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Welcome to Quantum Reality: Virtual worlds of imaginary particles: The dreams stuff is made of: Life, the eternal ghost in the machine...
This site is dedicated to the quest for knowledge and wisdom, through science, mathematics, philosophy, invention, and technology. 
length/time  [m] meter <*> [s] second  Count of the quantity of separation; Location of events in spacetime 
mass  [kg] kilogram  Count of the quantity of matter; (the "stuff" at an event) 
EMcharge  [C] Coulomb  Count of the quantity of electric charge; the Coulomb is more fundamental than the Ampere 
temperature  [ºK] Kelvin  Count of the quantity of heat (statistical) 
"Flat" SpaceTime η_{μν} = g_{μν}{SR} =  
t x y z  +1  0  0  0 
0  1  0  0  
0  0  1  0  
0  0  0  1 
g_{μα} g^{μβ} = δ_{α}^{β} =
(4 if α = β for Minkowski)
g =  Det[g_{μν}] = 1 (for Minkowski) not a scalar invariant
Sqrt[g]ρ: Scalar density
There are other ways of defining the metrics and 4vectors available in SR which lead to the same
results, but this particular notation has some nice qualities which place it above the others.
First, it shows the difference between time and space in the metric. We perceive time differently
than space, despite there being only spacetime. Also, this metric gives all of the SR relations
(frame transformations) without using the imaginary unit ( i ) in the transforms. This is
important, as ( i ) is absolutely essential for the complex wave functions once we get to
QM. It is not needed, and would only complicate and confuse matters in SR. This metric will allow us
to separate the "real" SR stuff from the "complex/imaginary" QM stuff easily. It
also allows for the possibility of complex components in SR 4vectors. The
choice of +1 for the time component simplifies the derived equations later on,
as it usually allows rest frame square magnitudes to be positive.


γ  β_{x}γ  β_{y}γ  β_{z}γ 
β_{x}γ  1+(γ1)(β_{x}/β)^{2}  ( γ1)(β_{x}β_{y}/β)^{2}  ( γ1)(β_{x}β_{z}/β)^{2} 
β_{y}γ  ( γ1)(β_{y}β_{x}/β)^{2}  1+( γ1)(β_{y}/β)^{2}  ( γ1)(β_{y}β_{z}/β)^{2} 
β_{z}γ  ( γ1)(β_{z}β_{x}/β)^{2}  ( γ1)(β_{z}β_{y}/β)^{2}  1+( γ1)(β_{z}/β)^{2} 
General Lorentz Boost Transform using just vectors & componentsThank you Jackson, Master of
Vectors! Chap. 11
β = v/c, β = β, γ = 1/√[1β^{2}]
a^{0}' = γ(a^{0}β·a)
a' = a+(γ1)/β^{2}(β·a)βγ
β a^{0}
a^{0}' = γ(a^{0}β·a) Temporal
component
a^{}' = γ(a^{}βa^{0}) Spatial
parallel component
a^{⊥}' = a^{⊥} Spatial
perpendicular components
We are also able to use the Rapidity
φ = Ln[γ(1+ β)]
e^{φ} = γ(1+β) = √[(1+β)/(1β)]
β = Tanh[φ], γ = Cosh[φ], γβ = Sinh[φ], φ =
Rapidity
(which remains additive in SR, unlike v)
Formally, this is like a rotation in 3space, but becomes a hyperbolic rotation through spacetime
for a Lorentz boost
Cosh[φ]  Sinh[φ]  0  0 
Sinh[φ]  Cosh[φ]  0  0 
0  0  1  0 
0  0  0  1 
Time t = γ t_{o} > Time Dilation (e.g. decay times of
unstable particles increase in a cyclotron)
Length L = L_{o}/γ > Length Contraction
4Vector  4Vector = (temporal comp, spatial comp)  Units  Description 
***Calculus***  
4Displacement = 4Delta  ΔR = (cΔt, Δr) 
[m] Δt = Temporal Displacement, Δr =
Spatial
Displacement, (Finite Differences) The 4vector prototype, the "arrow" linking two events 
4Differential  dR = (cdt, dr)  [m] dt = Temporal Differential, dr = Spatial Differential, (Infinitesimals) 
4Gradient = 4Del or 4Partial 
∂ = ∂/∂x_{μ} = (∂/c∂t,
del) = (∂/c∂t,
∇) = (∂/c∂t, ∂/∂x, ∂/∂y, ∂/∂z) 
[m^{1}] ∂ is the partial derivative, del =
(∂/∂x
i + ∂/∂y j + ∂/∂z k) This is a very important 4vector operator, often used to generate continuity equations ∂/∂x_{u} = (∂/c∂t, del) and ∂/∂x^{u} = (∂/c∂t, del) ∂·∂ is also known as the D'alembertian (Wave Operator Δ) I usually write out (del) because the nabla/del symbol (∇) is quite often not displayed correctly in various browsers. It should look like an inverted triangle when displayed correctly. Let g_{μ} = ∂_{μ} f = ∂f / ∂x^{μ} Using the chain rule, one can show: g'_{ν} = ∂f '/∂x'^{ν} = Σ ( ∂f / ∂x^{μ} )( ∂x^{μ} / ∂x'^{ν} ) = ∂'_{ν} f ' = ( ∂_{μ} f )( ∂'_{ν} x^{μ} ) = (g_{μ})( ∂'_{ν} x^{μ} ) However, this appears to be a standard Lorentz transform ∂'^{μ} = Λ^{μ}_{ν} ∂^{ν}[function argument] = ∂^{ν}[function argument] Λ^{μ}_{ν} 
***Particle Dynamics***  
4Position 
R = R^{μ} = (ct, r), eg. radial
coords = (ct,r,θ,z) X = X^{μ} = (ct, x), eg. cartesian coords = (ct,x,y,z) 
[m] t = Time (temporal), r or x
= 3Position
(spatial) Location of an Event, the most basic 4vector (when,where) This is just a 4Displacement with one of the events at the origin (0,0,0,0) of the chosen coordinate systsem c=SpeedofLight sometimes seen as X, other times as R 
4Velocity 
U = γ(c, u) = dR/dτ = γdR/dt = γ(c, u_{r}) = γ(c, u) = γ(c,u_{x},u_{y},u_{z}) = γ_{c}c(1,n), for lightlike/photonic U_{o} = (c,0) in rest frame 
[m s^{1}] u_{r} =
Relativistic 3Velocity, u = dr/dt = Newtonian 3Velocity u_{r} = (r)' = r' u = dr/dt = r' thus, u_{r} = u "U is historically used instead of V" U_{o} = (c,0), 4Velocity is always futurepointing timelike usually seen as U, sometimes as V only 3 independent components since U·U = c^{2} = constant 
4Acceleration 
A = γ(c dγ/dt, dγ/dt u+γ a) =
dU/dτ = γ dU/dt =d^{2}R/dτ^{2} = γ(c dγ/dt, a_{r}) = γ(c γ', a_{r}), where γ' = dγ/dt = γ(c γ', γ' u+γ a) = γ(a_{r}·u/c, a_{r}), because A·U = 0 A_{o} = (0,a_{o}) in rest frame 
[m s^{2}] a_{r} =
Relativistic 3Acceleration, a = du/dt
= Newtonian 3Acceleration a_{r} = (γu_{r})' = γ' u_{r} + γ u_{r}' = γ' u + γ a = (γ^{3}/c^{2})(u·a) u+γ a a = du/dt = u' γ' = dγ/dt = (γ^{3}/c^{2})(u·a) = a_{r}·u/c^{2} 
4Jerk 
J = dA/dτ = γ dA/dt =d^{3}R/dτ^{3} = γ( c(dγ/dt)^{2} + cγ(d^{2}γ/dt^{2}), dγ/dt a_{r}+γ da_{r}/dt ) = γ( c γ'^{2} + c γ γ'', γ' a_{r} + γ a_{r}' ) = γ( c γ'^{2} + c γ γ'', j_{r} ) where γ' = dγ/dt, γ'' = d^{2}γ/dt^{2}, a_{r}' = da_{r}/dt 
[m s^{3}] j_{r} =
Relativistic 3Jerk, j = da/dt
= Newtonian 3Jerk j_{r} = (γa_{r})' = γ' a_{r} + γ a_{r}' j = da/dt = a' γ' = dγ/dt = (γ^{3}/c^{2})(u·a) γ'' = dγ'/dt = d^{2}γ/dt^{2} = (γ^{3}/c^{2})*[(3γ^{2}/c^{2})(u·a)^{2} + (u'·a) + (u·a')] 
***Kinematics***  
4Momentum 
P = (E/c, p) = (mc, p) = m_{o} γ(c,u) = m_{o}U = h_{bar}K = (E_{o}/c^{2})U = (h_{bar}ω/c)(1, n) = (E/c)(1, n), for lightlike/photonic P_{o} = (E_{o}/c,0) = (m_{o}c,0) in rest frame P = ((E_{o} + p_{o}V_{o})/c^{2})U, taking into account pressure*volume terms where pressure p = p_{o} volume V = V_{o}/ γ 
[kg m s^{1}] E = Energy, p =
Relativistic 3Momentum m_{o} = RestMass( 0 for photons, + for massive ) 4Momentum used with single whole particles P·P = (m_{o}c)^{2} = (E_{o}/c)^{2} generally P·P = 0 for photonic 4Momentum is used with single whole particles *Note* It is only the 4Momentum of a closed system that transforms as a 4vector, not the 4momenta of its open subsystems. For example, for a charged capacitor, one must sum both the mechanical and EM momenta together to get an overall 4vector for the system. 
4MomentumDensity = 4MassFlux 
G = (u/c, g) = (p_{m}c, g) =
p_{o_m}
γ(c, u) = p_{o_m}U =(u_{o}/c^{2})U = (1/V_{o})P = (1/c^{2})S p_{o_m} = m_{o}n_{o}?? G = ((u_{o} + p_{o})/c^{2})U, pressure p = p_{o} 
[kg m^{2} s^{1}] u =
EnergyDen = ne, p_{m} = MassDen = u/c^{2} g = MomentumDen = (u/c^{2})u = (e_{o})ExB, f = g·u = MomentumFlux u = 3velocity, n = ParticleDen, e = EnergyPerParticle 4MomentumDensity is used with mass distributions *Note* It is only the 4Momentum of a closed system that transforms as a 4vector, not the 4momenta of its open subsystems. For example, for a charged capacitor, one must sum both the mechanical and EM momenta together to get an overall 4vector for the system. 
4Force or 4Minkowski Force 
F = γ(dE/cdt, f_{r}) = dP/dτ = (m_{o}dU/dτ) + (dm_{o}/dτ)U, generally = ( m_{o}A) + (dm_{o}/dτ)U, generally = m_{o}A , if m_{o} is stable = γ(f_{r}·u/c, f_{r}), if restmass preserving = γm_{o}(c dγ/dt, a_{r}) = γm_{o}(c γ', a_{r}) 
[kg m s^{2}] dE/dt = Power, f_{r} =
Relativistic 3Force, f = Newtonian 3
Force f_{r} = dp/dt = m_{o} d(γu)/dt = m_{o}(γ' u + γ u') = m_{o}(γ' u + γ a) f = m_{o}u' = m_{o}a a = du/dt = u' Sometimes known as the Minkowski Force 4Force is used with single whole particles 
4Force Density  F_{d} = γ(du/cdt, f_{dr}) = dG/dτ?? 
[kg m^{2} s^{2}] 4Force divided by volume 4ForceDensity is used with mass distributions 
***Connection to Waves***  
4WaveVector or 4AngWaveVector 
K = (ω/c, k) = (ω/c,ω/v_{phase }n) = (1/h_{bar})(E/c,p) = (ω_{o}/c^{2}) γ(c,u) = (1/h_{bar})P = (ω_{o}/c^{2})U = (1/h_{bar})P = (m_{o}/h_{bar})U = (ω/c)(1,_{ }n) , for lightlike/photonic 
[rad m^{1}] ω = AngularFrequency [rad/s], k = WaveNumber or WaveVector [rad/m] n = UnitWaveNormalVector, v_{phase} = phase_velocity ω_{o} = RestAngularFrequency( 0 for photons, + for massive ) ω = 2πν, k = 2π/λ k everywhere points in the direction orthogonal to planes of constant phase φ where phase φ = K·R= (ωt  k·r) = (k·r  ωt) 
4Frequency ***Break with standard notation*** better to use the 4WaveVector 
Ν = (ν, c/λ
n) =(c/2π)K = (ν, νn) = ν(1,n) , for lightlike/photonic 
[cyc s^{1}] ν = ω/2π, λ = 2π/k ω = 2πν, k = 2π/λ νλ = c, for photonic ***this is bad notation based on our 4vector naming convention*** the cfactor should be in the time component the 4vector name should reference the space component I simply include it here because it is common in the literature 
4CycWaveVector or 4InverseWaveLength 
Kcyc = (ν/c,1/λ
n) = (1/λ)(w/c, n) = (ν)(1/c, n/w) =(1/2π)K = (1/h)P = (ν_{o}/c^{2})U = (1/h)P = (m_{o}/h)U = (ν/c)(1,_{ }n) , for lightlike/photonic sometimes called L 
[cyc m^{1}] ν =
CyclicalFrequency [cyc/s], λ = WaveLength [m/cyc] ν = ω/2π = Frequency 1/λ = k/2π = Inverse WaveLength [cyc/m] h_{bar} = h/2π = Dirac's Const w = λν = (Phase) Velocity of Wave n = UnitWaveNormalVector 
***Flux 4Vectors***  
Flux 4Vectors all in form of : V = {rest_charge_density} U V = {rest_charge}n_{o} U where n = γn_{o } alternately, V = (cs,f) where s = source, f = flux vector and ∂·V = 0 for a conserved flux 
Flux 4Vectors all have units of [{charge} m^{2} s^{1}] = [{charge_density} m s^{1}] Flux is the amount of {charge} that flows through a unit area in a unit time Flux can also be thought of as {charge_density_velocity} = {current_density} {charge} [{charge_unit}] {charge_density} [{charge_units}/m^{3}] {flux} = {charge_density_velocity} = {current_density} [({charge_units}/m^{3})*(m/s)] = {charge per area per second}  
4NumberFlux "SR Dust" 
N = (cn, n_{f}) = n_{o} γ(c, u) = n(c, u) = n_{o}U 
[# m^{2} s^{1}] n_{o} =
RestNumberDensity [#/m^{3}],
n = γn_{o} = NumberDensity [#/m^{3}] n_{f} = nu = NumberFlux [(#/m^{3})*(m/s)] # of stable particles N = n_{o}V_{o} = nV This is the SR "Dust" 4Vector, which is valid for a perfect gas, i.e. noninteracting particles, no shear stresses, no heat conduction N = Σ_{a} [∫dτ δ^{4}(xx_{a}(τ))(dX_{a}/dτ)] = = Σ_{a} [∫dτ δ^{4}(xx_{a}(τ))(U_{a})] 
4VolumetricFlux  V = V_{o}U??  [(m^{3}) m^{2} s^{1}] V_{o} = RestVolume 
4ElectricCurrentDensity =4CurrentDensity = 4ElectricChargeFlux 
J = (cρ, j) = ρ_{o} γ(c, u) =
ρ(c, u) = ρ_{o}U = q_{o}n_{o}U = q_{o}N = J_{elec} 
[C m^{2} s^{1}] ρ_{o} =
RestElecChargeDensity [C/m^{3}],
ρ = γρ_{o} = ElecChargeDensity j = γ ρ_{o} u = ρu =ElecCurrentDensity = ElecChargeFlux [(C/m^{3})*(m/s)] j = α(E+uxB), α = Conductivity q_{o} = Electric Charge [C] ρ_{o} = q_{o}n_{o} [C/m^{3}] 
4MagneticCurrentDensity = 4MagneticChargeFlux = Zero (so far..) 
J_{mag} = (cρ_{mag}, j_{mag}) =
ρ_{o_mag}
γ(c, u) = ρ_{o_mag}U = q_{o_mag}n_{o}U = Zero (so far...) 
[MagCharge m^{2} s^{1}] ρ_{o_mag} =
RestMagChargeDensity,
ρ_{mag} = γρ_{o_mag = }MagChargeDensity j_{mag} = MagCurrentDensity = MagChargeFlux q_{o_mag} = Magnetic Charge to date: ρ_{o_mag} = 0 and j_{mag} = 0  no magnetic (monopole) charges yet discovered 
4ChemicalFlux  [(mol) m^{2} s^{1}]  
4MassFlux = 4MomentumDensity 
G = (u/c, g) = (cρ_{m}, g) = ρ_{o_m}
γ(c, u) = ρ_{o_m}U ρ_{o_m} = m_{o}n_{o}?? = (1/c^{2})S 
[(kg) m^{2} s^{1}] u =
EnergyDen = ne, p_{m} = MassDen = u/c^{2} g = MomentumDen = (u/c^{2})u = (e_{o})ExB, f = g·u = MomentumFlux u = 3velocity, n = ParticleDen, e = EnergyPerParticle Poincare' made the observation that, since the EM momentum of radiation is 1/c^{2} times the Poynting flux of energy, radiation seems to possess a mass density 1/c^{2} times its energy density 
4PoyntingVector = 4EnergyFlux = 4RadiativeFlux = 4MomentumDensity? 
S = (cu, s) = u_{o} γ(c, u) =
u_{o}U = c^{2}G ????u_{o} = E_{o}n_{o }?? 
[(J) m^{2} s^{1}] u =
EnergyDen = ne, s = EnergyFlux = PoyntingVector = uu = c^{2}g =
Ne u_{e} = (ε_{o}E·E+B·B/µ_{o})/2 = (E·D+B·H)/2 = EM energy density typically see ∂u/∂t =  del·s + J_{f} _{}· E which in 4vector notation would be ∂·S = j_{f} · E, where j_{f} is the current density of free charges, so not conserved generally however, make the following observation ∂(u_{m})/∂t = j(t,x)·E(t,x), where this is rate of change of kinetic energy of a charge then let u = u_{e}+u_{m}, s = s_{e}+s_{m}, ∂·S = ∂(u_{e}+u_{m})/∂t +del·(s_{e}+s_{m}) = 0 we have conservation/continuity again, by allowing energy to transform into different types. In the example, energy is passing back & forth between the physical charges and the EM field itself. Energy as a whole is still conserved. ε_{o} = Permittivity, µ_{o} = Permeability ε_{o}µ_{o} = 1/c^{2} s = (E x B)/µ_{o} = EnergyCurrentDensity u = 3velocity, n = ParticleDen, e = EnergyPerParticle, N = ParticleFlux = nu see also UmovPoynting Vector for generalization to mechanical systems S = (cu, s) = (c(u_{e}+u_{m}),s_{e}+s_{m}) ∂·S = ∂(u_{e}+u_{m})/∂t +del·(s_{e}+s_{m}) = 0 u_{m} = mechanical kinetic energy density s_{m} = mechanical Poynting vector, the flux of their energies The sum of mechanical and EM energies, as well as the sum of mechanical and EM momenta are conserved inside a closed system of fields and charges. Another way to say this is that only the fourmomentum of a closed system transforms as a 4vector, not the fourmomentums of its open subsystems. Since only the microscopic fields E and B are needed in the derivation of S = (1/µ_{o})(ExB), assumptions about any material possibly present can be completely avoided, and Poynting's vector as well as the theorem in this definition are universally valid, in vacuum as in all kinds of material. This is especially true for the electromagnetic energy density, in contrast to the case above Energy types: electrical, magnetic, thermal, chemical, mechanical, nuclear The density of supplied energy is restricted by the physical properties through which it flows. In a material medium, the power of energy flux U is restricted, (U < vF), where v is the deformation propagation velocity, usually the speed of sound, F may be any elastic or thermal energy, U is a vector. div U determines the amount of energy transformation into a different form. For a gaseous medium, U = a √[T] p, where A is a coefficient which depends on molecular composition, T is temperature, p is pressure. not be be confused with the 4EntropyFlux Vector S 
4EntropyFlux  S = (cs, s_{f}) = s_{o} γ(c, u) = s_{o}U 
[(J ºK^{1}) m^{2}
s^{1}] s_{o} = RestEntropyDensity =
q_{o}/T,
s_{f} = EntropyFlux Entropy S = ∫s_{o}dV = k_{B} ln Ω, where Ω = # of microstates for a given macrostate ∂·S >= 0 alternate def: S = s_{o}U + Q/T_{o} where Q is the Thermal Heat Flux 4vector 1st term is entropy carried convectively with mass 2nd term is entropy transported by flow of heat (generalization of dS = dQ/T) not be be confused with the 4Poynting Vector S 
***Thermodynamic*** 

4InverseTempFlux 
β = β_{o} U = (1/k_{B}T_{o}) U where β_{o} = 1/k_{B}T_{o} dS = β·dP, differential entropy 
Considered on Thermodynamic principles also known as a Killing vector "The proper relativistic temperature is not agreed upon by Einstein, Ott, and Landsberg, who respectively think that moving objects are colder, hotter and invariant. You can try reading these and seeing what each do, how they differ in their assumptions and why they disagree with each other. However, given the fact that there does seem to exist genuine disagreement, it is suspected that the matter has not been settled. Also since neither SR nor thermodynamics are complicated in their mathematical settings, the problem is likely to be that of a foundational nature  i.e. what does temperature mean for a moving object." 
4MomentumTemperature 
P_{T} = P/k_{B} = (p_{T}^{0}/c,
p_{T}) = (T/c, p_{T}) = ((E/k_{B})/c, p/k_{B}) p_{T}^{0} = T = Temperature (in ºK) simply dividing 4Momentum by Boltzmann's const. k_{B} which gives E/c = k_{B}T/c, or E = k_{B}T 
[ºK m^{1 }s] ºK = degree Kelvin Temperature k_{B} = 1.380 6504(24)×10^{−23} [J/ºK] Boltzmann's constant Not sure if this is valid, but perhaps useful as a gauge of photon temperature based entirely on dimensional considerations of k_{B} [J/ºK] energy/temperature similarly to c [m/s] being a fundamental constant relating length/time 
***Diffusion/Continuity based*** see Atomic Diffusion Brownian Motion Electron Diffusion Momentum Diffusion Osmosis Photon Diffusion Reverse Diffusion Thermal Diffusion  
4Potential Flux?? 
V = (cq, q_{f}) = q_{o} γ(c, u) =
q_{o}U?? where q_{o} = [1]?? = ( c (k/a)φ , q_{f})?? = ( c (k/a)φ , k del [φ])?? needs work 
Potential Flow for Velocity?? Velocity Potential "Velocity" Conduction Equation: v = k del [φ] "Velocity" Diffusion Equation: a del·del [φ] = ∂φ/∂t where del·v ~ ∂φ/∂t Continuity gives ∂·V = ∂[c(k/a)φ]/∂t +del·v = 0 Thus, [ (k/a)φ ] and [ v ] are components of a 4vector In fluid dynamics, a potential flow is described by means of a velocity potential , φ being a function of space and time. The flow velocity v is a vector field equal to the negative gradient, del, of the velocity potential φ:^{[1] } Incompressible flowIn case of an incompressible flow — for instance of a liquid, or a gas at low Mach numbers; but not for sound waves — the velocity v has zero divergence:^{[1]}
with the dot denoting the inner product. As a result, the velocity potential φ has to satisfy Laplace's equation^{[1]}
where Δ = ∇·∇ is the Laplace
operator. In this case the flow can be determined completely from its kinematics:
the assumptions of irrotationality and zero divergence of the flow. Dynamics
only have to be applied afterwards, if one is interested in computing pressures:
for instance for flow around airfoils through the use of Bernoulli's
principle. The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow:
Steady flowThe flow of a fluid is said to be steady if v does not vary with time. That is if
Incompressible flowA fluid is incompressible if the divergence of v is zero:
That is, if v is a solenoidal vector field.
Irrotational flowA flow is irrotational if the curl of v is zero:
That is, if v is an irrotational vector field.
VorticityThe vorticity, ω, of a flow can be defined in terms of its flow velocity by Thus in irrotational flow the vorticity is zero.
The velocity potentialIf an irrotational flow occupies a simplyconnected fluid region then there exists a scalar field φ such that
The scalar field φ is called the velocity
potential for the flow. (See Irrotational
vector field.) An lamellar vector field is a synonym for an irrotational vector field.[1] The adjective "lamellar" derives from the noun "lamella", which means a thin layer. In Latin, lamella is the diminutive of lamina (but do not confuse with laminar flow). The lamellae to which "lamellar flow" refers are the surfaces of constant potential. An irrotational vector field which is also solenoidal is called a Laplacian vector field. The fundamental
theorem of vector calculus states that any vector field can be expressed as
the sum of a conservative vector field and a solenoidal
field. The fundamental
theorem of vector calculus states that any vector field can be expressed as
the sum of a conservative
vector field and a solenoidal field. The condition of zero divergence is
satisfied whenever a vector field v has only a vector
potential component, because the definition of the vector potential A
as: automatically results in the identity
(as can be shown, for example, using Cartesian coordinates): The converse
also holds: for any solenoidal v there exists a vector potential A
such that v = del x A.
(Strictly speaking, this holds only subject to certain technical conditions on v,
see Helmholtz
decomposition.) In vector
calculus, a Laplacian vector field is a vector
field which is both irrotational
and incompressible.
If the field is denoted as v, then it is described by the following differential
equations: Since the curl
of v is zero, it follows that v can be expressed as the gradient
of a scalar
potential (see irrotational
field) φ : Then, since the divergence of v is also zero, it follows from equation (1) that del·del φ = 0 which is equivalent to Therefore, the potential of a Laplacian field satisfies Laplace's
equation. In case of an incompressible flow the velocity potential satisfies the Laplace's equation. However, potential flows have also been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows. Applications of potential flow are for instance: the outer flow field for aerofoils, water waves, and groundwater flow. For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable. A velocity potential is used in fluid
dynamics, when a fluid occupies a simplyconnected region and is irrotational.
In such a case, where u denotes the flow
velocity of the fluid. As a result, u can be represented as the gradient
of a scalar
function Φ: Φ is known as a velocity potential for u. A velocity potential is not unique. If a is a constant then Φ + a is also a velocity potential for u. Conversely, if Ψ is a velocity potential for u then Ψ = Φ + b for some constant b. In other words, velocity potentials are unique up to a constant. Unlike a stream function, a velocity potential can exist in threedimensional flow. 
4HeatFlux 4ThermalHeatFlux 4ThermalEnergyFlux 
Q = (cq, q_{f}) = q_{o} γ(c, u) =
q_{o}U = ( c (k/a)T , q_{f}) = ( c (k/a)T , k del [T]) 
[(J) m^{2} s^{1}] = [(W) m^{2}] Potential Flow for Heat k = Thermal Conductivity [W/(m °K)] *Note  this is not Boltzmann's const k_{B}* a = k/(ρc_{p}) = Thermal Diffusivity [m^{2}/s] ρ = Density [1/m^{3}] c_{p} = Specific heat capacity [J/ °K] ρc_{p} = Volumetric heat capacity [J/m^{3} °K] T = Absolute Temperature [°K] (k/a)T = Thermal energy density [W s / m^{3}] = [J/m^{3}], q = Heat Flux [W/m^{2}] = Thermal Energy Flux Heat Conduction Equation: q = k del [T] Thermal Diffusion Equation: a del·del [T] = ∂T/∂t where del·q ~ ∂T/∂t Continuity gives ∂·Q = ∂[c(k/a)T]/∂t +del·q = 0 Thus, [ (k/a)T ] and [ q ] are components of a 4vector Fourier's Law: q = k del [T] The minus sign ensures that heat flows down the temperature gradient see Hydraulic Analogies quantity: Heat Q [J] potential: Temperature T [°K] = [J/k_{B}] flux: Heat Xfer Rate Qdot [J/s] flux density: Heat Flux Qdot'' [W/m^{2}] linear model: Fourier's Law Qdot'' = k del [T] 
4DarcyFlux 4HydraulicFlux 
Q = (cq, q_{f}) = q_{o} γ(c, u) =
q_{o}U = ( c (βφ)P , q_{f}) = ( c (βφ)P , κ/μ del [P]) 
[(m^{3}) m^{2} s^{1}] =
[m s^{1}] Potential Flow for Volume/Hydraulics κ = Permeability [m^{2}] μ = Dynamic Viscosity [Pa s] P = Pressure [N m^{2}] β = Compressibility Coefficient φ = Porosity [1] q = Flux [m s^{1}] = Volumetric Flux <> particle velocity v = q/φ = pore velocity Darcy's Law Equation: q = κ/μ del [P] Pressure Diffusion Equation: del·q = κ/μ del·del [P] = βφ∂P/∂t, by imposing incompressibility giving Pressure Diffusion Waves Continuity gives ∂·Q = ∂[βφP]/∂t +del·q = 0 Thus, [ βφP ] and [ q ] are components of a 4vector Darcy's Law: q = κ/μ del [P] The minus sign ensures that flux flows down the pressure gradient Darcy's Law  derivable from NavierStokes see Fourier's law for heat conduction see Ohm's law for electrical conduction see Fick's law for diffusion see Hydraulic Analogies quantity: Volume V [m^{3}] potential: Pressure P [Pa] = [J/m^{3}] flux: Current φ_{V} [m^{3}/s] flux density: Velocity [m/s] linear model: Poiseuille's Law φ_{V} = ... 
4ElectricChargeFlux 
Q = (cq, q_{f}) = q_{o} γ(c, u) =
q_{o}U = ( c ρ, j) 
[(C) m^{2} s^{1}] Potential Flow for Charge acts differently, presumably because this is a "charged" field, where the particle interacts with the field. μ = mobility σ = specific conductivity = q n μ where n = concentration of carriers Conservative Potential: E = ( del [φ]) Ohm's Law: j = σ E = σ del [φ] Gauss Law: del·E = del·( del [φ]) =  del·del [φ] = ρ/ε_{0} Fick's 1st Law Diffusion: j =  D del ρ where D = μ k T / e = EinsteinSmoluchowski Relation Continuity independently gives ∂·J = ∂[ρ]/∂t +del·j = 0 Thus, [ ρ ] and [ j ] are components of a 4vector Continuity independently gives ∂·A = ∂[φ]/∂t +del·a_{EM} = 0 in the Lorenz Gauge Thus, [ [φ ] and [ a_{EM} ] are components of a 4vector Ohm's Law: j = σ del [φ] The minus sign ensures that current flows down the potential gradient see Hydraulic Analogies quantity: Charge Q [C] potential: Potential φ [V] = [J/C] flux: Current I [A] = [C/s] flux density: Current Density j [A/m^{2}] linear model: Ohm's Law j =  σ del [φ] 
***Angular Momentum/Spin/Polarization***  
4SpinMomentum or PauliLubanski 4vector 
W = (w_{0},w) = (u·w/c,w) because W·U = 0 W = (w^{0},w) = (p·Σ , P^{0}Σ + p x k) where Σ is the spin part of angular momentum j W = m_{o} S where S is the 4Spin 
[spinmomentum] W·W = (u·w/c,w)·(u·w/c,w) = (u·w/c)^{2}  w·w) =  w·w =  m^{2} s(s+1) W^{2} = ( w_{0}^{2}  w·w ) =  (w·w) =  (P_{0}^{2}Σ^{2}) =  m^{2}c^{2} h_{bar}^{2} s(s+1) where Σ is the spin part of angular momentum j (P_{0}^{2}) = (m^{2}c^{2}) (Σ^{2}) = h_{bar}^{2} s(s+1) W·W = 0 for photonic plays the role of covariant angular momentum see BargmannMichelTelegdi (BMT) dynamical eqn 
4Spin 
S = (s^{0},s) = (u·s/c,s) because S·U = 0 S = (γ β·s_{o} , s + [γ^{2}/(γ+1)](β·s_{o}) β) in moving frame S = (1/m_{o}) W= (U·U/P·U) W where m_{o} = √[P·P/U·U] = P·U/U·U Magnetic moment μ = (g/2)(e/mc) s S_{o} = (0,s_{o}) in rest frame 
[ J s] = [spin] Spin = IntrinsicAngMomentum, u·s/c =
component such that U·S = 0 4Spin is orthogonal to 4Velocity, so time component is zero in rest frame S_{o}=(0,s_{o}) This is an axial vector, or pseudovector 4Spin has only 3 independent components, not 4, due to U·S = 0 S_{o}=(0,s_{o}), 4Spin is always spacelike S·S = (u·s/c,s)·(u·s/c,s) = ((u·s/c)^{2}  s·s) =  s_{o}·s_{o} =  h_{bar}^{2} s_{o}(s_{o}+1) s·s s,m> = h_{bar}^{2} s(s+1) s,m> s_{z} s,m> = h_{bar} m s,m> for s = {0 , 1/2 , 1 , 3/2 , 2 , 5/2 , ...} for m = {s, s+1, ..., s1, s} SpinMultiplicity[m] = (2s+1) denotes the # of possible quantum states of a system with given principal s for s = 0, {m} = {0} singlet for s = 1/2, {m} = {1/2 , 1/2} doublet for s = 1, {m} = {1 , 0 , 1} triplet s = h_{bar} √[s(s+1)], [ S_{x} ,S_{y} ] = i h_{bar} ε_{xyz }S^{z} Spin raising/lowering operators: S_{±} s,m> = h_{bar} √[s(s+1)  m(m ± 1)] s,m> where S_{±} = S_{x} ± i S_{y} BargmannMichelTelegdi (BMT) dynamical eqn. for spin dS/dτ = (e/mc)[ (g/2) F^{μβ}S_{β} + (1/c2)(g/21))v^{μ}S_{α}F^{αβ}v_{β} ] which leads to Thomas precession in the rest frame 
4Polarization or 4JonesVector 
Ε = (ε^{0}, ε) =
(ε·u/c,ε)
for a massive particle = (ε^{0}, ε) = ((c/v_{phase}) ε·n,ε) for a wave = (ε^{0}, ε) = (ε·n,ε) ,for lightlike/photonic for photon travelling in zdirection using the Jones Vector formalism n = z / z E = (0,1,0,0) : xpolarized linear E = (0,0,1,0) : ypolarized linear E = √[1/2] (0,1,1,0) : 45 deg from xpolarized linear E = √[1/2] (0,1,i,0) : rightpolarized circular = spin 1 E = √[1/2] (0,1,i,0) : leftpolarized circular = spin 1 Generalpolarized E = (0,Cos[θ]Exp[iα_{x}],Sin[θ]Exp[iα_{y}]),0) E* = (0,Cos[θ]Exp[iα_{x}],Sin[θ]Exp[iα_{y}]),0) Angle θ describes the relation between the amplitudes of the electric fields in the x and y directions Angles α_{x} and α_{y} describe the phase relationship between the wave polarized in x and the wave polarized in y Ε·E* = = (+0^{2}  Cos[θ]Exp[iα_{x}]Cos[θ]Exp[iα_{x}]  Sin[θ]Exp[iα_{y}]Sin[θ]Exp[iα_{y}] 0^{2} ) = = (+0Cos[θ]^{2}Sin[θ]^{2}0) =  (Cos[θ]^{2}+Sin[θ]^{2}) = 1 Ε·E* = 1 
[1] ε = PolarizationVector **This 4vector has complex components in QM** Called helicity for massless particles Helicity is spin component along the direction of motion. "Helicity is the only truly measurable component of spin for a moving particle, but at low enough velocities (nonrelativistic), the spin component m along an external axis becomes an alternative observable." Like the 4Spin, Ε orthogonal to U, or K, so time component = 0 in rest frame This is cancellation of the "scalar" polarization This would again give only 3 independent components Ε·U = 0, Ε·K = 0, Additionally, Ε·E* = 1 (normalized to unity along a spatial direction) Normalization combined with u = c for photons is enough to reduce it to 2 independent components for photons. Ε·E* = 1 imposes (ε·u/c)^{2} ε·ε* = 1 (ε·u/c)^{2} 1 = 1 (ε·u/c)^{2} = 0 ε·u = 0, so that the spatial components must be orthogonal. For a massive particle, there is always a rest frame where u = 0, so ε can have 3 independent components. For a photonic particle, there is no rest frame. ( ε·u/c = ε·n = 0 ) is therefore an additional constraint, limiting ε to 2 independent components, with polarization ε orthogonal to direction of photon motion n. This is cancellation of the longitudinal polarization. According to WikipediaGauge Fixing, Many of the differences between classical electrodynamics and QED can be accounted for by the role that the longitudinal and timelike polarizations play in interactions between charged particles at microscopic distances. see Field Quantization, by Walter Greiner, Joachim Reinhardt 
4SpinPolarization 
In the rest frame, where K = (m,0), choose
a unit 3vector n as the quantization axis. In a frame in which the momentum is K = (k^{0},k) the spin polarization of a massive particle N = N^{μ} = ( k·n/m , n + (k·n) k / (m(m + k^{0})) ) N·N = 1, Normalized to spatial unity K·N = 0, Orthogonal to Wave Vector alternately, S = S^{μ} = ( p·s/m , s + (p·s) p / (m(m + p^{0})) ) which makes more sense, called the covariant spin vector W = (w^{0},w) = (p·Σ , P^{0}Σ + p x k) where Σ is the spin part of angular momentum j W^{2} = m^{2} Σ^{2} = m^{2} s(s+1) Σ = 0 represents a spin0 particle Σ = Dirac spinor represents a spin1/2 particle, with Σ^{2} = 3/4 the unit matrix I W·N = m Σ·n =  m s, the component of spin along n as measured in the rest frame. s is the spin component in the direction n that would be measured by an observer in the particle's rest frame Apparently this only works for massive particles, so the N 4vector is undefined for massless Instead, helicity h = Σ·k / k, h = +1/2 or 1/2 for spin 1/2 particle helicity is component of spin parallel to the 3vector momentum k w^{0} = (k·Σ) and k^{0} = k for a massless particle alternately N^{μ} = ( T^{μ}  P^{μ}(pt/m^{2}))(m/p) where T^{μ} is the unit timelike 4vector  
4PauliMatrix 
Σ = Σ^{μ} = (σ^{0}, σ) where Σ·Σ = 2σ^{0}  The components of this 4vector are actually the Pauli Matrices 
***Electromagnetic Field Potentials***  
4VectorPotential or 4VectorPotential_{EM}  A_{EM} = (Φ_{EM}/c, a_{EM}) 
[kg m C^{1} s^{1}] Φ_{EM} =
ScalarPotenial_{EM} a_{EM} = VectorPotenial_{EM} Electric Field E = del[Φ_{EM}]∂a_{EM}/∂t, Magnetic Field B = del x a_{EM} Electric Field E [N/C = kg·m·A^{−1}·s^{−3}], Magnetic Field B [Wb/m^{2} = kg·s^{−2}·A^{−1} = N·A^{−1}·m^{−1}] for 4VectorPotenial of a moving point charge (LiénardWiechert potential) A_{EM} = (q/c4πε_{0}) U / (R·U) where (R·R = 0, the definition of a light signal) 4VectorPotentail of an Ε polarized planewave A_{EM} ~ Ε Exp[i K·R] Homogeneous Maxwell/Lorentz Equation (if ∂·A_{EM} = 0 Lorenz Gauge) (∂·∂)A_{EM} = µ_{0} J 
4VectorPotentialMomentum or 4PotentialMomentum  Q_{EM} = (E_{EM}/c, p_{EM}) = q_{o} A_{EM} 
[kg m s^{1}] E_{EM} =
ScalarPotenialEnergy p_{EM} = VectorPotenialMomentum Energy and Momentum of the EM field itself, for a single charge 
4Potential or 4Potential_{EM } ***break with standard notation*** better to use the 4VectorPotential_{EM} 
Φ_{EM} = (Φ_{EM},c a_{EM}) = c A_{EM} 
[kg m^{2} C^{1} s^{2}]
Φ_{EM} =
ScalarPotenial_{EM} a_{EM} = VectorPotenial_{EM} ***this is bad notation based on our 4vector naming convention*** the cfactor should be in the time component the 4vector name should reference the space component I simply include it here because it is common in the literature 
4Momentum_{EM} 4CanonicalMomentum 4TotalMomentum 
P_{EM} = (E/c + qΦ_{EM}/c, p
+ qa_{EM}) =
γ
m_{o}(c,u) P_{EM} = Π = P + q A_{EM} 
[kg m s^{1}] **Momentum including effects of EM potentials** also known as Canonical Momentum where P is the Kinetic Momentum term where qA is the Potential Momentum term Total Momentum = Kinetic Part + Potential Part 
4Gradient_{EM} Gauge Covariant Derivative 
D_{EM} = (∂/c∂t + iq/h_{bar}
Φ_{EM}/c,
del + iq/h_{bar} a_{EM}) = ∂ + (iq/h_{bar})A_{EM} for electrons, commonly seen as D_{EM} = ∂  (ie/h_{bar})A_{EM} where e is the electric charge 
[m^{1}] **Gradient including effects of EM potentials** Minimal coupling based on principle of local gauge invariance 
***Position space & Momentum Space Differentials***  
4Differential  dX = (cdt, dx)  [m] dt = Temporal Differential, dx = Spatial Differential 
4Volume Element Flux 
dV = (c dv_{0},dv) = (dV_{o}) U = (dV_{o}) γ(c, u) = γ(dV_{o}) (c, u) = (dV)(c, u) = (c dV,dV u) so that, in a rest frame dV_{o} = (1)(c dV_{o},dV_{o} 0) = (c dV_{o},0) dV = γdV_{o} ???? V should be as follows V = V_{o}/γ Perhaps acts a little differently since this is from a "vectorvalued volume element", and not a straight volume. 
[m^{3}] A vectorvalued volume element is just a
4vector that is perpendicular to all spatial vectors in the volume element, and
has a magnitude that's proportional to the volume. Using Clifford Algebra one can represent an oriented volume element by a threeform. In a 4d spacetime, a 3form has a dual representation (Hodges Dual) which is a 1form, which is basically a vector. Basically this means that you define a volume element by the spacetime vector that's perpendicular to it, and you make the length of this spacetime vector proportional to the proper volume you wish to represent. Hodge Duality in SR n=4 Minkowski spacetime with metric signature (+,,,) and coordinates (t,x,y,z) gives *dt = dx^dy^dz (* is the Hodge star operator, ^ is the wedge product operator) alternately, dV = √[g]d^{4}x is an invariant volume element scalar?? c dt dV = dx^{0} dx^{1} dx^{2} dx^{3} = dx'^{0} dx'^{1} dx'^{2} dx'^{3} c dt = dx^{0} dV = dx^{1} dx^{2} dx^{3} = d^{3}x (c dt)(dV) = (dx^{0} )(dx^{1} dx^{2} dx^{3}) = d^{4}x d^{3}x = d^{3}x'/γ d^{3}p/p_{o} = d^{3}p'/p'_{o} p'_{o} = p_{o}/γ d^{3}x d^{3}p = d^{3}x' d^{3}p' dV_{o}·dX = (dV_{o},0)·(cdt,dx) = (dV_{o} cdt) = dxdydz cdt = d^{4}x dV·dX = d^{4}x Hence, the differential 4element is an invariant dq = ρ d^{3}x = j^{0}/c d^{3}x Interesting derivation: dq = ρ dV differential charge element (dq)dX = (ρ dV)dX = ρ dV (dt/dt) dX = ρ (dV dt) dX/dt = ρ (dV dt) (1/γ) dX/dτ = ρ (dV dt) (1/γ)U = γρ_{o} (dV dt) (1/γ) U = ρ_{o} (dV dt) U = (dV dt) ρ_{o}U = (dV dt) J = ( dV(c/c) dt)J = ( dV c dt/c) J = (d^{4}x/c) J dq dX = (d^{4}x/c) J Apparently, (dx^{1} dx^{2} dx^{3})/x^{0} is also an invariant, based on Jacobian also, d^{3}x Δt is an invariant (dV·ΔR) / (U·U) = d^{3}x Δt 
4Momentum Differential  dP = (dE/c, dp)  [kg m s^{1}] dE = Temporal Momentum Differential, dp = Spatial Momentum Differential 
4MomentumSpace Volume Element Flux 
dV_{p} = (c dv_{p0},dv_{p}) = (dV_{po}) U = (dV_{po}) γ(c, u) = γ(dV_{po}) (c, u) = (dV_{p})(c, u) = (c dV_{p},dV_{p} u) so that, in a rest frame dV_{po} = (1)(c dV_{po},dV_{po} 0) = (c dV_{po},0) 
[kg^{3} m^{3} s^{3}] A vectorvalued
MomentumSpace volume element is just a
4vector that is perpendicular to all spatial vectors in the MomentumSpace volume element, and
has a magnitude that's proportional to the MomentumSpace volume. Using the same Clifford Algebra idea from positionspace, I think this can be done (dE/c)(dV_{p}) = dp^{0} dp^{1} dp^{2} dp^{3} = dp'^{0} dp'^{1} dp'^{2} dp'^{3} dE/ c = dp^{0} dV_{p} = dp^{1} dp^{2} dp^{3} = d^{3}p (dE/ c)(dV_{p}) = (dp^{0} )(dp^{1} dp^{2} dp^{3}) = d^{4}p d^{3}x = d^{3}x'/γ d^{3}p/p_{o} = d^{3}p'/p'_{o} p'_{o} = p_{o}/γ d^{3}x d^{3}p = d^{3}x' d^{3}p' dV_{po}·dP = (dV_{po},0)·(dE/c,dp) = (dV_{po} dE/c) = dp_{x}dp_{y}dp_{z} dE/c = d^{4}p dV_{p}·dP = d^{4}p Hence, the differential momentum 4element is an invariant also, dV_{o}·dV_{po} = (dV_{o},0)·(dV_{po},0) = (dV_{o}dV_{po}) = (dx^{1} dx^{2} dx^{3})(dp^{1} dp^{2} dp^{3}) = d^{3}xd^{3}p in the rest frame Thus dV·dV_{p} = d^{3}x d^{3}p generally, so (d^{3}x d^{3}p) is a Lorentz scalar invariant Apparently, (dp^{1} dp^{2} dp^{3})/p^{0} is also an invariant, based on Jacobian 
***Special 4Vectors*** 

4Zero  Zero = (0,0) 
[*] All components are 0 in all reference frames, the only vector with this
property Square Magnitude = 0, Length = 0 = 0 
4Null 
Null = (a,a) = (a,an) = a(1,n) where n is the unit 3vector Null·Null = a(1,n)·a(1,n) = a^{2}(1*1  n·n) = a^{2}(0) = 0 Null·Null = 0 
[*] Any 4vector for which the temporal component
magnitude equals the spatial component magnitude a^{0} = a which leads to the magnitude being 0, or LightLike/Photonic ex. The 4Velocity of a Photon, the 4Momentum of a Photon 
4Unit Temporal 
T = (1,0) T·T = (1,0)·(1,0) = (1*1  0·0) = 1 
[*] The Unit Temporal 4Vector Square Magnitude = 1, Length = 1 = 1 
4Unit Null 
N = (1,n) N·N = (1,n)·(1,n) = (1*1  n·n) = 0 
[*] The Unit Null 4Vector n = unit 3vector, n = 1 Square Magnitude = 0, Length = 0 = 0 The Null Vector is "perpendicular" to itself. 
4Unit Spatial 
S = (0,n) S·S = (0,n)·(0,n) = (0*0  n·n) = 1 
[*] The Unit Spatial 4Vector n = unit 3vector, n = 1 Square Magnitude = 1, Length = 1 = 1 
4Basis Vectors (1 time + 3 space) 
B_{t} = (1,0,0,0) B_{x} = (0,1,0,0) B_{y} = (0,0,1,0) B_{z} = (0,0,0,1) 
A tetrad of 4 mutually orthogonal, unitlength,
linearlyindependent, basis vectors This is simply one basis, there are others 
4Basis Vectors (null tetrad) 
B_{n1} = √[1/2] (1,0,0,1) B_{n2} = √[1/2] (1,0,0,1) B_{n3} = √[1/2] (0,1,i,0) B_{n4} = √[1/2] (0,1,i,0) 
A tetrad of unitlength, linearlyindependent, null basis
vectors Note the complex components, since there can be only 2 real linearly independent null vectors This is simply one basis, there are others ** Need to doublecheck these *** see null tetrad, Sachs tetrad, NewmanPenrose tetrad 
Event R 
Mass m_{o} Energy E_{o} = m_{o}c^{2} 
WaveAngFreq ω_{o} 
MassDensity
ρ_{o_m} Mass m_{o} = ρ_{o_m}V_{o} EnergyDensity u_{o} = ρ_{o_m}c^{2} 
ChargeDensity ρ_{o} Charge Q_{o} = ρ_{o}V_{o} 
NumberDensity n_{o} ParticleNumber N_{o} = n_{o}V_{o} 
event  particle  wave  density  density  density 
pos: R = (ct, r)  m_{o} at R  ω_{o} at R  ρ_{o_m} at R  ρ_{o} at R  n_{o} at R 
vel: U = dR/dτ  P = m_{o}U = (E_{o}/c^{2})U  K = (ω_{o}/c^{2})U = (1/h_{bar})P  G = ρ_{o_m}U = (u_{o}/c^{2})U  J = ρ_{o}U  N = n_{o}U 
accel: A = dU/dτ  F = dP/dτ  F_{d} = dG/dτ  
jerk: J = dA/dτ 
Event(SR)  EventMovement  MassEnergy  ParticleWaveDuality  QuantumMechanics(QM)  SpaceTimeVariations 
R = (ct, r)  dR/dτ = U = γ(c,u)  U = P/m_{o}  P = h_{bar} K  *** K = i ∂ ***  ∂ = (∂/c∂t,del) 
or K = (ω_{o}/c^{2})U 
d/dτ[R] = (i h_{bar} / m_{o}) ∂
Event motion ~ spacetime structure  depends on i h_{bar} / m_{o}
So, the following assumptions within SRSpecial Relativity lead to
QMQuantum Mechanics:
R = (ct,r)  Location of an event (i.e. a particle) within spacetime 
U = dR/dτ  Velocity of the event is the derivative of position wrt. Proper Time 
P = m_{o}U  Momentum is just the Rest Mass of the particle times its velocity 
K = 1/h_{bar} P  A particle's wave vector is just the momentum divided by Dirac's constant, but uncertain by a phase factor 
∂ = i K  The change in spacetime corresponds to (i) times the wave vector, whatever that means... 
R·R = (Δ s)^{2} = (ct)^{2}r·r =
(ct)^{2}r^{2}
: dR·dR = (ds)^{2} = (c dt)^{2}dr·dr =
(c dt)^{2}dr^{2} :
Invariant Interval
U·U = c^{2}
P·P = (m_{o}c)^{2}
K·K = (m_{o}c / h_{bar})^{2} = (ω_{o}/c)^{2}
∂·∂ = (∂/c∂t,del)·(∂/c∂t,del) =
∂^{2}/c^{2}∂t^{2}del·del = (m_{o}c
/ h_{bar})^{2} : KleinGordon Relativistic Wave Eqn.
Each relation may seem simple, but there is a lot of complexity generated by each level.
*see QM from SR (Quantum Mechanics derived from Special Relativity)*
This can be further explored:
∂·∂ + (m_{o}c
/ h_{bar})^{2} = 0
(∂·∂ + (m_{o}c
/ h_{bar})^{2} ) Ψ = 0, where Ψ is a scalar
KleinGordon eqn for massive spin0 field
(∂·∂ + (m_{o}c
/ h_{bar})^{2} ) A = 0, where A is a
4vector Proca eqn for massive spin1 field
and let m_{o} > 0
(∂·∂) Ψ = 0, where Ψ is a scalar
Freewave eqn for massless spin0 field
(∂·∂) A = 0, where A is a 4vector
Maxwell eqn for massless spin1 field, no current sources
Interesting Note about Proca eqn.
"Massive charged vector field  represent with complex fourvector field
φ^{μ}(X) and impose "Lorenz condition"
(∂_{μ}φ^{μ}) = 0 so that _{}
φ^{0}(X) the scalar polarization , can be discarded and the
KleinGordon equations emerge for the other three components
φ^{i}(X)
(∂·∂ + (m_{o}c
/ h_{bar})^{2} ) A = 0, where A is a
4vector Proca eqn for massive spin1 field
rewrite in index notation
(∂_{μ}∂^{μ} + (m_{o}c
/ h_{bar})^{2} ) A^{ν} = 0 and combine with the
Lorenz gauge condition
(∂_{μ}A^{μ} = 0)
apparently, this conjunction is equivalent to
∂_{μ}( ∂^{μ} A^{ν}  ∂^{ν} A^{μ}
)+ (m_{o}c
/ h_{bar})^{2} A^{ν} = 0
which is the EulerLagrange equation for the Proca Action
see Conceptual Foundations of Modern Particle Physics, ~ pg. 100
P = i h_{bar} ∂ = ∂(S_{act})  ∂ = (∂/c∂t,del)  A_{EM} = (0,0) *special case* 
P_{EM} = P+qA_{EM} = i h_{bar} D_{EM}  D_{EM} = ∂+iq/h_{bar} A_{EM}  A_{EM} = (V_{EM}/c,a_{EM}) 
Relations involving the 4Position or 4Displacment:  
R·R = (Δs)^{2} = (ct)^{2}r·r =
(ct)^{2}r^{2}
R·R = 0 for photonic signal 
Spacetime position of an event wrt. an origin event 
dR·dR = (ds)^{2} = (c dt)^{2}dr·dr = (c dt)^{2}dr^{2}  Differential interval magnitude  the fundamental invariant differential form 
ΔR·ΔR = (ds)^{2} = (c Δt)^{2}Δr·Δr = (c Δt)^{2}Δr^{2}  Spacetime displacement interval magnitude  used to derive SR 
∂·R = 4  The divergence of open spacetime is equal to the number of independent dimensions (t,x,y,z) 
K·R = φ_{EM} = (ωtk·r)  Phase of a SR wave; Psi = a E e^{ iK·R} Photon Wave Equation (Solution to Maxwell Equation) 
R·U = (ct,r)·γ(c,u ) = γ(c^{2}t  r·u)  Part of expression used in LiénardWiechert potential 
Relations involving the 4Velocity:  
U·U = c^{2}  The magnitude of 4Velocity is always c^{2} 
U·A = 0  The 4Acceleration is always normal to a particles worldline 
P·U = m_{o}c^{2} = E_{o}  Rest Energy 
K·U = m_{o}c^{2}/h_{bar} = E_{o}/h_{bar} = ω_{o}  Rest Ang. Frequency 
F·U = (m_{o}A+(dm_{o}/dτ)U)·U
= c^{2}(dm_{o}/dτ) = γ^{}c^{2}(dm_{o}/dt) U·F = γ^{2}(dE/dtu·f) = γ dm_{o}/dt c^{2} (pure force if dm_{o}/dt = 0) 
Power Law 
U_{1}·U_{2} = γ[u_{1}]γ[u_{2}](c^{2}u_{1}·u_{2}) = γ[u_{r}]c^{2}  (The scalar product of two uniformly moving particles is proportional to the γ factor of their relative velocities) 
U·∂ = d/dτ = γ(∂/∂t + u·del) = γ d/dt  Relativistic Convective (Time) Derivative U·∂ = d/dτ = γ(∂/∂t + u·del) = γ d/dt let u << c, then γ ~ 1 then U·∂ = d/dτ = γ(∂/∂t + u·del) ~ (∂/∂t + u·del) = d/dt del_{t}(v) = (∂/∂t + u·del) v is the gauge covariant derivative of a fluid where v is a velocity vector field of a fluid 
∂·U = 0  The General Continuity Equation, one might
say the conservation of event flux. I believe this is true generally but it needs checking... 
Relations involving the 4Acceleration:  
A·A = a^{2}  Magnitude squared of acceleration 
U·A = 0  The 4Acceleration is always normal to
a particles worldline U·U = c^{2} d/dτ(U·U) = d/dτ(c^{2}) = 0 d/dτ(U·U) = 2*(U·dU/dτ) = 2*(U·A) = 0 
Others:  
P·P = (m_{o}c)^{2} = (E_{o}/c)^{2} = m^{2}(c^{2}u·u) = (m^{2}c^{2}p·p) = (E^{2}/c^{2}p·p) = 0 for lightlike/photonic 
Square Magnitude of the 4Momentum 
P_{1}·P_{2} = γ[u_{1}]γ[u_{2}]m_{o1} m_{o2}(c^{2}u_{1}·u_{2}) = γ[u_{r12}]m_{o1} m_{o2}c^{2} 
Relativistic Billiards... P + Q = P' + Q': Momenta before and after collision generally  Conservation of 4Momentum P·Q = P'·Q': Momenta in an elastic (restmass preserving) collision  Relative velocities conserved 
N·N = (n_{o}c)^{2}  Square Magnitude of the 4NumberCurrentDensity 
J·J = (p_{o}c)^{2} = (q_{o}n_{o}c)^{2}  Square Magnitude of the 4ElectricCurrentDensity 
K·K = (m_{o}c / h_{bar})^{2} =
(ω_{o}/c)^{2} = 0 for lightlike/photonic 
Square Magnitude of the 4WaveVector 
∂·∂ = (∂/c∂t,del)·(∂/c∂t,del) = ∂^{2}/c^{2}∂t^{2}del·del = (m_{o}c / h_{bar})^{2}  KleinGordon Relativistic Wave Eqn. ** ∂·∂ is also known as the D'alembertian (Wave Operator) ** 
∂·J = dp/∂t +del·j = 0  Continuity Equation  Conservation of
Electric Charge No sources or sinks Charge is neither created nor destroyed 
E·K = 0  The Polarization of a photon is orthogonal to direction of wave motion (cancellation of "scalar" polarization) 
E·E* = 1  The Polarization of a photon is always unit magnitude and spacelike (cancellation of the longitudinal polarization) 
A_{EM}·A_{EM} = (V_{EM}/c,a_{EM})·(V_{EM}/c,a_{EM}) = (V_{EM}/c)^{2}a_{EM}·a_{EM} = ????  Square Magnitude of the Electromagnetic field 
c = √[U·U]  Speed of Light: c (in vacuum) E ~ cp 
h = √[P·P/L·L] =
P·L / L·L h_{bar} = √[P·P/K·K] = P·K / K·K 
Planck's const: h
E ~ hν Dirac's const: h_{bar} E ~ h_{bar}ω_{} 
k_{B} = √[P·P/P_{T}·P_{T}] = P·P_{T} / P_{T}·P_{T}  Boltzmann's const: k_{B} E ~ k_{B}T 
γ_{rel} = U·V_{o}/U·V  Relative Relativistic Gamma Factor 
Δs = √[ΔR·ΔR] = √[c^{2}Δt^{2}  Δx^{2}  Δy^{2}  Δz^{2}] 
Displacement < 0 Outside LightCone, Acausal SpaceLike Separation Δs^{2} = 0 "On The LightCone", LightLike Signal Separation > 0 Inside LightCone, Causally TimeLike Separation 
ds = √[dR·dR] = √[c^{2}dt^{2}  dx^{2}  dy^{2}  dz^{2}] 
Differential Length of World Line Element ds^{2} = c^{2}dτ^{2} 
dτ = √[dR·dR/U·U]  Differential Proper Time, aka. the Eigentime differential 
d/dτ = U·∂ = γ(∂/∂t + u·del) = γ d/dt 
Derivative wrt Proper Time d/dτ d/dt = total time derivative, ∂/∂t = partial time derivative 
Δ = ∂·∂ = (∂/c∂t,del)·(∂/c∂t,del) = ∂^{2}/c^{2}∂t^{2}del·del  D'Alembertian/wave operator 
d^{4}x = dV·dX 
Spacetime positionspace differential "4volume"
element Note: may need a correction factor if not using the Minkowski Metric 
d^{4}p = dV_{p}·dP  Spacetime momentumspace differential "4volume" element 
d^{3}xd^{3}p = dV·dV_{p}  Spacetime positionmomentum differential 3volume element 
δ^{4}(xy)  4D Dirac Delta Function 
m_{o} = √[P·P/U·U] = P·U/U·U  RestMass of a Particle m_{o} ( 0 for photons, + for massive ) 
q_{o} = √[J·J/N·N] = J·N/N·N  RestElectricCharge of a Particle q_{o} 
3vector s_{o} = √[S·S] = h_{bar} √[s(s+1)] 
Spin s_{o} S·S =  s_{o}·s_{o} =  h_{bar}^{2} s_{o}(s_{o}+1) 
magnetic moment  
E_{o} = P·U = m_{o}c^{2}  RestEnergy of a Particle ( 0 for photons, + for massive ) 
ω_{o} = K·U = m_{o}c^{2}/h_{bar}  RestAngFrequency of a Particle ( 0 for photons, + for massive ) 
φ = K·R 
*** Phase of a wave ***, e.g. an EM wave, a plane wave However, could also be a de Broglie matter wave... 
Sact = P·R = Integral[dt L;t_{i},t_{f}]  Action Variable S of Action Integral 
γ L = ??  Relativistic Lagrangian 
Action Integral  Action Integral 
n_{o} = √[N·N/U·U] = N·U/U·U  Particle RestNumberDensity (for stat mech) 
s_{o} = √[S·S/U·U] = S·U/U·U  RestEntropyDensity (for stat mech) 
Ω_{o} = Ω  Ω = # of microstates = (N!) / (n_{0}!n_{1}!n_{2}!...) 
N_{o} = N  (Stable) Particle Number: N = nV = (n/γ)(γ V) = n_{o}V_{o} = N_{o} 
P_{o} = P  Pressure of system: P = P_{o} 
S_{o} = S = k_{B} ln Ω  Entropy: S = sV = (s/γ)(γ V) = s_{o}V_{o} = S_{o} , 
T_{o} = γ T  RestTemperature (according to Einstein/Planck def.) 
Q_{o} = γ Q  RestHeat 
V_{o} = γ V  RestVolume 
dS = k_{B} d(ln Ω) = δQ / T  Change in Entropy 
Π (p^{α},x^{α}) = = dN/(d^{3}x d^{3}P) = (2j+1)/h^{3}  exp[(P·u)/kT  θ]  ε 
Invariant equilibrium distribution function for
relativistic gas j = particle spin h = Planck's const u = mean 4velocity = 1 BoseEinstein statistics ε = 0 MaxwellBoltzmann statistics = 1 FermiDirac statistics kTθ = Chemical potential μ=(ρ+p)/n  Ts 
F_{uv}^{ }F^{uv} = 2(B^{2}  E^{2}/c^{2})  EM invariant 
G_{cd}^{ }F^{cd} = ε_{abcd}F^{ab}F^{cd} =(2/c)(B·E)  EM invariant 
∂·J = ∂p/∂t +del·j = 0 
Conservation of 4CurrentDensity (EM charge): p & j change in ChargeDen wrt. time balanced by flow of CurrentDen 
∂·N = ∂/∂t(γ n_{o})+del·(γ
n_{o}u) = ∂n/∂t+del·n_{f} = 0  Conservation of 4NumberFlux (Particle NumberDen, NumFlux): n_{ & }n_{f }change in NumberDen wrt. time balanced by flow of NumFlux 
∂·P = (1/c^{2})∂E/∂t +del·p =
0 Sum[P_{f}P_{i}] = Zero 
Conservation of 4Momentum (Energy~Mass, Momentum): E & p change in Energy wrt. time balanced by flow of Momentum Alternately, the Sum[(Final 4Momenta)  (Initial 4Momenta)] = Zero 4Vector Note: this conservation equation, while rarely used, is perfectly acceptable for single particles. It is only when a group of particles is treated as a continuous fluid that the EnergyMomentum (2,0)Tensor is required. Then, the diagonal pressure terms and offdiagonal shear terms are necessary, basically allowing statistical particle interaction. 
∂·K = ∂/c∂t(w/c)+del·k = (1/c^{2})∂w/∂t +del·k = 0. 
Conservation of 4WaveVec (AngFreq, WaveNum): w & k change in AngFreq wrt. time balanced by flow of WaveNum 
∂·A_{EM} = (1/c^{2})∂V_{EM}/∂t +del·a_{EM} = 0  Conservation of 4VectPotential_{EM} (applies in the Lorenz Gauge): V_{EM} & a_{EM }change in ScalarPotential wrt. time balanced by flow of VectorPotential 
∂·U = ∂/∂t(γ[u])+del·(γ[u]
u) = γ^{3} (u/c^{2} ∂u/∂t + del·u) = ∂·Uo? = 0 if event is in a conservative field or space 
Conservation of 4Velocity: (FluxGauss' Law)??: γ & γ u change in (γ) wrt. time balanced by flow of (γ u) If this quantity equals zero, then any physical quantity that is just a (constant* 4velocity) is conserved. For example ∂·P = ∂·(m_{o}U) = m_{o}(∂·U) = 0 Also from d/dτ (∂·R) = ∂·U = 0 see also Noether's Theorem 
Lorentz 4Tensors
η_{μν} = η^{μν}
= Diag[1,1,1,1] = +1 if μ = ν = 0 = 1 if μ = ν = 1,2,3 = 0 if μ ≠ ν 
Minkowski Metric (flat spacetime) (pseudoEuclidean) 

1 if a=b, δ^{a}_{b} = 0 if a≠b 
Kronecker Delta  
= +1 if {abcd} is an even permutation of {0123} ε_{abcd} = ε^{abcd} = 1 if {abcd} is an odd permutation of {0123} = 0 otherwise 
LeviCivita symbol technically a pseudotensor ε_{abcd} = g ε^{abcd} =  ε^{abcd} = since g = 1 for Minkowski 

F_{uv}^{ } =

EM Field Tensor F^{uv} = ∂^{u}A^{v}∂^{v}A^{u} Electromagnetic Field Tensor (F^{0i} = E^{i},F^{ij} = e^{ijk}B^{k}) 

G_{cd} = (1/2)ε_{abcd}F^{ab}
=

Dual EM Field Tensor  
T^{ab}
=

EnergyMomentum Stress Tensor W = Energy Density S = Energy Flux Density? The stressenergy
tensor of a relativistic fluid can be written in the form Here
The heat flux vector and viscous shear tensor are transverse to the world lines, in the sense that
This means that they are effectively threedimensional quantities, and since the viscous stress tensor is symmetric and traceless, they have respectively 3 and 5 linearly independent components. Together with the density and pressure, this makes a total of 10 linearly independent components, which is the number of linearly independent components in a fourdimensional symmetric rank two tensor. 
U·d(P) = γ(dEu·dp) = (T_{o}dS_{o}  P_{o}dV_{o} + µ_{o}dN_{o}) = const = ? 0 ? 
U·P = γ(Eu·p) = (T_{o} S_{o}  P_{o} V_{o} + µ_{o} N_{o}) = m_{o}c^{2 }? for a spatially homogeneous system: relativistic GibbsDuhem eqn. 
Invariants  P = Pressure = P_{o}  N = ParticleNum = N_{o}  S = Entropy = S_{o} 
Variables  V = Volume = (1/γ)Vol_{o}  µ = ChemPoten = (1/γ)µ_{o}  T = Temperature = (1/γ)Temp_{o} 
V*P (particle superstructure = Vol*Press)
µ*N (particle structure = ChemPoten*ParticleNum)
T*S (particle substructure = Temp*Entropy)
Time t = γ t_{o}
Length L = L_{o}/γ
Heat Q = Q_{o}/γ
dQ_{o} = T_{o}dS_{o}
InertialMassDen(of radiation field) q = P/vV = γ q_{o}
Total Particle Number N = N_{o} is an invariant, because the NumberDensity n varies as
n = γ n_{o},
but this is balanced by Volume V = V_{o}/γ
NumberDenstiy n = γ n_{o} where NumberFlux 4Vector N =
(cn,n_{f}) = n_{o}
γ(c, u) = n_{o}U, n_{o} = N_{o}/(Δ_x_{o}*Δ_y_{o}*Δ_z_{o})
N = n * V = (γ n_{o})*(V_{o}/γ) = n_{o}* V_{o} =
N_{o}
N·N = (n_{o}c)^{2}
Total Entropy S = S_{o} is an invariant,
because the EntropyDensity s varies as s = γ s_{o},
but this is balanced by Volume V = V_{o}/γ
EntropyDensity s = γ s_{o} where EntropyFlux 4Vector S =
(cs,s_{f}) = s_{o}
γ(c, u) = s_{o}U, s_{o} = S_{o}/(Δ_x_{o}*Δ_y_{o}*Δ_z_{o})
S = s * V = (γ s_{o})*(V_{o}/γ) = s_{o}* V_{o} =
S_{o}
S·S = (s_{o}c)^{2}
An lamellar vector field is a synonym for an irrotational vector field.[1] The adjective "lamellar" derives from the noun "lamella", which means a thin layer. In Latin, lamella is the diminutive of lamina (but do not confuse with laminar flow). The lamellae to which "lamellar flow" refers are the surfaces of constant potential.
An irrotational vector field which is also solenoidal is called a Laplacian vector field.
The fundamental
theorem of vector calculus states that any vector field can be expressed as
the sum of a conservative vector field and a solenoidal
field.
In vector
calculus a solenoidal vector field (also known as an incompressible
vector field) is a vector
field v with divergence
zero:
del·v =
0
The fundamental
theorem of vector calculus states that any vector field can be expressed as
the sum of a conservative
vector field and a solenoidal field. The condition of zero divergence is
satisfied whenever a vector field v has only a vector
potential component, because the definition of the vector potential A
as:
v = del x A
automatically results in the identity
(as can be shown, for example, using Cartesian coordinates):
del·v = del·(del x A) =
0
The converse
also holds: for any solenoidal v there exists a vector potential A
such that v = del x A.
(Strictly speaking, this holds only subject to certain technical conditions on v,
see Helmholtz
decomposition.)
In vector
calculus, a Laplacian vector field is a vector
field which is both irrotational
and incompressible.
If the field is denoted as v, then it is described by the following differential
equations:
del x v = 0
del·v =
0
Since the curl
of v is zero, it follows that v can be expressed as the gradient
of a scalar
potential (see irrotational
field) φ :
v = del φ
(1)
Then, since the divergence of v is also zero, it follows from equation (1) that
del·del φ = 0
which is equivalent to
del^{2} φ =
0
Therefore, the potential of a Laplacian field satisfies Laplace's
equation.
In fluid
dynamics, a potential flow is a velocity
field which is described as the gradient of a scalar function: the velocity potential. As a result, a potential flow is
characterized by an irrotational
velocity field, which is a valid approximation for several applications. The
irrotationality of a potential flow is due to the curl
of a gradient always being equal to zero (since the curl of a gradient is
equivalent to take the cross
product of two parallel vectors, which is zero).
In case of an incompressible flow the velocity potential satisfies the Laplace's equation. However, potential flows have also been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows.
Applications of potential flow are for instance: the outer flow field for aerofoils, water waves, and groundwater flow.
For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable.
A velocity potential is used in fluid
dynamics, when a fluid occupies a simplyconnected region and is irrotational.
In such a case,
del x u = 0
where u denotes the flow
velocity of the fluid. As a result, u can be represented as the gradient
of a scalar
function Φ:
u = del Φ
Φ is known as a velocity potential for u.
A velocity potential is not unique. If a is a constant then Φ + a is also a velocity potential for u. Conversely, if Ψ is a velocity potential for u then Ψ = Φ + b for some constant b. In other words, velocity potentials are unique up to a constant.
Unlike a stream function, a velocity potential can exist in threedimensional flow.
see Cosmological Physics
Relativistic Euler Equations:
dv/dt =  1/[γ^{2}(ρ + p/c^{2})](del p +
p'v/c^{2}): Conservation of Momentum
d/dt[γ^{2}(ρ + p/c^{2})] = p'/c^{2}  γ^{2}(ρ + p/c^{2})del·v:
Conservation of Energy
where p' = ∂ p/∂ t
∂·J = 0 where J = n_{o}U (J is the
Number Flux here)
Relativistic Enthalpy w = ^{}(ρ + p/c^{2})
d/dt[γ^{}w/n] = p'/γnc^{2}
Thus, in steady flow, γ * (enthalpy/particle) = const.
In nonrelativistic limit these reduce to
dv/dt =  1/[ρ](del p): Conservation of Momentum
d/dt[(ρ)] =  (ρ)del·v: Conservation of Mass
p = Pressure
ΔE =  p ΔV
E = ρ c^{2} V
ΔV / V = 
Δρ_{o}/ρ_{o}
Relativisitic Bernoulli's eqn.
γ^{} w / ρ_{o} = const

 c
\ future /
\  /
\  /  spacelike
interval()
\/now
/\
/  \
elsewhere
/  \
/ past \
 c
(0,0) ZeroNull Vector
(+a,0) Future Pointing Pure TimeLike
(a,0) Past Pointing Pure TimeLike
(0,b) Pure SpaceLike
(a,b) a>b TimeLike
(a,b) a =b PhotonicLightLike
(a,b) a<b SpaceLike
(0,0)  φ  scalar field 
(1/2,0) (+) (0,1/2)  ψ  Dirac spinor 
(1/2,1/2)  A_{μ}  vector field 
(1,0) (+) (0,1)  F^{uv} = ∂^{u}A^{v}∂^{v}A^{u}  EM field tensor 
Consider an arbitrary spacetime vector x^{μ}
Construct the 2 x 2 Hermitian matrix X = X^{†}
X = x^{μ}σ_{μ} =  ( x^{0} + x^{3}  x^{1}  i x^{2} ) 
( x^{1} + i x^{2}  x^{0}  x^{3} ) 
then Det[X] = x^{2} = x·x =
η_{}_{μν} x^{μ} x^{ν}
see Proceedings of the Third International Workshop on Contemporary Problems in
Physics, By Jan Govaerts, M. Norbert Hounkonnou, Alfred
Z. Msezane
see Conceptual Foundations of Modern Particle Physics, Robert Eugene Marshak
see Fundamentals of Neutrino Physics and Astrophysics, Carlo Giunti
see Kinematical Theory of Spinning Particles, Martin Rivas
Spin  Statistics  Eqn. Mass=0  Eqn. Mass<>0  Representation  Polarizations Mass=0 Mass<>0 
0  Boson: BoseEinstein 
FreeWave NG bosons 
KleinGordon Higgs bosons 
scalar = 0tensor  1? 1 
1/2  Fermion FermiDirac 
Weyl Matter Neutrinos 
Dirac Matter Leptons/Quarks 
spinor  2 2 
1  Boson BoseEinstein 
Maxwell Force/Gauge Fields Photons/Gluons 
Proca Force 
vector = 1tensor  2 (= 2 transverse) 3 (= 2 transverse + 1 longitutinal) 
3/2  Fermion FermiDirac 
Gravitino?  RaritaSchwinger  spinorvector  2 4 
2  Boson BoseEinstein 
Einstein Graviton 
tensor = 2tensor  2 5 
Dim  Type  Hodge Dual  
0  scalar  
1  vector  
2  tensor  
3  pseudovector  magnietic field, spin, torque, vorticity, angular momentum  
4  pseudoscalar  magnetic charge, magnetic flux, helicity 
In Minkowski space (4dimensions), the { 1 4 6 4 1} Hodge dual of an nrank
(n<=2) tensor will be an (4n) rank skewsymmetric pseudotensor
Hodge duals
*dt = dx ^ dy ^ dz
*dx = dt ^ dy ^ dz
*dy =  dt ^ dx ^ dz
*dz = dt ^ dx ^ dy
*(dt ^ dx) = dy ^ dz
*(dt ^ dy) = dx ^ dz
*(dt ^ dz) = dx ^ dy
*(dx ^ dy) = dt ^ dz
*(dx ^ dz) = dt ^ dy
*(dy ^ dz) = dt ^ dx
With a simple division, the Schrödinger equation for a single particle of mass m in the absence of any applied force field can be rewritten in the following way:
This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation:
Applying this transformation to the expressions of the Green functions determined in the case of particle diffusion yields the Green functions of the Schrödinger equation, which in turn can be used to obtain the wavefunction at any time through an integral on the wavefunction at t=0:
Remark: this analogy between quantum mechanics and diffusion is a purely formal one. Physically, the evolution of the wavefunction satisfying Schrödinger's equation might have an origin other than diffusion.
Some examples of equivalent electrical and hydraulic equations:
type  hydraulic  electric  thermal 

quantity  volume V [m^{3}]  charge q [C]  heatQ [J] 
potential  pressure p [Pa=J/m^{3}]  potential φ [V=J/C]  temperature T [K=J/k_{B}] 
flux  current Φ_{V} [m^{3}/s]  current I [A=C/s]  heat transfer rate [J/s] 
flux density  velocity v [m/s]  j [C/(m^{2}·s) = A/m²]  heat flux [W/m^{2}] 
linear model  Poiseuille's law  Ohm's law  Fourier's law 
This remains a work in progress.
Please, send comments/corrections to John