s\n-v^1-iv^2 & v^0+v^3 & 0 & 0\n\send{array} \srb\n\send{eqnarray}\nand [[spacetime pseudoscalar|Cl(1,3)]], $\sga = \ssi^P_3 \sotimes 1$. The $v_{L/R}$ are ''left and right chiral vector parts'' -- $2\stimes2$ Hermitian matrices projected out by the [[left/right chirality projector]]. (//They satisfy...//) Note that a vector is completely determined by one of its chiral parts.\n\n''Dirac representation'' of CL(1,3),\n\sbegin{eqnarray}\n\sga_0 &=& \ssi^P_3 \sotimes 1 \s\s\n\sga_\sva &=& i \ssi^P_2 \sotimes \ssi^P_\sva\n\send{eqnarray}\n\n(real) ''Majorana representation'' of Cl(3,1),\n\sbegin{eqnarray}\n\sga_0 &=& i \ssi^P_1 \sotimes \ssi^P_2 \s\s\n\sga_1 &=& \ssi^P_1 \sotimes \ssi^P_3 \s\s\n\sga_2 &=& \ssi^P_3 \sotimes \ssi^P_1 \s\s\n\sga_3 &=& \ssi^P_1 \sotimes \ssi^P_1\n\send{eqnarray}\nMultiplying these matrices by $i$ switches them between representations of Cl(1,3) and Cl(3,1).\n\nRef:\nhttp://en.wikipedia.org/wiki/Dirac_matrices
If $\sPsi$ is a [[spinor]] field and $\sf{A} \sin \sf{\srm Lie}(G)$ a [[principal bundle]] connection in a representation matched to the spinor, the [[covariant derivative]] of the spinor field is\n$$\n\sf{\sna} \sPsi = \slp \sf{d} + \sf{A} \srp \sPsi\n$$\nNote that $\sf{A}$ includes the [[spin connection]], $\sf{\som}$ (as the connection for the [[Clifford vector bundle]] subbundle of the full principal bundle) and usually other parts, which will be written as $\sf{G}$, so\n$$\n\sf{A} = \sha \sf{\som} + \sf{G}\n$$\nIf we write the [[frame]] over the base manifold as $\sve{e} = \sga^\smu \sve{e_\smu}$, the ''Dirac operator'' acting on the spinor is defined as\n$$\n\sna \sPsi = \sve{e} \sf{\sna} \sPsi = \sga^\smu \slp e_\smu \srp^i \slp \spa_i + \sfr{1}{4} \som_i{}^{\snu \srh} \sga_{\snu \srh} + G_i{}^B T_B \srp \sPsi\n$$\nusing the [[vector-form algebra]].
<<note HideTags>>What is done:\n*All [[gauge fields|connection]], [[gravity|spacetime]], and Higgs in ''one'' [[connection]], with fermions as [[BRST ghosts|BRST technique]].\n\nTo do:\n*Will particle assignments work with [[E8]]? (Get the CKMPMNS matrix?)\n*Why is the action what it is? (How does symmetry breaking happen?)\n*Is a four dimensional base [[manifold]] emergent?\n*How does this theory get quantized? (LQG methods should apply.)\n**Natural explanation for QM as a bonus?\n\nWhat this theory will mean, if it all works:\n*Gravitational [[frame]] and Higgs are intimately related.\n*Naturally combines standard model with gravity -- so it's a [[T.O.E.|theory of everything]]\n**(It's also a U.F.T., but I don't like to call it that.)\n*Our universe is a very pretty shape!\n\n@@display:block;text-align:center;Gar@Lisi.com\nhttp://deferentialgeometry.org $\sp{{}_{(}}$@@
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The rank $6$ exceptional group, ''E6'', is a real, [[simple]], compact, connected [[Lie group|Lie group]]. It may be described by [[exponentiating|exponentiation]] its $78$ dimensional [[Lie algebra]], [[e6]].
The rank $8$ exceptional group, ''E8'', is the largest of the real, [[simple]], compact, connected [[Lie groups]] -- and is often regarded as the most beautiful. It may be described by [[exponentiating|exponentiation]] its $248$ dimensional [[Lie algebra]], [[e8]].
<<note HideTags>>Build a real form of complex [[E8]] by using $Cl^2(1,7)=so(1,7)$ instead of $Cl^2(8)=so(8)$. Then ''E8 T.O.E. connection'' is:\n$$\n\sudf{A} = \sf{H} + \sf{G} + \sud{\sPs}{}_I + \sud{\sPs}{}_{II} + \sud{\sPs}{}_{III} = \n$$\n$$\n\stext{something like}_{\sp{\sbig(}}\n$$\n$$\n{\ssmall\n\sbegin{array}{c}\n\n\s!\s!\s! \slb \sbegin{array}{cccc}\n\sfrac{1}{2} \sf{\som_L} \s!+\s! i \sf{W^3} \s!&\s! i \sf{W^1} \s!+\s! \sf{W^2} \s!&\s! - \s! \sfrac{1}{4} \sf{e_R} \sph_0^* \s!& \sfrac{1}{4} \sf{e_R} \sph_+ \s! \s\s\n\ni \sf{W^1} \s!-\s! \sf{W^2} \s!&\s! \sfrac{1}{2} \sf{\som_L} \s!-\s! i \sf{W^3} \s!&\s! \sp{-} \sfrac{1}{4} \sf{e_R} \sph_+^* \s!& \sfrac{1}{4} \sf{e_R} \sph_0 \s! \s\s\n\n-\sfrac{1}{4} \sf{e_L} \sph_0 & \sfrac{1}{4} \sf{e_L} \sph_+ & \s!\s!\s!\s! \sfrac{1}{2} \sf{\som_R} \s!+\s! i \sf{B} \s!\s! \s!& & \s! \s\s\n\n\sp{-}\sfrac{1}{4} \sf{e_L} \sph_+^* & \sfrac{1}{4} \sf{e_L} \sph_0^* & &\s! \s!\s! \sfrac{1}{2} \sf{\som_R} \s!-\s! i \sf{B} \s!\s!\s!\s!\s!\n\send{array} \srb\n\n\s!\s!+\s!\s!\n\slb \sbegin{array}{cccc}\ni \sf{B} \s!\s! & & & \s\s\n&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!+\s! i \sf{G^{3+8}} \s!\s!\s!&\s!\s!\s! i\sf{G^1} \s!-\s! \sf{G^2} \s!\s!\s!&\s!\s!\s! i\sf{G^4} \s!-\s! \sf{G^5} \s\s\n&\s!\s!\s! i\sf{G^1} \s!+\s! \sf{G^2} \s!\s!\s!&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!-\s! i \sf{G^{3+8}} \s!\s!\s!&\s!\s!\s! i\sf{G^6} \s!-\s! \sf{G^7} \s\s\n&\s!\s!\s! i\sf{G^4} \s!+\s! \sf{G^5} \s!\s!\s!&\s!\s!\s! i\sf{G^6} \s!+\s! \sf{G^7} \s!\s!\s!&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!-\s!\s! \sfrac{2i}{\ssqrt{3}}\sf{G^8}\n\send{array} \srb\n\s\s\n\s; \s\s\n+\n\slb \sbegin{array}{cccc}\n\sud{\snu}{}^e_L & \sud{u}{}_L^r & \sud{u}{}_L^g & \sud{u}_L^b \s\s\n\sud{e}{}_L & \sud{d}{}_L^r & \sud{d}{}_L^g & \sud{d}{}_L^b \s\s\n\sud{\snu}{}^e_R & \sud{u}{}_R^r & \sud{u}{}_R^g & \sud{u}{}_R^b \s\s\n\sud{e}{}_R & \sud{d}{}_R^r & \sud{d}{}_R^g & \sud{d}{}_R^b\n\send{array} \srb\n\s;+\s;\n\slb \sbegin{array}{cccc}\n\sud{\snu}{}^\smu_L & \sud{c}{}_L^r & \sud{c}{}_L^g & \sud{c}_L^b \s\s\n\sud{\smu}{}_L & \sud{s}{}_L^r & \sud{s}{}_L^g & \sud{s}{}_L^b \s\s\n\sud{\snu}{}^\smu_R & \sud{c}{}_R^r & \sud{c}{}_R^g & \sud{c}{}_R^b \s\s\n\sud{\smu}{}_R & \sud{s}{}_R^r & \sud{s}{}_R^g & \sud{s}{}_R^b\n\send{array} \srb\n\s;+\s;\n\slb \sbegin{array}{cccc}\n\sud{\snu}{}^\sta_L & \sud{t}{}_L^r & \sud{t}{}_L^g & \sud{t}_L^b \s\s\n\sud{\sta}{}_L & \sud{b}{}_L^r & \sud{b}{}_L^g & \sud{b}{}_L^b \s\s\n\sud{\snu}{}^\sta_R & \sud{t}{}_R^r & \sud{t}{}_R^g & \sud{t}{}_R^b \s\s\n\sud{\sta}{}_R & \sud{b}{}_R^r & \sud{b}{}_R^g & \sud{b}{}_R^b\n\send{array} \srb_{\sp{(}}\n\send{array}\n}\n$$\n
*Quantization\n**Coupling constants run.\n***Large $\sLa$ compatible with UV fixed point.\n**Just a connection -- amenable to LQG, spin foams, etc.\n*Understand triality-generation relationship better\n**Possible collapse or mixing to graviweak $SL(2,\smathbb{C})$.\n**The role of $\sf{w}+\sf{x}\sPh$ and symmetry breaking.\n**Getting the CKMPMNS matrix would be nice.\n*Why is the action what it is?\n**Pulling $\sf{e}$ out and putting it into $\sff{F} \sff{*F}$ and $\sfff{\sod{B}}$ seems weird.\n***Why $\sf{e}\sph$ simple?\n***Four dimensional base manifold emergent?\n\nWhat this theory will mean, if it all works:\n*Combines standard model with gravity -- with LQG, it's a T.o.E.\n*Our universe is very pretty.\n\n@@display:block;text-align:center; http://deferentialgeometry.org &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Garrett Lisi@@\n<<note HideTags>>
Everything in an $E8$ principal bundle connection,\n$$\n\sudf{A} \sin \sudf{e8}\n$$\nPeriodic table of interactions (Feynman vertices) from curvature,\n$$\n\sudff{F} = \sf{d} \sudf{A} + {\sscriptsize \sfrac{1}{2}} \sbig[ \sudf{A}, \sudf{A} \sbig]\n$$\ndescribed by the $E8$ root polytope. Three generations through triality,\n$$\nT \s, e = \smu \sqquad T \s, \smu = \sta \sqquad T \s, \sta = e\n$$\nPati-Salam $SU(2)_L \stimes SU(2)_R \stimes SU(4)$ GUT and MM gravity together,\n$$\nS = \sint \sbig< \sff{\sod{B}} \sudff{F}\n+ {\sscriptsize \sfrac{\spi}{4}} \sff{B}{}_G \sff{B}{}_G \sga + \sff{B'} \sff{*B'} \sbig>\n$$\nNo free parameters -- masses from Higgs VEV's,\n$$\ng_1 = \ssqrt{\sfr{3}{5}} \sqquad g_2=1 \sqquad g_3=1 \sqquad \sLa=\sfr{3}{4}\sph^2 \sqquad \sph_0 , \sph_1, \sPh \sdots \n$$\nEverything is pure geometry, and it's very beautiful.\n<<note HideTags>>
$$\n\sudf{A} = \sf{H}{}_1 + \sf{H}{}_2 + \sud{\sPs}{}_{I} + \sud{\sPs}{}_{II} + \sud{\sPs}{}_{III} \squad \sin \s;\s; \sudf{e8} \svp{|_{\sbig(}}\n$$\n$$\n\sbegin{array}{rclcl}\n\sf{H}{}_1 \s!\s!&\s!\s!=\s!\s!&\s!\s! {\sscriptsize \sfrac{1}{2}} \sf{\som} + {\sscriptsize \sfrac{1}{4}} \sf{e}\sph + \sf{W} + \sf{B}{}_1 & \sin & \sf{so}(7,1) \s\s[-.1em]\n&& \sf{\som} & \sin & \sf{so}(3,1) \s\s[-.1em]\n&& \sf{e} \sph = (\sf{e}{}_1+\sf{e}{}_2+\sf{e}{}_3+\sf{e}{}_4)\stimes(\sph_{+/0}+\sph_{-/1}) & \sin & \sf{4} \stimes (2+\sbar{2}) \s\s\n&& \sf{W} + \sf{B}{}_1 & \sin & \sf{su}(2) + \sf{su}(2) \s\s\n\sf{H}{}_2 \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{w} + \sf{B}{}_2 + \sf{x} \sPh + \sf{g} & \sin & \sf{so}(8) \s\s[-.2em]\n&& \sf{w} + \sf{B}{}_2 & \sin & \sf{u}(1) + \sf{u}(1) \s\s[-.3em]\n&& \sf{x} \sPh = (\sf{x}{}_{1}+\sf{x}{}_{2}+\sf{x}{}_{3})\stimes(\sPh^{r/g/b} + {\sPh}{}^{\sbar{r}/\sbar{g}/\sbar{b}}) & \sin & \sf{3} \stimes (3+\sbar{3}) \s\s[-.1em]\n&& \sf{g} & \sin & \sf{su}(3) \s\s\n\sud{\sPsi}{}_{I} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sud{\snu}{}_e + \sud{e} + \sud{u} + \sud{d} & \sin & 8_{S+} \s!\stimes 8_{S+} \s\s\n\sud{\sPsi}{}_{II} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sud{\snu}{}_\smu + \sud{\smu} + \sud{c} + \sud{s} & \sin & 8_{V} \stimes 8_{V} \s\s \n\sud{\sPsi}{}_{III} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sud{\snu}{}_\sta + \sud{\sta} + \sud{t} + \sud{b} & \sin & 8_{S-} \s!\stimes 8_{S-} \s\s\n\send{array}\n$$\n<<note HideTags>>
$$\n\sudff{F} = \sf{d} \sudf{A} + \sudf{A} \sudf{A}\n= \sff{F}{}_1+\sff{F}{}_2+ \sf{D} \sbig( \sud{\sPs}{}_{I} + \sud{\sPs}{}_{II} + \sud{\sPs}{}_{III} \sbig) \squad \sin \s;\s; \sudff{e8} \svp{|_{\sBig(}}\n$$\n$$\n\sbegin{array}{rlcl}\n\sff{F}{}_1 \s!\s!\s!\s!&=\n\sha \sbig( \sff{R} - \sfr{1}{8} \sf{e} \sf{e} \sph^2 \sbig)\n+ \sfr{1}{4} \sbig( \sff{T} \sph - \sf{e} \sf{D} \sph \sbig)\n+ \sbig( \sff{F}{}_{B_1} + \sff{F}{}_W \sbig) & \sin & \sf{so}(7,1) \s\s[.1em]\n& \sff{R} = \sf{d} \sf{\som} + \sha \sf{\som} \sf{\som} & \sin & \sf{so}(3,1) \s\s[.1em]\n& \sff{T} \sph \s!-\s! \sf{e} \sf{D} \sph = \sbig( \sf{d} \sf{e} \s!+\s! \sha [ \sf{\som}, \sf{e} ] \sbig) \sph - \sf{e} \sbig( \sf{d} \sph \s!+\s! [ \sf{B}{}_1 \s!+\s! \sf{W}, \sph ] \sbig) & \sin & \sf{4} \stimes (2+\sbar{2}) \s\s[.2em]\n& \sff{F}{}_{B_1} + \sff{F}{}_W = (\sf{d} \sf{B}{}_1 + \sf{B}{}_1 \sf{B}{}_1) + (\sf{d} \sf{W} + \sf{W} \sf{W}) & \sin & \sf{su}(2) \s!+\s! \sf{su}(2) \s\s[.4em]\n\sff{F}{}_2 \s!\s!\s!\s!&=\n\sbig( \sff{F}{}_{w} + \sff{F}{}_{B_2} + \sf{x}\sPh\sf{x}\sPh \sbig)\n+ \sbig( (\sf{D} \sf{x}) \sPh - \sf{x} \sf{D} \sPh \sbig)\n+\sff{F}{}_{g}\n& \sin & \sf{so}(8) \s\s[.1em]\n& \sff{F}{}_{w} + \sff{F}{}_{B_2} = \sf{d} \sf{w} + \sf{d} \sf{B}{}_2 & \sin & \sf{u}(1) + \sf{u}(1) \s\s[.1em]\n& (\sf{D} \sf{x}) \sPh \s!-\s! \sf{x} \sf{D} \sPh \s!=\s! \n\sbig( \sf{d} \sf{x} \s!+\s! [ \sf{w} \s!+\s! \sf{B}{}_2, \s! \sf{x} ] \sbig) \sPh \s!-\s! \sf{x} \sbig( \sf{d} \sPh \s!+\s! [ \sf{g}, \s! \sPh ] \sbig) \s!\s!\s!\n & \sin & \sf{3} \stimes (3+\sbar{3}) \s\s[0em]\n& \sff{F}{}_{g} = \sf{d} \sf{g} + \sf{g} \sf{g} & \sin & \sf{su}(3)\n\send{array}\n$$\n$$\n\sf{D} \sud{\sPsi} = \sbig( \sf{d} + {\sscriptsize \sfrac{1}{2}} \sf{\som} + {\sscriptsize \sfrac{1}{4}} \sf{e}\sph \sbig) \sud{\sPs}\n+ \sf{W} \sud{\sPs}{}_L + \sf{B}{}_1 \sud{\sPs}{}_R - \sud{\sPs} \sbig( \sf{w} + \sf{B}{}_2 + \sf{x} \sPh \sbig) - \sud{\sPs}{}_q \s, \sf{g}\n\svp{|^{\sBig(}}\n$$\n<<note HideTags>>
<<note HideTags>>Build new ${\srm Lie}(E8)$ generators from old ones:\n$$\n\sbegin{array}{rclclcll}\nH_{\sal\sbe} \s!\s!&\s!=\s!&\s!\s! \sga^{\slp16\srp+}_{\sal\sbe} \s!\s!&\s!\s!=\s!&\s!\s! \sga^{(8)+}_{\sal\sbe} \sotimes 1 \s!\s!&\s!\s!\sin\s!&\s!\s! so(8)^+ \sotimes 1\n\s!\s!&\s!=\s, so(8)^H \s\s\nG_{\sal\sbe} \s!\s!&\s!=\s!&\s!\s! \sga^{\slp16\srp+}_{\slp\sal+8\srp\slp\sbe+8\srp} \s!\s!&\s!\s!=\s!&\s!\s! P^{\slp8\srp}_+ \sotimes \sga^{(8)}_{\sal\sbe} \s!\s!&\s!\s!\sin\s!&\s!\s! 1 \sotimes so(8) \n\s!\s!&\s!=\s, so(8)^G \s\s\n\sPs^I_{\sal\sbe} \s!\s!&\s!=\s!&\s!\s! \sga^{\slp16\srp+}_{\sal\slp\sbe+8\srp} \s!\s!&\s!\s!=\s!&\s! \sga^{(8)+}_\sal \sotimes \sga^{(8)}_\sbe \s!\s!&\s!\s!\sin\s!&\s!\s! v^{(8)+} \sotimes v^{(8)}\n\s!\s!&\s!=\s, S^I \s\s\n\sPs^{II}_{ab} \s!\s!&\s!=\s!&\s!\s! Q^+_{16\slp a-1\srp+b} \s!\s!&\s!\s!=\s!&\s!\s! q^+_a \sotimes q^+_b \s!\s!&\s!\s!\sin\s!&\s!\s! S^{(8)+} \sotimes S^{(8)+}\n\s!\s!&\s!=\s, S^{II} \s\s\n\sPs^{III}_{ab} \s!\s!&\s!=\s!&\s!\s! Q^+_{16\slp a-1\srp+b+8} \s!\s!&\s!\s!=\s!&\s!\s! q^+_a \sotimes q^-_b \s!\s!&\s!\s!\sin\s!&\s!\s! S^{(8)+} \sotimes S^{(8)-}\n\s!\s!&\s!=\s, S^{III}\n\send{array}\n$$\n\nWith these basis generators, the ${\srm Lie}(E8)$ elements are:\n\sbegin{eqnarray}\nE &=& H + G + \sPs_I + \sPs_{II} + \sPs_{III} \s\s\n&=& \sha h^{\sal\sbe} H_{\sal\sbe} + \sha g^{\sal\sbe} G_{\sal\sbe} + \sps_I^{\sal\sbe} \sPs^I_{\sal\sbe} + \sps_{II}^{ab} \sPs^{II}_{ab} + \sps_{III}^{ab} \sPs^{III}_{ab} \s\s\n&\sin& so(8)^H + so(8)^G + S^I + S^{II} + S^{III}_{\sp{(}}\n\send{eqnarray}\n\n
<<note HideTags>>@@display:block;text-align:center;[img[images/png/e8 periodic table.png]]@@\n//"E8 is perhaps the most beautiful structure in all of mathematics, but it's very complex."// &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -- Hermann Nicolai
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour.mov" width="602" height="602" controller="false" autoplay="false" loop="false"></embed>\n<!-- <embed src="talks/Perimeter07/anim/e8tour (om up)/p1.png" width="608" height="609"></embed> -->\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p1.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p21.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p101.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p181.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p201.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p236.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p243.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p256.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p280.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p320.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p361.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p391.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p410.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p422.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p430.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p482.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p562.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p642.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p662.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n
<<note HideTags>>@@display:block;text-align:center;${\srm Lie}(E8)$ has $(248-8)=240$ roots in 8D space -- vertices of $P4_{2,1}$:$\sp{{}_{\sbig(}}$\n<html><center><embed src="talks/FQXi07/video/e8anim.mov" width="510" height="510" controller="false" autoplay="false" loop="false"></embed></center></html>$E8$ T.O.E.: Each vertex corresponds to an elementary particle.$\sp{{}{\sBig(}^{(}}$@@
<<note HideTags>>The ${\srm Lie}(E8)$ brackets between elements in the various parts:\n$$\n\sbegin{array}{cc}\n\sbegin{array}{rcl}\n\sbig[ H_1, H_2 \sbig] \s!\s!&\s!=\s!&\s!\s! H_1 H_2 - H_2 H_1 \s\s\n\sbig[ G_1, G_2 \sbig] \s!\s!&\s!=\s!&\s!\s! G_1 G_2 - G_2 G_1 \s\s\n&&\s\s\n\sbig[ H, \sPs_I \sbig] \s!\s!&\s!=\s!&\s!\s! H \s, \sPs_I \s\s\n\sbig[ H, \sPs_{II} \sbig] \s!\s!&\s!=\s!&\s!\s! H^+ \s, \sPs_{II} \s\s\n\sbig[ H, \sPs_{III} \sbig] \s!\s!&\s!=\s!&\s!\s! H^+ \s, \sPs_{III} \s\s\n&&\s\s\n\sbig[ G, \sPs_I \sbig] \s!\s!&\s!=\s!&\s!\s! \sPs_I \s, G \s\s\n\sbig[ G, \sPs_{II} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sPs_{II} \s, G^+ \s\s\n\sbig[ G, \sPs_{III} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sPs_{III} \s, G^-\n\send{array}\n&\n\sbegin{array}{rcl}\n\sbig[ \sPs^1_I, \sPs^2_I \sbig] \s!\s!&\s!=\s!&\s!\s! -2 \sbig( \sPs^1_I \s, {\sPs^2_I}^T \sbig)_H \s\s\n&& -2 \sbig( {\sPs^1_I}^T \sPs^2_I \sbig)_{G_{\sp{(}}} \s\s\n\s\s\n\sbig[ \sPs^1_{II}, \sPs^2_{II} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sbig( \sPs^1_{II} \sGa^+ {\sPs^2_{II}}^T \sbig)_H \s\s\n&&\s!\s! - \sbig( {\sPs^1_{II}}^T \sGa^+ \sPs^2_{II} \sbig)_{G_{\sp{(}}} \s\s\n\sbig[ \sPs^1_{III}, \sPs^2_{III} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sbig( \sPs^1_{II} \sGa^+ {\sPs^2_{II}}^T \sbig)_H \s\s\n&&\s!\s! - \sbig( {\sPs^1_{II}}^T \sGa^- \sPs^2_{II} \sbig)_G \s\s\n&&\s\s\n\sbig[ \sPs_I, \sPs_{II} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sbig( \sPs_I \sGa^{++} \sPs_{II} \sbig)_{III} \s\s\n\sbig[ \sPs_I, \sPs_{III} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sbig( \sPs_I \sGa^{+-} \sPs_{III} \sbig)_{II} \s\s\n\sbig[ \sPs_{II}, \sPs_{III} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sbig( \sPs_{II} \sGa^{++} \sPs_{III} \sbig)_I\n\send{array}\n\send{array}\n$$\nNote: $H$ acts on $\sPs$'s from the left and $G$ acts from the right.$^{\sp{\sbig(}}_{\sp{(}}$
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The ''Ehresmann Cartan connection'', $\sf{\sve{\scal C}}$, is an [[Ehresmann principal bundle connection]] over the total space, $E_G$, of an [[Ehresmann Cartan geometry]]. In coordinates ($x$ over $M$ patches, $x_s$ over $G/H$ patches, and $y$ over $H$ patches) adapted to the reference sections, the Ehresmann Cartan connection may be written locally as\n\sbegin{eqnarray}\n\sf{\sve{\scal C}}(x, x_s, y) &=& \sf{C^J}(x) \s, \sve{\sxi^L_J}(x_s, y) + \sf{\sve{\scal I}} \s\s\n&=& \sf{C^J}(x) \s, L^I{}_J(x_s, y) \s, \sve{\sxi^R_I}(x_s, y) + \sf{\sve{\scal I}} \n\send{eqnarray}\nin which $\sve{\sxi^L_J}$ and $\sve{\sxi^R_J} \ssim T_J$ are the [[left and right action vector fields|Lie group geometry]] for the fibers, $G_x$, the [[left-right rotator]] is\n$$\nL^I{}_J(x_s, y) = \sve{\sxi^L_J} \sf{\sxi_R^I} = \slp T^I, g^-(x_s,y) \s, T_J \s, g(x_s,y) \srp\n$$\nthe [[Ehresmann-Maurer-Cartan form|Maurer-Cartan form]] (the [[identity projection|vector projection]] along the fibers) is\n$$\n\sf{\sve{\scal I}}(x_s,y) = \sf{\sxi_R^J} \sve{\sxi^R_J} = \sf{dx_s^a} \sve{\spa^s_a} + \sf{dy^p} \sve{\spa_p} = \sf{\sve{{\scal I}_{G/H}}} + \sf{\sve{{\scal I}_H}}\n$$\nand $\sf{C^J}(x)$ are the components of the [[Cartan connection|Cartan geometry]] over $M$. The Ehresmann Cartan connection is a projection, $\sf{\sve{\scal C}} \sf{\sve{\scal C}} = \sf{\sve{\scal C}}$, is [[right invariant]], $R_g^*\sf{\sve{\scal C}} = \sf{\sve{\scal C}}$, and has a natural [[spectral decomposition|spectral decomposition of the Ehresmann principal bundle connection]]. It may also be written as a [[Lieform]] over the total space by [[contracting|vector-form algebra]] it with the [[Maurer-Cartan form]], $\sf{\scal I}(x_s,y) = \sf{\sxi_R^J} T_J = g^- \sf{d} g$, over the total space to get the ''Ehresmann Cartan connection form'',\n\sbegin{eqnarray}\n\sf{\scal C}(x,x_s,y) &=& \sf{\sve{\scal C}} \sf{\scal I} = \sf{C^J}(x) \sve{\sxi^L_J} \sf{\sxi_R^I} T_I + \sf{\sve{\scal I}} \sf{\scal I} \s\s\n&=& \slp \sf{C^J} L^I{}_J(x_s,y) + \sf{\sxi_R^I} \srp T_I \s\s\n&=& g^-(x_s,y) \s, \sf{C}(x) \s, g(x_s,y) + g^-(x_s,y) \s, \sf{d} \s, g(x_s,y) \n\send{eqnarray}\nThis form [[pulls back|pullback]] along the canonical reference section, $\ssi_0^G$, to give the Cartan connection,\n$$\n\ssi_0^{G*} \sf{\scal C} = \sf{C}(x)\n$$\nand satisfies $R_g^* \sf{\scal C} = g^- \sf{\scal C} g$ under the right action.\n\nThe ''[[FuN curvature]] of the Ehresmann Cartan connection'' is\n\sbegin{eqnarray}\n\sff{\sve{\scal F}}(x,x_s,y) &=& - \sha \slb \sf{\sve{\scal C}}, \sf{\sve{\scal C}} \srb_L \s\s\n&=& \slp \sf{d} \sf{C^K} + \sha \sf{C^I} \sf{C^J} C_{IJ}{}^K \srp \sve{\sxi^L_K}(x_s,y)\n\send{eqnarray}\nwhich is vector valued in the vertical subspace and right invariant, $R_g^* \sff{\sve{\scal F}} = \sff{\sve{\scal F}}$. The ''FuN curvature form of the Ehresmann Cartan connection'' is a $Lie(G)$ valued 2-form over $E_G$,\n\sbegin{eqnarray}\n\sff{\scal F} &=& \sff{\sve{\scal F}} \sf{\scal I} = \slp \sf{d} \sf{C^K} + \sha \sf{C^I} \sf{C^J} C_{IJ}{}^K \srp g^-(x_s,y) \s, T_K \s, g(x_s,y) \s\s\n&=& g^-(x_s,y) \slp \sf{d} \sf{C} + \sha \slb \sf{C}, \sf{C} \srb \srp g(x_s,y) \s\s\n&=& g^-(x_s,y) \slp \sf{d} \sf{C} + \sf{C} \sf{C} \srp g(x_s,y)\n\send{eqnarray}\nThis form pulls back along the canonical reference section to give the [[Cartan geometry]] curvature,\n$$\n\ssi_0^{G*} \sff{\scal F} = \sf{d} \sf{C} + \sf{C} \sf{C} = \sff{F}(x)\n$$\nand satisfies $R_g^* \sff{\scal F} = g^- \sff{\scal F} g$ under the right action.
When $H$ is [[reductive]] in $G$ (which is usually assumed) the [[Cartan connection|Cartan geometry]] splits as\n$$\n\sf{C}(x) = \sf{e}(x) + \sf{A}(x) = \sf{e^A} K_A + \sf{A^P} H_P\n$$\nthe [[Ehresmann Cartan connection]] can be made to follow this split,\n\sbegin{eqnarray}\n\sf{\sve{\scal C}}(x,x_s,y) &=& \sf{C^J}(x) \s, L^I{}_J(x_s, y) \s, \sve{\sxi^R_I}(x_s, y) + \sf{\sve{\scal I}} \s\s\n&=& \sf{\sve{\scal E}} + \sf{\sve{\scal A}}\n\send{eqnarray}\nwith the ''Ehresmann Cartan frame'' and ''Ehresmann Cartan H-connection'' defined over patches of the total space, $E_G$, of the [[Ehresmann Cartan geometry]] as:\n\sbegin{eqnarray}\n\sf{\sve{\scal E}}(x, x_s, y) &=& \sf{e^A}(x) \s, (L^h)^I{}_K(y) \s, (L^r)^K{}_A(x_s) \s, \sve{\sxi^R_I}(x_s, y) \s\s\n\sf{\sve{\scal A}}(x, x_s, y) &=& \sf{A^P}(x) \s, (L^h)^I{}_K(y) \s, (L^r)^K{}_P(x_s) \s, \sve{\sxi^R_I}(x_s, y) + \sf{\sve{\scal I}}\n\send{eqnarray}\nwith the [[left-right rotator]] and [[Killing vector fields|Lie group geometry]] split over the [[reductive Lie group geometry]]. The [[Ehresmann Cartan connection form|Ehresmann Cartan connection]], $\sf{\scal C} = \sf{\sve{\scal C}} \sf{\scal I}$, also splits,\n\sbegin{eqnarray}\n\sf{\scal C}(x,x_s,y) &=& g^-(x_s,y) \s, \sf{C}(x) \s, g(x_s,y) + g^-(x_s,y) \s, \sf{d} \s, g(x_s,y) \s\s\n&=& \sf{\scal E} + \sf{\scal A}\n\send{eqnarray}\nwith the ''Ehresmann Cartan frame form'' and ''Ehresmann Cartan H-connection form'' defined over $E_G$ as:\n\sbegin{eqnarray}\n\sf{\scal E}(x, x_s, y) &=& \sf{\sve{\scal E}} \sf{\scal I} = \sf{{\scal E}^I} T_I = g^- \s, \sf{e} \s, g(x_s,y) \sin \sf{\srm Lie}(G) \s\s\n\sf{\scal A}(x, x_s, y) &=& \sf{\sve{\scal A}} \sf{\scal I} = \sf{{\scal A}^I} T_I = g^- \s, \sf{A} \s, g(x_s,y) + g^- \s, \sf{d} \s, g(x_s,y) \sin \sf{\srm Lie}(G)\n\send{eqnarray}\nwith $g(x_s,y) = r(x_s) \s, h(y)$.\n\nThis splitting is not a natural thing to do for the Ehresmann Cartan connection or connection form, for which the gauge group is $G$; however, it makes more sense when the Ehresmann Cartan conenction form is pulled back to the [[Cartan homogeneous space bundle]] or [[Cartan H-bundle]]. \n
An ''Ehresmann [[Cartan geometry]]'' modeled on an $n_K$ dimensional [[homogeneous space]], $S=G/H$, is described by an [[Ehresmann principal bundle connection]] (the [[Ehresmann Cartan connection]]), $\sf{\sve{\scal A}} = \sf{\sve{\scal C}}$, over a $(n_M + n_G)$ dimensional total space, $E_G \ssim M \stimes G$, built from an $n_M = n_S$ dimensional base, $M$, and $n_G$ dimensional fiber, $F = G$. This fiber of an ''Ehresmann Cartan geometry'' has a subgroup, $H \ssubset G$, so the bundle produces two [[associated]] bundles, the [[Cartan H-bundle]], $E_H \ssim M \stimes H$, and the [[Cartan homogeneous space bundle]], $E_S \ssim M \stimes S$. The Ehresmann Cartan connection gives the ''Ehresmann Cartan connection form'', $\sf{\scal C} = \sf{\sve{\scal C}} \sf{\scal I} \sin \sf{\srm Lie}(G)$, which gives the associated [[Cartan H-bundle connection form|Cartan H-bundle]], $\sf{{\scal C}_H}$, over $E_H$ and [[Cartan homogeneous space connection form|Cartan homogeneous space bundle]], $\sf{{\scal C}_S}$, over $E_S$. A ''generalized Ehresmann Cartan geometry'' has $n_M \sneq n_S$.\n\nThere is a convenient set of local coordinates for the total space. The $n_M$ coordinates, $x^a$, cover patches of the base manifold, $M$, the $n_H$ coordinates, $y^p$, correspond to elements $h(y) \sin H \ssubset G$, and the remaining $n_S$ homogeneous space coordinates, $x_s^a$, correspond to $x_s \sin S$. So the combined coordinates, $(x_s, y)$, cover patches of $G$ and the total combined coordinates, $(x,x_s,y)$, cover patches of $E_G$ -- so a point of $E_G$ may be written as\n$$\np \ssim (x,x_s,y) \ssim (x,x_s,h(y)) \ssim (x,g(x_s,y))\n$$\nThe chosen [[coset representative section|homogeneous space]], $r : S \sto G$, allows points of $G$ to be specified in terms of points of $G/H$ and $H$ via the right action, \n$$\ng(x_s, y) = R_{h(y)} r(x_s) = r(x_s) \s, h(y)\n$$\nThe ''Cartan geometry [[reference section|Ehresmann gauge transformation]]'', $\ssi^G : M \sto E_G$, is then determined by the reference section, $\ssi^H : M \sto E_H$, of the Cartan H-bundle and the reference section, $\ssi^S : M \sto E_S$, of the Cartan homogeneous space bundle. With $\ssi^H(x) = {\sbig (} x,h(y_\ssi(x)) {\sbig )}$ and $\ssi^S(x) = (x, x_{s\ssi}(x))$ we have:\n$$\n\ssi^G(x) = {\sbig (} x, r(x_{s\ssi}(x)) \s, h(y_\ssi(x)) {\sbig )} \n$$\nThe [[canonical reference section|Ehresmann principal bundle connection]], $\ssi_0^G(x) = (x,1) \ssim (x,0,0)$, of $E_G$ corresponds to the canonical reference section, $\ssi_0^H(x) = (x,1) \ssim (x,0)$, of $E_H$ and the zero point reference section, $\ssi_0^S(x) = (x, 0)$, of $E_S$. \n\nThe Ehresmann Cartan geometry total space, $E_G$, is not only a bundle over $M$ -- it is also a bundle over $E_H$ and over $E_S$. The fundamental bundle maps, $\spi^G_H : E_G \sto E_H$ and $\spi^G_S : E_G \sto E_S$, are given by $\spi^G_H(x,x_s,y)=(x,y)$ and $\spi^G_S(x,x_s,y) = (x,x_s)$. There are also reference sections, $\ssi'^S : E_H \sto E_G$ and $\ssi'^H : E_S \sto E_G$, over these bases, determined by the reference sections over their partner bundle, $\ssi'^S(x,y)=(x,x_{s\ssi}(x),y)$ and $\ssi'^H(x,x_s)=(x,x_s,y_\ssi(x))$. The complete web of bundle maps is summarized by:\n$$\n\sbegin{array}{ccc}\nE_G & \smatrix{\slower8mu {\soverset{\ssi'^S}{\slongleftarrow}}\s\s \sraise8mu {\sunderset{\spi^G_H}{\slongrightarrow}}} & E_H\s\s\n{}^{\spi^G_S} \s! {\sbig \sdownarrow} {\sbig \suparrow} \s! {}_{\ssi'^H} & {}_{\spi_G} \s! \s! \s! \s! \ssearrow \s! \s! \snwarrow \s! \s! \s! \s! {}^{\ssi^G} & {}^{\spi_H} \s! {\sbig \sdownarrow} {\sbig \suparrow} \s! {}_{\ssi^H}\s\s\nE_S & \smatrix{\slower8mu {\soverset{\ssi^S}{\slongleftarrow}}\s\s \sraise8mu {\sunderset{\spi_S}{\slongrightarrow}}} & M\n\send{array}\n$$\nand we have\n\sbegin{eqnarray}\n\spi_G &=& \spi_H \scirc \spi^G_H = \spi_S \scirc \spi^G_S \s\s\n\ssi^G &=& \ssi'^H \scirc \ssi^S = \ssi'^S \scirc \ssi^H\n\send{eqnarray}\n\nThe geometry of an Ehresmann Cartan geometry and its associated bundles is described by the [[Ehresmann Cartan connection]] and its curvature.
The geometry of a [[fiber bundle]] may be described via a [[connection]], $\sf{A}$, and [[covariant derivative]], $\sf{\sna}$, defined over the base manifold, $M$, or alternatively via an ''Ehresmann connection'', $\sf{\sve{\scal A}}$, defined over the total space, $E$, of the bundle. This [[vector valued form]] is a [[vector projection]], $\sf{\sve{\scal A}}\sf{\sve{\scal A}}=\sf{\sve{\scal A}}$, that succinctly describes the geometric structure of the bundle, including the [[Lie group]] symmetry. As a projection, it splits the tangent vector space at each point, $p$, of $E$, into range and kernel subspaces,\n$$\nT_p E = V_r + V_0\n$$\nThe range subspace of $\sf{\sve{\scal A}}$ is the ''vertical subspace'', $V_r=V_V$, and the collection of these vertical vector fields over $E$ is an involutive [[distribution]], $\sve{\sDe_V}=\sve{\sDe_r}$, of vectors tangent to the fibers of the bundle, $\spi_* \sve{\sDe_V} = 0$. In this way, the Ehresmann connection determines the fibers of the fiber bundle. The vector fields, $\sve{\sxi_A} \ssim T_A$, in $\sve{\sDe_V}$ are the flow fields of the group action on the fibers. They are in involution since\n$$\n\slb \sve{\sxi_A},\sve{\sxi_B} \srb_L = C_{AB}{}^C \sve{\sxi_C}\n$$\nThe kernel subspace of $\sf{\sve{\scal A}}$ is the ''horizontal subspace'', $V_0=V_H$, and the collection of these horizontal vector fields over $E$ form a ''horizontal distribution'', $\sve{\sDe_H}=\sve{\sDe_0}$, that may or may not be in involution (more on that further down). \n\nThe Ehresmann connection respects the symmetry of the structure group. If we take the group action on manifold points to be a right action $p \smapsto R_g p = pg$, the Ehresmann connection at different points along fibers are related by the [[pullback]],\n$$\nR_g^* \sf{\sve{\scal A}} (pg) = \sf{\sve{\scal A}} (p)\n$$\nIn this way the Ehresmann connection is related to the [[Ehresmann-Maurer-Cartan form|Maurer-Cartan form]] on the fiber. Note that the 1-form and vector parts of the Ehresmann connection are being pulled back -- in coordinates this could well be written as\n$$\n\slp R_g^* \sf{dz^i} \srp {\scal A}_i{}^j (pg) \sve{\spa_j} = \sf{dz^i} {\scal A}_i{}^j (p) \slp R_{g*} \sve{\spa_j} \srp\n$$\nThe vertical and horizontal distributions satisfy\n\sbegin{eqnarray}\n\sve{\sDe_V} \sf{\sve{\scal A}} &=& \sve{\sDe_V} \s\s\n\sve{\sDe_H} \sf{\sve{\scal A}} &=& 0\n\send{eqnarray}\nand so must also respect the symmetry of the structure group,\n\sbegin{eqnarray}\n\sve{\sDe_V} (p g) &=& R_{g*} \sve{\sDe_V} (p) \s\s\n\sve{\sDe_H} (p g) &=& R_{g*} \sve{\sDe_H} (p)\n\send{eqnarray}\nso knowing the distributions at any point $p$ in $E$ implies the distributions at any other point on the fiber containing $p$.\n\nIff the [[FuN curvature]] of the Ehresmann connection vanishes,\n$$\n\sff{\sve{{\scal F}}} = - \sha \slb \sf{\sve{\scal A}},\sf{\sve{\scal A}} \srb_L = 0\n$$\nthe horizontal distribution is also in involution and may be integrated to get ''horizontal section''s. \n\nRefs:\n*http://philsci-archive.pitt.edu/archive/00002133/01/geometrie.pdf\n*http://www.mat.univie.ac.at/~michor/gaubook.pdf\n*http://www.mat.univie.ac.at/~michor/listpubl.html\n*http://www.emis.de/monographs/KSM/index.html
For any [[vector valued form]] field, $\snf{\sve{\scal K}}$, on the total space of a [[fiber bundle]], a natural grade 1 [[derivation]] is provided by the [[FuN derivative]] with respect to the [[Ehresmann connection]], defining the ''Ehresmann covariant derivative'',\n$$\n\sf{\scal D} \snf{\sve{\scal K}} = - {\scal L}_{\sf{\sve{\scal A}}} \snf{\sve{\scal K}}\n$$\nOnce a choice of [[gauge|Ehresmann gauge transformation]] is made, the Ehresmann connection may be expressed in local coordinates as\n$$\n\sf{\sve{\scal A}}(x,y) = \sf{A^B}(x) \sve{\sxi_B}(y) + \sf{\sve{W}}(y)\n$$\nIf the vector valued form field is right invariant over the total space and may be written as\n$$\n\snf{\sve{\scal K}} = \snf{K^B}(x) \sve{\sxi_B}(y)\n$$\nthen, using a couple of [[FuN identities]], its Ehresmann covariant derivative is\n\sbegin{eqnarray}\n\sf{\scal D} \snf{\sve{\scal K}} &=& - \slb \sf{\sve{\scal A}}, \snf{\sve{\scal K}} \srb_L = - \slb \sf{A^B}(x) \sve{\sxi_B}(y), \snf{K^C}(x) \sve{\sxi_C}(y) \srb_L - \slb \sf{\sve{W}}(y), \snf{K^C}(x) \sve{\sxi_C}(y) \srb_L \s\s\n&=& - \sf{A^B} \snf{K^C} \slb \sve{\sxi_B}, \sve{\sxi_C} \srb_L - \sf{A^B} \slp {\scal L}_{\sve{\sxi_B}} \snf{K^C} \srp \sve{\sxi_C}\n+ \slp {\scal L}_{\sve{\sxi_C}} \sf{A^B} \srp \snf{K^C} \sve{\sxi_B}\n+ \slp \sf{d} \sf{A^B} \srp \sve{\sxi^B} \snf{K^C} \sve{\sxi_C}\n+ \slp \sve{\sxi^C} \sf{A^B} \srp \slp \sf{d} \snf{K^C} \srp \sve{\sxi_B} \s\s\n&-& \slp \sf{\sve{W}} \sf{\spa} \srp \snf{K^C} \sve{\sxi_C} + \slp -1 \srp^k \slp \snf{K^C} \sve{\sxi_C} \sf{\spa} \srp \sf{\sve{W}} + \slp \sf{\spa} \sf{\sve{W}} \srp \snf{K^C} \sve{\sxi_C} + \slp \sf{\spa} \snf{K^C} \sve{\sxi_C} \srp \sf{\sve{W}} \s\s\n&=& \slp \sf{d} \snf{K^C} \srp \sve{\sxi_C} - \sf{A^B} \snf{K^C} \slb \sve{\sxi_B}, \sve{\sxi_C} \srb_L\n\send{eqnarray}
An [[Ehresmann connection]] may be described in local coordinates by choosing a ''reference [[section|fiber bundle]]'', $\ssi_0$, that maps from some base manifold, $M$, to the total space, $E$. If coordinates $x^a$ are used in a local patch over $M$, and coordinates $y^p$ are used in patches over a typical fiber, these coordinates can be chosen so $y=0$ on the reference section, and the Ehresmann connection can be written locally over $E$ as\n$$\n\sf{\sve{\scal A}}(x,y) = \sf{dx^a} A_a{}^B(x) \sxi_B{}^p(y) \sve{\spa_p} + \sf{dy^p} \sve{\spa_p} = \sf{A^B} \sve{\sxi_B} + \sf{\sve{\scal I}}\n$$\nIn which $\sve{\sxi_B}$ are the [[right (or left) invariant vector fields|Lie group geometry]] on the fibers. Another section ([[gauge|gauge transformation]]), $\ssi:M \srightarrow E$, may be chosen by flowing the original section along a [[diffeomorphism]] along the fibers, $\sph(x,y) = (x,y_\sph(x,y))$, to $\ssi = \sph \scirc \ssi_0$. The new section is described in the original coordinates by $y^p_\ssi(x)$. Since the Ehresmann connection is valued in $TE$ it can't be [[pulled back|pullback]] along the section; however, the [[vector projection onto a section]],\n$$\n\sf{\sve{P_\ssi}} = \sf{dx^a} \sve{\spa_a} + \sf{dx^a} \sfr{\spa y_\ssi^p}{\spa x^a} \sve{\spa_p}\n$$\ncan be used to project to the TE valued 1-form on the section,\n$$\n\sf{\sve{{\scal A}_\ssi}} = \sf{\sve{P_\ssi}} \sf{\sve{\scal A}} = \sf{dx^a} \slp A_a{}^B(x) \sxi_B{}^p(y_\ssi) + \sfr{\spa y_\ssi^p}{\spa x^a} \srp \sve{\spa_p}\n$$\nThe Ehresmann connection everywhere in the total space is determined by the connection components, $A_a{}^B(x)$, on a chosen section. Changing to the connection on a different section is called a passive [[gauge transformation]].\n\nAn alternative way of effecting a gauge transformation is to flow the Ehresmann connection by the diffeomorphism, $\sf{\sve{\scal A'}} = \sphi^*\sf{\sve{\scal A}}$, along the fibers while projecting it onto the original section, $\ssi_0$. This is called an ''active gauge transformation'', and gives the same result,\n$$\n\sf{\sve{{\scal A'}_{\ssi_0}}} = \sf{\sve{P_{\ssi_0}}} \sf{\sve{\scal A'}} = \sf{\sve{P_{\ssi_0}}} \sph^* \sf{\sve{\scal A}} = \sf{\sve{P_{\sph \scirc \ssi_0}}} \sf{\sve{\scal A}} = \sf{\sve{P_\ssi}} \sf{\sve{\scal A}} = \sf{\sve{\scal A_\ssi}}\n$$\n\nRef:\n*[[Geometrical aspects of local gauge symmetry|papers/Geometrical aspects of local gauge symmetry.pdf]]
It was enlightening to consider the [[Ehresmann principal bundle connection]] as a construction in the entire space, $E$, of a [[principal bundle]] with base space, $M=S$. It is equally enlightening to consider the ''Ehresmann homogeneous space geometry'' as the analogous construction in the [[Lie group geometry]], $G$, with a base space that is a [[homogeneous space geometry]], $S = G/H$.\n\nPoints of the [[homogeneous space]], $x = [r(x)] \sin M = G/H$, are mapped to $G$, by the homogeneous reference section, $r: S \sto G$, and any point (element) in $G$, as a function of coordinates $x$ of $M$ and $y$ of $H$, may be specified by\n$$\ng(z) = g(x,y) = r(x) h(y) \sin G\n$$\nin which $h(y) \sin H$ operates on $r(x) \sin G$ via the [[right action|group]]. In these coordinates, adapted to the reference section, the reference section is the [[submanifold]] corresponding to $y=0$. The [[Maurer-Cartan form]], $\sf{\scal I} = \sf{{\scal I}^J} T_J$, over $G$, is\n\sbegin{eqnarray}\n\sf{\scal I}(z) &=& g^- \sf{d} g = h^- \slp r^- \sf{d} r \srp h + h^- \sf{d} h \s\s\n&=& h^- \sf{I} h + h^- \sf{d} h \s\s\n&=& \sf{{\scal E}_S} + \sf{{\scal A}_S}\n\send{eqnarray}\nWhen $H$ is [[reductive]] in $G$ (which is assumed) the [[Maurer-Cartan frame|homogeneous space geometry]],\n$$\n\sf{I}(x) = r^- \sf{d} r = \sf{e_S} + \sf{A_S}\n$$\nsplits into the homogeneous space frame, $\sf{e_S}(x) = \sf{e_S^A} K_A \sin \sf{\srm Lie}(G/H)$, and homogeneous H-connection, $\sf{A_S}(x) = \sf{A_S^P} H_P \sin \sf{\srm Lie}(H)$. These correspond to the ''Ehresmann homogeneous space frame form'' and ''Ehresmann homogeneous H-connection form'' over $G$,\n\sbegin{eqnarray}\n\sf{{\scal E}_S} = \sf{{\scal E}_S^A} K_A &=& h^- \sf{e_S} h \sin \sf{\srm Lie}(G/H) \s\s\n\sf{{\scal A}_S} = \sf{{\scal A}_S^P} H_P &=& h^- \sf{A_S} h + h^- \sf{d} h \sin \sf{\srm Lie}(H)\n\send{eqnarray}\nTheir 1-form components are computed using the [[Killing form]],\n\sbegin{eqnarray}\n\sf{{\scal E}_S^A}(z) &=& \sf{e_S^B} \slp K^A, h^- K_B h \srp = \slp L^h \srp^A{}_B \s, \sf{e_S^B}(x) \s\s\n\sf{{\scal A}_S^P}(z) &=& \sf{A_S^Q} \slp H^P, h^- H_Q h \srp + \slp H^P, h^- \sf{d} h \srp = \slp L^h \srp^P{}_Q \s, \sf{A_S^Q}(x) + \sf{{\scal I}_H^P}(y)\n\send{eqnarray}\nwith the appearance of the [[left-right rotator]]s, $\slp L^h \srp^I{}_J(y)$, and the Maurer-Cartan form, $\sf{{\scal I}_H}$, for $H$. These are the same as the frame components, $\sf{e^A}(z) = \sf{{\scal E}_S^A}(z)$ and $\sf{e^P}(z) = \sf{{\scal A}_S^P}(z)$, for a [[reductive Lie group geometry]]. Using the correspondence between the Lie algebra generators and the left invariant vector fields of the Lie group geometry, $T_I \ssim \sve{\sxi^R_I}$, allows us to write the [[Ehresmann-Maurer-Cartan vector valued form|Maurer-Cartan form]] as\n\sbegin{eqnarray}\n\sf{\sve{\scal I}} &=& \sf{\sve{{\scal E}_S}} + \sf{\sve{{\scal A}_S}} \s\s\n&=& \sf{{\scal E}_S^A} \sve{\sxi^R_A} + \sf{{\scal A}_S^P} \sve{\sxi^R_P} \s\s\n&=& \sf{e_S^B} \slp L^h \srp^A{}_B \s, \sve{\sxi^R_A} + \sf{A_S^Q} \slp L^h \srp^P{}_Q \s, \sve{\sxi^R_P} + \sf{\sve{{\scal I}_H}} \s\s\n&=& \sf{I^J} \slp L^r \srp_J{}^K \s, \sve{\sxi^L_K} + \sf{\sve{{\scal I}_H}}\n\send{eqnarray}\nwith the ''Ehresmann homogeneous space frame'', $\sf{\sve{{\scal E}_S}}$, and ''Ehresmann homogeneous H-connection'', $\sf{\sve{{\scal A}_S}}$, satisfying $\sf{\sve{{\scal E}_S}} \sf{\scal I} = \sf{{\scal E}_S}$ and $\sf{\sve{{\scal A}_S}} \sf{\scal I} = \sf{{\scal A}_S}$ using the Maurer-Cartan form, $\sf{\scal I} = \sf{\sxi_R^J}(z) T_J$, over $G$. The Ehresmann homogeneous H-connection, $\sf{\sve{{\scal A}_S}} = \sf{{\scal A}_S^P} \sve{\sxi^R_P}(z)$, is a [[Ehresmann principal bundle connection]] for an H-bundle and satisfies $\sf{\sve{{\scal A}_S}} \sf{\scal I_H} = \sf{{\scal A}_S}$, since $\sve{\sxi^R_P}(z) = \sve{\sxi^{HR}_P}(y)$ in a reductive Lie group geometry. Another way of looking at an Ehresmann homogeneous space geometry is as a [[Cartan H-bundle]] with $\sf{C} = \sf{I}$.\n\nSince the Ehresmann-Maurer-Cartan VVF is the identity projection, its [[FuN curvature]] vanishes, $\sff{\sve{\scal F}} = -\sha \slb \sf{\sve{\scal I}}, \sf{\sve{\scal I}} \srb_L = 0$.
from [[Ehresmann connection]]\n\nequivalent to [[parallel transport]]
[<img[images/png/fiber bundle.png]]A [[principal bundle]] consists of a total space, $E$, built locally from the direct product of a [[Lie group geometry]] (the typical fiber, $F=G$) over a base manifold, $M$. The same [[Lie group]], $G$, is the structure group, acting on the fibers, and hence on the total space, via left action. This group also acts on the fibers, and the total space, via right action. A [[connection]], $\sf{A}(x)=\sf{A^B}T_B=\sf{dx^a}A_a{}^BT_B$, a [[Lieform]] over the base space, describes principal bundle geometry. By choosing a [[reference section|Ehresmann gauge transformation]], $\ssi_0:M \sto E$, this connection may be related to an [[Ehresmann connection]], $\sf{\sve{\scal A}}$, over the total space.\n\nThere is a convenient set of local coordinates for the total space. The $n_M$ coordinates, $x^a$, with [[spacetime]] [[indices]], cover patches of the base manifold and the $n_G$ coordinates, $y^p$, are from the typical fiber. So a point of $E$ may be described by $p=(x,y)$ or equivalently by $p=(x,g)$ -- where $g(y)$ is the Lie group (fiber) element parameterized by $y$. The fiber bundle projection is then simply $\spi(x,y)=x$. The coordinates are chosen so that $y^p=0$, and hence $g=1$, on the ''canonical reference section'', $\ssi_0(x)=(x,y_0(x))=(x,0) \ssim (x,1)$, which provides the ''canonical local trivialization'', $(x,g) \sin E$. With these coordinates each fiber corresponds to a coordinate surface of constant $x$. The $y$ [[coordinate basis vectors]] are in the vertical subspace, $\sve{\spa_p} \sin \sve{\sDe_V}$, but the $x$ coordinate basis vectors are not necessarily in the horizontal subspace, $\sve{\spa_a} \snotin \sve{\sDe_H}$. We will abuse the use of the same label, $x^a$, for the coordinates on the base and some on the total space. Using these coordinates, the ''Ehresmann principal bundle connection'' (a [[vector projection]]) over the total space is\n$$\n\sf{\sve{\scal A}}(x,y) = \sf{A^B}(x) \sve{\sxi^L_B}(y) + \sf{\sve{\scal I}}\n$$\nin which $\sve{\sxi^L_B}$ are the [[left action vector fields|Lie group geometry]] for the Lie group geometry, defined by\n$$\n\sve{\sxi^L_B} \sf{\spa} g(y) = T_B g\n$$\nand\n$$\n\sf{\sve{\scal I}} = \sf{\sxi_L^B}(y) \sve{\sxi^L_B}(y) = \sf{\sxi_R^B}(y) \sve{\sxi^R_B}(y) = \sf{dy^p} \sve{\spa_p}\n$$\nis the [[Ehresmann-Maurer-Cartan form|Maurer-Cartan form]] (the [[identity projection|vector projection]] along the fibers). The Ehresmann connection is a projection, $\sf{\sve{\scal A}} \sf{\sve{\scal A}} = \sf{\sve{\scal A}}$, is [[right invariant]], $R_h^*\sf{\sve{\scal A}} = \sf{\sve{\scal A}}$, and has a natural [[spectral decomposition|spectral decomposition of the Ehresmann principal bundle connection]]. It may also be written as a Lie algebra valued 1-form over the total space by [[contracting|vector-form algebra]] it with the [[Maurer-Cartan form]], $\sf{\scal I}(y) = \sf{\sxi_R^B}(y)T_B = g^- \sf{d} g$, over the total space to get the ''Ehresmann connection form'',\n\sbegin{eqnarray}\n\sf{\scal A}(x,y) &=& \sf{\sve{\scal A}} \sf{\scal I} = \sf{A^B}(x) \sve{\sxi^L_B}(y) \sf{\sxi_R^C}(y) T_C + \sf{\sve{\scal I}} \sf{\scal I} \s\s\n&=& \slp \sf{A^B} L^C{}_B(y) + \sf{\sxi_R^C} \srp T_C \s\s\n&=& g^-(y) \sf{A}(x) g(y) + g^-(y) \sf{d^y} g(y) \n\send{eqnarray}\nusing the defining equation for the [[left-right rotator]],\n$$\nL^C{}_B(y) = \sve{\sxi^L_B}(y) \sf{\sxi_R^C}(y) = \slp T^C, g^-(y) T_B g(y) \srp \n$$\nThis form pulls back along the canonical reference section ($y=0$) to give the principal bundle connection,\n$$\n\ssi_0^*\sf{\scal A} = \sf{A}(x)\n$$\nand satisfies $R_h^* \sf{\scal A} = h^- \sf{\scal A} h$ under the right action.\n\nThe ''[[FuN curvature]] of the Ehresmann principal bundle connection'' is\n\sbegin{eqnarray}\n\sff{\sve{\scal F}}(x,y) &=& - \sha \slb \sf{\sve{\scal A}}, \sf{\sve{\scal A}} \srb_L = \slp 1 - \sf{\sve{\scal A}} \srp \slp \sf{\spa} \sf{\sve{\scal A}} \srp = \slp 1 - \sf{\sve{\scal A}} \srp \slp \slp \sf{\spa^x} + \sf{\spa^y} \srp \sf{\sve{\scal A}} \srp \s\s\n&=& \slp 1 - \sf{A^B}(x) \sve{\sxi^L_B}(y) - \sf{dy^p} \sve{\spa_p} \srp \n\slp \slp \sf{d^x} \sf{A^C} \srp \sve{\sxi^L_C}(y) - \sf{A^C} \sf{\spa^y} \sve{\sxi^L_C} \srp \s\s\n&=& \slp \sf{d^x} \sf{A^C} \srp \sve{\sxi^L_C} - \sf{A^B} \sf{A^C} \sve{\sxi^L_B} \sf{\spa^y} \sve{\sxi^L_C} \s\s\n&=& \slp \sf{d^x} \sf{A^D} + \sha \sf{A^B} \sf{A^C} C_{BC}{}^D \srp \sve{\sxi^L_D}(y)\n\send{eqnarray}\nusing the [[Lie bracket for left action vector fields|Lie group geometry]],\n$$\n\sve{\sxi^L_{\slb B \srd}} \slp \sf{\spa} \sve{\sxi^L_{\sld C \srb}} \srp = \sha \slb \sve{\sxi^L_B} , \sve{\sxi^L_C} \srb_L = - \sha C_{BC}{}^D \sve{\sxi^L_D}\n$$\nThis curvature is vector valued in the vertical subspace, and is right invariant, $R_h^* \sff{\sve{\scal F}} = \sff{\sve{\scal F}}$. The ''FuN curvature form of the Ehresmann connection'' is a $Lie(G)$ valued 2-form over $E$,\n\sbegin{eqnarray}\n\sff{\scal F} &=& \sff{\sve{\scal F}} \sf{\scal I} = \slp \sf{d^x} \sf{A^D} + \sha \sf{A^B} \sf{A^C} C_{BC}{}^D \srp g^-(y) T_D g(y) \s\s\n&=& g^-(y) \slp \sf{d^x} \sf{A} + \sha \slb \sf{A}, \sf{A} \srb \srp g(y) \s\s\n&=& g^-(y) \slp \sf{d^x} \sf{A} + \sf{A} \sf{A} \srp g(y)\n\send{eqnarray}\nThis form pulls back along the canonical reference section to give the [[principal bundle]] curvature,\n$$\n\ssi_0^*\sff{\scal F} = \sf{d} \sf{A} + \sf{A} \sf{A} = \sff{F}(x)\n$$\nand satisfies $R_h^* \sff{\scal F} = h^- \sff{\scal F} h$ under the right action.
The [[Ehresmann covariant derivative]] of a [[vector valued form]] field, $\snf{\sve{\scal K}}$, over the total space of a [[principal bundle]] using an [[Ehresmann principal bundle connection]] is\n$$\n\sf{\scal D} \snf{\sve{\scal K}} = - {\scal L}_{\sf{\sve{\scal A}}} \snf{\sve{\scal K}}\n$$\nusing the [[FuN derivative]]. The VVF will usually be right invariant and expressible as\n$$\n\snf{\sve{\scal K}} = \snf{K^B}(x) \sve{\sxi^L_B}(y)\n$$\ncorresponding to the [[Lieform]],\n$$\n\snf{\scal K} = \snf{\sve{\scal K}} \sf{\scal I} = \snf{K^B}(x) g^-(y) T_B g(y) \n$$\nobtained with the [[Maurer-Cartan form]], $\sf{\scal I}(y) = \sf{\sxi_R^B} T_B$, and the [[left-right rotator]]. The [[pullback]] of this form along a section, $\ssi_1=(x,g_1(x))$, gives the Lieform over the base,\n$$\n\snf{K_1}(x) = \ssi_1^* \snf{\scal K} = \snf{K^B}(x) g^-_1(x) T_B g_1(x) = g^-_1(x) \snf{K}(x) g_1(x)\n$$\nin which the form pulled back along the reference section is $\snf{K}(x) = \snf{K^B}(x) T_B$. The Ehresmann covariant derivative of the VVF,\n$$\n\sf{\scal D} \snf{\sve{\scal K}} = \slp \sf{d} \snf{K^C} \srp \sve{\sxi^L_C} - \sf{A^B} \snf{K^C} \slb \sve{\sxi^L_B}, \sve{\sxi^L_C} \srb_L\n= \slp \sf{d} \snf{K^D} + \sf{A^B} \snf{K^C} C_{BC}{}^D \srp \sve{\sxi^L_D}\n$$\ngives a definition for the ''Ehresmann covariant derivative of a Lieform'',\n$$\n\sf{\scal D} \snf{\scal K} = \slp \sf{\scal D} \snf{\sve{\scal K}} \srp \sf{\scal I} = \slp \sf{d} \snf{K^D} + \sf{A^B} \snf{K^C} C_{BC}{}^D \srp g^-(y) T_D g(y)\n$$\nwhich pulls back along any chosen section to give the [[covariant derivative|principal bundle]] of $\snf{K}$ on the base, \n\sbegin{eqnarray}\n\slp \sf{D_1} \snf{K_1} \srp(x) &=& \ssi_1^* \slp \sf{\scal D} \snf{\scal K} \srp \n= \slp \sf{d} \snf{K^D} + \sf{A^B} \snf{K^C} C_{BC}{}^D \srp g^-_1(x) T_D g_1(x) \s\s\n&=& g^-_1(x) \slp \sf{d} \snf{K} + \slb \sf{A}, \snf{K} \srb \srp g_1(x)\n= g^-_1(x) \slp \sf{\sna} \snf{K} \srp g_1(x)\n\send{eqnarray}\n(This should be equivalent to the usual definition of the Ehresmann covariant derivative of Lieform via\n$$\n\sve{v_1} \sve{v_2} \sdots \sve{v_{k+1}} \sf{\scal D} \snf{\scal K} = \sve{v^H_1} \sve{v^H_2} \sdots \sve{v^H_{k+1}} \sf{d} \snf{\scal K}(x,y)\n$$\nin which the vectors on the right are horizontal projections of the ones on the left, $\sve{v^H} = \sve{v}(1-\sf{\sve{\scal A}})$. //That definition needs to be checked.//)\n\nSimilarly, if a VVF is left invariant and may be expressed as\n$$\n\snf{\sve{\scal K}} = \snf{K^B}(x) \sve{\sxi^R_B}(y)\n$$\ncorresponding to the Lieform,\n$$\n\snf{\scal K} = \snf{\sve{\scal K}} \sf{\scal I} = \snf{K^B}(x) T_B \n$$\nThe [[pullback]] of this form along any section, $\ssi_1=(x,g_1(x))$, gives the same Lieform over the base,\n$$\n\snf{K_1}(x) = \ssi_1^* \snf{\scal K} = \snf{K^B}(x) T_B = \snf{K}(x)\n$$\nThe Ehresmann covariant derivative of this VVF,\n$$\n\sf{\scal D} \snf{\sve{\scal K}} = \slp \sf{d} \snf{K^C} \srp \sve{\sxi^R_C} - \sf{A^B} \snf{K^C} \slb \sve{\sxi^L_B}, \sve{\sxi^R_C} \srb_L\n= \slp \sf{d} \snf{K^C} \srp \sve{\sxi^R_C}\n$$\ngives the Lieform,\n$$\n\sf{\scal D} \snf{\scal K} = \slp \sf{\scal D} \snf{\sve{\scal K}} \srp \sf{\scal I} = \slp \sf{d} \snf{K^C} \srp T_C = \sf{d} \snf{\scal K}\n$$\nwhich pulls back along any chosen section to give the [[exterior derivative]] of $\snf{K}$ on the base, \n$$\n\slp \sf{D_1} \snf{K_1} \srp = \ssi_1^* \slp \sf{\scal D} \snf{\scal K} \srp\n= \sf{d} \snf{K}\n$$\n
A passive [[Ehresmann gauge transformation]] for an [[Ehresmann principal bundle connection]] corresponds to changing to a different choice of section along which to pull back the Ehresmann connection form. The choice of reference section is equivalent to the choice of a local trivialization for a [[fiber bundle]]. Once a reference section, $\ssi_0:M \sto E$, and principal bundle connection, $\sf{A}$, have been used to build the principal bundle Ehresmann connection, $\sf{\sve{\scal A}}$, a different section, $\ssi_1$, can be introduced and used to pull back a different principal bundle connection, $\sf{A'}$ -- this is a [[gauge transformation]]. Using coordinates adapted to the reference section, the Ehresmann connection is\n$$\n\sf{\sve{\scal A}}(x,y) = \sf{A^B} \sve{\sxi^L_B}(y) + \sf{\sve{\scal I}}\n$$\nand the Ehresmann connection form is\n$$\n\sf{\scal A} = \sf{\sve{\scal A}} \sf{\scal I} = g^-(y) \sf{A}(x) g(y) + g^- \sf{d^y} g(y)\n$$\nusing the [[Maurer-Cartan form]], $\sf{\scal I}(y) = \sf{\sxi_R^B} T_B$, and [[left-right rotator]]. The new section, $\ssi_1 = \sph \scirc \ssi_0$, may be obtained by flowing the reference section by an equivariant vertical [[diffeomorphism]], $\sph(x,y) = (x,y_\sph(x,y))$, satisfying $\sph(ph)=\sph(p)h$ and giving $\ssi_1(x) = (x,y_1(x)) = (x,y_\sph(x,0))$. This is equivalent to transforming the original section by right (//?//) action by an element of $G$ to $\ssi_1(x) = \ssi_0(x) \s, g(y_1(x)) = \ssi_0 \s, g_1(x)$. The [[vector projection onto a section]], $\ssi_1$, is\n$$\n\sf{\sve{P_1}} = \sf{dx^a} \sve{\spa_a} + \sf{dx^a} \sfr{\spa y_1^p}{\spa x^a} \sve{\spa_p}\n$$\nand is used to project the Ehresmann connection to\n$$\n\sf{\sve{{\scal A}_1}} = \sf{\sve{P_1}} \sf{\sve{\scal A}} = \sf{A^B}(x) \sve{\sxi^L_B}(y_1) + \sf{dx^a} \sfr{\spa y_1^p}{\spa x^a} \sve{\spa_p}\n$$\nand the Ehresmann connection form to\n$$\n\sf{{\scal A}_1} = \sf{\sve{P_1}} \sf{\scal A} = \sf{\sve{{\scal A}_1}} \sf{\scal I} = \sf{\sve{P_1}} \sf{\sve{\scal A}} \sf{\scal I} = g^-(y_1(x)) \sf{A}(x) g(y_1(x)) + g^-(y_1(x)) \sf{d^x} g(y_1(x)) \n$$\non the section. This, the gauge transformed connection, gives the [[pullback]] of the Ehresmann connection form along the section,\n$$\n\sf{A'}(x) = \ssi_1^* \sf{\scal A} = \ssi_1^* \sf{{\scal A}_1} = g^-_1(x) \sf{A} g_1(x) + g^-_1(x) \sf{d^x} g_1(x) \n$$\nidentified as the [[principal bundle gauge transformation|principal bundle]] with $g_1(x)=g^-(x)$.\n\nAn alternative way of effecting a gauge transformation is to flow the Ehresmann connection form by the diffeomorphism, $\sf{\scal A'} = \sph^*\sf{\scal A}$, then pull it back along the original section, $\ssi_0$. This ''active gauge transformation'' gives the same result,\n$$\n\ssi_0^* \sf{\scal A'} = \ssi_0^* \slp \sph^* \sf{\scal A} \srp\n= \slp \sph \scirc \ssi_0 \srp^* \sf{\scal A}\n= \ssi_1^* \sf{\scal A} = \sf{A'}\n$$\n\nRef:\n*[[Geometrical aspects of local gauge symmetry|papers/Geometrical aspects of local gauge symmetry.pdf]]
[[Ehresmann lift]] for an [[Ehresmann principal bundle connection]].
A [[vector bundle]] consists of a total space, $E$, built locally from the direct product of a [[vector space]] (the typical fiber, consisting of elements $v = v^\sal b_\sal \sin V = F$) and a base manifold, $M$. A [[vector bundle connection]], $\sf{A}{}_\sal{}^\sbe(x) = \sf{dx^i} A_{i \sal}{}^\sbe(x)$, a 1-form over the base space valued in some subgroup of the general linear group, describes the geometry of the vector bundle. By choosing a ''reference [[section|fiber bundle]]'', $\ssi_0:M \sto E$, this connection may be related to an [[Ehresmann connection]], $\sf{\sve{\scal A}}$, over the total space.\n\nThere is a convenient set of local coordinates for the total space. The $n$ coordinates, $x^a$, with [[spacetime]] [[indices]], are from the base manifold and the $K$ coordinates, $v^\sal$, are from the typical fiber (vector space). So a point of $E$ (above some patch of $M$) may be described by $p = (x,v)$. The fiber bundle projection is then simply $\spi(x,v) = x$. The coordinates are chosen so that $v^\sal = 0$ on the reference section, a ''canonical local trivialization'', $\ssi_0(x) = (x,v_0(x)) = (x,0)$. With these coordinates each fiber corresponds to a coordinate surface of constant $x$. The [[coordinate basis vectors]] for the $v^\sal$ coordinates, $\sve{\spa_\sal} \sin \sve{\sDe_V}$, are in the vertical subspace, but the $x$ coordinate basis vectors are not necessarily in the horizontal subspace, $\sve{\spa_a} \snotin \sve{\sDe_H}$. We will abuse the use of the same label, $x^a$, for the coordinates on the base and some on the total space. Using these coordinates, the linear ''Ehresmann vector bundle connection'' over the total space is\n$$\n\sf{\sve{\scal A}}(x,v) = \slp \sf{dx^i} A_{i \sal}{}^\sbe(x) v^\sal + \sf{dv^\sbe} \srp \sve{\spa_\sbe}\n$$\n\nIn analogy with the [[Maurer-Cartan form]], we build a vector ($V$) valued 1-form,\n$$\n\sf{\scal I} = \sf{dv^\sbe} b_\sbe\n$$\nand use it to define the ''Ehresmann vector bundle connection form'',\n$$\n\sf{\scal A} = \sf{\sve{\scal A}} \sf{\scal I} = \slp \sf{A}{}_\sal{}^\sbe v^\sal + \sf{dv^\sbe} \srp b_\sbe\n$$\nThis allows us to define the [[vector bundle covariant derivative|vector bundle connection]] of any section, $\ssi_1 : M \smapsto E$, $\ssi_1(x) = (x, v_1(x))$, as the [[pullback]] of $\sf{\scal A}$ along the section to M,\n$$\n\ssi_1^* \sf{\scal A} = \sf{A}{}_\sal{}^\sbe v_1^\sal(x) b_\sbe + \sf{dx^a} \sfr{\spa v_1^\sbe}{\spa x^a} b_\sbe = \sf{\sna} v_1(x)\n$$ \n\nThe ''[[FuN curvature]] of the Ehresmann vector bundle connection'' is\n\sbegin{eqnarray}\n\sff{\sve{\scal F}}(x,y) &=& - \sha \slb \sf{\sve{\scal A}}, \sf{\sve{\scal A}} \srb_L = \slp 1 - \sf{\sve{\scal A}} \srp \slp \sf{\spa} \sf{\sve{\scal A}} \srp \s\s\n&=& \slp 1 - \sf{A}{}_\sal{}^\sbe v^\sal \sve{\spa_\sbe} - \sf{dv^\sbe} \sve{\spa_\sbe} \srp \n\slp \slp \sf{d^x} \sf{A}{}_\sga{}^\sde \srp v^\sga \sve{\spa_\sde} - \sf{A}{}_\sga{}^\sde \sf{dv^\sga} \sve{\spa_\sde} \srp \s\s\n&=& \slp \sf{d^x} \sf{A}{}_\sal{}^\sde - \sf{A}{}_\sal{}^\sbe \sf{A}{}_\sbe{}^\sde \srp v^\sal \sve{\spa_\sde} \s\s\n&=& \sff{F}{}_\sal{}^\sde v^\sal \sve{\spa_\sde}\n\send{eqnarray}\nin which the [[vector bundle curvature]], $\sff{F}{}_\sal{}^\sde$, appears.\n\nThe [[Ehresmann covariant derivative]] of any [[vector valued form]] over the total space (such as the curvature above) that can be written as\n$$\n\snf{\sve{\scal K}} = \snf{K}{}_\sga{}^\sde(x) v^\sga \sve{\spa_\sde}\n$$\nis defined using the [[FuN derivative]] as\n\sbegin{eqnarray}\n\sf{\scal D} \snf{\sve{\scal K}} &=& - {\scal L}_{\sf{\sve{\scal A}}} \snf{\sve{\scal K}}\n= - \sf{\sve{\scal A}} \slp \sf{\spa} \snf{\sve{\scal K}} \srp + \slp -1 \srp^k \snf{\sve{\scal K}} \slp \sf{\spa} \sf{\sve{\scal A}} \srp + \sf{\spa} \slp \snf{\sve{\scal K}} \sf{\sve{\scal A}} \srp \s\s\n&=& - \slp \sf{A}{}_\sal{}^\sbe(x) v^\sal + \sf{dv^\sbe} \srp \sve{\spa_\sbe} \slp \slp \sf{d^x} \snf{K}{}_\sga{}^\sde \srp v^\sga \sve{\spa_\sde} + \slp -1 \srp^k \snf{K}{}_\sga{}^\sde \sf{dv^\sga} \sve{\spa_\sde} \srp \s\s\n& & + \slp -1 \srp^k \snf{K}{}_\sga{}^\sde v^\sga \sve{\spa_\sde} \slp \slp \sf{d^x} \sf{A}{}_\sla{}^\sbe \srp v^\sal \sve{\spa_\sbe} - \sf{A}{}_\sal{}^\sbe \sf{dv^\sal} \sve{\spa_\sbe} \srp\n+ \sf{\spa} \slp \snf{K}{}_\sga{}^\sde v^\sga \sve{\spa_\sde} \srp \s\s\n&=& \slp \sf{d^x} \snf{K}{}_\sga{}^\sde - \sf{A}{}_\sga{}^\sbe \snf{K}{}_\sbe{}^\sde + \sf{A}{}_\sbe{}^\sde \snf{K}{}_\sga{}^\sbe \srp v^\sga \sve{\spa_\sde} \s\s\n&=& \slp \sf{\sna} \snf{K}{}_\sga{}^\sde \srp v^\sga \sve{\spa_\sde}\n\send{eqnarray}\n\nAn [[Ehresmann gauge transformation]] corresponding to a change in local trivialization, $b_\sbe \smapsto b'_\sbe = g_\sbe{}^\sal(x) b_\sal$, gives changes in $\sf{A}{}_\sal{}^\sbe$ and other coefficients corresponding to a [[vector bundle gauge transformation]].
In a [[spacetime]], equivalent to a [[Cl(1,3)]] or [[Cl(3,1)]] [[Clifford vector bundle]], the [[Clifford-Ricci curvature]], $\sf{R}$, [[Clifford curvature scalar]], $R$, [[frame]], $\sf{e}$, ''cosmological constant'', $\sLa$, and ''Clifford energy-momentum tensor'', $\sf{T}$, for matter are dynamically related by ''Einstein's equation'',\n$$\n\sf{R} - \sha R \sf{e} = \set_{00} ( \sLa \sf{e} - 8 \spi G \sf{T} ) \n$$\nin which $\set_{00}$ specifies the [[Minkowski metric]] sign convention. In a vacuum, $\sf{T} = 0$, Einstein's equation contracted with the coframe, $\sve{e}$, gives\n$$\n\sve{e} \scdot \slp \sf{R} - \sha R \sf{e} \srp = R - \sha R n = \set_{00} \sLa n\n$$\nrequiring the curvature scalar to be constant, $R = - \sfr{2n}{n-2} \set_{00} \sLa = - 4 \set_{00} \sLa$ (with $n=4$), and giving the ''vacuum Einsten's equation'',\n$$\n\sf{R} = - \sfr{2}{n-2} \set_{00} \sLa \sf{e} = - \set_{00} \sLa \sf{e}\n$$\nAny spacetime that satisfies $\sf{R} = \sal \sf{e}$ for some constant, $\sal$, is an ''Einstein space''.\n\nEinstein's equation is derived by extremizing the ''Einstein-Hilbert action'',\n$$\nS = \sint \snf{e} \slp \sfr{1}{16 \spi G} \slp R + 2 \set_{00} \sLa \srp + L_M \srp\n$$\nwith respect to $\sve{e}$, in natural [[units]].
http://arxiv.org/abs/gr-qc/0606062\n*looks to be a good reveiw of ECT\n\nIn ECT...\nThe curvature picks up a contribution from torsion, and the Ricci curvature is no longer guaranteed to be symmetric in its indices. This change in the equation of motion allows matter with a spin component to couple to the angular momentum of the gravitational field.\n\nIn teleparallel theories of gravity the spin connection is purely torsional ($\sf{\snu}=0$, $\sf{d} \sf{e}=0$, $\sf{\ska} \sneq 0$) and the [[spacetime]] is, in that sense, flat, with the gravitational field a force represented solely by torsion.\n\nRef:\n[[Huang - Cosmological Solutions with Torsion in a Model of de Sitter Gauge Theory of Gravity|papers/Huang - Cosmological Solutions with Torsion in a Model of de Sitter Gauge Theory of Gravity.pdf]]
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<html>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td><div class="math">\n\sbegin{array}{c}\nW =\n\slb \sbegin{array}{cc}\n{\ssmall \sfrac{i}{2}} W^3 & W^+ \s\s\nW^- & {\ssmall - \s! \sfrac{i}{2}} W^3\n\send{array} \srb \svp{|_{\sBig(}}\n\squad\nB_1 =\n\slb \sbegin{array}{cc}\n{\ssmall \sfrac{i}{2}} B_1^3 & B_1^+ \s\s\nB_1^- & {\ssmall - \s! \sfrac{i}{2}} B_1^3\n\send{array} \srb \svp{|_{\sBig(}}\n\s\s\n\sbig[\n\slb \sbegin{array}{cc}\nW & \s\s\n& B_1\n\send{array} \srb\n,\n\slb \sbegin{array}{cc}\n & \sph_B \s\s\n\sph_W &\n\send{array} \srb\n\sbig] \svp{|_{\sBig(}}\n\s\s\n\sqquad \sqquad \sqquad \sqquad \squad\n\sph_{W/B} =\n\slb \sbegin{array}{cc}\n- \sph_{0/1} & \sph_+ \s\s\n\sph_- & \sph_{1/0}\n\send{array} \srb \svp{|_{\sBig(}}\n\s\s\n\slb \sbegin{array}{cc}\nW & \s\s\n& B_1\n\send{array} \srb\n\squad\n\slb \sbegin{array}{c}\n\snu_{eL} \s\s e_L \s\s \snu_{eR} \s\s e_R\n\send{array} \srb\n\squad\n\slb \sbegin{array}{c}\nu_L \s\s d_L \s\s u_R \s\s d_R\n\send{array} \srb \s\s[.5em]\n\sbig( \sfr{\ssqrt{3}}{\ssqrt{5}} B_1^3 - \sfr{\ssqrt{2}}{\ssqrt{5}} B_2 \sbig)\n= (\sfr{\ssqrt{3}}{\ssqrt{5}}) \sha Y \n \s;\sto\s; g_1=\sfr{\ssqrt{3}}{\ssqrt{5}}\n\send{array}\n</div></td>\n\n<td>&nbsp;&nbsp;&nbsp;</td>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th ALIGN=CENTER COLSPAN="2"><SPAN class="math">SO(4)</SPAN></th>\n<th></th>\n<th ALIGN=CENTER><SPAN class="math">W^3</SPAN></th>\n<th ALIGN=CENTER><SPAN class="math">B_1^3</SPAN></th>\n<th></th>\n<th ALIGN=CENTER><SPAN class="math">\sfr{\ssqrt{2}}{\ssqrt{3}} B_2</SPAN></th>\n<th ALIGN=CENTER><SPAN class="math">\sha Y</SPAN></th>\n<th ALIGN=CENTER><SPAN class="math">Q</SPAN></th>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#FFFF00} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">W^+</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#FFFF00} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">W^-</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">- 1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#FFFFFF} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">B_1^+</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n</tr>\n<tr class="butt">\n<td ALIGN=CENTER><SPAN class="math">\smcir{#FFFFFF} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">B_1^-</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">- 1</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smsqu{#B2B200} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sph_+</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smdia{#F2F200} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sph_-</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smsqu{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sph_0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td ALIGN=CENTER><SPAN class="math">\smdia{#4D4D4D}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sph_1</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#B2B200}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\snu_{eL}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#F2F200}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">e_L</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\snu_{eR}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">e_R</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\n\strip{\smtri{#BF6000}}{\smtri{#668000}}{\smtri{#8F00B2}}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">u_L</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{2}{3}</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\n\strip{\smtri{#F77C00}}{\smtri{#99BF00}}{\smtri{#AD00F7}}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">d_L</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\n\strip{\smtri{#990000}}{\smtri{#009900}}{\smtri{#0000B2}}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">u_R</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{2}{3}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{2}{3}</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\n\strip{\smtri{#D90000}}{\smtri{#00BF00}}{\smtri{#0000F7}}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">d_R</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n</table>\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>
\n<html>\n<center>\n<table class="ptable">\n<tr>\n<th ALIGN=CENTER COLSPAN="3"><SPAN>Forces (bosons)</SPAN></th>\n<th><SPAN class="math">\s;\s;\s;\s;</SPAN></th>\n<th ALIGN=CENTER COLSPAN="3"><SPAN>Matter (fermions)</SPAN></th>\n<th><SPAN class="math">\s;\s;</SPAN></th>\n<th ALIGN=CENTER COLSPAN="3"><SPAN>Second generation</SPAN></th>\n<th><SPAN class="math">\s;\s;</SPAN></th>\n<th ALIGN=CENTER COLSPAN="3"><SPAN>Third generation</SPAN></th>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#000000} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sp{{\sBig(}^{(}_{\sbig(}} \sga \sp{{\sBig(}^{(}_{\sbig(}} </SPAN></td>\n<td ALIGN=CENTER><SPAN>electromagnetism</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#F2F200} \sbtri{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">e</SPAN></td>\n<td ALIGN=CENTER><SPAN>electron</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\smtri{#F2F200} \smtri{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\smu</SPAN></td>\n<td ALIGN=CENTER><SPAN>muon</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sstri{#F2F200} \sstri{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\stau</SPAN></td>\n<td ALIGN=CENTER><SPAN>tau</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#FFFF00} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">W,Z</SPAN></td>\n<td ALIGN=CENTER><SPAN>weak</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sbutr{#F2F200} \sbutr{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{e}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sp{{\sBig(}^{\sBig(}}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\smutr{#F2F200} \smutr{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{\smu}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\ssutr{#F2F200} \ssutr{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{\stau}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">g</SPAN></td>\n<td ALIGN=CENTER><SPAN>strong</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#B2B200} \sbtri{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\snu_e</SPAN></td>\n<td ALIGN=CENTER><SPAN>electron <br> neutrino</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\smtri{#B2B200} \smtri{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\snu_\smu</SPAN></td>\n<td ALIGN=CENTER><SPAN>muon <br> neutrino</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sstri{#B2B200} \sstri{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\snu_\stau</SPAN></td>\n<td ALIGN=CENTER><SPAN>tau <br> neutrino</SPAN></td>\n</tr>\n<tr class="butt">\n<td ALIGN=CENTER><SPAN class="math">\smcir{#59FF59} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sp{{\sBig(}^{\sbig(}_{\sbig(}} \som \sp{{\sBig(}^{\sbig(}_{\sbig(}}</SPAN></td>\n<td ALIGN=CENTER><SPAN>gravity</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sbutr{#B2B200} \sbutr{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{\snu}{}_e</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sp{{\sBig(}_(}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\smutr{#B2B200} \smutr{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{\snu}{}_\smu</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\ssutr{#B2B200} \ssutr{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{\snu}{}_\stau</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\smdia{#F2F200} \s, \smdia{#BF6000} \s\s[-.5em]\n\smsqu{#B2B200} \s, \smsqu{#F77C00}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sph</SPAN></td>\n<td ALIGN=CENTER><SPAN>Higgs</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sbtri{#BF6000} \sbtri{#990000} \s\s[-.8em]\n\sbtri{#668000} \sbtri{#009900} \s\s[-.8em]\n\sbtri{#8F00B2} \sbtri{#0000B2}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">u</SPAN></td>\n<td ALIGN=CENTER><SPAN>up <br> quark</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\smtri{#BF6000} \smtri{#990000} \s\s[-.8em]\n\smtri{#668000} \smtri{#009900} \s\s[-.8em]\n\smtri{#8F00B2} \smtri{#0000B2}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">c</SPAN></td>\n<td ALIGN=CENTER><SPAN>charm <br> quark</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sstri{#BF6000} \sstri{#990000} \s\s[-.8em]\n\sstri{#668000} \sstri{#009900} \s\s[-.8em]\n\sstri{#8F00B2} \sstri{#0000B2}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">t</SPAN></td>\n<td ALIGN=CENTER><SPAN>top <br> quark</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sbutr{#BF6000} \sbutr{#990000} \s\s[-.8em]\n\sbutr{#668000} \sbutr{#009900} \s\s[-.8em]\n\sbutr{#8F00B2} \sbutr{#0000B2}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{u}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\smutr{#BF6000} \smutr{#990000} \s\s[-.8em]\n\smutr{#668000} \smutr{#009900} \s\s[-.8em]\n\smutr{#8F00B2} \smutr{#0000B2}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{c}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\ssutr{#BF6000} \ssutr{#990000} \s\s[-.8em]\n\ssutr{#668000} \ssutr{#009900} \s\s[-.8em]\n\ssutr{#8F00B2} \ssutr{#0000B2}\n\send{array}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{t}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sbtri{#F77C00} \sbtri{#D90000} \s\s[-.8em]\n\sbtri{#99BF00} \sbtri{#00BF00} \s\s[-.8em]\n\sbtri{#AD00F7} \sbtri{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">d</SPAN></td>\n<td ALIGN=CENTER><SPAN>down <br> quark</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\smtri{#F77C00} \smtri{#D90000} \s\s[-.8em]\n\smtri{#99BF00} \smtri{#00BF00} \s\s[-.8em]\n\smtri{#AD00F7} \smtri{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">s</SPAN></td>\n<td ALIGN=CENTER><SPAN>strange <br> quark</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sstri{#F77C00} \sstri{#D90000} \s\s[-.8em]\n\sstri{#99BF00} \sstri{#00BF00} \s\s[-.8em]\n\sstri{#AD00F7} \sstri{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">b</SPAN></td>\n<td ALIGN=CENTER><SPAN>bottom <br> quark</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sbutr{#F77C00} \sbutr{#D90000} \s\s[-.8em]\n\sbutr{#99BF00} \sbutr{#00BF00} \s\s[-.8em]\n\sbutr{#AD00F7} \sbutr{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{d}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\smutr{#F77C00} \smutr{#D90000} \s\s[-.8em]\n\smutr{#99BF00} \smutr{#00BF00} \s\s[-.8em]\n\smutr{#AD00F7} \smutr{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{s}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\ssutr{#F77C00} \ssutr{#D90000} \s\s[-.8em]\n\ssutr{#99BF00} \ssutr{#00BF00} \s\s[-.8em]\n\ssutr{#AD00F7} \ssutr{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{b}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n</tr>\n</table>\n</center>\n</html>
Images can be included by their filename or full URL. It's good practice to include a title to be shown as a tooltip, and when the image isn't available. An image can also link to another note or or a URL\n[img[Romanesque broccoli|images/fractalveg.jpg][http://www.flickr.com/photos/jermy/10134618/]]\n{{{\n[img[Romanesque broccoli|images/fractalveg.jpg]\n [http://www.flickr.com/photos/jermy/10134618/]]\n[img[title|filename]]\n[img[filename]]\n[img[title|filename][link]]\n[img[filename][link]]\n}}}\n[<img[Forest|images/forest.jpg][http://www.flickr.com/photos/jermy/8749660/]][>img[Field|images/field.jpg][http://www.flickr.com/photos/jermy/8749285/]]You can also float images to the left or right: the forest is left aligned with {{{[<img[}}}, and the field is right aligned with {{{[>img[}}}.\n@@clear(left):clear(right):display(block):You can use CSS to clear the floats@@\n{{{\n[<img[Forest|images/forest.jpg][http://www.flickr.com/photos/jermy/8749660/]]\n[>img[Field|images/field.jpg][http://www.flickr.com/photos/jermy/8749285/]]\nYou can also float images to the left or right:\n the forest is left aligned with {{{[<img[}}},\nand the field is right aligned with {{{[>img[}}}.\n@@clear(left):clear(right):display(block):\nYou can use CSS to clear the floats@@\n}}}
Decent intros using MaxEnt:\nhttp://arxiv.org/abs/cond-mat/0507388\nhttp://arxiv.org/abs/physics/9805024\n\nsome Q.A.Wang papers\n*http://arxiv.org/abs/cond-mat/0312329\n**I don't like his use of ergodicity in defining the long time average equal to the Bayesian expectation value.\n**nice: uses fixed average action and MaxEnt to get partition function with action per path instead of energy per state\n***I might like using Rovelli's Hamiltonian constraint dynamics better.\n*http://arxiv.org/abs/cond-mat/0407515\n**seems inferior to previous paper\n\nGood Hawking paper on it:\nhttp://arxiv.org/abs/gr-qc/9501014\n*boundary term in depth\n*Hamiltonian formulation\n*relationship between partition functions for static spacetimes (timelike Killing vector).\n\nrelated discussion on time and Tomita flow in Rovelli book
<<note HideTags>>$$\n\sbegin{array}{llcl}\n\n1918, \s!\s!&\s!\s! {\srm Weyl} \s!\s!&\s!\s! : & \sf{A} \sin \sf{\srm Lie}(G) \sp{{}_{\sbig(}} \s\s\n\n1954, \s!\s!&\s!\s! {\srm Y.M.} \s!\s!&\s!\s! : & \sf{A} = \sf{B} + \sf{W} + \sf{G}\n\s;\s; \sin \s; \sf{\srm Lie}(G) = \sf{su}(1) + \sf{su}(2) + \sf{su}(3) \sp{{}_{\sbig(}} \s\s\n\n1967, \s!\s!&\s!\s! {\srm F.P.} \s!\s!&\s!\s! : & \sudf{A} = \sf{A} + \sud{g} \sp{{}_{\sbig(}}\n\s;\s; \sin \s; \sudf{\srm Lie}(G) \s\s\n\n1977, \s!\s!&\s!\s! {\srm M.M.} \s!\s!&\s!\s! : & \sf{A} = \sf{\som} + \sf{e}\n\s;\s; \sin \s; \sf{\srm Lie}(G) = \sf{so}(1,4) \sp{{}_{\sBig(_(}} \s\s\n\n\n2002, \s!\s!&\s!\s! {\srm Y.T.} \s!\s!&\s!\s! : & \sud{\sps} = \sud{g} \sp{{}_{\sbig(}} \s\s\n\n2005, \s!\s!&\s!\s! {\srm Y.T.} \s!\s!&\s!\s! : & \sudf{A} = {\ssmall \sfrac{1}{2}} \sf{\som} + {\ssmall \sfrac{1}{4}} \sf{e} \sph + \sf{B} + \sf{W} + \sf{G} + \sud{\snu^e} + \sud{e} + \sud{u} + \sud{d} \s\s\n & & & \s;\s;\s;\s,\s, \sin \s; \sf{\srm Lie}(G) = \sf{Cl}(1,7) \sp{{}_{\sbig(}} \s\s\n\n{\srm now}, \s!&\s! {\srm Y.T.} \s!\s!&\s!\s! : & \sudf{A} = {\ssmall \sfrac{1}{2}} \sf{\som} + {\ssmall \sfrac{1}{4}} \sf{e} \sph + \sf{B} + \sf{W} + \sf{G} + \sud{\snu^e} + \sud{e} + \sud{u} + \sud{d} \s\s\n & & & \s;\s;\s;\s;\s;\s;\s;\s; + \sud{\snu^\smu} + \sud{\smu} + \sud{c} + \sud{s}\n+ \sud{\snu^\sta} + \sud{\sta} + \sud{t} + \sud{b} \s\s\n & & & \s;\s;\s;\s,\s, \sin \s; \sf{\srm Lie}(G) = \sf{e8} ? \sp{{}_{\sbig(_(}}\n\send{array}\n$$\n
<<note HideTags>>The $14$ Lie algebra elements of the smallest exceptional Lie group, $G2$:\n$$\n\sbegin{array}{rcccccccl}\ng2 \s!\s!&\s!\s!=\s!\s!&\s!\s! su(3) \s!\s!&\s!\s! + \s!\s!&\s!\s! 3 \s!\s!&\s!\s! + \s!\s!&\s!\s! \sbar{3} \s!\s!&& \s\s\n&&\s!\s! \sf{g} \s!\s!&\s!\s! + \s!\s!&\s!\s! \sud{q} \s!\s!&\s!\s! + \s!\s!&\s!\s! \sud{\sbar{q}} \s!\s!&\s! \sin \s! &\s! \sudf{g2} \n\send{array}\n$$\nStructure of $G2$ implies Lie bracket equivalent to fundamental action,\n$$\n[ g,q ] = \sbig[ g^A T_A,q^B T_B \sbig] = g \s, q =\n\slb\n\smatrix{\n\s! \sfr{i}{2} g^3 \s!+\s! {\sscriptsize \sfrac{i}{2\ssqrt{3}}} g^8 \s!\s! & g^{r\sbar{g}} & g^{r\sbar{b}} \s\s\ng^{\sbar{r}g} & \s!\s! {\sscriptsize -\s!\sfrac{i}{2}} g^3 \s!+\s! {\sscriptsize \sfrac{i}{2\ssqrt{3}}} g^8 \s!\s! & g^{g\sbar{b}} \s\s\ng^{\sbar{r}b} & g^{\sbar{g}b} & {\sscriptsize -\s!\sfrac{i}{\ssqrt{3}}} g^8\n}\n\srb\n\slb \smatrix{\nq^r \s\s q^g \s\s q^b\n} \srb\n$$\ncorresponding to the strong interactions, such as\n<html>\n<center>\n<table class="gtable">\n<tr>\n<td>\n<div class="math">\n\sbig[ g^{r\sbar{g}}, q^g \sbig] = q^r\n</div>\n</td>\n<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>\n<td ALIGN=CENTER><img SRC="images/png/quark gluon vertex.png" height=160px></td>\n</tr>\n</table>\n</center>\n</html>
''Bold''\n{{{''Bold''}}}\n==Strikethrough==\n{{{==Strikethrough==}}}\n__Underline__ \n{{{__Underline__}}}\n//Italic// \n{{{//Italic//}}}\n2^^3^^=8 \n{{{2^^3^^=8}}}\na~~ij~~ = -a~~ji~~ \n{{{a~~ij~~ = -a~~ji~~}}}\n@@highlight@@ \n{{{@@highlight@@}}}\n\n//The highlight can also accept CSS syntax to directly style the text://\n@@color:green;green coloured@@\n{{{@@color:green;green coloured@@}}}\n@@background-color:#ff0000;color:#ffffff;red coloured@@\n{{{@@background-color:#ff0000;color:#ffffff;red coloured@@}}}\n@@text-shadow:black 3px 3px 8px;font-size:18pt;display:block;margin:1em 1em 1em 1em;border:1px solid black;Access any CSS style@@\n{{{@@text-shadow:black 3px 3px 8px;font-size:18pt;display:block;margin:1em 1em 1em 1em;border:1px solid black;Access any CSS style@@}}}\n@@display:block;text-align:center;centered text or image@@\n{{{@@display:block;text-align:center;centered text or image@@}}}\n\n//For backwards compatibility, the following highlight syntax is also accepted://\n@@bgcolor(#ff0000):color(#ffffff):red coloured@@\n{{{\n@@bgcolor(#ff0000):color(#ffffff):red coloured@@\n}}}
The rank $4$ exceptional group, ''F4'', is a real, [[simple]], compact, connected [[Lie group|Lie group]]. It may be described by [[exponentiating|exponentiation]] its $52$ dimensional [[Lie algebra]], [[f4]].
$$\n\sbegin{array}{rcll}\nF4 &:& ( \sha \som_L^3, \sha \som_R^3, W^3, B_1^3 ) \s;\s;\s; \sp{(B_2, g^3, g^8)} &\sleft\s{ \n\sbegin{array}{l}\stext{graviweak interactions} \s\s \stext{three generations}\send{array} \sright.\n\s\s\nG2 &:& \sp{( \sha \som_L^3, \sha \som_R^3, W^3, B_1^3 ) \s;\s;\s;}(B_2, g^3, g^8) &\sleft\s{ \n\sbegin{array}{l}\stext{strong interactions} \s\s \stext{anti-particles}\send{array} \sright.\n\s\s[-.3em]\n\srlap{\shbox{@(hr noshade size="1" style="position:relative; left:-2em;\n width:30em; border:0px; border-top:1px solid black")}}\s\s[-.5em]\nE8 &:& ( \sha \som_L^3, \sha \som_R^3, W^3, B_1^3, w, B_2, g^3, g^8) & \s, \sleft\s{ \s; \stext{everything} \sright.\n\send{array}\n$$\n\nBreakdown of E8 to the standard model and gravity:\n\sbegin{eqnarray}\ne8 &=& f4 + g2 + 26 \s! \stimes \s! 7 \s\s\n&=& so(7,1) + su(3) + (8_{S+}\s!+\s!8_V+\s!8_{S-})\s!\stimes\s!(1\s!+\s!1\s!+\s!3\s!+\s!\sbar{3}) + 3\s!\stimes\s!(3\s!+\s!\sbar{3}) + 2 \s\s[.4em]\nA &=& \sbig( {\sscriptsize \sfrac{1}{2}} \som + {\sscriptsize \sfrac{1}{4}} e \sph + W + B_1 \sbig) + g + 3 \s! \stimes \s! \sPs + x \sPh + B_2 + w\n\send{eqnarray}\nTwo new quantum numbers and some non-standard particles:\n$$\n\s{ \s; w \squad (B_1^3\s!+\s!B_2) \squad B_1^\spm \squad x_{1/2/3} \sPh^{r/g/b} \squad x_{1/2/3} \sPh^{\sbar{r}/\sbar{g}/\sbar{b}} \s; \s} \svp{\sbig(}\n$$\n<<note HideTags>>
<<note HideTags>>\n@@display:block;text-align:center;\n<html><center><embed src="images/png/f4.png" width="510" height="510"></embed></center></html>\n@@
confirmed attendees:\n*[[Scott Aaronson|http://www.scottaaronson.com/]], QMI, quantum computing\n*[[Fred Adams|http://www.physics.lsa.umich.edu/department/directory/bio.asp?ID=1]]${}^*$, constant change, astrophysics\n*[[Anthony Aguirre|http://scipp.ucsc.edu/~aguirre/]], cosmology\n*[[Stephon Alexander|http://www.phys.psu.edu/people/display/index.html?person_id=4901]], astrophysics\n**Just put out a new paper: Isogravity\n**friends with James Bjorken (and everyone else, apparently)\n*[[Markus Aspelmeyer|http://homepage.univie.ac.at/Markus.Aspelmeyer/]]${}^*$, QM foundations\n*[[Paul A Benioff|http://www.phy.anl.gov/theory/staff/pab.html]], QM foundations, older guy\n*[[Caslav Brukner|http://homepage.univie.ac.at/Caslav.Brukner/index.htm]], QM foundations\n*[[Dmitry Budker|http://www.fqxi.org/aw-budker2.html]]${}^*$, constant change (experimental)\n*[[Gregory Chaitin|http://www.umcs.maine.edu/~chaitin/]], math, complexity, and philosophy of science\n*[[Hyung Choi|http://www.zoominfo.com/people/Choi_Hyung_78134925.aspx]], metanexus, QM foundations, science and religion (uh oh)\n*[[Louis Crane|http://www.fqxi.org/aw-crane2.html]]${}^*$, QGR, QM histories\n*[[Paul Davies|http://cosmos.asu.edu/]], QMI, astrophysics, popular author\n*[[John Donoghue|http://www.fqxi.org/aw-donoghue2.html]]${}^*$, emergent symmetry\n*[[Richard Easther|http://www.fqxi.org/aw-easther.html]]${}^*$, superstring cosmology\n*[[David Ritz Finkelstein|http://www.physics.gatech.edu/people/faculty/dfinkelstein.html#personal]], older particle physicist\n**Lie algebra expert. Proponent of stable Lie algebras.\n*[[Rodolfo Gambini|http://www.fqxi.org/aw-pullin2.html]]${}^*$, QM GR\n*[[Jaume Garriga|http://www.ffn.ub.es/gcg/personal/jaume.html]], cosmology, branes\n*[[Steven Gratton|http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+ea+gratton%2C+steven]], cosmology, inflation\n*[[Alan Guth|http://web.mit.edu/physics/facultyandstaff/faculty/alan_guth.html]], err, invented inflation\n*[[Lucien Hardy|http://www.perimeterinstitute.ca/index.phpindex.php?option=com_content&task=view&id=30&Itemid=7&view_directory=1&pi=1078]], Causaloids\n*[[Adrian Kent|http://www.damtp.cam.ac.uk/user/apak/]], QM foundations\n*[[Lawrence Krauss|http://www.phys.cwru.edu/~krauss/]], astrophysics and cosmology, popular author, dislikes KK and strings, lectures a LOT\n*[[Matthew Leifer|http://www.fqxi.org/aw-leifer2.html]], QM foundations\n*[[Eugene Lim|http://pantheon.yale.edu/~eal48/papers.html]]${}^*$, cosmology\n*A. [[Garrett Lisi]]${}^*$\n*[[Abraham Loeb|http://www.fqxi.org/aw-loeb2.html]]${}^*$, SETI, astronomy\n*[[Fotini Markopoulou|http://www.fqxi.org/aw-markopoulou2.html]]${}^*$, quantum graphity\n*[[Laura Mersini|http://en.wikipedia.org/wiki/Laura_Mersini]], cosmology\n*[[Farzad Nekoogar|http://www.fqxi.org/aw-nekoogar.html]]${}^*$, popularizer of theoretical physics -- [[multiversal journeys|http://www.multiversaljourneys.com/]]\n*[[Ken Olum|http://www.fqxi.org/aw-olum2.html]]${}^*$, GR, wants to rule out wormholes and other GR exotics\n*[[Maulik Parikh|http://www.fqxi.org/aw-khoury2.html]]${}^*$, GR boundaries, mach's principle, hep-th and strings\n*[[Philip Pearle|http://physerver.hamilton.edu/people/]], QM foundations, older guy\n*[[Ekkehard Peik|http://www.fqxi.org/aw-peik2.html]]${}^*$, constant change (experimental)\n*[[Simon Saunders|http://www.fqxi.org/aw-saunders2.html]]${}^*$, QM foundations\n*Lee Smolin (not going)\n*[[Robert Spekkens|http://www.fqxi.org/aw-spekkens2.html]]${}^*$, QM foundations\n*[[Max Tegmark|http://web.mit.edu/physics/facultyandstaff/faculty/max_tegmark.html]], astrophysics, cosmology, trouble maker...\n*[[Mark Trodden|http://physics.syr.edu/~trodden/]], cosmology, particle physics -- QFT\n*[[Roderich Tumulka|http://www.fqxi.org/aw-tumulka2.html]]${}^*$, Bohmian QM\n*[[Jos Uffink|http://www.phys.uu.nl/igg/jos/]], QM foundations\n*[[Vitaly Vanchurin|http://cosmos.phy.tufts.edu/~vitaly/]], cosmic strings \n*[[Xiao-Gang Wen|http://www.fqxi.org/aw-wen2.html]]${}^*$, gravity and light emergent from substrate :P\n*[[Serge Winitzki|http://www.theorie.physik.uni-muenchen.de/~serge/]], quantum cosmology\n**Likes wiki, and likes ToE.\n**Inflation expert -- says $R \sph^2$ term would be great, among others.\n***Strong constraints on these coefficients.\n*[[Toby Wiseman|http://schwinger.harvard.edu/~wiseman/]], string theory\n*[[Wojciech Zurek|http://public.lanl.gov/whz/]], cosmology and astrophysics, chaos, QM foundations\n\n${}^*$ grant winners (19)\n\nPress and Foundation people\n*[[Graham P Collins|http://www.sff.net/people/GPC/]], Scientific American Magazine\n*[[Valerie Jamieson|http://www.scienceinpublic.com/scienceweek/speakers.htm#Valerie%20Jamieson%20background]], New Scientist Magazine\n**particle physics background\n*[[Wade Davis|http://en.wikipedia.org/wiki/Wade_Davis]], National Geographic Explorer-in-Residence, ethnobiologist\n*[[Charles Harper|http://www.templeton.org/about_us/who_we_are/leadership_team/charles_harper/]], Senior Vice-President, John Templeton Foundation\n**He's paying, try not to insult him.\n*[[Amanda High|http://www.nptrust.org/about_npt/key_staff.asp#high]], Vice President, National Philanthropic Trust\n**What's she doing at the FQXi conference\n*[[Howard Burton|http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=index.php&e=Founding%20Executive%20Director&cat_id=53&cat_table=2]], Executive Director, Perimeter Institute for Theoretical Physics\n**Just got ousted from PI position, even though he founded it. Used to be main PI talent scout.\n*[[Christopher Liedel|http://executiveeducation.wharton.upenn.edu/fellows/feb_info/roster_detail.cfm?id=KRSM00000024468]], Executive Vice President & Chief Financial Officer, National Geographic Society\n*Robert Kuhn, Kuhn foundation -- makes science documentaries for PBS
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Choosing the anti-Grassmann 3-form to be $\sfff{\sod{B}} = \snf{e} \sod{\sPs} \sve{e} \s,$ gives the massive Dirac action in curved spacetime:\n\sbegin{eqnarray}\nS_f &=& \sint \sbig< \sfff{\sod{B}} \sudff{F} \sbig>\n= \sint \sbig< \sfff{\sod{B}} \sf{D} \sud{\sPs} \sbig> \s\s\n&=& \sint \sbig< \snf{e} \sod{\sPs} \sve{e} \sbig( \sf{d} \sud{\sPs} + \sf{H}{}_1 \sud{\sPs} - \sud{\sPs} \sf{H}{}_2 \sbig) \sbig> \s\s\n&=& \sint \sbig< \snf{e} \sod{\sPs} \sve{e} \sbig( ( \sf{d} + {\sscriptsize \sfrac{1}{2}} \sf{\som} + {\sscriptsize \sfrac{1}{4}} \sf{e}\sph + \sf{W} + \sf{B}{}_1 ) \sud{\sPs}\n- \sud{\sPs} ( \sf{w} + \sf{B}{}_2 + \sf{x} \sPh + \sf{g} ) \sbig) \sbig> \s\s\n&=& \sint \snf{d^4 x} \s, |e| \s, \sbig< \sod{\sPs} \sga^\smu (e_\smu)^i \sbig( \spa_i \sud{\sPs} + {\sscriptsize \sfrac{1}{4}} \som_i^{\sp{i} \smu \snu} \sga_{\smu \snu} \sud{\sPs} + W_i \sud{\sPs} + B_{1i} \sud{\sPs} \s\s\n&& \shphantom{\sint \snf{d^4 x} \s, |e| \s, \sbig< \sod{\sPs} \sga^\smu (e_\smu)^i \sbig(} + \sud{\sPs} w_i + \sud{\sPs} B_{2i} + \sud{\sPs} x_i \sPh + \sud{\sPs} g_i \sbig) + \sod{\sPs} \s, \sph \s, \sud{\sPs} \sbig>\n\send{eqnarray}\nThe $\sod{\sPs} \s, \sph \s, \sud{\sPs}$ is the standard Higgs mass term.$\svp{\sHuge(}$\nThe $\sod{\sPs} \sga^\smu \sud{\sPs} x_\smu \sPh$ term... I don't understand yet -- promising for CKM.\n<<note HideTags>>
The ''FuN curvature'' (//''Frölicher-Nijenhuis curvature''//), $\sff{\sve{\scal F}}=\sf{dx^i} \sf{dx^j} \sha {\scal F}_{ij}{}^k \sve{\spa_k}$, of a [[vector valued form]], $\sf{\sve{\scal A}}=\sf{dx^i} {\scal A}_i{}^j \sve{\spa_j}$, is its [[FuN bracket|FuN derivative]] with itself,\n\sbegin{eqnarray}\n\sff{\sve{\scal F}} &=& - \sha \slb \sf{\sve{\scal A}}, \sf{\sve{\scal A}} \srb_L\n= - \slb \sf{\sve{\scal A}}, \sf{\spa} \srb \sf{\sve{\scal A}} \s\s\n&=& - \sf{\sve{\scal A}} \slp \sf{\spa} \sf{\sve{\scal A}} \srp + \sf{\spa} \slp \sf{\sve{\scal A}} \sf{\sve{\scal A}} \srp \s\s\n&=& - \slp \sf{\sve{\scal A}} \sf{\spa} \srp \sf{\sve{\scal A}} + \slp \sf{\spa} \sf{\sve{\scal A}} \srp \sf{\sve{\scal A}} \n\send{eqnarray}\nIn components, this is\n$$\n{\scal F}_{ij}{}^k = - {\scal A}_i{}^m \spa_m {\scal A}_j{}^k + {\scal A}_j{}^m \spa_m {\scal A}_i{}^k + {\scal A}_m{}^k \spa_i {\scal A}_j{}^m - {\scal A}_m{}^k \spa_j A_i{}^m\n$$\n\nIf, as often happens, a vector valued form is a [[vector projection]], $\sf{\sve{\scal P}} = \sf{\sve{\scal P}} \sf{\sve{\scal P}}$, its FuN curvature is\n\sbegin{eqnarray}\n\sff{\sve{\scal F}} &=& \sf{\spa} \sf{\sve{\scal P}} - \sf{\sve{\scal P}} \slp \sf{\spa} \sf{\sve{\scal P}} \srp \s\s\n&=& \slp 1 - \sf{\sve{\scal P}} \srp \slp \sf{\spa} \sf{\sve{\scal P}} \srp \n\send{eqnarray}\nwhich satisfies $\sf{\sve{\scal P}} \sff{\sve{\scal F}}=0$ -- the form part of the FuN curvature is in the kernel (horizontal part) of the projection. If the vector projection is an [[Ehresmann connection]], any two vectors contracted with the FuN curvature give the vertical part of the [[Lie bracket|Lie derivative]] of the horizontal part of the vectors,\n$$\n\sve{u} \sve{v} \sff{\sve{\scal F}} = - \sha {\slb \sve{u_H} , \sve{v_H} \srb_L}_V = - \sha \slb \sve{u} \slp 1- \sf{\sve{\scal P}} \srp , \sve{v} \slp 1- \sf{\sve{\scal P}} \srp \srb_L \sf{\sve{\scal P}}\n$$\n//check that//
The ''Frölicher-Nijenhuis Lie derivative'' -- which we refer to as the //''FuN derivative''// -- is a [[natural]] operator generalizing the [[Lie derivative]] to handle [[vector valued form]] fields. The FuN derivative of a vector valued $k$-form field, $\snf{\sve{K}}$, with respect to a vector field, $\sve{v}$, is written terms of [[partial derivative]]s as\n\sbegin{eqnarray}\n{\scal L}_{\sve{v}} \snf{\sve{K}} &=& \slim_{t \sto 0} \sfr{\sph_t^*\snf{\sve{K}} - \snf{\sve{K}}}{t} = \sve{v} \slp \sf{\spa} \snf{\sve{K}} \srp + \sf{\spa} \slp \sve{v} \snf{\sve{K}} \srp - \slp \snf{\sve{K}} \sf{\spa} \srp \sve{v} \s\s\n&=& \slp \sve{v} \sf{\spa} \srp \snf{\sve{K}} + \slp \sf{\spa} \sve{v} \srp \snf{\sve{K}} - \slp \snf{\sve{K}} \sf{\spa} \srp \sve{v}\n\send{eqnarray}\nThis defines the ''Frölicher-Nijenhuis bracket'' (//''FuN bracket''//) for these fields, and enforcing antisymmetry defines the FuN derivative of a vector field with respect to a vector valued form.,\n$$\n{\scal L}_{\snf{\sve{K}}} {\sve{v}} = \slb \snf{\sve{K}} , \sve{v} \srb_L = - \slb \sve{v} , \snf{\sve{K}} \srb_L = - {\scal L}_{\sve{v}} \snf{\sve{K}} \n$$ \nSimilarly, generalizing Cartan's formula for the Lie derivative, the FuN derivative of a differential form is,\n\sbegin{eqnarray}\n{\scal L}_{\snf{\sve{K}}} \snf{F} &=& \slb \snf{\sve{K}} , \snf{F} \srb_L = \snf{\sve{K}} \slp \sf{d} \snf{F} \srp + \slp -1 \srp^k \sf{d} \slp \snf{\sve{K}} \snf{F} \srp \s\s\n&=& \slp \snf{\sve{K}} \sf{\spa} \srp \snf{F} + \slp -1 \srp^k \slp \sf{\spa} \snf{\sve{K}} \srp \snf{F}\n\send{eqnarray}\nwhich also defines the FuN bracket of these objects. The above expression for the FuN bracket, and Cartan's formula, can also be written using the natural [[exterior derivative]] in a [[commutator]] bracket,\n$$\n\slb \snf{\sve{K}} , \snf{F} \srb_L = \slb \snf{\sve{K}} , \sf{d} \srb \snf{F}\n$$\nthereby demonstrating the naturalness of the FuN derivative acting on forms. These definitions generalize furthest to give the glorious FuN bracket (and FuN derivative) between vector valued $k$ and $l$ forms \n\sbegin{eqnarray}\n\slb \snf{\sve{K}} , \snf{\sve{L}} \srb_L &=& {\scal L}_{\snf{\sve{K}}} \snf{\sve{L}} \s\s\n&=& \snf{\sve{K}} \slp \sf{\spa} \snf{\sve{L}} \srp - \slp -1 \srp^{kl} \snf{\sve{L}} \slp \sf{\spa} \snf{\sve{K}} \srp + \slp -1 \srp^k \sf{\spa} \slp \snf{\sve{K}} \snf{\sve{L}} \srp - \slp -1 \srp^{kl+l} \sf{\spa} \slp \snf{\sve{L}} \snf{\sve{K}} \srp \s\s\n&=& \slp \snf{\sve{K}} \sf{\spa} \srp \snf{\sve{L}} - \slp -1 \srp^{kl} \slp \snf{\sve{L}} \sf{\spa} \srp \snf{\sve{K}} + \slp -1 \srp^k \slp \sf{\spa} \snf{\sve{K}} \srp \snf{\sve{L}} - \slp -1 \srp^{kl+l} \slp \sf{\spa}\snf{\sve{L}} \srp \snf{\sve{K}} \s\s\n&=& \slb \snf{\sve{K}} , \sf{\spa} \srb \snf{\sve{L}} - \slp -1 \srp^{kl} \slb \snf{\sve{L}} , \sf{\spa} \srb \snf{\sve{K}}\n\send{eqnarray}\nwhich gives all the FuN brackets and Lie derivatives as special cases. This FuN bracket of vector valued $k$ and $l$ forms is a vector valued $(k+l)$-form, and is defined to satisfy\n$$\n{\scal L}_{\slb \snf{\sve{K}} , \snf{\sve{L}} \srb_L} = \slb {\scal L}_{\snf{\sve{K}}} , {\scal L}_{\snf{\sve{L}}} \srb\n$$\nwhen acting on vectors or forms.\n\nThe FuN derivative has a number of other nice [[properties|FuN identities]].\n\n//(Most everything here was learned from talking with [[Michael Edwards]] and reading [[Peter Michor]] et al. (Though the above explicit expression is mine, so if it's wrong, blame [[me|Garrett Lisi]]))//
The [[FuN derivative]] with respect to a [[vector valued k-form|vector valued form]], ${\scal L}_{\snf{\sve{K}}}$, is a grade $k$ [[derivation]] that combines with itself and other operators in a number of ways. Like the [[Lie bracket|Lie derivative identities]], it is linear in both arguments.\n\nThe FuN Lie bracket may or may not change sign under the exchange of its vector valued $k$-form and vector valued $l$-form arguments,\n$$\n\slb \snf{\sve{K}}, \snf{\sve{L}} \srb_L = - \slp -1 \srp^{kl} \slb \snf{\sve{L}}, \snf{\sve{K}} \srb_L\n$$\nAs a derivation, the FuN derivative operates on products of forms via the graded Liebniz rule,\n$$\n{\scal L}_{\snf{\sve{K}}} \slp \snf{F} \snf{G} \srp \n= \slp {\scal L}_{\snf{\sve{K}}} \snf{F} \srp \snf{G} + \slp -1 \srp^{kf} \snf{F} \slp {\scal L}_{\snf{\sve{K}}} \snf{G} \srp\n$$\nBut it is not a derivation over products of VVFs and forms. Using some [[vector valued form identities]] we get\n\sbegin{eqnarray}\n{\scal L}_{\snf{\sve{K}}} \slp \snf{\sve{L}} \snf{F} \srp &=& \slp \snf{\sve{K}} \sf{\spa} \srp \slp \snf{\sve{L}} \snf{F} \srp + \slp -1 \srp^k \slp \sf{\spa} \snf{\sve{K}} \srp \slp \snf{\sve{L}} \snf{F} \srp \s\s\n&=& \slp \slp \snf{\sve{K}} \sf{\spa} \srp \snf{\sve{L}} \srp \snf{F} \n+ \slp-1\srp^{k\slp l-1\srp} \slb \snf{\sve{L}} \slp \slp \snf{\sve{K}} \sf{\spa} \srp \snf{F} \srp\n- \slp \snf{\sve{L}} \slp \snf{\sve{K}} \sf{\spa} \srp \srp \snf{F} \srb \s\s\n& &\n+ \slp-1\srp^k \slp \slp \sf{\spa} \snf{\sve{K}} \srp \snf{\sve{L}} \srp \snf{F}\n+ \slp-1\srp^{kl} \snf{\sve{L}} \slp \slp \sf{\spa} \snf{\sve{K}} \srp \snf{F} \srp\n- \slp-1\srp^{kl} \slp \snf{\sve{L}} \slp \sf{\spa} \snf{\sve{K}} \srp \srp \snf{F} \s\s\n&=& \n\slp {\scal L}_{\snf{\sve{K}}} \snf{\sve{L}} \srp \snf{F} \n+ \slp-1\srp^{k\slp l-1\srp} \snf{\sve{L}} \slp {\scal L}_{\snf{\sve{K}}} \snf{F} \srp \n - \slp-1\srp^{k\slp l-1\srp} {\scal L}_{\snf{\sve{L}}\snf{\sve{K}}} \snf{F}\n\send{eqnarray}\nand\n\sbegin{eqnarray}\n{\scal L}_{\snf{\sve{L}}\snf{\sve{K}}} \snf{F} &=& \slp \slp \snf{\sve{L}} \snf{\sve{K}} \srp \sf{\spa} \srp \snf{F} - \slp-1\srp^{\slp l+k\srp} \slp \sf{\spa} \slp \snf{\sve{L}} \snf{\sve{K}} \srp \srp \snf{F} \s\s\n&=& \slp \snf{\sve{L}} \slp \snf{\sve{K}} \sf{\spa} \srp \srp \snf{F} - \slp-1\srp^k \slp\n\slp-1\srp^l \slp \sf{\spa} \snf{\sve{L}} \srp \snf{\sve{K}} - \snf{\sve{L}} \slp \sf{\spa} \snf{\sve{K}} \srp + \slp \snf{\sve{L}} \sf{\spa} \srp \snf{\sve{K}} \n\srp \snf{F} \s\s\n&=& \n\snf{\sve{L}} \slp {\scal L}_{\snf{\sve{K}}} \snf{F} \srp\n+ \slp-1\srp^{k\slp l-1\srp} \slp {\scal L}_{\snf{\sve{K}}} \snf{\sve{L}} \srp \snf{F}\n- \slp-1\srp^{k\slp l-1\srp} {\scal L}_{\snf{\sve{K}}} \slp \snf{\sve{L}} \snf{F} \srp \n\send{eqnarray}\nwhich are linked by the last lines of each -- they are the same equation (an equation that may be used to define the FuN bracket of two VVF's in terms of the FuN derivatives of forms). A similar identity exists for three VVF's:\n$$\n{\scal L}_{\snf{\sve{L}}\snf{\sve{K}}} \snf{\sve{M}}\n= \snf{\sve{L}} \slp {\scal L}_{\snf{\sve{K}}} \snf{\sve{M}} \srp\n+ \slp-1\srp^{k\slp l-1\srp} \slp {\scal L}_{\snf{\sve{K}}} \snf{\sve{L}} \srp \snf{\sve{M}}\n- \slp-1\srp^{k\slp l-1\srp} {\scal L}_{\snf{\sve{K}}} \slp \snf{\sve{L}} \snf{\sve{M}} \srp\n- \slp-1\srp^{m\slp k+ l-1\srp} \slp {\scal L}_{\snf{\sve{M}}} \snf{\sve{L}} \srp \snf{\sve{K}}\n$$\n\nWhen the two VVF's are written as $\snf{\sve{K}}=\snf{K^A} \sve{X_A}$ and $\snf{\sve{L}}=\snf{L^A} \sve{Y_A}$ their FuN bracket is\n\sbegin{eqnarray}\n\slb \snf{\sve{K}}, \snf{\sve{L}} \srb_L &=& \snf{K^A} \snf{L^B} \slb \sve{X_A}, \sve{Y_B} \srb_L + \snf{K^A} \slp {\scal L}_{\sve{X_A}} \snf{L^B} \srp \sve{Y_B}\n- \slp {\scal L}_{\sve{Y_B}} \snf{K^A} \srp \snf{L^B} \sve{X_A} \s\s\n&+& \slp -1 \srp^k \slp \sf{d} \snf{K^A} \srp \sve{X_A} \snf{L^B} \sve{Y_B}\n+ \slp -1 \srp^k \slp \sve{Y_B} \snf{K^A} \srp \slp \sf{d} \snf{L^B} \srp \sve{X_A}\n\send{eqnarray}\nand, for the FuN bracket of a vector valued 1-form with itself,\n\sbegin{eqnarray}\n\slb \sf{\sve{K}}, \sf{\sve{K}} \srb_L &=& \sf{K^A} \sf{K^B} \slb \sve{X_A}, \sve{X_B} \srb_L + 2 \sf{K^A} \slp {\scal L}_{\sve{X_A}} \sf{K^B} \srp \sve{X_B}\n- 2 \slp \sf{d} \sf{K^A} \srp \sve{X_A} \sf{K^B} \sve{X_B}\n\send{eqnarray}\n\nWhen acting on forms, the FuN derivative commutes with the [[exterior derivative]],\n$$\n0 = \slb {\scal L}_{\snf{\sve{K}}}, \sf{d} \srb = {\scal L}_{\snf{\sve{K}}} \sf{d} + \slp -1 \srp^k \sf{d} {\scal L}_{\snf{\sve{K}}} \n$$\nIn fact, the FuN derivative of a form with respect to the [[identity projection|vector projection]] is the exterior derivative,\n$$\n{\scal L}_{\snf{\sve{I}}} \snf{F} = \sf{d} \snf{F}\n$$\nand of a VVF is zero, ${\scal L}_{\snf{\sve{I}}} \snf{\sve{K}} = 0$.\n\nActing on itself twice, the FuN bracket satisfies the ''graded Jacobi identity'',\n$$\n\slb \snf{\sve{K}}, \slb \snf{\sve{L}} , \snf{\sve{M}} \srb_L \srb_L = \slb \slb \snf{\sve{K}}, \snf{\sve{L}} \srb_L, \snf{\sve{M}} \srb_L - \slp -1 \srp^{kl} \slb \snf{\sve{L}}, \slb \snf{\sve{K}} , \snf{\sve{M}} \srb_L \srb_L\n$$
The rank $2$ exceptional group, ''G2'', is a real, [[simple]], compact, connected [[Lie group|Lie group]]. It may be described by [[exponentiating|exponentiation]] its $14$ dimensional [[Lie algebra]], [[g2]].
<html>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>\n<th></th>\n<th><SPAN class="math">x</SPAN></th>\n<th><SPAN class="math">y</SPAN></th>\n<th><SPAN class="math">z</SPAN></th>\n<th></th>\n<th><SPAN class="math">g^3</SPAN></th>\n<th><SPAN class="math">g^8</SPAN></th>\n<th><SPAN class="math">\sfr{\ssqrt{8}}{\ssqrt{3}} B_2</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{g}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}g}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{g}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{g\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n\n<tr>\n<td><SPAN class="math">\sbtri{#D90000} </SPAN></td>\n<td><SPAN class="math">q^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#00BF00} </SPAN></td>\n<td><SPAN class="math">q^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#0000F7} </SPAN></td>\n<td><SPAN class="math">q^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#D90000} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#00BF00} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\sbutr{#0000F7} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">{\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\smutr{#0000F7}}{\smtri{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{II}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\smp 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\spm \sfr{2}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sstri{#0000F7}}{\ssutr{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{III}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\spm 1 \s;\s; \spm \s! 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\smp \sfr{4}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sbutr{#999999}}{\sbtri{#999999}}</SPAN></td>\n<td><SPAN class="math">l</SPAN></td>\n<td></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm 1</SPAN></td>\n</tr>\n</table>\n</td>\n\n<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>\n\n<td>\n<embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed>\n<!-- <embed src="talks/Perimeter07/anim/g2spin/p1.png" width="462" height="462"></embed> -->\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>
<html>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>\n<th></th>\n<th><SPAN class="math">x</SPAN></th>\n<th><SPAN class="math">y</SPAN></th>\n<th><SPAN class="math">z</SPAN></th>\n<th></th>\n<th><SPAN class="math">g^3</SPAN></th>\n<th><SPAN class="math">g^8</SPAN></th>\n<th><SPAN class="math">\sfr{\ssqrt{8}}{\ssqrt{3}} B_2</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{g}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}g}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{g}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{g\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n\n<tr>\n<td><SPAN class="math">\sbtri{#D90000} </SPAN></td>\n<td><SPAN class="math">q^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#00BF00} </SPAN></td>\n<td><SPAN class="math">q^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#0000F7} </SPAN></td>\n<td><SPAN class="math">q^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#D90000} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#00BF00} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\sbutr{#0000F7} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">{\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\smutr{#0000F7}}{\smtri{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{II}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\smp 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\spm \sfr{2}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sstri{#0000F7}}{\ssutr{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{III}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\spm 1 \s;\s; \spm \s! 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\smp \sfr{4}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sbutr{#999999}}{\sbtri{#999999}}</SPAN></td>\n<td><SPAN class="math">l</SPAN></td>\n<td></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm 1</SPAN></td>\n</tr>\n</table>\n</td>\n\n<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>\n\n<td>\n<!-- <embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed> -->\n<embed src="talks/Perimeter07/anim/g2spin/p1.png" width="462" height="462"></embed>\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>
<html>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>\n<th></th>\n<th><SPAN class="math">x</SPAN></th>\n<th><SPAN class="math">y</SPAN></th>\n<th><SPAN class="math">z</SPAN></th>\n<th></th>\n<th><SPAN class="math">g^3</SPAN></th>\n<th><SPAN class="math">g^8</SPAN></th>\n<th><SPAN class="math">\sfr{\ssqrt{8}}{\ssqrt{3}} B_2</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{g}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}g}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{g}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{g\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n\n<tr>\n<td><SPAN class="math">\sbtri{#D90000} </SPAN></td>\n<td><SPAN class="math">q^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#00BF00} </SPAN></td>\n<td><SPAN class="math">q^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#0000F7} </SPAN></td>\n<td><SPAN class="math">q^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#D90000} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#00BF00} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\sbutr{#0000F7} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">{\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\smutr{#0000F7}}{\smtri{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{II}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\smp 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\spm \sfr{2}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sstri{#0000F7}}{\ssutr{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{III}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\spm 1 \s;\s; \spm \s! 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\smp \sfr{4}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sbutr{#999999}}{\sbtri{#999999}}</SPAN></td>\n<td><SPAN class="math">l</SPAN></td>\n<td></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm 1</SPAN></td>\n</tr>\n</table>\n</td>\n\n<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>\n\n<td>\n<!-- <embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed> -->\n<embed src="talks/Perimeter07/anim/g2spin/p20.png" width="462" height="462"></embed>\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>
<html>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>\n<th></th>\n<th><SPAN class="math">x</SPAN></th>\n<th><SPAN class="math">y</SPAN></th>\n<th><SPAN class="math">z</SPAN></th>\n<th></th>\n<th><SPAN class="math">g^3</SPAN></th>\n<th><SPAN class="math">g^8</SPAN></th>\n<th><SPAN class="math">\sfr{\ssqrt{8}}{\ssqrt{3}} B_2</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{g}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}g}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{g}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{g\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n\n<tr>\n<td><SPAN class="math">\sbtri{#D90000} </SPAN></td>\n<td><SPAN class="math">q^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#00BF00} </SPAN></td>\n<td><SPAN class="math">q^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#0000F7} </SPAN></td>\n<td><SPAN class="math">q^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#D90000} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#00BF00} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\sbutr{#0000F7} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">{\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\smutr{#0000F7}}{\smtri{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{II}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\smp 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\spm \sfr{2}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sstri{#0000F7}}{\ssutr{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{III}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\spm 1 \s;\s; \spm \s! 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\smp \sfr{4}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sbutr{#999999}}{\sbtri{#999999}}</SPAN></td>\n<td><SPAN class="math">l</SPAN></td>\n<td></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm 1</SPAN></td>\n</tr>\n</table>\n</td>\n\n<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>\n\n<td>\n<!-- <embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed> -->\n<embed src="talks/Perimeter07/anim/g2spin/p46.png" width="462" height="462"></embed>\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>
<html>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>\n<th></th>\n<th><SPAN class="math">x</SPAN></th>\n<th><SPAN class="math">y</SPAN></th>\n<th><SPAN class="math">z</SPAN></th>\n<th></th>\n<th><SPAN class="math">g^3</SPAN></th>\n<th><SPAN class="math">g^8</SPAN></th>\n<th><SPAN class="math">\sfr{\ssqrt{8}}{\ssqrt{3}} B_2</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{g}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}g}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{g}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{g\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n\n<tr>\n<td><SPAN class="math">\sbtri{#D90000} </SPAN></td>\n<td><SPAN class="math">q^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#00BF00} </SPAN></td>\n<td><SPAN class="math">q^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#0000F7} </SPAN></td>\n<td><SPAN class="math">q^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#D90000} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#00BF00} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\sbutr{#0000F7} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">{\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\smutr{#0000F7}}{\smtri{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{II}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\smp 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\spm \sfr{2}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sstri{#0000F7}}{\ssutr{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{III}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\spm 1 \s;\s; \spm \s! 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\smp \sfr{4}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sbutr{#999999}}{\sbtri{#999999}}</SPAN></td>\n<td><SPAN class="math">l</SPAN></td>\n<td></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm 1</SPAN></td>\n</tr>\n</table>\n</td>\n\n<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>\n\n<td>\n<!-- <embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed> -->\n<embed src="talks/Perimeter07/anim/g2spin/p72.png" width="462" height="462"></embed>\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>
<html>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>\n<th></th>\n<th><SPAN class="math">x</SPAN></th>\n<th><SPAN class="math">y</SPAN></th>\n<th><SPAN class="math">z</SPAN></th>\n<th></th>\n<th><SPAN class="math">g^3</SPAN></th>\n<th><SPAN class="math">g^8</SPAN></th>\n<th><SPAN class="math">\sfr{\ssqrt{8}}{\ssqrt{3}} B_2</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{g}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}g}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{g}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{g\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n\n<tr>\n<td><SPAN class="math">\sbtri{#D90000} </SPAN></td>\n<td><SPAN class="math">q^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#00BF00} </SPAN></td>\n<td><SPAN class="math">q^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#0000F7} </SPAN></td>\n<td><SPAN class="math">q^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#D90000} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#00BF00} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\sbutr{#0000F7} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">{\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\smutr{#0000F7}}{\smtri{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{II}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\smp 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\spm \sfr{2}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sstri{#0000F7}}{\ssutr{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{III}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\spm 1 \s;\s; \spm \s! 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\smp \sfr{4}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sbutr{#999999}}{\sbtri{#999999}}</SPAN></td>\n<td><SPAN class="math">l</SPAN></td>\n<td></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm 1</SPAN></td>\n</tr>\n</table>\n</td>\n\n<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>\n\n<td>\n<!-- <embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed> -->\n<embed src="talks/Perimeter07/anim/g2spin/p92.png" width="462" height="462"></embed>\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>
[>img[Garrett at Burning Man, 2004|images/person/Garrett.jpg]]Homepage: http://interstice.com/~aglisi/\n*Email: garrett.lisi&#064;gmail.com\n*Location: Lake Tahoe or Maui, usually\n*CV: http://interstice.com/~aglisi/Physics/CVp.html\n*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Lisi_A/0/1/0/all/0/1\n\nSelected work:\n*You're looking at it.\n*[[An Exceptionally Simple Theory of Everything]]\n*[[Quantum mechanics from a universal action reservoir|http://arxiv.org/abs/physics/0605068]]\n*[[Clifford bundle formulation of BF gravity generalized to the standard model|papers/0511120.pdf]]\n\nTalks:\n*[[talk for ILQGS 07]]\n*[[talk for Perimeter Institute 07]]\n*[[talk for FQXi 07]]\n*[[talk for Loops 07]]\n\nPopular press coverage:\n*[[FQXi project|http://fqxi.org/aw-lisi.html]]\n*[[FQXi profile|http://www.fqxi.org/community/data/articles/Lisi_Garrett.pdf]]
The eight [[trace]]less, Hermitian, ''Gell-Mann matrices'', $\sla_A$, are\n$$\n\sbegin{array}{cccc}\n\sla_0 = \sla_8 = \sfr{1}{\ssqrt{3}} \sleft[\sbegin{array}{ccc}\n1 & 0 & 0\s\s\n0 & 1 & 0\s\s\n0 & 0 & -2\n\send{array}\sright]\n&\n\sla_1 = \sleft[\sbegin{array}{ccc}\n0 & 1 & 0\s\s\n1 & 0 & 0\s\s\n0 & 0 & 0\n\send{array}\sright]\n&\n\sla_2 = \sleft[\sbegin{array}{ccc}\n0 & -i & 0\s\s\ni & 0 & 0\s\s\n0 & 0 & 0\n\send{array}\sright]\n&\n\sla_3 = \sleft[\sbegin{array}{ccc}\n1 & 0 & 0\s\s\n0 & -1 & 0\s\s\n0 & 0 & 0\n\send{array}\sright]\n\s\s\n\sla_4 = \sleft[\sbegin{array}{ccc}\n0 & 0 & 1\s\s\n0 & 0 & 0\s\s\n1 & 0 & 0\n\send{array}\sright]\n&\n\sla_5 = \sleft[\sbegin{array}{ccc}\n0 & 0 & -i\s\s\n0 & 0 & 0\s\s\ni & 0 & 0\n\send{array}\sright]\n&\n\sla_6 = \sleft[\sbegin{array}{ccc}\n0 & 0 & 0\s\s\n0 & 0 & 1\s\s\n0 & 1 & 0\n\send{array}\sright]\n&\n\sla_7 = \sleft[\sbegin{array}{ccc}\n0 & 0 & 0\s\s\n0 & 0 & -i\s\s\n0 & i & 0\n\send{array}\sright]\n\send{array}\n$$\n
<<note HideTags>>Start with a [[Lie group manifold|Lie group geometry]] (//torsor//), $G$, coordinatized by $y^p$.\nTwo sets of invariant vector fields (//symmetries, [[Killing vector]] fields//):\n[>img[images/png/torsor.png]]$$\n\sve{\sxi^L_A}(y) \s, \sf{d} g = T_A \s, g(y) \s;\s;\s;\s;\s;\s;\s;\s; \sve{\sxi^R_A}(y) \s, \sf{d} g = g(y) \s, T_A\n$$\n[[Lie derivative]]: &nbsp;~~&nbsp;&nbsp;~~ $[ \sve{\sxi^R_A}, \sve{\sxi^R_B} ] = C_{AB}^{\sp{AB}C} \sve{\sxi^R_C}$\n[[Lie bracket|Lie algebra]]: &nbsp;&nbsp;~~&nbsp;~~&nbsp;&nbsp;&nbsp; $\slb T_A, T_B \srb = C_{AB}^{\sp{AB}C} T_C$\n[[Killing form]] (//[[Minkowski metric]]//): ^^&nbsp;^^ $g_{AB} = C_{AC}^{\sp{AC}D} C_{BD}^{\sp{BD}C}$\n[[Maurer-Cartan form]] (//[[frame]]//): ^^&nbsp;^^&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\sf{\scal I} = \sf{dy^p} ( \sxi^R_p )^A T_A$\nEntire space of a [[principal bundle]]: &nbsp;&nbsp; $E \ssim M \stimes G^{\sp{\sbig(}}$\n[[Ehresmann principal bundle connection]] over patches of $E$:\n[>img[images/png/fiber bundle.png]]$$\n\sve{\sf{\scal E}}(x,y) = \sf{dx^i} A_i^{\sp{a}B}(x) \s, \sve{\sxi^L_B}(y) + \sf{dy^p} \sve{\spa_p}\n$$\nGauge field [[connection]] over $M$:\n$$\n\sf{A}(x) = \ssi_0^* \sve{\sf{\scal E}} \sf{\scal I} = \sf{dx^i} A_i^{\sp{a}B}(x) \s, T_B\n$$
<<note HideTags>>$$\ng = g^A T_A = g^A \sfr{i}{2} \sla_A\n= \sfr{i}{2}\n\slb\n\sbegin{array}{ccc}\n\s! g^3 \s!+\s! {\sscriptsize \sfrac{1}{\ssqrt{3}}} g^8 \s! & g^1 \s!-\s! ig^2 \s!\s! & g^4 \s!-\s! ig^5 \s\s\ng^1 \s!+\s! i g^2 & \s!\s!\s! -\s! g^3 \s!+\s! {\sscriptsize \sfrac{1}{\ssqrt{3}}} g^8 \s!\s! & g^6 \s!-\s! ig^7 \s\s\ng^4 \s!+\s! i g^5 & \s!\s! g^6 \s!+\s! i g^7 & {\sscriptsize -\s!\sfrac{2}{\ssqrt{3}}} g^8 \s!\n\send{array}\n\srb\n$$\nCartan subalgebra: $\sqquad C = g^3 T_3 + g^8 T_8 \squad$ (the diagonal)\nRoots and root vectors:\n$$\n\sbig[ C , V_{g^{g\sbar{b}}} \sbig] = i \slp \sBig( -\s!\sfr{1}{2} \sBig) g^3 + \sBig( \sfr{\ssqrt{3}}{2} \sBig) g^8 \srp V_{g^{g\sbar{b}}}\n\sqquad\nV_{g^{g\sbar{b}}} =\n\slb \smatrix{\n0 & 0 & 0 \scr\n0 & 0 & 1 \scr\n0 & 0 & 0 \scr\n} \srb\n$$\nfor the $g^{g\sbar{b}}$ gluon. Weights and weight vectors:\n$$\nC \s, V_{q^r} = i \slp \sBig( \sfr{1}{2} \sBig) g^3 + \sBig( \sfr{1}{2\ssqrt{3}} \sBig) g^8 \srp V_{q^r}\n\sqquad\nV_{q^r} = [ 1,0,0 ]\n$$\nfor a red quark, $q^r$, and for their duals acted on by $-C^T$, the anti-quarks.
Real ''Grassmann numbers'', $\sud{a},\sud{b} \sin \smathbb{G}$, are like real numbers but they anti-commute with each other, $\sud{a} \sud{b} = - \sud{b} \sud{a}$, and commute with reals. The square of a Grassmann number is necessarily zero, $\sud{a} \sud{a} = 0$. The product of two real Grassman numbers is a real number (acording to [[Ramond|http://www.amazon.com/Field-Theory-Modern-Frontiers-Physics/dp/0201304503/ref=pd_bbs_sr_1/104-9709999-3726336?ie=UTF8&s=books&qid=1177293245&sr=8-1]]),\n$$\n\slp \sud{a} \sud{b} \srp^* = \sud{a}^* \sud{b}^* = \sud{a} \sud{b} \sin \smathbb{R}\n$$\nSince Grassmann numbers square to zero, the Taylor expansion of any function of Grassmann variables terminates at the first order,\n$$\nf(\sud{c}) = a + b \s, \sud{c} \n$$\nDerivatives work as for real numbers (but make sure to change the sign when commuting them past other Grassmann numbers). Using the example above,\n$$\n\sfr{\spa}{\spa \sud{c}} f(\sud{c}) = b\n$$\nIntegrals are effectively the same as derivatives,\n$$\n\sint{\sud{dc} \s, f(\sud{c})} = b\n$$\nUsing this rule, for two sets of Grassmann variables, $\sud{a^i},\sud{b^j}$, and a real matrix, $A$, integration gives the [[determinant]],\n$$\n\sint{\sud{da} \s, \sud{db} \s, \sexp(\sud{a^i} A_{ij} \sud{b^j}}) = \sdet A\n$$\n\nFor a compex Grassmann number, $\sud{z} = \sud{x} + i \sud{y}$, its square and norm are:\n\sbegin{eqnarray}\n\sud{z} \sud{z} &=& i \sud{x} \sud{y} + i \sud{y} \sud{x} = 0 \s\s\n\sud{z}^* \sud{z} &=& i \sud{x} \sud{y} - i \sud{y} \sud{x} = 2 i \sud{x} \sud{y} = - \sud{z} \sud{z}^* \sin \smathbb{I} \n\send{eqnarray}\n\nAn ''anti-Grassmann number'', $\sod{a}$, contracts with a Grassmann number to give a real, $\sod{a} \sud{b} \sin \smathbb{R}$, just like in [[vector-form algebra]]. In fact, a Grassmann number may be thought of as a [[1-form]] in the space of functions. With this interpretation, the product of two Grassmann numbers is not a real, but a ''Grassmann grade two number''.
<html>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td><div class="math">\n\sbegin{array}{l}\n\som = \sha {\som}{}^{\smu \snu} \sga_{\smu\snu} = \n\slb \sbegin{array}{cc}\n{\som}{}_L & \s\s\n & {\som}{}_R\n\send{array} \srb \svp{|_{\sBig(}}\n\s\s\n\sqquad \sqquad \sqquad \sqquad\n{\som}{}_{L/R} =\n\slb \sbegin{array}{cc}\ni {\som}{}_{L/R}^3 & {\som}{}_{L/R}^\swedge \s\s\n{\som}{}_{L/R}^\svee & -i {\som}{}_{L/R}^3\n\send{array} \srb \svp{|_{\sBig(_{\sbig(}}}\n\s\s\n{e} = {e}{}^\smu \sga_\smu\n=\n\slb \sbegin{array}{cc}\n & {e}{}_R \s\s\n{e}{}_L & \n\send{array} \srb \svp{|_{\sBig(}}\n\s\s\n\sqquad \sqquad \sqquad \sqquad\n{e}{}_{L/R} =\n\slb \sbegin{array}{cc}\n{e}{}_T^{\swedge/\svee} & \smp {e}{}_S^\swedge \s\s\n\smp {e}{}_S^\svee & {e}{}_T^{\svee/\swedge}\n\send{array} \srb \svp{|_{\sBig(_{\sbig(}}}\n\s\s\n{f} =\n\slb \sbegin{array}{c}\n{f}{}_L \s\s {f}{}_R\n\send{array} \srb\n\sqquad \squad \s;\n{f}{}_{L/R} =\n\slb \sbegin{array}{c}\n{f}{}_{L/R}^\swedge \s\s {f}{}_{L/R}^\svee\n\send{array} \srb\n\send{array}\n</div></td>\n\n<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th ALIGN=CENTER COLSPAN="2"><SPAN class="math">SO(3,1)</SPAN></th>\n<th></th>\n<th ALIGN=CENTER><SPAN class="math">\sha \som_L^3</SPAN></th>\n<th ALIGN=CENTER><SPAN class="math">\sha \som_R^3</SPAN></th>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#59FF59} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\som_L^\swedge</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#59FF59} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\som_L^\svee</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#59FF59} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\som_R^\swedge</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n</tr>\n<tr class="butt">\n<td ALIGN=CENTER><SPAN class="math">\smcir{#59FF59} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\som_R^\svee</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smsqu{#FF5959}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">e_S^\swedge</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smsqu{#FF5959}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">e_S^\svee</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sha</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smsqu{#FF5959}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">e_T^\swedge</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n</tr>\n<tr class="butt">\n<td ALIGN=CENTER><SPAN class="math">\smsqu{#FF5959}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">e_T^\svee</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sha</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#F2F200}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">f_L^\swedge</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#F2F200}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">f_L^\svee</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">f_R^\swedge</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">f_R^\svee</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sha</SPAN></td>\n</tr>\n</table>\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>
<<note HideTags>>\sbegin{eqnarray}\nS_s &=& \sint \sbig< \sff{B_s} \sff{F_s} + \sPh_s(\sff{B_s}) \sbig> \s,\n= \sint \sbig< \sff{B_s} {\sscriptsize \sfrac{1}{2}} \sbig( \sff{R} + {\sscriptsize \sfrac{1}{8}}M^2 \sf{e}\sf{e} \sbig) - {\sscriptsize \sfrac{1}{4}} \sff{B_s} \sff{B_s} \sga \sbig> \s\s\n&& \sde \sff{B_s} \srightarrow \sff{B_s} = \sbig( \sff{R} + {\sscriptsize \sfrac{1}{8}} M^2 \sf{e}\sf{e} \sbig) \sga^- \n\s;\s;\s;\s;\s;\s; \stext{pseudoscalar:} \s;\s; \sga = {\sga_0 \sga_1 \sga_2 \sga_3}{\sphantom{\sBigg(}} \s\s\n\nS_s &=& {\sscriptsize \sfrac{1}{4}} \sint \sbig< \sbig( \sff{R} + {\sscriptsize \sfrac{1}{8}} M^2 \sf{e}\sf{e} \sbig) \sbig( \sff{R} + {\sscriptsize \sfrac{1}{8}} M^2 \sf{e}\sf{e} \sbig) \sga^- \sbig>\n= \sint \sbig< \sff{F_s} \sff{F_s} \sga^- \sbig> \s\s\n&& \sbig< \sff{R} \sff{R} \sga^- \sbig> = \sf{d} \sbig< \sbig( \sf{\som} \sf{d} \sf{\som} + {\sscriptsize \sfrac{1}{3}} \sf{\som} \sf{\som} \sf{\som} \sbig) \sga^- \sbig> \n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s,\n \sleftarrow \stext{Chern-Simons} \s\s\n&& {\sscriptsize \sfrac{1}{4!}} \sbig< \sf{e}\sf{e} \sf{e} \sf{e} \sga^- \sbig> = \snf{e} \n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\n \sleftarrow \stext{volume element} \s\s\n&& \sbig< \sf{e}\sf{e} \sff{R} \s, \sga^- \sbig> = \snf{e} R_{\sp{\sBig(}}\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\n \sleftarrow \stext{curvature scalar} \s\s\n\nS_s &=& {\sscriptsize \sfrac{\sLa}{12}} \sint \snf{e} \slp R + 2 \sLa \srp\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\n\stext{cosmological constant:}_{\sp{\sbig(}} \s;\s; \sLa = {\sscriptsize \sfrac{3}{4}} M^2\n\send{eqnarray}
<<note HideTags>>\sbegin{eqnarray}\nS_G &=& \n\sint \sbig< \sff{B}{}_G \sff{F}{}_G + {\sscriptsize \sfrac{\spi G}{4}} \sff{B}{}_G \sff{B}{}_G \sga \sbig>\n\sqquad\n\sff{F}{}_G = {\sscriptsize \sfrac{1}{2}} \sbig( \sff{R} - {\sscriptsize \sfrac{1}{8}} \sf{e} \sf{e} \sph^2 \sbig)\n\sin \sff{so}(3,1)\n\s\s\n&& \sde \sff{B}{}_G \srightarrow \sff{B}{}_G = \sfr{1}{\spi G} \sbig( \sff{R} - {\sscriptsize \sfrac{1}{8}} \sf{e}\sf{e} \sph^2 \sbig) \sga\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\n\sga = {\sga_1 \sga_2 \sga_3 \sga_4}{\sphantom{\sBigg(}} \s\s\n\nS_G &=& \n{\sscriptsize \sfrac{1}{\spi G}} \sint \sbig< \sff{F}{}_G \sff{F}{}_G \sga \sbig>\n=\n{\sscriptsize \sfrac{1}{4 \spi G}} \sint \sbig< \sbig( \sff{R} - {\sscriptsize \sfrac{1}{8}} \sf{e}\sf{e} \sph^2 \sbig) \sbig( \sff{R} - {\sscriptsize \sfrac{1}{8}} \sf{e}\sf{e} \sph^2 \sbig) \sga \sbig> \s\s[.6em]\n&& \sbig< \sff{R} \sff{R} \sga \sbig> = \sf{d} \sbig< \sbig( \sf{\som} \sf{d} \sf{\som} + {\sscriptsize \sfrac{1}{3}} \sf{\som} \sf{\som} \sf{\som} \sbig) \sga \sbig> \n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s,\n \sleftarrow \stext{Chern-Simons} \s\s\n&& {\sscriptsize \sfrac{1}{4!}} \sbig< \sf{e}\sf{e} \sf{e} \sf{e} \sga \sbig> = - \snf{e} \n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s,\n \sleftarrow \stext{volume element} \s\s\n&& \sbig< \sf{e}\sf{e} \sff{R} \s, \sga \sbig> = - \snf{e} R_{\sp{\sBig(}}\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s,\n \sleftarrow \stext{curvature scalar} \s\s\n\nS_G &=& {\sscriptsize \sfrac{1}{16\spi G}} \sint \snf{e} \s, \sph^2 \slp R - {\sscriptsize \sfrac{3}{2}} \sph^2 \srp\n\s;\s;\s;\s;\s;\s;\n\stext{cosmological constant:}_{\sp{\sbig(}} \s;\s; \sLa = {\sscriptsize \sfrac{3}{4}} \sph^2\n\send{eqnarray}
<<note HideTags>>Using [[chiral]] (//Weyl//) $\smathbb{C}(4 \stimes 4)$ representation of [[Cl(1,3)]] [[Dirac matrices]]:\n$$\n\sbegin{array}{rclrcl}\n\sga_0 \s!\s!&\s!\s!=\s!\s!&\s!\s! \ssi_1 \sotimes 1\n=\n\slb \sbegin{array}{cc}\n & 1 \s\s\n1 &\n\send{array} \srb_{\sp{\sbig(}} \n& \s;\s;\s;\s;\n\sga_\spi \s!\s!&\s!\s!=\s!\s!&\s!\s! i \ssi_2 \sotimes \ssi_\spi\n=\n\slb \sbegin{array}{cc}\n & \ssi_\spi \s\s\n-\ssi_\spi &\n\send{array} \srb\n\s\s\n\n\sga_{0\sva} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sga_0 \sga_\sva\n=\n\slb \sbegin{array}{cc}\n-\ssi_\sva & \s\s\n & \ssi_\sva\n\send{array} \srb_{\sp{(}}\n& \s;\s;\s;\s;\n\sga_{\sva\spi} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sga_\sva \sga_\spi\n=\n\slb \sbegin{array}{cc}\n- i \sep_{\sva\spi\sta} \ssi_\sta & \s\s\n& - i \sep_{\sva\spi\sta} \ssi_\sta\n\send{array} \srb\n\n\send{array}\n$$\n[[Spacetime frame|spacetime frame]] and [[spin connection|spacetime spin connection]]:\n\sbegin{eqnarray}\n\sf{\som} + \sf{e} &=&\n\sf{dx^a} {\sscriptsize \sfrac{1}{2}} \som_a^{{\sp{a}}\smu\snu} \sga_{\smu\snu} + \sf{dx^a} ( e_a )^\smu \sga_\smu {}_{\sp{(}} \s\s\n&=&\n\slb \sbegin{array}{cc}\n( - \sf{\som^{0 \sva}} \ssi_\sva - {\ssmall \sfrac{i}{2}} \sf{\som^{\sva \spi}} \sep_{\sva \spi \sta} \ssi_\sta ) & ( \sf{e^0} + \sf{e^\spi} \ssi_\spi ) \s\s\n( \sf{e^0} - \sf{e^\spi} \ssi_\spi ) & ( \sf{\som^{0 \sva}} \ssi_\sva - {\ssmall \sfrac{i}{2}} \sf{\som^{\sva \spi}} \sep_{\sva \spi \sta} \ssi_\sta )\n\send{array} \srb_{\sp{(}}\n\s\s\n&=&\n\slb \sbegin{array}{cc}\n\sf{\som_L} & \sf{e_R} \s\s\n\sf{e_L} & \sf{\som_R}\n\send{array} \srb_{\sp{(}}\n\s;\s; \sin \s;\s; \sf{Cl}^{1+2}(1,3)\n\send{eqnarray}Note algebraic equivalence: &nbsp;&nbsp; $Cl^{1+2}(1,3) = Cl^2(1,4) = so(1,4)_{\sp{(}}$
<html><center><table class="gtable">\n<tr border=none>\n\n<td align="left">A <b>triality</b> rotation, <span class="math">T</span>, of <span class="math">D4</span>:\n<div class="math">\n{\ssmall\n\slb \sbegin{array}{c}\n\sha {\som'}^3_L \s\s \sha {\som'}^3_R \s\s {W'}^3 \s\s {B'}_1^3\n\send{array} \srb\n=\n\slb \sbegin{array}{cccc}\n0 & 1 & 0 & 0 \s\s\n0 & 0 & 0 & 1 \s\s\n0 & 0 & 1 & 0 \s\s\n1 & 0& 0 & 0\n\send{array} \srb\n\slb \sbegin{array}{c}\n\sha \som^3_L \s\s \sha \som^3_R \s\s W^3 \s\s B_1^3\n\send{array} \srb\n=\n\slb \sbegin{array}{c}\n\sfr{1}{2} \som^3_R \s\s B_1^3 \s\s W^3 \s\s \sha \som^3_L \n\send{array} \srb }\n</div>\n<div class="math">\nT \s, T \s, T \s, \som_R^\swedge = T \s, T \s, \som_L^\swedge = T \s, B_1^+ = \som_R^\swedge\n</div>\nRoots invariant under this <span class="math">T</span>:\n<div class="math">\n\s{\nW^+, \s,\nW^- , \s,\ne_S^\swedge\sph_+, \s,\ne_S^\swedge \sph_0, \s,\ne_S^\svee \sph_-, \s,\ne_S^\svee \sph_1\n\s}\n</div>\nRotations to triality-equivalent vector and negative chiral spinor representation spaces:\n<div class="math">\nT \s, 8_{S+} = 8_V \squad \s; T \s, 8_V = 8_{S-} \squad \s; T \s, 8_{S-} = 8_{S+}\n</div>\nThree generations, related by triality:\n<div class="math">\nT \s, e_L^\swedge = \smu_L^\swedge\n\squad \s;\nT \s, \smu_L^\swedge = \sta_L^\swedge\n\squad \s;\nT \s, \sta_L^\swedge = e_L^\swedge\n</div>\n</td>\n\n<td>&nbsp;&nbsp;&nbsp;</td>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">8_V</SPAN></th>\n<th></th>\n<th><SPAN class="math">\sha \som_L^3</SPAN></th>\n<th><SPAN class="math">\sha \som_R^3</SPAN></th>\n<th><SPAN class="math">W^3</SPAN></th>\n<th><SPAN class="math">B_1^3</SPAN></th>\n</tr>\n<tr>\n<th COLSPAN="2">&nbsp;&nbsp;&nbsp;&nbsp;tri&nbsp;&nbsp;&nbsp;</th>\n<th></th>\n<th><SPAN class="math">\sha \som_R^3</SPAN></th>\n<th><SPAN class="math">B_1^3</SPAN></th>\n<th><SPAN class="math">W^3</SPAN></th>\n<th><SPAN class="math">\sha \som_L^3</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\smtri{#B2B200}</SPAN></td>\n<td><SPAN class="math">\snu_{\smu L}^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smtri{#F2F200}</SPAN></td>\n<td><SPAN class="math">\smu_L^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smtri{#999999}</SPAN></td>\n<td><SPAN class="math">\snu_{\smu R}^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smtri{#D9D9D9}</SPAN></td>\n<td><SPAN class="math">\smu_R^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n</table>\n<br>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">8_{S-}</SPAN></th>\n<th></th>\n<th><SPAN class="math">\sha \som_L^3</SPAN></th>\n<th><SPAN class="math">\sha \som_R^3</SPAN></th>\n<th><SPAN class="math">W^3</SPAN></th>\n<th><SPAN class="math">B_1^3</SPAN></th>\n</tr>\n<tr>\n<th COLSPAN="2">&nbsp;&nbsp;&nbsp;&nbsp;tri&nbsp;&nbsp;&nbsp;</th>\n<th></th>\n<th><SPAN class="math">B_1^3</SPAN></th>\n<th><SPAN class="math">\sha \som_L^3</SPAN></th>\n<th><SPAN class="math">W^3</SPAN></th>\n<th><SPAN class="math">\sha \som_R^3</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\sstri{#B2B200}</SPAN></td>\n<td><SPAN class="math">\snu_{\sta L}^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sstri{#F2F200}</SPAN></td>\n<td><SPAN class="math">\sta_L^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sstri{#999999}</SPAN></td>\n<td><SPAN class="math">\snu_{\sta R}^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sstri{#D9D9D9}</SPAN></td>\n<td><SPAN class="math">\sta_R^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n</tr>\n</table>\n\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>
<html>\n<center>\n<table class="gtable">\n<tr border=none>\n<td><div class="math">\n\sbegin{array}{l}\nH_1 = (\sha \som + \sfr{1}{4} e \sph + W + B_1) \svp{|_{(}} \s\s[.5em]\n\squad \s;\s; \sin so(3,1) + 4 \stimes 4 + \sbig( su(2)+su(2) \sbig) \svp{|_{(}} \s\s[.5em]\n\squad \s;\s; = Cl^2(7,1) = so(7,1) = d4\s\s[3em]\n8_{S+} \squad \sto \sqquad \squad H_1 \s, (\snu_e + e)\s\s[.5em]\n\sqquad \sqquad \sqquad \sqquad \squad =\s\s[.5em]\n{\ssmall\n\slb \sbegin{array}{cccc}\n\s! \sfr{1}{2} \som_L \s!+\s! \sfr{i}{2} W^3 \s!\s!\s! & W^+ & - \s! \sfr{1}{4} e_R \sph_1 & \sfr{1}{4} e_R \sph_+ \s\s\nW^- & \s!\s!\s! \sfr{1}{2} \som_L \s!-\s! \sfr{i}{2} W^3 \s!\s!\s! & \sp{-} \sfr{1}{4} e_R \sph_- & \sfr{1}{4} e_R \sph_0 \s\s\n-\sfr{1}{4} e_L \sph_0 & \sfr{1}{4} e_L \sph_+ & \s!\s!\s! \sfr{1}{2} \som_R \s!+\s! \sfr{i}{2} B_1^3 \s!\s!\s! & B_1^+ \s\s\n\sp{-}\sfr{1}{4} e_L \sph_- & \sfr{1}{4} e_L \sph_1 & B_1^- & \s!\s!\s! \sfr{1}{2} \som_R \s!-\s! \sfr{i}{2} B_1^3 \s!\n\send{array} \srb\n\slb \sbegin{array}{c}\n\snu_{eL} \s\s e_L \s\s \snu_{eR} \s\s e_R\n\send{array} \srb }\n\send{array}\n</div></td>\n\n<td>&nbsp;&nbsp;</td>\n\n<td>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">D4</SPAN></th>\n<th></th>\n<th><SPAN class="math">\sha \som_L^3</SPAN></th>\n<th><SPAN class="math">\sha \som_R^3</SPAN></th>\n<th><SPAN class="math">W^3</SPAN></th>\n<th><SPAN class="math">B_1^3</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#59FF59} </SPAN></td>\n<td><SPAN class="math">\som_L^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\spm 1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#59FF59} </SPAN></td>\n<td><SPAN class="math">\som_R^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm 1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#FFFF00} </SPAN></td>\n<td><SPAN class="math">\ssmash{W^\spm}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm 1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#FFFFFF} </SPAN></td>\n<td><SPAN class="math">\ssmash{B_1^\spm}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm 1</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smsqu{#B2B200} </SPAN></td>\n<td><SPAN class="math">e_T^{\swedge/\svee} \sph_+</SPAN></td>\n<td></td>\n<td><SPAN class="math">\smp \sha</SPAN></td>\n<td><SPAN class="math">\spm \sha</SPAN></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\sha</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smsqu{#B2B200} </SPAN></td>\n<td><SPAN class="math">e_S^{\swedge/\svee} \sph_+</SPAN></td>\n<td></td>\n<td><SPAN class="math">\spm \sha</SPAN></td>\n<td><SPAN class="math">\spm \sha</SPAN></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\sha</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smdia{#F2F200} </SPAN></td>\n<td><SPAN class="math">e_T^{\swedge/\svee} \sph_-</SPAN></td>\n<td></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smdia{#F2F200} </SPAN></td>\n<td><SPAN class="math">e_S^{\swedge/\svee} \sph_-</SPAN></td>\n<td></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smsqu{#F77C00}</SPAN></td>\n<td><SPAN class="math">e_T^{\swedge/\svee} \sph_0</SPAN></td>\n<td></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smsqu{#F77C00}</SPAN></td>\n<td><SPAN class="math">e_S^{\swedge/\svee} \sph_0</SPAN></td>\n<td></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smdia{#BF6000}</SPAN></td>\n<td><SPAN class="math">e_T^{\swedge/\svee} \sph_1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\smdia{#BF6000}</SPAN></td>\n<td><SPAN class="math">e_S^{\swedge/\svee} \sph_1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#B2B200}</SPAN></td>\n<td><SPAN class="math">\snu_{eL}^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#F2F200}</SPAN></td>\n<td><SPAN class="math">e_L^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#999999}</SPAN></td>\n<td><SPAN class="math">\snu_{eR}^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#D9D9D9}</SPAN></td>\n<td><SPAN class="math">e_R^{\swedge/\svee}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n</tr>\n</table>\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>
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Every [[differential form]] [[field|cotangent bundle]] may be decomposed as\n$$\n\snf{F} = \snf{\sOm} + \sf{d} \snf{\sPhi} + \sve{*d*} \snf{\sPsi} \n$$\nin which $\snf{\sOm}$ is [[harmonic]], $\sf{d}$ is the [[exterior derivative]], and $\sve{*d*}$ is the [[codifferential]]. \n\nEvery [[closed]] differential form field, $\sf{d} \snf{F} = 0$, may be decomposed as\n$$\n\snf{F} = \snf{\sOm} + \sf{d} \snf{\sPhi} \n$$\nTherefore, the [[cohomology]] is the same as the space of harmonic forms -- Hodge's theorem. That's kind of strange, since cohomology doesn't require a [[metric]], while the [[Hodge dual]] does.\n\nRef:\n*http://en.wikipedia.org/wiki/De_Rham_cohomology\n*[[Vector Calculus and the Topology of Domains in 3-Space|papers/vectorcalc.pdf]]
There are duality transformations in and between the tangent and cotangent spaces similar to [[Clifford dual]]ity. For any [[differential form]] of grade $p$,\n$$\n\snf{a} = \sfr{1}{p!} a_{\sal \sdots \sbe} \sf{e^\sal} \sdots \sf{e^\sbe}\n$$\nits ''vector dual'' (-p)-form is\n$$\n\sbar{a} = \sfr{1}{p!} a_{\sal \sdots \sbe} \set^{\sal \sga} \sdots \set^{\sbe \sde} \sve{e_\sga} \sdots \sve{e_\sde}\n$$\nBy multiplying any p-form by the (-n)-form, $\sbar{e} = \sve{e_0} \sve{e_1} \sdots \sve{e_{n-1}}$, one gets its ''Hodge vector dual'' (p-n)-form,\n$$\n\sbar{* a} = \sbar{e} \snf{a} = \sve{e_0} \sve{e_1} \sdots \sve{e_{n-1}} \sfr{1}{p!} a_{\sal \sdots \sbe} \sf{e^\sal} \sdots \sf{e^\sbe} \n= \sfr{1}{p! \slp n-p \srp!} a_{\sal \sdots \sbe} \sep^{\sal \sdots \sbe \sga \sdots \sde} \sve{e_\sga} \sdots \sve{e_\sde}\n$$\nusing [[vector-form algebra]] and the [[permutation symbol]]. And finally, by taking the form dual to this, one gets the ''Hodge dual'' (n-p)-form,\n\sbegin{eqnarray}\n* \snf{a} = \snf{*a} &=& \sfr{1}{p! \slp n-p \srp!} a_{\sal \sdots \sbe} \sep^{\sal \sdots \sbe \sga \sdots \sde} \sf{e_\sga} \sdots \sf{e_\sde} \s\s\n&=& \sfr{\sll \set \srl}{\sll e \srl p! \slp n-p \srp!} a_{i \sdots j} \sva^{i \sdots jk \sdots l} g_{km} \sdots g_{ln} \sf{dx^m} \sdots \sf{dx^n}\n\send{eqnarray}\nThe Hodge dual, which is only defined in the presence of a [[frame]] or [[metric]], is quite useful and allows the construction of the n-form product of any two p-forms, $\snf{a}$ and $\snf{b}$,\n\sbegin{eqnarray}\n\snf{*a}\snf{b} &=& \sfr{1}{p! \slp n-p \srp!} a_{\sal \sdots \sbe} \sep^{\sal \sdots \sbe \sga \sdots \sde} \sf{e_\sga} \sdots \sf{e_\sde} \sfr{1}{p!} b_{\sep \sdots \sup} \sf{e^\sep} \sdots \sf{e^\sup}\s\s\n&=& \sfr{\sll \set \srl}{p! \slp n-p \srp!} \sfr{1}{p!} a_{\sal \sdots \sbe} b_{\sep \sdots \sup} \sep^{\sal \sdots \sbe \sga \sdots \sde} \sep_{\sga \sdots \sde}{}^{\sep \sdots \sup} \snf{e}\s\s\n&=& \snf{e} \sfr{1}{p!} a_{\sal \sdots \sbe} b^{\sal \sdots \sbe} = \snf{e} \sfr{1}{p!} a_{i \sdots j} b^{i \sdots j} = \snf{e} \slp \sbar{a} \snf{b} \srp = \snf{a} \snf{*b}\n\send{eqnarray}\nrelying on [[permutation identities]]. Just as the Clifford dual squares to $\spm 1$ depending on [[signature|Minkowski metric]], the Hodge dual of a p-form similarly squares to\n\s[ \snf{**a} = \sll \set \srl \slp -1 \srp^{p \slp n-p \srp} \snf{a} \s]\n\nThere is an example important enough to address specifically. If $\sff{F} = \sha \sf{e^\smu} \sf{e^\snu} F_{\smu \snu}$ is a 2-form over a four dimensional space (or spacetime), then its Hodge dual is:\n$$\n\sff{*F} = \sfr{1}{4} F_{\smu \snu} \sep^{\smu \snu \srh \ssi} \sf{e_\srh} \sf{e_\ssi} = \sff{\svv{\sep}} \sff{F}\n$$\nin which the ''Hodge dual projector'' is a 2-vector valued 2-form defined as\n$$\n\sff{\svv{\sep}} = - \sf{e^\srh} \sf{e^\ssi} \sep_{\srh \ssi}^{\sp{\srh \ssi} \smu \snu} \sve{e_\smu} \sve{e_\snu} = - \sf{e_\srh} \sf{e_\ssi} \sep^{\srh \ssi \smu \snu} \sve{e_\smu} \sve{e_\snu}\n$$\nwhich contracts with $\sff{F}$ via the [[vector-form algebra]]. Other Hodge dual projectors may be built corresponding to other cases.\n\nThere is a somewhat awkward but coordinate free expression for the Hodge dual, taking the angle brackets to group the enclosed Clifford elements and the parenthesis to group the form elements in\n\sbegin{eqnarray}\n\snf{*a} &=& \sfr{1}{p! \slp n-p \srp!} < \slp \sve{e} \srp^p \sga^- ( \slp \sf{e} \srp^{n-p} > \snf{a})\s\s\n&=& \sfr{1}{p! \slp n-p \srp!} \sve{e_\sal} \sdots \sve{e_\sbe} \slp \sf{e^\sga} \sdots \sf{e^\sde} \sf{e^\sep} \sdots \sf{e^\sup} \srp \sli \sga^\sal \sdots \sga^\sbe \sga^- \sga_\sga \sdots \sga_\sde \sri \sfr{1}{p!} a_{\sep \sdots \sup}\s\s\n&=& \sfr{\sll \set \srl}{p! \slp n-p \srp!} \sve{e_\sal} \sdots \sve{e_\sbe} \snf{e} \sep^{\sga \sdots \sde \sep \sdots \sup} \sep_{\sga \sdots \sde}{}^{\sal \sdots \sbe} a_{\sep \sdots \sup}\s\s\n&=& \sfr{1}{p! \slp n-p \srp!} \sf{e_\sal} \sdots \sf{e_\sbe} \sep^{\sga \sdots \sde \sal \sdots \sbe} a_{\sga \sdots \sde}\n\send{eqnarray}
For a [[tangent vector]], $\sve{v}$, and a grade $p$ [[differential form]], $\snf{f}$, the [[vector-form algebra]] contraction can be equated with an expression involving the [[Hodge dual]],\n$$\sve{v} \snf{f} = (-1)^{p(n-p)} * \slp \sf{v} \snf{*f} \srp$$\nin which $\sf{v} = v^\sal \set_{\sal \sbe} \sf{e^\sbe}$ is the form dual of $\sve{v}$.\n\n(//add more as needed//)
David Ritz Finkelstein\nhttp://arxiv.org/abs/gr-qc/0608086\n*proposes a "flexing" of [[Lie algebra]] structure constants to go from one level of physical theory (some struct const = 0) to another (some not 0, or all not 0 ([[simple]])).\n*hey, isn't this the same as Lie algebra deformation?
A horizontal dividing line.\n----\n{{{----}}}
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<<<\nA whole block\nof text to be quoted.\n<<<\nor\n>>>Multiple levels of indented quotes.\n>>Just like [[Bullet Points]].\n>yep\n>>or like [[Numbered Lists]]\nThat's what they said.\n{{{\n<<<\nA whole block\nof text to be quoted.\n<<<\nor\n>>>Multiple levels of indented quotes.\n>>Just like [[Bullet Points]].\n>yep\n>>or like [[Numbered Lists]]\nThat's what they said.\n}}}
TiddlyWiki lets you write ordinary HTML by enclosing it in {{{<html>}}} and {{{</html>}}}:\n<html>\n<a href="javascript:;" onclick="onClickNoteLink(event);" \ntiddlyLink="Welcome"\nstyle="background-color: yellow;">\nLink to Welcome constructed in HTML</a>\n</html>\n{{{\n<html>\n<a href="javascript:;" onclick="onClickNoteLink(event);" \ntiddlyLink="Welcome"\nstyle="background-color: yellow;">\nLink to Welcome constructed in HTML</a>\n</html>\n}}}\nHTML can enable some exotic new features (like [[embedding GMail and Outlook|http://groups.google.com/group/TiddlyWiki/browse_thread/thread/d363303aff5868d0/056269d8409d121f?lnk=st&q=embedding+gmail&rnum=1#056269d8409d121f]] in a TiddlyWiki). But, care needs to be taken with including things like JavaScript code.
[>img[images/person/John Baez.jpg]]Homepage: http://math.ucr.edu/home/baez/\n*Location: UCRiverside\n\nA wonderfully prolific mathematical physicist, and all around good guy.
Access keys are shortcuts to common functions accessed by typing a letter with either the 'alt' (PC) or 'control' (Mac) key:\n|!PC|!Mac|!Function|\n|Alt-F|Ctrl-F|Search|\n|Alt-J|Ctrl-J|NewJournal|\n|Alt-N|Ctrl-N|NewNote|\n|Alt-S|Ctrl-S|SaveChanges|\nThese access keys are provided by the associated internal [[Macros]] for the functions above. The macro needs to be used in an open note (or the [[MainMenu]] or SideBar) in order for the access keys to work.\n\nWhile editing a note:\n* ~Control-Enter or ~Control-Return accepts your changes and switches out of editing mode (use ~Shift-Control-Enter or ~Shift-Control-Return to stop the date and time being updated for MinorChanges)\n* Escape abandons your changes and reverts the note to its previous state\n\nIn the search box:\n* Escape clears the search term
A [[Lie algebra]] is a [[vector space]] as well as an algebra, spanned by the basis vectors, $T_A$, and one may use the structure constants to build a natural [[metric]] giving a scalar result from two Lie algebra elements,\n$$\n\slp B, C \srp = {\srm Tr}({\srm Ad}_B {\srm Ad}_C) = T^D \slb B, \slb C, T_D \srb \srb = B^A C^B g_{AB} \n$$\nusing the [[trace]] and Lie algebra adjoint action, ${\srm Ad}_B C = \slb B, C \srb$. The resulting metric coefficients are those of the ''Killing form'',\n$$\ng_{AB} = \slp T_A, T_B \srp = C_{AC}{}^D C_{BD}{}^C\n$$\nIt is always possible to transform to a new set of generators, $T'_A = L_A{}^B T_B$, producing a new set of structure constants and hence a new metric, $\set_{AB} = L_A{}^C L_B{}^D g_{CD}$. As long as the metric is non-degenerate, which happens iff $Lie(G)$ is semi-[[simple]], it is possible to transform to a set of generators such that this metric is unit diagonal (with $+1$ and $-1$ entries like in the [[Minkowski metric]]) via the methods of [[spectral decomposition|eigen]]. This has already been done for most common generator representations used in physics, up to a constant factor, so that often $g_{AB} = \set_{AB} = \sde_{AB}$. \n\nUsing the Jacobi identity, the Killing form is symmetric, $g_{AB} = g_{BC}$, and is ''adjoint invariant'',\n\sbegin{eqnarray}\n\slp \slb A, B \srb, C \srp &=& \slp A, \slb B, C \srb \srp \s\s\n\slp g A g^-, C \srp &=& \slp A, g^- C g \srp\n\send{eqnarray}\nwhich implies the structure constants are antisymmetric in the last two indices,\n$$\nC_{ABC} = -C_{ACB}\n$$\nwhen the Killing forms $g_{AB}$ (and $g^{AB}$) are used to lower (and raise) Lie algebra indices, $C_{ABC}=C_{AB}{}^D g_{DC}$. Since we always have $C_{AB}{}^C = - C_{BA}{}^C$, the structure constants are completely [[antisymmetric|index bracket]],\n$$\nC_{ABC} = C_{\slb ABC \srb} = C_{BCA} = C_{CAB} = -C_{BAC} = -C_{ACB} = -C_{CBA}\n$$\nwhich is useful enough to call the ''Killing form identity''.\n\nThe ''inverse Killing form'', $g^{AB}$, is used to define the ''generator duals'', $T^A = g^{AB} T_B$, satisfying $\slp T^A, T_B \srp = \sde^A_B$.\n\nFor some Lie algebras with [[Clifford algebra]] or matrix generators, the scalar part or trace gives the orthonormality relations\n$$\n\slp T_A, T_B \srp = g_{AB} \ssim \sli T_A T_B \sri = \sde_{AB}\n$$
A ''Killing spinor'' is a [[spinor]] field satisfying\n$$\n\sf{\sna} \sps = \sla \sf{e} \sps\n$$\nfor some constant ''Killing number'', $\sla$, in which $\sf{\sna}$ is the [[spinor covariant derivative]] and $\sf{e}$ is the [[coframe|frame]].\n\nThe [[tangent vector]] field corresponding to a Killing spinor is a [[Killing vector]],\n$$\n\sve{\sxi} = \sleft< \sbar{\sps} \sga^\sal \sps \sright> \sve{e_\sal}\n$$\nin which $\sbar{\sps}$ is the [[Clifford conjugate]] of the spinor and $\sve{e_\sal}$ are [[frame]] vectors.
A ''Killing vector'' field, $\sve{\sxi}(x)$, is the generator of a [[flow]], $\sph_t = e^{t {\scal L}_{\sve{\sxi}}}$, that leaves the geometry of a [[manifold]] invariant &mdash; constituting a symmetry of the geometry. It is a [[tangent vector field|tangent bundle]] satisfying ''Killing's equation'',\n$$\nL_{\sve{\sxi}} \sf{e} = B \stimes \sf{e}\n$$\nThe [[Lie derivative]] of the [[frame]] along a Killing vector field gives a [[rotation|Clifford rotation]] of the frame by some corresponding Clifford bivector field, $B (x) \sin Cl^2$,\n$$\nL_{\sve{\sxi}} \sf{e^\sal} = B_\sbe{}^\sal \sf{e^\sbe}\n$$\nwith ''Killing rotation coefficients'', $B_\sbe{}^\sal = - B^\sal{}_\sbe$. This version of Killing's equation, or equivalently,\n$$\nL_{\sve{\sxi}} \sve{e_\sal} = - B_\sal{}^\sbe \sve{e_\sbe}\n$$\nmatches the usual definition that the Lie derivative of the [[metric]] along a Killing vector field vanishes.\n\nAny set of Killing vector fields is related through the [[Lie bracket|Lie derivative]],\n$$\n\slb \sve{\sxi_A}, \sve{\sxi_B} \srb_L = C_{AB}{}^C \sve{\sxi_C}\n$$\nwith $C_{AB}{}^C$ the set of ''structure constants'' for the symmetries. The manifold [[diffeomorphism]]s (''isometries'') built from the flows generated by a set of Killing vector fields constitute a [[Lie group]].\n\nA Killing vector field has many nice [[properties|Killing vector identities]].
The defining equation of a [[Killing vector]] field, $\sve{\sxi}$, is\n\sbegin{eqnarray}\nB_\sbe{}^\sal \sf{e^\sbe} &=& {\scal L}_{\sve{\sxi}} \sf{e^\sal} = \sve{\sxi} \slp \sf{d} \sf{e^\sal} \srp + \sf{d} \sxi^\sal \s\s\n&=& \sve{\sxi} \slp \sf{w}{}_\sbe{}^\sal \sf{e^\sbe} + \sff{T^\sal} \srp + \sf{d} \sxi^\sal \s\s\n&=& \sf{e^\sbe} \slp \sxi^\sde w_{\sde \sbe}{}^\sal - \sxi^\sde w_{\sbe \sde}{}^\sal + 2 \sxi^\sde T_{\sde \sbe}{}^\sal + \spa_\sbe \sxi^\sal \srp\n\send{eqnarray}\nby virtue of the defining equations for the [[Lie derivative]], the [[cotangent bundle connection]], and the [[torsion]]. This gives a useful expression for the derivative of the Killing vector field coefficients,\n$$\n\spa_\sbe \sxi_\sal = B_{\sbe \sal} - \sxi^\sde w_{\sde \sbe \sal} + \sxi^\sde w_{\sbe \sde \sal} - 2 \sxi^\sde T_{\sde \sbe \sal}\n$$\nThe [[1-form dual|frame]] to the Killing vector field is $\sf{\sxi} = \sxi_\sal \sf{e^\sal} = \sxi^\sbe \set_{\sbe \sal} \sf{e^\sal}$. The cotangent bundle covariant derivative of this field is\n$$\n\sf{\sna} \sf{\sxi} = \sf{e^\sbe} \sna_\sbe \slp \sxi_\sal \sf{e^\sal} \srp\n= \sf{e^\sbe} \sf{e^\sga} \slp \spa_\sbe \sxi_\sga + \sxi_\sal w_{\sbe \sga}{}^\sal \srp\n= \sf{e^\sbe} \sf{e^\sga} \slp B_{\sbe \sga} - \sxi^\sde w_{\sde \sbe \sga} - 2 \sxi^\sde T_{\sde \sbe \sga} \srp\n$$\nSimilarly,\n$$\n\sf{\sna} \sve{\sxi} = \sf{e^\sbe} \sna_\sbe \slp \sxi^\sal \sve{e_\sal} \srp\n= \sf{e^\sbe} \sve{e_\sga} \slp \spa_\sbe \sxi^\sga + \sxi^\sal w_\sbe{}^\sga{}_\sal \srp\n= \sf{e^\sbe} \sve{e_\sga} \slp B_\sbe{}^\sga - \sxi^\sde w_{\sde \sbe}{}^\sga - 2 \sxi^\sde T_{\sde \sbe}{}^\sga \srp\n$$\nIf $\sve{v}$ is the velocity along a [[geodesic]],\n$$\n0 = \sve{v} \sf{\sna} \sve{v} = v^\sal \slp \spa_\sal v^\sde + v^\sbe w_\sal{}^\sde{}_\sbe \srp \sve{e_\sde}\n$$\nthe component of this velocity along any Killing vector field, $p = \slp \sve{v}, \sve{\sxi} \srp = \sve{v} \sf{\sxi}$, is constant along the geodesic,\n$$\n\sve{v} \sf{d} p = v^\sal \spa_\sal \slp v^\sbe \sxi_\sbe \srp \n= v^\sal \slp \spa_\sal v^\sbe \srp \sxi_\sbe + v^\sal v^\sbe \slp \spa_\sal \sxi_\sbe \srp\n= v^\sal \slp \spa_\sal v^\sde \srp \sxi_\sde + v^\sal v^\sbe \slp w_{\sal \sde \sbe} \sxi^\sde - 2 \sxi^\sde T_{\sde \sal \sbe} \srp\n= 0\n$$\nas long as the torsion vanishes, or at least $T_{\sde \slp \sal \sbe \srp} = 0$.\n\nIf the Killing vector field is of constant length, $\sve{\sxi} \sf{\sxi} = \sxi^\sal \sxi_\sal = c$, then\n$$\n0 = \spa_\sbe \slp \sxi^\sal \sxi_\sal \srp = 2 \sxi^\sal \slp \spa_\sbe \sxi_\sal \srp\n= 2 \sxi^\sal \slp B_{\sbe \sal} - \sxi^\sde w_{\sde \sbe \sal} - 2 \sxi^\sde T_{\sde \sbe \sal} \srp\n$$\nand the integral curves of the Killing vector field are geodesics,\n$$\n\sve{\sxi} \sf{\sna} \sve{\sxi} = \sxi^\sbe \sve{e_\sga} \slp B_\sbe{}^\sga - \sxi^\sde w_{\sde \sbe}{}^\sga \srp = 0\n$$\nas long as the torsion vanishes or $T_{\sde \slp \sal \sbe \srp} = 0$.
The ''Kronecker product'' (//''tensor product''//), $\sotimes$, of an $m$ by $n$ matrix, $A$, and a $p$ by $q$ matrix, $B$, is an $mp$ by $nq$ matrix, $C = A \sotimes B$,\n$$\nC_{\slp \slp a - 1 \srp p + x \srp}{}^{\slp \slp b - 1 \srp q + y \srp} = A_a{}^b B_x{}^y\n$$\nIt is a ''block'' matrix of $B$'s multiplied by the entries of $A$. For example,\n$$\n\slb \sbegin{array}{cc}\n1 & 2\s\s\n3 & 1\n\send{array} \srb\n\sotimes\n\slb \sbegin{array}{cc}\n0 & 3\s\s\n2 & 1\n\send{array} \srb\n=\n\slb \sbegin{array}{cc}\n1\n\slb \sbegin{array}{cc}\n0 & 3\s\s\n2 & 1\n\send{array} \srb\n & 2\n\slb \sbegin{array}{cc}\n0 & 3\s\s\n2 & 1\n\send{array} \srb\n\s\s\n3\n\slb \sbegin{array}{cc}\n0 & 3\s\s\n2 & 1\n\send{array} \srb\n & 1\n\slb \sbegin{array}{cc}\n0 & 3\s\s\n2 & 1\n\send{array} \srb\n\send{array} \srb\n=\n\slb \sbegin{array}{cccc}\n0 & 3 & 0 & 6\s\s\n2 & 1 & 4 & 2\s\s\n0 & 9 & 0 & 3\s\s\n6 & 3 & 2 & 1\n\send{array} \srb\n$$\n\nThis product spawns several identities, including:\n$$\n\slp A \sotimes B \srp \slp C \sotimes D \srp = A C \sotimes B D\n$$\n\nRef:\nhttp://en.wikipedia.org/wiki/Kronecker_product
//Use the first method in each example below, unless you have some reason not to.//\nMathematical symbols, such as \s(e^{x^2}\s), may be inserted inline.\n{{{\nMathematical symbols, such as $e^{x^2}$, may be inserted inline.\nMathematical symbols, such as \s(e^{x^2}\s), may be inserted inline.\n}}}\nOr as displayed math,$$e^{x^2}$$ on its own line.\n{{{\nOr as displayed math, \s[e^{x^2}\s] on its own line.\nOr as displayed math, $$e^{x^2}$$ on its own line.\nOr as displayed math, \sbegin{equation}e^{x^2}\send{equation} on its own line.\n}}}\nOr as an equation array,\n\sbegin{eqnarray}A &=& e^{x^2}\s\s&=&C\send{eqnarray}\n{{{\nOr as an equation array,\sbegin{eqnarray}A &=& e^{x^2}\s\s&=& C\send{eqnarray}\n}}}\n\nSome of the available TeX symbols can be found at [[jsMath|http://www.math.union.edu/~dpvc/jsMath/symbols/welcome.html]], the best method I could find for displaying TeX online. The small button in the lower right corner of this window opens its control planel. I'm not sure how many LaTeX and AMSTeX commands are supported -- play around.\n\nTeX substitution macros such as $\sf{A}$, ({{{$\sf{A}$}}}), may be inserted into the [[jsMathPlugin]] just before the jsMath.process call. See that plugin for abbreviated commands I've included.
[>img[images/person/Laurent Freidel.jpg]]Homepage: http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=Laurent_Freidel\n*Location: $\sPi$\n*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Freidel_L/0/1/0/all/0/1\n\nOften coauthors with: Artem Starodubtsev.\n\nSelected work:\n*[[Quantum gravity in terms of topological observables|papers/0501191.pdf]]\n*[[Particles as Wilson lines of gravitational field]]\n
[>img[images/person/Leonardo Castellani.jpg]]Homepage: http://www.mfn.unipmn.it/~castella/\n*Location: Turin\n*Papers: http://www-library.desy.de/cgi-bin/spiface/find/hep/www?rawcmd=find++fa+castellani%2Cl+and+not+a+martinelli+or+a+aschieri+and+castellani&FORMAT=WWW&SEQUENCE=\n\nSelected work:\n*[[Group Geometric Methods in Supergravity and Superstring Theories|papers/Group Geometric Methods in Supergravity and Superstring Theories.pdf]]\n**nice job on [[reductive Lie group geometry]]\n**BRST by adding a (Grassmann?) piece to the Lie group geometry\n*[[A Geometric Interpretation of BRST symmetry|papers/A Geometric Interpretation of BRST symmetry.pdf]]\n**just the BRST part from paper above\n**add generator, $Q$, and $\sf{d \sth} g^A T_A$.\n**This reminds me of the geometry of Lagrange multipliers\n*[[Gravity on Finite Groups|papers/9909028.pdf]]\n**differential geometry over disconnected spaces\n*lots of [[Kaluza-Klein]] stuff
A ''Lie algebra'' consists of elements of a $N$ dimensional [[vector space]], $B=B^A T_A \sin {\srm Lie}(G)$, composed of real (or complex, for a complex Lie algebra) coefficients, $B^A$, multiplying $N$ ''Lie algebra generators'', $T_A$. The elements are closed under a ''Lie algebra bracket'', equal to the Lie algebra ''adjoint action'' of one element on another,\n$$\n{\srm Ad}_A B = \slb A, B \srb = C\n$$\nand equivalent to the [[commutator]] relation. The Lie algebra brackets of the generators,\n$$\n\slb T_A , T_B \srb = T_A T_B - T_B T_A = C_{AB}{}^C T_C\n$$\ngive the real or complex ''structure constants'' (//''structure coefficients''//), $C_{AB}{}^C = - C_{BA}{}^C$, for the Lie algebra. The generators may be [[Clifford basis elements]] ($T_A=\sga_A$), square matrices, or abstract operators. The bracket must satisfy the ''Jacobi identity'',\n\sbegin{eqnarray}\n0 &=& \slb A, \slb B , C \srb \srb + \slb B, \slb C , A \srb \srb + \slb C, \slb A , B \srb \srb \s\s\n0 &=& C_{AD}{}^E C_{BC}{}^D + C_{BD}{}^E C_{CA}{}^D + C_{CD}{}^E C_{AB}{}^D \n\send{eqnarray}\nidentical to the [[Clifford Jacobi identity|Clifford product identities]].\n\nAlthough the choice of generators is somewhat arbitrary, every Lie algebra has a specific [[Lie algebra structure]].
It has been seen that an $n$ dimensional [[Clifford algebra]] can be interpreted as a $2^n$ dimensional [[Lie algebra]], with the [[Clifford basis elements]] identified as generators, $\sga_{\sal \sdots \sbe} \ssim T_A$, and the Lie bracket given by the [[Clifford basis product identities]]. It is also the case that any $n$ dimensional Lie algebra may be identified as a subalgebra of the bivector algebra of an $n$ dimensional Clifford algebra. This representation may be made as:\n$$\nT_A = - \sfr{1}{4} C_A{}^{BC} \sga_{BC}\n$$\nemploying the Lie algebra structure constants, the Clifford bivector basis elements, and using the unit diagonal [[Killing form]], $g^{BD}=\set^{BD}$, as the Clifford algebra metric and to raise the structure constant indices. This representation faithfully gives\n\sbegin{eqnarray}\n\slb T_A, T_D \srb &=& \sfr{1}{8} C_A{}^{BC} C_D{}^{EF} \sga_{BC} \stimes \sga_{EF} \s\s\n&=& \sha C_A{}^{BC} C_{DC}{}^E \sga_{BE} \s\s\n&=& - \sfr{1}{4} C_{AD}{}^C C_C{}^{BE} \sga_{BE} \s\s\n&=& C_{AD}{}^C T_C\n\send{eqnarray}\nvia the [[Jacobi identity|Lie algebra]] and Clifford basis identities. These Lie algebra generators are orthogonal,\n\sbegin{eqnarray}\n\sli T_A T_D \sri &=& \sfr{1}{16} C_A{}^{BC} C_D{}^{EF} \sli \sga_{BC} \sga_{EF} \sri \s\s\n&=& \sfr{1}{16} C_A{}^{BC} C_D{}^{EF} \slp \set_{BF} \set_{CE} - \set_{BE} \set_{CF} \srp \s\s\n&=& \sfr{1}{8} C_A{}^{BC} C_{DCB} \s\s\n&=& \sfr{1}{8} \set_{AD}\n\send{eqnarray}\nin the [[scalar part|Clifford grade]] (trace) operator. Note that the $2^{[n/2]}$ dimensional [[Clifford matrix representation]] gives a corresponding matrix representation for the Lie algebra.
The structure of every [[simple]] [[Lie algebra]] may be described by arranging the generators to span different related sub-spaces. The ''rank'', $R$, of a $N$ dimensional Lie algebra is the maximal number if inter-commuting generators, $T^C_a$,\n$$\n\sbig[ T^C_a, T^C_b \sbig] = 0 \s;\s;\s;\s; \sforall \s;\s; 1 \sle a,b \sle R\n$$\nThese generators span a $R$ dimensional [[vector space]], the ''Cartan subalgebra'', $C = c^a T^C_a$. [[Exponentiating|exponentiation]] the Cartan subalgebra gives the ''maximal torus'' -- a $R$ dimensional [[submanifold]] of the [[Lie group manifold|Lie group geometry]].\n\nEvery element of the Cartan subalgebra may be represented by a $(N \stimes N)$ matrix, $M$, related to the adjoint action, ${\srm Ad}_C$ , which acts on other Lie algebra elements, $B = b^A T_A$, as vectors,\n$$\n[ C , B ] = {\srm Ad}_C \s, B = c^a \sbig[ T^C_a , T_A \sbig] b^A = T_D \sbig( c^a C_{aA}^{\sp{aA}D} \sbig) b^A = T_D M^D_{\sp{D}A} b^A \n$$\nThe components of $M$ may be written in terms of the structure constants as $M^D_{\sp{D}A} = c^a C_{aA}^{\sp{aA}D}$. As a vector space, the Lie algebra structure may be understood in terms of the [[eigen]]values and eigenvectors of $M$.\n\nSince $M \s, C_1 = 0$ for any $C_1$ in the Cartan subalgebra, $R$ of the eigenvalues of $M$ will be $0$, with that null eigenspace equal to $C$. The remaining $(N-R)$ eigenvectors of $M$ come in pairs, $V^\spm_i$, with each corresponding to a unique complex eigenvalue,\n$$\nM(c) \s, V^\spm_i = \spm \sal_i(c) V^\spm_i\n$$\n(no sum over $i$) Since $M(c)$ depends linearly on the $c^a$, each of the eigenvalues, $\sal_i(c)$, also depends linearly on the $c^a$. Note that the eigenvectors, $V^\spm_i$, do not depend on $c^a$ -- they depend only on the whole Cartan subalgebra. The eigenvalues may be related to elements, $\sal_i \sin C^*$, of a dual vector space to the Cartan subalgebra, $\sal_i : C \sto \smathbb{C}$. Written out in terms of dual basis elements, $T^a$, they are $\sal_i = \sal_{ia} T^a$, so\n$$\n\sal_i(c) = \sal_i \s, C = \sal_{ia} T^a \s, c^b T_b = \sal_{ia} c^a \sin \smathbb{C}\n$$\nThese dual space elements, $\sal_i$, are called the ''roots''. Each root has a negative partner root, and may not be twice any other root, $\sal_i \sne 2 \sal_j$.\n\nSince a Lie algebra has a natural metric, the [[Killing form]], every element, $\sal \sin C^*$, of the dual space has a corresponding element in the Cartan subalgebra, $h_\sal = h_\sal^a T_a = ( g^{ab} \sal_b ) T_a \sin C$, such that\n$$\n\sal \s, C = \sal_a T^a \s, c^b T_b = \sal_a c^a = h_\sal^a c^b g_{ab} = h_\sal^a c^b \sbig( T_a , T_b) = \sbig( h_\sal , C \sbig)\n$$\nIn this way, the Killing form gives a ''root space metric'', $< \sal, \sbe > = (h_\sal, h_\sbe)$. Using this metric, the roots for any Lie algebra may be seen to have a nicely symmetric arrangement which describes the algebra.\n\nThe complete set of non-zero Lie brackets for the algebra are:\n\sbegin{eqnarray}\n\sbig[ C_1 , C_2 \sbig] &=& 0 \s\s\n\sbig[ C , V^\spm_i \sbig] &=& \spm \sal_i(c) V^\spm_i \s\s\n\sbig[ V^\spm_i , V^\smp_i \sbig] &=& \sbig( V^\spm_i , V^\smp_i \sbig) h_{\sal_i} \s\s\n\sbig[ V^\spm_i , V^\spm_j \sbig] &=& N_{ij} \s, V^\spm_k\n\send{eqnarray}\nin which $N_{ij}$ are normalization constants and it must be the case that $\sal_i + \sal_j = \sal_k$ in the last bracket above. This algebra has $R$ $su(2)$ subalgebras,\n\sbegin{eqnarray}\n\sbig[ h_{\sal_i} , V^+_i \sbig] &=& < \sal_i , \sal_i > V^+_i \s\s\n\sbig[ h_{\sal_i} , V^-_i \sbig] &=& - < \sal_i , \sal_i > V^-_i \s\s\n\sbig[ V^+_i , V^-_i \sbig] &=& \sbig( V^+_i , V^-_i \sbig) h_{\sal_i}\n\send{eqnarray}\n//(More on root diagrams, Dynkin, etc. -- use examples.)//\n\nRef:\n*Robert Cahn\n**[[Semi-Simple Lie Algebras and Their Representation|papers/Semi-Simple Lie Algebras and Their Representation.pdf]]
link from [[Maurer-Cartan form]]\n\nref:\nhttp://en.wikipedia.org/wiki/Lie_algebroid\n[[Differential Operators and Actions of Lie Algebroids|papers/0209337.pdf]]
The ''Lie derivative'' is the rate of change of any field perceived by an observer as she moves along a path with some [[velocity|tangent vector]], $\sve{v}$. Basically, the field where she is going is pulled back and compared with the field where she's at. This description is extended to give the Lie derivative, ${\scal L}_{\sve{v}}$, with respect to a [[flow]], $\sph_t$, giving the rate of change of any field perceived by observers at every manifold point as they move according to the velocity field, $\sve{v}(x)$. The parameterized flow is given to first order in $t$ by\n$$\n\sph_t^i(x) \ssimeq x^i + t v^i(x)\n$$\nThe Lie derivative of any field, $X$, is a [[natural]] operator defined as\n$$\n{\scal L}_{\sve{v}} X = \sfr{d}{d t} \sph_t^* X = \sfr{d}{d t} X(t) = \slim_{t \sto 0} \sfr{\sph_t^*X - X}{t} = \slim_{t \sto 0} \sfr{ \slp \sph_{-t} \srp_* X - X}{t}\n$$\nin which\n$$\n\sph_t^*X = \sph_t^*\slb X \srl_{\sph_t} \ssimeq \sph_t^* \slp \slb X \srl_x + t v^i \spa_i \slb X \srl_x \srp\n$$\nis the [[pullback]] of the field from where the flow is going back to the initial points, expanded to first order in $t$.\n\nIf the field is a [[function]] over the manifold, the Lie derivative of this field is the same as the [[directional derivative|tangent vector]] of this function with respect to the velocity field at every point,\n$$\n{\scal L}_{\sve{v}} f = \sfr{d}{d t} f(x) = \slim_{t \sto 0} \sfr{\sph_t^*f - f}{t} = v^i \spa_i f = \sve{v} \sf{d} f \n$$\nIf the field is a [[1-form|cotangent bundle]] field, $\sf{f} = \sf{dx^i} f_i(x)$, the pullback along the flow is\n$$\n\sph_t^*\sf{f} = \sf{dx^i} \slb \sfr{\spa \sph_t^j}{\spa x^i} \srb f_j(\sph_t) \n\ssimeq \sf{dx^i} \slb \sde^j_i + t \spa_i v^j \srb \slb f_j(x) + t v^k \spa_k f_j(x) \srb\n\ssimeq \sf{dx^i} \slp f_i + t v^k \spa_k f_i + t f_j \spa_i v^j \srp\n$$\nand the Lie derivative is thus\n\sbegin{eqnarray}\n{\scal L}_{\sve{v}} \sf{f} &=& \slim_{t \sto 0} \sfr{\sph_t^*\sf{f} - \sf{f}}{t}\n= \sf{dx^i} \slp v^k \spa_k f_i + f_j \spa_i v^j \srp \s\s\n&=& \slp \sve{v} \sf{\spa} \srp \sf{f} + \slp \sf{\spa} \sve{v} \srp \sf{f} \s\s\n&=& \sve{v} \slp \sf{d} \sf{f} \srp + \sf{d} \slp \sve{v} \sf{f} \srp\n\send{eqnarray}\nusing the [[exterior derivative]], [[partial derivative]], and [[vector-form algebra]]. For any differential form or [[Clifform]] field with form grade greater than zero this generalizes to give ''Cartan's formula'' for the Lie derivative,\n\sbegin{eqnarray}\n{\scal L}_{\sve{v}} \s, \snf{F}\n&=& \slp \sve{v} \sf{\spa} \srp \snf{F} + \slp \sf{\spa} \sve{v} \srp \snf{F} \s\s\n&=& \sve{v} \slp \sf{d} \snf{F} \srp + \sf{d} \slp \sve{v} \snf{F} \srp\n\send{eqnarray}\nand another nice formula (easier for computations) obtained via the power of vector-form algebra and the partial derivative operator.\nIf the field is a [[vector|tangent bundle]] field, $\sve{u}(x)$, the pushforward along the negative flow of the velocity at where the flow goes is\n$$\n\slp \sph_{-t} \srp_* \sve{u} = u^j(\sph_t) \slb \sfr{\spa \sph_{-t}^i}{\spa x^j} \srb \sve{\spa_i}\n\ssimeq \slb u^j + t v^k \spa_k u^j \srb \slb \sde^i_j - t \spa_j v^i \srb \sve{\spa_i}\n\ssimeq \slp u^j + t v^k \spa_k u^i - t u^j \spa_j v^i \srp \sve{\spa_i}\n$$\nand the the Lie derivative of a vector field is thus\n$$\n{\scal L}_{\sve{v}} \sve{u} = \slim_{t \sto 0} \sfr{ \slp \sph_{-t} \srp_* \sve{u} - \sve{u}}{t}\n= \slp v^k \spa_k u^i - u^j \spa_j v^i \srp \sve{\spa_i}\n= \sve{v} \sf{\spa} \sve{u} - \sve{u} \sf{\spa} \sve{v}\n$$\n\nThe Lie derivative of one velocity field with respect to another defines the ''Lie bracket'',\n$$\n\slb \sve{v}, \sve{u} \srb_L = {\scal L}_{\sve{v}} \sve{u} = - \slb \sve{u}, \sve{v} \srb_L\n$$\n\nThe Lie derivative has a number of nice [[properties|Lie derivative identities]].
The [[Lie derivative]] is a natural, fundamental derivative operator on any geometric field on a manifold.\n\nBy virtue of Cartan's formula, it commutes with the [[exterior derivative]] operator when acting on [[function]]s, [[differential form]]s or [[Clifform]]s,\n$$\n{\scal L}_{\sve{v}} \sf{d} \snf{F} = \sf{d} \slp \sve{v} \slp \sf{d} \snf{F} \srp \srp = \sf{d} {\scal L}_{\sve{v}} \snf{F} \n$$\nbut not always when acting on vector fields.\n\nIt is linear in both the velocity field,\n$$\n{\scal L}_{\sve{v} + \sve{u}} X = {\scal L}_{\sve{v}} X + {\scal L}_{\sve{u}} X\n$$\nand argument,\n$$\n{\scal L}_{\sve{v}} \slp X + Y \srp = {\scal L}_{\sve{v}} X + {\scal L}_{\sve{v}} Y\n$$\n\nThe Lie derivative is a grade $0$ [[derivation]], acting on various products of fields via the Liebniz rule,\n\sbegin{eqnarray}\n{\scal L}_{\sve{v}} \slp a \snf{F} \srp &=& \slp {\scal L}_{\sve{v}} a \srp \snf{F} + a \slp {\scal L}_{\sve{v}} \snf{F} \srp \s\s\n{\scal L}_{\sve{v}} \slp \sf{a} \snf{F} \srp &=& \slp {\scal L}_{\sve{v}} \sf{a} \srp \snf{F} + \sf{a} \slp {\scal L}_{\sve{v}} \snf{F} \srp \s\s\n{\scal L}_{\sve{v}} \slp \sve{a} \snf{F} \srp &=& \slp {\scal L}_{\sve{v}} \sve{a} \srp \snf{F} + \sve{a} \slp {\scal L}_{\sve{v}} \snf{F} \srp \s\s\n\send{eqnarray}\n\nScaling the velocity field by a function results in\n\sbegin{eqnarray}\n{\scal L}_{f \sve{v}} \snf{F} &=& f {\scal L}_{\sve{v}} \snf{F} + \slp \sf{d} f \srp \slp \sve{v} \snf{F} \srp \s\s\n{\scal L}_{f \sve{v}} \sve{u} &=& f {\scal L}_{\sve{v}} \sve{u} - \slp \sve{u} \sf{d} f \srp \sve{v}\n\send{eqnarray}\n\nThe [[commutator]] of Lie derivatives with respect to two velocity fields acting on anything is equal to the Lie derivative with respect to the Lie bracket of the two velocity fields,\n$$\n\slb {\scal L}_{\sve{v}}, {\scal L}_{\sve{u}} \srb = {\scal L}_{\sve{v}} {\scal L}_{\sve{u}} - {\scal L}_{\sve{u}} {\scal L}_{\sve{v}} = {\scal L}_{\slb \sve{v}, \sve{u} \srb_L}\n$$\nThis gives the [[Jacobi identity|Lie algebra]] for the Lie bracket when acting on a vector field,\n$$\n\slb \slb \sve{v}, \sve{u} \srb_L, \sve{w} \srb_L = - \slb \slb \sve{u}, \sve{w} \srb_L, \sve{v} \srb_L + \slb \slb \sve{v}, \sve{w} \srb_L, \sve{u} \srb_L\n$$\n\nThe Lie derivative of [[fiber bundle]] basis elements (other than tangent or cotangent bundle basis), such as [[Clifford basis elements]], is zero,\n$$\n{\scal L}_{\sve{v}} \sga_\sal = 0\n$$\n\nThe definition of the Lie derivative in terms of the flow allows the flow to be written as the [[exponentiation]] of the Lie derivative,\n$$\nX(t) = \sph_t^* X = e^{t {\scal L}_{\sve{v}}} X\n$$\n\nFor a surface, $\sSi_t$, carried along with the flow, the time derivative of an integral over that surface is\n$$\n\sfr{d}{d t} \sint_{\sSi_t} \snf{F} = \sint_{\sSi_t} {\scal L}_{\sve{v}} \snf{F}\n$$\nwhich combines consistently with [[Stoke's theorem|integration]],\n$$\n\sfr{d}{d t} \sint_{\spa \sSi_t} \snf{F}\n= \sfr{d}{d t} \sint_{\sSi_t} \sf{d} \snf{F}\n= \sint_{\sSi_t} {\scal L}_{\sve{v}} \sf{d} \snf{F} \n= \sint_{\spa \sSi_t} {\scal L}_{\sve{v}} \snf{F}\n$$
An $n$ dimensional ''Lie group'' is a [[group]] of infinitely many elements, $g(x) \sin G$, parametrized by $n$ real (or complex, for a complex Lie group) parameters, $x \sin \sRe^n$. A Lie group is also a [[manifold]], with points, $x$, corresponding to the parameters (in various patches).\n\nNear the identity, $1=g(0)$, group elements may be described in terms of coordinates multiplying the [[Lie algebra]] generators associated with the group,\n$$\ng(x) \ssimeq 1 + x^A T_A\n$$\nThe Lie algebra completely describes the local geometry of the group. For a ''connected'' manifold, and Lie group, all manifold points (and corresponding group elements) may be connected to the identity by smooth paths. [[Exponentiation|exponentiation]] of Lie algebra generators gives the ''universal cover'' of the corresponding connected Lie group,\n$$\ng(x)=e^{x^A T_A} \sin G\n$$\nPoints of a connected Lie group manifold may be described by the $x^A$ coordinates, with global group structure (manifold topology) determined by the ranges and matchings of $x^A$. A connected manifold, and Lie group, is ''simply connected'' if all paths are contractible. For example, a sphere is simply connected while a torus is not. The universal cover of a connected Lie group is simply connected.\n\nThe [[Lie group geometry]] is the Lie group manifold with geometry and symmetries corresponding to the action of the Lie algebra generators as vector fields. (Lie group geometry may alternatively be described as a [[Lie group bundle]], with the base manifold taken to be the Lie group manifold, the Lie group as typical fiber, and a special "identity" section, $g_I(x)$, defined.)
A [[Lie group]] is an $n$ dimensional group, $G$, with elements, $g$, that can be identified with points, $x$, on an $n$ dimensional manifold, $M$. Typically the group elements, $g(x) \sin G$, are understood to be written as a function of parameters; however, it is possible to consider this identification as a bijective section, the ''identity section'', $g_I(x)$, of a [[fiber bundle]]. This bundle clearly has $M$ as base and $G$ as typical fiber. But what is the structure group and action? It is desirable to preserve the group structure of the Lie group fiber: if any three fiber/group elements satisfy $g_1 g_2 = g_3$ the corresponding elements, after transformation by an element of the structure group, should satisfy $g'_1 g'_2 = g'_3$. The structure group and action is thus identified as the [[automorphism]] group and action for $G$, and the ''Lie group bundle'' is defined as the [[automorphism bundle]] for an $n$ dimensional $G$ over an $n$ dimensional base, along with an identity section.\n\nSince the Lie group bundle comes with a special identity section, $g_I(x)$, it has a particular automorphism bundle connection, the ''Lie group connection'', $\sf{A}$, similar to the [[Maurer-Cartan connection]], such that the identity section is horizontal,\n$$\n0 = \sf{\sna} g_I = \sf{d} g_I + \sha \sf{A} g_I - \sha g_I \sf{A}\n$$\nThis Lie group connection can be calculated in a few steps. Presuming the matrix of connection coefficients has an [[inverse|matrix inverse]], $A_B{}^i$, this set of vectors is identified as the adjoint action vectors of the [[Lie group geometry]],\n$$\n\sve{A_B} = \sve{\sxi^A_B} = \sha \sve{\sxi^L_B} - \sha \sve{\sxi^R_B}\n$$\nin which the left and right action vector field matrices may be calculated via the defining equations for their inverses, $\sf{\sxi_L^B} T_B = \slp \sf{d} g_I \srp g_I^-$ and $\sf{\sxi_R^B} T_B = g_I^- \slp \sf{d} g_I \srp$. The curvature of the Lie group connection vanishes. //(check that)// The identity section transforms under a gauge transformation to $g'_I = h g_I h^-$.\n\n//Will outer automorphisms make things more complicated?//
[<img[images/png/torsor.png]]A [[Lie group]] has elements $g(x) \sin G$ specified by the points, $x$ (corresponding to coordinates, $x^i$), of an $n$ dimensional [[manifold]]. This manifold naturally acquires a geometry corresponding to the structure of the Lie group.\n\nThe [[group action|group]] -- left, right, or adjoint -- of any group element, $h$, on all other elements of the group provides a set of [[autodiffeomorphism|diffeomorphism]]s, $\sleft\s{ \sph^L_h,\sph^R_h,\sph^A_h \sright\s}$, on the group manifold. However, only the adjoint action is a group [[automorphism]] as well as a manifold autodiffeomorphism. A group element, $h$, near the identity,\n$$h \ssimeq 1 + t v^B T_B$$\nis specified by a [[Lie algebra]] vector, $v=v^B T_B \sin Lie \s, G$, and a small parameter, $t$. These parameterized actions on the group correspond to [[flow]]s,\n$$\ne^{t \sve{\sxi} \sf{d}}\n$$\non the group manifold, with each flow corresponding to a vector field generator, $\sve{\sxi}$, over the manifold. For example, for the right action,\n\sbegin{eqnarray}\nR_h g &=& g h = g(\sph^R_h(x)) \s\s\n&\ssimeq& g + t v^B g T_B = g + t v^B \slp \sxi^R_B \srp^i \spa_i g(x)\n\send{eqnarray}\nSo, for each group action, $\sleft\s{ L, R, A \sright\s}$, and each Lie algebra generator, $T_B$, there is a corresponding vector field,\n\sbegin{eqnarray}\nT_B g &=& {\scal L}_{\sve{\sxi^L}} g = \sve{\sxi^L_B} \sf{d} g = \slp \sxi^L_B \srp^i \spa_i g(x) \s\s\ng T_B &=& {\scal L}_{\sve{\sxi^R}} g = \sve{\sxi^R_B} \sf{d} g = \slp \sxi^R_B \srp^i \spa_i g(x) \s\s\n\sha T_B g - \sha g T_B &=& {\scal L}_{\sve{\sxi^A}} g = \sve{\sxi^A_B} \sf{d} g = \slp \sxi^A_B \srp^i \spa_i g(x)\n\send{eqnarray}\nwhich acts via the [[Lie derivative]] and corresponds to a flow on the group manifold. The action of the Lie algebra generators and the flow by the corresponding right action vector field generators may be conceptually identified as the same, $T_B \ssim \sve{\sxi^R_B}$. When working with a specific group representation and coordinatization, the vector field component matrices, $\slp \sxi_B \srp^i$, may be found explicitly by solving the above defining equations. Using the defining equations, the [[Lie bracket|Lie derivative]]s between vector field generators are:\n\sbegin{eqnarray}\n\slb \sve{\sxi^R_B}, \sve{\sxi^R_C} \srb_L &=& C_{BC}{}^D \sve{\sxi^R_D} \s\s\n\slb \sve{\sxi^L_B}, \sve{\sxi^L_C} \srb_L &=& - C_{BC}{}^D \sve{\sxi^L_D} \s\s\n\slb \sve{\sxi^L_B}, \sve{\sxi^R_C} \srb_L &=& 0\n\send{eqnarray}\nwith the same structure constants as from the Lie algebra brackets. Also note that\n$$\sve{\sxi^A_B} = \sha \sve{\sxi^L_B} - \sha \sve{\sxi^R_B}$$\nallows the Lie brackets of adjoint action vector fields with the others to be easily determined. To make things more confusing, the ''left action vector fields'', $\sve{\sxi^L_B}$, are ''[[right invariant]] vector fields'', while the ''right action vector fields'', $\sve{\sxi^R_B}$, are ''[[left invariant]] vector fields'',\n\sbegin{eqnarray}\nR_{h*} \sve{\sxi^L_B}(x) &=& \sve{\sxi^L_B}(\sPhi^R_h(x)) \s\s\nL_{h*} \sve{\sxi^R_B}(x) &=& \sve{\sxi^R_B}(\sPhi^L_h(x))\n\send{eqnarray}\nThese vector fields have [[1-form]] duals, the ''right invariant 1-forms'', $\sf{\sxi_L^B}=\sf{dx^i} \slp \sxi^L_i \srp^B$, satisfying $\sve{\sxi^L_A} \sf{\sxi_L^B} = \sde_A^B$ and\n$$\n\sf{\sxi_L^B} T_B = \slp \sf{d} g \srp g^-\n$$\nand the ''left invariant 1-forms'', $\sf{\sxi_R^B} = \sf{dx^i} \slp \sxi^R_i \srp^B$, satisfying $\sve{\sxi^R_A} \sf{\sxi_R^B} = \sde_A^B$ and\n$$\n\sf{\sxi_R^B} T_B = g^- \sf{d} g = \sf{\scal I}\n$$\nin which $\sf{\scal I}$ is the [[Maurer-Cartan form]] over the Lie group manifold. (Note that the $L$ and $R$ are just labels that can move around, and not [[indices]] to be summed over.)\n\nThe left invariant vector fields, along with their natural [[Lie algebra metric|Lie algebra]], provide a natural [[Lie group tangent bundle geometry]], including a frame, connection, and curvature. Note that on the ''Lie group geometry'', a manifold with a collection of special vector fields on it, all points are identical since the Lie brackets are the same between the vector fields at every point. In particular, there is no special ''identity point'' on the Lie group geometry -- a Lie group geometry is also called a //''torsor''//, //''G-torsor''//, or //''principal homogeneous space''//.
A [[metric]] for the [[tangent bundle]] over the [[Lie group]] manifold may be defined such that the [[left invariant]] vector fields of a [[Lie group geometry]] have the same scalar product as the [[Lie algebra]] generators using the [[Killing form]]:\n$$\n\slp \sve{\sxi^R_B}, \sve{\sxi^R_C} \srp = \slp T_B, T_C \srp = g_{BC} = C_{BD}{}^E C_{CE}{}^D\n$$\nThe Lie algebra metric, $g_{BC}$, may be made diagonal (like the [[Minkowski metric]] and Kronecker delta) by transforming the generators by a constant matrix (as long as $G$ is semi-[[simple]]), found via the methods of [[spectral decomposition|eigen]]. In this way, the left invariant vector fields are identified as the set of [[orthonormal basis vector fields|frame]] on the Lie group manifold, $\sve{e_B} = \sve{\sxi^R_B}$, and the left invariant 1-form coefficients,\n\sbegin{eqnarray}\n\slp e_i \srp^B &=& \slp \sxi^R_i \srp^B = {\scal I}_i{}^B \s\s\n\sf{e^B} &=& \sf{\sxi_R^B} = \sf{{\scal I}^B} \n\send{eqnarray}\nare the coefficients of the frame 1-form and the [[Maurer-Cartan form]], $\sf{\scal I} = g^- \sf{d} g$. The resulting metric on the manifold is\n$$\ng_{ij} = \slp e_i \srp^B \slp e_j \srp^C g_{BC} \n$$\nThe right invariant vector fields, $\sve{\sxi_B} = \sve{\sxi^L_B}$, are [[Killing vector]]s for this Lie group geometry, since\n$$\n{\scal L}_{\sve{\sxi_B}} \sve{e_C} = \slb \sve{\sxi^L_B}, \sve{\sxi^R_C} \srb_L = 0\n$$\nAnd, since\n$$\n{\scal L}_{\sve{e_B}} \sve{e_C} = \slb \sve{\sxi^R_B}, \sve{\sxi^R_C} \srb_L = C_{BC}{}^A \sve{e_A}\n$$\nthe left invariant vector fields (the orthonormal basis vectors) are also Killing, by the [[Killing form]] identity, $\slp B_B \srp_C{}^A = C_{BC}{}^A = -C_B{}^A{}_C$.\n\nThe [[torsion]]less [[tangent bundle connection]], $\sf{w}{}^A{}_B$, for the Lie group manifold may be found by solving [[Cartan's equation]],\n$$\n0 = \sf{d} \sf{e^C} + \sf{w}^C{}_B \sf{e^B} \n$$\nWe can cheat a little by seeing that, since $\sf{e^B}=\sf{{\scal I}^B}$, this is the same as the Maurer-Cartan equation,\n$$\n0 = \sf{d} \sf{{\scal I}^C} + \sha \sf{{\scal I}^A} \sf{{\scal I}^B} C_{AB}{}^C\n$$\ngiving the ''Lie group tangent bundle connection'',\n$$\n\sf{w}^C{}_B = \sha \sf{e^A} C_{AB}{}^C = - \sha \sf{e^A} C_A{}^C{}_B\n$$\nwith the indices of the structure constants raised and lowered by the diagonal matrices, $g^{AB}$ and $g_{AB}$, and using the Killing form identity. The connection coefficients directly relate to the structure constants, $w_{AC}{}^B = - \sha C_{AC}{}^B$. Alternatively, a connection with torsion could be defined if desired.\n\nUsing one of the [[Killing vector identities]], the covariant derivative of any of the right invariant vector fields or some combination, $\sve{\sxi} = \sxi^B \sve{\sxi^L_B}$, along itself vanishes\n$$\n\sve{\sxi} \sf{\sna} \sve{\sxi} = \sve{\sxi} \sf{e^A} \sve{e_C} \slp \sxi^B \slp B_B \srp_A{}^C - \sxi^B w_{B A}{}^C \srp\n= \sfr{3}{2} \sxi^A \sxi^B \sve{e_C} C_{BA}{}^C \n= 0\n$$\nThis implies the integral curves of the [[flow]]s along any right invariant field are [[geodesic]]s. By another Killing vector identity, all the right invariant vector fields are constant length. \n\nThe [[Riemann curvature]] for the Lie group tangent bundle connection is\n\sbegin{eqnarray}\n\sff{R}^A{}_B &=& \sf{d} \sf{w}^A{}_B + \sf{w}^A{}_C \sf{w}^C{}_B \s\s\n&=& - \sha \slp \sf{d} \sf{e^C} \srp C^A{}_B{}_C + \sf{w}^A{}_C \sf{w}^C{}_B \s\s\n&=& \sha \slp \sf{w}^C{}_D \sf{e^D} \srp C^A{}_B{}_C + \sf{w}^A{}_C \sf{w}^C{}_B \s\s\n&=& \sfr{1}{4} \sf{e^F} \sf{e^D} \slp - C^C{}_{DF} C^A{}_{BC} + C^A{}_{CF} C^C{}_{BD} \srp \s\s\n&=& - \sfr{1}{4} \sf{e^F} \sf{e^D} C_{BCF} C_D{}^{AC}\n\send{eqnarray}\nusing the Jacobi identity. The [[Ricci curvature]] is\n$$\n\sf{R}{}_B = \sve{e_A} \sff{R}^A{}_B = - \sfr{1}{4} \sve{e_A} \sf{e^F} \sf{e^D} C_{BCF} C_D{}^{AC}\n= - \sfr{1}{4} \sf{e^D} C_{BCA} C_D{}^{AC} = - \sfr{1}{4} \sf{e^D} g_{BD} = - \sfr{1}{4} \sf{e}{}_B \n$$\nshowing that a Lie group geometry is an [[Einstein space|Einstein's equation]]. The [[curvature scalar]] is $R = \sve{e^B} \sf{R}{}_B = - \sfr{1}{4} n$.\n\nThe [[volume form]] over the Lie group manifold is the ''Haar measure'',\n$$\n\sf{e^1} \sdots \sf{e^n} = \snf{d^n x} \sleft| e \sright|\n$$
The complete list of real, [[simple]], compact, connected [[Lie group]]s was completed around 1890. They fall into four infinite families (the classical Lie groups) and five exceptional groups. Sorted by rank, $r$, they are:\n| !rank | !group | !a.k.a. | !dim | !name |\n| $r$ | $A_r$ | $SU(r+1)$ | $r(r+2)$ | [[special unitary group]] |\n| $r$ | $B_r$ | $SO(2r+1)$ | $r(2r+1)$ | odd [[special orthogonal group]] |\n| $r$ | $C_r$ | $Sp(2r)$ | $r(2r+1)$ | symplectic group |\n| $r>2$ | $D_r$ | $SO(2r)$ | $r(2r-1)$ | even [[special orthogonal group]] |\n| $2$ | $G_2$ | | $14$ | [[G2]] |\n| $4$ | $F_4$ | | $52$ | [[F4]] |\n| $6$ | $E_6$ | | $78$ | [[E6]] |\n| $7$ | $E_7$ | | $133$ | E7 |\n| $8$ | $E_8$ | | $248$ | [[E8]] |\n
A ''Lieform'' is a ''[[Lie algebra]] valued [[differential form]]'', having a single form grade, $p$. In terms of [[coordinate basis forms]] and Lie algebra generators, an arbitrary Lieform may be written as\n$$\n\snf{A} = \sf{dx^i} \sdots \sf{dx^k} \sfr{1}{p!} A_{i \sdots k}{}^B T_B \sin \snf{\srm Lie}(G)\n$$\nThe basis forms and Lie algebra generators act in different algebras. By convention, the form basis elements will be collected on the left and the Lie algebra generators on the right. The most common type of Lie form is a ''Lie algebra valued 1-form'', $\sf{A} = \sf{dx^i} A_i{}^B T_B \sin \sf{\srm Lie}(G)$.\n\nThe most common operation between Lieforms is the graded [[commutator]], equivalent to the graded [[Lie algebra bracket|Lie algebra]],\n$$\n\slb \snf{A}, \snf{B} \srb = \snf{A^C} \snf{B^D} \slb T_C, T_D \srb = \snf{A^C} \snf{B^D} C_{CD}{}^E T_E = \snf{A} \snf{B} - \slp -1 \srp^{pq} \snf{B} \snf{A}\n$$\nwhich produces a grade $(pq)$ Lieform from the bracket of grade $p$ and $q$ Lieforms.
Link to notes, such as [[Horizontal Rule]].\n{{{\nLink to notes, such as [[Horizontal Rule]].\n}}}\nLink to [[external sites|http://www.osmosoft.com]] or [[ordinary notes|Horizontal Rule]] with ordinary words,\nwithout the messiness of the full URL appearing.\n{{{\nLink to [[external sites|http://www.osmosoft.com]] or [[ordinary notes|Horizontal Rule]] with ordinary words,\nwithout the messiness of the full URL appearing.\n}}}\nOr just type out http://www.osmosoft.com and it will be automatically linkified.
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[[Loops '07|http://www.matmor.unam.mx/eventos/loops07/]] was held in Morelia, Mexico. My [[talk for Loops 07]] has links to my slides and the accompanying audio.\n\nHere are some talks I went to, and some personal impressions (these are just brief notes to myself, please don't take them too seriously)\n*Monday\n**[[Lucien Hardy|http://www.perimeterinstitute.ca/index.phpindex.php?option=com_content&task=view&id=30&Itemid=7&view_directory=1&pi=1078]]${}^*$, The causaloid formalism: a tentative framework for quantum gravity\n***Compression of measurement data\n***Obtain probabilistic distribution over temporal orderings of measurements maybe?\n**[[Rafael Sorkin|http://www.phy.syr.edu/~sorkin/]], Quantum reality and anhomomorphic logic\n***Wants to use a "quantum" version of probability by discarding preclusion or inference rules\n****discarding logical "and" (multiplication) and/or discarding logical "or" (addition)\n***Talked to him about complex probability distributions over paths.\n**[[John F. Donoghue|http://www.fqxi.org/aw-donoghue.html]]${}^*$, Effective field theory and quantum general relativity\n***He argued that the Effective Field Theory of gravity could be used to perturb around a Newtonian potential to get the classical (GR) corrections and the quantum GR correction proportional to $h$ -- in two different ways.\n***Quantum GR will have to reproduce this.\n**[[Rodolfo Gambini|http://www.fqxi.org/aw-pullin.html]]${}^*$, Relational physics with real rods and clocks, (A)\n***Works with Jorge Pullin\n***Err, it was hard to understand what he was saying, and his talk was kind of scattered.\n**[[Johannes Tambornino|http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=4735]], Taming observables in GR: A perturbative approach, (B)\n***Tries to quantize symmetry reduced theory the right way.\n**[[Frederic P. Schuller|http://www.nucleares.unam.mx/~f.p.schuller/]], Area metric gravity, (B)\n***Reformulate everything in terms of $G_{[ab][cd]}$ "area metric."\n**[[Merced Montesinos|http://www.fis.cinvestav.mx/~geogravi/gr_mphysics/Nuevos/faculty.html]], Cartan's equations define a topological field theory, (B)\n***Main idea: include Riemann curvature and Torsion as independent degrees of freedom. Them put extra "topological" terms, like Euler characteristic and Pontryagin characteristic, in the action to ensure $\sff{R}$ and $\sff{T}$ satisfy Cartan equations.\n***Possibly related paper: http://arxiv.org/abs/gr-qc/0603076\n**Marat Reyes, Generalized path dependent representations for gauge theories, (B)\n****Rats, I missed this one.\n**Yuya Sasai, Braided quantum field theories and their symmetries, (A)\n***Err, very hard to tell what the heck he was doing since his English wasn't very good.\n**[[Garrett Lisi|http://sifter.org/~aglisi/]]${}^*$, Deferential Geometry, (A)\n*** :) My talk went very well.\n***"This is very interesting." -- L.S. "It's not bullshit." -- S.H.\n*Tuesday\n**Thomas Thiemann, Elements of Loop Quantum Gravity\n***Has a new book coming out in September -- should be good.\n***Talk was packed full of good stuff.\n***Start wit Palatini formulation, combine constraints into master $M$, convert variables to $su(2)$ holonomies and fluxes, build solutions using coherent states, minimize expectation value of master constraint, $M$, rather than making it strictly $0$.\n**Abhay Ashtekar, LQG: Lessons from models\n***Symmetry reduced models. Spherically symmetric black holes, 1+1.\n**[[Carlo Rovelli]]${}^*$, Vertex amplitude and propagator in loop quantum gravity\n***Work in $so(4)$. Barrett/Crane model has problem with off diagonal (vertex) terms -- B.C. doesn't give intertwiners in perturbation calc. so it's wrong. Need to use GFT to get better model.\n**Jan Ambjørn, 4d quantum gravity as a sum over histories\n***His talk went long and he got cut off.\n***I'm kind of unimpressed with CDT -- it's an odd approximation that forces plausible numerical results.\n**Dan Christensen, Computations involving spin networks, spin foams, quantum gravity and lattice gauge theory (B)\n***Kind of odd: gets positive real values for QM amplitude of spinfoam areas.\n**[[Wade Cherrington|http://arxiv.org/abs/0705.2629]], Numerical Spin Foam Computation of Pure Yang-Mills Theory (B)\n***YM on a lattice, in an odd way. Holonomies. Expand amplitudes into group characters.\n**Seth Major\n***Kodama state, approximating flat spacetime, from exponentiating Chern-Simons action.\n**John Swain, Spin Networks and Simplicial Quantum Gravity (B)\n***Cool guy. 3+1 D Regge -> simplicial, areas and angles.\n*Wednesday\n**Moshe Rozali, Background Independence in String Theory\n***No action for $g$ in string theory. Have to choose a metric and perturb around that.\n**Klaus Fredenhagen, General covariance in quantum field theory and the background problem in perturbative quantum gravity\n***Sited [[new Stefan Hollands paper|http://arxiv.org/abs/0705.3340]] on BRST and Yang-Mills in curved spacetime.\n***Path integral is not covariant when you define it in detail.\n***Admissible embeddings are ones that preserve causal curves.\n**Alejandro Perez, Regulator dependence in quantum gravity and non perturvative renormalizability: possible new perspectives\n***missed it to talk to S.H.\n**Martin Reuter, Asymptotically safe quantum gravity and cosmology\n***Missed it. :( I came down with flu. But watched it online.\n***Using truncated action, with running coupling constants $G(k)=g(k)/k^2$ and $\sLa(k)=k^2 \sla(k)$, there is a Gaussian fixed point at $k \sto 0$ and a non-Gaussian fixed point, $\sbig( g(\sinfty), \sla(\sinfty) \sbig) \sto {\srm const}$.\n***Using more action terms, with many running parameters, the EH action appears to emerge as the unique fixed point at high energy. Just comes from fields and symmetries.\n***Large cosmo constant at high energy can drive inflation, without inflaton.\n***His recent [[paper|http://arxiv.org/abs/0708.1317]] on this just came out.\n**No parallel sessions\n**Quarantined myself because of flu.\n*Thursday\n**Feeling better -- think I'll venture out of hotel room again.\n**Daniele Oriti, Group field theory: spacetime from quantum discreteness to an emergent continuum\n***G's live at vertices. Graphically beautiful slides -- equations and figures.\n***My guess on what GFT is: start with many copies of a Lie gp, G. State is a collection of reps for each gp. When these states "match" these gps are linked, giving a spin network approximating a base manifold for a principal G-bundle.\n****No, I talked with Thomas Thiemann and he said this isn't how it works. There is only a G for each dimension of the spacetime (e.g. four, or three for just space), $(D-1)=4?$\n**Artem Starodubtsev, Some physical results from spinfoam models\n***According to a discussion at Physics Forums, this talk will actually be about BF gravity.\n***He didn't show up to the conference. Had some travel holdup in the US.\n**Martin Bojowald, Loop quantum cosmology and effective theory\n***Skipped it to talk with L and S.\n**Jorge Pullin${}^*$, Uniform discretizations and spherically symmetric loop quantum gravity\n**Sundance Bilson-Thompson, Braids, loops, and the emergence of the standard model (B)\n***Leader of the Braidy Bunch.\n***Poor guy had a plague of audio problems.\n***gluons are two stacked braids, with a plus and minus charge.\n**Jonathan Hackett, ribbon networks, (B)\n***reduced link invariants\n**Yidun Wan, ribbons, (B)\n***nice circle diagrams for tetra, and links could describe topology, but why are they braided?\n**Jonathan Engle, cosmology, (A)\n**Ileana Naish-Guzman, On the regularizability of the Ponzano-Regge model, (A)\n***Works with John Barrett. Soft Brittish accent. Twisted cohomology groups over a cell complex.\n**James Ryan, Aspects of Group Field Theory (A)\n***Excellent intro to GFT. with group su(1,1) or su(2)\n**Winston J. Fairbairn, Quantization of string-like sources coupled to BF theory: transition amplitudes and topological invariance, (A)\n***(d-3) branes.\n*Friday\n**Fotini Markopoulou${}^*$, Quantum gravity and emergent locality\n***Quantum Graphity\n**Lee Smolin${}^*$, Chiral excitations of quantum geometries as a possible route to unification\n***"New theories should include surprises."\n***Ribbons as framed graphs, consistent with a cosmological constant.\n***We don't really know the relationship between spin network Hamiltonian constraint and spin foam with evolution moves.\n***Working with Sundance's braids \n**Sabine Hossenfelder, Phenomenological Quantum Gravity\n***"Top down inspired bottom up approaches"\n***Pessimistic Freeman Dyson quote on QFT/GR independence.\n***Colider constraints on KK models. (Black hole production)\n***Zero black holes in standard setup (I'm not sure that's true).\n***Minimal length scale as UV cutoff. (Doesn't this relate to discrete deSitter modes?)\n**William Donnelly, Entanglement Entropy in Loop Quantum Gravity (B)\n***Black hole entropy. Spin networks as the boundary/horizon in two part spinfoam/spacetime.\n**Olaf Dreyer${}^*$, Internal Relativity: A progress report (A)\n***Start with something like Ising model on lattice with Lorentz group as internal structure.\n**Florian Girelli, 2-Groups and Topological Action (changed talk title!) (A)\n***parallel transp of strings. defined 2-group, 2-Lie algebra, 2-principal bundle. Need 2-Peter-Weil theorem.\n***one of not many DSR talks...\n**Roberto Pereira, The loop-quantum-gravity vertex-amplitude (A)\n***Works with Rovelli.\n**Emanuele Alesci, Graviton propagator: the non diagonal terms (A)\n***Works with Rovelli. (I get the impression this guy does the grind work of the calculations.) on {10J} propogator.\n***Had to make up a term so that intertwiners would be involved when calculating the off diagonal part of the propogator.\n**[[Isabeau Prémont-Schwarz|http://arxiv.org/abs/hep-th/0508168]], Quantum Evolution in an Expanding Hilbert Space (Talk title at last minute) (A)\n***Is this just using a non-square $U$ for evolution?\n*Saturday\n**John Stachel, Projective and Conformal Structures in General Relativity\n***Older guy. Einstein biographer.\n***Cecile deWitt has new book out, "Functional Integration"\n***Affine space, affine connection and curvature. Need to break geometric variables into smaller pieces before quantizing.\n***(His slides were messed up, missing most of the math symbols, which kinda wrecked it)\n**Michael Reisenberger, Canonical gravity with free null initial data\n***Free (unconstrained) gravitational initial data variables are known for initial hypersurfaces consisting of two intersecting null hypersurfaces. Recently the Poisson bracket on functions of such data has been obtained. This opens the prospect of a constraint free canonical formulation of general relativity.\n**David Rideout, Can the supercomputer provide new insights into quantum gravity?\n***Cactus. Causal sets and spin networks.\n\n${}^*$ FQXi member -- might see the same talk in Iceland\n(A,B) denotes which parallel session\n\n\nDuring last two hours, questions were asked of the Plenary speakers, based on a book that had been passed around the audience. Carlo Rovelli moderated.\n#You're at Loops '17, presenting your talk. Presuming your research program has been fully successful, describe your talk. What likelihood do you estimate for this happening? (This question was saved until the end, so speakers could prepare.)\n#Topology change in QG?\n##Abhay: not possible in canonical framework, but possible in LQG spin network.\n#Does QG say anything about QM? Do we need a deeper framework, or a new interpretation, or different GR?\n##Lucien: Need new math.\n##John S: Need process QMI. (use paths)\n##Thomas Thiemann: No. (just use conservative approach)\n##Bianca: Relational framework\n##John D: Path integrals OK, need to change GR.\n#QG has fluctuating causal structure -- would this have any measurable effect?\n##Sabine: photon spreading\n##Michael: Matter sees only one spacetime. (yep)\n#What is finite in spin foam models?\n##Alejandro Perez: There are ambiguities in the theory.\n#What would be a graph theory of nature? Spinfoam fundamental, or embedded in a manifold?\n##Thomas: Topology change, so not embedded. He thinks spinfoam.\n##Sundance: Braids.\n#The first question was answered last:\n##Lucien: Causaloids.\n##John D: Background dependent, GR + SM emergent from spin substrate, with perturbative breaking of general covariance. 60%\n##Thomas T: Complete description of QGR, as well as experiments done by himself. (ha) 5%\n##Ashtekar: Resolution of all ambiguities in LQG, and establishment of all relationships between branches.\n##A Perez: Thiemann was wrong. (ha!) "We have to make the road by walking."\n##M Reuter: Same underlying theory, with non-Gaussian fixed point. 20%\n##Daniele:\n###100% - statistical GFT, with low temperature equilibrium phase\n###80% - derive effective dynamics from GFT\n###50% - one model singled out as successful\n##Sabine: her talk would have the same title (ha) Measurements to support one model or another.\n###New, unexpected data. 50%\n###She'd find a permanent position. (ha)\n##John S: Quantized Conformal Structures. But odds were low he'd be with it in 10 years. :(\n##David Rideout: Causel sets give QM and GR. Probability: epsilon. (ha)\n##Carlo Rovelli: closing words. "Let each of a hundred flowers bloom" -- quote from Michael's talk.\n\nRandom notes:\n*Many people used [[Beamer|http://latex-beamer.sourceforge.net/]] for their slides.\n\nLoops '08 will be in Nottingham, England, in July (unless I misheard?)\nLoops '09 probably in Beijing (or maybe P.I.)\n\nI met a LOT of people\n**PI grad students\n***Chanda\n***Jean Christian Boileau\n***Jonathan Hackett (social guy at the end of table)\n***Joel Brownstein (inflation guy)\n***Bruno Hartmann (skinny sharp german guy with glasses, works with Thiemann)\n***Isabeau Prémont-Schwarz (funny german guy with glasses) \n***Sean (ultimate frisbee guy)\n***Cecilia\n***Alejandro Satz (http://realityconditions.blogspot.com/)\n***Joel Brownstein (inflation)\n***William Donnely (http://williamdonnelly.blogspot.com/)\n**other PI people\n***Lucien Hardy\n***Fotini\n***Olaf?\n***Lee Smolin\n***Sabine\n***Sundance\n***Hans Westman (big baldish german with glasses)\n***Rafael Sorkin\n**misc\n***John Stachel\n***Daniele Oriti\n****Alejandro Perez (kind of wild looking)\n****Florian Gireli\n***John Swain\n***Michael Reisenberger\n***Jonathon Engle\n***Wayne Bomstad (grad student, works with John Klauder in Florida, said he admires for being well rounded)\n
The generalized ''special orthochronous Lorentz group'', $\smbox{SO}{}^+(1,n-1)$, is a [[Lie group]] composed of [[Lorentz rotation]]s in $n$ dimensions, including one of time. The generalized ''Lorentz group'', $\smbox{O}(1,n-1)$, has four disconnected components &mdash; two are special ($\smbox{S}$) and two are orthochronous (${}^+$). The special orthochronous Lorentz group is the only group component containing the identity.\n\nThe Wikipedia article is quite thorough:\nhttp://en.wikipedia.org/wiki/Lorentz_group
A rotation is a smoothly operating linear transformation acting on vectors that leaves the scalar product between vectors invariant. ("Vectors" in this case may stand for [[tangent vector]]s, [[1-form]]s, [[Clifford vectors|Clifford element]], or any other appropriately [[indexed|indices]] object.) For example, two vectors with components $u^\sal$ and $v^\sal$ may have the scalar product, $u^\sal \set_{\sal \sbe} v^\sbe$. A linear transformation of the vector components by the ''Lorentz matrix'' maps the vector components to\n\sbegin{eqnarray}\n{u'}^\sal &=& L^\sal {}_\sbe u^\sbe\s\s\n{v'}^\sal &=& L^\sal {}_\sbe v^\sbe\n\send{eqnarray}\nwhich must preserves the scalar product,\n\s[ u^\sal \set_{\sal \sbe} v^\sbe = {u'}^\sal \set_{\sal \sbe} {v'}^\sbe = L^\sal{}_\sbe u^\sbe \set_{\sal \sga} L^\sga{}_\sde v^\sde \s]\nSo the Lorentz matrix must be ''orthogonal'',\n\s[ L^\sal{}_\sbe \set_{\sal \sga} L^\sga{}_\sde = \set_{\sbe \sde} \s]\nor, with the [[Minkowski metric]] raising and lowering indices, $L_{\sga \sbe} L^{\sga \sde} = \sde_\sbe^\sde$ or $L^T L = I$. A transformation satisfying this restriction is a ''Lorentz transformation''. But such a transformation could also include reflections, and would then not be "smoothly operating" (not connected to the identity). To exclude this possibility, $L$ is restricted to have positive determinant, $|L|=1$, in which case it is called "special" or "proper", and $L$ is also restricted to preserver the direction of time (no reflection of the $0$ components), in which case it is called "orthochronous". A special orthochronous Lorentz transformation is called a ''Lorentz rotation''. It is "smoothly operating" or "connected to the identity" in that it may be built up by many small rotations,\n\s[ L = \slim_{N \sto \sinfty} \slp I + \sfr{1}{N} l \srp^N \s]\nin which $l$ is an antisymmetric matrix, $l_{\sal \sbe} = l_{\slb \sal \sbe \srb}$. A Lorentz transformation built this way is special and orthochronous.\n\nThe group of Lorentz rotations forms the [[special orthochronous Lorentz group|Lorentz group]].\n\nAlthough the matrix representation is more standard, rotations are better described and carried out as [[Clifford rotation]]s.
Macros let you write notes containing more exotic objects than just text. Macros may be added as plugins. If so, they should be tagged<<tag plugin>>, and described in [[Configuration]].\n!These are some of the built-in macros:\nToday is <<today>>\n{{{\nToday is <<today>>\n}}}\nClick on <<tag editing>> to popup all notes tagged "editing".\n{{{\nClick on <<tag editing>> to popup all notes tagged "editing".\n}}}\nTransclude one note into another via\n<<note 'Horizontal Rule'>>\n{{{\nTransclude one note into another via\n<<note 'Horizontal Rule'>>\n}}}\n//There is no protection against inadvertently setting up endless loops. And this may have problems if the transcluded note isn't loaded.//\nSlider: <<slider chkTestSlider 'Horizontal Rule' 'press me»' "Click here to see the Horizontal Rule slide out">>\n{{{\nSlider: <<slider chkTestSlider 'Horizontal Rule' 'press me»' "Click here to see the Horizontal Rule slide out">>\n}}}\nThe slider parameters are:\n* cookie name to be used to save the state of the slider\n* name of the note to include in the slider\n* title text of the slider\n* tooltip text of the slider\n
<<note HideTags>>\sbegin{eqnarray}\n\slp D \s!\s!\s!\s! / + \sph \srp \sud{\sps} &=& \sga^\smu \slp e_\smu\srp^a \slp \spa_a + \sfr{1}{4} \som_a^{\sp{a}\snu\srh} \sga_{\snu\srh} + B,W,G_a^{\sp{a}A} T_A \srp \sud{\sps} + \sph \s, \sud{\sps} \s\s\n&=& \sga^\smu \slp e_\smu\srp^a \slp \spa_a + \sfr{1}{4} \som_a^{\sp{a}\snu\srh} \sga_{\snu\srh}\n+ \sfr{1}{4} \slp e_a \srp^\snu \sga_\snu \sph\n+ B,W,G_a^{\sp{a}A} T_A \srp \sud{\sps}\n\send{eqnarray}\n| $\s; \sga_\smu \s;$ |[[Clifford basis vectors]] for [[Cl(1,3)]] |\n| $\s; \sga_{\smu\snu} = \sga_\smu \sga_\snu \s;$ |[[Clifford basis bivectors|Clifford basis elements]] |\n| $\s; T_A \s;$ |[[Lie algebra]] basis elements (//generators//) |\n| $\s; ( e_\smu )^a \s;$ |[[orthonormal basis vector|frame]] components (//frame, vierbein//) |\n| $\s; \som_a^{\sp{a}\snu\srh} \s;$ |[[spin connection]] components |\n| $\s; B_a^{\sp{a}A}, W_a^{\sp{a}A}, G_a^{\sp{a}A} \s;$ |Yang-Mills [[gauge field|principal bundle]] components (//connections//) |\n| $\s; \sph \s;$ |Higgs scalar field multiplet |\n| $\s; \sud{\sps} \s;$ |[[Grassmann|Grassmann number]] valued [[spinor]] field multiplet |\n$$\n\sbegin{array}{rcl}\n{\srm Clifford \s; algebra} \s!\s!&\s!\s! \slongleftrightarrow \s!\s!&\s!\s! {\srm Lie \s; algebra}^{\sphantom{(}} \s\s\n\ssearrow \s!\s!\s!\s!\s!\s! \snwarrow \s!\s!&\s!\s! \s!\s!&\s!\s! \sswarrow \s!\s!\s!\s!\s!\s! \snearrow \s\s\n& {\srm Matrices} &\n\send{array}\n$$
<<note HideTags>>First $\smathbb{C}(8\stimes8)$ quadrant of a $\smathbb{C}(16\stimes16)$ [[chiral]] rep of $Cl(1,7)$ bivectors:\n\sbegin{eqnarray}\n\sf{H^+} &=& \sbig( \sha \sf{w} + \sfr{1}{4} \sf{e} \sph + \sf{W} + \sf{B} \sbig)^+ \s\s\n&=& \sfr{1}{4} \sf{w^{\smu\snu}} \sga^+_{\smu\snu} + \sfr{1}{4} \sf{e^\smu} \sph^\sph \sga^+_{\smu\sph} \s\s\n&& - \sf{W^\spi} \sfr{1}{4} \sbig( \sep_{\spi (\sph-4)(\sps-4)} \sga^+_{\sph \sps} + \sga^+_{(\spi+4)8} \sbig) \n+ \sf{B} \sha \sbig( \sga^+_{78} - \sga^+_{56} \sbig)_{\sphantom{\sBig(}} \s\s\n\n&=&\n\slb \sbegin{array}{cccc} \n\sha \sf{\som_L} \s!+\s! i \sf{W^3} & i \sf{W^1} \s!+\s! \sf{W^2} & - \sfr{1}{4} \sf{e_R} \sph_0^* & \sfr{1}{4} \sf{e_R} \sph_+ \s\s\ni \sf{W^1} \s!-\s! \sf{W^2} & \sha \sf{\som_L} \s!-\s! i \sf{W^3} & \sfr{1}{4} \sf{e_R} \sph_+^* & \sfr{1}{4} \sf{e_R} \sph_0 \s\s\n-\sfr{1}{4} \sf{e_L} \sph_0 & \sfr{1}{4} \sf{e_L} \sph_+ & \sha \sf{\som_R} \s!+\s! i \sf{B} & \s\s\n\sfr{1}{4} \sf{e_L} \sph_+^* & \sfr{1}{4} \sf{e_L} \sph_0^* & & \sha \sf{\som_R} \s!-\s! i \sf{B}\n\send{array} \srb^{\sphantom{\sbig(}}\n\send{eqnarray}\n\nwith $\sph_0 = (\sph^7 + i \sph^8)$ abd $\sph_+ = (-\sph^5 + i \sph^6)$.\n\n
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The Maurer-Cartan connection is like the [[Lie group bundle]] connection, but for a [[principal bundle]]. The relevant ''principal Lie group bundle'' is a fiber bundle with $n$ dimensional base manifold, $M$, and $n$ dimensional Lie group, $G$, as typical fiber and structure group acting on the fiber from the right -- it is a principal bundle. The structure group action (and maybe the structure group) differs from that of a Lie group bundle. Like for a Lie group bundle, a bijective identity section, $g_I(x)$, maps base manifold points to group/fiber elements. The ''Maurer-Cartan connection'', $\sf{M}$, is the connection such that the covariant derivative of the identity section is horizontal,\n$$\n0 = \sf{\sna} g_I = \sf{d} g_I - g_I \sf{M}\n$$\nwhich gives\n$$\n\sf{M} = g_I^- \sf{d} g_I\n$$\n\nThe [[exterior derivative]] of an inverse element comes from\n$$\n0 = \sf{d} \slp g g^- \srp = g \sf{d} g^- + \slp \sf{d} g \srp g^-\n$$\nand is $\sf{d} g^- = - g^- \slp \sf{d} g \srp g^-$. So the Maurer-Cartan connection satisfies the ''Maurer-Cartan equation'',\n$$\n\sf{d} \sf{M} = \sf{d} \slp g_I^- \slp \sf{d} g_I \srp \srp = \slp \sf{d} g_I^- \srp \slp \sf{d} g_I \srp = - \sf{M} \sf{M} = -\sf{M} \stimes \sf{M} \n$$\nand the related principal bundle curvature vanishes,\n$$\n\sff{F} = \sf{d} \sf{M} + \sf{M} \sf{M} = 0\n$$\n\nInterestingly, the Maurer-Cartan connection may also be related to the vielbein arising from [[Lie group geometry]],\n$$\n\sf{M} = \sf{\sxi_R^B} T_B = g_I^- \sf{d} g_I = \sf{e^B} T_B \n$$\nIn this way, the Maurer-Cartan connection is a frame as well as a connection for a Lie group.\n\nTo reiterate: The Maurer-Cartan connection is a connection for a $2n$ dimensional principal bundle. This is just something new to try playing with. It's not the [[Maurer-Cartan form]], which is a 1-form over the $n$ dimensional Lie group manifold.
The ''Maurer-Cartan form'' (//''M-C form''//) is a particular [[Lieform]] field, $\sf{\scal I}(x) = \sf{dx^i} {\scal I}_i{}^A T_A$, defined over a [[Lie group]] manifold. It arises in [[Lie group geometry]] from the equation for the right action vector field,\n$$\n\sve{\sxi_A^R} \sf{d} g = g T_A\n$$\nThe inverse matrix of this vector field's components gives the components of the Maurer-Cartan form, ${\scal I}_i{}^A = \slp \sxi^R_i \srp^A$, which are the same as the components of the natural Lie group geometry vielbein. However, unlike the vielbein, which is a set of 1-forms, the Maurer-Cartan form is Lie algebra valued. By playing with the above equation, at every manifold point, $x$, it equals the [[inverse]] of the group element corresponding to that point times the [[exterior derivative]] of the group element, \n$$\n\sf{\scal I} = g^- \sf{d} g\n$$\nThe components of the Maurer-Cartan form may be found by solving this equation, or equivalently by solving for the right action vector fields and inverting.\n\nThe M-C form is a sort of identity map from vectors on the Lie group manifold to Lie algebra elements, $\sf{\scal I} = \sf{{\scal I}^A} T_A = \sf{\sxi_R^A} T_A$ -- specifically, it maps right acting vector fields at a point to their corresponding Lie algebra element, $\sve{\sxi^R_A} \sf{\scal I} = T_A$. The M-C form provides an explicit isomorphism from vector valued fields to Lie algebra valued fields, $\sve{v} \sf{\scal I} = v \sin {\srm Lie}(G)$, and thus acts as the anchor of a [[Lie algebroid]]. If the equivalence between Lie algebra elements and their corresponding right acting vector fields is taken seriously, then the resulting ''Ehresmann-Maurer-Cartan [[vector valued form]]'' (//''E-M-C VVF''//), $\sf{\sve{\scal I}}$, is nothing but the [[identity projection|vector projection]] on the Lie group manifold:\n$$\n\sf{\sve{\scal I}} = \sf{\sxi_R^A} \sve{\sxi^R_A} = \sf{dx^i} \sve{\spa_i}\n$$ \nLooking at it in a weird way, the E-M-C VVF is the [[Ehresmann connection]] for a fiber bundle with the Lie group manifold as fiber and a single point as the base.\n\nThere is a manifold [[diffeomorphism]], $x \smapsto y = y_h(x)$, corresponding to any choice of right acting group element, $h \sin G$, according to $g(y_h(x)) = R_h g(x) = g(x) h$. The [[pullback]] of the M-C form under this diffeomorphism is\n$$\nR_h^* \sf{\scal I} = \sf{dx^i} \sfr{\spa y_h^j}{\spa x^i} {\scal I}_j{}^A(y_h(x)) T_A = \sf{dx^i} h^- g^-(x) \spa_i g(x) h = h^- \sf{\scal I}(x) h\n$$\nwhich is often taken as a defining property of the M-C form. Similarly, the pullbacks of the M-C form under the left and adjoint actions are $L_h^* \sf{\scal I} = h \sf{\scal I} h^-$ and $A_h^* \sf{\scal I} = h \sf{\scal I} h^-$. Note that this relates to the fact that, as the identity projection, the E-M-C VVF is invariant under any diffeomorphism, $\sphi^* \sf{\sve{\scal I}} = \sf{\sve{\scal I}}$. \n\nA formula for the exterior derivative of an inverse element, $\sf{d} g^- = - g^- \slp \sf{d} g \srp g^-$, comes from\n$$\n0 = \sf{d} \slp g g^- \srp = g \sf{d} g^- + \slp \sf{d} g \srp g^-\n$$\nSo the exterior derivative of the Maurer-Cartan form is\n$$\n\sf{d} \sf{\scal I} = \sf{d} \slp g_I^- \slp \sf{d} g_I \srp \srp = \slp \sf{d} g_I^- \srp \slp \sf{d} g_I \srp = - \sf{\scal I} \sf{\scal I} = -\sf{\scal I} \stimes \sf{\scal I} = - \sha \slb \sf{\scal I} , \sf{\scal I} \srb\n$$\nThis gives the ''Maurer-Cartan equation'',\n\sbegin{eqnarray}\n0 &=& \sf{d} \sf{\scal I} + \sha \slb \sf{\scal I}, \sf{\scal I} \srb = \sff{\scal F} \s\s\n0 &=& \sf{d} \sf{{\scal I}^C} + \sha \sf{{\scal I}^A} \sf{{\scal I}^B} C_{AB}{}^C\n\send{eqnarray}\nwhich gives vanishing [[curvature]] for the M-C form. Of course, the [[FuN curvature]] of the E-M-C VVF (which is the identity projection) also vanishes, $\sff{\sve{\scal F}} = - \sha \slb \sf{\sve{\scal I}}, \sf{\sve{\scal I}} \srb_L = 0$.
[>img[images/person/Max Tegmark.jpg]]Homepage: http://space.mit.edu/home/tegmark/index.html\n*Location: MIT\n\nSelected work:\n*[[The Mathematical Universe|papers/0704.0646v1.pdf]]\n**Computable Universe Hypothesis\n**Complexity based measure on space of possible mathematics
[>img[images/person/Michael Edwards.jpg]]\n*Location: Santa Cruz\n\nPersuaded me that the [[FuN derivative]] was worth thinking about in the context of [[Ehresmann connection]]s, and wrote a nice [[synopsis|papers/BRST2-6.pdf]].
The ''Minkowski metric'', $\set_{\smu \snu}$, is a $4 \stimes 4$ diagonal matrix with unit magnitude real entries. The choice of ''signature'' is somewhat arbitrary, and mostly a matter of taste. For positive time and negative space signature $(p=1, q=3)$, generally preferred by field theorists,\n\s[ \set_{\smu \snu}\n= \n\slb \sbegin{matrix}\n1& 0& 0& 0\s\s\n0& -1& 0& 0\s\s\n0& 0& -1& 0\s\s\n0& 0& 0& -1\n\send{matrix} \srb\n = \scases {\n 1&\stext{if $\smu=\snu=0$}\scr\n -1&\stext{if $\smu=\snu>0$}}\n= \set^{\smu \snu}\n\s]\nwhile for negative time and positive space signature $(p=3,q=1)$, generally preferred by relativists,\n\s[ \seta_{\smu \snu}\n= \n\slb \sbegin{matrix}\n-1& 0& 0& 0\s\s\n0& 1& 0& 0\s\s\n0& 0& 1& 0\s\s\n0& 0& 0& 1\n\send{matrix} \srb\n = \scases {\n -1&\stext{if $\smu=\snu=0$}\scr\n 1&\stext{if $\smu=\snu>0$}}\n= \set^{\smu \snu}\n\s]\n(This "matrix" notation is a bit sloppy, since $\set_{\smu \snu}$ is not really a matrix but just a collection of indexed coefficients.) All computations should be signature ambivalent. When they're not, the signature may be accommodated by including $\set_{00}= \spm 1$ in expressions.\n\nThe ''generalized Minkowski metric'' is $n \stimes n$ &mdash; accommodating extra spatial dimensions (of signature $-\set_{00}$).\n\nThe Minkowski metric may be used to raise or lower label [[indices]], such as in "$\sga^\sal = \set^{\sal \sbe} \sga_\sbe$" and "$v_\smu = \set_{\smu \snu} v^\snu$". The Minkowski metric with one index raised or lowered is just the Kronecker delta, $\set^\sal_\sbe = \set^{\sal \sga} \set_{\sga \sbe} = \sde^\sal_\sbe$.
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The ''Nieh-Yan density'' is an invariant closed 4-form,\n$$N = \sf{d} \slp \sf{e} \scdot \sff{T} \srp = \sff{T} \scdot \sff{T} - \sff{R} \scdot \sf{e} \sf{e}$$\nThis may make for an interesting KK action term...\n\nAnother invariant cloese 4-form is the ''Pontryagin density'',\n$$P = \sli \sff{R} \sff{R} \sri$$\n\nmentioned in http://arxiv.org/abs/gr-qc/0603134
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Creating numbered lists is simple.\n# Just add a pounds sign\n# at the beginning of a line.\n## If you want to create sub-lists\n## start the line with two pounds\n### And if you want yet another level\n### use three pounds\n# You can also do [[Bullet Points]]\n{{{\nCreating numbered lists is simple.\n# Just add a pounds sign\n# at the beginning of a line.\n## If you want to create sub-lists\n## start the line with two pounds\n### And if you want yet another level\n### use three pounds\n# You can also do [[Bullet Points]]\n}}}
*[[On a Covartiant Formulation of the Barbero-Immirzi Connection|papers/070134.pdf]]\n**New paper by [[Carlo Rovelli]] et. al. on a cleaner way of getting a $su(2)$ gravity connection from a $spin(4)$ connection.
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authors: [[Laurent Freidel]], J. Kowalski--Glikman, A. Starodubtsev\narxiv: http://arxiv.org/abs/gr-qc/0607014\nlocally: [[0607014|papers/0607014.pdf]]\nabstract:\n Since the work of Mac-Dowell-Mansouri it is well known that gravity can be written as a gauge theory for the de Sitter group. In this paper we consider the coupling of this theory to the simplest gauge invariant observables that is, Wilson lines. The dynamics of these Wilson lines is shown to reproduce exactly the dynamics of relativistic particles coupled to gravity, the gauge charges carried by Wilson lines being the mass and spin of the particles. Insertion of Wilson lines breaks in a controlled manner the diffeomorphism symmetry of the theory and the gauge degree of freedom are transmuted to particles degree of freedom. \n\n*Nice summation of BF.\n*Three topological terms: Euler, Pontryagin, Nieh-Yan\n*Interesting treatment of Imirzi parameter\n*Ahh, their main idea seems to be putting in a term and identifying it as a spinning particle. I guesss that's nice, but what's the big deal? Anyway, it's a cute way of putting in point particle matter, rather than QFT matter fields. The matter action is an integral along the parameterized path of the particle.\n*Path integral <--> Wilson line correspondence\n*matter as gravitational singularity.\n\nThis looks like the result you should get if you include the Dirac action and insert an arbitrarily boosted point particle solution for the Dirac field.\n
<<note HideTags>>@@display:block;text-align:center;[img[images/png/pati-salam table.png]]@@$$\n\sbig( SO(3,1) + 4\stimes4 + SU(2)_L + SU(2)_R \sbig) + \sbig( U(1) + SU(3) \sbig) \n$$
The three [[trace]]less, Hermitian, ''Pauli matrices'', $\ssi^P_A$, are\n$$\n\sbegin{array}{ccc}\n\ssigma_{1}^{P}=\sleft[\sbegin{array}{cc}\n0 & 1\s\s\n1 & 0\send{array}\sright] & \ssigma_{2}^{P}=\sleft[\sbegin{array}{cc}\n0 & -i\s\s\ni & 0\send{array}\sright] & \ssigma_{3}^{P}=\sleft[\sbegin{array}{cc}\n1 & 0\s\s\n0 & -1\send{array}\sright]\send{array}\n$$\nThe [[cross product|antisymmetric bracket]] of any two gives\n$$\n\ssi^P_A \stimes \ssi^P_B = \sha \slp \ssi^P_A \ssi^P_B - \ssi^P_B \ssi^P_A \srp = i \sep_{ABC} \ssi^P_C\n$$\nwith $\sep_{ABC}$ the [[permutation symbol]]. The symmetric product gives\n$$\n\ssi^P_A \scdot \ssi^P_B = \sha \slp \ssi^P_A \ssi^P_B + \ssi^P_B \ssi^P_A \srp = \sde_{AB} 1\n$$\nThe product of the three Pauli matrices is\n$$\n\ssi_1^P \ssi_2^P \ssi_3^P = \sleft[\sbegin{array}{cc}\ni & 0\s\s\n0 & i\send{array}\sright]\n= i 1\n$$\nwith the identity, $1 = \ssi_0^P$, often referred to as the (non-traceless) ''zero-eth Pauli matrix''.
<<note HideTags>>\n@@display:block;text-align:center;[img[images/png/standard model and gravity 2.png]]@@
[>img[images/person/Peter Michor.jpg]]Homepage: http://www.mat.univie.ac.at/~michor/\n*Location: Austria\n*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Michor_P/0/1/0/all/0/1\n\nBig on [[natural]]ness.\nI think I annoyed him with my questions.\n\nSelected work:\n*[[Topics in Differential Geometry|papers/Topics in Differential Geometry.pdf]]\n**Great introductory book to the tough stuff, and seems to contain the best from his other work\n**Haar measure on p124\n**[[Hodge dual]] p209\n**[[FuN derivative]] p215\n**grab theorem from p226 on relationship between [[Lie algebra]] and holonomy algebra\n**homogeneous space p230\n**gauge transformations p240\n**[[FuN curvature]] p245\n**covariant derivative p248, p260\n**[[holonomy]] p251\n**characteristic classes p263 (Wow!)\n**Hamiltonian mechanics p283\n*[[The Frölicher-Nijenhuis Bracket|papers/The Frolicher-Nijenhuis Bracket.pdf]]\n*[[Remarks on the Frolicher-Nijenhuis Bracket|papers/Remarks on the Frolicher-Nijenhuis Bracket.pdf]]\n*[[Gauge Theory for Fiber Bundles|papers/Gauge Theory for Fiber Bundles.pdf]]\n*[[Natural Operations in Differential Geometry|papers/Natural Operations in Differential Geometry.pdf]]
<<note HideTags>>Pirated from GS&W, [[Superstring Theory|http://www.amazon.com/Superstring-Cambridge-Monographs-Mathematical-Physics/dp/0521357527/ref=pd_bbs_sr_3/104-9709999-3726336?ie=UTF8&s=books&qid=1179001057&sr=8-3]]:\n\sbegin{eqnarray}\nE &=& B + \sPs = \sha b^{\sal\sbe} \sga^{\ssmall (16)+}_{\sal\sbe} + \sps^a Q^+_a \s\s\n&\sin& so(16) + S^{\ssmall (16)+} = {\srm Lie}(E8)\n\send{eqnarray}\n[[Lie brackets|Lie algebra]] between generators (structure constants):\n$$\n{\ssmall\n\sbegin{array}{rcl}\n\sbig[ \sga^{\ssmall (16)+}_{\sal \sbe}, \sga^{\ssmall (16)+}_{\sga \sde} \sbig] &=& 2 \s, \sbig\s{ - \set_{\sal \sga} \sga^{\ssmall (16)+}_{\sbe \sde} + \set_{\sal \sde} \sga^{\ssmall (16)+}_{\sbe \sga} + \set_{\sbe \sga} \sga^{\ssmall (16)+}_{\sal \sde} - \set_{\sbe \sde} \sga^{\ssmall (16)+}_{\sal \sga} \sbig\s}^{\sp{(}} \s\s\n\sbig[ \sga^{\ssmall (16)+}_{\sal \sbe}, Q^+_a \sbig] &=& \sbig( \sga^{\ssmall (16)+}_{\sal \sbe} \sbig)^b{}_c \sbig( Q^+_a \sbig)^c Q^+_b = \sga^{\ssmall (16)+}_{\sal \sbe} Q^+_a \s\s\n\sbig[ Q^+_a, Q^+_b \sbig] &=& - \sbig( {\sga^{\ssmall (16)+}}^{\sal \sbe} \sbig)_{ab} \sga^{\ssmall (16)+}_{\sal \sbe}\n\send{array}\n}\n$$\n${\srm Lie}(E8)$ brackets act as multiplication between $120$ dimensional [[Cl(16)]] [[Clifford|Clifford algebra]] [[bivector|Clifford basis elements]]s, $B$, and positive [[chiral]], $128$ dim column [[spinor]]s, $\sPs$:\n$$\n\sbegin{array}{rcll}\n\slb B_1, B_2 \srb \s!\s!&\s!\s!=\s!\s!&\s!\s! B_1 B_2 - B_2 B_1 & \sin \s; so(16) \s\s\n\slb B, \sPs \srb \s!\s!&\s!\s!=\s!\s!&\s!\s! B^+ \s, \sPs & \sin \s; S^{\ssmall (16)+} \s\s\n\slb \sPs_1, \sPs_2 \srb \s!\s!&\s!\s!=\s!\s!&\s!\s! -\sPs_1^\sdagger \sGa^+ \sPs_2 & \sin \s; so(16)_{{\sp{\sbig(}}_{\sp{(}}}\n\send{array}\n$$
<<note HideTags>>Work forwards, guess the answer, then work backwards.\n\nWork forwards towards unification:\n#[[Gauge fields|principal bundle]], [[gravity|spacetime]] and Higgs in one [[connection]].\n#Calculate its [[curvature]] to get the interactions.\n#Join fermions as ([[Grassmann|Grassmann number]] valued) [[BRST ghosts|BRST technique]] of a larger connection.\n#Correct [[standard model]] and gravitational interactions and charges from the curvature.\n\nGuess the answer:\n*Pure [[geometry of a principal bundle|Ehresmann principal bundle connection]] -- just vector fields.\n*One very large [[Lie group]] is a match!\n\nWork backwards:\n#All interactions from the [[structure|Lie algebra]] of this group, after symmetry breaking.\n#Explains exactly what and why [[spinor]]s are.\n#Gives three generations.\n#Calculating particle masses (CKM) is a possibility.\n
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http://arxiv.org/abs/quant-ph/0610204\nauthor: Rafael Sorkin\n*primacy of path integral history formulation
decent summary to look at for quantizing perturbed BF\nhttp://arxiv.org/pdf/hep-th/0610194
http://arxiv.org/abs/gr-qc/0404088\n*looks to be an excellent treatment of issues with path integral treatment of GR\n*justifies 3+1 dimensions from algebraic topology
\n| !rank | !group | !a.k.a. | !dim | !name |\n| $r$ | $A_r$ | $SU(r+1)$ | $r(r+2)$ | [[special unitary group]] |\n| $r$ | $B_r$ | $SO(2r+1)$ | $r(2r+1)$ | odd [[special orthogonal group]] |\n| $r$ | $C_r$ | $Sp(2r)$ | $r(2r+1)$ | symplectic group |\n| $r>2$ | $D_r$ | $SO(2r)$ | $r(2r-1)$ | &nbsp; even [[special orthogonal group]] &nbsp; |\n| $2$ | $G_2$ | | $14$ | G2 |\n| $4$ | $F_4$ | | $52$ | F4 |\n| $6$ | $E_6$ | | $78$ | [[E6]] |\n| $7$ | $E_7$ | | $133$ | E7 |\n| $8$ | $E_8$ | | $248$ | [[E8]] |\n<<note HideTags>>\n"E8 is perhaps the most beautiful structure in all of mathematics, but it's very complex."\n&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -- Hermann Nicolai\n
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<<note HideTags>>\nOne particularly interesting way $e8$ can be broken down:\n\n\sbegin{eqnarray}\ne8 &=& e6 + su(3) + 54 \s! \stimes \s! 3 \s\s\n &=& so(1,9) + u(1) + 32 + su(3) + 54 \s! \stimes \s! 3\s\s\n &=& so(1,3) + su(2) + su(2) + u(1) + 4 \s! \stimes \s! 8 + u(1) + 32 + su(3) + 54 \s! \stimes \s! 3 \s\s\n&\sto& {\sscriptsize \sfrac{1}{2}} \som + W + B + {\sscriptsize \sfrac{1}{4}} e \sph + G + 3 \s! \stimes \s! \sps + X? \sp{{}^{\sbig(}}\n\send{eqnarray}\n\nHow does this $e8$ breakdown relate to [[e8 triality decomposition]]?\n\n\sbegin{eqnarray}\ne8 &=& so(1,7) + so(8) + 3 \s! \stimes \s! 8 \s! \stimes \s! 8 \s\s\n &=& so(1,3) + so(4) + 4 \s! \stimes \s! 4 + so(6) + so(2) + 6 \s! \stimes \s! 2 + 3 \s! \stimes \s! 8 \s! \stimes \s! 8 \s\s\n &=& so(1,3) + su(2) + su(2) + 4 \s! \stimes \s! 4 + su(4) + u(1) + 6 \s! \stimes \s! 2 + 3 \s! \stimes \s! 8 \s! \stimes \s! 8 \sp{{}_{\sbig(}}\n\send{eqnarray}
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<<note HideTags>>''Cartan subalgebra'': $\squad C=C^a T_a \s;\s; \ssubset \s; {\srm Lie}(G) \svp{|_(}$\nBuilt from a maximal commuting set of $R$ generators,\n$$\n\sbig[ T_a, T_b \sbig] = T_a T_b - T_b T_a = 0 \sqquad \sforall \squad 1 \sle a,b \sle R\n$$\n''Root vectors'', $V_\sbe$, are eigenvectors of $C$ in the Lie bracket,\n$$\n[ C , V_\sbe ] = \sal_\sbe V_\sbe = \ssum_a i C^a\sal_{a\sbe} V_\sbe\n$$\n''Roots'', $\sal_{a\sbe}$, are the eigenvalue coefficients. The pattern of roots in $R$ dimensions corresponds to the Lie algebra,\n$$\n[ V_\sbe , V_\sga ] = V_\sde \squad \sLeftrightarrow \squad \sal_{\sbe} + \sal_{\sga} = \sal_{\sde}\n$$\n''Weight vectors'' and ''weights'' are eigenvectors and eigenvalue coefficients of $C$ acting on some representation space,\n$$\nC \s, V_\sbe = \sal_\sbe V_\sbe\n$$\nWeight vectors are particles, weights are their quantum numbers.
[[Contracting|vector-form algebra]] the [[coordinate basis vectors]] with the [[Riemann curvature]] gives the ''Ricci curvature'',\n$$\n\sf{R}{}_m = \sve{\spa_k} \sff{R}^k{}_m = \sve{\spa_k} \sf{dx^i} \sf{dx}^j \sha R_{ij}{}^k{}_m\n = \sf{dx}^j R_{ij}{}^i{}_m\n$$\nwith the components of the ''Ricci curvature tensor'' equaling a partial contraction of the Riemann curvature tensor,\n$$\nR_{jm} = R_{ij}{}^i{}_m = 2 \spa_{\slb i \srd} \sGa^i{}_{\sld j \srb m} + 2 \sGa^i{}_{\slb i \srd l} \sGa^l{}_{\sld j \srb m}\n$$\nThis tensor is symmetric if the [[torsion]] vanishes, $R_{jm}=R_{mj}$. In terms of the [[tangent bundle spin connection|tangent bundle connection]], the Ricci curvature is\n$$\n\sf{R}{}_\sal = \sf{e^\sde} R_{\sde \sal} = \sve{e_\sbe} \sff{R}^\sbe{}_\sal\n= \sf{dx^j} 2 \slp e_\sbe \srp^i \slp \spa_{\slb i \srd} w_{\sld j \srb}{}^\sbe{}_\sal + w_{\slb i \srd}{}^\sbe{}_\sga w_{\sld j \srb}{}^\sga{}_\sal \srp\n$$\nwith coefficients $R_{\sde \sal} = R_{\sbe \sal}{}^\sbe{}_\sde$. If the spin connection is torsionless, the Ricci curvature tensor may also be written as\n\sbegin{eqnarray}\nR_{\sga \sal} &=& R_{\sbe \sga}{}^\sbe{}_\sal = 2 \spa_{\slb \sbe \srd} w_{\sld \sga \srb}{}^\sbe{}_\sal + 2 w_{\slb \sbe \sga \srb}{}^\sep w_\sep{}^\sbe{}_\sal - 2 w_{\slb \sbe \srd}{}^{\sep \sbe} w_{\sld \sga \srb}{}_{\sep \sal} \s\s\n&=& 2 \spa_{\slb \sbe \srd} w_{\sld \sga \srb}{}^\sbe{}_\sal + w_{\sbe \sga}{}^\sep w_\sep{}^\sbe{}_\sal - w_\sbe{}^{\sep \sbe} w_{\sga \sep \sal}\n\send{eqnarray}
The [[vector bundle curvature]] for a [[tangent bundle]] describes the local geometry of the base manifold. Applying the [[tangent bundle covariant derivative|tangent bundle connection]] twice, and taking the [[antisymmetric|index bracket]] part, gives the tangent bundle curvature,\n$$\n\sna_{\slb i \srd} \sna_{\sld j \srb} \sve{v} = \sna_{\slb i \srd} \slp \spa_{\sld j \srb} v^k + \sGa^k{}_{\sld j \srb l} v^l \srp \sve{\spa_k} \n= \slp \spa_{\slb i \srd} \sGa^k{}_{\sld j \srb l} + \sGa^k{}_{\slb i \srd m} \sGa^m{}_{\sld j \srb l} \srp v^l \sve{\spa_k}\n= \sha R_{ij}{}^k{}_l v^l \sve{\spa_k}\n$$\nThe components of the ''Riemann curvature'' (//''tangent bundle curvature''//), $\sff{R}^k{}_l = \sha \sf{dx^i} \sf{dx^j} R_{ij}{}^k{}_l$, are the components of the conventional Riemann curvature tensor after rearrangement, $R_{ij}{}^k{}_l \sleftrightarrow R^k{}_{lij}$. (//The non-conventional Riemann index placement used here instead follows the conventional index placement for curvature tensors.//) The components are:\n$$\nR_{ij}{}^k{}_l = 2 \spa_{\slb i \srd} \sGa^k{}_{\sld j \srb l} + 2 \sGa^k{}_{\slb i \srd m} \sGa^m{}_{\sld j \srb l}\n$$\nWritten with fewer indices, this is:\n$$\n\sff{R}^k{}_l = \sf{d} \sf{\sGa}^k{}_l + \sf{\sGa}^k{}_m \sf{\sGa}^m{}_l\n$$\nA different expression for the tangent bundle curvature, $\sff{R}^\sbe{}_\sal = \sha \sf{dx^i} \sf{dx^j} R_{ij}{}^\sbe{}_\sal$, may also be written in terms of the [[tangent bundle spin connection|tangent bundle connection]],\n$$\n\sna_{\slb i \srd} \sna_{\sld j \srb} \sve{v} = \sna_{\slb i \srd} \slp \spa_{\sld j \srb} v^\sbe + w_{\sld j \srb}{}^\sbe{}_\sal v^\sal \srp \sve{e_\sbe} \n= \slp \spa_{\slb i \srd} w_{\sld j \srb}{}^\sbe{}_\sal + w_{\slb i \srd}{}^\sbe{}_\sga w_{\sld j \srb}{}^\sga{}_\sal \srp v^\sal \sve{e_\sbe}\n= \sha R_{ij}{}^\sbe{}_\sal v^\sal \sve{e_\sbe}\n$$\nwith components:\n$$\nR_{ij}{}^\sbe{}_\sal = 2 \spa_{\slb i \srd} w_{\sld j \srb}{}^\sbe{}_\sal + 2 w_{\slb i \srd}{}^\sbe{}_\sga w_{\sld j \srb}{}^\sga{}_\sal\n$$\nOr, with fewer indices:\n$$\n\sff{R}^\sbe{}_\sal = \sff{F}^\sbe{}_\sal = \sf{d} \sf{w}^\sbe{}_\sal + \sf{w}^\sbe{}_\sga \sf{w}^\sga{}_\sal\n$$\nIf the spin connection is [[torsion]]less, the Riemann curvature tensor may also be written, using the [[frame]], as\n$$\nR_{ \sde \sga}{}^\sbe{}_\sal = \slp e_\sde\srp^i \slp e_\sga\srp^j R_{ij}{}^\sbe{}_\sal\n= 2 \spa_{\slb \sde \srd} w_{\sld \sga \srb}{}^\sbe{}_\sal + 2 w_{\slb \sde \sga \srb}{}^\sep w_\sep{}^\sbe{}_\sal - 2 w_{\slb \sde \srd}{}^{\sep \sbe} w_{\sld \sga \srb}{}_{\sep \sal}\n$$\nin which $\spa_\sal = \slp e_\sal \srp^i \spa_i$ and $w_\sal{}^\sga{}_\sbe = \slp e_\sal \srp^i w_i{}^\sga{}_\sbe$.\n\nThe Riemann curvature may alternatively be obtained from the [[tangent bundle holonomy]].
GUT unification can be done using the $SU(5)$ subalgebra of $SO(10)$, but there is a (probably) better way. $SO(10)$ has $SU(2) \stimes SU(2) \stimes SU(4)$ as a maximal subalgebra. The $SU(2) \stimes SU(2)$ is $SO(4)$ and the $SU(4)$ is $SO(6)$, so the $SO(4) \stimes SO(6)$ are diagonal blocks of the $SO(10)$. The Dynkin diagram surgery for this reduction is the removal of the central $SU(2)$ node.\n\nRef:\n*Howard Georgi's book, p283:\n**http://www.amazon.com/gp/reader/0738202339/ref=sib_dp_pop_toc/104-9709999-3726336?ie=UTF8&p=S00E#\n**Also see p169 of his recent talk on GUT's for the 16 complex dim spinor rep:\n***[[GUTs|papers/yt100sym_georgi.pdf]]
The ''three dimensional [[special unitary group]]'' (//''special unitary group of order two''//), $G = SU(2)$, is the [[Lie group]] of [[unitary]] $2 \stimes 2$ complex matrices with unit [[determinant]]. Its elements, $g \sin G$, may be parameterized and obtained by [[exponentiating|exponentiation]] the [[su(2)]], $T_A = i \ssi_A^P$, [[Lie algebra]] generators,\n$$\ng(x) = e^{x^i T_i} = e^X\n$$\nwith $X=x^i T_i \sin su(2)$. It is possible to carry out this exponentiation explicitly, and do calculations in these coordinates. However, it is more instructive to convert to [[spherical coordinates]], with\n$$\nX = a^1 \ssin(a^2) \scos(a^3) T_1 + a^1 \ssin(a^2) \ssin(a^3) T_2 + a^1 \scos(a^2) T_3 \n= \sleft[\sbegin{array}{cc}\ni a^1 \scos(a^2) & i a^1 e^{-i a^3} \ssin(a^2)\s\s\ni a^1 e^{i a^3} \ssin(a^2) & -i a^1 \scos(a^2)\send{array}\sright]\n$$\nand perform the [[spectral decomposition|eigen]],\n$$\nX = U \sLa U^-\n= \sleft[\sbegin{array}{cc}\n- e^{-i a^3} \ssin(\sfr{a^2}{2}) & e^{-i a^3} \scos(\sfr{a^2}{2}) \s\s\n\scos(\sfr{a^2}{2}) & \ssin(\sfr{a^2}{2}) \send{array}\sright]\n\sleft[\sbegin{array}{cc}\n- i a^1 & 0 \s\s\n0 & i a^1 \send{array}\sright]\n\sleft[\sbegin{array}{cc}\n- e^{i a^3} \ssin(\sfr{a^2}{2}) & e^{i a^3} \scos(\sfr{a^2}{2}) \s\s\n\scos(\sfr{a^2}{2}) & \ssin(\sfr{a^2}{2}) \send{array}\sright]\n$$\nin order to exponentiate and get:\n\sbegin{eqnarray}\ng(a) &=& e^X = U e^\sLa U^-\n= \sleft[\sbegin{array}{cc}\n- e^{-i a^3} \ssin(\sfr{a^2}{2}) & e^{-i a^3} \scos(\sfr{a^2}{2}) \s\s\n\scos(\sfr{a^2}{2}) & \ssin(\sfr{a^2}{2}) \send{array}\sright]\n\sleft[\sbegin{array}{cc}\ne^{- i a^1} & 0 \s\s\n0 & e^{i a^1} \send{array}\sright]\n\sleft[\sbegin{array}{cc}\n- e^{i a^3} \ssin(\sfr{a^2}{2}) & e^{i a^3} \scos(\sfr{a^2}{2}) \s\s\n\scos(\sfr{a^2}{2}) & \ssin(\sfr{a^2}{2}) \send{array}\sright] \s\s\n&=&\n\sleft[\sbegin{array}{cc}\n\scos(a^1) + i \ssin(a^1) \scos(a^2) & i \ssin(a^1) e^{-i a^3} \ssin(a^2) \s\s\ni \ssin(a^1) e^{i a^3} \ssin(a^2) & \scos(a^1) - i \ssin(a^1) \scos(a^2) \send{array}\sright] \s\s\n&=& \scos(a^1) 1 + \sfr{\ssin(a^1)}{a^1} X\n\send{eqnarray}\nThis could have been found more easily by noting that $XX = - (a^1)^2$ -- but this won't be true for general Lie groups, while the above method generalizes nicely.\n\nThe [[Lie group geometry]] is described by the left and right acting (right and left invariant) vector fields, and their dual 1-form fields. Over most of the group manifold, the [[Maurer-Cartan form]],\n$$\n\sf{\scal I}(a) = g^-(a) \sf{d} g(a) = \sf{da^i} \slp\sxi^R_i\srp^A T_A = \sf{da^i} \slp e_i\srp^A T_A \n$$\nhas components (best computed using Mathematica or something):\n\sbegin{eqnarray}\n\slp e_i\srp^A &=& \slp T^A, g^-(a) \spa_i g(a) \srp \s\s\n&=&\n\sleft[\sbegin{array}{ccc}\n\ssin(a^2) \scos(a^3) & \ssin(a^2) \ssin(a^3) & \scos(a^2) \s\s\n\ssin(a^1) \slp \scos(a^1) \scos(a^2) \scos(a^3) - \ssin(a^1) \ssin(a^3) \srp & \ssin(a^1) \slp \ssin(a^1) \scos(a^3) + \scos(a^1) \scos(a^2) \ssin(a^3) \srp & - \sha \ssin(2 a^1) \ssin(a^2) \s\s\n\sha \slp - \ssin^2(a^1) \ssin(2 a^2) \scos(a^3) - \ssin(2 a^1) \ssin(a^2) \ssin(a^3) \srp & \ssin(a^1) \ssin(a^2) \slp \scos(a^1) \scos(a^3) - \ssin(a^1) \scos(a^2) \ssin(a^3) \srp & \ssin^2(a^1) \ssin^2(a^2)\n\send{array}\sright]\n\send{eqnarray}\nIdentifying these as the [[frame]] components for the [[Lie group tangent bundle geometry]], using the su(2) [[Killing form]], $g_{AB} = -8 \sde_{AB}$, gives the metric for the Lie group geometry,\n$$\ng_{ij}(a) = \slp e_i\srp^A g_{AB} \slp e_j \srp^B\n=\n\sleft[\sbegin{array}{ccc}\n-8 & 0 & 0 \s\s\n0 & -8 \ssin^2(a^1) & 0 \s\s\n0 & 0 & -8 \ssin^2(a^1) \ssin^2(a^2)\n\send{array}\sright]\n$$\n \n$SU(2) = Spin(3)$ may also be thought of as the group generated by the bivectors of the three dimensional Clifford algebra, [[Cl(3)]]. Under this representation, each group element, $g$, is a $Cl(3)$ scalar plus a bivector. This is also equivalent to representation by [[quaternions]]. The ''unitary group'', $\sleft\s{ U\sin GL(C)\smid UU^{\sdagger}=1\sright\s}$, corresponds to the unitary [[subgroup]] of the Clifford Algebra, $\sleft\s{ U\sin Cl\smid U\sgamma_{0}\swidetilde{U}=\sgamma_{0}\sright\s}$, with $\swidetilde{U}$ the [[Clifford reverse|Clifford conjugate]].
The ''eight dimensional [[special unitary group]]'' (//''special unitary group of order three''//), $G = SU(3)$, is the [[Lie group]] of [[unitary]] $3 \stimes 3$ complex matrices with unit [[determinant]]. Its elements, $g \sin G$, may be parameterized and obtained by [[exponentiating|exponentiation]] the [[su(3)]] [[Lie algebra]] generators,\n$$\ng(x) = e^{x^i T_i}\n$$
For [[loops|vector-form algebra]] and higher grade multivectors.\n\nhttp://www.mimuw.edu.pl/~pwit/TOK/sem4/online/node9.html
*[[John Baez]] has a nice recent writeup from his course on quantization:\n**[[path integrals|papers/w07week08a.pdf]]
The ''Schwarzschild solution'' gives the unique geometry of [[spacetime]] in the vicinity of an uncharged, non-rotating, spherically symmetric mass, $M$. This approximately describes spacetime around the sun, earth, or black holes. The solution is most concisely expressed by the [[frame]],\n$$\n\sf{e} = \sf{dt} \slp 1 - \sfr{R_s}{r} \srp^\sha \sga_0 + \sf{dr} \sfr{1}{c} \slp 1 - \sfr{R_s}{r} \srp^{-\sha} \sga_1\n+ \sf{d\sth} \sfr{r}{c} \sga_2 + \sf{d\sph} \sfr{r \ssin{\sth}}{c} \sga_3\n$$\nhaving diagonal frame matrix. The coordinates are $(x^0,x^1,x^2,x^3)=(t,r,\sth,\sph)$ and have [[units]] $(T,L,0,0)$. The solution has a coordinate singularity at $r=R_S=\sfr{2GM}{c^2}$, corresponding to the ''Schwarzchild radius'' -- the horizon beyond which light cannot escape. (An alternative set of coordinates and frame better suited for calculations near the horizon,\n$$\n\sf{e} = \sf{dt} \sga_0 + \sf{dt} \ssqrt{\sfr{R_S}{r}} \sga_1 + \sf{dr} \sfr{1}{c} \sga_1 + \sf{d\sth} \sfr{r}{c} \sga_2 + \sf{d\sph} \sfr{r \ssin{\sth}}{c} \sga_3\n$$\nis given by Chris Doran in http://xxx.lanl.gov/abs/gr-qc/9910099 ) The angular [[spherical coordinates]], $\sth$ and $\sph$, range from $0$ to $\spi$ and from $0$ to $2\spi$. The radial coordinate, $r$, is scaled so the area of the 2D surface at $r=R$ is\n$$\nA = c^2 \sint_{r=R} \sf{e^2} \sf{e^3} = \sint \sf{d\sth}\sf{d\sph} R^2 \ssin{\sth}=4\spi R^2\n$$\nfor any time, $t$.\n\nUsing Doran's frame, the coframe is\n$$\n\sve{e} = \sga^0 \sve{\spa_t} - \sga^0 c \ssqrt{\sfr{R_S}{r}} \sve{\spa_r} + \sga^1 c \sve{\spa_r} + \sga^2 \sfr{c}{r} \sve{\spa_\sth} + \sga^3 \sfr{c}{r \ssin(\sth)} \sve{\spa_\sph}\n$$\nThe [[torsion]]less [[spin connection]], found by solving [[Cartan's equation]], $0=\sf{d} \sf{e} + \sf{\som} \stimes \sf{e}$, is\n\sbegin{eqnarray}\n\sf{\som} &=& - \sve{e} \stimes \sf{d} \sf{e} + \sfr{1}{4} \slp \sve{e} \stimes \sve{e} \srp \slp \sf{e} \scdot \sf{d} \sf{e} \srp \s\s\n&=& - \sf{d t} \sfr{c R_S}{2 r^2} \sga_{01} - \sf{d r} \sfr{1}{2 r} \ssqrt{\sfr{R_S}{r}} \sga_{01}\n+ \sf{d \sth} \ssqrt{\sfr{R_S}{r}} \sga_{02} + \sf{d \sph} \ssqrt{\sfr{R_S}{r}} \ssin(\sth) \sga_{03}\n+ \sf{d \sph} \ssin(\sth) \sga_{13} + \sf{d \sph} \scos(\sth) \sga_{23} \n\send{eqnarray}\nThe [[Clifford vector bundle]] curvature is\n\sbegin{eqnarray}\n\sff{F} &=& \sf{d} \sf{\som} + \sha \sf{\som} \sf{\som} \s\s\n&=& - \sf{d t} \sf{d r} \sfr{c R_S}{r^3} \sga_{01} \n+ \sf{d t} \sf{d \sth} \sfr{c R_S}{2 r^2} \sga_{02}\n+ \sf{d t} \sf{d \sth} \sfr{c R_S}{2r^2} \ssqrt{\sfr{R_S}{r}} \sga_{12}\n+ \sf{d t} \sf{d \sph} \sfr{c R_S \ssin{\sth}}{2r^2} \sga_{03} \s\s\n&+& \sf{d t} \sf{d \sph} \sfr{c R_S \ssin{\sth}}{2r^2} \ssqrt{\sfr{R_S}{r}} \sga_{13}\n+ \sf{d r} \sf{d \sth} \sfr{R_S}{2r^2} \sga_{12}\n+ \sf{d r} \sf{d \sph} \sfr{R_S \ssin{\sth}}{2r^2} \sga_{13}\n- \sf{d \sth} \sf{d \sph} \sfr{R_S \ssin{\sth}}{r} \sga_{23}\n\send{eqnarray}\nThe [[Clifford-Ricci curvature]] is\n\sbegin{eqnarray}\n\sf{R} &=& \sve{e} \stimes \sff{F} = 0\n\send{eqnarray}\nshowing that the Schwarzschild solution satisfies the vacuum [[Einstein's equation]] away from the curvature singularity at $r=0$.\n\nRef:\n*http://en.wikipedia.org/wiki/Schwarzschild_coordinates
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Deferential Geometry
http://interstice.com/~aglisi/dg/index.cgi
$$\n\sbegin{array}{rcl}\n\sudf{A} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{H}{}_1 + \sf{H}{}_2 + \sud{\sPs} = {\ssmall \sfrac{1}{2}} \sf{\som} + {\ssmall \sfrac{1}{4}} \sf{e} \sph + \sf{W} + \sf{B}{}_1 + \sf{w} + \sf{B}{}_2 + \sf{x} \sPh + \sf{g} + \sud{\snu^e} + \sud{e} + \sud{u} + \sud{d}\n\s; \sin \s; \sudf{\srm Lie}(E8) = \sudf{e8}\n\s\s[.5em]\n\n\s!\s!&\s!\s!=\s!\s!&\s!\s!\n{\ssmall\n\slb \sbegin{array}{cccccccc}\n\sfrac{1}{2} \sf{\som}{}_L \s!+\s! i \sf{W}{}^3 \s!&\s! i \sf{W}{}^1 \s!+\s! \sf{W}{}^2 \s!&\s! - \s! \sfrac{1}{4} \sf{e}{}_R \sph_1 \s!&\s! \sfrac{1}{4} \sf{e}{}_R \sph_+ \s!&\n\s; \sud{\snu}{}_L &\s!\s! \sud{u}{}_L^r \s!\s!&\s!\s! \sud{u}{}_L^g \s!\s!&\s!\s! \sud{u}{}_L^b \s\s\n\ni \sf{W}{}^1 \s!-\s! \sf{W}{}^2 \s!&\s! \sfrac{1}{2} \sf{\som}{}_L \s!-\s! i \sf{W}{}^3 \s!&\s! \sp{-} \sfrac{1}{4} \sf{e}{}_R \sph_- \s!&\s! \sfrac{1}{4} \sf{e}{}_R \sph_0 \s!&\n\s; \sud{e}{}_L &\s!\s! \sud{d}{}_L^r \s!\s!&\s!\s! \sud{d}{}_L^g \s!\s!&\s!\s! \sud{d}{}_L^b \s\s\n\n-\sfrac{1}{4} \sf{e}{}_L \sph_0 & \sfrac{1}{4} \sf{e}{}_L \sph_+ & \s! \sfrac{1}{2} \sf{\som}{}_R \s!+\s! i \sf{B}{}_1^3 \s! \s!&\s! i \sf{B}{}_1^1 \s!+\s! \sf{B}{}_1^2 \s!&\n\s; \sud{\snu}{}_R &\s!\s! \sud{u}{}_R^r \s!\s!&\s!\s! \sud{u}{}_R^g \s!\s!&\s!\s! \sud{u}{}_R^b \s\s\n\n\sp{-}\sfrac{1}{4} \sf{e}{}_L \sph_- & \sfrac{1}{4} \sf{e}{}_L \sph_1 &\s! i \sf{B}{}_1^1 \s!-\s! \sf{B}{}_1^2 \s!&\s! \s! \sfrac{1}{2} \sf{\som}{}_R \s!-\s! i \sf{B}{}_1^3 \s! &\n\s; \sud{e}{}_R &\s!\s! \sud{d}{}_R^r \s!\s!&\s!\s! \sud{d}{}_R^g \s!\s!&\s!\s! \sud{d}{}_R^b \s\s\n\n& & & & \s; i \sf{B}{}_2 &\s!\s! \s!\s!&\s!\s! \s!\s!&\s!\s! \s\s\n& & & & &\s!\s!\s! \sfrac{-i}{3} \s! \sf{B}{}_2 \s!+\s! i \sf{g}{}^{3+8} \s!\s!\s!&\s!\s!\s! i\sf{g}{}^1 \s!-\s! \sf{g}{}^2 \s!\s!\s!&\s!\s!\s! i\sf{g}{}^4 \s!-\s! \sf{g}{}^5 \s\s\n& & & & &\s!\s!\s! i\sf{g}{}^1 \s!+\s! \sf{g}{}^2 \s!\s!\s!&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B}{}_2 \s!-\s! i \sf{g}{}^{3+8} \s!\s!\s!&\s!\s!\s! i\sf{g}{}^6 \s!-\s! \sf{g}{}^7 \s\s\n& & & & &\s!\s!\s! i\sf{g}{}^4 \s!+\s! \sf{g}{}^5 \s!\s!\s!&\s!\s!\s! i\sf{g}{}^6 \s!+\s! \sf{g}{}^7 \s!\s!\s!&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B}{}_2 \s!-\s!\s! \sfrac{2i}{\ssqrt{3}}\sf{g}{}^8\n\send{array} \srb\n}\n\s\s[1.5em]\n\sudff{F} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{d} \sudf{A} + \sudf{A} \sudf{A} = \n( \sf{d} \sf{H}{}_1 + \sf{H}{}_1 \sf{H}{}_1 ) + ( \sf{d} \sf{H}{}_2 + \sf{H}{}_2 \sf{H}{}_2 ) + ( \sf{d} \sud{\sPs} + \sf{H}{}_1 \sud{\sPs} - \sud{\sPs} \sf{H}{}_2 ) \s; \sin \s; \sudff{e8} \s\s[.5em]\n\n\s!\s!&\s!\s!=\s!\s!&\s!\s!\n\sha \sbig( (\sf{d} \sf{\som} + \sha \sf{\som} \sf{\som}) - \sfr{1}{8} \sf{e} \sf{e} \sph^2 \sbig)\n+ \sfr{1}{4} \sbig( ( \sf{d} \sf{e} \s!+\s! \sha [ \sf{\som}, \sf{e} ] ) \sph - \sf{e} ( \sf{d} \sph \s!+\s! [ \sf{W} \s!+\s! \sf{B}{}_1, \sph ] ) \sbig)\n+ (\sf{d} \sf{W} + \sf{W} \sf{W})\n\s\s\n&&\n\s!\s!+\s, (\sf{d} \sf{B}{}_1 + \sf{B}{}_1 \sf{B}{}_1) + \sf{d} \sf{w} + \sf{d} \sf{B}{}_2 + \sf{x}\sPh\sf{x}\sPh\n+ \sbig( ( \sf{d} \sf{x} \s!+\s! [ \sf{w} \s!+\s! \sf{B}{}_2, \s! \sf{x} ] ) \sPh \s!-\s! \sf{x} ( \sf{d} \sPh \s!+\s! [ \sf{g}, \s! \sPh ] ) \sbig)\n+ (\sf{d} \sf{g} + \sf{g} \sf{g}) \s\s\n&&\n\s!\s!+\s, \sbig( ( \sf{d} + {\sscriptsize \sfrac{1}{2}} \sf{\som} + {\sscriptsize \sfrac{1}{4}} \sf{e}\sph ) \sud{\sPs}\n+ \sf{W} \sud{\sPs}{}_L + \sf{B}{}_1 \sud{\sPs}{}_R - \sud{\sPs} ( \sf{w} + \sf{B}{}_2 + \sf{x} \sPh ) - \sud{\sPs}{}_q \s, \sf{g} \sbig) \s\s[.5em]\n\n\s!\s!&\s!\s!=\s!\s!&\s!\s!\n\sha \sbig( \sff{R} - \sfr{1}{8} \sf{e} \sf{e} \sph^2 \sbig)\n+ \sfr{1}{4} \sbig( \sff{T} \sph - \sf{e} \sf{D} \sph \sbig)\n+ \sff{F}{}_W + \sff{F}{}_{B_1} \n+ \sff{F}{}_{w} + \sff{F}{}_{B_2} + \sf{x}\sPh\sf{x}\sPh\n+ \sbig( (\sf{D} \sf{x}) \sPh - \sf{x} \sf{D} \sPh \sbig)\n+ \sff{F}{}_{g}\n+ \sf{D} \sud{\sPsi}\n\send{array}\n$$\n$$\nS \s,= \sint \sbig< \sff{\sod{B}} \sudff{F}\n+ {\sscriptsize \sfrac{\spi G}{4}} \sff{B}{}_G \sff{B}{}_G \sga - \sff{B'} \sff{*B'} \sbig>\n= \sint \sbig< \sfff{\sod{B}} \sf{D} \sud{\sPs}\n+ \snf{e} {\sscriptsize \sfrac{1}{16 \spi G}} \sph^2 \sbig( R - \sfr{3}{2} \sph^2 \sbig) - \sfr{1}{4} \sff{F'} \sff{*F'} \sbig>\n\svp{{\sBig(}_{\sBig(}^{\sBig(}}\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\n$$\n$$\nZ = \sint D A \s, e^{\sfr{i}{\shbar} S[A]} \s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s; p[A] = \sfrac{1}{Z} \s, e^{\sfr{i}{\shbar} S[A]} \s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\n$$\n
<<note HideTags>>@@display:block;text-align:center;[img[images/png/standard model and gravity.png]]\n$\sp{{}_{\ssmall (}^{(}}$http://deferentialgeometry.org &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [[Garrett Lisi]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; FQXi@@
<<note HideTags>>$$\n\sudf{A} = \sf{H} + \sf{G} + \sud{\sps}\n=\n{\ssmall\n\slb \sbegin{array}{cc}\n\sf{H^+} & \sud{\sps}^- \s\s\n& \sf{G^-}\n\send{array} \srb\n}\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s; \sin \s;\s; \sf{so}(1,7) + \sf{so}(8) + \sud{\smathbb{C}}(8 \stimes 8)\n$$\n$$\n{\ssmall\n\s!\s! = \s!\s! \slb \sbegin{array}{cccccccc}\n\sfrac{1}{2} \sf{\som_L} \s!+\s! i \sf{W^3} \s!&\s! i \sf{W^1} \s!+\s! \sf{W^2} \s!&\s! - \s! \sfrac{1}{4} \sf{e_R} \sph_0^* \s!&\s! \sfrac{1}{4} \sf{e_R} \sph_+ \s!&\n\s; \sud{\snu}{}_L &\s!\s! \sud{u}{}_L^r \s!\s!&\s!\s! \sud{u}{}_L^g \s!\s!&\s!\s! \sud{u}{}_L^b \s\s\n\ni \sf{W^1} \s!-\s! \sf{W^2} \s!&\s! \sfrac{1}{2} \sf{\som_L} \s!-\s! i \sf{W^3} \s!&\s! \sp{-} \sfrac{1}{4} \sf{e_R} \sph_+^* \s!&\s! \sfrac{1}{4} \sf{e_R} \sph_0 \s!&\n\s; \sud{e}{}_L &\s!\s! \sud{d}{}_L^r \s!\s!&\s!\s! \sud{d}{}_L^g \s!\s!&\s!\s! \sud{d}{}_L^b \s\s\n\n-\sfrac{1}{4} \sf{e_L} \sph_0 & \sfrac{1}{4} \sf{e_L} \sph_+ & \s! \sfrac{1}{2} \sf{\som_R} \s!+\s! i \sf{B} \s! \s!& &\n\s; \sud{\snu}{}_R &\s!\s! \sud{u}{}_R^r \s!\s!&\s!\s! \sud{u}{}_R^g \s!\s!&\s!\s! \sud{u}{}_R^b \s\s\n\n\sp{-}\sfrac{1}{4} \sf{e_L} \sph_+^* & \sfrac{1}{4} \sf{e_L} \sph_0^* & &\s! \s! \sfrac{1}{2} \sf{\som_R} \s!-\s! i \sf{B} \s! &\n\s; \sud{e}{}_R &\s!\s! \sud{d}{}_R^r \s!\s!&\s!\s! \sud{d}{}_R^g \s!\s!&\s!\s! \sud{d}{}_R^b \s\s\n\n& & & & \s; i \sf{B} &\s!\s! \s!\s!&\s!\s! \s!\s!&\s!\s! \s\s\n& & & & &\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!+\s! i \sf{G^{3+8}} \s!\s!\s!&\s!\s!\s! i\sf{G^1} \s!-\s! \sf{G^2} \s!\s!\s!&\s!\s!\s! i\sf{G^4} \s!-\s! \sf{G^5} \s\s\n& & & & &\s!\s!\s! i\sf{G^1} \s!+\s! \sf{G^2} \s!\s!\s!&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!-\s! i \sf{G^{3+8}} \s!\s!\s!&\s!\s!\s! i\sf{G^6} \s!-\s! \sf{G^7} \s\s\n& & & & &\s!\s!\s! i\sf{G^4} \s!+\s! \sf{G^5} \s!\s!\s!&\s!\s!\s! i\sf{G^6} \s!+\s! \sf{G^7} \s!\s!\s!&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!-\s!\s! \sfrac{2i}{\ssqrt{3}}\sf{G^8}\n\send{array} \srb\n}\n$$\nCorrect interactions and charges from [[curvature]]:\n$$\sbegin{array}{rcl}\n\sudff{F} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{d} \sudf{A} + \sudf{A} \sudf{A} \s\s\n\s!\s!&\s!\s!=\s!\s!&\s!\s! ( \sf{d} \sf{H} + \sf{H} \sf{H} ) + ( \sf{d} \sf{G} + \sf{G} \sf{G} ) + ( \sf{d} \sud{\sps} + \sf{H} \sud{\sps} + \sud{\sps} \sf{G} )\n\send{array}$$
<html>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">G2</SPAN></th>\n<th></th>\n<th><SPAN class="math">V_\sbe</SPAN></th>\n<th></th>\n<th><SPAN class="math">g^3</SPAN></th>\n<th><SPAN class="math">g^8</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{g}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">(T_2 - i T_1)</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}g}</SPAN></td>\n<td></td>\n<td><SPAN class="math">(-T_2 - i T_1)</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">(T_5-i T_4)</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">(-T_5-i T_4)</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{g}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">(-T_7-i T_6)</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{g\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">(T_7-i T_6)</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#D90000} </SPAN></td>\n<td><SPAN class="math">q^r</SPAN></td>\n<td></td>\n<td><SPAN class="math">[1,0,0]</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#00BF00} </SPAN></td>\n<td><SPAN class="math">q^g</SPAN></td>\n<td></td>\n<td><SPAN class="math">[0,1,0]</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#0000F7} </SPAN></td>\n<td><SPAN class="math">q^b</SPAN></td>\n<td></td>\n<td><SPAN class="math">[0,0,1]</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{\ssqrt{3}}}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#D90000} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^r</SPAN></td>\n<td></td>\n<td><SPAN class="math">[1,0,0]</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#00BF00} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^g</SPAN></td>\n<td></td>\n<td><SPAN class="math">[0,1,0]</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#0000F7} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^b</SPAN></td>\n<td></td>\n<td><SPAN class="math">[0,0,1]</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">{\sfr{1}{\ssqrt{3}}}</SPAN></td>\n</tr>\n</table>\n</td>\n\n<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>\n\n<td>\n<img SRC="images/png/g2.png">\n<br><br>\n<img SRC="images/png/dynkin g2.png">\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>
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http://arxiv.org/abs/hep-th/0610039\nSuper coset spaces play an important role in the formulation of supersymmetric theories. The aim of this paper is to review and discuss the geometry of super coset spaces with particular focus on the way the geometrical structures of the super coset space G/H are inherited from the super Lie group G. The isometries of the super coset space are discussed and a definition of Killing supervectors - the supervectors associated with infinitesimal isometries - is given that can be easily extended to spaces other than coset spaces.
|!Symbol|![[LaTeX]]|!Use|\n| $\smathbb{R} \s;\s; \smathbb{C} \s;\s; n \s;\s; \sud{a}$ | {{{ \smathbb{R} \smathbb{C} n \sud{a} }}} |[[real numbers|http://en.wikipedia.org/wiki/Real_numbers]], complex numbers, dimension, [[Grassmann number]] |\n| $M \s; \s; T_p M \s; \s; T_p^* M$ | {{{M T_p M T_p^* M }}} |[[manifold]], [[tangent space to M at point p|coordinate basis vectors]], [[cotangent space to M at point p|coordinate basis 1-forms]] |\n| $x^i \s; \s; \sve{\spa_i} \s; \s; \sve{v} \s; \s; \svv{l}$ | {{{x^i \sve{\spa_i} \sve{v} \svv{l} }}} |[[coordinates|manifold]] and [[coordinate basis vectors]] with coordinate [[indices]], [[tangent vector]], [[loop|vector-form algebra]] |\n| $t \s; \s; \sta$ | {{{ t \sta }}} |parameter time, [[proper time]] |\n| $\sf{dx^i} \s;\s; \sf{a} \s;\s; \sff{b} \s;\s; \sfff{c} \s;\s; \snf{f}$ | {{{\sf{dx^i} \sf{a} \sff{b} \sfff{c} \snf{f} }}} |[[coordinate basis 1-forms]], [[1-form]], [[2-form|differential form]], 3-form, [[differential form]] of high or unspecified form grade |\n| $\spa_i \s;\s; \sf{\spa} \s;\s; \sf{d}$ | {{{\spa_i \sf{\spa} \sf{d} }}} |[[partial derivative]], partial derivative, [[exterior derivative]] |\n| $\sph \s; \s; \sph^* \s; \s; \sph_*$ | {{{\sph \sph^* \sph_* }}} |[[diffeomorphism]], [[pullback]], pushforward |\n| ${\scal L}_{\sve{v}} \s;\s; \slb\sve{v},\sve{u}\srb_L \s;\s; \sve{\sDe}$ | {{{{\scal L}_{\sve{v}} \slb\sve{v},\sve{u}\srb_L \sve{\sDe} }}} |[[Lie derivative]], [[Lie bracket|Lie derivative]] of two [[vector fields|tangent bundle]], [[distribution]] |\n| $\snf{\sve{A}} \s;\s; {\scal L}_{\snf{\sve{K}}} \s;\s; \slb\snf{\sve{K}},\snf{\sve{L}}\srb_L \s;\s; \sf{\sve{P}}$ | {{{ \sf{\sve{A}} {\scal L}_{ \snf{\sve{K}} } \slb\snf{\sve{K}},\snf{\sve{L}}\srb_L \sf{\sve{P}} }}} |[[vector valued form]], [[FuN derivative]], FuN bracket, [[vector projection]] |\n| $\sf{\sve{\scal A}} \s;\s; \sff{\sve{\scal F}} \s;\s; \sf{\scal D}$ | {{{ \sf{\sve{\scal A}} \sff{\sve{\scal F}} \sf{\scal D} }}} |[[Ehresmann connection]], [[FuN curvature]], [[Ehresmann covariant derivative]] |\n| $\sde_i^j \s;\s; \set_{\sal \sbe} \s; \s; \sep_{\sal \sdots \sbe} \s; \s; \sotimes$ | {{{ \sde_i^j \set_{\sal \sbe} \sep_{\sal \sdots \sbe} \sotimes }}} |[[Kronecker delta|http://en.wikipedia.org/wiki/Kronecker_delta]], [[Minkowski metric]], [[permutation symbol]], [[Kronecker product]] |\n| $G \s;\s; g^- \s;\s; T_A \s;\s; \slb{T_A,T_B}\srb$ | {{{G g^- T_A \slb{T_A,T_B}\srb }}} |[[Lie group]], [[inverse]] of a group element, [[Lie algebra]] generators, [[commutator]] bracket |\n| $\sf{\sna} \s;\s; \sf{A} \s;\s; \sff{F}$ | {{{\sf{\sna} \sf{A} \sff{F} }}} |[[covariant derivative]], [[connection]], [[curvature]] |\n| $\sf{\scal I} \s;\s; \sf{\sve{\scal I}} \s;\s; \sve{\sxi^L_A} \s;\s; \sve{\sxi^R_A}$ | {{{\sf{\scal I} \sf{\sve{\scal I}} \sve{\sxi^L_A} \sve{\sxi^R_A} }}} |[[Maurer-Cartan form]], Ehresmann-Maurer-Cartan form, [[left and right action vector fields|Lie group geometry]] |\n| $Cl \s; \s; Cl^*$ | {{{Cl Cl^* }}} |[[Clifford algebra]], [[Clifford group]] |\n| $\sga_\sal \s; \s; \sga_{\sal \sdots \sbe} \s; \s; \sga$ | {{{\sga_\sal \sga_{\sal \sdots \sbe} \sga }}} |[[Clifford basis vectors]], [[Clifford basis elements]], Clifford [[pseudoscalar]] |\n| $\shat{A} \s; \s; \stilde{A} \s; \s; \sbar{A} \s; \s; A^\sdagger \s; \s; \soverline{A}$ | {{{\shat{ \stilde{ \sbar{ \soverline{A}^\sdagger } } } }}} |[[Clifford involution, reverse, conjugate, Hermitian conjugate, Dirac conjugate|Clifford conjugate]] |\n| $\scdot \s; \s; \stimes$ | {{{\scdot \stimes }}} |symmetric and antisymmetric [[Clifford algebra]] product |\n| $\slb{A,\sdots,B}\srb_A \s;\s; a_{\slb{\sal\sdots\sbe}\srb}$ | {{{ \slb{A,\sdots,B}\srb_A a_{\slb{\sal\sdots\sbe}\srb} }}} |[[antisymmetric bracket]], [[index bracket]] |\n| $\sli{A}\sri_q \s; \s; \sli{A}\sri$ | {{{ \sli{A}\sri_q \sli{A}\sri }}} |[[Clifford grade]] $q$ part, [[scalar part|Clifford grade]] |\n| $\sf{A} \s; \s; \sff{b} \s; \s; \sve{e}$ | {{{ \sf{A} \sff{b} \sve{e} }}} |[[Lieform]]s or [[Clifform]]s |\n| $\slp{e_i}\srp^\sal \s;\s; \slp{e_\sal}\srp^i \s;\s; g_{ij} \s;\s; \slp\sve{u},\sve{v}\srp$ | {{{ \slp{e_i}\srp^\sal \slp{e_\sal}\srp^i g_{ij} \slp\sve{u},\sve{v}\srp }}} |co[[frame]] matrix, frame matrix, [[metric]], scalar product |\n| $\sf{e^\sal} \s;\s; \sve{e_\sal}$ | {{{ \sf{e^\sal} \sve{e_\sal} }}} |co[[frame]] 1-forms, orthonormal basis vectors |\n| $\sf{e} \s;\s; \sve{e}$ | {{{ \sf{e} \slp{e_i}\srp^\sal \sve{e} \slp{e_\sal}\srp^i g_{ij} }}} |co[[frame]], frame |\n| $\snf{e} \s;\s; \sll{e}\srl$ | {{{ \snf{e} \sll{e}\srl }}} |[[volume form]], frame [[determinant]] |\n| $\snf{*f} \s;\s; \sff{\svv{\sep}}$ | {{{ \snf{*f} \sff{ \svv{\sep} } }}} |[[Hodge dual]], Hodge dual projector |\n| $\sf{e^s} \s;\s; \slp{e^s_i}\srp^\sal \s;\s; s$ | {{{ \sf{e^s} \s;\s; \slp{e^s_i}\srp^\sal \s;\s; s }}} |[[special frame]], special coframe matrix, conformal scalar |\n| $TM \s;\s; T^*M$ | {{{ TM T^*M }}} |[[tangent bundle]], [[cotangent bundle]] |\n| $\sGa^k{}_{ij} \s;\s; \sf{\sGa}^k{}_j \s;\s; \sff{R}^k{}_j$ | {{{ \sGa^k{}_{ij} \sf{\sGa}^k{}_j \sff{R}^k{}_j }}} |[[Christoffel symbols]], [[tangent bundle connection]], [[Riemann curvature]] |\n| $\sf{R}{}_j \s;\s; R$ | {{{ \sf{R}{}_j R }}} |[[Ricci curvature]], [[curvature scalar]] |\n| $L^\sbe{}_\sal \s;\s; \sf{w}^\sbe{}_\sal \s;\s; \sff{F}^\sbe{}_\sal$ | {{{ L^\sbe{}_\sal \sf{w}^\sbe{}_\sal \sff{F}^\sbe{}_\sal }}} |[[Lorentz rotation]], [[tangent bundle spin connection|tangent bundle connection]], [[Riemann curvature]] |\n| $ClM \s;\s; Cl^1M$ | {{{ ClM Cl^1M }}} |[[Clifford bundle]], [[Clifford vector bundle]] |\n| $\sf{A} \s;\s; \sf{\som} \s;\s; \sff{R}$ | {{{ \sf{A} \sf{\som} \sff{R} }}} |[[Clifford connection]], [[spin connection]], [[Clifford-Riemann curvature]] |\n| $\sf{R} \s;\s; R$ | {{{ \sf{R} R }}} |[[Clifford-Ricci curvature]], [[Clifford curvature scalar]] |\n| $\sff{T} \s;\s; \sf{\ska}$ | {{{ \sff{T} \sf{\ska} }}} |[[torsion]], contorsion |\n| $\sud{C} \s;\s; \snf{\sod{B}} \s;\s; \sudf{A} \s;\s; \sudff{F}$ | {{{ \sud{C} \snf{\sod{B}} \sudf{A} \sudff{F} }}} |[[BRST|BRST technique]] ghost, anti-ghost, extended connection, extended curvature |
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[>img[images/person/Thanu Padmanabhan.jpg]]Homepage: http://www.iucaa.ernet.in/~paddy/\n*Location: Pune, India\n*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Padmanabhan_T/0/1/0/all/0/1\n\nSelected work:\n*[[Holographic Gravity and the Surface term in the Einstein-Hilbert Action|http://arxiv.org/abs/gr-qc/0412068]]\n**Einstein's equation from the extrinsic curvature surface term alone!\n**related paper by Sotiriou: http://arxiv.org/abs/gr-qc/0603096\n*http://arxiv.org/abs/gr-qc/0309053\n**Horizons, for collections (congruences) of observers, at coordinate singularities for the corresponding coords.\n**Observers only have access to fields and dynamics inside this region, bounded by horizons.\n**Upon a Wick rotation, the horizon and region beyond disappears into the origin of Euclidean space -- a conical singularity.\n***The resulting Euclidean space region is naturally periodic in $t$ (in example, it resembles angular coord on a cone).\n**Boundary area quantization in order to avoid quantum entanglement across the boundary.\n**Action/Entropy is related to the degrees of freedom hidden behind the boundary/horizon. \n**Lots of good stuff in appendices. explicit calculations.\n***extrinsic curvature and boundary term\n***derivation of unique EH action\n*http://arxiv.org/abs/gr-qc/0204019\n**"action is the free energy of the horizon"\n*http://arxiv.org/abs/gr-qc/0311036\n**How to relate stat mech, QM, and GR\n**Introductory level, good survey\n**conical singulartity regularization (again)
<<<\nLet G be a connected symmetry group of the S matrix, and let the following five conditions hold: (1) G contains a subgroup locally isomorphic to the Poincaré group. (2)... Then, we show that G is necessarily locally isomorphic to the direct product of an internal symmetry group and the Poincaré group.\n<<<\nE8 Theory does not allow a subgroup locally isomorphic to the Poincaré group. The S matrix exists as an approximation, in which the theorem is satisfied.\n\n*K. Cahill, "[[On the unification of the gravitational and electronuclear forces|papers/Cahill - On the unification of the gravitational and electronuclear forces.pdf]]," Phys. Rev. D 26, 1916 (1982).\n*T. Love, "Geometry of Unification," Int. J. Th. Phys., 801 (1984).\n*...\n*F. Nesti and R. Percacci, "[[GraviWeak Unification|http://arxiv.org/abs/0706.3307]]" arxiv{0706.3307}\n*S. Alexander, "[[Isogravity: Toward an Electroweak and Gravitational Unification|http://arxiv.org/abs/0706.4481]]," arxiv{0706.4481}.\n<<note HideTags>>
Wow, I'm [[This Week's Find in Mathematical Physics|http://math.ucr.edu/home/baez/week253.html]]! Here's a quick note I wrote that week when I found out:\n\n[[John Baez]] discusses my work on describing all fields of the standard model and gravity as parts of a [[superconnection]] for the [[E8]] [[principal bundle]] over a four dimensional base manifold. And he discusses a LOT of other, related mathematics that will be keeping me busy over the next year, at least.\n\nOn 6/25/08 I delivered my [[talk for Loops 07]] in Morelia. (That links to a wiki note including links to all my talk slides (printed out from this web page), as well as supporting links, AND the audio files for the talk as I practiced it (high bandwidth recommended). The slides are also available as [[this pdf file|talks/Loops07/Loops2007.pdf]]. The talk as I actually gave it is available at the [[Loops '07|http://www.matmor.unam.mx/eventos/loops07/cont_abs.html]] site under my name. It includes many excellent questions asked by Lee Smolin and others afterwards. (An excellent question being one a speaker has already thought through and wants to talk about anyway.) This talk was VERY well received -- I've had so many physics conversations with LQG people in the past two days that my head is ready to explode. They are such great and friendly people! Sadly, on the morning of the third day I started having flu symptoms, and pulled myself away from the discussion and into hotel room quarantine -- I didn't want to infect new friends with viruses, just ideas. I'm here in my room trying to get better, and being sad, when up pops a link to This Week's Finds.... happy and sick is a bizarre combination.\n\n!Summary of the proposed [[E8]] [[T.o.E.|theory of everything]] for physicists, from the top down:\nThe universe is described by a Yang-Mills theory, with [[E8]] as the gauge group over a four dimensional base [[manifold]]. The field is a non-compact [[e8]] valued [[connection]] [[1-form]] which breaks up into different parts of the [[Lie algebra]],\n\sbegin{eqnarray}\n\sf{A} &=& \sf{H} + \sf{G} + \sf{\sPs}{}_I + \sf{\sPs}{}_{II} + \sf{\sPs}{}_{III} \s\s\n&& \sin \sf{so}(1,7) \soplus \sf{so}(1,7) \soplus \sf{End}(V^{(1,7)}) \soplus \sf{End}(S^{(1,7)+}) \soplus \sf{End}(S^{(1,7)-}) \n\send{eqnarray}\nThe action for this connection is presumed to be invariant under gauge transformations taking the $\sPs$'s to zero -- we gauge away the $\sPs$'s. The [[BRST technique]] (Faddeev-Popov) of standard QFT is used to replace the $\sPs$'s with blocks of [[Grassmann|Grassmann number]] valued ghost fields in the same part of the Lie agebra, giving the BRST extended connection,\n$$\n\sudf{A} = \sf{H} + \sf{G} + \sud{\sPs}{}_I + \sud{\sPs}{}_{II} + \sud{\sPs}{}_{III} \n$$\nComputing the [[curvature]] of this extended connection gives\n$$\n\sudff{F} = \sbig( \sf{d} \sf{H} + \sf{H} \sf{H} \sbig) + \sbig( \sf{d} \sf{G} + \sf{G} \sf{G} \sbig) + \sbig( \sf{d} \sud{\sPs}{}_{I-III} + \sf{H} \sud{\sPs}{}_{I-III} + \sud{\sPs}{}_{I-III} \sf{G} \sbig)\n$$\nHere, in the beautiful structure of $E8$, the $so(1,7)$ parts of the connection act as [[Clifford bivectors|Clifford algebra]] multiplying a set of three blocks of Grassmann valued [[spinor]]s from the left and from the right. These ghost fields have precisely the transformations and charges of the fermions -- so I go ahead and interpret them as the physical fermions. (If you don't like this derivation, you're welcome to just start with a superconnection and go from there.) The first $so(1,7)$ part of the connection, $\sf{H}$, is broken up into the gravitational [[spin connection]], $\sf{\som} \sin \sf{\srm so}(1,3)$, the electroweak fields, $\sf{W} + \sf{B} \sin \sf{\srm su}(2) \soplus \sf{\srm u}(1) \ssubset \sf{\srm su}(2) \soplus \sf{\srm su}(2) = \sf{\srm so}(4)$, and the combined [[frame]] and Higgs are assigned to the rest of $\sf{H}$, giving\n$$\n\sf{H} = \sha \sf{\som} + \sfr{1}{4}\sf{e}\sph + \sf{B} + \sf{W}\n$$\nThe $\sf{G_{\srm strong}} \sin \sf{\srm su}(3) \ssubset \sf{\srm su}(4) = \sf{\srm so}(6)$ gluons and part of the $\sf{B}$ go in the second $\sf{\srm so}(1,7)$ part of the connection, $\sf{G}$, with $19$ of $28$ generators left unused. (We could just as well have started with $\sf{\srm so}(8)$ for this part of $E8$ -- I'm not sure which is better yet.) The curvature, $\sudff{F}$, gives all the correct interactions of the standard model and [[modified BF gravity]]. This is very beautiful -- the entire structure of the standard model and gravity, including interactions, fits snugly in $E8$, considered by some to be the most beautiful structure in mathematics. Exactly what you want for a T.O.E. But it doesn't work perfectly yet -- I haven't figured out if the Higgs can correctly mix the generations; ideally, I want to get the CKMPMNS (mass) matrix out of $E8$. This will be what I work on from here until I get it or find it doesn't work -- and it will very distinctly be one or the other. Since we have room (unassigned generators) in $\sf{G}$, we can steal some, reducing that block to $\sf{G} \sin \sf{\srm so}(6)$ while still fitting the gluons, and using those stolen generators to have more Higgs fields in a larger $\sf{H} \sin \sf{\srm so}(1,9)$. These Higgs terms mix between the generations, and I will be trying to use this new gauge field reassignment to fit the fermions in what's left over and get the mass matrix out. Depite how well this picture has come together so far, it may just not work, but it's what I'm after.\n\nFrom my perspective, fitting the standard model into the structure of $E8$ came as a complete shock: I did not initially build this theory from the top down, but from the bottom up -- by spending years massaging the standard model and gravity into the most elegant (but weird!) and minimalistic mathematical framework possible. If you look at [[this paper|http://arxiv.org/abs/gr-qc/0511120]], you will find this strange Lie algebra block structure of $\sudf{A} = \sf{H} + \sf{G} + \sud{\sPs}$, with a big empty space I couldn't explain. (Note that E8 is not mentioned in this paper!) When I bumped into $E8$, and found this same algebraic block structure but with two more generations of fermions... what can I say, you get only one or two moments like that in life, if you're very lucky.\n\n!!Summary of the summary:\nEverything is described by a broken [[e8]] valued [[superconnection]] over our four dimensional base manifold,\n\sbegin{eqnarray}\n\sudf{A} &=& {\ssmall \sfrac{1}{2}} \sf{\som} + {\ssmall \sfrac{1}{4}} \sf{e} \sph + \sf{B} + \sf{W} + \sf{G} + \sud{\snu^e} + \sud{e} + \sud{u} + \sud{d} \s\s\n&& + \sud{\snu^\smu} + \sud{\smu} + \sud{c} + \sud{s}\n+ \sud{\snu^\sta} + \sud{\sta} + \sud{t} + \sud{b}\n\send{eqnarray}\nAll standard model interactions (and gravity) come from the curvature of this connection.\n\nFor the current version of how this work is playing out, check out [[the big picture]].
What to do next.\n\nNew notes and changes:\n*[[Cl(3,1)]]\n*[[energy-momentum tensor]]\n*tag the slides?\n*If $so(8)$ and [[su(3)]] are embedded in [[E8]] as in [[the big picture]], then the coupling constants for GR and EW at that ToE scale should be the same, and the [[su(3)]] coupling constant should be larger by a factor of $\ssqrt{2}$ because of how the $su(3)$ root hexagon is scaled compared to the [[Gell-Mann matrices]].\n**Maybe pull running SM coupling constants from Frank Wilczek's [[paper|http://arxiv.org/abs/hep-th/9803075]].\n*clean up [[the big picture]]\n**improve Higgs and torsion part in action\n*fix [[Cartan geometry]] -- take out "G going wavy"\n*[[Cl(1,7)]] or [[Cl(7,1)]] modeled on [[Cl(8)]]\n*[[SO(1,7)]] or [[SO(7,1)]] modeled on [[SO(8)]]\n*[[Cl(1,15)]] modeled on [[Cl(16)]]\n*[[standard model polytope]]\n**Mathematica\n***Start with big matrix from BF paper.\n***label roots as particles\n***plot roots in 15 planes\n***get things ready for Troy\n*[[e8]]\n**Mathematica\n***Build generators from [[Cl(16)]].\n***Calculate e8 roots\n***group and label them\n***plot roots in 28 planes\n**[[e8 triality decomposition]]\n**[[e(7,1)]]\n*[[representation]]\n**add links from others\n*[[broken SU(3)]]\n*[[CP2]]\n*[[doubly homogeneous space]] -- [[normalizer]] $H \striangleleft N_G(H) \ssubset G$\n**Baez TWF on double coset space\n*[[Kaluza-Klein]]\n*[[Cartan tangent bundle curvature]]\n*[[calculus of variations]]\n*[[Hamiltonian]]\n\nNew [[Tags]] and hierarchy adjustment:\n*[[e8]] under sym\n*[[toe]] under gr and sm\n*[[conf]] conferences and talks, under meta\n\nNew Illustrations:\n*[[submanifold]]\n*[[Killing vector]]\n*[[Ehresmann Cartan geometry]]\n\nPapers to read:\n*Frederick Witt, on [[triality]] (includes spinor valued 1-forms)\n**[[Special metrics and Triality|papers/0602414.pdf]]\n**[[Special metric structures and closed forms|papers/0502443.pdf]]\n***thesis\n\nChange referenced paper files to "[author - title|author - title.pdf]"?\n\nNew features:\n\nAnd<<slider chkSliderTDM [[To Do Maybe]] 'Maybe do these»' 'things I maybe want to do'>> ([[To Do Maybe]])\n
http://arxiv.org/abs/gr-qc/0608135\nAuthors: Etera Livine\nWe review the canonical analysis of the Palatini action without going to the time gauge as in the standard derivation of Loop Quantum Gravity. This allows to keep track of the Lorentz gauge symmetry and leads to a theory of Covariant Loop Quantum Gravity. This new formulation does not suffer from the Immirzi ambiguity, it has a continuous area spectrum and uses spin networks for the Lorentz group. Finally, its dynamics can easily be related to Barrett-Crane like spin foam models.\n\n*looks like a good intro to recent developments
<<note HideTags>>[[John Baez]] in [[TWF90|http://math.ucr.edu/home/baez/week90.html]]:\n<<<\nwe now look at the vector space\n$$\nso(8) + so(8) + end(V) + end(S^+) + end(S^-)\n$$\n...Since $so(8)$ has a representation as linear transformations of $V$, it has two representations on $end(V)$, corresponding to ''left and right matrix multiplication''; glomming these two together we get a representation of $so(8) + so(8)$ on $end(V)$. Similarly we have representations of $so(8) + so(8)$ on $end(S^+)$ and $end(S^-)$. Putting all this stuff together we get a Lie algebra, if we do it right - and it's $E8$.\n<<<\n$$\nE = H + G + \sPs_I + \sPs_{II} + \sPs_{III} \s;\s;\s;\s; \sin {\srm Lie}(E8)\sp{{}_{(}}\n$$\n$$\n\sbegin{array}{rclcrcl}\n[ H , \sPs_I ] \s!\s!&\s!\s!=\s!\s!&\s!\s! H \s, \sPs_I & & [ H , \sPs_{II} ] \s!\s!&\s!\s!=\s!\s!&\s!\s! H^+ \s, \sPs_{II} & & [ H , \sPs_{III} ] \s!\s!&\s!\s!=\s!\s!&\s!\s! H^- \s, \sPs_{III} \s\s\n[ G , \sPs_I ] \s!\s!&\s!\s!=\s!\s!&\s!\s! \sPs_I \s, G & & [ G , \sPs_{II} ] \s!\s!&\s!\s!=\s!\s!&\s!\s! \sPs_{II} \s, G^+ & & [ G , \sPs_{III} ] \s!\s!&\s!\s!=\s!\s!&\s!\s! \sPs_{III} \s, G^-\sp{{}_{(}}\n\send{array}\n$$
easy to invert
<<note HideTags>>Work forwards, guess the answer, then work backwards.\n\nWork forwards:\n#[[Gauge fields|principal bundle]], [[gravity|spacetime]] and Higgs in one [[connection]].\n#Calculate its [[curvature]] to get the interactions.\n#Join fermions as ([[Grassmann|Grassmann number]] valued) [[BRST ghosts|BRST technique]] of a larger connection.\n#Correct [[standard model]] and gravitational interactions and charges from the curvature.\n\nGuess the answer:\n*Pure [[geometry of a principal bundle|Ehresmann principal bundle connection]] -- just vector fields.\n*One very large [[Lie group]] is a match!\n\nWork backwards:\n#All interactions from the group [[structure|Lie algebra]], after symmetry breaking.\n#Explains exactly what and why [[spinor]]s are.\n#Gives three generations.\n#Lots still to do, but do-able.~~&nbsp;~~
<<note HideTags>>\sbegin{eqnarray}\n\slp D \s!\s!\s!\s! / + \sph \srp \sud{\sps} &=& \sve{e} \s, \sf{\sna} \sud{\sps}\n= \sve{e} \slp \sf{d} + \sf{H} \srp \sud{\sps} \s\s\n\sf{H} &=& \sha \sf{\som} + \sfr{1}{4}\sf{e}\sph + \sf{B} + \sf{W} \s\s\n&\sin& \sf{\srm Lie}(H) = \sf{Cl}^2(1,7) = \sf{so}(1,7) \ssubset \sf{\smathbb{C}}(8\stimes8)\n\send{eqnarray}\n\n$$\n\sbegin{array}{ccc}\n\sbegin{array}{rcl}\n\sve{e} \s!\s! &\s!=\s!& \s!\s! \sga^\smu \slp e_\smu \srp^a \sve{\spa_a} \s\s\n\sf{e} \s!\s! &\s!=\s!& \s!\s! \sf{dx^a} \slp e_a \srp^\smu \sga_\smu \s\s\n\sf{\som} \s!\s! &\s!=\s!& \s!\s! \sf{dx^a} \sha \som_a^{\sp{a}\snu \srh} \sga_{\snu \srh} \s\s\n\sf{B},\sf{W} \s!\s! &\s!=\s!& \s!\s! \sf{dx^a} \sha W_a^{\sp{a}\sph\sps} \sga_{\sph\sps} \s\s\n\sph \s!\s! &\s!=\s!& \s!\s! \sph^\sph \sga_\sph \s\s\n\sfr{1}{4} \sf{e} \sph \s!\s! &\s!=\s!& \s!\s! \sfr{1}{4} \sf{dx^a} \slp e_a \srp^\smu \sph^\sph \sga_{\smu \sph}\n\send{array}\n&\n\s;\s;\s;\s;\n&\n\sbegin{array}{l}\n{\srm My \s; funny \s; notation \s! :} \s\s\n\sve{\spa_a} \s, \sf{dx^b} = {\sbf i}_{\spa_a} dx^b = \sde_a^b \s\s\n\sve{e} \sf{e} = \sga^\smu \slp e_\smu \srp^a \sve{\spa_a} \s, \sf{dx^b} \slp e_b \srp^\snu \sga_\snu = 4 \s\s\n\s; \s\s\n{\srm Higgs \s; "vector" \s; constrained \s! :} \s\s\n\sph \scdot \sph = \sph^\sph \sph^\sps \set_{\sph\sps} = -M^2\n\send{array}\n\send{array}\n$$\n@@display:block;text-align:center;[[indices]]: ^^&nbsp;^^spacetime coordinates: $0 \sle a,b \sle 3$\nClifford labels: (lower, spacetime) $0 \sle \smu,\snu,\srh \sle 3$, (higher) $5 \sle \sph,\sps \sle 8$@@
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This is a personal wiki notebook in theoretical physics.\n\nEach note describes some bit of mathematical physics, while linking it to everything else. The notes are organized by [[Tags]] in an expandable list to the left of this window. Or you can look up geometric objects by [[symbol|Symbols]], or type any phrase to search for in the field to the left. From any note you can follow links to others or click in the bottom list of notes that link to it. (These are wiki-links, so tabbed browsing won't work the usual way.) You can also see which notes have been edited recently, under "[[Latest|TabTimeline]]." That's about it for basic orientation &mdash; you can read more [[About]] what you're looking at, or pick it up as you go. The math on this site will look even better if you install [[some fonts|http://deferentialgeometry.org/download/DG-ttf-TeX-fonts.zip]].\n\nI recently posted a paper to the arXiv, [[An Exceptionally Simple Theory of Everything]]. As with any new theory, it may be wrong, but even in an incomplete state it looks pretty good and is getting a bit of attention.\n\nWelcome to [[my|Garrett Lisi]] brain, have fun looking around.\n\nThe last two notes I edited are below.
A [[spinor]], $\sPsi$, of the [[spacetime]] [[Cl(1,3)]] [[Clifford algebra]] may be written, using the [[Weyl representation|Dirac matrices]], as a sum of ''left [[chiral]]'' and ''right chiral'' parts,\n$$\n\sPsi = \sPsi_L + \sPsi_R =\n\slb \sbegin{array}{c}\n\sps_L \s\s\n0\n\send {array} \srb\n+\n\slb \sbegin{array}{c}\n0 \s\s\n\sps_R\n\send {array} \srb\n=\n\slb \sbegin{array}{c}\n\sps_L \s\s\n\sps_R\n\send {array} \srb\n=\n\slb \sbegin{array}{c}\n\sps_L^\swedge \s\s\n\sps_L^\svee \s\s\n\sps_R^\swedge \s\s\n\sps_R^\svee\n\send {array} \srb\n$$\nThese parts are the ''left handed Weyl spinor'',\n$$\n\sps_L = \slb \sbegin{array}{c}\n\sps_L^\swedge \s\s\n\sps_L^\svee\n\send {array} \srb\n$$\nand ''right handed Weyl spinor'', $\sps_R$ -- each represented by a column of 2 complex (or complex [[Grassmann|Grassmann number]]) numbers -- the ''spin up'' and ''spin down'' components. These weyl spinors may be projected out,\n$$\n\sPsi_{L/R} = P_{L/R} \sPsi\n$$\nby the [[left/right chirality projector]],\n$$\nP_{L/R} = \sha \slp 1 \spm \sga \srp\n$$\n(In this equation, the four component column with two zero entries is equated to a two component column.)\n
<<note HideTags>>$$\n\sudf{A} = \sf{H} + \sf{G} + \sud{\sps}\n=\n{\ssmall\n\slb \sbegin{array}{cc}\n\sf{H^+} & \sud{\sps}^- \s\s\n& \sf{G^-}\n\send{array} \srb\n}\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\n$$\n$$\n{\ssmall\n\s!\s! = \s!\s! \slb \sbegin{array}{cccccccc}\n\sfrac{1}{2} \sf{\som_L} \s!+\s! i \sf{W^3} \s!&\s! i \sf{W^1} \s!+\s! \sf{W^2} \s!&\s! - \s! \sfrac{1}{4} \sf{e_R} \sph_0^* \s!&\s! \sfrac{1}{4} \sf{e_R} \sph_+ \s!&\n\s; \sud{\snu}{}_L &\s!\s! \sud{u}{}_L^r \s!\s!&\s!\s! \sud{u}{}_L^g \s!\s!&\s!\s! \sud{u}{}_L^b \s\s\n\ni \sf{W^1} \s!-\s! \sf{W^2} \s!&\s! \sfrac{1}{2} \sf{\som_L} \s!-\s! i \sf{W^3} \s!&\s! \sp{-} \sfrac{1}{4} \sf{e_R} \sph_+^* \s!&\s! \sfrac{1}{4} \sf{e_R} \sph_0 \s!&\n\s; \sud{e}{}_L &\s!\s! \sud{d}{}_L^r \s!\s!&\s!\s! \sud{d}{}_L^g \s!\s!&\s!\s! \sud{d}{}_L^b \s\s\n\n-\sfrac{1}{4} \sf{e_L} \sph_0 & \sfrac{1}{4} \sf{e_L} \sph_+ & \s! \sfrac{1}{2} \sf{\som_R} \s!+\s! i \sf{B} \s! \s!& &\n\s; \sud{\snu}{}_R &\s!\s! \sud{u}{}_R^r \s!\s!&\s!\s! \sud{u}{}_R^g \s!\s!&\s!\s! \sud{u}{}_R^b \s\s\n\n\sp{-}\sfrac{1}{4} \sf{e_L} \sph_+^* & \sfrac{1}{4} \sf{e_L} \sph_0^* & &\s! \s! \sfrac{1}{2} \sf{\som_R} \s!-\s! i \sf{B} \s! &\n\s; \sud{e}{}_R &\s!\s! \sud{d}{}_R^r \s!\s!&\s!\s! \sud{d}{}_R^g \s!\s!&\s!\s! \sud{d}{}_R^b \s\s\n\n& & & & \s; i \sf{B} &\s!\s! \s!\s!&\s!\s! \s!\s!&\s!\s! \s\s\n& & & & &\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!+\s! i \sf{G^{3+8}} \s!\s!\s!&\s!\s!\s! i\sf{G^1} \s!-\s! \sf{G^2} \s!\s!\s!&\s!\s!\s! i\sf{G^4} \s!-\s! \sf{G^5} \s\s\n& & & & &\s!\s!\s! i\sf{G^1} \s!+\s! \sf{G^2} \s!\s!\s!&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!-\s! i \sf{G^{3+8}} \s!\s!\s!&\s!\s!\s! i\sf{G^6} \s!-\s! \sf{G^7} \s\s\n& & & & &\s!\s!\s! i\sf{G^4} \s!+\s! \sf{G^5} \s!\s!\s!&\s!\s!\s! i\sf{G^6} \s!+\s! \sf{G^7} \s!\s!\s!&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!-\s!\s! \sfrac{2i}{\ssqrt{3}}\sf{G^8}\n\send{array} \srb\n}\n$$\nNote: Only one generation, and fermion masses not quite right.${\sp{\sbig(}}_{\sp{(}}$\nFor three generations: &nbsp; $\sudf{A} \s; \sin \s; \sf{so}(1,7) + \sf{so}(8) + 3 * \sud{\smathbb{R}}(8 \stimes 8) \s; = \s; \s;\s;\s; ?{\sp{\sBig(}}$ \nBIG Lie algebra: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $n \s;\s, = \s;\s;\s;\s;\s;\s, 28 \s;\s;\s, + \s;\s; 28 \s;\s;\s, + \s;\s;3 \s; * \s; 64 \s;\s;\s;\s;\s;\s; = \s; 248{\sp{\sbig(}}$
An ''almost complex structure'', $\sf{\sve{J}}$, on a [[manifold]] is a [[vector projection]] that satisfies,\n$$\n\sf{\sve{J}} \sf{\sve{J}} = - \sf{\sve{I}}\n$$\nThe product of an almost complex structure with vectors is equivalent to multiplication by $i=\ssqrt{-1}$. The manifold is ''complex'', admitting a complex coordinatization, iff the [[FuN curvature]] of the almost complex structure vanishes,\n$$\n\slb \sf{\sve{J}} , \sf{\sve{J}} \srb_L = 0\n$$\nand $\sf{\sve{J}}$ is then called a ''complex structure''.\n
Refs:\n*Jeffrey A. Harvey\n**[[TASI 2004 Lectures on Anomalies|papers/0509097.pdf]]\n***anomalies from the particle physics point of view
The ''antisymmetric bracket'' is an operation on a list of arbitrary [[Lie algebra]] generators or [[Clifford element]]s. The antisymmetric bracket of two elements,\n\s[ \slb A, B \srb_A = A \stimes B = \sha \slb A, B \srb = \sha \slp A B - B A \srp \s]\nequivalent to the ''cross product'', is equivalent to the [[commutator]] bracket with a multiplier of $\sha$. The antisymmetric bracket of three elements is\n\s[ \slb A, B, C \srb_A = \sfr{1}{3!} \slp ABC + BCA + CAB - ACB - CBA - BAC \srp \s]\nand so on for more elements. An antisymmetric bracket changes sign under the exchange of any two neighboring elements.\n\nThe antisymmetric bracket does not commute [[coordinate basis 1-forms]] -- these must be taken out of the bracket first. For example, for the cross product of two [[Clifform]]s,\n$$\n\sf{A} \sti \sf{B} = \slb \sf{A},\sf{B} \srb_A = \sf{dx^i} \sf{dx^j} \slb A_i, B_j \srb_A\n= \sf{dx^i} \sf{dx^j} \sha \slp A_i B_j - B_j A_i \srp = \sha \slp \sf{A} \sf{B} + \sf{B} \sf{A} \srp\n$$\nIn general, for two $p$ and $k$ forms,\n$$\n\snf{A} \sti \snf{B} = \slb \snf{A}, \snf{B} \srb_A = \sha \slp \snf{A} \snf{B} - \slp -1 \srp^{pk} \snf{B} \snf{A} \srp \n$$
Two [[fiber bundle]]s are ''associated'' if they have the same structure group and the same transition functions.
An ''automorphism'' is a structure preserving map from an object to itself.\n\nIf the object is a [[group]], with elements satisfying $g_1 g_2 = g_3$, then after a ''group automorphism'', $Aut:G \sto G, g \smapsto g' = \sph(g)$, the new group elements must satisfy $g'_1 g'_2 = g'_3$ -- that's what's meant by "structure preserving". The group of all automorphisms of a group, $G$, is called the ''automorphism group'' of $G$, $Aut(G)$. The typical group automorphism is an ''inner automorphism'',\n$$\ng' = \sph_h(g) = A_h g = h g h^-\n$$\nwith an element $h \sin G$ acting on $G$ itself through conjugation (the [[adjoint action|group]]). These form the ''inner automorphism group'', $Inn(G)$. In some cases there may be group automorphisms that are not inner automorphisms. The automorphisms of $G$ which are not inner are called ''outer automorphism''s, and the [[coset]] is labeled $Out(G)=Aut(G)/Inn(G)$.
An ''automorphism bundle'' is a [[fiber bundle]] with a [[Lie group]], $G$, or algebra as the typical fiber and the [[automorphism]] group, $Aut(G)$, acting on $G$ as the structure group.\n\nFor most Lie groups the automorphism group is the same as the group itself, $Aut(G)=Inn(G)=G$, with all automorphisms represented by inner automorphisms, and the group action given by the [[adjoint action|group]] of $G$ on the $G$ fiber:\n$$\ng' = A_h g = h g h^-\n$$\nfor any $h \sin G$ in the structure group and $g \sin G$ in the fiber. When not all automorphisms of $G$ are inner automorphisms things can get more interesting! But we will first handle the cases for when they are. Note that this bundle is different than a [[principal bundle]], for which the structure group action is the left action -- but there are many similar expressions.\n\nFor a section, $g(x)$, transforming under the adjoint action [[gauge transformation]], $g \smapsto g'=h g h^-$, the [[covariant derivative]] is\n$$\n\sf{\sna} g = \sf{d} g + \sf{A} g - g \sf{A} = \sf{d} g + \slb \sf{A} , g \srb\n$$\nwith the ''automorphism bundle [[connection]]'', $\sf{A} = \sf{dx^i} A_i{}^B T_B$, a 1-form over M valued in the [[Lie algebra]] of $G$.\n\nAny fiber element, $g$, at $t=0$ may be [[parallel transport]]ed to $g(t)=h(t)gh^-$ along a path on the base by a parameter dependent element, $h \sin G$, the path holonomy, $h(t) = Pe^{-\sint_0^t \sf{A}}$, satisfying the [[path holonomy]] equation,\n$$\n\sfr{d}{dt} h(t) = - \sve{v} \sf{A} h\n$$\n\nApplying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),\n\sbegin{eqnarray}\n\sf{\sna} \sf{\sna} g &=& \sf{d} \slp \sf{d} g + \sf{A} g - g \sf{A} \srp + \sf{A} \slp \sf{d} g + \sf{A} g - g \sf{A} \srp + \slp \sf{d} g + \sf{A} g - g \sf{A} \srp \sf{A} \s\s\n&=& \slp \sf{d} \sf{A} \srp g - \sf{A} \sf{d} g - \slp \sf{d} g \srp \sf{A} - g \sf{d} \sf{A} \n + \sf{A} \slp \sf{d} g + \sf{A} g - g \sf{A} \srp + \slp \sf{d} g + \sf{A} g - g \sf{A} \srp \sf{A} \s\s\n&=& \slb \sff{F} , g \srb\n\send{eqnarray}\ngives the ''automorphism bundle [[curvature]]'',\n$$\n\sff{F} = \sf{d} \sf{A} + \sf{A} \sf{A}\n$$\na Lie algebra valued 2-form. This expression for the curvature may alternatively be obtained from the [[holonomy]].\n\nUnder a gauge transformation, $g(x) \smapsto g'(x) = h(x) g(x) h^-(x)$, the covariant derivative changes to\n\sbegin{eqnarray}\n\sf{\sna'} g' &=& h \slp \sf{\sna} g \srp h^-\s\s\n\sf{d} \slp h g h^- \srp + \sf{A'} h g h^- - h g h^- \sf{A'} &=& h \slp \sf{d} g \srp h^- + h \sf{A} g h^- - h g \sf{A} h^-\n\send{eqnarray}\ngiving the transformation law for the connection,\n$$\n\sf{A'} = h \sf{A} h^- - \slp \sf{d} h \srp h^- = h \sf{A} h^- + h \slp \sf{d} h^- \srp \n$$\nAn infinitesimal transformation, $h \ssimeq 1 + H$, changes the connection to\n$$\n\sf{A'} \ssimeq \sf{A} - \sf{d} H - \sf{A} H + H \sf{A} = \sf{A} - \sf{\sna} H\n$$\nThe curvature consequently transforms under a gauge transformation to\n$$\n\sff{F'} = \sf{d} \sf{A'} + \sf{A'} \sf{A'} = h \sff{F} h^- \ssimeq \sff{F} + \slb H , \sff{F} \srb\n$$\n\nThe covariant derivative acting on a Lie algebra valued [[differential form]] such as the curvature, transforming under the adjoint action, $\sff{F'} = h \sff{F} h^-$, is still\n$$\n\sf{\sna} \sff{F} = \sf{d} \sff{F} + \slb \sf{A} , \sff{F} \srb \n$$\n\nIt is worth repeating that when the automorphism group for $G$ includes automorphisms that are not inner, things are going to get more complicated...\n\n//I'm starting to think that constructing this bundle just doesn't work. For one thing, I don't think I can make an [[Ehresmann connection]] for it since I haven't been able to build automorphism invariant vector fields. For another thing, automorphisms leave the identity point in the same place -- and singling out a point in the fiber like that would be an odd thing to do. I'll shelve the idea for now.//
review:\nhttp://arxiv.org/abs/hep-ph/9512245
<<ListTagged bohm>>
As a nice warm up, before tackling the standard model, it's instructive to see how the [[SU(3)]] [[Lie group geometry]] might go wobbly and become a [[Cartan geometry]]. It has a U(2) [[subgroup]], constructed from the [[simple]] [[SU(2)]] and U(1) Lie groups. The [[homogeneous space]], [[CP2]], is\n$$\nCP2 = \sfr{SU(3)}{U(2)} = \sfr{SU(3)}{SU(2) \stimes U(1)}\n$$
<<ListTagged brst>>
<<ListTagged cartan>>
The ''center'', $Z(G) \striangleleft G$, of a [[group]], $G$, is an abelian [[normal subgroup]] consisting of all elements of $G$ that commute with all other elements,\n$$\nZ(G) = \slc z \sin G \s; | \s; gz = zg \s; \sforall \s, g \sin G \src\n$$
The ''centralizer'', $C_G(a) \ssubset G$, of an element $a$ of a [[group]], $G$, is a [[subgroup]] consisting of all elements, $c \sin G$, that commute with $a$,\n$$\nC_G(a) = \slc c \sin G \s; | \s; ca = ac \src \n$$\nThe centralizer is the largest subgroup of $G$ having $a$ in its [[center]], $a \sin Z(C_G(a))$. The centralizer of a subset, $S$, in $G$ is the subgroup consisting of all elements commuting with the elements of $S$,\n$$\nC_G(S) = \slc c \sin G \s; | \s; cs = sc \s; \sforall s \sin S \src\n$$\nThe centralizer of $G$ in $G$ is the center, $C_G(G) = Z(G)$. The centralizer of a subset in a subgroup, $H \ssubset G$, is\n$$\nC_H(S) = \slc c \sin H \s; | \s; cs = sc \s; \sforall s \sin S \src\n$$
Refs:\n*Jeffrey A. Harvey\n**[[TASI 2004 Lectures on Anomalies|papers/0509097.pdf]]\n***good brief intro
A [[Clifford matrix representation]] is ''chiral'' if all [[Clifford basis vectors]] are represented by matrices non-zero only in the second and third quadrant blocks. This is the case iff the first [[Pauli matrix|Pauli matrices]] in the [[Kronecker product]] expression of each vector is either $\ssi^P_1$ or $\ssi^P_2$,\n$$\n\sga_\sal = \ssi^P_{1 \s, {\srm or} \s, 2} \sotimes \sdots\n$$\nA representation may be said to be chiral if this holds for the first Pauli matrices in the product, or it may be chiral for the level where the [[spin connection]] lives -- to be concrete, a rep should be called ''n'th level chiral'' if the n'th Kronecker product matrices are all $\ssi^P_1$ or $\ssi^P_2$. In a (1st level) chiral representation, all odd [[Clifford grade]] elements are represented by matrices non-zero only in the second and third quadrants, while all even elements are represented by matrices only non-zero in the first and fourth quadrant,\n$$\nA =\n\slb \sbegin{array}{cc}\nA^e & A^o \s\s\nA^o & A^e\n\send {array} \srb\n\sin Cl\n$$\n\nA ''chiral spinor'' is half of a [[spinor]]. Expressed in a chiral rep, it is either ''positive chiral'',\n$$\n\sPsi^+ = \n\slb \sbegin{array}{c}\n\sps^+ \s\s\n0\n\send {array} \srb\n$$\nor ''negative chiral'',\n$$\n\sPsi^- = \n\slb \sbegin{array}{c}\n0 \s\s\n\sps^-\n\send {array} \srb\n$$\non the level of chirality. A full spinor may be built by adding two chiral spinors, $\sPsi = \sPsi^+ + \sPsi^-$, such as [[Weyl spinor]]s. A spinor may also be broken up into n'th level chiral pieces.\n\nThe ''chirality projector'', $P_\spm$, is one of a pair of Clifford elements that operate on the suitable level to project out the desired chiral degrees of freedom,\n$$\n\sbegin{array}{cc}\nP_+ =\n\slb \sbegin{array}{cc}\n1 & 0 \s\s\n0 & 0\n\send {array} \srb &\nP_- =\n\slb \sbegin{array}{cc}\n0 & 0 \s\s\n0 & 1\n\send {array} \srb &\n\send{array}\n$$\nIt is often built using the [[pseudoscalar]].
*<<slider chkSliderdiracF diracF 'dirac »' 'Dirac operators, Dirac equation'>>\n<<ListTagged clifford>>
A [[differential form]] [[field|cotangent bundle]], $\snf{f}(x)$, over a [[manifold]], $M$, is ''closed'' iff its [[exterior derivative]] vanishes,\n$$\n\sf{d} \snf{f} = 0\n$$\nThe [[vector space]] of closed $p$-forms over $M$ is labeled $C^p$.
see Nakahara, p253\n//''adjoint exterior derivative''//\nFrankel, p\n\nLaplacian...\n\nRef:\nhttp://en.wikipedia.org/wiki/Codifferential
The $p$-th (de Rham) ''cohomology'' of a [[manifold]], $M$, is the [[vector space]],\n$$\nH^p(M) = C^p / E^p\n$$\nequal to the [[coset]] of all [[closed]] $p$-forms over $M$ that are not [[exact]]. An element of the cohomology is specified by a closed coset representative, $[\snf{f^C}] \sin H^p(M)$, with $\snf{f^C} \sin C^p$.
The ''commutator bracket'' (or simply //''commutator''//) of any two arbitrary [[Lie algebra]] generators, [[Clifford element]]s, or operators is another generator, element, or operator equal to\n$$\n\slb A, B \srb = A B - B A \n$$\nemploying the appropriate product or operator composition. It relates to the [[antisymmetric bracket]] (and cross product) by a factor of $\sha$,\n$$\nA \stimes B = \slb A, B \srb_A = \sha \slb A, B \srb = \sha \slp A B - B A \srp \n$$\n\nThe commutator (called more precisely the ''graded commutator'') does not commute [[coordinate basis 1-forms]] -- these must be taken out of the bracket first. For example, for the commutator of two grade 1 [[Lieform]]s or [[Clifform]]s,\n$$\n\slb \sf{A},\sf{B} \srb = \sf{dx^i} \sf{dx^j} \slb A_i, B_j \srb\n= \sf{dx^i} \sf{dx^j} \slp A_i B_j - B_j A_i \srp = \sf{A} \sf{B} + \sf{B} \sf{A}\n$$\nSo, in general, for two $p$ and $k$ forms,\n$$\n\slb \snf{A}, \snf{B} \srb = \snf{A} \snf{B} - \slp -1 \srp^{pk} \snf{B} \snf{A} \n$$\nand [[tangent vector]]s are considered $k$ forms with $k=-1$.
A ''connection'' completely encodes the local geometry of a [[fiber bundle]]. Specifically, it describes how the local trivializations change as one moves around on the base manifold. The group of these changes is the same as the structure group, $G$, of the fiber bundle. From any point, the infinitesimal change of a local trivialization when moving in any direction is described by the operation of a [[Lie algebra]] element. These changes may be described via a [[Lie algebra]] valued [[1-form]] over the base, the connection,\n$$\n\sf{A} = \sf{dx^i} A_i{}^B(x) T_B\n$$\nwith the appropriate action on the fiber elements. Using this connection, the [[covariant derivative]] of a section, $\ssi(x)$, (valued in the fiber) is\n$$\n\sf{\sna} \ssi = \sf{d} \ssi + \sf{A} \ssi = \sf{dx^i} \slp \spa_i \ssi + A_i{}^B T_B \ssi \srp\n$$\nin which the Lie algebra basis elements, $T_B$, act on the fiber. The connection changes under a [[gauge transformation]] so as to keep this derivative covariant.
The ''coordinate basis 1-forms'', $\sf{dx^i}$, are [[1-form]]s dual to the [[coordinate basis vectors]],\n\s[ \sf{dx^{i}}(\sve{\spartial_j}) = \sve{\spartial_j} \sf{dx^{i}} = \sdelta_{j}^{i} \s]
The ''coordinate basis forms'' (//''coordinate basis p-forms''//) are constructed by taking the [[wedge product]] of (p) [[coordinate basis 1-forms]],\n$$\n\snf{dx^{i \sdots j}} = \sf{dx^i} \sdots \sf{dx^j}\n$$\nFor example, the ''coordinate basis 2-forms'' are\n$$\n\sff{dx^{ij}} = \sf{dx^i} \sf{dx^j}\n$$\nThe wedge product between basis 1-forms is implied but never written, as the antisymmetric nature of the form product is assumed in the [[vector-form algebra]]. The coordinate basis forms are [[antisymmetric|index bracket]], changing sign under the interchange of any two adjacent indices, $\sff{dx^{ij}}=-\sff{dx^{ji}}$. On an $n$ dimensional manifold, the highest grade coordinate basis form is the ''coordinate basis $n$-form'',\n\s[ \snf{d^n x} = \sf{dx^0} \sdots \sf{dx^{n-1}} \s]\nTechnically, there is also a coordinate basis $0$-form,\n$$\n1\n$$\n\nFor any grade, $p$, there are $\sfrac{n!}{\sleft(n-p\sright)!p!}$ distinct coordinate basis $p$-forms. Adding these up over the $n+1$ possible grades, including the basis 0-form, there are $2^n$ distinct coordinate basis forms.
The coordinate basis vectors, \s[\n\sve{\spa_i} = \sve{\sfrac{\spartial}{\spartial x^i}} \sin T_p M\n\s] with coordinate [[index|indices]], $i$, span the space, $T_p M$, of [[tangent vector]]s to possible curves passing through each point, $p$, of a [[manifold]], $M$. The basis vectors may not be colinear, but are not otherwise inherently related &mdash; unlike the [[Clifford basis vectors]], they are not necessarily orthogonal or of unit length.\n\nThe coordinate basis vectors may be visualized as little arrows pointing along each coordinate curve. Each ''coordinate basis vector'', $\sve{\spa_i}$, is the [[tangent vector]] to the curve formed by varying the $x^i$ coordinate while holding the others fixed.
The coordinates, $x^i$, used to describe points in a [[manifold]] patch may always be abandoned in favor of a new set of coordinates, $y^i$. Since coordinates in the old and new set describe the same manifold points, the new coordinates may be written as functions of the old, $y^i(x)$, and the old as functions of the new, $x^i(y)$. Similarly, two such sets of coordinates must be used in coordinate patch overlaps on the manifold.\n\nThe [[coordinate basis vectors]] are different in the two sets of coordinates, and are related by the partial derivatives of the old and new coordinates as functions of each other:\n$$\n\sve{\spa^y_i} = \sve{\sfr{\spa}{\spa y^i}} = \sfr{\spa x^j}{\spa y^i} \sve{\sfr{\spa}{\spa x^j}} = \sfr{\spa x^j}{\spa y^i} \sve{\spa^x_j}\n\squad \squad \squad\n\sve{\spa^x_i} = \sve{\sfr{\spa}{\spa x^i}} = \sfr{\spa y^j}{\spa x^i} \sve{\sfr{\spa}{\spa y^j}} = \sfr{\spa y^j}{\spa x^i} \sve{\spa^y_j}\n$$\nWith the partial derivative matrices satisfying\n$$\n\sfr{\spa x^j}{\spa y^i} \sfr{\spa y^i}{\spa x^k} = \sde_k^j\n$$\nSince $x$ and $y$ are coordinates for the same point, the partial derivative matrices may equivalently be considered functions of $x$ or $y$ as necessary. Similarly, the [[coordinate basis 1-forms]] are related by\n$$\n\sf{dy^i} = \sfr{\spa y^i}{\spa x^j} \sf{dx^j}\n\squad \squad \squad\n\sf{dx^i} = \sfr{\spa x^i}{\spa y^j} \sf{dy^j}\n$$\nand the [[partial derivative]]s, $\spa_i$, of a function (or field components), $f(x)$, in different coordinate systems are related by\n$$\n\spa^x_i f(x) = \sfr{\spa}{\spa x^i} f(x) = \sfr{\spa y^j}{\spa x^i} \sfr{\spa}{\spa y^j} f(x(y)) = \sfr{\spa y^j}{\spa x^i} \spa^y_j f(y)\n$$\n\nA [[natural]] geometric object is invariant under coordinate change. For example, [[tangent vector]]s and [[1-form]]s may be expressed in terms of either set of coordinate basis vectors and forms,\n\sbegin{eqnarray}\n\sve{v} &=& v^i \sve{\sfr{\spa}{\spa x^i}} = v^i \sfr{\spa y^j}{\spa x^i} \sve{\sfr{\spa}{\spa y^j}} = v'^j \sve{\sfr{\spa}{\spa y^j}} = \sve{v'}\s\s\n\sf{f} &=& f_i \sf{dx^i} = f_i \sfr{\spa x^i}{\spa y^j} \sf{dy^j} = f'_j \sf{dy^j} = \sf{f'}\n\send{eqnarray}\nIn old terminology, tangent vectors are described by "contravariant" (upper) indexed components transforming as $v'^j = v^i \sfr{\spa y^j}{\spa x^i}$ and forms are described by "covariant" (lower) indexed components transforming as $f'_j = f_i \sfr{\spa x^i}{\spa y^j}$ under coordinate change. Any indexed object transforming this way under coordinate change is called a ''tensor''. \n\nAnother way of looking at coordinate change is as the identity map from the manifold to itself -- technically a [[diffeomorphism]]. However, a coordinate change is a ''passive diffeomorphism'' as it does not move the manifold points, but only mixes (re-labels) their coordinates.
A collection of elements called a ''coset'', $G/H$, can be formed by modding a [[group]], $G$, by a [[subgroup]], H. Specifically, a ''left coset'' element, $[g] \sin G/H$, consists of all elements of $G$ related by the right action of elements of $H$,\n$$\n[g] = gH = \sleft\s{ gh : \sforall \s; h \sin H \sright\s}\n$$\nA coset is not a group unless $H$ is a [[normal subgroup]], in which case $G/H$ is called the ''quotient group''.\n\nAn example, let $G$ be the set of integers,\n$$\nG = \sleft\s{ \sdots, -2, -1, 0, 1, 2, \sdots \sright\s}\n$$\nwith addition, $+$, as the group product. Choose the subgroup, $H$, to be all elements of $G$ that are multiples of $4$,\n$$\nH = \sleft\s{ \sdots, -8, -4, 0, 4, 8, \sdots \sright\s}\n$$\nThe left coset consists of four ( $=$ the ''index'' of $H$ in $G$) elements,\n\sbegin{eqnarray}\nG/H &=& \sleft\s{\n\slb \sdots, -3, 1, 5, \sdots \srb,\n\slb \sdots, -2, 2, 6, \sdots \srb,\n\slb \sdots, -1, 3, 7, \sdots \srb,\n\slb \sdots, 0, 4, 8, \sdots \srb \sright\s} \s\s\n&=& \sleft\s{ [1], [2], [3], [0] \sright\s}\n\send{eqnarray}\nThe notation "$[g]$" means that $g \sin G$ is a ''coset representitive'' -- any other representative related by $h \sin H$ is equivalent, $[g]=[gh]$. Every element of $G$ is in exactly one of the coset elements, and each coset element is isomorphic to $H$ -- in fact, one of the coset elements, $[0]$, is $H$. There is a product between coset elements determined by the product of their representatives and representative equivalence. For example, $[1]+[3] = [4] = [0]$. And, in this example, the coset does form a group, since $H$ is normal in $G$,\n$$\nghg^- = g + h - g = h \sin H\n$$\n\nA ''right coset'' element, $[g] \sin G/H$, consists of all elements of $G$ related by the left action of elements of $H$,\n$$\n[g] = Hg = \sleft\s{ hg : \sforall \s; h \sin H \sright\s}\n$$
<<ListTagged cosmo>>
The ''cotangent bundle'' (//''1-form bundle''//), $T^* M = \sOm^1 M$, with $n$ dimensional base [[manifold]], $M$, is a [[vector bundle]] with $n$ fiber basis elements identified as the [[coordinate basis 1-forms]], $\sf{dx^i}$, for the manifold. It is the dual bundle to the [[tangent bundle]]. The fiber at a base manifold point, $p$, is the $n$ dimensional cotangent space, $T_p^* M$, spanned by the basis 1-forms, and the cotangent bundle is the union of all cotangent spaces, $T^* M = \sbigcup_{p \sin M} T_p^* M$. The transition functions for the basis elements, $\sf{dx^i_2} = \slp t^{21} \srp_j{}^i \sf{dx^j_1}$, over overlapping patches, $U_1$ and $U_2$, are given by the ''Jacobian matrix'',\n$$\slp t^{21} \srp_j{}^i = \sfr{\spa x_2^i}{\spa x_1^j}$$\nThe structure group is thus the group of general linear transformations, $G = GL(n,\sRe)$. A ''covector field'' (//''1-form field''//), $\sf{f} = \sf{f}(x) = f_i(x) \sf{dx^i}$, over the manifold is a section of the cotangent bundle, and consists of a [[1-form]] at each manifold point.\n\nWhen a [[metric]] exists for the tangent bundle the [[frame]] basis forms, $\sf{e^\sal} = \sf{dx^i} \slp e_i \srp^\sal$, may alternatively be used as local fiber basis elements for the cotangent bundle. The transition functions are then [[Lorentz transformations|Lorentz rotation]], $\sf{e^\sal_2} = \slp L^{21} \srp_\sbe{}^\sal \sf{e^\sbe_1}$. This [[reduction of the structure group]] is the same as for the tangent bundle. Similarly, through equating the Lorentz transition functions and using the [[frame]], $\sve{e} \sf{e^\sal} = \sga^\sal$, the cotangent bundle may be [[associated]] to the [[Clifford vector bundle]].\n\nSince the cotangent bundle is dual to the tangent bundle, its geometric elements -- including the [[cotangent bundle connection]], holonomy, curvature, etc. -- are the dual constructions to those for the tangent bundle, and provide no new geometric insight.\n\nGrade $p$ [[differential form]] fields are sections of the ''p-form bundle'', $\sOmega^p M$, which has the $\sfrac{n!}{\sleft(n-p\sright)!p!}$ [[coordinate basis p-forms|coordinate basis forms]], $\snf{dx^{i \sdots j}}=\sf{dx^i} \sdots \sf{dx^j}$, as basis. The combined collection of these p-form product bundles is the ''differential form bundle'', $\sOmega M = \sbigoplus_{p=0}^{n} \sOmega^{p} M$, having dimension $2^{n}$.
The [[vector bundle connection]] for the [[cotangent bundle]] is defined through the operation of the suitable [[vector bundle covariant derivative|vector bundle connection]] on [[coordinate basis 1-forms]],\n\sbegin{eqnarray}\n\sna_i \sf{dx^j} &=& -\sGa^j{}_{ik} \sf{dx^k} \s\s\n\sf{\sna} \sf{dx^j} &=& -\sf{\sGa}^j{}_k \sf{dx^k} \n\send{eqnarray}\nThe coefficients, $\sGa^j{}_{ik}$, of the ''cotangent bundle connection'', $\sf{\sGa}^j{}_k = \sf{dx^i} \sGa^j{}_{ik}$, are referred to as the [[Christoffel symbols]], and the index positions are arranged to agree with convention rather than having the usual order for connection coefficients. These are the same Christoffel symbols that arise in the [[tangent bundle connection]] since the basis elements are duals,\n$$\n0 = \sna_i \sde_j^k = \sna_i \slp \sve{\spa_j} \sf{dx^k} \srp\n= \slp \sGa^m{}_{ij} \sve{\spa_m} \srp \sf{dx^k} - \sve{\spa_j} \slp \sGa^k{}_{im} \sf{dx^m} \srp\n= \sGa^k{}_{ij} - \sGa^k{}_{ij}\n$$\n\nAn alternative expression for the cotangent bundle connection may be found by calculating the covariant derivative of the [[vielbein 1-forms|frame]],\n\sbegin{eqnarray}\n\sna_i \sf{e^\sal} &=& \slp \spa_i \slp e_j \srp^\sal - \slp e_k \srp^\sal \sGa^k{}_{ij} \srp \sf{dx^j} = w_{i\sbe}{}^\sal \sf{e^\sbe} \s\s\n\sf{\sna} \sf{e^\sal} &=& \sf{d} \sf{e^\sal} - \sf{dx^i} \sf{dx^j} \slp e_k \srp^\sal \sGa^k{}_{ij} = \sf{d} \sf{e^\sal} - \sff{T^\sal} = \sf{w}{}_\sbe{}^\sal \sf{e^\sbe}\n\send{eqnarray}\nin which the [[tangent bundle spin connection|tangent bundle connection]] coefficients, $w_{i\sbe}{}^\sal$, appear. This last equation, involving the [[torsion]] coefficients, $T^\sal{}_{ij} = 2 \sGa^\sal{}_{\slb ij \srb}$, may be solved for the spin connection coefficients by solving [[Cartan's equation]].\n\nThe cotangent bundle covariant derivative extends via the [[distributive rule|derivation]] to act on [[differential form]]s of higher order,\n\sbegin{eqnarray}\n\sna_i \sf{dx^j} \sf{dx^k} \sdots \sf{dx^l} &=& \slp \sna_i \sf{dx^j} \srp \sf{dx^k} \sdots \sf{dx^l} + \sf{dx^j} \slp \sna_i \sf{dx^k} \srp \sdots \sf{dx^l} + \sf{dx^j} \sf{dx^k} \sdots \slp \sna_i \sf{dx^l} \srp \s\s\n&=& - \sGa^j{}_{im} \sf{dx^m} \sf{dx^k} \sdots \sf{dx^l} - \sf{dx^j} \sGa^k{}_{im} \sf{dx^m} \sdots \sf{dx^l} - \sf{dx^j} \sf{dx^k} \sdots \sGa^l{}_{im} \sf{dx^m}\n\send{eqnarray}
The ''covariant derivative'' operator is a grade $1$ [[derivative|derivation]] of a [[fiber bundle]] section (field) that accounts for the local trivialization (change of basis) via the appropriate [[connection]]. It may be written as a [[1-form]] operator, or as a derivative with respect to a specific coordinate direction,\n$$\n\sf{\sna} = \sf{dx^i} \sna_i\n$$\nIt is defined to have the following properties,\n$$\n\sf{\sna} \slp \snf{B} + f \snf{C} \srp = \sf{\sna} \s,\snf{B} + f \sf{\sna} \s, \snf{C} + \slp \sf{d} f \srp \snf{C}\n$$\nwhere $f$ is any scalar [[function]] over the base manifold, $\sf{d}$ is the [[exterior derivative]], and $\snf{B}$ and $\snf{C}$ are any tangent vector, differential form, Clifford element, or generally any fiber bundle section or direct product of sections. Using the [[partial derivative]] and connection, the covariant derivative of a section is\n\sbegin{eqnarray}\n\sna_i B &=& \spa_i B + A_i B \s\s\n\sf{\sna} B &=& \sf{\spa} B + \sf{A} B\n\send{eqnarray}\nA section is ''horizontal'' at a point iff its covariant derivative vanishes, $\sf{\sna} B = 0$.\n\nIn general, the covariant derivative may be defined for any fiber valued form that varies under a [[gauge transformation]] as $\snf{B'} = g \s, \snf{B}$. Using the [[exterior derivative]],\n$$\n\sf{\sna} \snf{B} = \sf{d} \snf{B} + \sf{A} \snf{B}\n$$\nThis operator generalizes further to a covariant derivative operating as forms that transform arbitrarily under the gauge group and aren't necessarily sections. For example, operating on the the [[curvature]], which transforms as $\sff{F'} = g \s, \sff{F} \s, g^-$, the covariant derivative is\n$$\n\sf{\sna} \sff{F} = \sf{d} \sff{F} + \sf{A} \sff{F} - \sff{F} \sf{A}\n$$\n"Covariance" refers to the property that the covariant derivative of any field transforms under the same group action as the field. The covariant derivative thus plays an essential role in constructing gauge invariant objects, and this restriction provides the rule for the behavior of the connection under gauge transformation. When it is not obvious, the covariant derivative should be labeled with the symbol(s) of the connection(s) for the bundle for which it is covariant,\n$$\n\sf{\sna^A} C = \sf{dx^i} \sna^A_i C = \sf{dx^i} \slp \spa_i C + A_i C \srp = \sf{d} C + \sf{A} C\n$$
<<ListTagged cp2>>
The curvature is perhaps the most important object characterizing the local geometry of a [[fiber bundle]] and [[connection]]. Its expression and action depends on the action of the structure group. Taking, for example, the structure group to act from the left, the ''curvature'' is then a local, [[Lie algebra]] (of the structure group) valued [[2-form|differential form]] defined as\n\sbegin{eqnarray}\n\sff{F}(x) &=& \sha \sf{dx^i} \sf{dx^j} F_{ij}{}^B T_B \s\s\n&=& \sf{d} \sf{A} + \sf{A} \sf{A} \s\s\n&=& \sf{d} \sf{A} + \sf{A} \stimes \sf{A} \s\s\n&=& \sf{d} \sf{A} + \sha \slb \sf{A}, \sf{A} \srb\n\send{eqnarray}\nThe curvature coefficients are\n$$\nF_{ij}{}^C = \spa_i A_j{}^C - \spa_j A_i{}^C + A_i{}^A A_j{}^B C_{AB}{}^C \n$$\nin which $C_{AB}{}^C$ are the [[structure constants|Lie algebra]].\n\nThe curvature is most intuitively derived in terms of the [[holonomy]] of infinitesimal loops.\n\nIt may also be derived by applying the [[covariant derivative]] twice to any fiber bundle section,\n$$\n\sf{\sna} \sf{\sna} C = \slp \sf{d} + \sf{A} \srp \slp \sf{d} + \sf{A} \srp C =\n\sf{d} \sf{d} C + \sf{d} \sf{A} C + \sf{A} \sf{d} C + \sf{A} \sf{A} C\n = \slp \sf{d} \sf{A} + \sf{A} \sf{A} \srp C = \sff{F} C\n$$ \n\nThe curvature changes under [[gauge transformation]]s, $C \smapsto C'=gC$, as $\sff{F} \smapsto \sff{F'}=g \sff{F} g^-$.\n\nNote again that the expression of the curvature, and its action, depends on the form of the group action.
[[Contracting|vector-form algebra]] the [[coordinate basis vectors]] with the [[Ricci curvature]] and the [[tangent bundle]] [[metric]] gives the ''curvature scalar'' (//''Ricci scalar''//),\n$$\nR = g^{im} \sve{\spa_i} \sf{R}{}_m = R^i{}_i\n$$\nThis equals a full contraction of the [[Riemann curvature]] tensor,\n$$\nR = g^{mj} R_{ij}{}^i{}_m = 2 g^{mj} \slp \spa_{\slb i \srd} \sGa^i{}_{\sld j \srb m} + \sGa^i{}_{\slb i \srd l} \sGa^l{}_{\sld j \srb m} \srp\n$$\nIn terms of the [[tangent bundle spin connection|tangent bundle connection]] and [[frame]], the curvature scalar is\n$$\nR = \sve{e^\sal} \sf{R}{}_\sal = \sve{e^\sal} \sve{e_\sbe} \sff{R}^\sbe{}_\sal\n= 2 \slp e_\sal \srp^j \slp e_\sbe \srp^i \slp \spa_{\slb i \srd} w_{\sld j \srb}{}^\sbe{}_\sal + w_{\slb i \srd}{}^\sbe{}_\sga w_{\sld j \srb}{}^\sga{}_\sal \srp\n$$\nIf the spin connection is torsionless, the curvature scalar may also be written as\n\sbegin{eqnarray}\nR = R_\sal{}^\sal &=& 2 \spa_\sbe w_\sal{}^{\sbe \sal} + w_{\sbe \sal}{}^\sep w_\sep{}^{\sbe \sal} - w_\sbe{}^{\sep \sbe} w_{\sal \sep}{}^\sal\n\send{eqnarray}
''De Sitter spacetime'' is the unique geometry of a [[spacetime]], $M$, satisfying [[Einstein's equation]] with no matter and a positive cosmological constant, $\sLa$. It may be embedded in the five dimensional, flat, Lorentzian $(\set_{00}=-1)$ spacetime, $\smathbb{R}(1,4)$ -- in which it is a [[hyperboloid|http://en.wikipedia.org/wiki/Hyperboloid]] of one sheet:\n$$\nx^0 x^0 - x^w x^w = - \sal^2\n$$\nThe spaces corresponding to each time, $x^0=t$, are [[3-sphere]]s -- growing larger for $t>0$ and for $t<0$. De Sitter spacetime may also be described as a [[homogeneous space]], $M = SO(1,4)/SO(1,3)$.\n\nThe geometry is concisely expressed by the [[frame]],\n$$\n\sf{e} = \sf{dt} \sga_0 + \sf{d a^1} \sal \scosh(\sfr{t}{\sal}) \sga_1 + \sf{d a^2} \sal \scosh(\sfr{t}{\sal}) \ssin(a^1) \sga_2 \n+ \sf{d a^3} \sal \scosh(\sfr{t}{\sal}) \ssin(a^1) \ssin(a^2) \sga_3\n$$\nemploying the angular coordinates of the spatial 3-spheres, $(x^0,x^1,x^2,x^3)=(t,a^1,a^2,a^3)$. The coframe is\n$$\n\sve{e} = \sga^0 \sve{\spa_t} + \sga^1 \sfr{1}{\sal \scosh(\sfr{t}{\sal})} \sve{\spa_1} + \sga^2 \sfr{1}{\sal \scosh(\sfr{t}{\sal}) \ssin(a^1)} \sve{\spa_2} + \sga^3 \sfr{1}{\sal \scosh(\sfr{t}{\sal}) \ssin(a^1) \ssin(a^2)} \sve{\spa_3} \n$$\nThe [[torsion]]less [[spin connection]] for the [[Clifford vector bundle]], found by solving [[Cartan's equation]], $0=\sf{d} \sf{e} + \sf{\som} \stimes \sf{e}$, is\n\sbegin{eqnarray}\n\sf{\som} &=& - \sve{e} \stimes \sf{d} \sf{e} + \sfr{1}{4} \slp \sve{e} \stimes \sve{e} \srp \slp \sf{e} \scdot \sf{d} \sf{e} \srp \s\s\n&=& -\sf{d a^1} \ssinh(\sfr{t}{\sal}) \sga_{01}\n-\sf{d a^2} \ssinh(\sfr{t}{\sal}) \ssin(a^1) \sga_{02}\n+\sf{d a^2} \scos(a^1) \sga_{12} \s\s\n&-&\sf{d a^3} \ssinh(\sfr{t}{\sal}) \ssin(a^1) \ssin(a^2) \sga_{03}\n+\sf{d a^3} \scos(a^1) \ssin(a^2) \sga_{13}\n+\sf{d a^3} \scos(a^2) \sga_{23}\n\send{eqnarray}\nThe [[Clifford-Riemann curvature]] is\n\sbegin{eqnarray}\n\sff{R} &=& \sf{d} \sf{\som} + \sha \sf{\som} \sf{\som} \s\s\n&=& - \sf{d t} \sf{d a^1} \sfr{1}{\sal} \scosh(\sfr{t}{\sal}) \sga_{01}\n- \sf{d t} \sf{d a^2} \sfr{1}{\sal} \scosh(\sfr{t}{\sal}) \ssin(a^1) \sga_{02}\n- \sf{d t} \sf{d a^3} \sfr{1}{\sal} \scosh(\sfr{t}{\sal}) \ssin(a^1) \ssin(a^2) \sga_{03} \s\s\n&-& \sf{d a^1} \sf{d a^2} \scosh^2(\sfr{t}{\sal}) \ssin(a^1) \sga_{12}\n- \sf{d a^1} \sf{d a^3} \scosh^2(\sfr{t}{\sal}) \ssin(a^1) \ssin(a^2) \sga_{13}\n- \sf{d a^2} \sf{d a^3} \scosh^2(\sfr{t}{\sal}) \ssin^2(a^1) \ssin(a^2) \sga_{23} \s\s\n&=& - \sfr{1}{2 \sal^2} \sf{e} \sf{e}\n\send{eqnarray}\nThe [[Clifford-Ricci curvature]] is\n$$\n\sf{R} = \sve{e} \stimes \sff{R} = - \sfr{1}{2 \sal^2} \sve{e} \stimes \sf{e} \sf{e} = - \sfr{3}{\sal^2} \sf{e} = - \sLa \sf{e}\n$$\nshowing that the de Sitter spacetime satisfies the vacuum [[Einstein's equation]] with positive cosmological constant, $\sLa = \sfr{3}{\sal^2}$.\n\nSince our universe appears to have a positive cosmological constant, at large $t$ this dominates the matter content and our universe is well approximated by a de Sitter spacetime at large $t$. In such a universe, the distant galaxies will accelerate away from us until, at the ''de Sitter horizon'', they are receding from us faster than the speed of light -- so their light cannot reach us. Our galaxy appears to have a lonely future.\n\nRef:\n*http://en.wikipedia.org/wiki/De_Sitter_space\n*[[The Case for a Gravitational de Sitter Gauge Theory|papers/9610068.pdf]]\n**Overview of how a Poincare gauge theory description of gravity, which has no Lagrangian formulation, needs to have terms added in order to be renormalizeable. These terms turn out to produce de Sitter gauge theory, with Lagrangian.\n*[[Some Implications of the Cosmological Constant to Fundamental Physics|papers/0702065.pdf]]
Any graded operator, $\snf{D}$, that acts distributively over [[products|vector-form algebra]] of [[differential form]]s of grades $f$ and $g$ according to the ''graded Liebniz rule'',\n$$\n\snf{D} \snf{F} \snf{G} = \slp \snf{D} \snf{F} \srp \snf{G} + \slp -1 \srp^{df} \snf{F} \snf{D} \snf{G}\n$$\nis a ''graded derivation'' (or simply //''derivation''//) of grade $d$.\n\nThe most general grade $d$ derivation operator may be written as\n$$\n\snf{D} = {\scal L}_{\snf{\sve{K}}} + \snf{\sve{L}}\n$$\nin which $\snf{\sve{K}}$ is a vector valued $d$-form (a [[vector valued form]] of total grade $(d-1)$) and $\snf{\sve{L}}$ is a vector valued $(d+1)$-form (a vector valued form of grade $d$) and ${\scal L}$ is the [[FuN derivative]].
The ''determinant'' of a real $n\stimes n$(square) matrix, such as the [[frame]] matrix, $A_i{}^\sal$, is a real number equal to\n\sbegin{eqnarray}\n\sll A \srl &=& \sdet ( A_i{}^\sal ) = \sva^{ij\sdots k} A_i{}^0 A_j{}^1 \sdots A_k{}^{n-1} \s\s\n&=& \sva^{ij\sdots k} A_i{}^\sal A_j{}^\sbe \sdots A_k{}^\sga \sfr{1}{n!} \sep_{\sal \sbe \sdots \sga} \s\s\n&=& A_0{}^\sal A_1{}^\sbe \sdots A_{n-1}{}^\sga \sep_{\sal \sbe \sdots \sga}\n\send{eqnarray}\n(using [[permutation symbol]]s)\n\nThe determinant of an [[exponentiated|exponentiation]] matrix is the exponential of the [[trace]] of the matrix,\n$$\n\sll e^A \srl = e^{\sli A \sri}\n$$
*<<slider chkSliderfbF fbF 'fb »' 'fiber bundles'>>\n<<ListTagged dg>>
A ''diffeomorphism'' is a smooth invertible map from one [[manifold]] to another, or to itself. A diffeomorphism from a manifold to itself is also an [[automorphism]], so should probably be called an autodiffeomorphism -- but it's usually just called a diffeomorphism of a manifold.\n\nAn ''autodiffeomorphism'', $\sph : x \smapsto y$, typically maps points, $x$, of a manifold, $M$, to different points, $y$, of $M$. Typically each of these points is identified by coordinates, $x^i$ and $y^i$, on manifold patches, and a diffeomorphism is written as $y^i(x)$ or as $\sphi^i(x)$. If these points are all the same, $x=y \sforall x$, it is a passive diffeomorphism, doing nothing but [[changing the coordinates|coordinate change]].\n\nDifferential forms [[pull back|pullback]] and tangent vectors push forward under a diffeomorphism; both pull back and push forward under an autodiffeomorphism.
A ''differential form'', or //''p-form''//, or //''grade p form''// is a geometric object acting antisymmetrically on $p$ [[tangent vector]]s at a point to give a real number. It generalizes [[1-form]]s, and may be visualized as a $p$ dimensional volume element sitting at a [[manifold]] point. A ''2-form'' may be written in terms of real coefficients times the [[coordinate basis forms]] as\n\s[ \sff{a} = \sha a_{ij} \sf{dx^i}\sf{dx^j} \s]\nSuch a 2-form may be visualized as an infinitesimal area element. The coefficients are [[antisymmetric|index bracket]] in the indices,\n\s[ a_{ij} = a_{\slb ij \srb} = \sha \slp a_{ij} - a_{ji} \srp \s]\nA general p-form may be written as\n\s[ \snf{b} = \sfr{1}{p!} b_{i \sdots k} \sf{dx^i} \sdots \sf{dx^k} \s]\nThe ''vector-form decoration'' convention requires that [[tangent vector]]s have an over-arrow, while grade $p$ forms have $p$ under-arrows or an under-bar if $p$ is unspecified or greater than 2. Multi-tangent vectors of grade $p$, which arise in [[vector-form algebra]], may be referred to as ''(-p)-form''s and are decorated with $p$ over-arrows or an over-bar. Unlike the case for [[Clifford element]]s, no use is made of differential forms of mixed grade. In an $n$ dimensional manifold, the highest grade form is an $n$-form,\n\s[ \snf{z} = z_v \sf{dx^0} \sdots \sf{dx^{n-1}} = z_v \snf{d^n x} \s]\nin which $\snf{d^n x}$ is the coordinate basis n-form. Also, technically, a real number at a manifold point is a 0-form.\n\nAny differential form may also be written in terms of the [[frame]] basis forms as\n\s[ \snf{b} = \sfr{1}{p!} b_{\sal \sdots \sbe} \sf{e^\sal} \sdots \sf{e^\sbe} \s]
*<<slider chkSliderspinF spinF 'spin »' 'spin odds and ends (more theoretical then dirac)'>>\n<<ListTagged dirac>>
Consider an $n$ dimensional [[manifold]] and its [[tangent bundle]]. At each manifold point, $x$, the tangent bundle fiber, $V_x = T_x M$, is a vector space spanned by the $n$ [[coordinate basis vectors]], $\sve{\spa_i} \sin V_x$. An $m$ dimensional subspace, $V^s_x \ssubset V_x$, can be spanned at each manifold point by a set of $m$ linearly independent basis vectors, $\sve{s_a}$. A collection of such subspaces, one defined at each manifold point, is a ''distrubution'', \n$$\n\sve{\sDe} = \sleft\s{ \sve{s_1}, \sve{s_2}, \sdots, \sve{s_m} \sright\s}\n$$\nspecified by $m$ linearly independent vector fields, $\sve{s_a}(x)$. A distribution is a subbundle of the tangent bundle.\n\nA distribution is ''involutive'' (//in ''involution''//) iff the basis vector fields of the distribution close under the [[Lie bracket|Lie derivative]],\n$$\n\slb \sve{s_a}, \sve{s_b} \srb_L \sin V^s\n$$\nThis is sometimes written as $\slb \sve{\sDe}, \sve{\sDe} \srb_L = \sve{\sDe}$. An involutive distribution may be integrated to give the ''foliation'' of the manifold by the collection of $m$ dimensional [[submanifold]]s which have $V^s_x$ as their tangent vector space at each point.\n\n
The ''divergence'' of a [[vector field|tangent bundle]], $\sve{v}$ is a function describing how much the vector field is spreading away from (or converging towards) each point. The divergence operator takes a vector field as argument and returns a real valued field, and is defined implicitly by the [[Lie derivative]] of the [[volume form]] along the vector field,\n$$\n{\scal L}_{\sve{v}} \snf{e} = \sf{d} \slp \sve{v} \snf{e} \srp = \snf{e} \s, \smathrm{div}(\sve{v})\n$$\nIn terms of the [[tangent bundle covariant derivative|tangent bundle connection]] it is\n$$\n\smathrm{div}(\sve{v}) = D_i v^i = \spa_i v^i + v^j \sGa^i{}_{ij} = D_\sal v^\sal = \spa_\sal v^\sal + v^\sal \som_\sbe{}^\sbe{}_\sal \n$$\nThis involves the [[trace]] of the [[Christoffel symbols]] or [[spin connection]], which can be used to produce an important formula relating the divergence to the [[partial derivative]] of the [[frame determinant|volume form]],\n$$\n\spa_i \sll e \srl v^i = \sll e \srl D_i v^i\n$$\n
The idea that Lorentz transformations should be modified to preserve a constant minimum length as well as the speed of light.\n\nThis makes sense, since it's basically a modification of special relativity to take place in a [[de Sitter spacetime]] instead of Minkowski space -- which is appropriate in a universe with a positive cosmological constant. But I don't see how the effect is going to be measurable in QFT, since this [[spacetime]] curvature is usually so small. Proponents of "DSR" claim the effect is increased locally by large local energy density. But it seems to me like a hack -- and what we really want to do is QFT in an arbitrarily curved spacetime.\n\nGood introductory paper:\nhttp://arxiv.org/abs/gr-qc/0207085\n\nSpeculation:\nA momentum cutoff... maybe the momentum is a closed manifold rather than a plane, similar to how the tangent space in [[Cartan geometry]] is a curved surface rather than a plane. Ah, this is supported by these papers:\nhttp://arxiv.org/abs/hep-th/0207279\nhttp://arxiv.org/abs/gr-qc/0612093\nand mentioned here:\nhttp://math.ucr.edu/home/baez/week232.html\nHey, does Derek's paper on [[Cartan geometry]] mention that?\n\nah, here, relation to [[de Sitter gravity]]:\n*[[de Sitter special relativity|paper/0606122.pdf]]\n**wow, the connection between this and a minimal length is rather tenuous -- it relies on the assumption that a high energy process will change the local value of $\sLa$. Why would that happen? Should consider non-constant $\sLa$...\n**basically, approximate the whole universe by de Sitter spacetime instead of by a flat Minkowski spacetime, and do particle physics in this universe the way it's normally done in Minkowski. This is less general than our reality, which is a bumpy spacetime.\n\nHmm, Lorentz transformations shouldn't need to be modified to preserve a minimal finite [[proper time]].
The rank $6$ exceptional [[Lie group]], [[E6]], is described by its $78$ dimensional [[Lie algebra]], ''e6''. This Lie algebra may be decomposed as a $45$ dimensional [[symmetric|symmetric space]] [[subgroup]], the [[special orthogonal group]] Lie algebra, $so(10)$, acting on the $32$ dimensional space of, real, positive, [[Cl(10)]] [[spinor]]s, $S^{\slp10\srp}$, and a $u(1)$,\n$$\ne6 = so(10) \soplus u(1) \soplus S^{\slp10\srp}\n$$\nAlso, using [[f4]] and its fundamental representation,\n$$\ne6 = f4 + 26\n$$\nThe fundamental representation of $e6$ is $27$.
The rank $8$ exceptional [[Lie group]], [[E8]], is described by its $248$ dimensional [[Lie algebra]], ''e8''. This Lie algebra may be decomposed as a $120$ dimensional [[symmetric|symmetric space]] [[subgroup]], the [[special orthogonal group]] Lie algebra, $so(16)$, acting on the $128$ dimensional space of, real, positive [[chiral]], [[Cl(16)]] [[spinor]]s, $S^{\slp16\srp+}$. In this way, any $e8$ element may be written in terms of basis generators as:\n\sbegin{eqnarray}\nE &=& B + \sPs = \sha b^{\sal\sbe} \sga^{(16)+}_{\sal\sbe} + \sps^a Q^+_a \s\s\n&& \sin so(16)^+ + S^{(16)+} = {\srm Lie}(E8)\n\send{eqnarray}\nExplicitly, a $so(16)$ element is expressed above as the first (upper left) quadrant of a Cl(16) bivector, $B = \sha b^{\sal \sbe} \sga^{\slp16\srp+}_{\sal \sbe}$, in a real, [[chiral]] [[Clifford matrix representation]], with $1 \sle \sal,\sbe \sle 16$. This is a $128\stimes128$ real, antisymmetric matrix that is part of a $256\stimes256$ dimensional matrix of the Cl(16) rep. In terms of matrix components, with matrix indices $1 \sle a,b \sle 128$, this positive chiral part of the bivector is\n$$\n\slp B \srp^a{}_b = \sha b^{\sal \sbe} \slp \sga^{\slp16\srp+}_{\sal \sbe} \srp^a{}_b\n$$\nThe $120$ unique, positive chiral, basis bivectors, $\sga^{\slp16\srp+}_{\sal \sbe} \ssim T_A$, are Lie algebra generators of $so(16)$ and of $e8$, represented as $128\stimes128$ matrices. A positive chiral, real spinor, $\sPs = \sps^a Q^+_a$, is a column of $128$ real numbers on which these bivectors act. In terms of matrix components, these generators are $\slp Q^+_a \srp^b = \sde_a^b$ and this spinor is $\slp \sPs \srp^b = \sps^b$. The action of a bivector on a spinor, written three different ways, is:\n\sbegin{eqnarray}\n\sps'^{a} &=& \slp B \srp^a{}_b \spsi^b \s\s\n\sps'^{d} \slp Q^+_d \srp^a &=& \sha b^{\sal \sbe} \slp \sga^{\slp16\srp+}_{\sal \sbe} \srp^a{}_b \spsi^{c} \slp Q^+_c \srp^b \s\s\n\sPs' = \sps'^c Q^+_c &=& B \sPs = \sha b^{\sal \sbe} \spsi^{c} \sga^{\slp16\srp+}_{\sal \sbe} Q^+_c\n\send{eqnarray} \nThe $248$ dimensional Lie algebra, e8, is spanned by these two sets of generators. The Lie brackets between bivectors, and between bivector generators, are determined by a [[Clifford basis product identity|Clifford basis product identities]],\n\sbegin{eqnarray}\n\slb B_1, B_2 \srb &=& B_1 B_2 - B_2 B_1 \s\s\n\slb \sga^{\slp16\srp+}_{\sal \sbe}, \sga^{\slp16\srp+}_{\sga \sde} \srb &=& 2 \sleft\s{ - \set_{\sal \sga} \sga^{\slp16\srp+}_{\sbe \sde} + \set_{\sal \sde} \sga^{\slp16\srp+}_{\sbe \sga} + \set_{\sbe \sga} \sga^{\slp16\srp+}_{\sal \sde} - \set_{\sbe \sde} \sga^{\slp16\srp+}_{\sal \sga} \sright\s}\n\send{eqnarray}\ngiving the same structure constants as for the special orthogonal group,\n$$\nC_{\slb\sal\sbe\srb\slb\sga\sde\srb}{}^{\slb\sep\sup\srb} = 2 \sleft\s{ - \set_{\sal \sga} \sde^{\slb\sep \sup\srb}_{\sbe \sde} + \set_{\sal \sde} \sde^{\slb\sep \sup\srb}_{\sbe \sga} + \set_{\sbe \sga} \sde^{\slb\sep \sup\srb}_{\sal \sde} - \set_{\sbe \sde} \sde^{\slb\sep \sup\srb}_{\sal \sga} \sright\s}\n$$\nin which the appropriate Clifford algebra metric for Cl(16,0) is $\set_{\sal \sbe} = \sde_{\sal \sbe}$. The Lie brackets between bivector and spinor, and between their generators, are\n\sbegin{eqnarray}\n\slb B, \sPs \srb &=& B \sPs \s\s\n\slb \sga^{\slp16\srp+}_{\sal \sbe}, Q^+_a \srb &=& \sga^{\slp16\srp+}_{\sal \sbe} Q^+_a = \slp \sga^{\slp16\srp+}_{\sal \sbe} \srp^b{}_c \slp Q^+_a \srp^c Q^+_b\n\send{eqnarray}\ngiving structure constants:\n$$\nC_{\slb\sal\sbe\srb a}{}^{b} = \slp \sga^{\slp16\srp+}_{\sal \sbe} \srp^b{}_a = - \slp \sga^{\slp16\srp+}_{\sal \sbe} \srp_a{}^b\n$$\nFinally, the e8 Lie algebra description is completed by letting the structure constants be completely antisymmetric -- the [[Killing form]] identity,\n$$\nC_{ab}{}^{\slb\sal\sbe\srb} = C^{\slb\sal\sbe\srb}{}_{a b} = \slp {\sga^{\slp16\srp+}}^{\sal \sbe} \srp_{ba} = - \slp {\sga^{\slp16\srp+}}^{\sal \sbe} \srp_{ab}\n$$\ngiving the Lie brackets between spinor generators and elements,\n\sbegin{eqnarray}\n\slb Q^+_a, Q^+_b \srb &=& - \slp {\sga^{\slp16\srp+}}^{\sal \sbe} \srp_{ab} \sga^{\slp16\srp+}_{\sal \sbe} \s\s\n\slb \sPs_1, \sPs_2 \srb &=& - \sps_1^a \sps_2^b \slp {\sga^{\slp16\srp+}}^{\sal \sbe} \srp_{ab} \sga^{\slp16\srp+}_{\sal \sbe}\n\send{eqnarray}\n\nTo summarize the above expressions, if elements of $e8$ are expressed as a combinations of $16\stimes16$ antisymmetric matrices of coefficients, $b = - b^T$, and $128$ elements columns, $\sps$, then the $e8$ Lie brackets can be defined heuristically in terms of matrix operations between these elements as:\n\sbegin{eqnarray}\n\slb b_1, b_2 \srb_{e8} &=& 2 \slp b_1 \set b_2 - b_2 \set b_1 \srp \s\s\n\slb b, \sps \srb_{e8} &=& \sbig< \sha b \sga^{(16)+} \sbig> \sps \s\s\n\slb \sps_1, \sps_2 \srb_{e8} &=& - 2 \sbig( \sps_1^T \sga^{(16)+} \sps_2 \sbig)_B\n\send{eqnarray}\n\nThe Killing form for e8 is\n\sbegin{eqnarray}\ng_{\slb \sal \sbe \srb \slb \sga \sde \srb} &=&\nC_{\slb \sal \sbe \srb \slb \sep \sup \srb}{}^{\slb \sze \set \srb} C_{\slb \sga \sde \srb \slb \sze \set \srb}{}^{\slb \sep \sup \srb}\n+ C_{\slb \sal \sbe \srb a}{}^b C_{\slb \sga \sde \srb b}{}^a\n= 240 \slp \set_{\sal \sde} \set_{\sbe \sga} - \set_{\sal \sga} \set_{\sbe \sde} \srp \s\s\ng_{ab} &=& 2 C_{a \slb \sal \sbe \srb}{}^c C_{b c}{}^{\slb \sal \sbe \srb}\n= 2 \slp \sga^{\slp16\srp+}_{\sal \sbe} \srp_a{}^c \slp {\sga^{\slp16\srp+}}^{\sal \sbe} \srp_{cb} = 2 \slp \sde_\sal^\sbe \sde_\sbe^\sal - \sde_\sal^\sal \sde_\sbe^\sbe \srp \sde_{ab} = - 480 \s, \sde_{ab} \n\send{eqnarray}\n\nSince we use a real representation for Cl(16,0), the above describes the compact real form of E8. If we use a complex rep for Cl(16,0), we get a complex form of e8. And if we use a Clifford algebra of different signature, like [[Cl(1,15)]], we get a (real or complex) non-compact E8. For all of these choices, the above structure constants remain symbolically the same, with the appropriate choice of $\set_{\sal \sbe}$. Note though that it is always necessary to choose a chiral rep for Cl.\n\nThe $E8$ Lie group has many subgroups other than the [[SO(16)]] which has been discussed above. Two maximal subgroups of $E8$ are $(E7 \stimes SU(2))/(\smathbb{ Z}/2 \smathbb{ Z})$ and $(E6 \stimes SU(3))/(\smathbb{ Z}/3 \smathbb{ Z})$ -- involving the other exceptional groups, [[E7]] and [[E6]], and the special unitary groups, [[SU(3)]] and [[SU(2)]]. One particularly interesting way $e8$ can be broken down is:\n\sbegin{eqnarray}\ne8 &=& e6 + su(3) + 54 \stimes 3 \s\s\n &=& so(10) + u(1) + 32 + su(3) + 54 \stimes 3\s\s\n &=& so(4) + su(2) + su(2) + u(1) + 4 \stimes 8 + u(1) + 32 + su(3) + 54 \stimes 3\n\send{eqnarray}\nYet another way $e8$ can be broken up is via the [[e8 triality decomposition]]:\n\sbegin{eqnarray}\ne8 &=& so(8) + so(8) + 3 \stimes 8 \stimes 8 \s\s\n &=& so(4) + so(4) + 4 \stimes 4 + so(6) + so(2) + 6 \stimes 2 + 3 \stimes 8 \stimes 8 \s\s\n &=& so(4) + su(2) + su(2) + 4 \stimes 4 + su(4) + u(1) + 6 \stimes 2 + 3 \stimes 8 \stimes 8\n\send{eqnarray}\nI'm currently trying to use these to build a [[T.O.E.|theory of everything]]\n\nRef:\n*[[http://en.wikipedia.org/wiki/E8_(mathematics)|http://en.wikipedia.org/wiki/E8_(mathematics)]]\n*G,S,&W, [[Superstring Theory|http://www.amazon.com/Superstring-Cambridge-Monographs-Mathematical-Physics/dp/0521357527/ref=pd_bbs_sr_3/104-9709999-3726336?ie=UTF8&s=books&qid=1179001057&sr=8-3]]\n*J. F. Adams, [[Lectures on Exceptional Lie groups|http://www.amazon.com/gp/reader/0226005267/ref=sib_dp_pt/104-6593454-7361512#reader-link]]\n*S. Adler\n**[[Should E8 SUSY Yang-Mills be Reconsidered as a Family Unification Model?|http://arxiv.org/abs/hep-ph/0201009]]\n*P. Ramond\n**[[Exceptional Groups and Physics|http://arxiv.org/abs/hep-th/0301050]]
[[e8]] [[Lie algebra structure]]\n\n@@display:block;text-align:center;[img[images/png/e8 root system.png]]@@\n\nRef:\n*[[David Richter|http://homepages.wmich.edu/~drichter/]]\n**[[Triacontagonal coordinates for the E8 root system|papers/0704.3091.pdf]]\n**[[Gosset's figure in a Clifford algebra|papers/gossetfigurecliffordalgebra2004.pdf]]\n*Mark W Hopkins? (sci.physics.research poster)\n**[[Standard Model|papers/MH Standard Model.pdf]]\n*Richard Koch\n**[[HyperSolids|http://www.uoregon.edu/~koch/hypersolids/hypersolids.html]]
The [[e8]] [[Lie algebra]], $e8={\srm Lie}(E8)$, corresponding to the [[Lie group]], [[E8]], breaks into a $120$ dimensional $so(16)$ [[special orthogonal group]] [[symmetric|symmetric space]] [[subalgebra|subgroup]] acting on a $128$ dimensional [[chiral]] [[Cl(16)]] [[spinor]], $S^{\slp16\srp+}$. However, this $so(16)$ can be decomposed into two $28$ dimensional $so(8)$'s and a $64$ dimensional piece, related to two pieces of the $128=64+64$ spinor through [[triality]] -- a decomposition described by [[John Baez]] in [[TWF90|http://math.ucr.edu/home/baez/week90.html]]:\n<<<\nEmboldened with our success, we now look at the vector space\n\nso(8) + so(8) + end(S+) + end(S-) + end(V)\n\nHere end(S+) is the space of all linear transformations of the vector space S+, so if you like, it's just the space of 8x8 matrices. Similarly for end(S-) and end(V). Now the dimension of this space is\n\n28 + 28 + 64 + 64 + 64 = 248\n\nHey! This is just the dimension of E8! Maybe this space is E8!\n\nYes indeed. Again, you can cook up a bracket operation on this space using all the stuff we've got. Here's the basic idea. end(S+), end(S-), and end(V) are already Lie algebras, where the bracket of two guys x and y is just the commutator [x,y] = xy - yx, where we multiply using matrix multiplication. Since so(8) has a representation as linear transformations of V, it has two representations on end(V), corresponding to left and right matrix multiplication; glomming these two together we get a representation of so(8) + so(8) on end(V). Similarly we have representations of so(8) + so(8) on end(S+) and end(S-). Putting all this stuff together we get a Lie algebra, if we do it right - and it's E8. At least that's what Kostant said; I haven't checked it.\n<<<\nWe can build this by breaking up the $so(16)^+ + S^{\slp16\srp+}$ generators and structure constants into the new ones. Letting the indices run $1 \sle \sal,\sbe \sle 7$ and $1 \sle a,b \sle 8$, and using the [[chiral Cl(16) bivector|Cl(16)]] decomposition into [[Cl(8)]] elements using the [[Kronecker product]], we define the new set of e8 generators in terms of the old:\n$$\n\sbegin{array}{rclcccl}\nH_{\sal\sbe} &=& \sga^{\slp16\srp+}_{\sal\sbe} &=& \sGa^+_{\sal\sbe} \sotimes 1 &\sin& so(8)^+ \sotimes 1 \s\s\nG_{\sal\sbe} &=& \sga^{\slp16\srp+}_{\slp\sal+8\srp\slp\sbe+8\srp} &=& P^{\slp8\srp}_+ \sotimes \sGa_{\sal\sbe} &\sin& 1 \sotimes so(8) \s\s\n\sPs^I_{\sal\sbe} &=& \sga^{\slp16\srp+}_{\sal\slp\sbe+8\srp} &=& -\sGa^+_\sal \sotimes \sGa_\sbe &\sin& v^{\slp8\srp+} \sotimes v^{\slp8\srp}\s\s\n\sPs^{II}_{ab} &=& Q^+_{16\slp a-1\srp+b} &=& q^+_a \sotimes q^+_b &\sin& S^{\slp8\srp+} \sotimes S^{\slp8\srp+}\s\s\n\sPs^{III}_{ab} &=& Q^+_{16\slp a-1\srp+b+8} &=& q^+_a \sotimes q^-_b &\sin& S^{\slp8\srp+} \sotimes S^{\slp8\srp-}\n\send{array}\n$$\nin which $P^{\slp8\srp}_+ = \sha \slp 1 + \sGa \srp$ is the positive chirality projector for Cl(8), giving $\sGa^+_{\sal\sbe} = P^{\slp8\srp}_+ \sGa_{\sal\sbe}$ and $\sGa^+_{\sal} = P^{\slp8\srp}_+ \sGa_{\sal}$, and $q^\spm_a$ are positive and negative chiral Cl(8) spinors. Since this is just a re-labeling, the new Lie brackets (and structure constants) come from the old structure constants:\n''//(danger, some of this calculation is wrong -- need to use a chiral rep for Cl(16), not the one use here now.)//''\n\sbegin{eqnarray}\n\slb H_{\sal\sbe}, H_{\sga\sde} \srb &=& C^{so(8)}_{\slb\sal\sbe\srb\slb\sga\sde\srb}{}^{\slb\sep\sup\srb} H_{\sep\sup} \s\s\n\n\slb G_{\sal\sbe}, G_{\sga\sde} \srb &=& C^{so(8)}_{\slb\sal\sbe\srb\slb\sga\sde\srb}{}^{\slb\sep\sup\srb} G_{\sep\sup} \s\s\n\n\slb H_{\sal\sbe}, G_{\sga\sde} \srb &=& 0 \s\s\n\n\slb H_{\sal\sbe}, \sPs^I_{\sga\sde} \srb &=& C_{\slb\sal\sbe\srb\slb\sga\slp\sde+8\srp\srb}{}^{\slb\sep\slp\sup+8\srp\srb} \sPs^I_{\sep\sup} \s\s\n&=& 2 \sleft\s{ - \set_{\sal \sga} \sde^{\slb\sep \slp\sup+8\srp\srb}_{\sbe \slp\sde+8\srp} + \set_{\sal \slp\sde+8\srp} \sde^{\slb\sep \slp\sup+8\srp\srb}_{\sbe \sga} + \set_{\sbe \sga} \sde^{\slb\sep \slp\sup+8\srp\srb}_{\sal \slp\sde+8\srp} - \set_{\sbe \slp\sde+8\srp} \sde^{\slb\sep \slp\sup+8\srp\srb}_{\sal \sga} \sright\s} \sPs^I_{\sep\sup} \s\s\n&=& \sleft\s{ - \set_{\sal \sga} \sde^{\sep\sup}_{\sbe \sde} + \set_{\sbe \sga} \sde^{\sep\sup}_{\sal\sde} \sright\s} \sPs^I_{\sep\sup} \s\s\n\n\slb H_{\sal\sbe}, \sPs^{II}_{ab} \srb &=& C_{\slb\sal\sbe\srb\slp16\slp a-1\srp+b\srp}{}^{\slp16\slp c-1\srp+d\srp} \sPs^{II}_{cd} \n= - (\sga^{\slp16\srp+}_{\sal \sbe})_{\slp16\slp a-1\srp+b\srp}{}^{\slp16\slp c-1\srp+d\srp} \sPs^{II}_{cd}\n= - (\sGa^+_{\sal\sbe})_a{}^c \sde_b^d \sPs^{II}_{cd} \s\s\n\n\slb H_{\sal\sbe}, \sPs^{III}_{ab} \srb &=& C_{\slb\sal\sbe\srb\slp16\slp a-1\srp+b+8\srp}{}^{\slp16\slp c-1\srp+d+8\srp} \sPs^{III}_{cd} \n= - (\sga^{\slp16\srp+}_{\sal \sbe})_{\slp16\slp a-1\srp+b+8\srp}{}^{\slp16\slp c-1\srp+d+8\srp} \sPs^{III}_{cd}\n= - (\sGa^+_{\sal\sbe})_a{}^c \sde_b^d \sPs^{III}_{cd} \s\s\n\n\slb G_{\sal\sbe}, \sPs^I_{\sga\sde} \srb &=& C_{\slb\slp\sal+8\srp\slp\sbe+8\srp\srb\slb\sga\slp\sde+8\srp\srb}{}^{\slb\sep\slp\sup+8\srp\srb} \sPs^I_{\sep\sup} \s\s\n&=& 2 \sleft\s{ - \set_{\slp\sal+8\srp \sga} \sde^{\slb\sep \slp\sup+8\srp\srb}_{\slp\sbe+8\srp \slp\sde+8\srp} + \set_{\slp\sal+8\srp \slp\sde+8\srp} \sde^{\slb\sep \slp\sup+8\srp\srb}_{\slp\sbe+8\srp \sga} + \set_{\slp\sbe+8\srp \sga} \sde^{\slb\sep \slp\sup+8\srp\srb}_{\slp\sal+8\srp \slp\sde+8\srp} - \set_{\slp\sbe+8\srp \slp\sde+8\srp} \sde^{\slb\sep \slp\sup+8\srp\srb}_{\slp\sal+8\srp \sga} \sright\s} \sPs^I_{\sep\sup} \s\s\n&=& \sleft\s{ \set_{\sal\sde} \sde^{\sep \sup}_{\sbe \sga} - \set_{\sbe\sde} \sde^{\sep\sup}_{\sal \sga} \sright\s} \sPs^I_{\sep\sup} \s\s\n\n\slb G_{\sal\sbe}, \sPs^{II}_{ab} \srb &=& C_{\slb\slp\sal+8\srp\slp\sbe+8\srp\srb\slp16\slp a-1\srp+b\srp}{}^{\slp16\slp c-1\srp+d\srp} \sPs^{II}_{cd} \n= - (\sga^{\slp16\srp+}_{\slp\sal+8\srp\slp\sbe+8\srp})_{\slp16\slp a-1\srp+b\srp}{}^{\slp16\slp c-1\srp+d\srp} \sPs^{II}_{cd}\n= - \sde_a^c (\sGa^+_{\sal\sbe})_b{}^d \sPs^{II}_{cd} \s\s\n\n\slb G_{\sal\sbe}, \sPs^{III}_{ab} \srb &=& C_{\slb\slp\sal+8\srp\slp\sbe+8\srp\srb\slp16\slp a-1\srp+b+8\srp}{}^{\slp16\slp c-1\srp+d+8\srp} \sPs^{III}_{cd} \n= - (\sga^{\slp16\srp+}_{\slp\sal+8\srp\slp\sbe+8\srp})_{\slp16\slp a-1\srp+b+8\srp}{}^{\slp16\slp c-1\srp+d+8\srp} \sPs^{III}_{cd}\n= - \sde_a^c (\sGa^-_{\sal\sbe})_b{}^d \sPs^{III}_{cd} \s\s\n\n\slb \sPs^I_{\sal\sbe}, \sPs^I_{\sga\sde} \srb &=& C_{\slb\sal\slp\sbe+8\srp\srb\slb\sga\slp\sde+8\srp\srb}{}^{\slb\sep\sup\srb} H_{\sep\sup}\n+ C_{\slb\sal\slp\sbe+8\srp\srb\slb\sga\slp\sde+8\srp\srb}{}^{\slb\slp\sep+8\srp\slp\sup+8\srp\srb} G_{\sep\sup} \s\s\n&=& 2 \sleft\s{ - \set_{\sal \sga} \sde^{\slb\sep \sup\srb}_{\slp\sbe+8\srp \slp\sde+8\srp} + \set_{\sal \slp\sde+8\srp} \sde^{\slb\sep \sup\srb}_{\slp\sbe+8\srp \sga} + \set_{\slp\sbe+8\srp \sga} \sde^{\slb\sep \sup\srb}_{\sal \slp\sde+8\srp} - \set_{\slp\sbe+8\srp \slp\sde+8\srp} \sde^{\slb\sep \sup\srb}_{\sal \sga} \sright\s} H_{\sep\sup} \s\s\n&+& 2 \sleft\s{ - \set_{\sal \sga} \sde^{\slb\slp\sep+8\srp \slp\sup+8\srp\srb}_{\slp\sbe+8\srp \slp\sde+8\srp} + \set_{\sal \slp\sde+8\srp} \sde^{\slb\slp\sep+8\srp \slp\sup+8\srp\srb}_{\slp\sbe+8\srp \sga} + \set_{\slp\sbe+8\srp \sga} \sde^{\slb\slp\sep+8\srp \slp\sup+8\srp\srb}_{\sal \slp\sde+8\srp} - \set_{\slp\sbe+8\srp \slp\sde+8\srp} \sde^{\slb\slp\sep+8\srp \slp\sup+8\srp\srb}_{\sal \sga} \sright\s} G_{\sep\sup} \s\s\n&=& - 2 \set_{\sbe\sde} \sde^{\slb\sep \sup\srb}_{\sal \sga} H_{\sep\sup} - 2 \set_{\sal \sga} \sde^{\slb\sep\sup\srb}_{\sbe \sde} G_{\sep\sup} \s\s\n\n\slb \sPs^{II}_{ab}, \sPs^{II}_{cd} \srb &=& C_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d\srp}{}^{\sal \sbe} H_{\sal\sbe}\n+ C_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d\srp}{}^{\sal \slp\sbe+8\srp} \sPs^I_{\sal\sbe} \n+ C_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d\srp}{}^{\slp\sal+8\srp \slp\sbe+8\srp} G_{\sal\sbe} \s\s\n&=& - ({\sga^{\slp16\srp+}}^{\sal \sbe})_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d\srp} H_{\sal\sbe}\n- ({\sga^{\slp16\srp+}}^{\sal \slp\sbe+8\srp})_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d\srp} \sPs^I_{\sal\sbe}\n- ({\sga^{\slp16\srp+}}^{\slp\sal+8\srp \slp\sbe+8\srp})_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d\srp} G_{\sal\sbe} \s\s\n&=& - ({\sGa^+}^{\sal \sbe})_{ac} \sde_{bd} H_{\sal\sbe} - \sde_{ac} ({\sGa^+}^{\sal \sbe})_{bd} G_{\sal\sbe} \s\s\n\n\slb \sPs^{III}_{ab}, \sPs^{III}_{cd} \srb &=& C_{\slp16\slp a-1\srp+b+8\srp\slp16\slp c-1\srp+d+8\srp}{}^{\sal \sbe} H_{\sal\sbe}\n+ C_{\slp16\slp a-1\srp+b+8\srp\slp16\slp c-1\srp+d+8\srp}{}^{\sal \slp\sbe+8\srp} \sPs^I_{\sal\sbe} \n+ C_{\slp16\slp a-1\srp+b+8\srp\slp16\slp c-1\srp+d+8\srp}{}^{\slp\sal+8\srp \slp\sbe+8\srp} G_{\sal\sbe} \s\s\n&=& - ({\sga^{\slp16\srp+}}^{\sal \sbe})_{\slp16\slp a-1\srp+b+8\srp\slp16\slp c-1\srp+d+8\srp} H_{\sal\sbe}\n- ({\sga^{\slp16\srp+}}^{\sal \slp\sbe+8\srp})_{\slp16\slp a-1\srp+b+8\srp\slp16\slp c-1\srp+d+8\srp} \sPs^I_{\sal\sbe}\n- ({\sga^{\slp16\srp+}}^{\slp\sal+8\srp \slp\sbe+8\srp})_{\slp16\slp a-1\srp+b+8\srp\slp16\slp c-1\srp+d+8\srp} G_{\sal\sbe} \s\s\n&=& - ({\sGa^+}^{\sal \sbe})_{ac} \sde_{bd} H_{\sal\sbe} - \sde_{ac} ({\sGa^-}^{\sal \sbe})_{bd} G_{\sal\sbe} \s\s\n\n\slb \sPs^I_{\sal\sbe}, \sPs^{II}_{ab} \srb &=& C_{\slb\sal\slp\sbe+8\srp\srb\slp16\slp a-1\srp+b\srp}{}^{\slp16\slp c-1\srp+d+8\srp} \sPs^{III}_{cd} \n= - (\sga^{\slp16\srp+}_{\sal\slp\sbe+8\srp})_{\slp16\slp a-1\srp+b\srp}{}^{\slp16\slp c-1\srp+d+8\srp} \sPs^{III}_{cd}\n= - (\sGa^+_\sal)_a{}^c (\sGa^+_\sbe)_b{}^d \sPs^{III}_{cd} \s\s\n\n\slb \sPs^I_{\sal\sbe}, \sPs^{III}_{ab} \srb &=& C_{\slb\sal\slp\sbe+8\srp\srb\slp16\slp a-1\srp+b+8\srp}{}^{\slp16\slp c-1\srp+d\srp} \sPs^{II}_{cd} \n= - (\sga^{\slp16\srp+}_{\sal\slp\sbe+8\srp})_{\slp16\slp a-1\srp+b+8\srp}{}^{\slp16\slp c-1\srp+d\srp} \sPs^{II}_{cd}\n= - (\sGa^+_\sal)_a{}^c (\sGa^-_\sbe)_b{}^d \sPs^{II}_{cd} \s\s\n\n\slb \sPs^{II}_{ab}, \sPs^{III}_{cd} \srb &=& C_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d+8\srp}{}^{\sal \sbe} H_{\sal\sbe}\n+ C_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d+8\srp}{}^{\sal \slp\sbe+8\srp} \sPs^I_{\sal\sbe} \n+ C_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d+8\srp}{}^{\slp\sal+8\srp \slp\sbe+8\srp} G_{\sal\sbe} \s\s\n&=& - ({\sga^{\slp16\srp+}}^{\sal \sbe})_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d+8\srp} H_{\sal\sbe}\n- ({\sga^{\slp16\srp+}}^{\sal \slp\sbe+8\srp})_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d+8\srp} \sPs^I_{\sal\sbe}\n- ({\sga^{\slp16\srp+}}^{\slp\sal+8\srp \slp\sbe+8\srp})_{\slp16\slp a-1\srp+b\srp\slp16\slp c-1\srp+d+8\srp} G_{\sal\sbe} \s\s\n&=& - ({\sGa^+}^\sal)_{ac} ({\sGa^+}^\sbe)_{bd} \sPs^I_{\sal\sbe} \s\s\n\send{eqnarray} \n//(not sure about those last three)//\n//check factor of 2 in $\sPs^I$//\n\nAny element of e8 can be written as\n$$\nA = A^B T_B = H + G + \sPs_I + \sPs_{II} + \sPs_{III} \n= \sha h^{\sal \sbe} H_{\sal \sbe} + \sha g^{\sal \sbe} G_{\sal \sbe} + \sps_I^{\sal \sbe} \sPs^I_{\sal \sbe} + \sps_{II}^{a b} \sPs^{II}_{a b} + \sps_{III}^{a b} \sPs^{III}_{a b}\n$$\nin which $h^{\sal \sbe}, g^{\sal \sbe}, \sps_I^{\sal \sbe}, \sps_{II}^{a b}, \sps_{III}^{a b}$ are real, $8\stimes8$ matrix components. Written as matrices, these are $h,g,\sps_I,\sps_{II},\sps_{III}$, with $h$ and $g$ antisymmetric. Remarkably, the e8 Lie brackets can be written in terms of matrix multiplication with these matrices. We can use this to define an ''e8 Lie bracket'', taking these $8\stimes8$ matrices as inputs, that knows and depends on which group of algebra elements matrices are coming from. For some of these brackets we get:\n|$\slb h_1, h_2 \srb_{e8} = 2 \slp h_1 \set h_2 - h_2 \set h_1 \srp_H$ |$\slb \sha h_1^{\sal\sbe}H_{\sal\sbe},\sha h_2^{\sga\sde}H_{\sga\sde} \srb = \sha 2 \slp h_1^{\sep\sbe} \set_{\sbe\sga} h_2^{\sga\sup} - h_2^{\sep\sbe} \set_{\sbe\sga} h_1^{\sga\sup} \srp H_{\sep\sup}$ |\n|$\slb g_1, g_2 \srb_{e8} = 2 \slp g_1 \set g_2 - g_2 \set g_1 \srp_G$ |$\slb \sha g_1^{\sal\sbe}G_{\sal\sbe},\sha g_2^{\sga\sde}G_{\sga\sde} \srb = \sha 2 \slp g_1^{\sep\sbe} \set_{\sbe\sga} g_2^{\sga\sup} - g_2^{\sep\sbe} \set_{\sbe\sga} g_1^{\sga\sup} \srp G_{\sep\sup}$ |\n|$\slb h, \sps_I \srb_{e8} = \sli h \slb \sde \set \srb \sri \sps_I$ |$\slb \sha h^{\sal\sbe}H_{\sal\sbe}, \sps_I^{\sga\sde}\sPs^I_{\sga\sde} \srb = \slp h^{\sal\sbe} \sha \slp \sde_\sal^\sep \set_{\sbe\sga} - \sde_\sbe^\sep \set_{\sal\sga} \srp \srp \sps_I^{\sga\sde} \sPs^I_{\sep\sde}$ |\n|$\slb h, \sps_{II} \srb_{e8} = \sli h \sha \sGa^+ \sri \sps_{II}$ |$\slb \sha h^{\sal\sbe}H_{\sal\sbe}, \sps_{II}^{ab}\sPs^{II}_{ab} \srb = \slp h^{\sal\sbe} \sha \slp \sGa^+_{\sal\sbe} \srp^c{}_a \srp \sps_{II}^{ab} \sPs^{II}_{cb}$ |\n|$\slb h, \sps_{III} \srb_{e8} = \sli h \sha \sGa^+ \sri \sps_{III}$ |$\slb \sha h^{\sal\sbe}H_{\sal\sbe}, \sps_{III}^{ab}\sPs^{III}_{ab} \srb = \slp h^{\sal\sbe} \sha \slp \sGa^+_{\sal\sbe} \srp^c{}_a \srp \sps_{III}^{ab} \sPs^{III}_{cb}$ |\n|$\slb g, \sps_I \srb_{e8} = \sps_I \sli g \slb \set \sde \srb \sri$ |$\slb \sha g^{\sal\sbe}G_{\sal\sbe}, \sps_I^{\sga\sde}\sPs^I_{\sga\sde} \srb = \sps_I^{\sga\sde} \slp g^{\sal\sbe} \sha \slp \set_{\sbe\sde} \sde_\sal^\sep - \set_{\sal\sde} \sde_\sbe^\sep \srp \srp \sPs^I_{\sep\sde}$ |\n|$\slb g, \sps_{II} \srb_{e8} = - \sps_{II} \sli g \sha \sGa^+ \sri$ |$\slb \sha g^{\sal\sbe}G_{\sal\sbe}, \sps_{II}^{ab}\sPs^{II}_{ab} \srb = - \sps_{II}^{ab} \slp g^{\sal\sbe} \sha \slp \sGa^+_{\sal\sbe} \srp_b{}^c \srp \sPs^{II}_{ac}$ |\n|$\slb g, \sps_{III} \srb_{e8} = - \sps_{III} \sli g \sha \sGa^- \sri$ |$\slb \sha g^{\sal\sbe}G_{\sal\sbe}, \sps_{III}^{ab}\sPs^{III}_{ab} \srb = - \sps_{III}^{ab} \slp g^{\sal\sbe} \sha \slp \sGa^-_{\sal\sbe} \srp_b{}^c \srp \sPs^{III}_{ac}$ |\n|$\slb \sps^1_I, \sps^2_I \srb_{e8} = -4 \slp \sps^1_I \set {\sps^2_I}^T \srp_H -4 \slp {\sps^1_I}^T \set \sps^2_I \srp_G$ |$\sbegin{array}{rl}\slb \sps_I^{1\sal\sbe} \sPs^I_{\sal\sbe}, \sps_I^{2\sga\sde} \sPs^I_{\sga\sde} \srb = \s!\s!& \sha \sbig( -4 \sps_I^{1\sal\sbe} \set_{\sbe\sde} \sps_I^{2\sga\sde} \sbig) H_{\sal\sga} \s\s & + \sha \sbig( -4 \sps_I^{1\sal\sbe} \set_{\sal\sga} \sps_I^{2\sga\sde} \sbig) G_{\sbe\sde}\send{array}$ |\n|$\slb \sps^1_{II}, \sps^2_{II} \srb_{e8} = -2 \sli \slp \sps^1_{II} {\sps^2_{II}}^T \srp \sGa^+ \sri_H -2 \sli \slp {\sps^1_{II}}^T \sps^2_{II} \srp \sGa^+ \sri_G$ |$\sbegin{array}{rl}\slb \sps_{II}^{1ab} \sPs^{II}_{ab}, \sps_{II}^{2cd} \sPs^{II}_{cd} \srb =\s!\s!& \sha \slp -2 \slp \sps_{II}^{1ab} \sde_{bd} \sps_{II}^{2cd} \srp \slp \sGa^{+\sal\sbe} \srp_{ac} \srp H_{\sal\sbe} \s\s & + \sha \slp -2 \slp \sps_{II}^{1ab} \sde_{ac} \sps_{II}^{2cd} \srp \slp \sGa^{+\sal\sbe} \srp_{bd} \srp G_{\sal\sbe}\send{array}$ |\n|$\slb \sps^1_{III}, \sps^2_{III} \srb_{e8} = -2 \sli \slp \sps^1_{II} {\sps^2_{II}}^T \srp \sGa^+ \sri_H -2 \sli \slp {\sps^1_{II}}^T \sps^2_{II} \srp \sGa^- \sri_G$ |$\sbegin{array}{rl}\slb \sps_{II}^{1ab} \sPs^{II}_{ab}, \sps_{II}^{2cd} \sPs^{II}_{cd} \srb =\s!\s!& \sha \slp -2 \slp \sps_{II}^{1ab} \sde_{bd} \sps_{II}^{2cd} \srp \slp \sGa^{+\sal\sbe} \srp_{ac} \srp H_{\sal\sbe} \s\s & + \sha \slp -2 \slp \sps_{II}^{1ab} \sde_{ac} \sps_{II}^{2cd} \srp \slp \sGa^{-\sal\sbe} \srp_{bd} \srp G_{\sal\sbe}\send{array}$ |\n|$\slb \sps_I, \sps_{II} \srb_{e8} = - \sbig<\sbig< \sps_I \sps_{II} \sGa^{++} \sbig>\sbig>_{III}$ |$\slb \sps_I^{\sal\sbe} \sPs^I_{\sal\sbe}, \sps_{II}^{ab} \sPs^{II}_{ab} \srb = \slp - \sps_{I}^{\sal\sbe} \sps_{II}^{ab} (\sGa^+_\sal)_a{}^c (\sGa^+_\sbe)_b{}^d \srp \sPs^{III}_{cd}$ |\n|$\slb \sps_I, \sps_{III} \srb_{e8} = - \sbig<\sbig< \sps_I \sps_{III} \sGa^{+-} \sbig>\sbig>_{II}$ |$\slb \sps_I^{\sal\sbe} \sPs^I_{\sal\sbe}, \sps_{III}^{ab} \sPs^{III}_{ab} \srb = \slp - \sps_{I}^{\sal\sbe} \sps_{III}^{ab} (\sGa^+_\sal)_a{}^c (\sGa^-_\sbe)_b{}^d \srp \sPs^{II}_{cd}$ |\n|$\slb \sps_{II}, \sps_{III} \srb_{e8} = - \sbig<\sbig< \sps_{II} \sps_{III} \sGa^{++} \sbig>\sbig>_I$ |$\slb \sps_{II}^{ab} \sPs^{II}_{ab}, \sps_{III}^{cd} \sPs^{III}_{cd} \srb = \slp - \sps_{II}^{ab} \sps_{III}^{cd} ({\sGa^+}^\sal)_{ac} ({\sGa^+}^\sbe)_{bd} \srp \sPs^I_{\sal\sbe}$ |\nwith the corresponding equation also written out in the original notation, to make it clear what is being done here and why (note the matrix index positions). This table confirms and adds in the details of John Baez's description above. The elements, $H$ and $G$, of the two $so(8)$'s, represented by their $8\stimes8$ antisymmetric matrices of coefficients, $h$ and $g$, have Lie brackets acting as left and right matrix multiplication of related matrices (dependent on the [[Clifford matrix representation]] of Cl(8)) multiplying the three $\sps$'s. \n\nRef:\n*[[John Baez]]\n**http://math.ucr.edu/home/baez/week90.html\n**[[Octonions|papers/oct.pdf]]\n*Barton and Sudbery\n**[[Magic Squares and Matrix Models of Lie algebras|papers/0203010v2.pdf]]\n***dang, too hard for my little physics brain
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Any real $n \stimes n$ (square) matrix, $A_i{}^j$, has a corresponding set of $n$ complex ''eigenvalues'', $\sla_\sal$, ''right eigenvectors'', $\slp r_i \srp^\sal$, and ''left eigenvectors'', $\slp l_\sal \srp^i$, satisfying the ''eigenequations'',\n\sbegin{eqnarray}\nA_i{}^j \slp r_j \srp^\sbe &=& \slp r_i \srp^\sal \sLa_\sal{}^\sbe\s\s\n\slp l_\sal \srp^i A_i{}^j &=& \sLa_\sal{}^\sbe \slp l_\sbe \srp^j\n\send{eqnarray}\nin which $\sLa_\sal{}^\sbe$ is the diagonal matrix with the eigenvalues, $\sla_\sal$, on the diagonal,\n$$\n\sLa = \slb\n\sbegin{array}{cccc}\n\sla_1 & 0 & \sdots & 0\s\s\n0 & \sla_2 & \sdots & 0\s\s\n\svdots & \svdots & \sddots & \svdots\s\s\n0 & 0 & \sdots & \sla_n\s\s\n\send{array}\n\srb\n$$\nand the left eigenvectors, $l_\sal = \slp l_\sal \srp^i b_i$, are presumed to be elements of some [[vector space]]. From the the above equations, the eigenvalues are the roots of the ''characteristic polynomial'',\n$$\n0 = p_A(\sla) = \sll \sla - A \srl = \sdet \slp \sla \sde_\sal^\sbe - A_\sal{}^\sbe \srp\n$$\nusing the [[determinant]]. The matrix is called ''singular'' iff one of the eigenvalues is $0$ and ''degenerate'' iff two or more of the eigenvalues are the same, with each eigenvalue having a ''multiplicity'', $m$. If $A$ is real and symmetric, $A^T = A$, ($A_i{}^j = A^j{}_i$), or complex and Hermitian, $A^\sdagger = A$, ($A^*_i{}^j = A^j{}_i$), the eigenvalues are real and the left and right eigenvectors are dual and orthonormal. If $A$ is real but not symmetric, the eigenvalues and eigenvectors are complex. The eigenvectors are only determined up to a scaling factor by the eigenequations. Or, if $A$ is degenerate, the $m$ eigenvectors corresponding to an eigenvalue need only span that $m$ dimensional ''eigenspace''.\n\nA real matrix, $A$, is not ''defective'' iff the left or right eigenvectors span the entire $n$ dimensional vector space, in which case the left and right eigenvectors can be scaled to satisfy the ''normality conditions'',\n$$\n\slp l_\sal \srp^i \slp r_i \srp^\sbe = \sde_\sal^\sbe\n$$\nWith that scaling done, the matrix may be written as\n$$\nA_i{}^j = \slp r_i \srp^\sbe \sLa_\sbe{}^\sal \slp l_\sal \srp^j\n$$\nthe ''spectral decomposition''. A symmetric or Hermitian matrix is never defective, and has $l = r^\sdagger$. The determinant of a matrix equals the product of its eigenvalues, and its [[trace]] is their sum,\n$$\n\sll A \srl = \sll \sLa \srl = \sprod_\sal \sla_\sal \s;\s;\s;\s;\s;\s;\s;\s;\s; \sli A \sri = \sli \sLa \sri = \ssum_\sal \sla_\sal\n$$\n\nIf we are dealing with a [[vector valued 1-form|vector valued form]], $\sf{\sve{A}} = \sf{dx^i} A_i{}^j \sve{\spa_j}$, then the [[vector-form algebra]] gives the eigenequations for the ''eigenforms'' and //''eigenvectors''//,\n\sbegin{eqnarray}\n\sf{\sve{A}} \sf{r^\sbe} &=& \sf{r^\sal} \sLa_\sal{}^\sbe\s\s\n\sve{l_\sal} \sf{\sve{A}} &=& \sLa_\sal{}^\sbe \sve{l_\sbe}\n\send{eqnarray}\nand, if the eigenvectors and eigenforms can be scaled to satisfy $\sve{l_\sal} \sf{r^\sbe} = \sde_\sal^\sbe$, the spectral decomposition is\n$$\n\sf{\sve{A}} = \sf{r^\sbe} \sLa_\sbe{}^\sal \sve{l_\sal}\n$$
A [[differential form]] [[field|cotangent bundle]], $\snf{f}(x)$, over a [[manifold]], $M$, is ''exact'' iff it is the [[exterior derivative]] of some other differential form field,\n$$\n\snf{f} = \sf{d} \snf{g}\n$$\nThe [[vector space]] of exact $p$-forms over $M$ is labeled $E^p$. Since the exterior derivative is nilpotent, all exact forms are [[closed]], $E^p \ssubset C^p$.
Any algebraic element may be ''exponentiated'',\n\s[ e^A = 1 + A + \sha A A + \sfr{1}{3!} A A A + \sdots \s]\nOr, equivalently, ''exponentiation'' may be defined as\n\s[ e^A = \slim_{N \sto \sinfty} \slp 1 + \sfr{1}{N} A \srp^N \s]\n\nThe derivative of a parameterized exponential is\n\s[ \sfr{d}{dt} e^{tA} = A + t A^2 + \sfr{1}{2!} t^2 A^3 + \sdots = A e^{tA} = e^{tA} A \s]\nMore rigorously, the solution of any set of first order ODE's,\n$$\n\sfr{d}{dt} E = A E\n$$\nin which $A$ is a linear operator, is used to define the exponentiation of that operator,\n$$\nE(t) = e^{t A} E(0)\n$$\n\nIf $A$ may be written in terms of an [[adjoint|Clifford adjoint]] operator and a diagonal matrix of [[eigen]]values, $A = U \sLa U^-$, then\n\s[ e^A = U e^\sLa U^- \s]\nis easily computed by exponentiating the eigenvalues.
The ''exterior derivative'' operator is the [[partial derivative]] operator, $\sf{d}=\sf{\spa}$, applied to [[differential form]]s,\n$$\n\sf{d} \snf{A} = \sf{\spa} \snf{A} = \slp \sf{dx^i} \spa_i \srp \slp \sf{dx^j} \sdots \sf{dx^k} \sfr{1}{p!} A_{j \sdots k} \srp = \sf{dx^i} \sf{dx^j} \sdots \sf{dx^k} \sfr{1}{p!} \spa_i A_{j \sdots k} \n$$\nThis operation is conventionally written as $dA$ but is written in this work using the under-arrow since it has a form grade of $1$. Even though it is defined using the un-natural partial derivative it is a [[natural]] (coordinate independent) operator since the non-tensor terms arising from [[coordinate change]], $x \smapsto x(y)$, vanish by the symmetry of partial derivation and antisymmetry of collections of forms,\n$$\n\sf{d} \sf{f} = \sf{dx^i} \sf{dx^j} \spa^x_i f_j = \sf{dy^k} \sf{dy^m} \sfr{\spa x^i}{\spa y^k} \sfr{\spa x^j}{\spa y^m} \sfr{\spa y^n}{\spa x^i} \spa^y_n f_j = \sf{dy^k} \sf{dy^m} \slp \spa^y_k \sfr{\spa x^j}{\spa y^m} f_j - \sfr{\spa^2 x^j}{\spa y^k \spa y^m} f_j \srp = \sf{dy^k} \sf{dy^m} \spa^y_k f'_m = \sf{d'} \sf{f'}\n$$\nThis doesn't work for the partial derivative applied to [[tangent vector]]s, so there is no such thing as the exterior derivative of a [[vector valued form]]. The exterior derivative of a scalar function, $\sf{d} f = \sf{dx^i} \spa_i f$, is called the ''gradient'' in old fashioned vector calculus, while the exterior derivative of a 1-form, $\sf{d} \sf{f}$, is associated to the ''curl'' in three dimensional space.\n\nThe operator is nilpotent,\n$$\n\sf{d} \sf{d} = \sf{dx^i} \spa_i \sf{dx^j} \spa_j = \sf{dx^i} \sf{dx^j} \spa_i \spa_j = 0 \n$$\nand, as a grade $1$ [[derivation]], distributes over the [[product|vector-form algebra]] of a $f$-form, $\snf{F}$, and $g$-form, $\snf{G}$, via the graded Liebniz rule,\n$$\n\sf{d} \slp \snf{F} \snf{G} \srp = \slp \sf{d} \snf{F} \srp \snf{G} + \slp -1 \srp^f \snf{F} \slp \sf{d} \snf{G} \srp\n$$\nBut it does not distribute over the product of vectors and forms, since instead\n$$\n\sf{d} \slp \sve{v} \snf{G} \srp = \sf{\spa} \slp \sve{v} \snf{G} \srp = \slp \sf{\spa} \sve{v} \srp \snf{G} - \sve{v} \slp \sf{d} \snf{G} \srp + \slp \sve{v} \sf{\spa} \srp \snf{G} = {\scal L}_{\sve{v}} \snf{G} - \sve{v} \slp \sf{d} \snf{G} \srp \n$$\nin which the pair of unnatural terms reassemble into the natural [[Lie derivative]].
The rank $4$ exceptional [[Lie group]], [[F4]], is described by its $52$ dimensional [[Lie algebra]], ''f4''. This Lie algebra may be decomposed as a $36$ dimensional [[symmetric|symmetric space]] [[subgroup]], the [[special orthogonal group]] Lie algebra, $so(9)$, acting on the $16$ dimensional space of, real, positive, [[Cl(9)]] [[spinor]]s, $S^{\slp9\srp}$,\n$$\nf4 = so(9) \soplus S^{(9)} = so(8) \soplus V^{(8)} \soplus S^{(8)+} \soplus S^{(8)-}\n$$\nwhich breaks up further into $28$ dimensional $so(8)$ and three $8$ dimensional elements: the vector, $V^{(8)}$, positive [[chiral]] [[spinor]], $S^{(8)+}$, and negative chiral spinor, $S^{(8)-}$ -- all related through [[triality]].\n\nThe smallest irreducible representation of f4 is 26 dimensional. An explicit construction, just as I'd do it, can be found in:\nRef:\n*[[Cerchiai - Mapping the geometry of the F4 group|papers/Cerchiai - Mapping the geometry of the F4 group.pdf]]\n
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A ''fiber bundle'' is a [[manifold]], $E$, the ''total space'' (''//entire space//''), along with a ''defining map'', $\spi$, to a separate ''base manifold'', $M$. Locally, a patch of the total space, $E_{U_a} \ssim U_a \sotimes F$, is the product of a base patch, $U_a$, and a ''typical fiber'', $F$ -- and $\spi : E_{U_a} \srightarrow U_a$ is a projection, $\spi : z \smapsto x$. A fiber bundle may be visualized as the base manifold with a copy of the fiber attached at each base manifold point -- the fiber is said to be "over" the base manifold. There are thus two equivalent ways of describing a fiber bundle: as things happening in fibers over the base space, $M$, or as things in the total space, $E$. Each fiber is a [[submanifold]] of $E$, but there is not necessarily any submanifold of $E$ associated with $M$.\n\nThe best we can usually do is specify the explicit maps from all the $E_{U_a}$ to $U_a \sotimes F$ via a ''local trivialization'', $\sphi_a : z \smapsto (x,f)$. By inverting these maps, each typical fiber element, $f \sin F$, gives a local section, $\sph_a^-(x,f)$, in $E_{U_a}$ over each patch. A ''local section'', $\ssi$, is a map from base manifold patches to the total space patches, $\ssi:U_a \srightarrow E_{U_a}$, which projects trivially, $\spi(\ssi(x))=x \s; \sforall \s; x \sin U_a$. The local trivializations over the patches are glued together such that the ''transition functions'',\n$$\nt_{ab}(z) = \sph^-_a \scirc \sph_b(z) \n$$\nat each overlap point, $x=\spi(z)$, are [[autodiffeomorphism|diffeomorphism]]s of the fiber there, $\sld t_{ab} \srl_x : F \sto F$, corresponding to the action of an element of the ''structure group'', $G$, of the fiber bundle on the fibers, $G:F \sto F$. Another way of writing this, letting $z=\sph^-_b(x,f)$, is\n$$\nt_{ab} \scirc \sph^-_b(x,f) = \sph^-_a(x,f) = \sph^-_b(x,g_{ab}(x) f)\n$$\nin which $g_{ab}(x) \sin G$, acts on each fiber element at each overlap point via the [[left action|group]] of the [[Lie group]], $G$. When describing a fiber bundle it is necessary to specify the base, the fiber, the structure group, and the group action on the fiber.\n\nWith a local trivialization in hand, a local section, $\ssi(x) = \sph_a(x,f_\ssi(x))$, can be specified by choosing a typical fiber element, $f_\ssi(x)$, at each $x \sin U_a$. The collection of fiber valued functions, $f_\ssi(x) \sin F$ (which is sometimes just written as $\ssi(x) \sin F$), is refered to by physicists as a ''field'' over the base manifold. A complete collection of local sections can be glued together to give a ''global section'' iff\n$$\n\sph^-_a(x,f^a_\ssi(x)) = \sph^-_b(x,g_{ab}(x) f^a_\ssi(x))\n$$\nand hence iff\n$$\nf^b_\ssi(x) = g_{ab}(x) f^a_\ssi(x)\n$$\nIn this way, a ''global section'' (//''section''//) associates $M$ with a particular submanifold, $\ssi$, of $E$. This change of how a section is represented when the local trivialization is changed, $\ssi'(x) = g(x) \ssi(x)$, is the most basic type of [[gauge transformation]].\n\nThe [[partial derivative]] is zero when acting on a ''constant'' section, $f_\ssi(x) = \ssi(x) = \ssi$, that is specified locally by a constant field, $\spa_i \ssi=0$. This derivative doesn't properly keep track of the local trivialization or gluing between patches. To remedy this, a [[covariant derivative]] is introduced which, via a [[connection]], keeps track of how the local trivialization changes over the base when taking the derivative of a section -- it co-varies with a gauge transformation. Using the covariant derivative, any fiber element may be [[parallel transport]]ed along any path on the base to obtain a new fiber element at any point along the path. For a closed path, the parallel transport of a fiber element is represented by a [[holonomy]] -- an element of the structure group which acts on the fiber element. For a small closed path, or loop, the holonomy is given approximately by the [[curvature]] -- an important geometric descriptor of the fiber bundle and connection.\n\nThe above description, employing local trivializations, treats fiber bundle geometry as something happening over a base space. Fiber bundle geometry may be described more naturally over the total space by employing an [[Ehresmann connection]]. The defining map, $\spi$, of a fiber bundle gives an involutive [[distribution]], $\sve{\sDe_\spi}$, and foliation of the total space, $E$, by fibers. This distribution is the kernel of the [[pushforward|pullback]] of the map, $\spi_* \sve{\sDe_\spi}=0$, and is tangent to the foliating fibers of $E$.\n\nRefs:\n*A more thorough description is available at http://en.wikipedia.org/wiki/Fiber_bundle\n*[[A Route Towards Gauge Theory|papers/A Route Towards Gauge Theory.pdf]]\n*[[Geometrical aspects of local gauge symmetry|papers/Geometrical aspects of local gauge symmetry.pdf]]\n*[[Preparation for Gauge Theory|papers/9902027.pdf]]\n**excellent mathematical review of the basics
Consider a [[vector field|tangent bundle]], $\sve{v}(x)$ over a manifold. There are unique [[path]]s, $x(t)$, called ''integral curves'' of the vector field, such that the [[tangent vector]] at each point along the path is equal to the vector of the vector field at that point,\n$$\n\sfr{d x^i(t)}{d t} = v^i(x(t))\n$$\nThis relation may be thought of, and solved, as a set of ODE's. Consider the integral curves, with $\sve{v}$ as tangent vectors, starting from each manifold point, $x$, with these paths parameterized such that $t=0$ at these points. From these initial conditions, the point of each path, $y(t,x)$, is determined as a function of parameter and starting point, $x$. This is a ''flow'' &mdash; a parameterized [[autodiffeomorphism|diffeomorphism]], $\sph_t(x)=y(t,x)$, satisfying the "time symmetry" rule:\n$$\n\sph_t(\sph_{t'}(x)) = y(t,y(t',x)) = y(t+t',x) = \sph_{t+t'}(x) \n$$\nA flow may be visualized as a movement of the manifold points beneath any overlying geometric elements. Any chosen initial point, $x$, is carried along by the flow along the path, $\sph_t(x)$, defined by the ''flow equation'',\n$$\n\slb \sfr{\spa}{\spa t} \sph_t^i(x) \srb \sve{\spa_i} = \sve{v}(x)\n$$\nfor all $t$. The flow is completely determined by the vector field, $\sve{v}(x)$ &mdash; the ''vector field generator'' of the flow. An observer attached to such a point carried by the flow may either consider herself to be moving through a field of geometric elements or, alternatively, to be having the geometric elements change over her. For short times, the flow is (in terms of coordinates)\n$$\n\sph_t^i(x) \ssimeq x^i + t v^i(x)\n$$\nThe solution to the flow equation may be written heuristically as the [[exponentiation]] of the flow,\n$$\n\sph_t(x) = e^{t\sve{v}\sf{d}} x = e^{t {\scal L}_{\sve{v}}} x\n$$\nand any geometric object may be [[pushed forward|pullback]] along the flow by exponentiating the [[Lie derivative]], with\n$$\n\sph^*_t X = e^{t {\scal L}_{\sve{v}}} X\n$$\n\nIt is possible to have a parameterized autodiffeomorphism, $\sph_t(x)$, that satisfies $\sph_0(x)=x$ but does not satisfy the time symmetry rule. This is a ''time dependent flow'', and produces two distinct, time dependent velocity fields. The first, the ''Lagrangian flow field'', is the velocity of each initial manifold point, $x_0$, wherever it might be carried on the manifold: \n$$\n\sve{v_t}(x_0) = \slb \sfr{\spa}{\spa t} \sph_t^i(x_0) \srl_t \sve{\spa_i}\n$$\nThe second, the ''Euler flow field'', is the velocity at each manifold point at time $t$:\n$$\n\sld \sve{v_t}(x) \srl_{x=\sph_t(x_0)} = \slb \sfr{\spa}{\spa t} \sph_t^i(x_0) \srl_t \sve{\spa_i}\n$$\nIf the Euler flow field is constant in time, it is the vector field generator for the corresponding flow.
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The ''equivalence principle'' of General Relativity states that physics in a sufficiently small region around each point in an $n$ dimensional curved [[spacetime]] is locally indistinguishable from physics in a flat spacetime around that point. The mathematical implication is that at each point there is a set of $n$ ''orthonormal basis vectors'' (a.k.a. //''frame vectors''//), which can be written in terms of the [[coordinate basis vectors]] as\n\s[ \sve{e_\sal} = \slp e_\sal \srp^i \sve{\spa_i} \s]\nThey are orthonormal, $\slp \sve{e_\sal},\sve{e_\sbe} \srp = \set_{\sal \sbe}$, under use of a [[metric]], in which $\set_{\sal \sbe}$ is a [[Minkowski metric]]. The set of their [[1-form]] duals constitute the ''coframe 1-forms'' (a.k.a. //''coframe''//, //''vielbein''//, //''tetrad''//, //''frame 1-forms''//, or sometimes also just called the //''frame''//),\n\sbegin{eqnarray}\n\sf{e^\sal} &=& \sf{dx^i} \slp e_i \srp^\sal\s\s\n\sve{e_\sal} \sf{e^\sbe} &=& \slp e_\sal \srp^i \slp e_j \srp^\sbe \sve{\spa_i} \sf{dx^i} = \slp e_\sal \srp^i \slp e_i \srp^\sbe = \sde_\sal^\sbe\n\send{eqnarray}\nAs the set of "rulers" on the manifold, the coframe matrix components have [[units]] of time, $T$. The ''orthonormal basis vector matrix'', or //''frame matrix''//, $\slp e_\sal \srp^i$, is the inverse of the ''coframe matrix'', $\slp e_i \srp^\sal$ &mdash; they satisfy $\slp e_\sal \srp^i \slp e_i \srp^\sbe = \sde_\sal^\sbe$ and $\slp e_i \srp^\sal \slp e_\sal \srp^j = \sde_i^j$. (The ${}^-$ in $\slp e^-_\sal \srp^i$ is not written but is implied from the position of the [[indices]].)\n\nPhysically, at every point the coframe encodes a map from [[tangent vector]]s to vectors in a [[rest frame]]. It is very useful to employ the [[Clifford basis vectors]] as the fundamental geometric basis vector elements of this rest frame, $\sve{e_\sal} \sleftrightarrow \sga_\sal$. The ''coframe'', as a Clifford vector valued 1-form, is a map from the [[tangent bundle]] to the [[Clifford vector bundle]] &mdash; a map from tangent vectors to Clifford vectors &mdash; and is written as\n\s[ \sf{e} = \sf{e^\sal} \sga_\sal = \sf{dx^i} \slp e_i \srp^\sal \sga_\sal \s]\nIt is a [[Clifform]]. Using the coframe, any tangent vector, $\sve{v}$, on the manifold may be mapped to its corresponding Clifford vector, $v$, via [[vector-form algebra]],\n$$\n\sve{v} = v^\sal \sve{e_\sal} \sleftrightarrow v = \sve{v} \sf{e} = v^i \sve{\spa_i} \sf{dx^j} \slp e_j \srp^\sal \sga_\sal = v^i \slp e_i \srp^\sal \sga_\sal = v^\sal \sga_\sal\n$$\nTaking the [[Clifford algebra]] dot product of this Clifford vector with the frame gives the ''1-form dual'' of the vector,\n$$\n\sf{v} = v \scdot \sf{e} = v^\sal \sf{e^\sbe} \sga_\sal \scdot \sga_\sbe = \slp v^\sal \set_{\sal \sbe} \srp \sf{e^\sbe} = v_\sbe \sf{e^\sbe} = \slp \sve{v} \sf{e} \srp \scdot \sf{e}\n$$\nsatisfying $\sve{v} \sf{v} = v^\sal v_\sal = \slp \sve{v}, \sve{v} \srp$. The frame or coframe matrices multiply indexed objects and change their indices between coordinate indices and orthonormal basis labels, $T_{\sal i} \slp e_j \srp^\sal \slp e_\sga \srp^i = T_{j \sga}$. The Clifford vectors corresponding to the orthonormal basis vectors are the Clifford basis vectors,\n$$\n\sve{e_\sal} \sleftrightarrow \sve{e_\sal} \sf{e} = \sga_\sal\n$$\nThe ''frame'', as a Clifford vector valued vector, is defined as\n$$\sve{e}=\sga^\sal \sve{e_\sal}=\sga^\sal \slp e_\sal \srp^i \sve{\spa_i}$$\nand satisfies\n$$\sve{e} \sf{e} = \sga^\sal \slp e_\sal \srp^i \sve{\spa_i} \sf{dx^j} \slp e_j \srp^\sbe \sga_\sbe\n= \sga^\sal \slp e_\sal \srp^i \slp e_i \srp^\sbe \sga_\sbe\n= \sga^\sal \sga_\sal\n= n$$\nUsing the frame, any [[differential form]], $\sf{f}$, on the manifold may be mapped to its corresponding Clifford vector,\n$$\n\sf{f} = \sf{e^\sal} f_\sal \sleftrightarrow f = \sve{e} \sf{f} = \sga^\sal \slp e_\sal \srp^i \sve{\spa_i} \sf{dx^j} f_j = \sga^\sal \slp e_\sal \srp^i f_i = f_\sal \sga^\sal\n$$\n(Since the frame and coframe are used similarly and, as one is the inverse of the other, carry the same information, they are both often collectively referred to as the //''frame''//.)\n\nThe frame (or coframe), along with the Minkowski metric encoding the signature, completely determines a metric on the manifold. But the converse is not true &mdash; the metric only determines a frame up to a [[Lorentz transformation|Lorentz rotation]]. So the frame (or coframe) should be considered the fundamental object encoding the geometry of the manifold. It is possible to make the correspondence between frame and metric one-to-one by imposing a coordinate dependent restriction on the form of the frame matrix &mdash; such as restricting to the use of a [[UT frame]].
[[geodesic]]\n\n\n''momentum'' conserved along [[Killing vector]] directions.\n\nRovelli p122 for free particle in Minkowski space.
A ''function'' over a [[manifold]], $M$, is usually written as $f(x)$ &mdash; a function of the corresponding coordinates, $f \scirc x_a^{-} : \sRe^n \srightarrow M \srightarrow \sRe$. In this case it is a map from the points of the manifold, in the various charts, into the real numbers. The $x$ in $f(x)$ is shorthand for the set of coordinates, $x^i$, over each manifold patch.\n\nA function is more thoroughly described as a section of a [[fiber bundle]].
The rank $2$ exceptional [[Lie group]], [[G2]], is described by its $14$ dimensional [[Lie algebra]], ''g2''. This Lie algebra may be decomposed as a $8$ dimensional [[symmetric|symmetric space]] [[subgroup]], the [[special unitary group]] Lie algebra, [[su(3)]], acting on the standard $3$ representation and its dual, $\sbar{3}$,\n$$\ng2 = su(3) + 3 + \sbar{3}\n$$\nThis also relates to\n$$\ng2 \ssubset so(7) = so(6) + 6\n$$\n$$\nso(7) = g2 + 7\n$$\nwhich gives the fundamental $7$ rep.\n\nAn explicit construction of the Lie algebra and the group can be found in:\n\nRef:\n*[[Cerchiai - Euler angles for G2|papers/Cerchiai - Euler angles for G2.pdf]]\n
A ''gauge transformation'' (//''passive gauge transformation''//) is a transformation of the section (//''gauge''//), $\ssi(x)$, of a [[fiber bundle]] by an element, $g \sin G$, of the structure [[group|Lie group]] to another gauge, $\ssi'(x) = g \s, \ssi(x)$. A gauge transformation is ''local'' if it has a position dependent $g(x)$ and ''global'' if it isn't. Alternatively, a ''passive coordinate gauge transformation'' may be considered -- and treated equivalently -- that is nothing but a description of how the representation of the section changes under a change of local trivialization. \n\nThe [[covariant derivative]], written using a [[connection]], varies covariantly,\n\sbegin{eqnarray}\n\sf{\sna'} \ssi' &=& g(x) \sf{\sna} \ssi\s\s\n\slp \sf{d} g \srp \ssi + g \sf{d} \ssi + \sf{A'} g \ssi &=& g \sf{d} \ssi + g \sf{A} \ssi\s\s\n\send{eqnarray}\ngiving the transformation law for the connection under a gauge transformation,\n$$\n\sf{A'} = g \sf{A} g^- - \slp \sf{d} g \srp g^- = g \sf{A} g^- + g \slp \sf{d} g^- \srp\n$$\nAn infinitesimal transformation, $g \ssimeq 1 + G^A T_A = 1 + G$, changes the connection to\n\sbegin{eqnarray}\n\sf{A'} &\ssimeq& \sf{A} - \sf{d} G - \sf{A} G + G \sf{A} = \sf{A} - \sf{\sna} G \s\s\n\sde_G \sf{A} &=& - \sf{\sna} G\n\send{eqnarray}\nUnder a gauge transformation, the [[curvature]] changes to\n\sbegin{eqnarray}\n\sff{F'} &=& \sf{d} \sf{A'} + \sf{A'} \sf{A'} \s\s\n&=& \sf{d} \slb g \sf{A} g^- - \slp \sf{d} g \srp g^- \srb + \slb g \sf{A} g^- - \slp \sf{d} g \srp g^- \srb \slb g \sf{A} g^- - \slp \sf{d} g \srp g^- \srb \s\s\n&=& g \slp \sf{d} \sf{A} + \sf{A} \sf{A} \srp g^- \s\s \n&=& g \sff{F} g^- = A_g \sff{F} \s\s\n&\ssimeq& \sff{F} + G \sff{F} - \sff{F} G = \sff{F} + \slb G , \sff{F} \srb \s\s\n\sde_G \sff{F} &=& \slb G , \sff{F} \srb\n\send{eqnarray}\n\nAll physical, measurable quantities in physics are invariant under gauge transformations.\n\nThe above description of gauge transformations presumes the connection to be a field over the base manifold. A gauge transformation may also be described from the viewpoint of structures over the total space of the fiber bundle, as an [[Ehresmann gauge transformation]]. In this space, an ''active gauge transformation'' is a vertical [[autodiffeomorphism|diffeomorphism]] -- this gauge transformation, which transforms the connection over the total space while leaving the local sections fixed, is equivalent to the passive gauge transformation, which transforms the local sections while leaving the connection fixed.
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A ''geodesic'' on a [[manifold]] with a [[metric]] or [[frame]] is a [[path]], $x(t)$, of extremal length (or time). This [[proper time]], in temporal [[units]] such as seconds, is the integral of the speed between two points,\n$$S = \sDelta \sta = \sint \sf{dt} \sll v \srl = \sint \sf{dt} \ssqrt{\sll v \scdot v \srl} = \sint \sf{dt} \ssqrt{\sll \slp \sve{v} \sf{e} \srp \scdot \slp \sve{v} \sf{e} \srp \srl} = \sint \sf{dt} \ssqrt{\sll \sfr{d x^i}{d t} \sfr{d x^j}{d t} g_{ij}(x) \srl}$$\nwith respect to parameter time, $t$. Extremizing this length with respect to variation of $x^i(t)$ gives a geodesic equation invariant with respect to reparameterization. Alternatively, a geodesic also extremizes the action, \n$$S = \sint \sf{d \sta} \sha v^2 = \sint \sf{d \sta} \sha \sfr{d x^i}{d \sta} \sfr{d x^j}{d \sta} g_{ij}(x) = \sint \sf{d \sta} \sha \sfr{d x^i}{d \sta} \sfr{d x^j}{d \sta} g_{ij}(x)$$\nwhich is a simpler variation and produces the affine geodesic equation, with the parameter set equal to the proper time along the curve, $t=\sta$. The variation is\n\sbegin{eqnarray}\n\sde S &=& \sint \sf{d \sta} \slc \sha \sfr{d x^i}{d \sta} \sfr{d x^j}{d \sta} \sde x^k \spa_k g_{ij} + \sfr{d \sde x^i}{d \sta} \sfr{d x^j}{d \sta} g_{ij} \src \s\s\n&=& \sint \sf{d \sta} \sde x^k \slc \sha \sfr{d x^i}{d \sta} \sfr{d x^j}{d \sta} \spa_k g_{ij} - \sfr{d}{d \sta} \slp \sfr{d x^j}{d \sta} g_{kj} \srp \src\n\send{eqnarray}\nand gives the geodesic equation,\n$$\sfr{d^2 x^k}{d \sta^2} = \sfr{d x^i}{d \sta} \sfr{d x^j}{d \sta} \slp \sha g^{mk} \spa_m g_{ij} - g^{mk} \spa_i g_{mj} \srp \n= - \sfr{d x^i}{d \sta} \sfr{d x^j}{d \sta} \sGa^k{}_{ij}$$\nThis determines the path on a manifold followed by any freely falling body, given its initial position and velocity. This motion of a free particle also has a [[Hamiltonian formulation|free particle Hamiltonian]]. The geodesic dependends on derivatives of the metric via the [[Christoffel symbols|tangent bundle connection]], and may be expressed by derivatives of the frame via the spin connection. (This relationship also gives a quick way of calculating connection coefficients by varying the metric or frame.) Note that geodesics aren't influenced by [[torsion]] since only the symmetric part of the Christoffel symbols, $\sGa^k{}_{\slp ij \srp}$, enter the geodesic equation.\n\nThe [[tangent vector]] to a geodesic is [[parallel transport|tangent bundle parallel transport]]ed along the path,\n$$0 = \slp \sve{v} \sf{\snabla} \srp \sve{v} = v^i \snabla_i v^k \sve{\spa_k} = \slp \sfr{d v^k}{d \sta} + v^i v^j \sGa^k{}_{ij} \srp \sve{\spa_k}$$\nby virtue of the geodesic equation. If the velocity is expressed as a [[Clifford element]] using the frame, $v = \sve{v} \sf{e}$, then\n$$0 = \sve{v} \sf{\snabla} v = v^i \snabla_i v^\sal \sga_\sal = \slp \sfr{d v^\sal}{d \sta} - v^i v^\sbe \som_i{}_\sbe{}^\sal \srp \sga_\sal$$\nin terms of the [[spin connection]]. The speed along a geodesic with respect to proper time is constant, and its square, $v^2$, is either $1$, $0$, or $-1$ depending on whether the velocity is timelike, null, or spacelike.\n\nThe component of a geodesic's velocity along any [[Killing vector]] is constant along the path,\n$$\n...\n$$\n//add equation//
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\nRef:\n*Fabrizio Nesti\n**[[Standard Model and Gravity from Spinors|http://arxiv.org/abs/0706.3304]]\n***Uses $so(3,1)$ spin connection. Split this into self-dual part for gravity and anti-self-dual part for either electroweak $su(2)_L$ or $su(2)_R$. Nesti breaks both up out of $SO(3,1,\smathbb{C})$. He also goes on to talk about Pati-Salam, but his frame and Higgs are messed up and he doesn't use MacDowell-Mansouri. Ooh, he is getting close to what I'm doing though -- mentions embedding in non-orthogonal groups in a footnote. Mirror fermion problem. [[Coleman-Mandula]] doesn't apply because the [[frame]] doesn't have Poincare symmetry.\n**[[Gravi-Weak Unification|http://arxiv.org/abs/0706.3307]]\n***Uses $Cl(4,\smathbb{C})$ for a Pati-Salam model.\n*Stephon Alexander\n**[[Isogravity: Toward an Electroweak and Gravitational Unification|http://arxiv.org/abs/0706.4481]]
A ''group'' is a collection of elements, $g \sin G$, along with an ordered group product by which one element times a second equals a third, $g_1 g_2 = g_3$. A group has the following properties:\n*it includes the ''identity element'', $g 1 = 1 g = g$\n*every element has an [[inverse]], $g g^- = g^- g = 1$\n*its product is ''associative'', $a(bc)=(ab)c$\n*its product is ''closed'', $ab \sin G$\n\nOne group element, $h$, may act on another, $g$, via three different ''group action''s:\n*''left action'': $L_h g = h g$\n*''right action'': $R_h g = g h$\n*''conjugation'', also known as the inner [[automorphism]] or ''adjoint action'': $A_h g = L_h R_{h^-} g = h g h^-$\n\nA group is ''abelian'' iff $gh = hg$ for all $h, g \sin G$.\n\nAn example of a group is the set of integers, $G = \sleft\s{ \sdots, -3, -2, -1, 0, 1, 2, 3, \sdots \sright\s}$, with addition, $+$, as the group product. For this abelian group group $0$ is the identity element.
*James Ryan\n**[[A new proposal for group field theory I: the 3d case|http://arxiv.org/abs/gr-qc/0611080]]
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The ''holonomy'' is the [[path holonomy]], $U$, for an arbitrary closed path on the base manifold of a [[fiber bundle]]. It may be written heuristically as\n$$\nU = Pe^{-\soint \sf{A}}\n$$\nin which the [[connection]] is integrated all the way around the path. (The only real meaning of this expression is that it is a solution for the path holonomy at the end point (which equals the initial point) of the path.) \n\nIt is enlightening to calculate the approximate holonomy for a small, square-ish path. Such a path may be specified by choosing two orthonormal vectors, $\sve{u}$ and $\sve{v}$, at a point $x_{0}$ and making a closed path by going $\sva$ in the $\sve{u}$ direction, then $\sva$ along $\sve{v}$, $\svarepsilon$ along $-\sve{u}$, then $\sva$ along $-\sve{v}$ back to $x_{0}$. These four path segments, each parameterized by $0 \sleq t \sleq \sva$, are given by\n$$\n\sva_{1}^{i}=tu^{i}, \squad\n\sva_{2}^{i}=\sva u^{i}+tv^{i}, \squad\n\sva_{3}^{i}=\sva u^{i}+\sva v^{i}-tu^{i},\squad\n\sva_{4}^{i}=\sva v^{i}-tv^{i}\n$$\nand produce an anti-symmetric [[second order path dependence|path holonomy]],\n$$\n\sva^{ij} = \sint_{0}^{\sva}\sf{dt}\s,\sfr{d\sva_{1}^{i}}{dt}\sva_{1}^{j}\n+\sint_{0}^{\sva}\sf{dt}\s,\sfr{d\sva_{2}^{i}}{dt}\sva_{2}^{j}\n+\sint_{0}^{\sva}\sf{dt}\s,\sfr{d\sva_{3}^{i}}{dt}\sva_{3}^{j}\n+\sint_{0}^{\sva}\sf{dt}\s,\sfr{d\sva_{4}^{i}}{dt}\sva_{4}^{j}\n=\sva^{2} \slp v^{i}u^{j}-v^{j}u^{i} \srp\n$$\nimplying a [[loop|vector-form algebra]] described by a tangent 2-vector,\n$$\n\svv{L}=\sha L^{ij}\sve{\spa_i}\sve{\spa_j}\n=\sha \sva^{ij}\sve{\spa_i}\sve{\spa_j}\n=\sva^{2}v^{i}u^{j}\sve{\spa_i}\sve{\spa_j}\n=\sva^{2}\sve{v}\s,\sve{u}\n$$\nThe holonomy around this small loop is approximately the [[path holonomy]] to second order,\n$$\nU \ssimeq 1 + \sva^{ij} \slb - \spa_{j} A_{i} + A_{i} A_{j}\srb\n=1 + \sha \sva^{ij} \slb \spa_i A_j - \spa_j A_i + 2 A_i \stimes A_j \srb\n=1 + \sha \sva^{ij} F_{ij}\n=1 - \svv{L} \sff{F}\n$$\nwith the (defining) appearance of the [[curvature]],\n$$\n\sff{F} = \sf{d} \sf{A} + \sf{A} \sf{A} = \sf{dx^i} \sf{dx^j} \slp \spa_i A_j + A_i \stimes A_j \srp = \sha \sf{dx^i} \sf{dx^j} F_{ij} \n$$\nThe contraction of the loop with the curvature, $\svv{L} \sff{F}$, is a nice example of [[vector-form algebra]]. Any fiber element, $C$, parallel transported around a small loop, $\svv{L}$, is transformed to\n$$\nC \smapsto C' = UC \ssimeq C - \svv{L} \sff{F} C\n$$\nto first order in loop area, $\sva^2$. This provides a nice alternative definition of curvature in terms of parallel transport around small closed paths.
A ''homogeneous space'' (a.k.a. //''coset space''//, //''quotient space''//, or //''Klein geometry''//), $S=G/H$, is both a left [[coset]] and [[manifold]] formed by modding a [[Lie group geometry]], $G$, by a [[subgroup]], $H$. The points, $x \sin S$, of a homogeneous space are specified by their coset representatives, $r(x) \sin G$, up to right action by $H$,\n$$\nx \ssim \slb r(x) \srb = \slb r(x) \s, h(y) \srb = r(x) \s, H = \sleft\s{ r(x) \s, h(y) : \sforall \s, h(y) \sin H \sright\s}\n$$\nThe homogeneous space is the base space, $M = S = G/H$, of a [[principal bundle]] with total space $E=G$ and $F=H$ as the structure group and fiber. The defining map is $\spi : g \smapsto [g]$ and the choice of ''coset representative section'' (//''homogeneous reference section''//), $r : S \srightarrow G$, (a function of points on the base space, valued in the total space) serves as a [[reference section|Ehresmann principal bundle connection]] and provides the [[local trivialization|fiber bundle]], $\sph : (x,h) \smapsto r(x) \s, h \sin G$. A homogeneous space has a natural ''zero point'' corresponding to the equivalence class of the identity in $G$, so the coset representative section and coordinates are chosen so $r(0) = 1$. The [[Maurer-Cartan form]], $\sf{\scal I} = g^- \sf{d} g$, over $G$ [[pulls back|pullback]] along the reference section, $g=r(x)$, to give the ''Maurer-Cartan frame'' over the homogeneous space,\n$$\n\sf{I}(x) = r^* \sf{\scal I} = r^- \sf{d} r \sin \sf{\srm Lie}(G)\n$$\nwhich leads to a description of the [[homogeneous space geometry]] over $S$ or to the [[Ehresmann homogeneous space geometry]] over $G$. Note that a homogeneous space, $S=G/H$, is only itself a Lie group iff $H$ is a [[normal subgroup]] of $G$.
A [[principal bundle]] gauge transformation corresponds to a transformation of a reference section, $r(x) \sin G$ of a [[homogeneous space]],\n$$\nr'(x) = r(x) \s, h(x)\n$$\nby $h(x) \sin H$. This transformation doesn't move the points of the homogeneous space, since $[r'(x)] = [r(x) \s, h(x)] = [r(x)]$, but just moves the section up or down the fibers at those points by a [[diffeomorphism]], $\sph(x,y)=(x,y_\sph(x,y))$, in accordance with the point of view of the [[Ehresmann principal bundle gauge transformation]]. The [[Maurer-Cartan frame|homogeneous space]] over $S=G/H$ transforms to\n$$\n\sf{I'}(x) = \slp r h \srp^* \sf{\scal I} = h^- r^- \sf{d} \slp r h\srp = h^- \sf{I} h + h^- \sf{d} h = \sf{e'_S} + \sf{A'_S}\n$$\ncorresponding to the [[homogeneous space frame|homogeneous space geometry]], $\sf{e_S} = \sf{e_S^A} K_A$, and [[homogeneous H-connection|homogeneous space geometry]], $\sf{A_S} = \sf{A_S^P} H_P$, changing to\n\sbegin{eqnarray}\n\sf{e'_S} &=& h^- \sf{e_S} h \s\s\n\sf{A'_S} &=& h^- \sf{A_S} h + h^- \sf{d} h\n\send{eqnarray}\nsince $H$ is taken to be [[reductive]] in $G$, which implies $A_{h^-} {\srm Lie}(G/H) \sin {\srm Lie}(G/H)$, $A_{h^-} {\srm Lie}(H) \sin {\srm Lie}(H)$, and $h^- \spa_a h \sin {\srm Lie}(H)$. The gauge transformation of the H-connection is familiar as the [[gauge transformation]] of a principal bundle connection, and the transformation of the homogeneous space frame, $\sf{e'_S^A} = \slp K^A , h^- K_B h \srp \sf{e_S^B} = L^A{}_B \sf{e_S^B}$, is a [[Lorentz rotation]], familiar as the co[[tangent bundle gauge transformation]]. \n\nFor an infinitesimal gauge transformation, $h \ssimeq 1 + h^P H_P = 1 + H$, these transformations are\n\sbegin{eqnarray}\n\sf{e'_S} &\ssimeq& (1 - H) \sf{e_S^B} K_B (1 + H) \ssimeq \sf{e_S} + \slb \sf{e_S} , H \srb \s\s\n\sf{A'_S} &\ssimeq& (1 - H) \sf{A_S^R} H_R (1 + H) + (1 - H) \sf{d} (1 + H) \ssimeq \sf{A_S} + \sf{d} H + \slb \sf{A_S} , H \srb \n\send{eqnarray}\nto first order in the gauge parameters, $h^P$.\n\nA more general possible gauge transformation is\n$$\nr'(x) = r(x) \s, g(x)\n$$\nby $g(x) \sin G$. This transformation may move the points of the homogeneous space, with diffeomorphism $\sph(x,y)=(x_\sph(x,y),y_\sph(x,y))$. The Maurer-Cartan frame over $S$ transforms to\n$$\n\sf{I'}(x) = g^- \sf{I} g + g^- \sf{d} g = \sf{e'_S} + \sf{A'_S}\n$$\nwhich possibly mixes the homogeneous space frame and H-connection. For an infinitesimal gauge transformation,\n$$\ng \ssimeq 1 + g^A K_A + g^P H_P = 1 + K + H\n$$\nthis transformation is\n\sbegin{eqnarray}\n\sf{I'} &\ssimeq& \slp 1 - K - H \srp \slp \sf{e_S^B} K_B + \sf{A_S^Q} H_Q \srp \slp 1 + K + H \srp + \sf{d} K + \sf{d} H \s\s\n&\ssimeq& \sf{I} + \slb \sf{e_S} , K \srb + \slb \sf{A_S} , K \srb + \slb \sf{e_S} , H \srb + \slb \sf{A_S} , H \srb + \sf{d} K + \sf{d} H \s\s\n&=& \sf{I} - g^A \sf{e_S^B} \slp C_{AB}{}^C K_C + C_{AB}{}^P H_P \srp - g^A \sf{A_S^Q} C_{AQ}{}^C K_C - g^P \sf{e_S^B} C_{PB}{}^C K_C - g^P \sf{A_S^Q} C_{PQ}{}^R H_R + \sf{d} g^A K_A + \sf{d} g^P H_P\n\send{eqnarray}\ngiving\n\sbegin{eqnarray}\n\sf{e'_S} &\ssimeq& \sf{e_S} + \slp - g^A \sf{e_S^B} C_{AB}{}^C - g^A \sf{A_S^Q} C_{AQ}{}^C - g^P \sf{e_S^B} C_{PB}{}^C + \sf{d} g^C \srp K_C \s\s\n&=& \sf{e_S} + \sf{d} K + \slb \sf{A_S} , K \srb + \slb \sf{e_S} , H \srb + \slb \sf{e_S} , K \srb_K \s\s\n\sf{A'_S} &\ssimeq& \sf{A_S} + \slp - g^A \sf{e_S^B} C_{AB}{}^R - g^P \sf{A_S^Q} C_{PQ}{}^R + \sf{d} g^R \srp H_R \s\s\n&=& \sf{A_S} + \sf{d} H + \slb \sf{A_S} , H \srb + \slb \sf{e_S} , K \srb_H\n\send{eqnarray}\n\sbegin{eqnarray}\nto first order in the gauge parameters, $g^I$. There is, though, a potential problem with this type of gauge transformation: If we choose $g(x)=r^-(x)$ then the section transforms to $r'(x) = r(x) \s, g(x) = r \s, r^- = 1$, which is no longer a section since $\spi \scirc r'$ is not the identity map on $S$. It is still interesting to consider though, as this choice results in $\sf{I'} = r'^- \sf{d} r' = 0$, but such a gauge transformation may not be allowed as it may not come from a diffeomorphism, unless the space is contractable.
A [[homogeneous space]], $S=G/H$, built from a [[Lie group]], $G$, and a [[subgroup]], $H$, inherits a geometry from the [[Lie group tangent bundle geometry]] of $G$ and how the $H$ subgroup -- a [[submanifold]] of $G$ -- sits in $G$. The flows on the [[Lie group manifold|Lie group geometry]] are described by the [[Lie algebra]] generators, $T_I \sin {\srm Lie}(G)$, with [[index|indices]] $I$ running from $1$ to $n_G$ -- the dimension of $G$. These generators are presumed to be rotated so the [[Killing form]],\n$$\n\slp T_I, T_J \srp = g_{IJ} = C_{IK}{}^L C_{JL}{}^K \n$$\nis diagonal and the structure constants satisfy $C_{IKL} = - C_{ILK}$. The subgroup, $H$, is taken to be [[reductive]] in $G$, with generators $H_P = T_P \sin {\srm Lie}(H)$, (with $P$-series indices running from $1$ to $n_H$). The $n_S = (n_G - n_H)$ remaining generators are the ''coset generators'', $K_A = T_A$, spanning the [[vector space]], ${\srm Lie}(G/H)$. We take $z^i$ to be coordinates for $G$, $y^p$ to be coordinates for $H$, and $x^a$ to be coordinates for $S$. A good choice for [[coset representative|homogeneous space]]s is the [[exponentiation]] of the coset generators,\n$$\nr(x) = e^{x^a K_a} \sin G\n$$\nA coset representative is not necessarily a subgroup of $G$, but it is a section and a [[submanifold]]. If $G$ has a matrix representation, the exponentiation above gives the explicit form of $r(x)$ as a matrix -- a good way to think of it. Whatever the choice of coset representative, the [[Maurer-Cartan form]] over $G$ [[pulls back|pullback]] to give the [[Maurer-Cartan frame|homogeneous space]] over $S$,\n$$\n\sf{I}(x) = \sf{e_S} + \sf{A_S} = r^- \sf{d} r\n$$\nwhich splits into the ''homogeneous space frame'', $\sf{e_S} = \sf{e_S^A} K_A$, and the ''homogeneous H-connection'', $\sf{A_S} = \sf{A_S^P} H_P$. The coefficients are determined by the Lie group geometry and may be computed explicitly using the Killing form,\n\sbegin{eqnarray}\n\sf{e_S^A} &=& \slp K^A, \sf{I} \srp \s\s\n\sf{A_S^P} &=& \slp H^P, \sf{I} \srp\n\send{eqnarray}\nThese homogeneous space frame 1-forms may be used as the [[frame]] 1-forms for the [[tangent bundle]] over $S$, with resulting ''homogeneous space [[metric]]'',\n$$\n\slp \sve{u}, \sve{v} \srp = \slp \sve{u} \sf{e_S}, \sve{v} \sf{e_S} \srp = u^A v^B \slp K_A, K_B \srp = u^A v^B g_{AB} = u^a v^b \slp e^S_a \srp^A \slp e^S_b \srp^B g_{AB} = u^a v^b g_{ab}\n$$\nNote that this metric, $g_{ab} = \slp e^S_a \srp^A \slp e^S_b \srp^B g_{AB}$, over $S$ does NOT correspond to the natural [[submanifold geometry]] of the coset representative in $G$ -- that metric would be:\n$$\ng'_{ab} = \slp e^S_a \srp^A \slp e^S_b \srp^B g_{AB} + \slp A^S_a \srp^P \slp A^S_b \srp^Q g_{PQ}\n$$\nRather, the homogeneous space metric is independent of the choice of coset representative, and thus necessarily independent of the homogeneous H-connection. The homogeneous H-connection, $\sf{A_S}(x)$, is a particular (constrained to be part of the Maurer-Cartan frame) [[principal bundle]] connection when the homogeneous space is viewed as the base of a principal $H$-bundle.\n\nWith the homogeneous space frame in hand it is straightforward to calculate the [[homogeneous space tangent bundle geometry]] based on a choice of [[torsion]]; and to calculate the [[homogeneous space geometry symmetries]] -- the [[Killing vector]]s of the homogeneous space frame.\n\nRefs:\n*Roberto Camporesi\n**http://calvino.polito.it/~camporesi/\n*Leonardo Castellani\n**http://www.mfn.unipmn.it/%7ecastella/\n**[[On G/H geometry and its use in M-theory compactifications|papers/9912277.pdf]]\n**[[Symmetries of Coset Spaces and Kaluza-Klein Supergravity|papers/Symmetries of Coset Spaces and Kaluza-Klein Supergravity.pdf]]\n*[[Super coset space geometry|papers/0610039.pdf]]\n*Excellent new paper:\n**[[Heat Kernel on Homogeneous Bundles over Symmetric Spaces|papers/0701489.pdf]]
Not surprisingly, a [[homogeneous space geometry]] has a large symmetry group, described by [[Killing vector]] fields, $\sve{\sxi}(x)$, over $S=G/H$. Most of the symmetries, $\sve{\sxi_I}$, correspond to the [[left action|group]] of the [[Lie group]], $G$, through its [[Lie algebra]] generators, $T_I$. However, more symmetries, $\sve{\sxi'_X}$, come from the right action of a different group, though some of these may correspond to some of the $\sve{\sxi_I}$.\n\nAn element $g \sin G$, acts from the left on the coset element, $[r(x)] \sin S$, to give another coset element, $[g \s, r(x)] = [r(x')]$. This implies that the left action of $g$ on the coset representative, $r(x)$, is\n$$\ng \s, r(x) = r(x') h\n$$\nfor some $h \sin H$. If the group element is approximated near the identity by $g \ssimeq (1 + \sep^I T_I)$, with a corresponding [[flow]] on the coset manifold of $x' \ssimeq x + \sep^I \sxi_I$, and $h \ssimeq (1 + \sep^I h_I^P(x) H_P)$, the above equation,\n$$\n(1 + \sep^I T_I) \s, r \ssimeq (r + \sep^I \sve{\sxi_I} \sf{d} r) (1 + \sep^I h_I^P H_P)\n$$\ngives, to first order in $\sep^I$,\n$$\nT_I r = \sve{\sxi_I} \sf{d} r + h_I^P r H_P\n$$\nMultiplying on the left by $r^-$ and using the Maurer-Cartan frame over $S$,\n$$\n\sf{I} = r^- \sf{d} r = \sf{e_S^A} K_A + \sf{A_S^P} H_P\n$$\ngives\n$$\nr^- T_I r = \slp \sxi_I \srp^a \slp e^S_a \srp^A K_A + \slp \sxi_I \srp^a \slp A^S_a \srp^P H_P + h_I^P H_P\n$$\nplugging this into the ${\srm Lie}(G)$ [[Killing form]] with the generator duals of the $K_A \sin {\srm Lie}(G/H)$ and $H_P \sin {\srm Lie}(H)$ gives explicit expressions for the coefficients of the ''left Killing vector fields on the homogeneous space geometry'',\n$$\n\slp \sxi_I \srp^a(x) = \slp e^S_A \srp^a \slp K^A , r^- T_I r \srp\n$$\nand the ''H-compensator'',\n$$\nh_I{}^P(x) = -\slp \sxi_I \srp^a \slp A^S_a \srp^P + \slp H^P , r^- T_I r \srp\n$$\nThese $n$ Killing vector fields comprise the left action flows of $G$ on $S$. They are demonstrably Killing. Using the definition of the [[Lie derivative]],\n$$\n{\scal L}_{\sve{\sxi_I}} \sf{e_S^A} = \sve{\sxi_I} \slp \sf{d} \sf{e_S^A} \srp +\sf{d} \slp \sve{\sxi_I} \sf{e_S^A} \srp\n$$\nand an expression from the [[homogeneous space tangent bundle geometry]],\n\sbegin{eqnarray}\n\sve{\sxi_I} \slp \sf{d} \sf{e_S^A} \srp &=& \sve{\sxi_I} \slp - \sha \sf{e_S^C} \sf{e_S^B} C_{CB}{}^A - \sf{A_S^P} \sf{e_S^B} C_{PB}{}^A \srp \s\s\n&=& - \slp \sxi_I \srp^C \sf{e_S^B} C_{CB}{}^A - \slp \sve{\sxi_I} \sf{A_S^P} \srp \sf{e_S^B} C_{PB}{}^A + \sf{A_S^P} \slp \sxi_I \srp^B C_{PB}{}^A\n\send{eqnarray}\ncombined with the coset representative relation, a Killing form identity, and the commutation relations for the [[reductive]] coset,\n\sbegin{eqnarray}\n\sf{d} \slp \sve{\sxi_I} \sf{e_S^A} \srp &=& \sf{d} \slp K^A , r^- T_I r \srp = \slp K^A , \slb r^- T_I r, \sf{I} \srb \srp = - \slp \slb K^A , \sf{I} \srb, r^- T_I r \srp \s\s\n&=& - \sf{e_S^B} C^A{}_B{}^P \slp H_P, r^- T_I r \srp - \slp \sf{e_S^B} C^A{}_B{}^C + \sf{A_S^P} C^A{}_P{}^C \srp \slp K_C, r^- T_I r \srp \s\s\n&=& - \sf{e_S^B} C^A{}_B{}^P \slp h_{IP} + \slp \sxi_I \srp^a \slp A^S_a \srp_P \srp - \slp \sf{e_S^B} C^A{}_B{}^C + \sf{A_S^P} C^A{}_P{}^C \srp \slp \sxi_I \srp_C\n\send{eqnarray}\ngives, after happy cancellation,\n$$\n{\scal L}_{\sve{\sxi_I}} \sf{e_S^A} = - \sf{e_S^B} h_I{}^P C_P{}^A{}_B\n$$\nwith nice Killing rotation coefficients, $\slp B_I \srp_B{}^A = h_I{}^P C_{PB}{}^A$.\n\nThe right action of $g \sin G$ on a coset element, $[r] = r H$, only makes sense if $g$ "gets past" the $H$ so that $R_g \s, [r] = r H g = r g H = [r g]$, which is true iff $g$ is in the [[normalizer]], $g \sin N_G(H) = N(H)$. Of course, if $g \sin H$ its right action has no effect on $[r]$, so the ''right action'' group of symmetries is the group $N(H)/H$. The right Killing vectors, $\sve{\sxi'_X}$, and H-compensators, $h'$, corresponding to a $g \sin N(H)/H$ come from\n\sbegin{eqnarray}\nr \s, g &=& r(x') h' \s\s\nr \s, (1 + \sep^X K_X) &\ssimeq& (r + \sep^X \sve{\sxi'_X} \sf{d} r) (1 + \sep^X h'_X^P H_P) \s\s\nK_X &=& \slp \sxi'_X \srp^a \slp e^S_a \srp^A K_A + \slp \sxi'_X \srp^a \slp A^S_a \srp^P H_P + h'_X^P H_P\n\send{eqnarray}\n(in which $K_X \sin {\srm Lie}(N(H)/H) \ssubset {\srm Lie}(G/H)$ are a reduced linear combination of $K_A$) and are\n\sbegin{eqnarray}\n\slp \sxi'_X \srp^a(x) &=& \slp e^S_X \srp^a \s\s\nh'_X{}^P(x) &=& -\slp \sxi'_X \srp^a \slp A^S_a \srp^P + \sde_I^P\n\send{eqnarray}\nSince the right Killing vectors are a reduced linear combination of the orthonormal basis vectors, the Killing equation is\n\sbegin{eqnarray}\n{\scal L}_{\sve{\sxi'_X}} \sf{e_S^A} &=& {\scal L}_{\sve{e_X}} \sf{e_S^A} = \sve{e_X} \slp \sf{d} \sf{e_S^A} \srp \s\s\n&=& \sve{e^S_X} \slp - \sha \sf{e_S^C} \sf{e_S^B} C_{CB}{}^A - \sf{A_S^P} \sf{e_S^B} C_{PB}{}^A \srp \s\s\n&=& \sf{e_S^B} \slp -C_{XB}{}^A - \slp A^S_X \srp^P C_{PB}{}^A + \slp A^S_B \srp^P C_{PX}{}^A \srp\n\send{eqnarray}\nshowing that $\sve{\sxi'_X}$ is Killing, with Killing rotation coefficients $\slp B'_X \srp_B{}^A = \slp -C_{XB}{}^A - \slp A_X \srp^P C_{PB}{}^A \srp$, if the last term above vanishes -- which it does since the reductivity of $H$ in $G$ implies $\slb {\srm Lie}(H), {\srm Lie}(G/H) \srb \ssubset {\srm Lie}(G/H)$, and that $H$ is normal in $N(H)/H$ implies $\slb {\srm Lie}(H), {\srm Lie}(N(H)/H) \srb \ssubset {\srm Lie}(H)$, and together these imply $\slb {\srm Lie}(H), {\srm Lie}(N(H)/H) \srb = 0$, which gives $C_{PX}{}^A = 0$.\n\nAfter this analysis one might think the full symmetry group of the homogeneous space geometry is $G \stimes N(H)/H$, and that is the biggest it could be, but it might be smaller since some of the right Killing vectors, $\sve{\sxi'_X}$, may not be independent of the left Killing vectors, $\sve{\sxi_I}$.
An analysis of [[reductive]] [[homogeneous space geometry]] resulted in the Maurer-Cartan frame,\n$$\n\sf{I} = \sf{e_S} + \sf{A_S} = r^- \sf{d} r\n$$\nover the [[homogeneous space]], $S=G/H$, which split into the homogeneous space frame, $\sf{e_S} = \sf{dx^a} \slp e^S_a \srp^A K_A$, and homogeneous H-connection, $\sf{A_S} = \sf{dx^a} \slp A^S_a \srp^P H_P$. As the [[pullback]] of the [[Maurer-Cartan form]] [[curvature]], the ''homogeneous space curvature'' vanishes,\n$$\n0 = \sff{F}(x) = \sf{d} \sf{I} + \sha \slb \sf{I}, \sf{I} \srb = \sf{d} \sf{e_S} + \sf{d} \sf{A_S} + \sha \slb \sf{e_S}, \sf{e_S} \srb + \slb \sf{A_S}, \sf{e_S} \srb + \sha \slb \sf{A_S}, \sf{A_S} \srb \n$$\nwhich, via the [[reductive]] homogeneous space commutation relations, gives\n\sbegin{eqnarray}\n0 &=& \sf{d} \sf{e_S^A} + \sf{A_S^P} \sf{e_S^B} C_{PB}{}^A + \sha \sf{e_S^C} \sf{e_S^B} C_{CB}{}^A \s\s \n0 &=& \sf{d} \sf{A_S^P} + \sha \sf{A_S^Q} \sf{A_S^R} C_{QR}{}^P + \sha \sf{e_S^C} \sf{e_S^D} C_{CD}{}^P\n\send{eqnarray}\nThis implies the ''[[principal bundle]] curvature of the homogeneous H-connection'' is\n$$\n\sff{F_H} = \sff{F_H^P} H_P = \sf{d} \sf{A_S} + \sha \slb \sf{A_S}, \sf{A_S} \srb = - \sha \sf{e_S^F} \sf{e_S^E} C_{FE}{}^P H_P\n$$\nBut what [[tangent bundle connection]], $\sf{w}^A{}_B$, do we choose to build over $S$ when treating $\sf{e_S^A}$ as the [[frame]] 1-forms? I.e., what do we choose for the [[torsion]],\n$$\n\sff{T^A} = \sf{d} \sf{e_S^A} + \sf{\som}^A{}_B \sf{e_S^B} = ?\n$$\nThere are two decent looking choices:\n\nIf we choose a torsion of $\sff{T^A} = - \sha \sf{e_S^C} \sf{e_S^B} C_{CB}{}^A$, the ''torsionful homogeneous space connection'' is $\sf{w}^A{}_B = \sf{A_S^P} C_{PB}{}^A$, which produces a nice [[Riemann curvature]] of\n\sbegin{eqnarray}\n\sff{R}^A{}_B &=& \sf{d} \sf{w}^A{}_B + \sf{w}^A{}_C \sf{w}^C{}_B \s\s\n&=& \sf{d} \sf{A_S^P} C_{PB}{}^A + \sf{A_S^Q} C_{QC}{}^A \sf{A_S^R} C_{RB}{}^C \s\s\n&=& - \sha \sf{e_S^F} \sf{e_S^E} C_{FE}{}^P C_{PB}{}^A + \sf{A_S^Q} \sf{A_S^R} \slp - \sha C_{QR}{}^P C_{PB}{}^A + C_{QC}{}^A C_{RB}{}^C \srp \s\s\n&=& - \sha \sf{e_S^F} \sf{e_S^E} C_{FE}{}^P C_{PB}{}^A\n\send{eqnarray}\nby the [[Jacobi identity|Lie algebra]]. This is nice because it matches the H-connection curvature,\n$$\n\sff{R}^A{}_B = \sff{F_H^P} C_{PB}{}^A\n$$\n\nBut we often want torsion to vanish, $\sff{T^A} = 0$. This choice results in the ''torsionless homogeneous space connection'',\n$$\n\sf{w}^A{}_B = \sha \sf{e_S^C} C_{CB}{}^A + \sf{A_S^P} C_{PB}{}^A\n$$\nwhich produces the Riemann curvature,\n\sbegin{eqnarray}\n\sff{R}^A{}_B &=& \sf{d} \sf{w}^A{}_B + \sf{w}^A{}_C \sf{w}^C{}_B \s\s\n&=& \sha \sf{d} \sf{e_S^C} C_{CB}{}^A + \sf{d} \sf{A_S^P} C_{PB}{}^A + \slp \sha \sf{e_S^F} C_{FC}{}^A + \sf{A_S^Q} C_{QC}{}^A \srp \slp \sha \sf{e_S^E} C_{EB}{}^C + \sf{A_S^R} C_{RB}{}^C \srp \s\s\n&=& \sha \sf{e_S^F} \sf{e_S^E} \slp - \sfr{1}{4} C_{FE}{}^C C_{CB}{}^A - C_{FE}{}^P C_{PB}{}^A \srp \s\s\n&=& \sha \sf{e_S^F} \sf{e_S^E} R_{FE}{}^A{}_B\n\send{eqnarray}\nand a [[Ricci curvature]] of\n\sbegin{eqnarray}\n\sf{R}{}_B &=& \sve{e^S_A} \sff{R}^A{}_B = \sf{e_S^E} R_{AE}{}^A{}_B \s\s\n&=& \sf{e_S^E} \slp - \sfr{1}{4} C_{AE}{}^C C_{CB}{}^A - C_{AE}{}^P C_{PB}{}^A \srp \s\s\n&=& - \sha \sf{e_S^E} \slp g_{EB} - \sha C_{EA}{}^C C_{BC}{}^A \srp\n\send{eqnarray}\nwhich gives a [[curvature scalar]] of\n$$\nR = \sve{e^S_B} \sf{R}{}^B = - \sha \slp n_S - \sha C_{BA}{}^C C^B{}_C{}^A \srp\n$$\n\nFor a [[symmetric space]], $C_{AB}{}^C=0$, there is no choice -- the torsion vanishes and the ''symmetric space connection'' is $\sf{w}^A{}_B = \sf{A_S^P} C_{PB}{}^A$.
The analogue of [[spherical coordinates]] in $n$ dimensions are given in terms of $r$ and the $(n-1)$ ''angular coordinates'', $a^w$, by\n\sbegin{eqnarray}\nx^1 &=& r \scos(a^1) \s\s\nx^2 &=& r \ssin(a^1) \scos(a^2) \s\s\nx^3 &=& r \ssin(a^1) \ssin(a^2) \scos(a^3) \s\s\n& \svdots \s\s\nx^{n-1} &=& r \ssin(a^1) \sdots \ssin(a^{n-2}) \scos(a^{n-1}) \s\s\nx^n &=& r \ssin(a^1) \sdots \ssin(a^{n-2}) \ssin(a^{n-1})\n\send{eqnarray}\n\nRef:\n*http://en.wikipedia.org/wiki/Hypersphere#Hyperspherical_coordinates
<<ListTagged illus>>
An ''antisymmetric index bracket'' is used to produce an indexed quantity (tensor) that is antisymmetric in its indices. For example, for a tensor with two [[indices]], $a_{ij}$, the antisymmetric part of this tensor is \n\s[ a_{\slb ij \srb} = \sha \slp a_{ij} - a_{ji} \srp \s]\nA tensor is antisymmetric in its indices iff it equals its corresponding antisymmetric part, $a_{ij} = a_{\slb ij \srb}$. Such an antisymmetric tensor changes sign under the interchange of any two neighboring indices, $a_{ij}=-a_{ji}$.\n\nFor a tensor with three indices, $b_{ijk}$, the antisymmeterized part is\n\s[ b_{\slb ijk \srb} = \sfr{1}{3!} \slp b_{ijk}+b_{jki}+b_{kij}-b_{jik}-b_{ikj}-b_{kji} \srp \s]\n\nThe ''antisymmeterized index bracket'' is similar in operation to the [[antisymmetric bracket]].\n\nA ''symmetric index bracket'' is used to produce an indexed quantity (tensor) that is symmetric in its indices. For example, for a tensor with two indices, $a_{ij}$, the symmetric part of this tensor is \n\s[ a_{\slp ij \srp} = \sha \slp a_{ij} + a_{ji} \srp \s]\nA tensor is symmetric in its indices iff it equals its corresponding symmetric part, $a_{ij} = a_{\slp ij \srp}$. Such a symmetric tensor is invariant under the interchange of any two neighboring indices, $a_{ij}=a_{ji}$.\n\nFor a tensor with three indices, $b_{ijk}$, the symmeterized part is\n\s[ b_{\slp ijk \srp} = \sfr{1}{3!} \slp b_{ijk}+b_{jki}+b_{kij}+b_{jik}+b_{ikj}+b_{kji} \srp \s]
Unless stated otherwise, repeated indices in expressions are summed &mdash; for example,\n\s[ \sve{v} \sf{f} = v^i f_i = \ssum_{i=0}^{n-1} v^i f_i \s]\nThis, Einstein's summation convention, loses the information on the range of the sum. To remedy this deficiency, indices from different parts of the alphabets are taken to range over different integers corresponding to the spaces they coordinatize or label:\n| !Latin index | !Greek index | !Range over |!For |\n| $i,j,k,l,m,n$ | $\sal,\sbe,\sga,\sde,\sep,\sup$ | $0 \sdots (n-1) \s, {\srm or} \s, 1 \sdots n$ |all $n$, or any appropriate subset |\n| $a,b,c,d$ | $\smu,\snu,\ska,\sla$ | $0 \s, {\srm or} \s, 4,1,2,3$ |[[spacetime]] |\n| $e,f,g,h$ | $\sva,\sze,\sta,\sio$ | $1,2,3$ |space |\n| $w,x,y,z$ | $\spi,\srh,\ssi,\sxi$ | $1 \sdots (n-1)$ |spatial |\n| $p,q,r,u$ | $\sth,\sph,\sch,\sps$ | $4 \sdots (n-1) \s, {\srm or} \s, 5 \sdots n$ |Kaluza-Klein or fiber coordinates &mdash; i.e. other than spacetime |\n| $A,B,C$ | | $1 \sdots$ ? |Lie algebra elements |\nLower case Latin indices are for [[coordinates|manifold]], Greek indices are [[Clifford algebra]] basis element labels, and upper case Latin indices are [[Lie algebra]] generator element labels.
The integral over a volume, $V$, of the [[exterior derivative]] of a [[differential form]] equals the integral of that form over the boundary, $\spa V$, of that volume,\n$$\n\sint_V \sf{d} \snf{F} = \sint_{\spa V} \snf{F}\n$$\n\nThis, ''Stoke's theorem'', may be used to evaluate integrals by finding an ''antiderivative'' of the integrand. For example,\n$$\n\sint_{\slb 0,1 \srb} \sf{dx} \s, x = \slb \sha x^2 \srl_0^1 = \sha\n$$\nin which $F = \sha x^2$ is the antiderivative of $\sf{d} F = \sf{dx} \s, x$, and $\spa V$ consists of the boundary points $0$ and $1$. The "integral" over two points is simply the ordered sum of the integrand evaluated at those points. \n\nStoke's theorem is a generalization of the ''fundamental theorem of calculus''.\n\n//need to patch together simply connected regions for this definition to work. Betti number? DeRham chains?//
[[Donald Knuth|http://en.wikipedia.org/wiki/Knuth]] suggested the use of a minus sign for group inverses,\n\sbegin{eqnarray}\ng^- &=& g^{-1}\s\s\ngg^- &=& g^- g = 1\n\send{eqnarray}\nduring a talk on notation, http://scpd.stanford.edu/scpd/students/Dam_ui/pages/ArchivedVideoList56K.asp?Include=musings. It's more compact and makes sense, since raising a group element to a power is not always natural, but the inverse is.
Using the definition for the [[determinant]] of a matrix, such as the [[frame]] matrix, $\slp e_i\srp^\sal$, and its [[matrix inverse]], $\slp e^-_\sal \srp^i = \slp e_\sal \srp^i$, with the [[permutation symbol]] gives\n\sbegin{eqnarray}\n\sep^{\sal \sbe \sdots \sga} &=& \sep^{ij \sdots k} \slp e_i\srp^\sal \slp e_j\srp^\sbe \sdots \slp e_k\srp^\sga\s\s\n\sep^{\sal \sbe \sdots \sde \sga} \slp e_\sga \srp^k &=& \sep^{ij \sdots m k} \slp e_i\srp^\sal \slp e_j\srp^\sbe \sdots \slp e_m\srp^\sde\s\s\n\sep^{\sal \sbe \sdots \sep \sde \sga} \slp e_\sde \srp^m \slp e_\sga \srp^k &=& \sep^{ij \sdots n m k} \slp e_i\srp^\sal \slp e_j\srp^\sbe \sdots \slp e_n\srp^\sep\n\send{eqnarray}\nCombining these identities with the [[permutation identities]] allows the [[matrix inverse]] to be written explicitly, as well as giving other expressions, such as\n\sbegin{eqnarray}\n\slp e_{\slb \sga \srd}\srp^k \slp e_{\sld \sde \srb} \srp^m &=& \sfr{\sll \set \srl}{2 \slp n-2 \srp!} \sep^{ij\sdots nmk} \slp e_i \srp^\sal \slp e_j \srp^\sbe \sdots \slp e_n \srp^\sep \sep_{\sal \sbe \sdots \sep \sga \sde} \s\s\n&=& \sfr{1}{2 \sll e \srl \slp n-2 \srp!} \sva^{ij\sdots nmk} \slp e_i \srp^\sal \slp e_j \srp^\sbe \sdots \slp e_n \srp^\sep \sep_{\sal \sbe \sdots \sep \sga \sde}\n\send{eqnarray}
Jet spaces are spaces of derivatives of sections.\n\nRefs:\n*Gennadi Sardanashvily\n**[[Ten Lectures on Jet Manifolds in Classical and Quantum Field Theory|papers/0203040.pdf]]
/***\n|Name|Plugin: jsMath|\n|Created by|BobMcElrath (edited by Garrett)|\n|Email|my first name at my last name dot org|\n|Location|http://bob.mcelrath.org/tiddlyjsmath-2.0.3.html|\n|Version|1.3.g|\n|Requires|[[TiddlyWiki|http://www.tiddlywiki.com]] &ge; 2.1, [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]] &ge; 3.0|\n!Description\n[[LaTeX]] is the world standard for specifying, typesetting, and communicating mathematics among scientists, engineers, and mathematicians. For more information about LaTeX itself, visit the [[LaTeX Project|http://www.latex-project.org/]]. This plugin typesets math using [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]], which is an implementation of the TeX math rules and typesetting in javascript, for your browser. Notice the small button in the lower right corner which opens its control panel.\n!Installation\nIn addition to this plugin, you must also [[install jsMath|http://www.math.union.edu/~dpvc/jsMath/download/jsMath.html]] on the same server as your TiddlyWiki html file. If you're using TiddlyWiki without a web server, then the jsMath directory must be placed in the same location as the TiddlyWiki html file.\n!Examples\n|!Source|!Output|h\n|{{{The variable $x$ is real.}}}|The variable $x$ is real.|\n|{{{The variable \s(y\s) is complex.}}}|The variable \s(y\s) is complex.|\n|{{{This \s[\sint_a^b x = \sfrac{1}{2}(b^2-a^2)\s] is an easy integral.}}}|This \s[\sint_a^b x = \sfrac{1}{2}(b^2-a^2)\s] is an easy integral.|\n|{{{This $$\sint_a^b \ssin x = -(\scos b - \scos a)$$ is another easy integral.}}}|This $$\sint_a^b \ssin x = -(\scos b - \scos a)$$ is another easy integral.|\n|{{{Block formatted equations may also use the 'equation' environment \sbegin{equation} \sint \stan x = -\sln \scos x \send{equation} }}}|Block formatted equations may also use the 'equation' environment \sbegin{equation} \sint \stan x = -\sln \scos x \send{equation}|\n|{{{Equation arrays are also supported \sbegin{eqnarray} a &=& b \s\s c &=& d \send{eqnarray} }}}|Equation arrays are also supported \sbegin{eqnarray} a &=& b \s\s c &=& d \send{eqnarray} |\n|{{{I spent \s$7.38 on lunch.}}}|I spent \s$7.38 on lunch.|\n|{{{I had to insert a backslash (\s\s) into my document}}}|I had to insert a backslash (\s\s) into my document|\n!Code\n***/\n//{{{\n\n// Define wikifers for latex\nconfig.formatterHelpers.mathFormatHelper = function(w) {\n var e = document.createElement(this.element);\n e.className = this.className;\n var endRegExp = new RegExp(this.terminator, "mg");\n endRegExp.lastIndex = w.matchStart+w.matchLength;\n var matched = endRegExp.exec(w.source);\n if(matched) {\n var txt = w.source.substr(w.matchStart+w.matchLength, \n matched.index-w.matchStart-w.matchLength);\n if(this.keepdelim) {\n txt = w.source.substr(w.matchStart, matched.index+matched[0].length-w.matchStart);\n }\n e.appendChild(document.createTextNode(txt));\n w.output.appendChild(e);\n w.nextMatch = endRegExp.lastIndex;\n }\n}\n\nconfig.formatters.push({\n name: "displayMath1",\n match: "\s\s\s$\s\s\s$",\n terminator: "\s\s\s$\s\s\s$\s\sn?",\n element: "div",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nconfig.formatters.push({\n name: "inlineMath1",\n match: "\s\s\s$", \n terminator: "\s\s\s$",\n element: "span",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nvar backslashformatters = new Array(0);\n\nbackslashformatters.push({\n name: "inlineMath2",\n match: "\s\s\s\s\s\s\s(",\n terminator: "\s\s\s\s\s\s\s)",\n element: "span",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nbackslashformatters.push({\n name: "displayMath2",\n match: "\s\s\s\s\s\s\s[",\n terminator: "\s\s\s\s\s\s\s]\s\sn?",\n element: "div",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\nbackslashformatters.push({\n name: "displayMath3",\n match: "\s\s\s\sbegin\s\s{equation\s\s}",\n terminator: "\s\s\s\send\s\s{equation\s\s}\s\sn?",\n element: "div",\n className: "math",\n handler: config.formatterHelpers.mathFormatHelper\n});\n\n// These can be nested. e.g. \sbegin{equation} \sbegin{array}{ccc} \sbegin{array}{ccc} ...\nbackslashformatters.push({\n name: "displayMath4",\n match: "\s\s\s\sbegin\s\s{eqnarray\s\s}",\n terminator: "\s\s\s\send\s\s{eqnarray\s\s}\s\sn?",\n element: "div",\n className: "math",\n keepdelim: true,\n handler: config.formatterHelpers.mathFormatHelper\n});\n\n// The escape must come between backslash formatters and regular ones.\n// So any latex-like \scommands must be added to the beginning of\n// backslashformatters here.\nbackslashformatters.push({\n name: "escape",\n match: "\s\s\s\s.",\n handler: function(w) {\n w.output.appendChild(document.createTextNode(w.source.substr(w.matchStart+1,1)));\n w.nextMatch = w.matchStart+2;\n }\n});\n\nconfig.formatters=backslashformatters.concat(config.formatters);\n\n/* G updated this */\nwindow.wikify = function(source,output,highlightRegExp,note)\n{\n if(source && source != "") {\n var wikifier = new Wikifier(source,getParser(note),highlightRegExp,note);\n wikifier.subWikifyUnterm(output);\n jsMath.Process();\n }\n}\n\n/* insert jsMath LaTeX macros here */\n\n/* jsMath.Extension.Require("AMSmath"); */\n/* jsMath.Extension.Require("AMSsymbols"); */\n/* jsMath.Extension.Require('underset-overset'); */\n\n/* Greek */\njsMath.Macro('al','\s\salpha');\njsMath.Macro('be','\s\sbeta');\njsMath.Macro('ga','\s\sgamma');\njsMath.Macro('de','\s\sdelta');\njsMath.Macro('ep','\s\sepsilon');\njsMath.Macro('va','\s\svarepsilon');\njsMath.Macro('ze','\s\szeta ');\njsMath.Macro('et','\s\seta');\njsMath.Macro('th','\s\stheta');\njsMath.Macro('io','\s\siota');\njsMath.Macro('ka','\s\skappa');\njsMath.Macro('la','\s\slambda');\njsMath.Macro('rh','\s\srho');\njsMath.Macro('si','\s\ssigma');\njsMath.Macro('ta','\s\stau');\njsMath.Macro('up','\s\supsilon');\njsMath.Macro('ph','\s\sphi');\njsMath.Macro('ch','\s\schi');\njsMath.Macro('ps','\s\spsi');\njsMath.Macro('om','\s\somega');\njsMath.Macro('Ga','\s\sGamma');\njsMath.Macro('De','\s\sDelta');\njsMath.Macro('Th','\s\sTheta');\njsMath.Macro('La','\s\sLambda');\njsMath.Macro('Si','\s\sSigma');\njsMath.Macro('Up','\s\sUpsilon');\njsMath.Macro('Ph','\s\sPhi');\njsMath.Macro('Ps','\s\sPsi');\njsMath.Macro('Om','\s\sOmega');\n\n/* misc */\njsMath.Macro('pa','\s\spartial');\njsMath.Macro('na','\s\snabla');\njsMath.Macro('ti','\s\stimes');\njsMath.Macro('lb','\s\sleft[');\njsMath.Macro('rb','\s\sright]');\njsMath.Macro('lp','\s\sleft(');\njsMath.Macro('rp','\s\sright)');\njsMath.Macro('li','\s\sleft<');\njsMath.Macro('ri','\s\sright>');\njsMath.Macro('ll','\s\sleft|');\njsMath.Macro('rl','\s\sright|');\njsMath.Macro('lc','\s\sleft\s\s{');\njsMath.Macro('rc','\s\sright\s\s}');\njsMath.Macro('ld','\s\sleft.');\njsMath.Macro('rd','\s\sright.');\njsMath.Macro('ha','{\s\ssmall \s\sfrac{1}{2}}');\njsMath.Macro('fr','{\s\ssmall \s\sfrac{#1}{#2}}',2);\njsMath.Macro('p','\s\sphantom{#1}',1);\njsMath.Macro('vp','\s\svphantom{#1}',1);\n\n/* accents */\njsMath.Macro('f','{\s\sunderset{\s\sraise4mu{\s\ssmash{-}}}{{#1}}}',1);\njsMath.Macro('ff','{\s\sunderset{\s\sraise3mu{\s\ssmash{=}}}{{#1}}}',1);\njsMath.Macro('fff','{\s\sunderset{\s\sraise3mu{\s\ssmash{\s\sequiv}}}{{#1}}}',1);\njsMath.Macro('nf','{\s\sunderset{\s\sraise4mu{\s\ssmash{\s\ssim}}}{{#1}}}',1);\njsMath.Macro('ud','{\s\sunderset{\s\sraise4mu{\s\ssmash{\s\scdot}}}{{#1}}}',1);\njsMath.Macro('od','{\s\soverset{\s\slower4mu{.}}{{#1}}}',1);\njsMath.Macro('udf','{\s\sunderset{\s\sraise4mu{\s\ssmash{- \s\scdot}}}{{#1}}}',1);\njsMath.Macro('udff','{\s\sunderset{\s\sraise3mu{\s\ssmash{= \s\scdot}}}{{#1}}}',1);\njsMath.Macro('ve','{\s\soverset{\s\slower4mu{\s\smoveright1mu{\s\srightharpoonup}}}{{#1}}}',1);\njsMath.Macro('vv','{\s\soverset{\s\slower4mu{\s\soverset{\s\smoveleft.1mu{\s\slower4mu{\s\sLarge \s\srightharpoonup}}}{\s\srightharpoonup}}}{{#1}}}',1);\n\n/* particle shapes */\n\njsMath.Macro('scir','\s\slower.1em{\s\srlap{\s\scolor{#1}{\s\sLarge \s\sbullet}}{\s\sLarge \s\scirc}}',1);\njsMath.Macro('ssqu','\s\srlap{\s\scolor{#1}{\s\sscriptsize \s\sblacksquare}}{{\s\sscriptsize \s\ssquare}}',1);\njsMath.Macro('sdia','\s\srlap{\s\scolor{#1}{\s\ssmall \s\sblacklozenge}}{\s\ssmall \s\slozenge}',1);\njsMath.Macro('stri','\s\sraise.08em{\s\srlap{\s\scolor{#1}{\s\ssmall \s\sblacktriangle}}{\s\ssmall \s\svartriangle}}',1);\njsMath.Macro('sutr','\s\slower.08em{\s\srlap{\s\scolor{#1}{\s\ssmall \s\sblacktriangledown}}{\s\ssmall \s\striangledown}}',1);\n\njsMath.Macro('mcir','\s\slower.1em{\s\srlap{\s\scolor{#1}{\s\sLARGE \s\sbullet}}{\s\sLARGE \s\scirc}}',1);\njsMath.Macro('msqu','\s\srlap{\s\scolor{#1}{\s\ssmall \s\sblacksquare}}{{\s\ssmall \s\ssquare}}',1);\njsMath.Macro('mdia','\s\srlap{\s\scolor{#1}{\s\sblacklozenge}}{\s\slozenge}',1);\njsMath.Macro('mtri','\s\sraise.08em{\s\srlap{\s\scolor{#1}{\s\sblacktriangle}}{\s\svartriangle}}',1);\njsMath.Macro('mutr','\s\slower.08em{\s\srlap{\s\scolor{#1}{\s\sblacktriangledown}}{\s\striangledown}}',1);\n\njsMath.Macro('bcir','\s\slower.1em{\s\srlap{\s\scolor{#1}{\s\shuge \s\sbullet}}{\s\shuge \s\scirc}}',1);\njsMath.Macro('bsqu','\s\srlap{\s\scolor{#1}{\s\sblacksquare}}{{\s\ssquare}}',1);\njsMath.Macro('bdia','\s\srlap{\s\scolor{#1}{\s\slarge \s\sblacklozenge}}{\s\slarge \s\slozenge}',1);\njsMath.Macro('btri','\s\sraise.08em{\s\srlap{\s\scolor{#1}{\s\slarge \s\sblacktriangle}}{\s\slarge \s\svartriangle}}',1);\njsMath.Macro('butr','\s\slower.08em{\s\srlap{\s\scolor{#1}{\s\slarge \s\sblacktriangledown}}{\s\slarge \s\striangledown}}',1);\n\njsMath.Macro('trip','\s\srlap{\s\sraise.15em{\s\skern.17em{{#1}}}}{\s\slower.1em{\s\srlap{{#2}}{{\s\skern.30em{{#3}}}}}}',3);\n\n/* end of jsMath macros */\n//}}}
*<<slider chkSliderssF ssF 'ss »' 'homogeneous spaces'>>\n*<<slider chkSlidercartanF cartanF 'cartan »' 'Cartan geometry'>>\n<<ListTagged kk>>
The [[left action|group]] of one [[Lie group]] element on all others induces a [[diffeomorphism]], $\sph_h(x)$ on the [[Lie group manifold|Lie group geometry]],\n$$\nL_h g(x) = h g(x) = g(\sph_h(x))\n$$\nA vector field on the Lie group manifold is ''left invariant'' iff it is invariant under the [[pushforward|pullback]] of this diffeomorphism for arbitrary $h$,\n$$\nL_{h*} \sve{v}(x) = \sve{v} \sf{\spa} \sph_h(x) = \sve{v}(\sph_h(x))\n$$\nThe partial derivative of the diffeomorphism in the above expression is computed explicitly by using the [[chain rule|partial derivative]],\n$$\n\sf{\spa} g(\sph_h(x)) = \slp \sf{\spa} \sve{\sph_h}(x) \srp \sf{\spa} g(\sph_h) = h \sf{\spa} g(x)\n$$\nand the defining equation for the [[right action vector fields and 1-forms|Lie group geometry]],\n$$\n\sf{\spa} g = g T_B \sf{\sxi_R^B}\n$$\nto write\n$$\n\slp \sf{\spa} \sve{\sph_h}(x) \srp h g T_B \sf{\sxi_R^B}(\sph_h) = h g T_B \sf{\sxi_R^B}(x)\n$$\nand get\n$$\n\sf{\spa} \sve{\sph_h}(x) = \sf{\sxi_R^B}(x) \sve{\sxi^R_B}(\sph_h(x))\n$$\nThis implies the right action vector fields are left invariant,\n$$\nL_{h*} \sve{\sxi^R_C}(x) = \sve{\sxi^R_C}(x) \sf{\spa} \sph_h(x) = \sve{\sxi^R_C}(x) \sf{\sxi_R^B}(x) \sve{\sxi^R_B}(\sph_h) = \sve{\sxi^R_C}(\sph_h(x))\n$$\n\nA [[differential form]] is left invariant iff it is invariant under the [[pullback]], $L_h^* \snf{F}(\sphi_h(x)) = \snf{F}(x)$. The 1-form duals to left invariant vector fields, such as the duals to the right action vector fields, are left invariant. Since autodiffeomorphisms are invertible, these statements may be summarized by defining any form or [[vector valued form]] to be left invariant iff it is invariant under the pushforward, $L_{h*} \snf{\sve{K}}(x) = \snf{\sve{K}}(\sphi_h(x))$, or pullback.
The defining equations for the left and right action Killing vector fields, $\sve{\sxi^L_B}$ and $\sve{\sxi^R_B}$, over a [[Lie group geometry]],\n\sbegin{eqnarray}\n\sve{\sxi^L_B} \sf{d} g &=& T_B g \s\s\n\sve{\sxi^R_B} \sf{d} g &=& g T_B\n\send{eqnarray}\nand for their [[1-form]] duals,\n\sbegin{eqnarray}\n\sf{\sxi_L^B} T_B &=& \slp \sf{d} g \srp g^- \s\s\n\sf{\sxi_R^B} T_B &=& g^- \sf{d} g\n\send{eqnarray}\nsatisfying $\sve{\sxi^L_B} \sf{\sxi_L^C} = \sde_B^C$ and $\sve{\sxi^R_B} \sf{\sxi_R^C} = \sde_B^C$, combine with the [[Killing form]] to give the ''left-right rotator'',\n$$\nL^C{}_B = \sve{\sxi^L_B} \sf{\sxi_R^C} = \sve{\sxi^L_B} \slp T^C , g^- \sf{d} g \srp = \slp T^C, g^- T_B g \srp \n$$\nThis is a [[Lorentz rotation]],\n\sbegin{eqnarray}\nL^A{}_B L^C{}_D g_{AC} &=& \slp T^A, g^- T_B g \srp \slp T^C, g^- T_D g \srp g_{AC} \s\s\n&=& \slp g T^A g^-, T_B \srp \slp g T_A g^-, T_D \srp \s\s\n&=& \slp T'^A, T_B \srp \slp T'_A, T_D \srp \s\s\n&=& \slp T'^B, T_D \srp\n= g_{BD}\n\send{eqnarray}\ngiving one set of Killing vector fields in terms of the other,\n\sbegin{eqnarray}\n\sve{\sxi^L_B} &=& L^C{}_B \sve{\sxi^R_C} \s\s\n\sve{\sxi^R_B} &=& L_B{}^C \sve{\sxi^L_C}\n\send{eqnarray}\n\nA Lorentz rotation of the structure constants by the left-right rotator leaves them invariant,\n\sbegin{eqnarray}\nL^C{}_D L^B{}_E C_{CB}{}^A L_A{}^F &=& \slp T^C, g^- T_D g \srp \slp T^B, g^- T_E g \srp \slp \slb T_C,T_B \srb, T^A \srp \slp T_A, g^- T^F g \srp \s\s\n&=& \slp \slb g^- T_D g,g^- T_E g \srb, g^- T^F g \srp = \slp g^- \slb T_D , T_E \srb g, g^- T^F g \srp \s\s\n&=& C_{DE}{}^F\n\send{eqnarray}\n\nThe [[exterior derivative]] of the left-right rotator is\n\sbegin{eqnarray}\n\sf{d} L^C{}_B &=& \sf{d} \slp T^C, g^- T_B g \srp \s\s\n&=& \slp T^C, \slb g^- T_B g \s, , \s, g^- \sf{d} g \srb \srp \s\s\n&=& \slp \slb \sf{\sxi_R^A} T_A , T^C \srb , g^- T_B g \srp \s\s\n&=& \sf{\sxi_R^A} C_A{}^C{}_D L^D{}_B\n\send{eqnarray}
The ''left/right [[chiral]]ity projector''s,\n$$\nP_{L/R} = \sha \slp 1 \spm \sga \srp\n$$\nare built using the spacetime Clifford algebra, [[Cl(1,3)]], [[pseudoscalar]], $\sga = \sga_0 \sga_1 \sga_2 \sga_3$. In the [[Weyl representation|Dirac matrices]], they are\n\sbegin{eqnarray}\nP_L &=&\n\slb \sbegin{array}{cccc}\n1 & 0 & 0 & 0 \s\s\n0 & 1 & 0 & 0 \s\s\n0 & 0 & 0 & 0 \s\s\n0 & 0 & 0 & 0\n\send{array} \srb \s\s\nP_R &=&\n\slb \sbegin{array}{cccc}\n0 & 0 & 0 & 0 \s\s\n0 & 0 & 0 & 0 \s\s\n0 & 0 & 1 & 0 \s\s\n0 & 0 & 0 & 1\n\send{array} \srb\n\send{eqnarray}
ref:\n[[Little Higgs Review|papers/0502182.pdf]]\nnice review
<<ListTagged lqg>>
[<img[images/png/manifold.png]]An oriented $n$ dimensional differentiable ''manifold'', $M$, may be visualized as a curved $n$ dimensional surface embedded in a higher dimensional, pseudo-Euclidean space. A manifold is described mathematically by a collection of coordinate charts (patches), $\sleft\s{ \sleft( U_a, \s: x_a \sright) \sright\s}$, with the open sets, $U_a$, labeled by $a$, covering $M$, and the coordinates, $x_a : U_a \srightarrow \smathbb{R}^n$, homeomorphic maps into open subsets of $\smathbb{R}^n$ such that overlap maps, $x_a \scirc x_b^{-} : \smathbb{R}^n \srightarrow M \srightarrow \smathbb{R}^n$, defined on $x_b( U_a \scap U_b)$, are infinitely differentiable. So, every point, $x$, on the manifold is labeled by a set of $n$ real ''coordinates'', $x_a^i(x)$, in some chart, $U_a$, with coordinate [[indices]], $i$, typically running from $1$ to $n$ or from $0$ to $(n-1)$. In most practical cases the chart label, $a$, is not written and the coordinates are simply written as $x^i$ with some chart implied.\n\nFor more on manifolds, see http://en.wikipedia.org/wiki/Manifold
*<<slider chkSliderdgF dgF 'dg »' 'basics of differential geometry'>>\n*<<slider chkSlidertopoF topoF 'topo »' 'topology, branching manifolds, morse theory'>>\n<<ListTagged math>>
The [[inverse]] of a square matrix, such as the [[frame]] matrix, $\slp e_i \srp^\sal$, may be written explicitly as\n\sbegin{eqnarray}\n\slp e^-_\sga \srp^k &=& \sfr{\sll \set \srl}{\slp n-1 \srp!} \sep^{ij\sdots mk} \slp e_i \srp^\sal \slp e_j \srp^\sbe \sdots \slp e_m \srp^\sde \sep_{\sal \sbe \sdots \sde \sga} \s\s\n&=& \sfr{1}{\sll e \srl \slp n-1 \srp!} \sva^{ij\sdots mk} \slp e_i \srp^\sal \slp e_j \srp^\sbe \sdots \slp e_m \srp^\sde \sep_{\sal \sbe \sdots \sde \sga}\n\send{eqnarray}\nby using the [[inverse matrix identities]] and [[permutation identities]]. It satisfies $\slp e^-_\sal \srp^i \slp e_i \srp^\sbe = \sde_\sal^\sbe$ and $\slp e_i \srp^\sal \slp e^-_\sal \srp^j = \sde_i^j$.
*<<slider chkSlidereditingF editingF 'editing »' 'tips on editing and authoring notes, including all sorts of tools'>>\n*<<slider chkSlider0F 0F '0 »' 'a note that is linked to but is empty or needs editing'>>\n*<<slider chkSlidersystemF systemF 'system »' 'control how the site operates and is layed out'>>\n*<<slider chkSliderillusF illusF 'illus »' 'notes containing illustrations'>>\n*<<slider chkSliderslideF slideF 'slide »' 'presentation slide (start note title with ".")'>>\n<<ListTagged meta>>
A ''metric'', $g$, for a [[vector space]] is a an object that takes two vectors and spits out a number -- it determines the symmetric ''scalar product'',\n$$\n\slp \sve{u}, \sve{v} \srp = u^i v^j \slp \sve{\spa_i}, \sve{\spa_j} \srp = v^i u^j g_{ij} \sin \smathbb{R}\n$$\nA metric on a manifold gives the scalar product between any two [[coordinate basis vectors]] at any point, $g_{ij} = \slp \sve{\spa_i}, \sve{\spa_j} \srp$. When a [[frame]] exists on the manifold it determines this scalar product and metric, using the [[Clifford algebra]] dot product and the [[vector-form algebra]], as\n\s[ \slp \sve{u}, \sve{v} \srp = \slp \sve{u} \sf{e} \srp \scdot \slp \sve{v} \sf{e} \srp = u^\sal \sga_\sal \scdot v^\sbe \sga_\sbe = u^\sal v^\sbe \set_{\sal \sbe} = u^i \slp e_i \srp^\sal v^j \slp e_j \srp^\sbe \set_{\sal \sbe} = u^i v^j g_{ij} \s]\nwith the use of frame coefficients and the [[Minkowski metric]] replacing the use of a metric if desired. Using component [[indices]], the ''metric matrix'' (often just abbreviated as "metric") in terms of the frame matrix is\n\s[ g_{ij} = \slp e_i \srp^\sal \slp e_j \srp^\sbe \set_{\sal \sbe} \s]\nThe metric is invariant under [[Lorentz transformations|Lorentz rotation]] of the frame,\n\s[ g'_{ij} = \slp e'_i \srp^\sal \slp e'_j \srp^\sbe \set_{\sal \sbe} = \slp e_i \srp^\sga L^\sal{}_\sga \slp e_j \srp^\sde L^\sbe{}_\sde \set_{\sal \sbe} = \slp e_i \srp^\sga \slp e_j \srp^\sde \set_{\sga \sde} = g_{ij} \s]\nAnother way of seeing this is that the scalar product of two tangent vectors is invariant under [[Clifford adjoint]] transformations of the frame,\n$$\sf{e} \smapsto \sf{e'} = U \sf{e} U^-$$\n$$\slp \sve{u} \sf{e'} \srp \scdot \slp \sve{v} \sf{e'} \srp = \slp \sve{u} U \sf{e} U^- \srp \scdot \slp \sve{v} U \sf{e} U^- \srp =\n\slp \sve{u} \sf{e} \srp \scdot \slp \sve{v} \sf{e} \srp$$\nwith Lorentz transformations forming a subset of these.
The combined spacetime curvature is:\n$$\n\sff{F_s} = \sha \slp \sff{R} + \sfr{\sLa}{6} \sf{e} \sf{e} \srp\n$$\nin which $\sff{R}$ is the [[Clifford vector bundle]] curvature, $\sLa$ is the [[cosmological constant|Einstein's equation]], and $\sf{e}$ is the [[frame]]. The ''modified BF action for gravity'' over a four dimensional base [[spacetime]] is:\n$$\nS_s = \sint \sli \sff{B_s} \sff{F_s} - \sfr{g \sLa}{48} \sff{B_s} \sff{B_s}) \sga \sri\n$$\nin which $\sff{B_s}$ is the ''dual bivector valued 2-form'', $g$ is some small coupling constant, and $\sga$ is the spacetime [[pseudoscalar]]. Insisting that $\sde S_s =0$ under $\sde \sff{B_s}$ gives:\n$$\n\sff{B_s} = \sfr{12}{g \sLa} \slp \sff{R} + \sfr{\sLa}{6} \sf{e} \sf{e} \srp \sga^-\n$$\nPlugging this back into the action gives:\n$$\nS_s = \sfr{3}{g \sLa} \sint \sli \slp \sff{R} + \sfr{\sLa}{6} \sf{e} \sf{e} \srp \slp \sff{R} + \sfr{\sLa}{6} \sf{e} \sf{e} \srp \sga^- \sri\n$$\nMultiplying this out gives a Chern-Simons boundary term,\n$$\n\sli \sff{R} \sff{R} \sga^- \sri = \sf{d} \sli \slp \sf{\som} \sf{d} \sf{\som} + \sfr{1}{3} \sf{\som} \sf{\som} \sf{\som} \srp \sga^- \sri\n$$\nas well as the [[Clifford curvature scalar]],\n$$\n\sli \sf{e}\sf{e} \sff{R} \s, \sga^- \sri = \snf{e} R\n$$\nand a [[volume form]] term,\n$$\n\sli \sf{e}\sf{e} \sf{e} \sf{e} \sga^- \sri = 4! \s, \snf{e}\n$$\nDropping the boundary term, the action is the Einstein-Hilbert action,\n$$\nS_s = \sfr{1}{g} \sint \snf{e} \slp R + 2 \sLa \srp\n$$\n\nVarying the frame in the modified BF action for gravity gives the equation of motion:\n$$\n0 = \sf{e} \scdot \sff{B_s} = \sha \slp \sf{e} \scdot \sff{R} + \sfr{\sLa}{6} \sf{e} \sf{e} \sf{e} \srp\n$$\nTaking the [[Hodge dual]] and [[Clifford dual]] of these trivector valued 3-forms gives\n\sbegin{eqnarray}\n* \slp \sf{e} \scdot \sff{R} \srp \sga^- &=& \slp * \sf{e^\smu} \sf{e^\snu} \sf{e^\srh} \srp \sfr{1}{4} R_{\snu\srh}{}^{\ska\sla} \slp \sga_{\smu\ska\sla} \sga^- \srp \s\s\n&=& \slp \sfr{1}{3!} \sep^{\smu\snu\srh\sde} \sf{e_\sde} \srp \sfr{1}{4} R_{\snu\srh}{}^{\ska\sla} \slp \sfr{1}{3!} \sep_{\smu\ska\sla\sga} \sga^\sga \srp \s\s\n&=& \sf{e_\sde} \sfr{\sll \set \srl}{3! \s,4} \sde^{\snu\srh\sde}_{\slb \ska\sla\sga \srb} R_{\snu\srh}{}^{\ska\sla} \sga^\sga \s\s\n&=& - \sfr{\sll \set \srl}{3! \s, 3!} \slp \sf{R} - \sfr{1}{2}\sf{e} R \srp\n\send{eqnarray}\nand\n$$\n* \slp \sf{e} \sf{e} \sf{e} \srp \sga^- = \slp \sfr{1}{3!} \sep^{\smu\snu\srh\sde} \sf{e_\sde} \srp \slp \sfr{1}{3!} \sep_{\smu\ska\sla\sga} \sga^\sga \srp\n= \sfr{\sll \set \srl}{3!} \sf{e}\n$$\nand we see that this equation of motion is [[Einstein's equation]],\n$$\n\sf{R} - \sfr{1}{2} \sf{e} R = \sLa \sf{e}\n$$\n\nRef:\n*K. Krasnov\n**[[Non-metric gravity: A status report|http://arxiv.org/abs/0711.0697]]\n***This looks interesting, have only read some and need to finish.
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When we consider a [[manifold]] there are ''natural'' geometric objects that arise "for free" -- without the addition of any further algebraic structure. [[Path|path]]s on the manifold lead to the definition of [[tangent vector]]s and the [[tangent bundle]], their dual [[1-form]]s and the [[cotangent bundle]] lead to [[differential form]]s and [[vector valued form]]s. Such objects are invariant under [[coordinate change]]. Explicitly, a natural object has coordinate indexed components that all transform as a [[tensor|coordinate change]]. Since coordinates and algebraic objects are a computational artifice, only natural objects may be expected to be physically meaningful.\n\nA ''natural operator'', such as the [[exterior derivative]], [[Lie derivative]], [[FuN derivative]], [[covariant derivative]], or [[vector-form algebra]] product acts on natural objects and produces natural objects as a result. Another way of understanding this is that natural operators commute with [[diffeomorphism]]s.\n\nTo be more explicit, a tangent vector is natural since between two coordinate systems\n$$\n\sve{v} = v^j \sve{\spa^x_j} = v^j \sfr{\spa y^p}{\spa x^j} \sve{\spa^y_p} = v'^p \sve{\spa^y_p} = \sve{v'}\n$$\nAn example of an operation that is not natural (an ''unatural'' operator) is the [[partial derivative]] acting on tangent vectors, since between two coordinate systems\n\sbegin{eqnarray}\n\sf{\spa} \sve{v} &=& \sf{dx^i} \spa^x_i v^j \sve{\spa_j} = \sf{dy^m} \sfr{\spa x^i}{\spa y^m} \slp \sfr{\spa y^k}{\spa x^i} \spa^y_k v^j \srp \sfr{\spa y^p}{\spa x^j} \sve{\spa^y_p}\n= \sf{dy^m} \slp \spa^y_m v^j \srp \sfr{\spa y^p}{\spa x^j} \sve{\spa^y_p} \s\s\n\sneq \sf{\spa'} \sve{v'} &=& \sf{dy^m} \spa^y_m \slp v^j \sfr{\spa y^p}{\spa x^j} \srp \sve{\spa^y_p} = \sf{dy^m} \slp \spa^y_m v^j \srp \sfr{\spa y^p}{\spa x^j} \sve{\spa^y_p} + \sf{dy^m} v^j \slp \spa^y_m \sfr{\spa y^p}{\spa x^j} \srp \sve{\spa^y_p}\n\send{eqnarray}\nThe resulting object, $\sf{\spa} \sve{v}$, is not natural because that last term does not vanish. However, unnatural objects can sometimes be assembled into natural objects if such terms are made to cancel. For example, subtracting\n$$\n\sve{u'} \sf{\spa'} \sve{v'} = u^q \sfr{\spa y^m}{\spa x^q} \slp \spa^y_m v^j \srp \sfr{\spa y^p}{\spa x^j} \sve{\spa^y_p} + u^q v^j \slp \sfr{\spa y^m}{\spa x^q} \spa^y_m \sfr{\spa y^p}{\spa x^j} \srp \sve{\spa^y_p}\n$$\nfrom\n$$\n\sve{v'} \sf{\spa'} \sve{u'} = v^q \sfr{\spa y^m}{\spa x^q} \slp \spa^y_m u^j \srp \sfr{\spa y^p}{\spa x^j} \sve{\spa^y_p} + v^q u^j \slp \sfr{\spa y^m}{\spa x^q} \spa^y_m \sfr{\spa y^p}{\spa x^j} \srp \sve{\spa^y_p}\n$$\nthe last terms of each cancel to give a natural object, the Lie bracket,\n$$\n\sve{v'} \sf{\spa'} \sve{u'} - \sve{u'} \sf{\spa'} \sve{v'} = \sve{v} \sf{\spa} \sve{u} - \sve{u} \sf{\spa} \sve{v} = \slb \sve{v} , \sve{u} \srb_L \n$$
Geometry over a disconnected manifold?\n\nThis seems to give an elementary picture:\nhttp://arxiv.org/abs/hep-th/9401145
A [[subgroup]], $N \ssubset G$, of a [[group]], $G$, is called a ''normal subgroup'', $N \striangleleft G$, iff it is invariant under [[conjugation|group]]; that is, for each $n \sin N$ and all $g \sin G$ the element $A_g n = gng^− \sin N$. Another way of saying this is that for each $n \sin N$ and all $g \sin G$ there is an $n' \sin N$ such that $gn = n'g$. Or, equivalently, $gN=Ng$ for all $g \sin G$.\n\nIf $N$ and $G$ are [[Lie group]]s, their elements near the identity may be approximated by [[Lie algebra]] elements,\n\sbegin{eqnarray}\ng &\ssimeq& 1 + x^I T_I \s\s \nn &\ssimeq& 1 + n^P N_P\n\send{eqnarray}\nin which $T_I$ and $N_P$ are the ${\srm Lie}(G)$ and ${\srm Lie}(N)$ generators. Iff $N$ is a normal subgroup, $N \striangleleft G$, then, collecting orders of $x^I$ gives\n\sbegin{eqnarray}\ngng^− &=& n' \s\s\n\slp 1 + x^I T_I \srp \slp 1 + n^P N_P \srp \slp 1 - x^J T_J \srp &\ssimeq& \slp 1 + n^P N_P + x^I n_I^P N_P \srp \s\s\n\slb T_I, n^P N_P \srb &=& n_I^P N_P\n\send{eqnarray}\nand so the Lie algebras satisfy\n$$\n\slb {\srm Lie}(G), {\srm Lie}(N) \srb \ssubset {\srm Lie}(N)\n$$
The ''normalizer'' of a subset, $S$, in $G$ is the [[subgroup]] consisting of all elements of $G$ that leave $S$ invariant under conjugation,\n$$\nN_G(S) = \slc n \sin G \s; | \s; nSn^- = S \src\n$$\nAnother way of thinking of this is that $N_G(S)$ consists of all elements, $n \sin G$, satisfying $ns=sn'$ for each $s \sin S$ and some $n' \sin N_G(S)$. The normalizer of a single element, $N_G(a) = C_G(a)$, is the [[centralizer]], and in general the centralizer is a [[normal subgroup]] of the normalizer, $C_G(S) \striangleleft N_G(S)$. If $H$ is a subgroup of $G$, the normalizer, $N_G(H)$, is the largest subgroup of $G$ having $H$ as a normal subgroup, $H \striangleleft N_G(H)$.\n\nIf $H$ and $N = N_G(H)$ are [[Lie group]]s, their [[Lie algebra]] generators satisfy\n$$\n\slb {\srm Lie}(N), {\srm Lie}(H) \srb \ssubset {\srm Lie}(H)\n$$
I am happy to announce that Deferential Geometry has been chosen as a sponsored project by the [[Foundational Questions Institute|http://www.fqxi.org/aw-lisi.html]]. \n\nThe mathematical basis for the project are being incrementally loaded into the wiki, from the bottom up. It's currently about half way there. From this partial foundation now in place, several physics ideas are being actively extended and pursued. Since this is an ongoing process, with changes occurring daily, things may look a bit chaotic. For a traditional, linear introduction to the physics content of the Deferential Geometry project, interested readers may wish to look at two recent papers:\n*[[An Exceptionally Simple Theory of Everything]]\n*[[Quantum mechanics from a universal action reservoir|http://arxiv.org/abs/physics/0605068]]\n\nI recently attended the very fun [[FQXi 07 conference]], where I had a great time and managed to give this [[talk for FQXi 07]] in ten minutes.
A matrix, $L$, is ''orthogonal'', iff its [[inverse]], $L^- = L^T$, is its transpose.
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A geometric object is parallel transported if it is perceived to be unmoving by an observer traveling along with it. Equivalently, a [[fiber bundle]] section, $C(x)$, having values along a [[path]], $x(t)$, is ''parallel transport''ed iff it is [[horizontal|covariant derivative]] along the path,\n$$\n0 = \sve{v} \sf{\sna} C = v^i \spa_i C + v^i A_i{}^B T_B C = \sfr{d}{d t} C(x(t)) + \sve{v} \sf{A} C\n$$\nIn this equation $\sve{v} = \sfr{dx^i}{dt} \sve{\spa_i}$ is the path velocity and $\sf{A}C$ represents the [[connection]] acting via the left action on the fiber section -- in practice the connection will act appropriately to the specific case.\n\nThe solution to the parallel transport equation may be written via the [[path holonomy]].
The ''partial derivative'', $\spartial_i$, of a [[function]], $f(x)$, over a [[manifold]] is a [[derivative|derivation]] taken with respect to one manifold coordinate, while holding the other coordinates constant,\n$$\n\spa_i f = \sfr{\spa f}{\spa x^i} = \spa^x_i f\n$$\nThis derivative is explicitly dependent on the choice of coordinates, and must be glued together over different manifold patches.\n\nPartial derivatives may also be taken of geometric objects (sections of [[fiber bundle]]s), with care taken to keep track of which elements are coordinate dependent and which are constant. As an example,\n$$\n\spa_i \sf{A} = \spa_i \sf{dx^j} A_j{}^B T_B = \sf{dx^j} \slp \spa_i A_j{}^B \srp T_B\n$$\nThe partial derivatives of [[fiber basis elements|vector bundle]], including [[coordinate basis vectors]], [[coordinate basis 1-forms]], and [[Lie algebra]] basis elements, vanish,\n$$\n\spa_i \sve{\spa_j}=0 \sqquad \spa_i \sf{dx^j} = 0 \sqquad \spa_i T_A=0\n$$\n\nThe partial derivative may be combined with [[coordinate basis 1-forms]] to produce the ''partial derivative operator'',\n$$\n\sf{\spa}=\sf{dx^i} \spa_i = \sf{dx^i} \sfr{\spa}{\spa x^i}\n$$\n(usually also just referred to as the //''partial derivative''//). This is NOT a [[natural]] operator on [[vector valued form]]s (or [[vectors|tangent bundle]]) but it is a natural operator on [[differential form]]s -- for which it is the [[exterior derivative]],\n$$\n\sf{\spa} \sf{f} = \sf{dx^i} \sf{dx^j} \spa_i f_j = \sf{d} \sf{f}\n$$\nEven though it is not a natural operator on VVF's, and therefore does not by itself produce geometrically meaningful objects, it may still be used on them,\n$$\n\sf{\spa} \sf{\sve{A}} = \sf{dx^i} \sf{dx^j} \slp \spa_i A_j{}^k \srp \sve{\spa_k}\n$$\nand provides a useful, coordinate dependent but index free calculational device when combined with other terms using [[vector-form algebra]] to build coordinate invariant geometric objects.\n\nAs a useful example, the partial derivative operator satisfies the ''chain rule'',\n$$\n\sf{\spa} f(y(x)) = \sf{dx^i} \slp \spa_i y^j(x) \srp \spa_j f(y) = \slp \sf{\spa} \sve{y}(x) \srp \sf{\spa} f(y)\n$$\nwith the partial derivatives taken with respect to the functional dependencies as written.\n
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A ''path'' or //''curve''//, $c$, on a [[manifold]] is a connected set of manifold points, with these path points usually labeled by a continuous, monotonically increasing real parameter, $t$.\n$$\nc:\smathbb{R} \sto M\n$$\n$$\nc(t)=x(t) \sin M\n$$\nPaths are often written in terms of the $n$ coordinates (in some manifold patch) corresponding to the path points, $x^i(t)=x_c^i(t)=c^i(t)$, for any parameter value. Physics is invariant under arbitrary reparameterizations of the path. Each path point, labeled by $t$, has a [[tangent vector]], $\sve{v}(t)$, with amplitude dependent on the parameter. Any "initial" path point, $x(0)$, corresponding to parameter $t=0$, has nearby path points given approximately by\n$$\nx^i(t) \ssimeq x^i(0) + t v^i(0)\n$$\nto first order in $t$. A path, or technically a path segment, can be defined to exist for a subset of the reals, such as $t \sin \slb 0,1 \srb$. Such a path is called a ''closed path'' or //''loop''// iff its ends meet, $x(0) = x(1)$, and otherwise is called an ''open path''. Paths, other than loops which intersect in one place, are usually restricted to be non-self-intersecting.
Any [[typical fiber|fiber bundle]] element at an initial point, $C_0 = C(x(0))$, may be [[parallel transport]]ed along a path, $x(t)$, by solving the (set of) first order ODE's,\n$$\n\sfr{d}{dt} C = - \sve{v} \sf{A} C\n$$\nSince the [[connection]] is in the Lie algebra of the structure group the solution may be expressed as $C(x(t)) = U(t) C_0$, in which $U(t) \sin G$ is the ''path holonomy''. (We use the left action throughout this example, which may be adapted for the appropriate structure group action.) $U(t)$ is an element of the structure group that acts on any initial fiber element to give the solution to the parallel transport equation along a path. Plugging this form for the solution into the parallel transport equation, the path holonomy is the solution to the resulting ''holonomy equation'',\n$$\n\sfr{d}{d t} U(t) = - \sve{v} \sf{A} U\n$$\nfrom an initial condition of $U(0)=1$. (once again, the actual action of the connection on the holonomy will depend on the specific group action -- here taken to be the left action.) This equation may be readily converted to an integral equation,\n$$\nU(t) - 1 = - \sint_0^t \sf{dt} \sfr{dx^i}{dt} A_i U(t) \n$$\nFor small displacements along the path, $x^i = x^i_0 + \sva^i(t)$, the solution may be found to any order. To first order,\n$$\nU(t) \ssimeq 1 - \sint_0^t \sf{dt} \sfr{d \sva^i}{dt} A_i(x_0) U(0) = 1 - \sva^i A_i\n$$\nand to second order,\n\sbegin{eqnarray}\nU(t) &\ssimeq& 1 - \sint_0^t \sf{dt} \sfr{d \sva^i}{dt} \slb A_i + \sva^j \spa_j A_i \srb \slb 1 - \sva^k A_k \srb \s\s\n&\ssimeq& 1 - \sva^i A_i + \sva^{ij} \slb - \spa_j A_i + A_i A_j \srb\n\send{eqnarray}\nwith the ''second order path dependence'' above defined as\n$$\n\sva^{ij} = \slb \sint_0^t \sf{dt} \sfr{d \sva^i}{dt} \sva^j \srb\n$$\n\nThe solution to the holonomy equation may be written heuristically as\n$$\nU(t) = Pe^{-\sint \sf{A}} = Pe^{-\sint_0^t \sf{dt} v^i A_i}\n$$\nin which $P$ stands for "''path ordered''", and is there to make sure it's understood that this isn't a proper [[exponentiation]], but rather a way of heuristically writing the solution to the holonomy equation.\n\nThe path holonomy is the [[holonomy]] for a path that isn't necessarily closed.
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Contracting components of $n$ dimensional [[permutation symbol]]s gives\n\sbegin{eqnarray}\n\sep^{\sal \sbe \sdots \sga} \sep_{\sal \sbe \sdots \sga} &=& \sll \set \srl n! \s\s\n\sep^{\sal \sbe \sdots \sga} \sep_{\sde \sbe \sdots \sga} &=& \sll \set \srl \slp n-1 \srp! \sde^\sal_\sde \s\s\n\sep^{\sal \sbe \sga \sdots \sde} \sep_{\sep \sup \sga \sdots \sde} &=& \sll \set \srl 2! \slp n-2 \srp! \sde^{\sal \sbe}_{\slb \sep \sup \srb} \s\s\n\sep^{\sal \sdots \sbe \sga \sdots \sde} \sep_{\sep \sdots \sup \sga \sdots \sde} &=& \sll \set \srl p! \slp n-p \srp! \sde^{\sal \sdots \sbe}_{\slb \sep \sdots \sup \srb}\n\send{eqnarray}
The ''permutation symbol for label [[indices]]'' ranging over $n$ dimension is\n$$\n\sep_{\sal \sdots \sde} = n! \sde^0_{\slb \sal \srd} \sdots \sde^{(n-1)}_{\sld \sde \srb} = n! \sde^{0 \sdots (n-1)}_{\slb \sal \sdots \sde \srb}\n$$\nusing the antisymmetric [[index bracket]]. Alternatively, the indices may range from $1$ to $n$, or over any collection of $n$ numbers. It is antisymmetric in all indices &mdash; returning $1$ for positive permutations, $-1$ for negative permutations, and $0$ if any indices are repeated. The indices may be raised with the [[Minkowski metric]] to get\n$$\n\sep^{\sal \sbe \sdots \sga} = \set^{\sal \sde} \set^{\sbe \sep} \sdots \set^{\sga \sup} \sep_{\sde \sep \sdots \sup} = \sll \set \srl \sep_{\sal \sbe \sdots \sga}\n$$\nSince [[Clifford basis vectors]] anti-commute, the permutation symbol arises geometrically as\n\s[ \sep_{\sal \sbe \sdots \sga} = \sli \sga_\sal \sga_\sbe \sdots \sga_\sga \sga^- \sri \s]\nusing the inverse of the [[pseudoscalar]], $\sga^- = \sga^{n-1} \sdots \sga^1 \sga^0$, and the [[scalar part|Clifford grade]] operator.\n\nThe ''permutation symbol for coordinate [[indices]]'' ranging over $n$ dimension is\n$$\n\sva^{i \sdots j} = n! \sde_{0 \sdots (n-1)}^{\slb i \sdots j \srb}\n$$\nThese indices may be lowered with a [[metric]] to get\n$$\n\sva_{i \sdots j} = g_{ik} \sdots g_{jl} \sva^{k \sdots l}\n$$\n\nLabel and coordinate indices may be changed using the [[frame]] and its inverse,\n$$\n\sbegin{array}{rclcrcl}\n\sep^{i \sdots j} &=& \sep^{\sal \sdots \sbe} \slp e_\sal \srp^i \sdots \slp e_\sbe \srp^j & \s;\s;\s;\s;\s;\s; &\n\sva^{\sal \sdots \sbe} &=& \sva^{i \sdots j} \slp e_i \srp^\sal \sdots \slp e_j \srp^\sbe \s\s\n\sep_{i \sdots j} &=& \slp e_i \srp^\sal \sdots \slp e_j \srp^\sbe \sep_{\sal \sdots \sbe} & \s;\s;\s;\s;\s;\s; &\n\sva_{\sal \sdots \sbe} &=& \slp e_\sal \srp^i \sdots \slp e_\sbe \srp^j \sva_{i \sdots j}\n\send{array}\n$$\nThe two different permutation tensors contract to give [[determinant]]s,\n\sbegin{eqnarray}\n\sep^{\sal \sdots \sbe} \sep_{\sal \sdots \sbe} &=& \sll \set \srl n! \s\s\n\sva^{i \sdots j} \sva_{i \sdots j} &=& \sll g \srl n! = \sll e \srl^2 \sll \set \srl n! \s\s\n\sep^{\sal \sdots \sbe} \sva_{\sal \sdots \sbe} &=& \sep^{i \sdots j} \sva_{i \sdots j}\n= \sep_{\sal \sdots \sbe} \sva^{\sal \sdots \sbe} = \sep_{i \sdots j} \sva^{i \sdots j}\n= \sll e \srl n!\n\send{eqnarray}\nand they are related by:\n$$\n\sva_{i \sdots j} = \sll e \srl \sll \set \srl \sep_{i \sdots j}\n$$\n//Yes, it's non-standard to have two permutation symbols -- but I didn't like all the factors of $\sll e \srl$ floating around. With the way I've defined them, the permutation tensors, $\sep_{\sal \sdots \sbe}$ and $\sva^{i \sdots j}$, actually ARE permutation symbols in these indices, which may be raised, lowered, or converted at will.//
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A ''principal bundle'' or //''principal $G$-bundle''// is a [[fiber bundle]] with arbitrary base, $M$, and structure group, $G$, acting via [[left action|Lie group]] on typical fibers -- [[Lie group geometries|Lie group geometry]] homeomorphic to $G$. Unlike the case for other kinds of bundles, the [[Lie group]], $G$, may also act [[transitive]]ly on the fibers via the [[right action|Lie group]]. For a section, $C(x)$, transforming under the left action [[gauge transformation]], $C \smapsto C' = g(x) C$, the [[covariant derivative]] is\n$$\n\sf{\sna} C = \sf{d} C + \sf{A} C\n$$\nwith the ''principal bundle [[connection]]'' (//''gauge field''//), $\sf{A} = \sf{dx^i} A_i{}^B T_B$, a 1-form over $M$ valued in the [[Lie algebra]] of $G$.\n\nAny fiber element, $C$, at $t=0$ may be [[parallel transport]]ed to $C(t) = U(t) C$ along a [[path]] on the base by a parameter dependent element $U \sin G$, the [[path holonomy]], $U=Pe^{-\sint_0^t \sf{A}}$, satisfying the path holonomy equation,\n$$\n\sfr{d}{dt} U(t) = - \sve{v} \sf{A} U\n$$\n\nApplying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),\n$$\n\sf{\sna} \sf{\sna} C = \sf{d} \slp \sf{d} C + \sf{A} C \srp + \sf{A} \slp \sf{d} C + \sf{A} C \srp \n= \slp \sf{d} \sf{A} + \sf{A} \sf{A} \srp C\n= \sff{F} C\n$$\ngives the ''principal bundle [[curvature]]'',\n$$\n\sff{F} = \sf{d} \sf{A} + \sf{A} \sf{A} = \sf{d} \sf{A} + \sha \slb \sf{A}, \sf{A} \srb \n$$\na Lie algebra valued 2-form with components,\n$$\n\sff{F^B} = \sf{d} \sf{A^B} + \sha \sf{A^C} \sf{A^D} C_{CD}{}^B\n$$\nThis expression for the curvature may alternatively be obtained from the [[holonomy]].\n\nUnder a gauge transformation, $C(x) \smapsto C'(x) = g(x) C(x)$, the covariant derivative changes to\n\sbegin{eqnarray}\n\sf{\sna'} C' &=& g \slp \sf{\sna} C \srp\s\s\n\slp \sf{d} g \srp C + g \sf{d} C + \sf{A'} g C &=& g \sf{d} C + g \sf{A} C\s\s\n\send{eqnarray}\ngiving the transformation law for the connection,\n$$\n\sf{A'} = g \sf{A} g^- + g \sf{d} g^-\n$$\nAn infinitesimal transformation, $g \ssimeq 1 + G^A T_A = 1 + G$, changes the connection to\n$$\n\sf{A'} \ssimeq \sf{A} - \sf{d} G - \sf{A} G + G \sf{A} = \sf{A} - \sf{\sna} G\n$$\nThe curvature consequently transforms under a gauge transformation to\n$$\n\sff{F'} = \sf{d} \sf{A'} + \sf{A'} \sf{A'} = g \sff{F} g^- \ssimeq \sff{F} + \slb G, \sff{F} \srb\n$$\n\nThe covariant derivative acting on a Lie algebra valued field (rather than a section) such as the curvature, transforming under the adjoint action, $\sff{F'} = g \sff{F} g^-$, is \n$$\n\sf{\sna} \sff{F} = \sf{d} \sff{F} + \sf{A} \sff{F} - \sff{F} \sf{A} = \sf{d} \sff{F} + \slb \sf{A}, \sff{F} \srb \n$$
A ''principle bundle'' is a [[principal bundle]] with strong moral fiber.
For a [[path]] designated by coordinates, $x^i(t)$, and parametrized by $t$ the [[velocity|tangent vector]] along the path with respect to this parameter is\n\s[ \sve{v} = v^i \sve{\spa_i} = \sfr{dx^i}{dt} \sve{\spa_i} \s]\nand the magnitude of the velocity is\n\s[ \sll v \srl = \ssqrt{\sll v \scdot v \srl} = \ssqrt{\sll \slp \sve{v} \sf{e} \srp \scdot \slp \sve{v} \sf{e} \srp \srl} = \ssqrt{\sll v^i v^j g_{ij} \srl} \s]\nusing the [[frame]], $\sf{e}$, to map the velocity into a [[rest frame]]. The change in ''proper time'', in seconds or other time [[units]], $T$, along the path is the integral along the path,\n\s[ \sDe \sta = \sint \sf{dt} \sll v \srl \s]\nThe proper time describes how much time passes for a particle or observer moving along that path &mdash; in contrast, parameter time is not necessarily physically meaningful. In terms of parameter time, the proper time changes as $\sfr{d\stau}{dt} = \sll v \srl$. If the path is parameterized (or reparameterized) by proper time so that $t(\stau)=\stau$, then the magnitude of the velocity with respect to proper time is one, $|v|=1$, and the parameter value, in $T$ units, marks out how a clock would read, traveling that path.\n\nIt is only possible to parameterize paths by proper time if their velocity is never null. For positive [[signature|Minkowski metric]], $\set_{00} = +1$, a ''timelike'' path satisfies $v \scdot v > 0$, a ''null'' path satisfies $v \scdot v = 0$, and a ''spacelike'' path satisfies $v \scdot v < 0$. This is reversed for negative signature. A null path, $|v|=0$, is lightlike and time does not pass for a particle or observer on that path. For a spacelike path, the proper time gives the ''spatial distance'' along the path, in "light seconds" or other spatial distance units, such as meters if the proper time is multiplied by the speed of light, $c$. Massive particles and observers only travel timelike paths, and massless particles only travel null paths.\n\nA path between two points that extremizes proper time is a [[geodesic]].
The Clifford ''pseudoscalar'', or ''//Clifford volume element//'', is the grade $n$ [[Clifford basis element|Clifford basis elements]], formed by the (antisymmetric) product of the $n$ [[Clifford basis vectors]],\n$$\n\sga = \sga_0 \sga_1 \sdots \sga_{n-1} = \sfr{\sll \set \srl}{n!} \sep^{\sal \sbe \sdots \sga} \sga_{\sal \sbe \sdots \sga}\n$$\n(giving the relation to the [[permutation symbol]]). For a [[Clifford algebra]] of [[signature|Minkowski metric]] $(p,q)$ the pseudoscalar squares to\n\s[ \sga \sga = \slp -1 \srp^q \slp -1 \srp^{\sfr{n \slp n+1 \srp}{2}} \s]\nand so has the inverse\n\s[ \sga^- = \slp -1 \srp^q \slp -1 \srp^{\sfr{n \slp n+1 \srp}{2}} \sga\s]\nThe pseudoscalar commutes with all even [[Clifford grade]]d elements, $A^e \sga = \sga A^e$, and commutes or anticommutes with odd graded elements, dependent on overall dimension, $A^o \sga = (-1)^{n+1} \sga A^o$.\n\nFor [[Cl(1,3)]], $\sga \sga = -1$ and so $\sga^- = -\sga$. And, since $n=1+3=4$, this [[spacetime]] pseudoscalar anticommutes with odd Clifford grade elements.
A map, $\sphi:x \smapsto y$, takes a point $x$ on a [[manifold]] $M$ to a point $y$ on the manifold $N$ (which may be $M$ itself).These points have coordinates $x^i$ and $y^j = \sph^j(x)$ in some local patches. When the map is smooth (continuously differentiable) the [[partial derivative]], $\sfr{\spa y^j}{\spa x^i} = \spa_i \sph^j(x)$, is well defined. In this way $\sphi$ induces a map, $\sphi^*$, from any [[differential form]], $\sf{a}$, at $y$ to a form at $x$ -- the ''pullback'' of $\sf{a}$ along $\sphi$,\n$$\n\sf{\sphi^*a} = \slb \sf{\sphi^*a} \srl_x = \sphi^* \slb \sf{a} \srl_y = \sphi^* \sf{a}= \sphi^* \sf{dy^j} a_j(y) = \sf{dx^i} \slb \sfr{\spa y^j}{\spa x^i} \srl_x a_j(y) = \slp \sf{\spa} \sve{\sph} \srp \sf{a}\n$$\nusing the [[vector-form algebra]]. This generalizes to forms of any grade. $\sphi$ also induces a map, $\sphi_*$, from any [[tangent vector]], $\sve{v}$, at $x$ to a vector at $y$ -- the ''pushforward'' of $\sve{v}$ along $\sphi$,\n$$\n\sve{\sphi_*v} = \slb \sve{\sphi_*v} \srl_y = \sphi_* \slb \sve{v} \srl_x = \sphi_* \sve{v}= \sphi_* \sve{\sfr{\spa}{\spa x^i}} v_i(x) = \sve{\sfr{\spa}{\spa y^j}} \slb \sfr{\spa y^j}{\spa x^i} \srl_x v_i(x) = \slp \sve{v} \sf{\spa} \srp \sve{\sph} \n$$\nIt is easy to confirm that $\sve{v} \slp \sphi^* \sf{a} \srp = \slp \sphi_* \sve{v} \srp \sf{a}$. Note that forms pull back and vectors push forward even if the map is not invertible or bijective. It is more natural for forms to pull back and for vectors to push forward under a map.\n\nHowever, if the map is smooth and invertible it is a [[diffeomorphism]] and its inverse partial derivative, $\sfr{\spa x^i}{\spa y^j} = \spa_j \sph^-i(y)$, is also well defined. For this kind of map, forms also push forward and vectors also pull back,\n\sbegin{eqnarray}\n\sf{\sphi_*a} &=& \slb \sf{\sphi_*a} \srl_y = \sphi_* \slb \sf{a} \srl_x = \sphi_* \sf{a} = \sphi_* \sf{dx^i} a_i(x) = \sf{dy^j} \slb \sfr{\spa x^i}{\spa y^j} \srl_y a_i(x) = \slp \sf{\spa} \sve{\sph^-} \srp \sf{a} \s\s\n\sve{\sphi^*v} &=& \slb \sve{\sphi^*v} \srl_x = \sphi^* \slb \sve{v} \srl_y = \sphi^* \sve{v} = \sphi^* \sve{\sfr{\spa}{\spa y^j}} v_j(y) = \sve{\sfr{\spa}{\spa x^i}} \slb \sfr{\spa x^i}{\spa y^j} \srl_y v_j(y) = \slp \sve{v} \sf{\spa} \srp \sve{\sph^-} \n\send{eqnarray}\nThis also allows the pushforward and pullback to be defined for [[vector valued form]]s,\n\sbegin{eqnarray}\n\sf{\sve{\sphi_*A}} &=& \slb \sf{\sve{\sphi_*A}} \srl_y = \sphi_* \slb \sf{\sve{A}} \srl_x = \sphi_* \sf{\sve{A}} = \sphi_* \slp \sf{dx^i} A_i{}^j(x) \sve{\sfr{\spa}{\spa x^j}} \srp =\n \sf{dy^j} \slb \sfr{\spa x^i}{\spa y^j} \srl_y A_i{}^k(x) \slb \sfr{\spa y^m}{\spa x^k} \srl_x \sve{\sfr{\spa}{\spa y^m}}\n= \slp \sf{\spa} \sve{\sph^-} \srp \slp \sf{\sve{A}} \sf{\spa} \srp \sve{\sph} \s\s\n\sf{\sve{\sphi^*A}} &=& \slb \sf{\sve{\sphi^*A}} \srl_x = \sphi^* \slb \sf{\sve{A}} \srl_y = \sphi^* \sf{\sve{A}} = \sphi^* \slp \sf{dy^i} A_i{}^j(y) \sve{\sfr{\spa}{\spa y^j}} \srp =\n \sf{dx^i} \slb \sfr{\spa y^j}{\spa x^i} \srl_x A_j{}^k(y) \slb \sfr{\spa x^m}{\spa y^k} \srl_y \sve{\sfr{\spa}{\spa x^m}}\n= \slp \sf{\spa} \sve{\sph} \srp \slp \sf{\sve{A}} \sf{\spa} \srp \sve{\sph^-}\n\send{eqnarray}\nPullbacks and/or pushforwards can also be done for vector and form fields over manifolds by extending the operation over every manifold point.
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[[fiber bundle]]\n\n[[tangent bundle]]\n\n$$\nGL \smapsto O\n$$\nmetric\n\n$$\nGL \smapsto U\n$$\nHermitian metric\n\nRef:\nhttp://en.wikipedia.org/wiki/Principal_bundle#Reduction_of_the_structure_group
A [[Lie group]], $G$, has a [[Lie algebra]], ${\srm Lie}(G)$, spanned by $n$ generators, $T_I$. A [[subgroup]], $H \ssubset G$, has its own set of $n_H$ generators, $H_P \sin {\srm Lie}(H)$, which can be chosen from some of $G$'s, $H_P = T_P$ (with $P$-series indices running from $1$ to $n_H$). The $n_S = n_G - n_H$ remaining generators, $K_A = T_A$, are the ''coset generators'' -- so $\s{T_I\s} = \s{H_P\s} \soplus \s{K_A\s}$. Let the [[vector space]] spanned by the coset generators be labeled ${\srm Lie}(G/H)$ (even though the [[homogeneous space]], $S=G/H$, isn't necessarily a group). The subgroup $H$ is ''reductive'' in $G$ iff the Lie algebra decomposition,\n$$\n{\srm Lie}(G) = {\srm Lie}(H) \soplus {\srm Lie}(G/H)\n$$\nis invariant under the adjoint action of ${\srm Lie}(H)$,\n\sbegin{eqnarray}\n{\srm Ad}_{{\srm Lie}(H)} {\srm Lie}(H) &=& \slb {\srm Lie}(H), {\srm Lie}(H) \srb = {\srm Lie}(H) \s\s\n{\srm Ad}_{{\srm Lie}(H)} {\srm Lie}(G/H) &=& \slb {\srm Lie}(H), {\srm Lie}(G/H) \srb = {\srm Lie}(G/H)\n\send{eqnarray}\nSpecifically, iff $H$ is reductive in $G$ the commutation relations are:\n\sbegin{eqnarray}\n\slb H_P, H_Q \srb &=& C_{PQ}{}^R H_R \s\s\n\slb H_P, K_A \srb &=& C_{PA}{}^B K_B \s\s\n\slb K_A, K_B \srb &=& C_{AB}{}^C K_C + C_{AB}{}^R H_R\n\send{eqnarray}\n\nHere's another way of looking at what this means: By choosing a particular set of generators, $T_I$, for $G$, the [[Killing form]] for ${\srm Lie}(G)$,\n$$\ng_{IJ} = C_{IK}{}^L C_{JL}{}^K\n$$\ncan be diagonalized, with the structure constants then satisfying $C_{IJK} = -C_{IKJ}$ after lowering indices with $g$. A subgroup, $H$, is ''reductive'' in $G$ iff its generators can be chosen from these, $H_P = T_P$, that diagonalized the Killing form.
When a [[subgroup]], $H \ssubset G$, is [[reductive]] in $G$ the [[Lie group tangent bundle geometry]], represented by the [[frame]] 1-forms,\n$$\n\sf{E^J} = \sf{\sxi_R^J} = \sf{{\scal I}^J} = \slp T^J , \sf{\scal I} \srp = \slp T^J , \sf{\scal I} \srp = \slp T^J , g^-(z) \sf{d} g(z) \srp\n$$\nsplits into two parts. The coordinates, $z^i$, over the [[Lie group manifold|Lie group geometry]] split into two sets of coordinates: the coordinates, $y^p$, over $H$, and the "leftover" coordinates, $x^a$, over a base manifold, $M$, of dimension $n_S = (n_G - n_H)$. So a point (element) of $G$ is specified by\n$$\ng(z) = g(x,y) = r(x) \s, h(y)\n$$\nwith $h(y) \sin H$ acting on $r(x) \sin G$ via the [[right action|group]]. The arbitrarily chosen ''reference section'', $r : M \sto G$, corresponds to the [[submanifold]] in $G$ corresponding to $y=0$. The base manifold may be thought of as the [[homogeneous space]], $M=S=G/H$. With this choice of coordinates, and reductivity assumed, the frame 1-forms over $G$ split as:\n\sbegin{eqnarray}\n\sf{E^A}(z) &=& \slp K^A , h^- \sf{e_S} h(y) \srp = \sf{e_S^B}(x) \s, \slp L^h\srp^A{}_B(y) \s\s\n\sf{E^P}(z) &=& \slp H^P , h^- \sf{A_S} h(y) + h^- \sf{d} h(y) \srp = \sf{A_S^Q}(x) \s, \slp L^h \srp^P{}_Q(y) + \sf{e_H^P}(y)\n\send{eqnarray}\nin which $\sf{e_S}(x) = \sf{e_S^B} K_B$ is the [[homogeneous space frame|homogeneous space geometry]], $\sf{A_S}(x) = \sf{A_S^Q} H_Q$, is the [[homogeneous H-connection|homogeneous space geometry]], $\sf{e_H^P}$ are the frame 1-forms over $H$. In this way, a reductive Lie group geometry is equivalent to an [[Ehresmann homogeneous space geometry]]. The above [[left-right rotator]] for $H$ in $G$ is,\n$$\n\slp L^h \srp^I{}_J = \slp T^I , h^- T_J h(y) \srp\n$$\nand the left-right rotator for $G$ splits as\n$$\nL^I{}_J = \slp T^I , g^- T_J g(z) \srp = \slp T^I , h^- r^- T_J r(x) h(y) \srp = \slp L^h \srp^I{}_K \s, \slp L^r \srp^K{}_J\n$$\nin which the left-right rotator for $r$ is\n$$\n\slp L^r \srp^K{}_J = \slp T^I , r^- T_J r(x) \srp\n$$\nRotating the frame 1-forms gives the frame of a particular [[Kaluza-Klein]] spacetime,\n\sbegin{eqnarray}\n\sf{E'^A}(z) &=& \slp L^h \srp_B{}^A \s, \sf{e^B} = \sf{e_S^B}(x) \s\s\n\sf{E'^P}(z) &=& \slp L^h \srp_Q{}^P \s, \sf{e^Q} = \sf{A_S^P}(x) + \sf{e_H^Q} \slp L^h \srp_Q{}^P\n\send{eqnarray}\nwith [[spacetime]] frame 1-forms, $\sf{e_S^B}(x)$, over $M$, and $\sf{e'_H^P}(y) = \sf{e_H^Q} \slp L^h \srp_Q{}^P$ identified as the frame 1-forms over the small compact Kaluza-Klein manifold, $H$.\n\nIt is straightforward to calculate the inverse to the matrix of frame 1-form components, and get the [[orthornormal basis vector fields|frame]] over $G$,\n\sbegin{eqnarray}\n\sve{E_A}(z) &=& \slp L^h \srp_A{}^B \s, \sve{e^S_B}(x) - \slp L^h \srp_A{}^C \slp \sve{e^S_C} \sf{A_S^Q} \srp \slp L^h \srp^P{}_Q \s, \sve{e^H_P} \s\s\n\sve{E_P}(z) &=& \sve{e^H_P}(y)\n\send{eqnarray}\ncorresponding to the [[left invariant Killing vector fields|Lie group geometry]], $\sve{\sxi^R_J} = \sve{E_J}$, and satisfying $\sve{E_I} \sf{E^J} = \sde_I^J$. Note that, in the coordinates we have chosen, the Killing vector fields over $G$ corresponding to the [[Lie algebra]] generators of $H$ equal the Killing vectors over $H$,\n$$\n\sve{\sxi^R_P}(z) = \sve{\sxi^{HR}_P}(y) \n$$\na fact that is true iff $H$ is reductive in $G$.\n\nThe [[reductive Lie group tangent bundle geometry]], including the connection and curvature, also splits in an interesting way.
A ''reductive Lie group tangent bundle geometry'' is a [[Lie group tangent bundle geometry]] for a [[reductive Lie group geometry]]. The [[frame]] 1-forms, $\sf{E^J}=\sf{{\scal I}^J}$, over the Lie group manifold, $G$, split in adapted coordinates as\n\sbegin{eqnarray}\n\sf{E^A}(x,y) &=& \sf{e_S^B}(x) \s, \slp L^h\srp^A{}_B(y) \s\s\n\sf{E^P}(x,y) &=& \sf{A_S^Q}(x) \s, \slp L^h \srp^P{}_Q(y) + \sf{e_H^P}(y)\n\send{eqnarray}\nin which $\sf{e_S^B}$ and $\sf{A_S^Q}$ are the [[homogeneous space frame|homogeneous space geometry]] forms and [[homogeneous H-connection|homogeneous space geometry]] forms, $\slp L^h \srp^A{}_B = \slp H^A, h^- H_B h(y) \srp$ is the [[left-right rotator]] over $H$, and $\sf{e_H^P}$ are the frame 1-forms over $H$. Using the [[Maurer-Cartan equation|Maurer-Cartan form]] over $G$,\n$$\n0 = \sf{d} \sf{E^J} + \sha \sf{E^I} \sf{E^K} C_{IK}{}^J\n$$\nand insisting that the [[torsion]] vanish over $G$,\n$$\n\sff{T^J} = 0 = \sf{d} \sf{E^J} + \sf{W}^J{}_K \sf{E^K}\n$$\ngives the same [[tangent bundle spin connection|tangent bundle connection]],\n$$\n\sf{W}^J{}_K = \sha \sf{E^I} C_{IK}{}^J\n$$\nover $G$, as for a Lie group tangent bundle geometry. These split to:\n\sbegin{eqnarray}\n\sf{W}^B{}_C &=& \sha \sf{e_S^D} \s, \slp L^h\srp^A{}_D C_{AC}{}^B + \sha \slp \sf{A_S^Q} \s, \slp L^h \srp^P{}_Q + \sf{e_H^P} \srp C_{PC}{}^B \s\s\n\sf{W}^B{}_R &=& \sha \sf{e_S^D} \s, \slp L^h\srp^A{}_D C_{AR}{}^B \s\s\n\sf{W}^Q{}_R &=& \sha \slp \sf{A_S^Q} \s, \slp L^h \srp^P{}_Q + \sf{e_H^P} \srp C_{PR}{}^Q\n\send{eqnarray}\nNote that these are the same values taken by the [[Cartan tangent bundle spin connection]] when $\sf{e^A}=\sf{e_S^A}$ and $\sf{A^P}=\sf{A_S^P}$, with\n\sbegin{eqnarray}\nF^{HS}_{DEQ} &=& \sve{e^S_E} \sve{e^S_D} \slp \sf{d} \sf{A^S_Q} + \sha \sf{A_S^P} \sf{A_S^R} C_{PRQ} \srp = -C_{DEQ} \s\s\n\sf{\snu^S}_{EF} &=& - \sha \sf{e_S^D} C_{DEF} - \sf{A_S^Q} C_{QEF}\n\send{eqnarray}\nThe [[Riemann curvature]] is also the same as for the Lie group tangent bundle geometry, $\sff{R}{}^J{}_K = - \sfr{1}{4} \sf{E^I} \sf{E^M} C_{KLI} C_M{}^{JL}$, which splits as:\n\sbegin{eqnarray}\n\sff{R}{}^B{}_C &=& - \sfr{1}{4} \sf{e_S^E} \s, \slp L^h\srp^A{}_E \s, \sf{e_S^F} \s, \slp L^h\srp^D{}_F \s, C_{CLA} C_D{}^{BL} \s\s\n\sff{R}{}^B{}_R &=& - \sfr{1}{4} \slp \sf{A_S^Q} \s, \slp L^h \srp^P{}_Q + \sf{e_H^P} \srp \sf{e_S^F} \s, \slp L^h\srp^D{}_F \s, C_{RLP} C_D{}^{BL} \s\s\n\sff{R}{}^Q{}_R &=& - \sfr{1}{4} \slp \sf{A_S^U} \s, \slp L^h \srp^P{}_U + \sf{e_H^P} \srp \slp \sf{A_S^V} \s, \slp L^h \srp^T{}_V + \sf{e_H^T} \srp C_{RLP} C_T{}^{QL}\n\send{eqnarray}\nThe [[Ricci curvature]] is $\sf{R}{}_J = - \sfr{1}{4} \sf{E}{}_J$ and the [[curvature scalar]] is $R = -\sfr{1}{4} n_G$.
[[Carlo Rovelli]] has a nice paper out on a local interpretation of EPR setup:\n[[Relational EPR|http://arxiv.org/abs/quant-ph/0604064]]\n\nConventionally, an observer, $A$, at $\sal$ measures a state,\n$$\sll \sps \sri = \sfr{1}{\ssqrt{2}} \slp \sll + \sri^z_\sal \sll - \sri^z_\sbe - \sll - \sri^z_\sal \sll + \sri^z_\sbe \srp$$\nand thus collapses the wave function for the spin partner at spatially distant $\sbe$. Rovelli says $A$ only measures and determines the new information locally, with an observation $S^z_{A, \sal} = \spm$, adding to the known square of total spin, $S^2_{A, \sal + \sbe} = 0$, and thus allowing $A$ to infer the state that will be measured at $\sbe$ once it is back in causal contact.\n\nSuggests the wave function, $\spsi$, should be static and represent state of information, while observables (operators) should evolve in time -- i.e. the Heisenberg picture.\n\nPerhaps the wavefunction, and its collapse, is a poor descriptor? Since what's really going on with "collapse" is just new information being acquired by an observer.\n\nHmm, this seems similar to Von Neuman's treatment of QM using a density matrix.
[[Lie algebra]]\n\n\nRef:\n*Clara Loeh\n**[[Representation Theory of Lie Algebras|papers/Loeh - Representation Theory of Lie Algebras.pdf]]
A ''rest frame'', or //''Minkowski space''// is an inertial reference frame. It is flat, having no curvature, and has cartesian coordinates, $x^\sal$, which multiply orthonormal vectors, $\sga_\sal$, to designate points, $x=x^\sal \sga_\sal$, in the space. Since the space is flat and cartesian, the basis vectors, $\sga_\sal$, [[Clifford basis vectors]], serve as both unit rulers on the space and as unit vectors at every space point. The coordinates carry [[units]] of time, $[x^\sal]=T$, which may be in seconds or other time units for $x^0$, and in ''light seconds'' (with the same unit, $T$) for [[spatial|indices]] coordinates, $x^\spi$. (spatial distances in meters, or other spatial length units, are obtained by multiplication with the speed of light, $c$.)\n\nFor a [[path]] in this space, $x(t)$, parameterized by $t$, the [[velocity|tangent vector]] along the path, with respect to this parameter, is\n\s[ v = \sfr{dx}{dt} = \sfr{dx^\sal}{dt} \sga_\sal = v^\sal \sga_\sal \s]\nAlong a path segment, the [[proper time]], $\stau$, in seconds or other time units, steps forward as\n\s[ \sf{d\stau} = \ssqrt{\sll \sf{dx^\sal} \sf{dx^\sbe} \set_{\sal \sbe} \srl} \s]\nin which $\set_{\sal \sbe}$ is the [[Minkowski metric]]. In terms of parameter time, the proper time changes as\n\s[ \sfr{d\stau}{dt} = \ssqrt{\sfr{dx^\sal}{dt} \sfr{dx^\sbe}{dt} \set_{\sal \sbe}} = \ssqrt{v^\sal v^\sbe \set_{\sal \sbe}} = \ssqrt{v \scdot v} = \sll v \srl \s]\nIf the path is parameterized (or reparameterized) by proper time, $t(\stau)=\stau$, then the magnitude of the velocity with respect to proper time is one, $|v|=1$. The proper time describes how time is experienced by a particle or observer moving along that path. The time coordinate, $x^0$, in the rest frame is the proper time for a path along that coordinate line &mdash; a line having constant velocity $v=\sga_0$. ''Null lines'', paths followed by massless particles such as light, are ''null paths'', $|v|=0$, of constant velocity, $\sfr{dv}{dt}=0$.\n\nA rest frame is only a local, flat approximation to a curved [[manifold]] at a point. The frame coordinates, $x^\sal$, do not have a meaningful mapping to manifold coordinates, $x^i$, which have no units. However, the tangent vectors and [[differential form]]s on a manifold do map back and forth to vectors and multivectors in a rest frame via the [[frame]],\n\s[ v = \sve{v} \sf{e} \s]\nwhich has temporal units, $[\sf{e}]=T$. Since all physics can be described by local interactions, physics described locally in a rest frame can be mapped back and forth to physics on a curved manifold via the frame. In a heuristic sense, the frame coefficients are $\slp e_i \srp^\sal = \sfr{\spa x^\sal}{\spa x^i}$ even though the frame coordinates, $x^\sal$, are not well defined functions of manifold coordinates, $x^i$, for an arbitrarily curved manifold.
The [[right action|group]] of one [[Lie group]] element on all others induces a [[diffeomorphism]], $\sph_h(x)$ on the [[Lie group manifold|Lie group geometry]],\n$$\nR_h g(x) = g(x) h = g(\sph_h(x))\n$$\nA vector field on the Lie group manifold is ''right invariant'' iff it is invariant under the [[pushforward|pullback]] of this diffeomorphism for arbitrary $h$,\n$$\nR_{h*} \sve{v}(x) = \sve{v} \sf{\spa} \sph_h(x) = \sve{v}(\sph_h(x))\n$$\nThe partial derivative of the diffeomorphism in the above expression is computed explicitly by using the [[chain rule|partial derivative]],\n$$\n\sf{\spa} g(\sph_h(x)) = \slp \sf{\spa} \sve{\sph_h}(x) \srp \sf{\spa} g(\sph_h) = \sf{\spa} g(x) h\n$$\nand the defining equation for the [[left action vector fields and 1-forms|Lie group geometry]],\n$$\n\sf{\spa} g = \sf{\sxi_L^B} T_B g\n$$\nto write\n$$\n\slp \sf{\spa} \sve{\sph_h}(x) \srp \sf{\sxi_L^B}(\sph_h) T_B g h = \sf{\sxi_L^B}(x) T_B g h\n$$\nand get\n$$\n\sf{\spa} \sve{\sph_h}(x) = \sf{\sxi_L^B}(x) \sve{\sxi^L_B}(\sph_h(x))\n$$\nThis implies the left action vector fields are right invariant,\n$$\nR_{h*} \sve{\sxi^L_C}(x) = \sve{\sxi^L_C}(x) \sf{\spa} \sph_h(x) = \sve{\sxi^L_C}(x) \sf{\sxi_L^B}(x) \sve{\sxi^L_B}(\sph_h) = \sve{\sxi^L_C}(\sph_h(x))\n$$\n\nA [[differential form]] is right invariant iff it is invariant under the [[pullback]], $R_h^* \snf{F}(\sphi_h(x)) = \snf{F}(x)$. The 1-form duals to right invariant vector fields, such as the duals to the left action vector fields, are right invariant. Since autodiffeomorphisms are invertible, these statements may be summarized by defining any form or [[vector valued form]] to be right invariant iff it is invariant under the pushforward, $R_{h*} \snf{\sve{K}}(x) = \snf{\sve{K}}(\sphi_h(x))$, or pullback.
\nRef:\n*Jeffrey D. Olson\n**[[Instantons and Self-Dual Gauge Fields|papers/selfdual.ps]]
A connected [[Lie group]], $G$, is ''simple'' iff it has no [[normal subgroup]]s. In this sense, other Lie groups can have simple Lie groups as "prime factors." A Lie algebra, ${\srm Lie}(G)$, is ''simple'' iff it's only [[ideal|spinor]] is itself -- i.e. there is no other ${\srm Lie}(H) \sin {\srm Lie}(G)$ such that ${\srm Lie}(G) {\srm Lie}(H) = {\srm Lie}(H)$. A Lie group is simple iff its Lie algebra is simple.\n\nA Lie algebra is ''semi-simple'' iff it is the direct sum of simple Lie algebras. A connected Lie group is ''semi-simple'' iff its Lie algebra is semi-simple.
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''Spacetime'' (//''Lorentzian spacetime''//) is a four dimensional [[manifold]], $M$, with a [[metric]], $g_{ab}$. This metric may be derived from four ''spacetime [[orthonormal basis vectors|frame]]'', $\sve{e_\smu} = \slp e_\smu \srp^a \sve{\spa_a}$ (spanning the ''spacetime [[tangent bundle]]''), with appropriate [[indices]], along with the [[Minkowski metric]] (chosen to have positive time signature, unless stated otherwise). The ''spacetime [[torsion]]'', usually taken to be zero, determines the ''spacetime [[tangent bundle spin connection|tangent bundle connection]]'', $\sf{w}^\smu{}_\snu = \sf{dx^a} w_a{}^\smu{}_\snu$, which in turn determines the ''spacetime [[Riemann curvature]]''. The spacetime orthonormal basis vectors have an inverse, the ''spacetime [[frame]] 1-forms'', $\sf{e^\smu} = \sf{dx^a} \slp e_a \srp^\smu$.\n\nThis structure matches that of a [[Clifford vector bundle]] with the spacetime manifold as its base. The Clifford algebra fiber of this bundle is the ''spacetime [[Clifford algebra]]'', [[Cl(1,3)]], generated by four [[Clifford basis vectors]], or with the other choice of signature, Cl(3,1). For this bundle, the [[spacetime frame]] is $\sf{e} = \sf{dx^a} \slp e_a \srp^\smu \sga_\smu$ and the [[spacetime spin connection]] is $\sf{\som} = \sf{dx^a} \sha w_a{}^{\smu \snu} \sga_{\smu \snu}$, which is determined by the spacetime torsion,\n$$\n\sff{T} = \sf{d} \sf{e} + \sf{\som} \stimes \sf{e}\n$$\nThe spacetime spin connection determines the ''spacetime [[Clifford-Riemann curvature]]'',\n$$\n\sff{R} = \sf{d} \sf{\som} + \sha \sf{\som} \sf{\som}\n$$\nwhich has coefficients equal to the spacetime Riemann curvature tensor, $R_{ab}{}^{\smu \snu}$. This, along with the spacetime frame, determines the ''spacetime [[Clifford-Ricci curvature]]'' and ''spacetime [[Clifford curvature scalar]]''.\n\nSometimes //''spacetime''// is used to refer to manifolds of dimension higher than four, along with a metric. In these cases the word should be used with qualifiers such as "any" or "generalized". A ''Riemannian spacetime'' is like a Lorentzian spacetime, but with a positive definite metric.
A [[frame]] over [[spacetime]] is a [[Cl(1,3)]] vector valued 1-form (a [[Clifform]]),\n$$\n\sf{e} = \sf{dx^a} \slp e_a \srp^\smu \sga_\smu\n$$\nwhich may be written as a matrix valued 1-form, using the [[Weyl representation|Dirac matrices]], as\n\sbegin{eqnarray}\n\sf{e} &=& \sf{e^\smu} \sga_\smu =\n\slb \sbegin{array}{cc}\n0 & \sf{e_R} \s\s\n\sf{e_L} & 0\n\send{array} \srb\n=\n\slb \sbegin{array}{cc}\n0 & \sf{e^0} + \sf{e^\sva} \ssi^P_\sva \s\s\n\sf{e^0} - \sf{e^\sva} \ssi^P_\sva & 0\n\send{array} \srb\n\s\s\n&=& \n\slb \sbegin{array}{cccc}\n0 & 0 & \sf{e^0}+\sf{e^3} & \sf{e^1}-i\sf{e^2} \s\s\n0 & 0 & \sf{e^1}+i\sf{e^2} & \sf{e^0}-\sf{e^3} \s\s\n\sf{e^0}-\sf{e^3} & -\sf{e^1}+i\sf{e^2} & 0 & 0 \s\s\n-\sf{e^1}-i\sf{e^2} & \sf{e^0}+\sf{e^3} & 0 & 0\n\send{array} \srb\n\send{eqnarray}\nwith left and right [[chiral]] parts, $\sf{e_{L/R}}$, represented by $2\stimes2$ complex matrix (or [[quaternion]]) valued 1-forms.
A [[spin connection]] over [[spacetime]] is a [[Cl(1,3) bivector]] valued 1-form (a [[Clifform]]),\n$$\n\sf{\som} = \sf{dx^a} \sha \som_a{}^{\smu \snu} \sga_{\smu \snu}\n$$\nwhich may be written as a matrix valued 1-form, using the [[Weyl representation|Dirac matrices]], as\n\sbegin{eqnarray}\n\sf{\som} &=& \sha \sf{\som^{\smu \snu}} \sga_{\smu \snu} =\n\slb \sbegin{array}{cc}\n\sf{\som_L} & 0 \s\s\n0 & \sf{\som_R}\n\send{array} \srb\n=\n\slb \sbegin{array}{cc}\n- \sf{\som^{0 \sva}} \ssi^P_\sva - i \sha \sf{\som^{\sva \sze}} \sep_{\sva \sze \sta} \ssi^P_\sta & 0 \s\s\n0 & \sf{\som^{0 \sva}} \ssi^P_\sva - i \sha \sf{\som^{\sva \sze}} \sep_{\sva \sze \sta} \ssi^P_\sta\n\send{array} \srb \s\s\n&=&\n\slb \sbegin{array}{cccc}\n-\sf{\som^{03}}- i \sf{\som^{12}} & -\sf{\som^{01}}+\sf{\som^{13}}+i \sf{\som^{02}}- i \sf{\som^{23}} & 0 & 0 \s\s\n-\sf{\som^{01}}- \sf{\som^{13}}-i \sf{\som^{02}}-i \sf{\som^{23}} & \sf{\som^{03}}+i \sf{\som^{12}} & 0 & 0 \s\s\n0 & 0 & \sf{\som^{03}}- i \sf{\som^{12}} & \sf{\som^{01}}+\sf{\som^{13}}-i \sf{\som^{02}}- i \sf{\som^{23}} \s\s\n0 & 0 & \sf{\som^{01}}- \sf{\som^{13}}+i \sf{\som^{02}}-i \sf{\som^{23}} & \sf{\som^{03}}+i \sf{\som^{12}}\n\send{array} \srb\n\send{eqnarray}\nwith left and right [[chiral]] parts, $\sf{\som_{L/R}}$, projected out by the [[left/right chirality projector]]. These $2\stimes2$ complex matrix (or [[quaternion]]) valued 1-forms satisfy $\sf{\som_L}^\sdagger = - \sf{\som_R}$, using Hermitian conjugation.
A rotation is particularly easy to express as a [[Clifford rotation]] in three dimensions using the three dimensional Clifford algebra, [[Cl(3)]]. First, consider the result of crossing a vector with a bivector. Starting with an arbitrary Clifford vector,\n$$\nv = v^i \ssigma_i = v^1 \ssigma_1 + v^2 \ssigma_2 + v^3 \ssigma_3\n$$\nand, for example, a "small" bivector in the $\ssigma_1 \ssigma_2$ plane,\n$$\nB = \sepsilon \ssigma_{12}\n$$\ntheir [[cross product|antisymmetric bracket]] gives\n\sbegin{eqnarray}\nB \stimes v &=& \sepsilon \slp v^1 \ssigma_{12} \stimes \ssigma_1 + v^2 \ssigma_{12} \stimes \ssigma_2 + v^3 \ssigma_{12} \stimes \ssigma_3 \srp \s\s\n &=& \sepsilon \slp - v^1 \ssigma_2 + v^2 \ssigma_1 \srp\n\send{eqnarray}\nThis new vector, $B \stimes v$, is perpendicular to $v$, and in the plane of $B$. This "small" vector is the one that needs to be added to $v$ in order to rotate it a small amount clockwise in the plane of $B$:\n$$\nv' \ssimeq v + B \stimes v \ssimeq \slp 1 + \sfrac{\sep}{2} \ssigma_{12} \srp v \slp 1 - \sfrac{\sep}{2} \ssigma_{12} \srp\n$$\nwhere the "$\ssimeq$" holds to first order in $\sepsilon$. Infinitesimal rotations like this one can be combined, with $\sep$ equated to an amplitude devided by a large integer, to give a finite rotation,\n\sbegin{eqnarray}\nv' &=& \slim_{N \srightarrow \sinfty} \slp 1+ \sfrac{\sth}{2N} \ssigma_{12} \srp^N v \slp 1- \sfrac{\sth}{2N} \ssigma_{12} \srp^N \s\s\n &=& e^{\sfrac{\sth}{2} \ssigma_{12}} v e^{-\sfrac{\sth}{2} \ssigma_{12}} = U v U^-\n\send{eqnarray}\nusing the "limit" definition for [[exponentiation]]. This is an exact expression for the rotation of a vector by a bivector. In three dimensions an arbitrary bivector, $B$, can be written as\n$$\nB = \stheta b\n$$\nwith a scalar amplitude, $\stheta$, multiplying a unit bivector encoding the orientation, $bb=-1$. The exponential can then be written, via exponentiation of the bivector, as:\n$$\nU = e^{\sfrac{1}{2} B} = \scos(\sfrac{\sth}{2}) + b \ssin(\sfrac{\sth}{2})\n$$\nAn arbitrary rotation in any plane can be expressed efficiently as $v' = UvU^-$. For example, a rotation of an arbitrary vector by $B=\stheta \ssigma_{12}$ gives (using some trig identities) :\n\sbegin{eqnarray}\nv' &=& e^{\sfrac{1}{2} B} v e^{-\sfrac{1}{2} B} \s\s\n&=& \slp \scos(\sfrac{\sth}{2}) + \ssigma_{12} \ssin(\sfrac{\sth}{2}) \srp \slp v^1 \ssigma_1 + v^2 \ssigma_2 + v^3 \ssigma_3 \srp \slp \scos(\sfrac{\sth}{2}) - \ssigma_{12} \ssin(\sfrac{\sth}{2}) \srp \s\s\n&=& \slp v^1 \scos(\stheta) + v^2 \ssin(\stheta) \srp \ssigma_1 + \slp v^2 \scos(\stheta) - v^1 \ssin(\stheta) \srp \ssigma_2 + v^3 \ssigma_3\n\send{eqnarray}\nThis is widely considered to be pretty neat, and useful as a general method of expressing and calculating rotations. It is equivalent to employing rotation matrices, but generally more intuitive.\n\nA rotation matrix is a 3x3 special [[orthogonal]] matrix (an element of the [[special orthogonal group]], $SO(3)$) that transforms one set of basis vectors into another. This equates to the Clifford way of doing a rotation as:\n$$\n\ssigma'_i = \ssigma_j L^j{}_i = U \ssigma_i U^-\n$$\nFor any rotation encoded by $U$ (which, as the exponential of a bivector, also represents an arbitrary [[SU(2)]] element), the corresponding rotation matrix elements may be explicitly calculated using the [[scalar part operator|Clifford grade]] as\n$$\nL^j{}_i = \sleft< \ssigma^j U \ssigma_i U^- \sright>\n$$\n$SU(2)$ elements of inequivalent sign, $U$ and $-U$, generate equivalent rotations. In this way, $SU(2)$ is a double cover (and the universal cover) of $SO(3)$.
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