New X'μ | = | 4-Tensor Antisymmetric Lorentz Transform Mμν Rotations j = Mab Boosts k = M0b = -Mb0 3 + 3 = 6 |
Original Xμ | + | 4-Vector SpaceTime Translation ΔXμ ~ e^Pμ Time Translation H = P0 Space Translation p = Pi 1 + 3 = 4 | ||||||||||||||||||||||||||||
| = e^ |
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| +e^ |
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Total of 6 + 4 = 10 parameters Poincaré Transform = Lorentz Transform + SpaceTime Translation Xμ' = Λμ'ν Xν + ΔXμ' with Λμ'ν = exp[1/2 ωαβMαβ]μ’ν ΔXμ' = exp[X∙P]μ’ where colors indicate
|
Translational Operator |
∂μ | Kμ | Pμ |
Equivalent | = ∂μ | = i∂μ | = iћ∂μ |
Normal Commutator |
[∂μ, Xν] = ημν | [Kμ,Xν] = iημν | [Pμ, Xν] = iћημν |
Reversed Commutator |
[Xν, ∂μ] = -ημν | [Xν, Kμ] = -iημν | [Xν, Pμ] = -iћημν |
Rotational Momentum Operator M |
Mμν |
Mμν |
Mμν | Dimensionless Rotational Operator O |
Oμν | Oμν | Oμν |
Equivalent | = iћ(Xμ∂ν - Xν∂μ) | = ћ(XμKν - XνKμ) | def. = (XμPν - XνPμ) |
Equivalent | def. = (Xμ∂ν - Xν∂μ) |
= (1/i)(XμKν - XνKμ) | = (1/iћ)(XμPν - XνPμ) |
Normal Commutator |
[Mμν, ∂ρ] = iћ(ηρν∂μ - ηρμ∂ν) |
[Mμν, Kρ] = iћ(ηρνKμ - ηρμKν) |
[Mμν, Pρ] = iћ(ηρνPμ - ηρμPν) |
Normal Commutator |
[Oμν, ∂ρ] = (ηρν∂μ - ηρμ∂ν) |
[Oμν, Kρ] = (1/i)(ηρνKμ - ηρμKν) |
[Oμν, Pρ] = (1/iћ)(ηρνPμ - ηρμPν) |
Reversed Commutator |
[∂ρ, Mμν] = -iћ(ηρν∂μ - ηρμ∂ν) = iћ(ηρμ∂ν - ηρν∂μ) |
[Kρ, Mμν] = -iћ(ηρνKμ - ηρμKν) = iћ(ηρμKν - ηρνKμ) |
[Pρ, Mμν] = -iћ(ηρνPμ - ηρμPν) = iћ(ηρμPν - ηρνPμ) | Reversed Commutator |
[∂ρ, Oμν] = -(ηρν∂μ - ηρμ∂ν) = (ηρμ∂ν - ηρν∂μ) | [Kρ, Oμν] = -(1/i)(ηρνKμ - ηρμKν) = (1/i)(ηρμKν - ηρνKμ) | [Pρ, Oμν] = -(1/iћ)(ηρνPμ - ηρμPν) = (1/iћ)(ηρμPν - ηρνPμ) |
Standard Model Elementary Particle |
Relativistic Wave Equations (RWE) | Relativistic Wave Equations (RWE) | Newtonian Limit ( |v| << c ) | |||
Particle Type | Spin | Statistics | Field | RQM Massless (mo = 0) | RQM Massive (mo > 0) | QM Massive (mo > 0) |
Fundamental | 0 | Boson | Lorentz Scalar ψ |
Scalar Wave (∂·∂)ψ = 0 |
Klein-Gordon Equation (∂·∂ + (moc/ћ)2)ψ = 0 |
Schrödinger Equation (iħ∂t)ψ ~ [(moc2) - (ħ∇)2/2mo]ψ |
Fundamental | 1/2 | Fermion | Spinor Ψ |
Weyl Equation [(iγμ∂μ)]Ψ = 0 → [(σμ∂μ)]Ψ = 0 |
Dirac Equation, Majorana Equation [(iγμ∂μ) - (moc/ћ)]Ψ = 0 (ΓμPμ)Ψ = (moc)Ψ iћ(Γμ∂μ)Ψ = (moc)Ψ |
Pauli Equation (iħ∂t)Ψ ~ [(moc2) + (σ·p)2/2mo]Ψ |
Fundamental | 1 | Boson | 4-Vector A |
Maxwell Equation (∂·∂)A = 0 |
Proca Equation (∂·∂ + (moc/ћ)2)A = 0 |
? |
Composites | 3/2 | Fermion | Spinor-Vector | Majorana Rarita-Schwinger | Rarita-Schwinger Equation | |
?? | 2 | Boson | (2,0)-Tensor | Graviton?? |
Standard Model Elementary Particle |
Relativistic Wave Equations (RWE) | Relativistic Wave Equations (RWE) | Newtonian Limit ( |v| << c ) | |||
Particle Type | Spin | Statistics | Field | RQM Massless (mo = 0) | RQM Massive (mo > 0) | QM Massive (mo > 0) |
Fundamental | 0 | Boson | Lorentz Scalar ψ |
Scalar Wave (D·D)ψ = 0 |
Klein-Gordon Equation (D·D + (moc/ћ)2)ψ = 0 |
Schrödinger Equation (iħ∂tT)ψ ~ [qφ + (moc2) + (-iħ∇T -qa)2/2mo]ψ (iħ∂tT)ψ ~ [V + (-iħ∇T -qa)2/2mo]ψ : with [V = qφ + (moc2)] |
Fundamental | 1/2 | Fermion | Spinor Ψ |
Weyl Equation ? |
Dirac Equation, Majorana Equation Γμ(Pμ-qAμ)Ψ = (moc)Ψ Γμ(iћ∂μ-qAμ)Ψ = (moc)Ψ | Pauli Equation (iħ∂tT)Ψ ~ [qφ + (moc2) + [σ·(pT -qa)]2/(2mo)]Ψ (iħ∂tT)Ψ ~ [qφ + (moc2) + ([(pT -qa)]2 - ћq[σ·B])/(2mo)]Ψ |
Fundamental | 1 | Boson | 4-Vector A |
Maxwell Equation (∂·∂)A = 0 (∂·∂)Aν = μoJν: Classical source (∂·∂)Aν = q(ψ̅ γν ψ): QED source |
Proca Equation | ? |
Composites | 3/2 | Fermion | Spinor-Vector | Majorana Rarita-Schwinger | Rarita-Schwinger Equation | |
?? | 2 | Boson | (2,0)-Tensor | Graviton?? |
Written" on the papers Einstein is holding: =================================== Rμν - (1/2)gμνR = κTμν (the theory of GR) eV = hν - A (the PhotoElectric Effect) E = mc2 (the Equivalence of Energy and Matter) =================================== |
Correlates to: ========== GR QM SR ========== |
V0 | V1 |
V2 |
V3 |
temporal part = V0 |
spatial part = Vi |
γ | -βxγ | 0 | 0 |
-βxγ | γ | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
cosh[ζ] | -sinh[ζ] | 0 | 0 |
-sinh[ζ] | cosh[ζ] | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
0 | cos[θ] | -sin[θ] | 0 |
0 | sin[θ] | cos[θ] | 0 |
0 | 0 | 0 | 1 |
γ | -βxγ | -βyγ | -βzγ |
-βxγ | 1 + (γ-1)(βx/β)2 | ( γ-1)(βxβy)/(β)2 | ( γ-1)(βxβz)/(β)2 |
-βyγ | ( γ-1)(βyβx)/(β)2 | 1 + ( γ-1)(βy/β)2 | ( γ-1)(βyβz)/(β)2 |
-βzγ | ( γ-1)(βzβx)/(β)2 | ( γ-1)(βzβy)/(β)2 | 1 + ( γ-1)(βz/β)2 |
Basis | 3-position Representation |
Lower Metric gij | Upper Metric gij | Line Element dx·dx = dxigijdxj = (dl)2 |
Euclidean Space Independent | x | ηij | ηij | (dl)2 = dx·dx = dxiηijdxj |
Euclidean Cartesian/Rectangular | x→(x,y,z) | ηij → Diag[+1,+1,+1] = δij = I |
ηij → Diag[+1,+1,+1] | (dl)2 = dx2 + dy2 + dz2 |
Euclidean Cylindrical/Polar | x→(r,θ,z) | ηij → Diag[+1,+r2,+1] | ηij → Diag[+1,+1/r2,+1] | (dl)2 = dr2 + r2dθ2 + dz2 |
Euclidean Spherical | x→(r,θ,φ) x→(r,{Ω}) |
ηij → Diag[+1,+r2,+(r·sin[θ])2] | ηij → Diag[+1,+1/r2,+1/(r·sin[θ])2 | (dl)2 = dr2 + r2dθ2 + (r·sin[θ])2dφ2 (dl)2 = dr2 + r2dΩ2 |
Basis | 4-Position Representation |
Lower Metric gμν | Upper Metric gμν | WorldLine Element dX·dX = dXμgμνdXν = (cdτ)2 |
Minkowski SpaceTime Independent | X | ημν | ημν | (cdτ)2 = dX·dX = dXμημνdXν |
Minkowski Time-Space | X→(ct,x) | ημν → Diag[+1,-I] | ημν → Diag[+1,-I] | (cdτ)2 = (cdt)2 - dx·dx |
Minkowski Cartesian/Rectangular | X→(ct,x,y,z) | ημν → Diag[+1,-1,-1,-1] | ημν → Diag[+1,-1,-1,-1] | (cdτ)2 = (cdt)2 - dx2 - dy2 - dz2 |
Minkowski Cylindrical/Polar | X→(ct,r,θ,z) | ημν → Diag[+1,-1,-r2,-1] | ημν → Diag[+1,-1,-1/r2,-1] | (cdτ)2 = (cdt)2 - dr2 - r2dθ2 - dz2 |
Minkowski Spherical | X→(ct,r,θ,φ) X→(ct,r,{Ω}) |
ημν → Diag[+1,-1,-r2,-(r·sin[θ])2] | ημν → Diag[+1,-1,-1/r2,-1/(r·sin[θ])2 | (cdτ)2 = (cdt)2 - dr2 - r2dθ2 - (r·sin[θ])2dφ2 (cdτ)2 = (cdt)2 - dr2 - r2dΩ2 |
others... | ||||
Newtonian Gravity Cartesian/Rectangular {weak gravity limiting-case |φ|<<1} {becomes Minkowski for φ→0} |
X→(ct,x,y,z) | gμν → Diag[+(1+2φ),-1,-1,-1] | gμν → Diag[+1/(1+2φ),-1,-1,-1] | (cdτ)2 = (1+2φ)(cdt)2 - dx2 - dy2 - dz2 |
Schwartzschild Spherical {becomes Minkowski for RS→0 or r→∞} |
X→(ct,r,θ,φ) | gμν → Diag[+(1-RS/r),-1/(1-RS/r),-r2,-r2sin(θ)] | gμν → Diag[+1/(1-RS/r),-(1-RS/r),-1/r2,-1/r2sin(θ)] | (cdτ)2 = (1-RS/r)(cdt)2 - 1/(1-RS/r)dr2 - r2dθ2 - (r·sin[θ])2dφ2 |
FLRW (or FRW) Spherical {assumes homogeneity & isotropy} a[t] is "scale factor" k is uniform curvature constant typically k={-,0,+} |
X→(ct,r,θ,φ) | gμν → Diag[+1,1/(a[t])2{-1/(1-kr2),-r2,-(r·sin[θ])2}] | gμν → Diag[+1,(a[t])2{-(1-kr2),-1/r2,-1/(r·sin[θ])2}] | (cdτ)2 = (cdt)2 - (a[t])2{1/(1-kr2)dr2 + r2dθ2 + (r·sin[θ])2dφ2} (cdτ)2 = (cdt)2 - (a[t])2{1/(1-kr2)dr2 + r2dΩ2} (cdτ)2 = (cdt)2 - (a[t])2{dΣ2} |
Tensor Type | Representation | Index Type | Upper Index Count |
Lower Index Count |
Alt Name | Further Definitions | |
(0,0)-Tensor | S | N/A | 0 | 0 | (Lorentz) (4-)Scalar | Invariant component | |
(1,0)-Tensor (0,1)-Tensor |
Vμ Vμ |
Contravariant Covariant |
1 0 |
0 1 |
4-Vector 4-Covector |
1 temporal, 3 spatial components | |
(2,0)-Tensor (1,1)-Tensor (0,2)-Tensor |
Tμν Tμν or Tμν Tμν |
Contravariant Mixed Covariant |
2 1 0 |
0 1 2 |
4-Tensor |
1 temporal, 9 spatial, 6 mixed time-space components Independent Components: Symmetric: 10 Anti-Symmetric: 6 Generic: 16 possible |
S |
V0 | V1 | V2 | V3 |
T00 | T01 | T02 | T03 |
T10 | T11 | T12 | T13 |
T20 | T21 | T22 | T23 |
T30 | T31 | T32 | T33 |
4-Acceleration A = (A0,Ai) = γ(cγ̇ , γ̇u + γu̇) = γ(cγ̇ , γ̇u + γa) | = ( γ4(a·u)/c , γ4(a·u)u/c2 + γ2a ) | = γ4( (a·u)/c , a + u x (a x u)/c2 ) |
= ( γ4(a·β) , γ4(a·β)β + γ2a ) | = γ4( (a·β) , a + β x (a x β) ) | |
= γ4( (a·β) , (a·β)β + a/γ2 ) |
Particle Count | Mass_Energy | (d/dτ)[Mass_Energy] | Entropy | EM Charge | WaveAngFreq | EM Potential | |
(Lorentz Scalar) <Potential> |
Ω = -X·U (free worldline) |
Sact = -X·P (free particle action) |
Φ = -X·K (free wave phase) |
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-d/dτ[<Potential>] <Charge>*c2 |
U·U = c2 |
Eo = U·P = -U·∂[S] = -d/dτ[S] Eo = moc2 |
ωo = U·K = -U·∂[Φ] = -d/dτ[Φ] ωo = (ωo/c2)c2 |
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<Charge> | N (usually 1) |
mo = (Eo/c2) | (d/dτ)[mo] | Sent | q | (ωo/c2) | (φo/c2) |
Particle 4-Vector <Charge>U |
U | P = -∂[S] P = moU = (Eo/c2)U |
F = (d/dτ)[mo]U + moA | Jq = qU |
K = -∂[Φ] K = (ωo/c2)U |
A = (φo/c2)U | |
Density 4-Vector Flux 4-Vector <Charge>N <ChargeDensity>U |
N = Uden = noU |
G = Pden = noP G = umoU = monoU = moN G = U·Tμν/c2 |
Fd = Fden = noF Fd = -∂·Tμν |
S = soU = SentnoU = SentN |
J = Jqden = noJq J = ρoU = qnoU = qN |
? = (ωo/c2)N | ? = (φo/c2)N |
<ChargeDensity> | no | umo = (ueo/c2) = nomo |
(d/dτ)[umo] |
so = noSEnt |
ρo = noq |
no(ωo/c2) | no(φo/c2) |
4-Divergence = 0 Conservation Law |
∂·N = 0 Conservation of Particle Count N |
∂·G = 0 Conservation of Mass_Energy mo |
∂·Fd = 0 Conservation of Power?? |
∂·S = 0 Conservation of Entropy Sent |
∂·J = 0 Conservation of Charge q |
∂·K = 0 Conservation of Wave_Freq? |
∂·A = 0 Conservation of EM Potential (Lorenz Gauge) |
4-WaveVector 4-AngularWaveVector |
K = (ω/c,k) = (ω/c,n̂ω/vphase) = (ω/c,ωu/c2) = (ω/c)(1,β) = (1/c |
Atomic # | 1 | 2 |
Element | H | He |
Electron Config |
1s1 | 1s2 |
Orbital Added |
1st↑ ~ +t | 1st↓ ~ -t |
Atomic # | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Element | Li | Be | B | C | N | O | F | Ne |
Electron Config |
[He]2s1 | [He]2s2 | [He]2s22p1 | [He]2s22p2 | [He]2s22p3 | [He]2s22p4 | [He]2s22p5 | [He]2s22p6 |
Orbital Added |
2st↑ ~ +t | 2st↓ ~ -t | 2px↑ ~ +x | 2px↓ ~ -x | 2py↑ ~ +y | 2py↓ ~ -y | 2pz↑ ~ +z | 2pz↓ ~ -z |
Alkali Metals Group 1 S-Block |
Alkaline Earth Metals Group 2 S-Block |
Icosagens Group 13 P-Block |
Crystallogens Group 14 P-Block |
Pnictogens Group 15 P-Block |
Chaocogens Group 16 P-Block |
Halogens Group 17 P-Block |
Aerogens - Noble Gases Group 18 P-Block |
|
Atomic # | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
Element | Na | Mg | Al | Si | P | S | Cl | Ar |
Electron Config |
[Ne]3s1 | [Ne]3s2 | [Ne]3s23p1 | [Ne]3s23p2 | [Ne]3s23p3 | [Ne]3s23p4 | [Ne]3s23p5 | [Ne]3s23p6 |
Orbital Added |
3st↑ ~ +t | 3st↓ ~ -t | 3px↑ ~ +x | 3px↓ ~ -x | 3py↑ ~ +y | 3py↓ ~ -y | 3pz↑ ~ +z | 3pz↓ ~ -z |
∂[τ] = ∂[X·U/c2] = U/c2 = U/(U·U)
An interesting relation, which can find some use in the Hamilton-Jacobi relation and the relativistic Action
=============================
Event R | Mass mo = ρmoVo Energy Eo = moc2 |
MassDensity ρmo = nomo EnergyDensity ueo = ρmoc2 |
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Derivative of 4-Position |
dnR/dτn | Event 4-Vector |
particle | density |
0th | R d0R/dτ0 |
pos: R = (ct,r) | mo at R | ρmo at R |
1st | dR/dτ d1R/dτ1 |
vel:U = dR/dτ | P = modR/dτ P = moU = (Eo/c2)U |
G = ρmodR/dτ G = ρmoU = (ueo/c2)U |
2nd | d2R/dτ2 | accel: A = dU/dτ | F = dP/dτ | Fd = dG/dτ |
3rd | d3R/dτ3 | jerk: J = dA/dτ jolt, surge, lurch: alt names |
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4th | d4R/dτ4 | snap: S = dJ/dτ jounce: alt name |
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5th | d5R/dτ5 | crackle: C = dS/dτ | ||
6th | d6R/dτ6 | pop: P = dC/dτ |
U1·U2 = γ12(c2) = γrel(c2) | U·U = (c)2 |
T1·T2 = γ12 = γrel | T·T = 1 |
Particle Count | Mass_Energy | (d/dτ)[Mass_Energy] | Entropy | EM Charge | WaveAngFreq | EM Potential | |
(Lorentz Scalar) <Potential> |
Ω = -X·U (free worldline) |
Sact = -X·P (free particle action) |
Φ = -X·K (free wave phase) |
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-d/dτ[<Potential>] <Charge>*c2 |
U·U = c2 | Eo = U·P = -U·∂[S] = -d/dτ[S] Eo = moc2 |
ωo = U·K = -U·∂[Φ] = -d/dτ[Φ] ωo = (ωo/c2)c2 |
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<Charge> | N (usually 1) | mo = (Eo/c2) | (d/dτ)[mo] | Sent | q | (ωo/c2) | (φo/c2) |
Particle 4-Vector <Charge>U |
U | P = -∂[S] P = moU = (Eo/c2)U |
F = (d/dτ)[mo]U + moA | Jq = qU | K = -∂[Φ] K = (ωo/c2)U |
A = (φo/c2)U | |
Density 4-Vector Flux 4-Vector <Charge>N <ChargeDensity>U |
N = Uden = noU |
G = Pden = noP G = umoU = monoU = moN G = U·Tμν/c2 |
Fd = Fden = noF Fd = -∂·Tμν |
S = soU = SentnoU = SentN | J = Jqden = noJq J = ρoU = qnoU = qN | ? = (ωo/c2)N | ? = (φo/c2)N |
<ChargeDensity> | no | umo = (ueo/c2) = nomo | (d/dτ)[umo] | so = noSEnt |
ρo = noq |
no(ωo/c2) | no(φo/c2) |
4-Divergence = 0 Conservation Law |
∂·N = 0 Conservation of Particle Count N | ∂·G = 0 Conservation of Mass_Energy mo | ∂·Fd = 0 Conservation of Power?? | ∂·S = 0 Conservation of Entropy Sent | ∂·J = 0 Conservation of Charge q | ∂·K = 0 Conservation of Wave_Freq? | ∂·A = 0 Conservation of EM Potential (Lorenz Gauge) |
Traditional Style | Projection Tensor Style |
Tperfectfluidμν = (ρeo + po)UμUν/c2 - poημν | Tperfectfluidμν = (ρeo)Vμν - (po)Hμν |
Contract with the 4-Velocity | Contract with the 4-Velocity |
TμνUν = (ρmo + po/c2)UμUνUν - poημνUν | TμνUν = (ρeo)VμνUν - (po)HμνUν |
TμνUν = (ρmo + po/c2)Uμc2 - poUμ | TμνUν = (ρeo)Uμ - (po)(0μ) |
TμνUν = (c2ρmo + po)Uμ - poUμ | TμνUν = (ρeo)Uμ |
TμνUν = c2ρmoUμ | TμνUν = c2ρmoUμ |
TμνUν = c2Gμ = c2G | TμνUν = c2Gμ = c2G |
c2ρmo = ρeo = (1/2)εo(e2 + c2b2) | cg = cεo(e x b) |
cg = cεo(e x b) | σij = -εo[eiej + c2bibj - (1/2)δij(e2 + c2b2)] |
c2ρmo = ρeo = (1/2)εo(e2 + c2b2) | cg = cεo(e x b) |
cg = cεo(e x b) | σij = -εo[eiej + c2bibj - (1/2)δij(e2 + c2b2)] |
Invariants | P = Pressure = Po | N = ParticleNum = No | S = Entropy = So |
Variables | V = Volume = (1/γ)Vo | μ = ChemPoten = (1/γ)μo | T = Temperature = (1/γ)To |
γ | [Ė/c] | = d/dτ | [E/c] | = qγ | [c] | · | [0 | -ei/c] | = qγ | [c*0 + u·e/c] |
[ f ] | [p] | [u] | [ei/c | -εijkbk] | [e + u⨯b] |
0 | -ex/c | -ey/c | -ez/c |
ex/c | 0 | -bz | by |
ey/c | bz | 0 | -bx |
ez/c | -by | bx | 0 |
0 | -ei/c |
+ej/c | -εijkbk |
0 | +ei/c |
-ej/c | -εijkbk |
Faraday Electromagnetic Tensor | Magnetization-Polarization Tensor | Electromagnetic Displacement Tensor | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Fαβ = (∂αAEMβ - ∂βAEMα) |
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Fαβ =
| Mαβ =
| Dαβ
=
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
e = ei = electric field b = bk = magnetic field | p = pi = electric polarization (polarization) m = mk = magnetic polarization (magnetization) |
d = di = electric displacement field h = hk = auxiliary magnetic field | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Fαβ:SI Units [T] = [kg/A·s2] = [kg/C·s] e: SI Units [kg·m/A·s3] = [kg·m/C·s2] e/c: SI Units [kg/C·s] b: SI Units [T] = [kg/A·s2] = [kg/C·s] | Mαβ: SI Units [C/m·s] p: SI Units [C/m2] pc: SI Units [C/m·s] m: SI Units [C/m·s] | Dαβ: SI Units [C/m·s] d: SI Units [C/m2] dc: SI Units [C/m·s] h: SI Units [C/m·s] | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
∂β Fαβ = μo Jα | ∂β Mαβ = Jboundα | ∂β Dαβ = Jfreeα |
Time-Space Components | Purely Spatial Components |
(-dc) = (1/μo)(-e/c) - (pc) (-dc) = (c2εo)(-e/c) - (pc) (-dc) = (εo)(-ec) - (pc) (-d) = (εo)(-e) - (p) (d) = (εo)(e) + (p) d = εoe + p | (h) = (1/μo)(b) - (m) h = (1/μo)b - m b = μo(h + m) |
0 | dx/c | dy/c | dz/c |
-dx/c | 0 | -μz/c | μy/c |
-dy/c | μz/c | 0 | -μx/c |
-dz/c | -μy/c | μx/c | 0 |
0 | +di/c |
-dj/c | -εijkμk |
Maxwell Eqn | ∂α(∂αAEMν - ∂νAEMα) = μoJν | ∂·Fαν = (μo)J | Divergence of Faraday EM Tensor |
Lorentz Force Eqn | Uα(∂νAEMα - ∂αAEMν) = (1/q)Fν | U·Fαν = (-1/q)F | Invariant Temporal Component of Faraday EM Tensor |
Hamiltonian | Lagrangian | Combo |
H | L | H + L = pT·u |
γ(PT·U) | -(PT·U)/γ | γ(PT·U)β2 |
γHo | Lo/γ | γHo + Lo/γ |
Rest Hamiltonian |
Rest Lagrangian |
Rest Combo = 0 |
Ho | Lo | Ho + Lo = 0 = pT·0 |
(PT·U) | -(PT·U) | (1)(PT·U)(0)2 = 0 |
H/γ | γL | 0 |
Projection Tensor | Alt Name/Mnemonic | Definition / Representation | 4-Divergence | Trace |
SpaceTime ημν | "(n)ow,here" Worldline Event |
ημν = ∂μ[Xν] = Vμν + Hμν → Diag[1,-1,-1,-1] | ∂μημν = 0ν | Tr[ημν] = 4 |
Temporal Vμν | "(V)ertical" Worldline Tangent |
Vμν = TμTν → Diag[1,0,0,0] | ∂νVμν = = ∂νTμTν = Tν∂ν[Tμ] + Tμ∂ν[Tν] = (d/cdτ)[Tμ] + Tμ(∂·T) = Aμ/c2 + Tμ(∂·T) |
Tr[Vμν] = 1 |
Spatial Hμν | "(H)orizontal" Worldline Normal Hyperplanes orthogonal to Worldline |
Hμν = ημν - TμTν → Diag[0,-1,-1,-1] | ∂μHμν = -∂μVμν |
Tr[Hμν] = 3 |
Null Nμν | "(N)ull" LightPath |
Nμν = NμNν = Vμν - (1/3)Hμν → Diag[1,1/3,1/3,1/3] | ∂μNμν = = ∂μ(Vμν - (1/3)Hμν) = ∂μ(Vμν + (1/3)Vμν) = (4/3)∂μVμν |
Tr[Nμν] = 0 |
Projection N̂μν |
(P)rojection to Hyperplanes orthogonal to N̂ |
N̂μν = ημν - N̂μN̂ν → ?? |
T00 | T01 | T02 | T03 |
T10 | T11 | T12 | T13 |
T20 | T21 | T22 | T23 |
T30 | T31 | T32 | T33 |
Aμ = A∥μ + A⟂μ ημν = Vμν + Hμν ημν = TμTν + SμSν |
SpaceTime 4-Vector Aμ |
Temporal 4-Vector A∥μ |
Spatial 4-Vector A⟂μ |
SpaceTime Projection ημν | ημνAν = Aμ | ημνA∥ν = A∥μ | ημνA⟂ν = A⟂μ |
Temporal Projection Vμν | VμνAν = A∥μ = Tμ(T·A) | VμνA∥ν = A∥μ | VμνA⟂ν = 0μ |
Spatial Projection Hμν | HμνAν = A⟂μ = Sμ(S·A) | HμνA∥ν = 0μ | HμνA⟂ν = A⟂μ |
Null Projection Nμν | NμνAν = A∠μ = Nμ(N·A) |
ημν = Vμν + Hμν | SpaceTime Projection "(n)ow" ημν |
Temporal Projection "(V)ertical" Vμν |
Spatial Projection "(H)orizontal" Hμν = ημν - Vμν |
Null Projection "(N)ull" Nμν = Vμν - (1/3)Hμν |
SpaceTime Tensor ημν | Tr[ημν] = ημνημν = 4 | Vμνημν = 1 | Hμνημν = 3 | Nμνημν = 0 |
Temporal Tensor Vμν | Tr[Vμν] = ημνVμν = 1 | VμνVμν = 1 | HμνVμν = 0 | NμνVμν = 1 |
Spatial Tensor Hμν | Tr[Hμν] = ημνHμν = 3 | VμνHμν = 0 | HμνHμν = 3 | NμνHμν = -1 |
Null Tensor Nμν | Tr[Nμν] = ημνNμν = 0 | VμνNμν = 1 | HμνNμν = -1 | NμνNμν = 4/3? Related I believe to the 4/3 problem of Electromagnetic Mass |
Tensor Form | AμημνBν | AμVμνBν | AμHμνBν |
4-Vector Form | A·B | (A·B)∥ | (A·B)⟂ |
Component Form | a0b0 - a·b | a0b0 | -a·b |
SpaceTime Projection | Tr[ημαηνβ] = ημαημνηνβ = ηαβ |
Temporal Projection | Tr[VμαVνβ] = VμαημνVνβ = Vαβ |
Spatial Projection | Tr[HμαHνβ] = HμαημνHνβ = Hαβ |
Energy Density ρmc2 = ρe time-time T00
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Energy Flux/c s/c = cg time-space T0j
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Momentum Density*c cg = s/c space-time Ti0
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Momentum Flux = Spatial Stress -σij space-space Tij
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ρeo | sx/c | sy/c | sz/c |
sx/c | -σxx | -σxy | -σxz |
sy/c | -σyx | -σyy | -σyz |
sz/c | -σzx | -σzy | -σzz |
γ2(ρeo + po) - po | γ2(ρeo + po)ux/c | γ2(ρeo + po)uy/c | γ2(ρeo + po)uz/c |
γ2(ρeo + po)ux/c | γ2(ρeo + po)uxux/c2 + po | γ2(ρeo + po)uxuy/c2 | γ2(ρeo + po)uxuz/c2 |
γ2(ρeo + po)uy/c | γ2(ρeo + po)uxuy/c2 | γ2(ρeo + po)uyuy/c2 + po | γ2(ρeo + po)uyuz/c2 |
γ2(ρeo + po)uz/c | γ2(ρeo + po)uxuz/c2 | γ2(ρeo + po)uyuz/c2 | γ2(ρeo + po)uzuz/c2 + po |
ρeo | |||
po | |||
po | |||
po |
T00 = ρeo | T0j = 0 |
Ti0 = 0 | Tij = poδij |
Perfect Fluid Stress-Energy Tensor Tμν = (ρeo)Vμν - (po)Hμν |
Invariant Tr[Tμν] = (ρeo - 3po) |
Pressure po = wρeo = wρmoc2 |
Eqn. of State (EoS )Parm w = (po/ρeo) = (n/3)-1 |
n = 3(w + 1) |
Energy Density ρeo falls off as a-n = a-3(w + 1) |
a[t] = a0t2/n | Cosmological Solution eg. Matter Dominated, Radiation Dominated, etc. |
Gravitational Pressure | Speed of Sound cs = (speed of sound) = c * Sqrt[w]? |
Dimensional Type |
< -2ρeo | > ρeo | >1 | >6 | Speed of Sound>Speed of Light = Unphysical | ||||||
(ρeo)Vμν - (3ρeo/3)Hμν | -2ρeo | ρeo | 1 | 6 | ρeo ∝ a-6 | ~Stiff Equation of State (Neutron Stars) ?? | ||||
(ρeo)Vμν - (2ρeo/3)Hμν | -1ρeo | 2ρeo/3 | 2/3 | 5 | ρeo ∝ a-5 | ??? | ||||
< 0ρeo | > ρeo/3 | > 1/3 | >4 | "Ultralight", meaning ultra-photonic... | Unknown | |||||
(ρeo)Vμν - (1ρeo/3)Hμν = Null-Dust = (ρeo)Nμν = (po)(4Vμν - ημν) |
0ρeo | ρeo/3 | 1/3 | 4 | ρR = ρeo ∝ a-4 | a[t] ∝ t1/2 | Radiation/~Ultra-Relativistic Matter/Soft Equation of State Null-Dust/Photon Gas/Hot Dust/Relativistic Neutrinos Tμν = (ρeo)Nμν |
Relativistic Point |
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(ρeo)Vμν - ((v/c)2ρeo/3)Hμν | [1-(v/c)2]ρeo = (γ-2)ρeo |
(v/c)2ρeo/3 = v2ρmo/3 = ρmoRT |
{0..1/3} | {3..4} | Perfect Gas (|v|<<c) = Warm Dust v = vth = √[3RT] = √[3KBT/m] = {0..c} = characteristic rms 3D thermal speed of molecules essentially this smoothly varies from Matter-Dust (v~0) to Null-Dust (v~c) |
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(ρeo)Vμν - (0ρeo/3)Hμν = Matter-Dust = (ρeo)Vμν |
1ρeo | 0 | 0 | 3 | ρM = ρeo ∝ a-3 | a[t] ∝ t2/3 | (Cold) Dust = (Incoherent) Matter/CDM/Baryons Normal Matter Einstein-de Sitter (EdS) solution Tμν = (ρeo)Vμν |
Point | ||
(ρeo)Vμν - (-1ρeo/3)Hμν | 2ρeo | -ρeo/3 | -1/3 | 2 | ρeo ∝ a-2 | a[t] ∝ t | Curvature = Einstein Static Universe/?Cosmic Strings? | Replusive? | Imaginary, Instabilities | Line |
> 2ρeo | < -ρeo/3 | < -1/3 | < 2 | Everything Below has Accelerating Expansion of Universe | Repulsive | Imaginary, Instabilities | ||||
(ρeo)Vμν - (-2ρeo/3)Hμν | 3ρeo | -2ρeo/3 | -2/3 | 1 | ρeo ∝ a-1 | a[t] ∝ t2 | ??? ?Domain Walls? | Replusive | Imaginary, Instabilities | Sheet |
(ρeo)Vμν - (-3ρeo/3)Hμν = Vacuum Energy = (ρeo)ημν = -(po)ημν |
4ρeo | -ρeo | -1 | 0 | ρΛ = ρeo ∝ a0 = constant |
a[t] ∝ eHt with H = Hubble Const |
(Quantum) Vacuum Energy/Dark Energy /Cosmological Constant Λ/de Sitter/(Inflation~ -1) Tμν = (ρeo)ημν |
Replusive | Imaginary, Instabilities | Volume |
> 4ρeo | < -ρeo | < -1 | < 0 | Big Rip = Phantom Energy | Replusive | Imaginary, Instabilities Speed of Sound>Speed of Light = Unphysical |
Unknown |
ρeo | |||
p = 0 | |||
p = 0 | |||
p = 0 |
ρeo | |||
p = -ρeo | |||
p = -ρeo | |||
p = -ρeo |
ρeo | |||
p = ρeo/3 | |||
p = ρeo/3 | |||
p = ρeo/3 |
c2ρmo = ρeo = = (1/2)εo(e2 + c2b2) = (1/2)(εoe2 + b2/μo) |
c g = sj/c = cεo(e x b) = (e x b)/(cμo) |
c g = si/c = cεo(e x b) = (e x b)/(cμo) |
-σij = -εo[eiej + c2bibj - (1/2)δij(e2 + c2b2)] = -[εoeiej + bibj/μo - (1/2)δij(εoe2 + c2b2/μo)] = The Maxwell Stress Tensor |
c2ρmo = ρeo | c g = sj/c |
c g = si/c | -σij = -εo[eiej + c2bibj - (1/2)δij(e2 + c2b2)] |
Temporal Projection | Spatial Projection |
Vσμ∂νTμν = Vσμ( ∂ν[ρeo]Vμν + (ρeo + po)∂ν[Vμν] - ∂ν[po]Hμν ) ∂ν[ρeo]VσμVμν + (ρeo + po)Vσμ∂ν[Vμν] - ∂ν[po]VσμHμν ∂ν[ρeo]Vσν + (ρeo + po)T∥σ(∂·T) - ∂ν[po](0) T∥σTν∂ν[ρeo] + (ρeo + po)T∥σ(∂·T) T∥σ[(d/cdτ)[ρeo] + (ρeo + po)(∂·T)] T∥σ[∂·[ρeoT] + (po)(∂·T)] T∥σ[∂·[ρeoU] + (po)(∂·U)]/c γ[∂ν[ρeoUν] + (po)(∂νUν)]/c |
Hσμ∂νTμν = Hσμ( ∂ν[ρeo]Vμν + (ρeo + po)∂ν[Vμν] - ∂ν[po]Hμν ) ∂ν[ρeo]HσμVμν + (ρeo + po)Hσμ∂ν[Vμν] - ∂ν[po]HσμHμν ∂ν[ρeo](0) + (ρeo + po)(A⊥σ/c2) - ∂ν[po]Hσν (ρeo + po)(A⊥σ/c2) - ∂⊥σ[po] ((ρeo + po)/c2)γ(cγ̇,γ̇u + γu̇)⊥ - (∂t/c, -∇)⊥[po] ((ρeo + po)/c2)γ(γ̇u + γu̇) - (-∇)[po] γ((ρeo + po)/c2)(γ̇u + γu̇) + ∇[po] |
Cool-Warm Dust Condition (po) << (ρeo) Uμ∂νTμν = Temporal Component ∂·(ρeoU) (∂t[γρeo] + ∇·[γρeou]) (∂t[ρe] + ∇·[ρeu]) = 0 if conserved |
Cool-Warm Dust Condition (po) << (ρeo) Hσμ∂νTμν = Spatial Components γ((ρeo)/c2)(γ̇u + γu̇) + ∇[po] (ρm)(γ̇u + γu̇) + ∇[po] = 0 if conserved |
Newtonian Limit: |u| << c (∂t[ρe] - ∇·[ρeu]) Same as Warm Dust = 0 if conserved |
Newtonian Limit: |u| << c, γ→1, γ̇→0 (ρm)(u̇) + ∇[po] (ρm)(a) + ∇[po] = = 0 if conserved |
A = Aμ | hTTμν = kAμAν = kAμ⊗Aν |
(∂·A) = (∂μAμ) = 0 | (∂·hTTμν) = (∂νhTTμν) = (∂νkAμAν) = k(∂νAν)Aμ = k(0)Aμ = 0μ |
(U·A) = (UνAν) = 0 for a photonic wave Generally AEM·U = (φ/c,a)·γ(c,u) = γ(φ - a·u) = φo As we will see, this is a photonic wave and the rest potential φo will be zero in the same way that the rest mass mo of a photon is zero In other words: There is no "at-rest" frame for light-like (U·A) = 0 = γ(c,u)·(φ/c,a) = γ[φ - u·a] = 0 Therefore, φ = u·a Therefore A = (u·a/c,a) To an at-rest observer (u=0), A appears spatial A → (0,a) To an â-null observer (u=câ), A appears null A → (|a|,a) A 3rd situation is that u·a=0 via 3D orthogonality, in which case A appears spatial A → (0,a) | (U·hTTμν) = (UνhTTμν) = (UνkAμAν) = k(UνAν)Aμ = k(0)Aμ = 0μ |
Generally A·A = (AμημνAν) = (φ/c,a)·(φ/c,a) = (φ/c)2 - a·a From above, A = (u·a/c,a) A·A = (u·a/c,a)·(u·a/c,a) = [(u·a/c)2 - a·a] To an at-rest observer (u=0), A·A = (- a·a) appears spatial To an â-null observer (u=câ), A·A = (0) appears null A 3rd situation is that u·a=0 via 3D orthogonality, in which case A appears spatial A → (0,a) and A·A = (- a·a) appears spatial | Tr[ hTTμν ] = (ημνhTTμν) = k(ημνAμAν) = k(AμημνAν) = k[(u·a/c)2 - a·a] If indeed (hTTμν = kAμAν = kAμ⊗Aν) decomposes this way, we should get Tr[ hTTμν ] = 0μ for the â-null observer Let me think on this a bit... |
*Lorentz Scalar* <Potential> | 4-Vector = -Gradient[<Potential>] | Rest value Temporal Component | |
SR Phase (Φ) | Φ = -(K·R) | 4-WaveVector K = -∂[Φ] | U·K = -U·∂[Φ] = -d/dτ[Φ] = ωo |
SR TotalPhase (ΦT) | ΦT = -(KT·R) | 4-TotalWaveVector KT = -∂[ΦT] | U·KT = -U·∂[ΦT] = -d/dτ[ΦT] = ωTo |
SR Action (Sact) | Sact = -(PT·R) | 4-TotalMomentum PT = -∂[Sact] | U·PT = -U·∂[Sact] = -d/dτ[Sact] = Ho |
U·PTden = -Ldeno = no(PT·U) = Hdeno | |||
SR Stress-Energy (Tμν) | eg. Tperfectfluidμν = (ρeo)Vμν - (po)Hμν | 4-ForceDensity Fdenμ = -∂ν[Tμν] | UμFdenμ = -Uμ∂ν[Tμν] = γfĖo = γfṁoc2 |
The magic behind the EM curtain... | ∂ν[PTμ] = q∂μ[AEMν] |
0 | -ex/c | -ey/c | -ez/c |
ex/c | 0 | -bz | by |
ey/c | bz | 0 | -bx |
ez/c | -by | bx | 0 |
0 | -ei/c |
+ej/c | -εijkbk |
∂X = ±iK |
[∂X,X] = ημν = [∂K,K] [∂X,X] = [∂K,K] [∂X,X] = -[K,∂K] [±iK,X] = -[K,∂K] ±i[K,X] = -[K,∂K] ±i2[K,X] = -i[K,∂K] ±(-1)[K,X] = -i[K,∂K] ±[K,X] = i[K,∂K] [K,X] = ±i[K,∂K] [K,X] = [K,±i∂K] |
X = ±i∂K ∂K = ∓iX |
*Relativistic* P = (E/c,p) = ħK = iħ∂ = iћ(∂t/c,-∇) |
*Classical* = limiting-case using { √[1 + x] ~ (1 + x/2 + ...O[x2] ) } |
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4-Momentum | Einstein Energy Relation P·P = (E/c)2 - p2 = (moc)2 |
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solved for temporal component | E = √[(moc2)2 + c2p2] | Newtonian Energy Relation E ~ [(moc2) + p2/2mo] |
4-Gradient | Free Particle Klein-Gordon RQM Equation ∂·∂ = (∂t/c)2 -∇2 = (-imoc/ћ)2 iћ∂·iћ∂ = (iћ∂t/c)2 - (-iћ∇)2 = (moc)2 |
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solved for temporal component | (iћ∂t) = √[(moc2)2 + c2(-iћ∇)2] (iћ∂t) = √[(moc2)2 - c2(ћ∇)2] |
Free Particle Schrödinger QM Equation (iħ∂t) ~ [(moc2) + (-iħ∇)2/2mo] (iħ∂t) ~ [(moc2) - (ħ∇)2/2mo] |
*Relativistic* P = (E/c,p) = ħK = iħ∂ = iћ(∂t/c,-∇) PT = P + qA |
*Classical* = limiting-case using { √[1 + x] ~ (1 + x/2 + ...O[x2] ) } |
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4-Momentum | Einstein Energy Relation P·P = (E/c)2 - p2 = (moc)2 = (ET/c - qφ/c)2 - (pT -qa)2 = (moc)2 |
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solved for temporal component | E = √[(moc2)2 + c2p2] (ET -qφ) = √[(moc2)2 + c2(pT -qa)2] |
Newtonian Energy Relation E ~ [(moc2) + p2/2mo] (ET -qφ) ~ [(moc2) + (pT -qa)2/2mo] |
4-Gradient | Free Particle Klein-Gordon RQM Equation ∂·∂ = (∂t/c)2 -∇2 = (-imoc/ћ)2 iћ∂·iћ∂ = (iћ∂t/c)2 - (-iћ∇)2 = (moc)2 Klein-Gordon RQM Equation w/Potential (iħ∂tT -qφ)2 = (moc2)2 + c2(-iħ∇T -qa)2: |
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solved for temporal component | (iћ∂t) = √[(moc2)2 + c2(-iћ∇)2] (iћ∂t) = √[(moc2)2 - c2(ћ∇)2] (iħ∂tT -qφ) = √[(moc2)2 + c2(-iħ∇T -qa)2] (iħ∂tT) = qφ + √[(moc2)2 + c2(-iħ∇T -qa)2] |
Free Particle Schrödinger QM Equation (iħ∂t) ~ [(moc2) + (-iħ∇)2/2mo] (iħ∂t) ~ [(moc2) - (ħ∇)2/2mo] Schrödinger QM Equation w/Potential (iħ∂tT -qφ) ~ [(moc2) + (-iħ∇T -qa)2/2mo] (iħ∂tT) ~ [qφ + (moc2) + (-iħ∇T -qa)2/2mo] (iħ∂tT) ~ [V + (-iħ∇T -qa)2/2mo] : with [V = qφ + (moc2)] (iħ∂tT) ~ [V - (ħ∇T)2/2mo]: with a = 0 the Standard way it is usually seen |
(∂·∂ + (moc/ћ)2 )Ψ = 0 | Ψ is a scalar, Klein-Gordon eqn for massive spin-0 field, ex. the Higgs Boson |
(∂·∂ + (moc/ћ)2 )A = 0 | A is a 4-Vector, Proca eqn for massive spin-1 field, Lorenz Gauge |
(∂·∂)Ψ = 0 | Ψ is a scalar, Free-wave eqn for massless (mo = 0) spin-0 field |
(∂·∂)A = 0 | A is a 4-Vector, Maxwell eqn for massless (mo = 0) spin-1 field, no current sources, Lorenz Gauge |
(∂·∂)A = μoJ = ρoμoU = qnoμoU = qμoN | A is a 4-Vector, Maxwell eqn for massless (mo =
0) spin-1 field, current source J, Lorenz Gauge Classical EM, does not include effects of particle spin in the current source J |
(∂·∂)A = μoJ = μo(qΨ ̅ γΨ) (∂·∂)Aμ = μoJ = μo(qΨ ̅ γμΨ) where Ψ ̅ γμΨ has units of flux (#/m2·s) |
QED, A is a 4-Vector, Maxwell eqn for massless (mo =
0) spin-1 field, current source J, Lorenz Gauge Quantum EM, does include effects of particle spin in the current source J = Ψ ̅ γμΨ |
Just a note: The classical Maxwell EM equations do not have Spin included (∂·∂)AEM = μoJ = μoρoU = μoqnoU = μoqN = μo(q/Vo)U = μoq(c/Vo)T Once spin is included, the equations for QED emerge: (∂·∂)AEM = μoqψ Γψ not sure if the μo factor is included or not |
Full Equation (un-gauged) | Lorenz Gauge (∂·A = 0) | Field Type |
(∂·∂ + (moc/ћ)2 )Ψ = 0 | Ψ is a scalar, Klein-Gordon eqn for massive spin-0 field | |
(∂·∂ + (moc/ћ)2 )A = 0 | A is a 4-Vector, Proca eqn for massive spin-1 field | |
(∂·∂)Ψ = 0 | Ψ is a scalar, Free-wave eqn for massless (mo = 0) spin-0 field | |
∂νFνμ = 0μ ∂ν(∂νAμ - ∂μAν) = 0μ ∂ν∂νAμ - ∂μ∂νAν = 0μ (∂·∂)Aμ - ∂μ(∂·A) = 0μ |
(∂·∂)A = Z (∂·∂)Aμ = 0μ |
A is a 4-Vector, Maxwell eqn for massless (mo = 0) spin-1 field, no current sources |
(∂·∂)A = μoJ = ρoμoU = qnoμoU = qμoN | A is a 4-Vector, Maxwell eqn for massless (mo = 0) spin-1 field, current source J | |
∂νFνμ = qΨ ̅ γμΨ ∂ν(∂νAμ - ∂μAν) = qΨ ̅ γμΨ ∂ν∂νAμ - ∂μ∂νAν = qΨ ̅ γμΨ (∂·∂)Aμ - ∂μ(∂·A) = qΨ ̅ γμΨ |
(∂·∂)A = qΨ ̅ γΨ (∂·∂)Aμ = qΨ ̅ γμΨ |
QED, A is a 4-Vector, spin-1 field, current source Ψ ̅ γμΨ |
(∂·∂)hTTμν = 0μν | Gravitational Waves, hTTμν is a (T)ranverse (T)raceless 2-Tensor representing gravitational radiation in the weak-field limit far from the source |
∂X = ±iK |
[∂X,X] = ημν = [∂K,K] [∂X,X] = [∂K,K] [∂X,X] = -[K,∂K] [±i K,X] = -[K,∂K] ±i[K,X] = -[K,∂K] ±i2[K,X] = -i[K,∂K] ±(-1)[K,X] = -i[K,∂K] ±[K,X] = i[K,∂K] [K,X] = ±i[K,∂K] [K,X] = [K,±i∂K] |
X = ±i∂K ∂K = ∓iX |
dX/dτ = (U·∂)[X] = U | dX/dθ = (K·∂)[X] = K |
U·U = c2 U1·U2 = (γ12)c2 U·Uo = (γrel)c2 |
K·K = (ωo/c)2 |
∂[U·U] = 2*U·∂[U] = ∂[c2] = Z ∂[U1·U2] = U1·∂[U2] +U2·∂[U1] = ∂[(γ12)c2] = c2 ∂[γ12] ∂[U·Uo] = U·∂[Uo] +Uo·∂[U] = (0) +Uo·∂[U] = ∂[(γrel)c2] = c2 ∂[γrel] |
∂[K·K] = 2* K·∂[K] = ∂[(ωo/c)2] = Z, if ωo is constant |
d/dτ[U·U] = 2*U·d/dτ[U] = 2*U·A = d/dτ[c2] = 0 d/dτ[U1·U2] = U1·d/dτ[U2] +U2·d/dτ[U1] = U1·A2 +U2·A1 = d/dτ[(γ12)c2] = c2d/dτ[γ12] d/dτ[U·Uo] = U·d/dτ[Uo] +Uo·d/dτ[U] = (0) +Uo·A = d/dτ[(γrel)c2] = c2d/dτ[γrel] |
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d2 X/dτ2 = dU/dτ = ? = (U·∂)[U] = U·∂[U] = Z but should be A instead try d2 X/dτ2 = dU/dτ = (Uo·∂)[U] = Uo·∂[U] = A = ? = c2 ∂[γrel] |
d2 X/dθ2 = dK/dθ = (K·∂)[K] = K·∂[K] = 0 ? |
R = (ct,r) | particle/location |
U = dR/dτ | movement/velocity |
P = moU | mass/momentum |
K = (1/ћ)P | wave/particle duality |
∂ = -iK | SpaceTime/wave structure |
∂t2/c2 = ∇·∇ - (moc/ћ)2 |
(∂·∂ + (moc/ћ)2 )Ψ = 0 | Ψ is a scalar, Klein-Gordon eqn for massive spin-0 field |
(∂·∂ + (moc/ћ)2 )A = 0 | A is a 4-Vector, Proca eqn for massive spin-1 field, Lorenz Gauge |
(∂·∂)Ψ = 0 | Ψ is a scalar, Free-wave eqn for massless (mo = 0) spin-0 field |
(∂·∂)A = 0 | A is a 4-Vector, Maxwell eqn for massless (mo = 0) spin-1 field, no current sources, Lorenz Gauge |
(∂·∂)A = μoJ = ρoμoU = qnoμoU = qμoN | A is a 4-Vector, Maxwell eqn for massless (mo =
0) spin-1 field, current source 4-Vector J, Lorenz Gauge Classical EM, does not include effects of particle spin in the source J = ρoU |
(∂·∂)A = μoJ = μoqΨ ̅ ΓΨ (∂·∂)Aν = μoJν = μoqΨ ̅ γνΨ |
QED, A is a 4-Vector, Maxwell eqn for massless (mo =
0) spin-1 field, current source Spinor J, Lorenz Gauge Quantum EM, does include effects of particle spin in the source J = qΨ ̅ γμΨ |
[a - b] | [X] | = | [c 0] | [X] |
[b - a] | [Y] | [0 c] | [Y] |
([1 0] | a + | [0 -1] | b | )[X] | = c | [1 0] | [X] |
([0 -1] | [1 0] | )[Y] | [0 1] | [Y] |
([1 0] | a + | [0 -1] | b | )[X] | = c | [1 0] | [X] |
([0 -1] | [1 0] | )[Y] | [0 1] | [Y] |
([1 0] | ps0 + | [0 -1] | ps | )[X] | = (moc)I2 | [X] |
([0 -1] | [1 0] | )[Y] | [Y] |
([1 0] | σ0p0 + | [0 -1] | σ·p | )[X] | = (moc)I2 |
[X] |
([0 -1] | [1 0] | )[Y] | [Y] |
([σ0 0] | p0 + | [0 -σ] | ·p | )[X] | = (moc)I2 |
[X] |
([0 -σ0] | [σ 0] | )[Y] | [Y] |
let Spinor Ψ = | [X] | and note that σ0 = I2 |
[Y] |
([I2 0] | p0 + | [0 -σ] | ·p | ) | Ψ= (moc)IΨ |
([0 -I2] | [σ 0] | ) |
Relativistic Hamiltonian H = γ(PT·U) |
Relativistic Lagrangain L = -(PT·U)/γ |
pT·u = ( γβ2)(PT·U) = H + L = γ(PT·U) + -(PT·U)/γ |
H = γ(PT·U) H = γ((P + Q)·U) H = γ(P·U + Q·U) H = γP·U + γQ·U H = γmoU·U + γqA·U H = γmoc2 + qγφo H = γmoc2 + qφ H = ( γβ2 + 1/γ )moc2 + qφ H = ( γmoβ2c2 + moc2/γ) + qφ H = ( γmou2 + moc2/γ) + qφ H = p·u + moc2/γ + qφ H = E + qφ H = ±c√[mo2c2 + p2] + qφ H = ±c√[mo2c2 + (pT -qa)2] + qφ H = ±moc2√[1 + (pT -qa)2/(mo2c2)] + qφ |
L = -(PT·U)/γ L = -((P + Q)·U)/γ L = -(P·U + Q·U)/γ L = - P·U/γ - Q·U/γ L = -moU·U/γ - qA·U/γ L = -moc2/γ - qA·U/γ L = -moc2/γ - q(φ/c,a)·γ(c,u)/γ L = -moc2/γ - q(φ/c,a)·(c,u) L = -moc2/γ - q(φ - a·u) L = -moc2/γ - qφ + qa·u L = -moc2/γ - qφo/γ L = -(moc2 + qφo)/γ |
H + L = γ(PT·U) - (PT·U)/γ H + L = (γ - 1/γ)(PT·U) H + L = ( γβ2)(PT·U) H + L = ( γβ2)((P + Q)·U) H + L = ( γβ2)(P·U + Q·U) H + L = ( γβ2)(moc2 + qφo) H + L = (γmoβ2c2 + qγφoβ2) H + L = (γmou·uc2/c2 + qφoγu·u/c2) H + L = (γmou·u + qa·u) H + L = (p·u + qa·u) H + L = pT·u |
Rest Hamiltonian Ho = (PT·U) = H/γ |
Rest Lagrangian Lo = -(PT·U) = γL |
Ho + Lo = 0 |
Probability | | ↑ ⟩ State | | ↓ ⟩ State | Ket Tensor Product Representaion |
1/4 | 1, 2 | | ↑ ⟩| ↑ ⟩ | |
1/4 | 1 | 2 | | ↑ ⟩| ↓ ⟩ |
1/4 | 2 | 1 | | ↓ ⟩| ↑ ⟩ |
1/4 | 1, 2 | | ↓ ⟩| ↓ ⟩ |
Probability | | ↑ ⟩ State | | ↓ ⟩ State | Ket Tensor Product Representation |
1/3 | x,x | | ↑ ⟩| ↑ ⟩ = | 1,1 ⟩ | |
1/3 | x | x | 1/√[2]*( | ↑ ⟩| ↓ ⟩ + | ↓ ⟩| ↑ ⟩ ) = | 1,0 ⟩ |
1/3 | x,x | | ↓ ⟩| ↓ ⟩ = | 1,-1 ⟩ |
Probability | | ↑ ⟩ State | | ↓ ⟩ State | Ket Tensor Product Representaion |
1 | x | x | 1/√[2]*(| ↑ ⟩| ↓ ⟩ - | ↓ ⟩| ↑ ⟩ ) = | 0,0 ⟩ |
Particles | Statistics | Energy Occupation | Principle | Canonical Commutation | Both (+) | Both (-) | One each ( + ,-) |
---|---|---|---|---|---|---|---|
Bosons | Bose-Einstein | <Ni> = gi/(e^[(εi -μ)/kT] - 1) | Agglutination or Congregation |
[bα,bβ] = [b†α,b†β] = 0 [bα,b†β] = bαb†β - b†βbα = δαβ |
1/3 | 1/3 | 1/3 |
Distinguishable | Maxwell-Boltzmann | <Ni> = gi/(e^[(εi -μ)/kT] + 0) | Simple Random | 1/4 | 1/4 | 1/2 | |
Fermions | Fermi-Dirac | <Ni> = gi/(e^[(εi -μ)/kT] + 1) | Pauli Exclusion | {fα,fβ} = {f†α,f†β} = 0 {fα,f†β} = fαf†β + f†βfα = δαβ |
0 | 0 | 1 |
Particles | Field Operator | Annihilation,Annihilation | Creation,Creation | Annihilation,Creation |
---|---|---|---|---|
Bosons | Φb(r) = Σj[e^(kj·r)bj] | [Φb(r),Φb(r')] = 0 or [Φb(r),Φb(r')] - = 0 |
[Φ†b(r),Φ†b(r')] = 0 or [Φ†b(r),Φ†b(r')] - = 0 |
[Φb(r),Φ†b(r')] = ⟨ r|r'⟩ = δ3(r-r') or [Φb(r),Φ†b(r')] - = ⟨ r|r'⟩ = δ3(r-r') |
Fermions | Φf(r) = Σj[e^(kj·r)fj] | {Φf(r),Φf(r')} = 0 or [Φf(r),Φf(r')] + = 0 |
{Φ†f(r),Φ†f(r')} = 0 or [Φ†f(r),Φ†f(r')] + = 0 |
{Φf(r),Φ†f(r')} = ⟨ r|r'⟩ = δ3(r-r') or [Φf(r),Φ†f(r')] + = ⟨ r|r'⟩ = δ3(r-r') |
time-time T00
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time-space T0j
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space-time Ti0
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space-space Tij
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T00 | T01 | T02 | T03 |
T10 | T11 | T12 | T13 |
T20 | T21 | T22 | T23 |
T30 | T31 | T32 | T33 |
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| = |
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(Vacuum) Field Equations | (Sourced) Field Equations Minkowski Metric Lorentz Gauge |
Equations of Motion | Potential Φ | Independent Parameters |
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Newton CM | gijΦ,ij = 0 | gijΦ,ij = ∇·∇Φ = 4πGρm | d2/dt2[Xi] = -gijΦ, j = -∂Φ/∂Xi d2/dt2[x] = a = -∇Φ |
Scalar = (0-Tensor) | 1 |
Maxwell SR | gμνΦρ,μν = 0 | gμνΦρ,μν = (∂·∂)Φρ = μoJρ (∂·∂)A = μoJ (∂·∂)(φ/c,a) = μo(ρec,j) (∂·∂)φ = μoρec2 = ρe/εo ∇·∇φ = -ρe/εo {in time-independent potential) |
(these assume constant restmass mo) d2/dτ2[Xμ] = -(q/cmo)gμα(Φα,β - Φβ,α)(dXβ/dτ) Aμ = -(q/cmo)gμα(Φα,β - Φβ,α)Uβ Fμ = qUν(∂μAEMν - ∂νAEMμ) = qUνFμν |
4-Vector = (1-Tensor) | 4 |
Einstein GR | gμνΦρσ,μν + ... = 0 | d2/dτ2[Xμ] = -(1/2)gμα(gαβ,γ + gαγ,β - gβγ,α) (dXβ/dτ)(dXγ/dτ) | Tensor = (2-Tensor) | 10 |
Approximation Level | Equation of Motion (Positions) | Equation of Motion (Velocities) | Limiting Case | ||
Einstein GR (base/fundamental) |
d2Xσ/dτ2 + (Γσμν)(dXμ/dτ)(dXν/dτ) = 0 | dUσ/dτ + (Γσμν)(Uμ)(Uν) = 0 | Geodesic Motion - no Symmetry/Charge Forces | ||
Einstein SR | d2Xσ/dτ2 = 0 | dUσ/dτ = d/dτ[Uσ] = 0 γdUσ/dt = γd/dt[Uσ] = 0 |
Geodesic Motion - no Symmetry/Charge Forces "Flat" Minkowski SpaceTime (Γσμν) → 0 |
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Newton CM | d2Xσ/dt2 = 0 | dUσ/dt = 0 | Geodesic Motion - no Symmetry/Charge Forces "Flat" Minkowski SpaceTime (Γσμν) → 0 Low Velocity (|v| << c; γ → 1, τ → t) |
Field Equations in Lorenz Gauge (Divergence of Basic Field = 0) |
Full Field Equations | CurrentDensity | Higher Field Construction | Basic Field | |
SR EM 4-Vector Style |
(∂·∂)AEM = μoJ with (∂·AEM) = 0 |
∂[Fρσ] = (∂·∂)AEM - ∂(∂·AEM) = μoJ | 4-CurrentDensity J | 4-VectorPotential AEM | |
SR EM Tensor Style |
(∂μ∂μ)AEMν = μoJν with (∂νAEMν) = 0 |
∂μFμν = ∂μ(∂μAEMν - ∂νAEMμ) = μoJν | 4-CurrentDensity Jν |
SR Faraday Tensor Fμν = (∂μAEMν - ∂νAEMμ) |
4-VectorPotential AEMμ |
SR EM (,) Style |
Fμν,μ = μoJν | ||||
GR (;) Style |
(∂·∂)Habc = (∂μ∂μ)Habc = Jabc -2RcdHabd+RadHbcd+RbdHacd +(Hdbegac-Hdaegbc)Rde+RHabc/2 |
(Jordan Formulation) Cabcd;d = (some constant)Jabc |
Cotton Tensor (~Matter Current) Jabc = Rca;b-Rcb;a +(gcbR;a-gcaR;b)/6 |
Weyl Tensor Cabcd = Habc;d-Habd;c+Hcda;b-Hcdb;a - (gac(Hbd+Hdb)-gad(Hbc+Hcb)+gbd(Hac+Hca)-gbc(Had+Hda))/2 +2Hefe;f(gacgbd-gadgbc)/3 where Hbd=Hbed;e-Hbec;d |
Lanczos TensorPotential Habc |
4-Vector(s) | Type | Relativistic Law | Newtonian Limit Low Velocity (v<<c) or Low Energy (E<<moc2) Basically, β → 0, γ → 1 |
R = (ct,r) | 4-Position | (ct,r) is single 4-vector entity t and r related by Lorentz transform |
t independent from r t is independent scalar, r is independent 3-vector |
ΔR = (cΔt,Δr) | 4-Displacement | Relative Simultaneity Δt' = γ(Δt - β·Δr/c) |
Absolute Simultaneity Δt' = Δt |
U = dR/dτ | 4-Velocity | Relativistic Composition of Velocities urel = =[u1+u2]/(1+β1·β2) =[u1+u2]/(1+u1·u2/c2) Imposes Universal Speed Limit of c |
Additive Velocities u12 = u1 + u2 Unlimited Speed |
A = dU/dτ | 4-Acceleration | Relativistic Larmor Formula Power radiated by moving charge P = = -( q2/ 6πεoc3)(A·A) = -(μoq2/6πc)(A·A) = (μoq2/6πc) γ6/ (a2 - (|u x a|)2/c2) |
Newtonian Larmor Formula Power radiated by a non-relativistic moving charge P = (μoq2/6πc)(a2) |
P = moU | 4-Momentum | Einstein Energy-Mass Relation E = γ moc2 = Sqrt[ mo2c4 + p·p c2 ] |
Total Energy = Rest Energy + Kinetic Energy E = moc2 + (p2/2mo) |
∂·P | Divergence of 4-Momentum | Local? Conservation of 4-Momentum | Conservation of Energy, Conservation of Momentum |
P1·P2 | Particle Interaction | Conservation of 4-Momentum | Conservation of Energy, Conservation of Momentum, sometimes Conservation of Kinetic Energy |
K = (ω/c,k) = (1/ћ)P = (mo/ћ)U = (ωo/c2)U |
4-WaveVector and 4-Velocity |
Relativistic Doppler Effect, inc. Transverse Doppler Effect ao_obs = = ao_emit / γ(1 - (n·v/c)) = ao_emit / γ(1 - (n·β)) = ao_emit √[1+|β|]√[1-|β|] / (1 - (n·β)) Relativistic Aberration Effect cos(ø_obs) = [cos(ø_emit)-β]/[1-βcos(ø_emit)] Relativistic Wave Speed, all elementary particles, matter or photonic λf = c/β = vphase |
Regular Doppler Effect ao_obs = ao_emit √[1+|β|]√[1-|β|] Newtonian Aberration = None cos(ø_obs)= cos(ø_emit) Newtonian Wave Speed, only photonic particles (a rare case when the lightspeed case is chosen for Newtonian description) λf = c |
P and K | 4-Momentum and 4-WaveVector |
Compton Scattering (λ'-λ) = (h/moc)(1-cos[ø]) (moc2)(1/E'-1/E) = (1-cos[ø]) Ratio of photon energy after/before collision P[E,ø] = 1/[1+(E/moc2)(1-cos[ø])] see also Klein-Nishina formula |
Thompson Scattering Ratio of photon energy after/before collision: E<<moc2 P[E,ø] → 1 |
∂ = -iK | 4-Gradient | D'Alembertian & Klein-Gordon Equation ∂t2/c2 = ∇·∇-(moc/ћ)2 |
Schrödinger Equation (i ћ)( ∂t ) = - (ћ)2(∇)2/2mo |
∂·J | Divergence of 4-Current | Conservation of 4-EM_CurrentDensity ∂·J = ∂/c∂t(cp)+∇·j = ∂p/∂t +∇·j = 0 |
Conservation of 4-EM_CurrentDensity ∂·J = ∂/c∂t(cp)+∇·j = ∂p/∂t +∇·j = 0 |
Jprob | Probability CurrentDensity | Conservation of ProbabilityCurrentDensity ρ = (iћ/2moc2)(ψ* ∂t[ψ]-∂t[ψ*] ψ) j = (-iћ/2mo)(ψ* ∇[ψ]-∇[ψ*] ψ) ∂·Jprob = ∂ρ/∂t +∇·j = 0 ρ = γ(ψ*ψ) for time separable wave functions Relativistically, this is conservation of the number of worldlines thru a given SpaceTime event |
Conservation of Probability ∂·Jprob = ∂ρ/∂t +∇·j = 0 ρ = (ψ*ψ) for time separable wave functions Typically set so that the sum over all quantum states in space = 1 At low energies/velocities, this appears as the conservation of probability of a given wavefunction at a given SpaceTime event - In other words, the probability interpretation of a wavefunction is just a Newtonian approximation to the more correctly stated conservation of relativistic worldlines. This is why the problem of positive definite probabilities and probabilities >1 vanishes once you consider anti-particles and conservation of charged currents. |
AEM = (ΦEM/c, aEM) | 4-VectorPotential | 4-VectorPotenial of a moving point charge (Lienard-Wiechert potential) AEM = (q/4πεoc) U / [R·U]ret [..]ret implies (R·R = 0, the definition of a light signal) ΦEM = (γΦo) = (γq/4πεor) aEM = (γΦo/c2)u = (γqμo/4πr)u |
Scalar Potential and Vector Potential of a stationary point charge ΦEM = (q/4πεor) aEM = 0 Scalar Potential and Vector Potential of a slowly moving point charge (|v|<<c implies γ-->1) ΦEM = (Φo) = (q/4πεor) aEM = (Φo/c2)u = (qμo/4π r)u |
QEM = (EEM/c, pEM) = q AEM = q (ΦEM/c, aEM) |
4-VectorPotentialMomentum |
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PEM = (E/c + qΦEM/c, p +
qaEM) = γ mo(c,u) PEM = Π = P + qAEM = moU + qAEM =(H/c,pEM) = (γmoc+q ΦEM/c,γmou+q aEM) |
4-MomentumEM 4-CanonicalMomentum 4-TotalMomentum |
Minimal Coupling ============= Total 4-Momentum = Particle 4-Momentum + Potential(Field) 4-Momentum |
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D = ∂ + iq/ћ AEM | Minimal Coupling Prescription |
KG equation, with minimal coupling to an EM
potential D·D = = -(moc/ћ)2 (∂ + iq/ћ AEM)·(∂ + iq/ћ AEM) + (moc/ћ)2 = 0 |
Schrödinger Equation (with standard scalar external potential) (i ћ)( ∂t ) = V[x] - (ћ)2(∇)2/2mo |
SpaceTime Symmetry | Internal Particle Symmetries | ||
SL(2,ℂ) ⋉ ℝ1,3 | U(1) | SU(2) | SU(3) |
Gravity | EM | Weak | Color |
GR | Standard Model (QM) |