### A Theoretically Simple Exception of Everything

Garrett Lisi, who was featured in our inspiration series back in August, has a new paper on the arxiv about his recent work

*An Exceptionally Simple Theory of Everything*

arXiv: 0711.0770

I met Garrett at the Loops '07 in Morelia, and invited him to PI. He gave a talk here in October, which confirmed my theory that the interest in a seminar is inversely proportional to the number of words in the abstract. In his case the abstract read:

*"All fields of the standard model and gravity are unified as an E8 principal bundle connection,"*and during my time at PI it was the best attended Quantum Gravity seminar I've been at.

Anyway, since I've spend some time trying to understand what he's been doing (famously referred to as 'kicking his baby in the head') here is a brief summary of my thoughts on the matter.

__Preliminaries__In the 50's physicists were faced with a confusing, and still growing multitude of particles. By introducing new quantum numbers, it was clear that this particle zoo exhibited some kind of pattern. Murray Gell-Mann realized the particles could be classified using the mathematics of Lie-groups. More specifically, he found that the baryons with spin 3/2 known at this time correspond to the weight diagram of the ten-dimensional representation of the group SU(3) [1].

A similar prediction could later be made for the baryon octet, where the center of the diagram should be doubly occupied. The existence of the missing Σ

^{0}was later experimentally confirmed.

After this, the use of symmetry groups to describe nature has repeatedly proven to be an enormously powerful and successful tool. Besides being useful, it is also aesthetically appealing since the symmetry of these diagrams is often perceived as beautiful [3].

**GUTs and TOEs**Today we are again facing a confusing multitude of particles, though on a more elementary level. The number of what we now believe are elementary particles hasn't grown for a while, but who knows what the LHC will discover? Given the previous successes with symmetry principles, it is only natural to try to explain the presently known particles in the standard model - their families, generations, and quantum numbers - as arising from some larger symmetry group in a Grand Unified Theory (GUT). One can do so in many ways; typically these models predict new particles, and so far unobserved features like proton decay and lepton number violation. This larger symmetry has to be broken at some high mass scale, leaving us with our present day observations.

Today's Standard Model of particle physics (SM) is based on a local SU(3)xSU(2)xU(1) gauge symmetry (with some additional complications like chirality and symmetry breaking). Unifying the electroweak and strong interaction would be great to begin with, but even then there is still gravity, the mysterious outsider. A theory which would also achieve the incorporation of gravity is often modestly called a 'Theory of Everything' (TOE). Such a theory would hopefully answer what presently is the top question in theoretical physics: how do we quantize gravity? It is also believed that a TOE would help us address other problems, like the observed value of the cosmological constant, why the gravitational interaction is so weak, or how to deal with singularities that classical general relativity (GR) predicts.

Commonly, gravity is thought of as an effect of geometry - the curvature of the space-time we live in. The problem with gravity is then that its symmetry transformations are tied to this space-time. A gauge transformations is 'local' with respect to the space-time coordinates (they are a function of x), but the transformations in space-time are not 'local' with respect to the position in the fibre, i.e. the Lie-Group. That is to say, usually a gauge transformation can be performed without inducing a Lorentz transformation. But besides this, the behavior of particles under rotations and boosts - depending on whether dealing with a vector, spinor or tensor - looks pretty much like a gauge transformation.

Therefore, people have tried to base gravity on an equal footing with the other interactions by either describing both as geometry, both as a gauge theory, or both as something completely different. Kaluza-Klein theory e.g. is an approach to unify GR with gauge theories. This works very nicely for the vector fields, but the difficulty is to get the fermions in. So far I thought there are two ways out of this situation. Either add dimensions where the coordinates have weird properties and make your theory supersymmetric to get a fermion for every boson. Or start by building up everything of fermionic fields.

__Exceptional Simplicity__On the algebraic level the problem is that fermions are defined through the fundamental representation of the gauge group, whereas the gauge fields transform under the adjoint representation. Now I learned from Garrett that the five exceptional Lie-groups have the remarkable property that the adjoint action of a subgroup

*is*the fundamental subgroup action on other parts of the group. This then offers the possibility to arrange both, the fermions as well as the gauge fields, in the Lie algebra and root diagram of a single group. Thus, Garret has a third way to address the fermionic problem, using the exceptionality of E8.

His paper consists of two parts. The first is an examination of the root diagram of E8. He shows in detail how this diagram can be decomposed such that it reproduces the quantum numbers of the SM, plus quantum numbers that can be used to label the behaviour under Lorentz transformations. He finds a few additional particles that are new, which are colored scalar fields. This is cute, and I really like this part. He unifies the SM with gravity while causing only a minimum amount of extra clutter. Plus, his plots are pretty. Note how much effort he put in the color coding!

Garrett calls his particle classification the "periodic table of the standard model". The video below shows projections of various rotations of the E8 root system in eight dimensions (see here for a Quicktime movie with better resolution ~10.5 MB)

[Each root of the E8 Lie algebra corresponds to an elementary particle field, including the gravitational (green circles), electroweak (yellow circles), and strong gauge fields (blue circles), the frame-Higgs (squares), and three generations of leptons (yellow and gray triangles) and quarks (rbg triangles) related by triality (lines). Spinning this root system in eight dimensions shows the F4 and G2 subalgebras.]

However, just from the root diagram alone it is not clear whether the additional quantum numbers actually have something to do with gravity, or whether they are just some other additional properties. To answer this question, one needs to tie the symmetry to the base manifold and identify part of the structure with the behaviour under Lorentz transformations. A manifold can have a lot of bundles over it, but the tangential bundle is a special one that comes with the manifold, and one needs to identify the appropriate part of the E8 symmetry with the local Lorentz symmetry in the tangential space. The additional complication is that Garrett has identified an SO(3,1) subgroup, but without breaking the symmetry one doesn't have a direct product of this subgroup with additional symmetries - meaning that gauge transformations mix with Lorentz-transformations.

Garrett provides the missing ingredient in the second part of the paper where he writes down an action that does exactly this. After he addressed the algebraical problem of the fermions being different in the first part, he now attacks the dynamical problem with the fermions: they are different because their action is - unlike that of the gauge fields - not quadratic in the derivatives. As much as I like the first part, I find this construction neither simple nor particularly beautiful. That is to say, I admittedly don't understand why it works. Nevertheless, with the chosen action he is able to reproduce the adequate equations of motion.

This is without doubt cool: He has a theory that contains gravity as well as the other interactions of the SM. Given that he has to choose the action by hand to reproduce the SM (see also update below), one can debate how natural this actually is. However, for me the question remains which problem he can address at this stage. He neither can say anything about the quantization of gravity, renormalizability, nor about the hierarchy problem. When it comes to the cosmological constant, it seems for his theory to work he needs it to be the size of about the Higgs vev, i.e. roughly 12 orders of magnitude too large. (And this is not the common problem with the too large quantum corrections, but actually the constant appearing in the Lagrangian.)

To make predictions with this model, one first needs to find a mechanism for symmetry breaking which is likely to become very involved. I think these two points, the cosmological constant and the symmetry breaking, are the biggest obstacles on the way to making actual predictions [4].

__Bottomline__Now I find it hard to make up my mind on Garrett's model because the attractive and the unattractive features seem to balance each other. To me, the most attractive feature is the way he uses the exceptional Lie-groups to get the fermions together with the bosons. The most unattractive feature are the extra assumptions he needs to write down an action that gives the correct equations of motion. So, my opinion on Garrett's work has been flip-flopping since I learned of it.

So far, I admittedly can't hear what Lee referred to in his book as

*'the ring of truth'*. But maybe that's just because my BlackBerry is beeping all the time. And then there's all the algebra clogging my ears. I think Garrett's paper has the potential to become a very important contribution, and his approach is worth further examination.

Aside: I've complained repeatedly, and fruitlessly, about the absence of coupling constants throughout the paper, and want to use the opportunity to complain one more time.

For more info: Check Garrett's Wiki or his homepage.

Update Nov. 10th: See also Peter Woit's post

Update Nov. 27th: See also the post by Jacques Distler, who objects on the reproduction of the SM.

[1] Note that this SU(3) classification is for quark flavor (the three lightest ones: up, down and strange), and not for color.

[2] For more historical details see Stefan's post

*The Omega-Minus gets a spin (part 1)*which is still patiently waiting for a part 2.

[3] See also my earlier post on

*The Beauty of it All*.

[4] If one were to find another action, the cc problem might vanish.

TAGS: PHYSICS, MATHEMATICS, TOE, GUT, E8

Labels: Best of Physics, Papers, Physics

## 284 Comments:

Hi Sabine, you've written an excellent summary.

The coupling constants are there -- they're all one. Then they run.

Well, Thanks to you for patiently answering my questions :-)

I am missing the coupling to gravity, not the gauge couplings. You've defined it into the fields and I keep looking for the Planck scale. It is confusing because it is useful for the question of derivatives in the action. The action is dimensionless, the gauge fields enter via F^2 meaning A has mass dimension (4-2)/2, whereas the fermions enter as \psi D \psi meaning they have mass dimension (4-1)/2. Usually you can't just add them.

btw, why did you order the names in the acknowledgements in reverse alphabetical order? So Peter is the first? Best,

B.

Hi bee,

This blog is one of my favorites.

I really like your posts and summary's...I haven't commented yet because they've always kept me thinking. I was wondering if you're a teacher? If not, you'd make a really good one.

My nickname is also

b.Hi b, thanks for the nice words. If I got you thinking, I rate that as success :-) No, I'm not a teacher. I occasionally feel like a preacher, but I don't particularly like it. Best, B.

Smolin's "Ring of Truth"

I might of missed that one in the book. What page was that?:)

The complexity of the table of elements by Mendeleev is a far cry from this topic, but imagine being able to place the particle constituents within the framework as Mendeleev did with new elements.

This indeed would be a success.

Hi Plato... ooohm, given that the 'search inside the book' option at amazon is missing, I actually don't know. Hmmm. I think he used it repeatedly. Somewhere in the first some chapters, about unification, theories making new predictions? Or something about scientific revolutions, changing our view of the world? Apologies, it's been more than I year I read the book, I can't recall. Maybe it actually wasn't from Lee's book *scratchhead*...

Random browsing resulted in ...

"But we are also fairly sure that we do not yet have all the pieces. Even with the recent successes, no idea yet has that absolute ring of truth."p. 255 (US hardcover). Sounds like a reference to a previous mentioning, so at least you can now be sure it's the right book I was referring to. Best,B.

It's a very nice paper indeed, and a lot of people with the relevant skills [not me!] should work on it to see how far it can be pushed.

Having said that, I'd just like to echo Bee's

"The additional complication is that Garrett has identified an SO(3,1) subgroup, but without breaking the symmetry one doesn't have a direct product of this subgroup with additional symmetries - meaning that gauge transformations mix with Lorentz-transformations."

Yes, the problem is that in gravity one does not have just any old O(1,3) bundle, it's a special one. [Failure to understand this simple fact is part of the reason that a lot of string theorists don't understand general relativity properly.] So the process of symmetry breaking not only has to produce O(1,3), it must at the same time explain why that sub-bundle miraculously turns into the bundle of orthonormal frames. Unless that is built in from the start, which would be cheating......

bee:

With the acknowledgments, I thought about ordering them according to how much people had helped, but that seemed too political. So I was going to go alphabetical, but it occurred to me that these people had been dealing with their alphabetical placement their whole lives. So I tried to help balance this a little by reversing the order.

Units... The unit issue is trickier than particle physicists usually deal with because the metric or vierbein is floating around too -- one needs to work out the units in curved spacetime. It doesn't make sense for the coordinates to have units, since they're just labels. The length unit (-1 mass dimension) is more sensibly in the vierbein, or the metric if you like, [g_ij] = -2. Since the partial derivative, d_i, is now also dimensionless, this means the connection is dimensionless as well,

[d_i] = [A_i] = [om_i] = 0

as is the curvature [F_ij] = [d_i A_j] = 0. (If you hate the idea of a dimensionless connection, we could contract it with the vielbein, but then we'd have to take this back out before calculating it's curvature -- so that's probably a bad idea.) Since we use the metric to raise indices, [F^ij] = 4, and the volume form has [sqrt(g)] = -4, so the "F^2" action term in curved spacetime is dimensionless. For the fermions, the action term in curved spacetime is

sqrt(g) psi e D psi

and since [e]=1 and [D]=0 we have [psi] = 3/2. So this does encounter the problem you're complaining about, as our superconnection is A+psi. I'm not sure if this is bad though, since the A's are 1-forms and the psi's are Grassmann numbers -- I'll have to go see how these dimensions get juggled in the BRST approach before I have a good answer for you.

plato:

Did you have a look at the periodic tables in the paper I wrote? Also, I initially opened the paper with a nice quote from Poincare that I stole from you, but my reviewers deemed such a flourish inappropriate -- party poopers.

Garrett, very pretty.

When I saw equation (2.4) I of course thought of the neutrino mixing matrix under the "tribimaximal" assumption, which of course is a cross generational thing. See the matrix in equation (3.2) of Koide's paper, and references, or my recent note on the charged lepton masses.

However, it didn't seem to me, at a first quick read, that this matrix is related to how the generations work in your model. I wonder if, in one of your other versions of this model, you ended up with that matrix having something to do with the generation structure.

Are the problems with the cosmological constant and symmetry-breaking any worse for this model than for others?

George

Carl:

That is a great clue. I remember looking at Koide's mass matrix idea several years ago, but somehow I had missed this correspondence. This rotation does in fact relate to how generations are related through triality. I'm not sure yet exactly how the masses are going to come out, but the pieces (like this one) seem to be falling into place nicely. Thanks for the hint -- I'll have another look at the Koide paper and yours.

george:

The symmetry breaking question is the same as for others. The cosmological constant is a little better in this model, because it needs to be there. But, for the most part, this model just unifies the existing problems.

Hi Carl, Hi Garrett:

Indeed, I too was reminded of that matrix. Though I wouldn't get over excited about it. It's not really hard to get. It popped out in some rather simplistic model that I've been working with a while ago, but the details (mass splittings) didn't. I thought about this again last year, after I read that paper by Kovtun & Zee. If I recall that correctly they say somewhere the mass splitting isn't a problem. I asked them about it, but couldn't clarify this. If somebody could let me know? I.e. in my model it turned out one gets this matrix only if two of the masses are identical. More generally, the mass-splittings are related to the deviations from the tribimaximal mixing, and this relation turned out to be fatal (i.e. in conflict with data). Best,

B.

Hi George, Hi Garrett:

The cosmological constant is a little better in this model, because it needs to be there. But, for the most part, this model just unifies the existing problems.I'd say this is a matter of perspective. Your model doesn't work without the CC to begin with. However, you have chosen the action by hand. If you'd want to claim this is an advantage, you'd have to argue there is no way to do it without a CC. Saying there is a way to it with a CC isn't sufficient.

Also, correct me when I'm wrong, but the way the CC is related to the VEV is not usually present, and I'd call that problem worse. I.e. the CC is evidently not the order of the Higgs VEV, and one can't just ignore this mismatch. In your case this is not the vacuum energy problem - which is commonly ignored - but actually the constant appearing in the Lagrangian. Best,

B.

Hi Other,

Thanks for your comment, you expressed that more clearly than I did!

Hi Garrett,

Reg. the dimensions of the fields/ and or coordinates. Sure, one can shift them elsewhere, I just find it very confusing. Let me know if you find an intuitive explanation for the gauge field + fermion question. I'm still hoping I'll wake up one morning thinking 'Oh THAT was what he did!'

Best,

B.

Thanks Bee for the Clarifications on Ring of Truth.

I think that "sense of ringing" can be profound on different people, of course depending on How Lee supposedly mean that terminology to be expressed. How people receive it. How they remember it?

Ultimately, I know that deep down all of us want to be based on a good foundation "to progress from" when we are dealing with the world. Finding the basis of that truth is very important. Philosophically, as well as mathematically and with just plain dealing with reality, what ever that is?:)

I think Lee was talking about "paradigmatic changes" in regards to Kuhn.

This post has been removed by the author.

Hi Garret,

"

The weights of these 222 elements corresponding to the quantum numbers of all gravitational and standard model exactly match 222 roots out of the 240 of the largest simple exceptional Lie group, E8."What I am seeing in my mind ultimately needs to be seen in relation to the complexity of the E8 "dimensional complexity." So it would have to ultimately lead to the elements in question.

Now, if you had looked at Grace satellite( I know your busy) and how it is measuring the planet, there is a response to the gravitational consideration by their tether?

So observationally, "densities of the elements," have a affect on that tether? Oui Non?

I am glad to see progression on the E8. Wonderful information. As a layman, I needed to see how your model would have been attached to the reality we live.

Seeing the "vibrational nature of all elements," E8 would serve for me and help me to understand that elemental structure being defined to it's roots. Escher and Poincaré?

Mendeleev when he developed his table, allowed for "prediction to be inserted" within the confines of the nature of that elemental table by it's "signatures."

Your "particles must fit within the complete rotations." So you are saying this. From a universal prospective, ultimately the 4 dimensional expression had to be reached. So gravity "is included" all the way down?

Ultimately, I am seeing the gravitational inclination of each of these elements within the context of your model. But I am a layman, so those testing your model have to do a good job of tearing it apart:)

Yes, party poopers indeed. But hey, your in the arxiv now. :)

bee:

Yes, at first I considered the large value of the cosmological constant in this model to be a worrisome bug. But now this idea is in agreement with current theories of a large cosmological constant at high energy (ultraviolet fixed point) running to the tiny value we experience at low energies. So the bug now looks to be a feature.

There may be other ways of understanding it, but if you want to have the realization of 'Oh THAT was what he did!' then you'll have to have a dream about how these fermionic fields could emerge as BRST ghosts.

Two quick questions:

1. What is the loophole in the Coleman-Mandula theorem used in this construction? note that the theorem allows constructing theories where internal and spacetime symmetries are unified, as long as those theories are free.

2. When packaging bosons and fermions together, at least one set of fields will have the wrong spin statistics relation. In addition to violating unitarity etc., this definitely is not what is going on in the standard model.

Hi moshe,

1. Yes, the Coleman-Mandula theorem assumes a background spacetime with Poincare symmetry, but this theory doesn't have this background spacetime -- with a cosmological constant, the vacuum spacetime is deSitter. So this theory avoids one of the necessary assumptions of the theorem, and is able to unify gravity with the other gauge fields. On small scales though, Poincare symmetry is a good approximation, and on those scales gravity and the other gauge feels are separate, in accordance with the theorem. (I'm not the first person to dodge C-M this way.)

2. The cool thing about the exceptional groups is that the femions come out, algebraically, with the correct spins -- so they do satisfy the spin statistics relation as Grassmann fields.

other:

The spin connection and gravitational frame are both parts of the E8 connection.

Hint 2: Garrett, for the benefit of string theorists, I was wondering what you thought of the reduction from the heterotic group, as discussed by kneemo.

A comment to Bee and a question to Garrett:

Bee,

the su(3) structure of the baryons was found independently by Neeman and Gellmann, but Neeman released his work a little earlier than Gellmann.

Garrett,

what is the mathematical identity of a connection containing both, Grassmanian and non-Grassmanian objects? is this construction of yours a direct sum? Am I missing something?

Hi Almida:

Yes, thanks. I know. I just tried to keep it brief and referred to details in footnote [2] to Stefan's earlier post.

Best,

B.

kea:

Are there benefits for string theorists now? I didn't know things had gotten THAT bad for them. ;)

almida:

A connection composed of Grassmannian and 1-form elements is called a superconnection, or a Z2 graded connection. It may be more familiar to physicists as a BRST extended connection. And no, I didn't make it up -- though I did make up this one.

Just googled "BRST extended connection" and the only results I got point to Garret Lisi's papers/talks.

On the other hand, the term "super connection" is indeed well known and simply denotes a superfield expansion. However, in that case, the fermion fields are all multiplied by the corresponding power of the Grassmann variable \theta so that all the terms in the sum have the same spin dimension.

What Mr. Lisi is doing is complete nonsense as he simply adds fields of different spins.

Hi Anonymous:

If you're referring to my earlier comment you're twisting words in my mouth. This was not what I said. I a priori have no problem with adding fields of different spin in some algebra, provided the addition is well defined. If one can do this for n-forms, why not also for spins and vectors? What I am confused about is if you treat them so similarly, how can you nevertheless get them to appear differently in the equations of motions (thus my comment about the different order of derivatives in the Lagrangians). This seems possible, but the construction Garrett currently has just looks to me too ad hoc to be really pretty. Might be an observer independent statement, but I'd like to see some more investigation of this point. Best,

B.

besides this, your comment

Just googled "BRST extended connection" and the only results I got point to Garret Lisi's papers/talks.Is pathetic and confirms (again) my concerns about the negative influence of the internet on scientific discussions.

-B

Anonymous, your google-fu is poor, try again. This kind of extended connection, combining Grassmann number and 1-form fields valued in a Lie algebra, is common in the BRST literature, though it often goes by different names.

bee:

I agree that the action is assembled by hand, in order to be in agreement with the standard model. This does cry out for a more elegant description. But the "modified BF theory" action I wrote down isn't so bad. This action, including gravity, gauge fields, and fermions, only has one derivative, so I'm not clear on what's making you unhappy?

I'm not unhappy. I just don't like it. It's supposed to be a TOE, so if you say "that the action is assembled by hand, in order to be in agreement with the standard model." it just doesn't sound particularly compelling to me. But what me or you do or don't like doesn't say anything about whether it could actually be a description of nature. Best, B.

bee:

OK, good, I agree with your dissatisfaction. I think the "true" action will be slightly different, but don't know what it is yet.

OK, so I used your google search and founf this paper:

"Modified Maurer-Cartan condition and fiber bundle structure of the gauge theory of gravity".

By Nakamura, Kikukawa and Kikukawa, Published in Prog.Theor.Phys.91:611-624,1994.

So, I downloaded the KEK scanned version of their paper from SPIRES to see the definition of BRST extended connection.

Indeed, as I'd expected it is just a kind of a superfield expansion in terms of Grassmann variables \xi^A.

The explicit expression for the connection is given by eq. 3.33 of their paper.

However, this expression is very different from what you call "extended connection". Note that in eq. 3.33

all the terms in the first line in the sum inside the parenthesis are bosonic. Indeed, since both d\xi^D and \partial_D are fermionic, their product is bosonic and therefore the sum makes perfect sense.

In contrast, when I look at your formula (1.1) I see nothing like the BRST extended connection I've found in the above paper. In fact, your formula in 1.1 makes absolutely no sense as the terms in the sum have different dimensions.

I'm very surprised that Smolin did not say anything to you about this nonsense.

Actually, just looking at eq. 1.1 in your paper I keep wondering what happened to the Lorentz spinor index of the fermion fields.

You wrote that u=U^AT_A but U^A also has another index. Where did it go?

Bee said "...and confirms (again) my concerns about the negative influence of the internet on scientific discussions."

I think there is a real problem here. When physics blogs started, they were a tremendous boon for people like me who are working in isolation --- at last I could feel that I was a member of a community, and could get all the latest news etc. But gradually I find that reading most blogs [not yours!] is actually having a very bad effect on me. It's not just the nastiness [though that is bad enough, it just reflects reality]. It's the dogmatic way people talk. Even in face-to-face discussions over lunch, very few people use words like "nonsense" as freely as they do in a lot of blogs. Also, it worries me that young people may get the wrong impression from reading some of this stuff. I'm not referring so much to Lubos Motl's stuff, which is so extreme and vicious that anyone can see that they ought to take his words with a very large grain of salt; I'm thinking of blogs like Peter Woit's, with his anti-Landscape propaganda. I really wish that people would stick to technical comments and avoid trying to cultivate a "everyone knows that that is nonsense, snicker snicker" kind of culture.

Anonymous:

Rather than following either of the two references cited in my paper for a clear and conventional definition of "BRST extended connection," you have deliberately chosen a paper that confuses the issue. The point of the paper you cite is to describe the Grassmann valued fields of a BRST extended connection as 1-forms in the vertical part of a principal bundle. This idea is consistent with the established fact (that I did not make up!) that Grassmann numbers may consistently be treated similarly to 1-forms, including the formal addition of the two in a superconnection. However, treating Grassmann fields this way has been tried many times before and it doesn't quite work right, though I agree with the sentiment that motivates this approach.

If you wish to understand the conventional definition of what a BRST extended connection is, try looking in the review paper I cited, in which it is called a "generalized connection" and defined on p70. Or, if you're unhappy with that paper for whatever reason, I'd be happy to provide other references. But I suspect your desire is not to understand.

Anonymous:

The double indices are described on page 9 of my paper.

Dear Mr. Lisi,

After looking at the definition on page 70 of the reference you suggested I don't see how this is related to your eq. 1.1.

In eq. 4.49 of the reference quantity C(\xi) is not a Lorentz spinor. Instead, it's a left-invariant 1-form. In addition, the Grassmann variable c=c^aT_a is a ghost and therefore does not transform as a Lorentz spinor, even though it is anticommuting.

I don't understand how the double indices on page 9 help me understand what happened to the Lorentz spinor index of the fermions in your eq. 1.1.

If this index in contracted, then how is it contracted? I think that this is a very simple question to which you should be able to give a quick answer instead of referring me to page 9.

Tank you!

Hi Other:

I agree with you that there is a really problem here, but it doesn't originate in the blogosphere, it just becomes more apparent. Also, it depends on who you go to lunch with, I've certainly heard worse than 'nonsense'. The worst being somebody mentioning person X - his new paper or latest talk - and everybody just laughing. This usually doesn't happen on blogs though. We've had a previous post on the problem of online communication, in case you're interested, I do in fact think that online communication poses a significant challenge to constructive argumentation. The medium and the problem is fairly new, and unfortunately people are not sufficiently aware of the issue. In a certain sense, the internet seems to amplify bad habits. Best,

B.

Ah, anonymous, maybe I gave up on you too quickly!

As you correctly point out, the C on page 70 of van Holten is a Lie algebra valued Grassmann number, C = C^A T_A. This is formally added to a Lie algebra valued 1-form, H = H^A T_A, to create what I am referring to as a BRST extended connection, A = H + C.

So, how can I be crazy enough to call C a fermion, which (as a spinor) should algebraically be in the fundamental representation space? That is a beautiful thing about the exceptional Lie algebras -- some of their basis Lie algebra elements, T_A, behave algebraically as basis elements of a fundamental representation space! The spinors we need are built into the Lie algebra structure of the exceptional groups! This is explained on page five, and used throughout the rest of the paper. It's a key idea that unlocks everything. Sadly, I was not the first to realize E8 has this structure -- I got the idea from He Who Shall Not Be Named, and it's been known to mathematicians for a very long time.

Dear Mr. Lisi,

I'm not sure if I follow your line of thought. You write that the u quark is u=u^AT_A. What are the T_A's in this case? Do they run over the generators of the color

SU(3)? But then, u is in the adjoint of SU(3). The u quark is both a fundamental of SU(3) and a bi-spinor of the Lorentz group, so ignoring the SU(2), it's gotta have those two indices at least. So far, I see only one index - "A" standing for the gauge group. So, again my question is, where is the Lorentz spinor index?

I'm sorry but I did not find an explanation to my question on page 5.

The other thing I wanted to ask is about the avoidance of Coleman Mandula by not being Poincare.

Do you suppose that spacetime is 4+1? And that you can pick up 3+1 quantum field theory from 4+1 classical statistical mechanics? And can the non Poincare symmetry be be tied to Euclidean relativity? And finally, does this give a decent explanation for the utility of Wick rotations?

Dear Mr. Lisi,

I guess, my question is obvious once you look at eq. 1.1 and compare the expressions for g and u. For g_i^A one clearly sees both the gauge group index - "A" and the Lorentz vector index - "i". On the other hand, u^A carries only the gauge group index but the Lorentz bi-spinor index is somehow not there. How is it contracted?

Bee, do you know the answer? Am I asking a stupid question?

anonymous

The quarks, such as u = u^A T_A, are in the fundamental 3 of su(3), and are in the adjoint of g2. The su(3) subalgebra of g2 acts on the quarks, under the g2 adjoint, as the su(3) acting on the fundamental 3. This is described, in complete detail, on pages 5 and 6. The fermions, including the quarks, are in the fundamental spinor 4 of so(3,1), and they are in the adjoint of f4. When f4 and g2 are combined as parts of e8, the quarks are in the adjoint, with u = u^A T_A an element of e8, and the so(3,1) and su(3) subalgebras of e8 act on the quarks, under the e8 adjoint action, as if they were in the fundamental spinor 4 and color 3. This remarkable fact is what the paper is about. The entire standard model fits in e8 this way.

The i index of g_i^A is a 1-form index. There is no similar index for the fermions. They are Grassmann numbers valued in the algebra, and A is their algebraic index, corresponding, ultimately, to the basis elements, T_A, of e8.

This algebraic equivalence is not obvious, but it is true.

carl:

I treat spacetime as 3+1. However, with a positive cosmological constant, the vacuum solution is de Sitter spacetime, which has so(4,1) as its symmetry algebra. At short distances, the so(4,1) algebra looks like the Poincare algebra. Since you're a Clifford algebra guy, I can describe these algebras in excruciating detail. Using Cl(3,1) basis vectors, ga_a, and bivectors, ga_ab, their algebra under the anti-symmetric product is the same as the algebra of Cl(4,1) bivectors, so(4,1) = Cl^2(4,1) = Cl^{2+1}(3,1). The Poincare algebra is the same, except that [ga_a, ga_b] = 0 for the Poincare algebra (translations commute), but [ga_a, ga_b] = 2 ga_ab in so(4,1), and therein lies all the difference. Poincare symmetry is the number 1 assumption of the C-M theorem, and by not satisfying this assumption we dodge the implications of the theorem and unify gravity with the other gauge fields using this so(4,1) = Cl^{2+1}(3,1) spacetime algebra of bivectors plus vectors.

To the degree that spacetime is locally flat -- which is to a large degree -- the Coleman-Mandula theorem applies, and gravity is separate from the other gauge fields. But, at very high energy, it's all E8.

"The i index of g_i^A is a 1-form index. There is no similar index for the fermions."

In which representation of the Lorentz group are g_i^A and which index "i" or "A" denotes this Lorentz representation?

Garrett:He Who Shall Not Be Namedheh heh, that's funny.

Donald (H. S. M.) Coxeter :)

There are two reasons that having mapped E8 is so important. The practical one is that E8 has major applications: mathematical analysis of the most recent versions of string theory and supergravity theories all keep revealing structure based on E8. E8 seems to be part of the structure of our universe.And to think strings is being saved by your adventure? No, they see this in another way. :)

5. Regular polytope: If you keep pulling the hypercube into higher and higher dimensions you get a polytope. Coxeter is famous for his work on regular polytopes. When they involve coordinates made of complex numbers they are called complex polytopes.Looking at the "false vacuum" to the "true," it seems to me, as if, "the complete rotation" much like the belt trick, could have so much complexity in the "E8 overall design."

Greg Egan's animations are always very helpful :)

Quite a few of you have spent lots of your professional lives trying to understand representations of reductive groups, so I won't say too much about why that's interesting. A little of the background for this particular story is that it's been known for more than twenty years that the unitary dual of a real reductive group can be found by a finite calculation.I hope good science people never tire of the layman's pursuit for understanding and good foundational beginnings?

All of this is way too advanced and technical for me, but the animation is pretty. :-)

anonymous:

"In which representation of the Lorentz group are g_i^A and which index "i" or "A" denotes this Lorentz representation?"

That is not a good way to think about it, but if you must... The A is the Lie algebra index, and the i is the 1-form index. Under a coordinate transformation, 1-forms are invariant, but their mathematical expression changes, as do tensors with this i index, by a Lorentz coordinate transformation matrix. Spinors are also invariant under a coordinate transformation, and their expression may change due to the change of variables, but they do not have any indices that get contracted with a transformation matrix. Spinors are collections of scalar fields with respect to coordinate transformations. Spinors do have indices that change under a change (Lorentz rotation) of the gravitational frame (vierbein), but this is a separate consideration from coordinate transformations, and happens in the A index. I realize this is confusing, but this is the conventional description of spinors in differential geometry. Do not confuse coordinate transformations with physical transformations of the frame.

plato:

Coxeter had it right.

And every physicist begins life as a layman.

Hi Anonymous:

Bee, do you know the answer? Am I asking a stupid question?Well, no, you are asking the natural question. In fact I've asked very similar questions. But I sense a certain amount of hostility in your comments, and I would appreciate if you could at least consider what Mr. Lisi says makes sense instead of being preoccupied by the conviction it is nonsense. I don't feel really qualified to talk algebra, it's somewhat outside where I've worked the last 12 years, but let me give it a simplistic try (Garrett, please correct me if I am crudely wrong).

You're starting with a large algebra with two hundred something dimensions. You can label elements in there with indices whatever you wish. Now you can write down all stuff that is in the algebra. So far that doesn't have anything to do with LORENTZ vectors, spinors or whatever.

Next thing is to notice the elements of the algebra act on the algebra itself, and to understand how they do so. It's here where the exceptionality of E8 plays a role that allows you to get the fundamental representation (usually for the fermions) togehter with the adjoint (for the gauge fields). The representation of the algebra induces a representation of the group. Now Garrett shows that you can decompose the elements such that some of them act on others in the way you'd expect for generators of lorentz trafos. Work needs to be done to show this transformation behavior has something to do with the rotations/boost in a base manifold - that's the point I and Other above have mentioned. So far there is no base manifold present whatsoever, thus no vector (coordinate) index.

I think what you are asking with the indices is how you get these transformations to say something about vector/spinor character. For this identify the SO(3,1) with the tangential space, and analyze which field transforms how under the SO(3,1) generators. Some of them do like vectors do, some like fermions do. There's no reason all of them need to have the same behaviour. That's nice. However, getting the gauge fields and the fermions together in this way comes now back to my problem with the action. If you are looking for the gauge-covariant derivative then you need the gauge fields in the connection. If you treat the fermions similarly, they should appear in the derivative the same way if the action was just E8 invariant. That however is not the cased for the SM, thus one needs to come up with some other way to get an action. I don't think you can blame that on symmetry breaking, because that wouldn't address this issue.

Does that help somehow?

Best,

B.

bee:

Yes, this is an accurate description.

Let me sketch (roughly) an answer to your questions about the fermionic part of the action. If we start out with a connection that is just 1 forms,

A = H + C

And a modified BF Lagrangian that is invariant under arbitrary gauge transformation of the C part of the Lie algebra,

L = B F + B_H*B_H

in which B = B_H + B_C is a Lagrange multiplier field that lives in both (H and C) parts of the algebra, and the curvature is

F = d A + A A

= (d H + H H + C C) + (d C + H C + C H)

Then we can replace the C part of the connection with BRST ghosts, Psi, and write the new effective action for this theory as

Leff = Bg D Psi + F_H*F_H

in which

D Psi = (d C + H C + C H)

is the Dirac derivative, Bg are BRST anti-ghosts (anti-fermions and frame), and the H part of the curvature is

F_H = d H + H H

The C C part you mention goes away because C is replaced by ghosts, and there's no ghost-ghost (Psi Psi) term in the effective action.

This is my guess as to what is going on, and it's in a previous paper I wrote. However, in the current paper I just started with a superconnection, A = H + Psi, without justifying why, since the BRST path I choose to get there may not be what other people choose.

P.S. I'm impressed and flattered that you're spending so much time on this while simultaneously running a conference.

Hi Garrett:

Given that I interpret correctly what you write above, the part that bothers me is not the CC in F, but the dCdC in the Lagrangian (I guess I could live with a CCCC). From what you wrote above, you've just put the second dC into what you call Bg - similarly to what you do in the paper, but why should that be?

P.S. I'm impressed and flattered that you're spending so much time on this while simultaneously running a conference.German organization pays off, the conference is mostly running on its own and very nicely so. Best,

B.

bee:

Yes, you understand, great. And good question. I'm not sure, but I think the modified BF action may be chosen by nature because the BF action is very natural -- it just says, classically, the curvature vanishes. And it uses a Lagrange multiplier to do it. Now, why, if it's so nice, is there a modifying term that breaks symmetry? Hell if I know.

If I ever organize a conference, it will be ruled by chaos.

I must be missing something.

You have a TOE. Didn't you just win physics?I mean, why isn't this, like, front-page news? Why aren't academics all across the world going "wow, we're done! Better drop what we're working on now because the TOE has been found, and all the fun is there!"

Clearly there's some reason that this isn't as revolutionary as it seems to be?

Thanks,

-Domenic-

Domenic:

As with any new theory, it might be wrong. One needs to be cautious about these things. Also, there are parts of it, such as Sabine and I have been discussing, that aren't perfectly well understood. And even I don't feel confident enough in this theory to make solid predictions with it yet; though they will certainly come out as it develops. Nevertheless, it looks pretty good, more people are working on it than just me, and you can expect articles about it to start showing up in the popular press in a week or two.

And yes, I am staring your sarcasm straight in the eye. ;)

Domenic:

I've been around long enough to see a lot of people run about waving ToE's that didn't work. This makes me somewhat jaded, so I guessed you were being sarcastic (as I might be, towards someone claiming a new ToE), but you're young and might of been asking these questions honestly. Either way, my answer is the same. But I apologize for mislabeling your post as sarcasm if it wasn't.

Also, I like your last blog post.

Hi Garrett,

D might have been both, asking honestly, and with a healthy dose of sarcasm.

Great the way you don't lose your rag easily, clearly you are not on a short fuse - even if 'jaded'.

Both The Post, and the interaction or debate in the comments section have made interesting reading.

I mean, why isn't this, like, front-page news? Why aren't academics all across the world going "wow, we're done! Better drop what we're working on now because the TOE has been found, and all the fun is there!"

Clearly there's some reason that this isn't as revolutionary as it seems to be?

I just want to mention that the paper has been on the arxiv now for about 48 hours.

Hmm, I thought I posted a reply, but it hasn't shown up yet... trying again.

Yeah, I was asking honestly, and was more... exaggerating than being sarcastic. I mean, I

wasmissing something, so I just kind of exaggerated my views to make it clear what I was missing. :)Bee, with respect to Garrett's paper about "... E8 ... over a four dimensional base manifold ...",

and

your remark "... Kaluza-Klein theory ... is an approach to unify GR with gauge theories. This works very nicely for the vector fields, but the difficulty is to get the fermions in ...",

and

"... the exceptional Lie-groups ... offers the possibility to arrange both, the fermions as well as the gauge fields, in the Lie algebra and root diagram of a single group ...":

Instead of just putting E8 and 4-dim spacetime together by hand,

what about

taking the 248 dimensions of E8

and

combining them with a 4+4 = 8 dimensional Kaluza-Klein Spacetime similar to that described by N. A. Batakis in Class. Quantum Grav. 3 (1986) L99-L105, which was successful with respect to the Standard Model gauge bosons and was not generally accepted because of difficulties of adding fermions.

Perhaps the boson-fermion structure of E8 could solve those difficulties of the Batakis model.

By combining the 248 E8 dimensions with the 8 Batakis Kaluza-Klein spacetime dimensions,

you could form the 256 dimensional Cl(1,7) Clifford algebra,

whose structure could be used as a guide for naturally constructing a Lagrangian that could reproduce the successes of the Standard Model and Gravity based (as Garrett does) on MacDowell-Mansouri.

Tony Smith

PS - Maybe I should have said that structurally the 248 dimensions of E8 break down into

128 Spin(16) half-spinor fermionic dimensions

and

120 dimensions of the Spin(16) adjoint bosonic dimensions,

and

the other 8 dimensions (Batakis Kaluza-Klein spacetime) of Cl(1,7) could correspond to the 8-dimensional root vector space of the Spin(16) Lie algebra, which is the same root vector space as that of E8.

In short,

256 Cl(1,7) generators = 248 E8 generators + 8 root vector space dimensions,

and the 8 root vector space dimensions correspond to 4+4 dimensional K-K spacetime.

Hi Tony,

Thanks for picking up that line of thought! Indeed, I had suggested something similar to Garrett (and at least two other people), but nobody seemed to be really excited about it. If possible I would even be more extreme and just take E8 (as a group) and deform it/break the symmetry, identify part of it as the 4-d spacetime with the appropriate remaining local Lorentz + gauge symmetries, instead of tying it to an additional space. Admittedly I don't know if that makes any sense. I would find it interesting, because such a large number of extra dimensions could allow one to address the hierarchy problem that lurks somewhere in the background, and so far nothing in that respect can be said whatsoever. Either way, I am just babbling around, and I myself am not excited enough about the model to work myself into all the nasty details of E8. One reason for me writing this post though is my hope to inspire people to look into the matter, cause I am curious to see what comes out of it. Best,

B.

Tony (and bee):

I'm very fond of Kaluza-Klein theory, but I don't think it's a necessary part of this E8 theory. I could change my opinion on this in the future, but right now I don't have a reason to.

Also, I'm pretty sure the E8 Lie algebra is not a subalgebra of Cl(8). However, I'd not be surprised if E8 were a subalgebra of Cl(16).

The idea of starting with just a non-compact E8 manifold, and letting part of it go wavy and be our spacetime base manifold, is very pretty. I believe this construction is called a Cartan geometry. The idea is worth thinking about, and I have a little, but I don't think it will work because I don't think there's a large enough subgroup of E8 to pull it off -- though it may work in some way I haven't considered. Thinking about constructing a Cartan geometry with E8 as a subgroup of a larger group is also interesting, but as nice as it would be if true, I don't think Cl(8) has E8 as a subalgebra.

Garrett, you say that you are "... pretty sure the E8 Lie algebra is not a subalgebra of Cl(8). However, [you would] not be surprised if E8 were a subalgebra of Cl(16). ...".

A set of 128 Cl(16) half-spinors accounts for 128 of the 248 generators of E8,

and

the Cl(16) bivectors form (but with the Spin(16) Lie bracket product instead of the Clifford product) the other 120 of the 248 generators of E8,

so

you can say (if it is OK with you to use two types of products, the Lie bracket and the Clifford) that E8 is contained in the structure of Cl(16),

and

then, due to 8-periodicity of real Clifford algebras,

Cl(16) = Cl(8) (x) Cl(8)

where (x) denotes tensor product,

so

you should be able to write all the E8 generators in terms of Cl(8).

Tony Smith

PS - Of course, if you are willing to be free with using various product rules, you might also be able to fit E8 inside Cl(8) itself.

I'm still not convinced you can bypass Coleman Mandula that way.

Appealing to DS might evade it near the gravity regime, but from the point of view of the effective theory near standard model scales, you will still have highly suppressed but still apparent unitarity violating terms.

If it was that easy, we could always bypass it, for any theory (since presumably we live in a world with nonvanishing cc)

Tony:

I'm not willing to alter the products, because then we'd not be using the algebra we're claiming to use. By your parenthetical remarks I see you acknowledge this opinion of mine -- and I appreciate that, and thank you for playing by this rule.

So, is it possible to embed E8 in Cl(16), associating twice the antisymmetric Clifford product with the Lie bracket? The so(16) = Cl^2(16) part works just fine, but I'm not sure if the 128_S+ spinor works.

If it does, then I agree we could associate E8 elements with elements of two copies of Cl(8) as you describe (but not with one, unless (as you suggest) we mess with the products). Tangentially: I do think so(16) is a subalgebra of Cl(8).

Hmm, I suspect Sabine's going to pull the plug if we inundate her blog with an algebra discussion -- unless she's interested, maybe we should continue via email.

anonymous:

Please don't take my word for it. The first person I know of to point out this loophole in Coleman-Mandula was Thomas Love in his 1987 dissertation. There is also a discussion of this loophole in this recent paper by F. Nesti and R. Percacci: Graviweak Unification. Or you can go to the source and look at Coleman and Mandula's paper, in which their condition (1) for the theorem is "G contains a subgroup locally isomorphic to the Poincare group." The G = E8 I am using does not contain a subgroup locally isomorphic to the Poincare group, it contains the subgroup SO(4,1) -- the symmetry group of deSitter spacetime.

Dear Bee, Garrett Lisi's paper has been removed from hep-th as the main archive and reclassified as gen-ph, thanks God.

I am amazed how uncritical and stupid things you're ready to write about this manifestly crackpot paper. Couldn't you sometimes approach science not by asking whether something is attractive, unattractive, cool, etc., but whether it is right or wrong? I have never seen you thinking in this way.