An 8x8 Matrix Rep Of Spacetime Symmetries
Author
Richard Shurtleff
Title
An 8x8 Matrix Rep Of Spacetime Symmetries
Description
This notebook describes an 8x8 matrix representation of the Poincare group and algebra of spacetime symmetries
Category
Working Material
Keywords
spacetime symmetries, Lie algebra, Poincare, matrix rep, rotation, boost, translation
URL
http://www.notebookarchive.org/2020-02-396rs6i/
DOI
https://notebookarchive.org/2020-02-396rs6i
Date Added
2020-02-07
Date Last Modified
2020-02-07
File Size
263.69 kilobytes
Supplements
Rights
CC BY 4.0
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Title: An8x8MatrixRepOfSpacetimeSymmetries
Operating System: Mathematica 12.0 running on Windows 10
File Size: 253 kbytes
Max Memory Used In Session: 124 Mbytes
Today’s Date: February 7, 2020
Operating System: Mathematica 12.0 running on Windows 10
File Size: 253 kbytes
Max Memory Used In Session: 124 Mbytes
Today’s Date: February 7, 2020
This notebook describes an 8x8 matrix representation of the Poincare group and algebra.
Poincare Group:
The Poincare group consists of transformations of spacetime that preserves the scalar product of 4-vectors. There are rotations, boosts, and translations.
A rotation changes coordinates in one 2-D plane in space, leaving a 2-D spacetime plane unchanged. A boost changes the velocity of the rest frame by changing coordinates in one 2-D plane (time and the direction of the boost), leaving one 2-D plane in space unchanged. Once time is treated on an equal footing with 3-D space, rotations and boosts act similarly: 2Ds change, 2Ds don't change. Rotations and boosts are "spacetime rotations" or just "rotations", if the spacetime context is understood.
A translation moves the origin of the rest frame, displacing the coordinates of all the points in spacetime by the same amount.
The Poincare algebra consists of the commutation relations between the generators of the transformations of the group.
The rotations and translations described above make the "continuous representation" of the Poincare group. It is well known and can be looked up in standard texts or knowledge storage centers (currently, Wikipedia is popular).
The 8x8 matrix representation(rep):
There are many matrix reps of the Poincare group. This is one.
The 8-dimensional rep of the Poincare group has 8x8 matrix generators that obey the same commutation relations as the generators of the continuous rep. The commutation relations form the algebra.
Shameless plug:
A discussion of the 8x8 matrix rep can be found in the paper `Intrinsic vector potential and electromagnetic mass', by Richard Shurtleff.
Poincare Group:
The Poincare group consists of transformations of spacetime that preserves the scalar product of 4-vectors. There are rotations, boosts, and translations.
A rotation changes coordinates in one 2-D plane in space, leaving a 2-D spacetime plane unchanged. A boost changes the velocity of the rest frame by changing coordinates in one 2-D plane (time and the direction of the boost), leaving one 2-D plane in space unchanged. Once time is treated on an equal footing with 3-D space, rotations and boosts act similarly: 2Ds change, 2Ds don't change. Rotations and boosts are "spacetime rotations" or just "rotations", if the spacetime context is understood.
A translation moves the origin of the rest frame, displacing the coordinates of all the points in spacetime by the same amount.
The Poincare algebra consists of the commutation relations between the generators of the transformations of the group.
The rotations and translations described above make the "continuous representation" of the Poincare group. It is well known and can be looked up in standard texts or knowledge storage centers (currently, Wikipedia is popular).
The 8x8 matrix representation(rep):
There are many matrix reps of the Poincare group. This is one.
The 8-dimensional rep of the Poincare group has 8x8 matrix generators that obey the same commutation relations as the generators of the continuous rep. The commutation relations form the algebra.
Shameless plug:
A discussion of the 8x8 matrix rep can be found in the paper `Intrinsic vector potential and electromagnetic mass', by Richard Shurtleff.
Table of Contents
1. Symbols
2. Definitions
3. The 8x8 matrix rep of the Poincare group
4. Parity
5. C, a discrete transformation used with charge conjugation
6. Additional Parity results
7. Additional results for C
8. Appendix: Some calculations found in the paper
1. Symbols
2. Definitions
3. The 8x8 matrix rep of the Poincare group
4. Parity
5. C, a discrete transformation used with charge conjugation
6. Additional Parity results
7. Additional results for C
8. Appendix: Some calculations found in the paper
1. Symbols: The table has the symbol from the paper, its Mathematica notation, and its definition x[[μ]] the four Minkowski coordinates of a point (event) δxμ[μ] the component of a coordinate interval xSQUARED[x] = = + + - ; scalar product of x with itself b[[μ]] 4-vector displacement parameters for a translation ω[[μ,ν]] antisymmetric tensor of parameters for a spacetime rotation (Lorentz transformation), ημν[[μ,ν]] the spacetime metric, flat, a 4x4 diagonal matrix with diagonal = {+1,+1,+1,-1} δij[[i,j]] Identity matrix, one along the diagonal i = j and zero otherwise an 8x8 matrix with nonzero components only in the ij-block with 4x4 blocks ij = 1,2. 1 IdentityMatrix[8] the unit 8x8 matrix oneij an 8x8 matrix with the unit 4x4 matrix in the block ZeroMatrix[n] an nxn matrix of zeros τμ[[μ]] the four Pauli 2x2 spin matrices with μ = 1,2,3,4 and time is μ = 4. γDμ[[μ]] the four Dirac 4x4 gamma matrices γD5 the product = , note that time is first. γijμ[[μ]] four 8x8 matrices with the 4x4 matrices in the ij block γμ[[μ]] four 8x8 matrices with in the 11 and 22 diagonal blocks σμν[[μ,ν]] the angular momentum 8x8 matrix πμ[[μ]] the linear momentum 8x8 matrix k kc scale factor for the matrices bμ[[μ]] the components of a coordinate displacement (ω) the Lorentz transformation 8x8 matrix, representing a spacetime rotation(Λ,b) a Poincare transformation, a rotation Λ(ω) followed by a translation along .D(Λ,b) DΛb[ω,b] the 8x8 matrix representing the transformation (Λ,b) (Λ,b) D1bDagger[b] hermitian conjugate of D(Λ,b); transpose of the complex conjugateD(Λ,0) DΛ0[ω] the 8x8 matrix representing a pure spacetime rotation (no translation)D(1,b) D1b[b] the 8x8 matrix representing a pure translation (no spacetime rotation) βd spacial inversion, i.e. a parity changing matrix with “d” for diagonal cTC the 8x8 matrix for a discrete transformation associated with charge conjugationψ ψ8[[i]] 8-spinor with component index i = 1,2,...,8 ψ1[[i]] the first 4-spinor in ψ, i = 1,2,3,4 with =0 for i = 5,6,7,8 ψ2[[i]] the second 4-spinor in ψ, i = 5,6,7,8 and =0 for i = 1,2,3,4
μ
x
μ
δx
th
μ
2
x
2
x
x
μ
μ
x
2
1
x
2
2
x
2
3
x
2
4
x
μ
b
μν
ω
μν
η
η
μν
δ
ij
M
ij
1
ij
th
ij
μ
τ
μ
γ
D
5
γ
D
5
γ
D
4
γ
D
1
γ
D
2
γ
D
3
γ
D
4
γ
D
μ
γ
ij
μ
γ
D
μ
γ
μ
γ
D
μν
σ
th
μν
μ
π
th
μ
μ
π
μ
b
μν
Λ
th
μν
μ
b
†
D
β
ψ
1
ψ
1
ψ
2
ψ
2
2. Definitions
In[]:=
x={x1,x2,x3,x4};b={b1,b2,b3,b4};y={y1,y2,y3,y4};ω={{0,ω12,ω13,ω14},{-ω12,0,ω23,ω24},{-ω13,-ω23,0,ω34},{-ω14,-ω24,-ω34,0}};ψ8={ψ81,ψ82,ψ83,ψ84,ψ85,ψ86,ψ87,ψ88};ψ1={ψ81,ψ82,ψ83,ψ84,0,0,0,0};ψ2={0,0,0,0,ψ85,ψ86,ψ87,ψ88};δij=IdentityMatrix[50];(*50=∞*)ημν={{+1,0,0,0},{0,+1,0,0},{0,0,+1,0},{0,0,0,-1}};
In[]:=
xSQUARED[x_]:=Sum[ημν[[μ1,μ2]]x[[μ1]]x[[μ2]],{μ1,4},{μ2,4}]ZeroMatrix[n_]:=ZeroMatrix[n]=IdentityMatrix[n]-IdentityMatrix[n]
In[]:=
τμ={{{0,1},{1,0}},{{0,-},{,0}},{{1,0},{0,-1}},{{1,0},{0,1}}};γDμ=-TableArrayFlatten{0,τμ[[μ]]},(-ημν〚μ,ν〛)τμν,0,{μ,4};γD5=γDμ[[4]].γDμ[[1]].γDμ[[2]].γDμ[[3]];
4
∑
ν
In[]:=
γμ11=Table[ArrayFlatten[{{γDμ[[μ]],ZeroMatrix[4]},{0,ZeroMatrix[4]}}],{μ,4}];γμ22=Table[ArrayFlatten[{{ZeroMatrix[4],0},{0,γDμ[[μ]]}}],{μ,4}];γμ21=Table[ArrayFlatten[{{0,ZeroMatrix[4]},{γDμ[[μ]],0}}],{μ,4}];γμ12=Table[ArrayFlatten[{{0,γDμ[[μ]]},{ZeroMatrix[4],0}}],{μ,4}];one11=ArrayFlatten[{{IdentityMatrix[4],0},{0,ZeroMatrix[4]}}];one22=ArrayFlatten[{{ZeroMatrix[4],0},{0,IdentityMatrix[4]}}];one12=ArrayFlatten[{{0,IdentityMatrix[4]},{ZeroMatrix[4],0}}];one21=ArrayFlatten[{{0,ZeroMatrix[4]},{IdentityMatrix[4],0}}];
In[]:=
γij[μ_]:={{γμ11[[μ]],γμ12[[μ]]},{γμ21[[μ]],γμ22[[μ]]}}oneij={{one11,one12},{one21,one22}};
In[]:=
γμ=Table[γμ11[[μ]]+γμ22[[μ]],{μ,4}];
In[]:=
γ5=γμ[[4]].γμ[[1]].γμ[[2]].γμ[[3]];γ5//MatrixForm
Out[]//MatrixForm=
-1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
In[]:=
σμν=Table(γμ[[μ]].γμ[[ν]]-γμ[[ν]].γμ[[μ]]),{μ,4},{ν,4};πμ=Table[kcγμ21[[μ]],{μ,4}];
-
4
In[]:=
DΛ0[ω_]:=Simplify[MatrixExp[Sum[+ημν[[μ1,μ2]]ημν[[ν1,ν2]]ω[[μ1,ν1]]σμν[[μ2,ν2]]/2,{μ1,4},{μ2,4},{ν1,4},{ν2,4}]],ω∈Reals]D1b[b_]:=Simplify[MatrixExp[Sum[-ημν[[μ1,μ2]]b[[μ1]]πμ[[μ2]],{μ1,4},{μ2,4}]]];D1bDagger[b_]:=FullSimplify[Transpose[Conjugate[(IdentityMatrix[8]-Sum[ημν[[μ1,μ2]]b[[μ1]]πμ[[μ2]],{μ1,4},{μ2,4}])]],{kc,b1,b2,b3,b4}∈Reals]DΛb[ω_,b_]:=D1b[b].DΛ0[ω](*Eqn.12,definitionofD(Λ,b).*)
In[]:=
βd=ArrayFlatten[{{γDμ[[4]],ZeroMatrix[4]},{0,γDμ[[4]]}}];βd//MatrixForm;γμ11[[4]]+γμ22[[4]]//MatrixForm;Print[" β = + : ",{0}Union[Flatten[βd-(γμ11[[4]]+γμ22[[4]])]]]
t
γ
11
t
γ
22
β = + : True
t
γ
11
t
γ
22
In[]:=
Cd=γμ11[[2]].γμ11[[4]]+γμ22[[2]].γμ22[[4]];Cd//MatrixForm;Print[" C = + : ",{0}Union[Flatten[+Cd-(γμ11[[2]].γμ11[[4]]+γμ22[[2]].γμ22[[4]])]]]
2
γ
11
t
γ
11
2
γ
22
t
γ
22
C = + : True
2
γ
11
t
γ
11
2
γ
22
t
γ
22
3. The 8x8 matrix rep of the Poincare group (See Sec. 2 in the paper)
In[]:=
(*Diracgammas*)Print["check the required property of Dirac 4x4 gammas, + = 2 1 : ",{0}==Union[Flatten[Table[(γDμ[[μ]].γDμ[[ν]]+γDμ[[ν]].γDμ[[μ]])-2ημν[[μ,ν]]IdentityMatrix[4],{μ,4},{ν,4}]]]]
μ
γ
D
ν
γ
D
ν
γ
D
μ
γ
D
μν
η
check the required property of Dirac 4x4 gammas, + = 2 1 : True
μ
γ
D
ν
γ
D
ν
γ
D
μ
γ
D
μν
η
In[]:=
Print"The auxilliary 8x8 gamma has in the 4x4 ij-block."Print"check the required property of auxilliary 8x8 gammas , + = 2 : ",{0}==Union[Flatten[Table[Table[γij[μ][[i,j]].γij[ν][[k,l]]+γij[ν][[i,j]].γij[μ][[k,l]]-(2IdentityMatrix[8][[j,k]]ημν[[μ,ν]]oneij[[i,l]]),{i,2},{j,2},{k,2},{l,2}],{μ,4},{ν,4}]]]
μ
γ
ij
μ
γ
D
μ
γ
ij
μ
γ
ij
ν
γ
kl
ν
γ
ij
μ
γ
kl
δ
jk
μν
η
1
il
The auxilliary 8x8 gamma has in the 4x4 ij-block.
μ
γ
ij
μ
γ
D
check the required property of auxilliary 8x8 gammas , + = 2 : True
μ
γ
ij
μ
γ
ij
ν
γ
kl
ν
γ
ij
μ
γ
kl
δ
jk
μν
η
1
il
In[]:=
Print["The 8x8 gammas are the `official' gammas."]Print["check definition of the official 8x8 gammas , = + : ",{0}==Union[Flatten[Table[γμ[[μ]]-(γμ11[[μ]]+γμ22[[μ]]),{μ,4}]]]]
μ
γ
μ
γ
μ
γ
μ
γ
11
μ
γ
22
The 8x8 gammas are the `official' gammas.
μ
γ
check definition of the official 8x8 gammas , = + : True
μ
γ
μ
γ
μ
γ
11
μ
γ
22
In[]:=
Print["check that spin (angular momentum) generators are reducible,"]Print" = -( - )-( - ): ",{0}==UnionFlattenTableσμν[[μ,ν]]--(γμ11[[μ]].γμ11[[ν]]-γμ11[[ν]].γμ11[[μ]])-(γμ22[[μ]].γμ22[[ν]]-γμ22[[ν]].γμ22[[μ]]),{μ,4},{ν,4}
μν
σ
4
μ
γ
11
ν
γ
11
ν
γ
11
μ
γ
11
4
μ
γ
22
ν
γ
22
ν
γ
22
μ
γ
22
4
4
check that spin (angular momentum) generators are reducible,
μν
σ
4
μ
γ
11
ν
γ
11
ν
γ
11
μ
γ
11
4
μ
γ
22
ν
γ
22
ν
γ
22
μ
γ
22
In[]:=
Print["check the definition of linear momentum, = : ",{0}==Union[Flatten[Table[πμ[[μ]]-(kcγμ21[[μ]]),{μ,4}]]]]
μ
π
μ
kγ
21
check the definition of linear momentum, = : True
μ
π
μ
kγ
21
In[]:=
Print["check Poincare algebra, [,] = --+ : ",{0}==Union[Flatten[Table[(σμν[[μ,ν]].σμν[[ρ,λ]]-σμν[[ρ,λ]].σμν[[μ,ν]])-(ημν[[ν,ρ]]σμν[[μ,λ]]-ημν[[μ,ρ]]σμν[[ν,λ]]-ημν[[ν,λ]]σμν[[μ,ρ]]+ημν[[μ,λ]]σμν[[ν,ρ]]),{μ,4},{ν,4},{ρ,4},{λ,4}]]]]Print["check Poincare algebra, [,] = - : ",{0}==Union[Flatten[Table[(σμν[[μ,ν]].πμ[[ρ]]-πμ[[ρ]].σμν[[μ,ν]])-(ημν[[ν,ρ]]πμ[[μ]]-ημν[[μ,ρ]]πμ[[ν]]),{μ,4},{ν,4},{ρ,4}]]]]Print["check Poincare algebra, [,] = 0 : ",{0}==Union[Flatten[Table[(πμ[[μ]].πμ[[ν]]-πμ[[ν]].πμ[[μ]]),{μ,4},{ν,4}]]]]
μν
σ
ρλ
σ
νρ
η
μλ
σ
μρ
η
νλ
σ
νλ
η
μρ
σ
μλ
η
νρ
σ
μν
σ
ρ
π
νρ
η
μ
π
μρ
η
ν
π
μ
π
ν
π
check Poincare algebra, , = --+ : True
μν
σ
ρλ
σ
νρ
η
μλ
σ
μρ
η
νλ
σ
νλ
η
μρ
σ
μλ
η
νρ
σ
check Poincare algebra, [,] = - : True
μν
σ
ρ
π
νρ
η
μ
π
μρ
η
ν
π
check Poincare algebra, [,] = 0 : True
μ
π
ν
π
In[]:=
Print"check Poincare transformation definition, D(Λ,b) = : ",{0}==Union[Flatten[DΛb[ω,b]-(Simplify[MatrixExp[Sum[-ημν[[μ1,μ2]]b[[μ1]]πμ[[μ2]],{μ1,4},{μ2,4}]]].Simplify[MatrixExp[Sum[+ημν[[μ1,μ2]]ημν[[ν1,ν2]]ω[[μ1,ν1]]σμν[[μ2,ν2]]/2,{μ1,4},{μ2,4},{ν1,4},{ν2,4}]],ω∈Reals])]]
-
b
μ
μ
π
+
2
ω
μν
μν
σ
check Poincare transformation definition, D(Λ,b) = : True
-
b
μ
μ
π
+
2
ω
μν
μν
σ
In[]:=
Print["check momentum products vanish, = 0 : ",{0}==Union[Flatten[Table[πμ[[μ]].πμ[[ν]],{μ,4},{ν,4}]]]]Print["Hence, translations are linear in the displacement ,"]Print[" D(1,b) = 1 - : ",{0}Union[Flatten[Simplify[D1b[b]-(IdentityMatrix[8]-Sum[ημν[[μ1,μ2]]b[[μ1]]πμ[[μ2]],{μ1,4},{μ2,4}]),b∈Reals]]]]
μ
π
ν
π
μ
b
ib
μ
μ
π
check momentum products vanish, = 0 : True
μ
π
ν
π
Hence, translations are linear in the displacement ,
μ
b
D(1,b) = 1 - : True
ib
μ
μ
π
In[]:=
Print["The effect of a translation on an 8-spinor, "]Print" D(1,b)ψ8 = {}, - : ",{0}Union[Flatten[Simplify[D1b[b].ψ8-(ψ8-Sum[ημν[[μ1,μ2]]b[[μ1]]πμ[[μ2]].ψ8,{μ1,4},{μ2,4}]),b∈Reals]]]
ψ
1
ψ
2
kb
μ
μ
γ
D
ψ
1
The effect of a translation on an 8-spinor,
D(1,b)ψ8 = {}, - : True
ψ
1
ψ
2
kb
μ
μ
γ
D
ψ
1
4. Parity
The discrete transformation β acting on a matrix X, , gives the effects of parity on X.Parity is synonymous with “spacial inversion”, (additive inversion, that is, not multiplicative).Spacetime Explanation: The metric η = diag(+1,+1,+1,-1) inverts time, so -η inverts space coordinates. The effect of +η is to move a vector index from up-index to down index, i.e. switching covariant and contravariant. Parity on coordinates , for example, gives = = = = . So parity, -η, lowers the index and changes the sign when applied to a vector . Parity for 8x8 Poincare rep is an 8x8 matrix β, not -η. The negative metric -η is for spacetime, i.e. the continuous rep of the Poincare group. The 8x8 matrix β is defined above and applied below.
-1
βXβ
μ
x
μ
Px
P(,,,)
1
x
2
x
3
x
4
x
-,-,-,
1
x
2
x
3
x
4
x
-
η
μλ
λ
x
-
x
μ
μ
v
In[]:=
Print["Check parity acting on gammas, = - : ",{0}Union[Flatten[Table[βd.γμ[[μ]].βd-(-Sum[+ημν[[μ,μ1]]γμ[[μ1]],{μ1,4}]),{μ,4}]]]]Print"Check parity effects, = -1 = + : ",{0}Union[Flatten[{Table[βd.σμν[[μ,ν]].βd-(Sum[+ημν[[μ,μ1]]ημν[[ν,ν1]]σμν[[μ1,ν1]],{μ1,4},{ν1,4}]),{μ,4},{ν,4}]}]]Print["Check parity effects, = - : ",{0}Union[Flatten[Table[βd.πμ[[μ]].βd-(-Simplify[Sum[ημν[[μ,μ1]]πμ[[μ1]],{μ1,4}],{ca,kc,m,y}∈Reals]),{μ,4}]]]]
μ
βγ
-1
β
γ
μ
μν
βσ
-1
β
2
)
σ
μν
σ
μν
μ
βπ
-1
β
π
μ
Check parity acting on gammas, = - : True
μ
βγ
-1
β
γ
μ
Check parity effects, = -1 = + : True
μν
βσ
-1
β
2
)
σ
μν
σ
μν
Check parity effects, = - : True
μ
βπ
-1
β
π
μ
5. C, a discrete transformation used with charge conjugation
The discrete transformation acting on a matrix X, , gives the negative transpose of X.
The symbol is due to its use with charge conjugation. (Also the letter “T” is busy with time inversion, transpose, possibly more)
-1
X
The symbol is due to its use with charge conjugation. (Also the letter “T” is busy with time inversion, transpose, possibly more)
In[]:=
Print["Check effects of , = - : ",{0}Union[Flatten[Table[Cd.γμ[[μ]].(-Cd)-(-Transpose[γμ[[μ]]]),{μ,4}]]]]Print["Check effects of , = (-1 = - : ",{0}Union[Flatten[Table[Cd.σμν[[μ,ν]].(-Cd)-(-Transpose[σμν[[μ,ν]]]),{μ,4},{ν,4}]]]]Print["There are three (-1) factors because there are two indices, one (-1) each. Also, the transpose exchanges matrix factors which gives the third (-1) since is antisymmteric."]Print"Check effects of , = - =: ",{0}Union[Flatten[Table[Table[(Cd.πμ[[μ]].(-Cd))[[i2,j2]],{i2,5,8},{j2,4}]-(-Transpose[Table[πμ[[μ]][[i1,j1]],{i1,5,8},{j1,4}]]),{μ,4}]]]Print["Note that the transpose with π applies to just the 21-block, not the entire matrix."]
μ
γ
-1
μT
γ
μν
σ
-1
3
)
μνT
σ
μνT
σ
μν
σ
μ
π
-1
)
(21)
μ
π
T
)
(21)
Check effects of , = - : True
μ
γ
-1
μT
γ
Check effects of , = (-1 = - : True
μν
σ
-1
3
)
μνT
σ
μνT
σ
There are three (-1) factors because there are two indices, one (-1) each. Also, the transpose exchanges matrix factors which gives the third (-1) since is antisymmteric.
μν
σ
Check effects of , = - =: True
μ
π
-1
)
(21)
μ
π
T
)
(21)
Note that the transpose with π applies to just the 21-block, not the entire matrix.
6. Additional results with parity matrix .
β
In[]:=
{βd//MatrixForm,ConjugateTranspose[βd]//MatrixForm};Print[" The parity matrix β is Hermitian: = β : ",{0}Union[Flatten[βd-ConjugateTranspose[βd]]]]βd.βd//MatrixForm;Print[" β is its own inverse, β = : ",{0}Union[Flatten[βd.βd-IdentityMatrix[8]]]]Print[" βd.ψ8 = ",βd.ψ8]
†
β
-1
β
The parity matrix β is Hermitian: = β : True
†
β
β is its own inverse, β = : True
-1
β
βd.ψ8 = {ψ83,ψ84,ψ81,ψ82,ψ87,ψ88,ψ85,ψ86}
In[]:=
Conjugate[];Table[γμ[[μ]]//MatrixForm,{μ,4}];Table[ConjugateTranspose[γμ[[μ]]]//MatrixForm,{μ,4}];Table[βd.ConjugateTranspose[γμ[[μ]]].βd//MatrixForm,{μ,4}];Print["Check Weinberg Vol 1, page 218, (5.4.30), = - : ",{0}Union[Flatten[Table[βd.ConjugateTranspose[γμ[[μ]]].βd-(-γμ[[μ]]),{μ,4}]]]]Print["Check Weinberg Vol 1, page 215, (5.4.14), = - and = +: ",{0}Union[Flatten[{Table[βd.γμ[[i]].βd-(-γμ[[i]]),{i,3}],βd.γμ[[4]].βd-(+γμ[[4]])}]]]Print["Check = - : ",{0}Union[Flatten[Table[βd.γμ[[μ]].βd-(-Sum[+ημν[[μ,μ1]]γμ[[μ1]],{μ1,4}]),{μ,4}]]]]
μ†
βγ
-1
β
μ
γ
i
βγ
-1
β
i
γ
t
βγ
-1
β
t
γ
μ
βγ
-1
β
γ
μ
Check Weinberg Vol 1, page 218, (5.4), = - : True
μ†
βγ
-1
β
μ
γ
Check Weinberg Vol 1, page 215, (5.4), = - and = +: True
i
βγ
-1
β
i
γ
t
βγ
-1
β
t
γ
Check = - : True
μ
βγ
-1
β
γ
μ
In[]:=
Print["Check Weinberg Vol 1, page 218, (5.4.33), = - : ",{0}Union[Flatten[βd.ConjugateTranspose[γ5].βd-(-γ5)]]]Print["Check Weinberg Vol 1, page 218, (5.4.34), β( = - : ",{0}Union[Flatten[Table[βd.ConjugateTranspose[γ5.γμ[[μ]]].βd-(-γ5.γμ[[μ]]),{μ,4}]]]]
†
βγ
5
-1
β
γ
5
γ
5
μ
γ
†
)
-1
β
γ
5
μ
γ
Check Weinberg Vol 1, page 218, (5.4), = - : True
†
βγ
5
-1
β
γ
5
Check Weinberg Vol 1, page 218, (5.4), β( = - : True
γ
5
μ
γ
†
)
-1
β
γ
5
μ
γ
In[]:=
Table[{μ,ν,σμν[[μ,ν]]//MatrixForm,βd.σμν[[μ,ν]].βd//MatrixForm},{μ,3},{ν,μ+1,4}];Print["Check Weinberg Vol 1, page 215, (5.4.15,16), = + and = -: ",{0}Union[Flatten[{Table[βd.σμν[[i,j]].βd-(+σμν[[i,j]]),{i,3},{j,3}],Table[βd.σμν[[4,j]].βd-(-σμν[[4,j]]),{j,3}]}]]]Print["Check Weinberg Vol 1, page 218, (5.4.31), = + : ",{0}Union[Flatten[Table[βd.ConjugateTranspose[σμν[[μ,ν]]].βd-(σμν[[μ,ν]]),{μ,4},{ν,4}]]]]Print["Check = + : ",{0}Union[Flatten[{Table[βd.σμν[[μ,ν]].βd-(Sum[+ημν[[μ,μ1]]ημν[[ν,ν1]]σμν[[μ1,ν1]],{μ1,4},{ν1,4}]),{μ,4},{ν,4}]}]]]
ij
βσ
-1
β
ij
σ
tj
βσ
-1
β
tj
σ
μν†
βσ
-1
β
μν
σ
μν
βσ
-1
β
σ
μν
Check Weinberg Vol 1, page 215, (5.4,16), = + and = -: True
ij
βσ
-1
β
ij
σ
tj
βσ
-1
β
tj
σ
Check Weinberg Vol 1, page 218, (5.4), = + : True
μν†
βσ
-1
β
μν
σ
Check = + : True
μν
βσ
-1
β
σ
μν
In[]:=
Table[{μ,βd.πμ[[μ]].βd//MatrixForm,-Simplify[Sum[ημν[[μ,μ1]]πμ[[μ1]],{μ1,4}],{ca,kc,m,y}∈Reals]//MatrixForm},{μ,4}];Print["Check = - : ",{0}Union[Flatten[Table[βd.πμ[[μ]].βd-(-Simplify[Sum[ημν[[μ,μ1]]πμ[[μ1]],{μ1,4}],{ca,kc,m,y}∈Reals]),{μ,4}]]]]
μ
βπ
-1
β
π
μ
Check = - : True
μ
βπ
-1
β
π
μ
7. Additional results for discrete transformations with matrix C
In[]:=
Cd.Cd//MatrixForm;Print["The negative of is its inverse, = - : ",{0}Union[Flatten[Cd.(-Cd)-IdentityMatrix[8]]]]
-1
The negative of is its inverse, = - : True
-1
In[]:=
Print["Verify Weinberg Vol 1, p. 219, (5.4.36) by inspection: ",Cd//MatrixForm," , compare - = ",-τμ[[2]]//MatrixForm]
τ
2
Verify Weinberg Vol 1, p. 219, (5.4.36) by inspection:
, compare - =
0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
τ
2
0 | -1 |
1 | 0 |
In[]:=
Print["Check Weinberg Vol 1, page 218, (5.4.38); = + : ",{0}Union[Flatten[Transpose[γ5]-(Cd.γ5.(-Cd))]]]
T
γ
5
γ
5
-1
Check Weinberg Vol 1, page 218, (5.4); = + : True
T
γ
5
γ
5
-1
In[]:=
Print"Check Weinberg Vol 1, page 218, (5.4.39); = + : ",{0}Union[Flatten[Table[Transpose[γ5.Sum[ημν[[μ,μ1]]γμ[[μ1]],{μ1,4}]]-(+Cd.γ5.Sum[ημν[[μ,μ2]]γμ[[μ2]],{μ2,4}].(-Cd)),{μ,4}]]]
γ
5
γ
μ
T
)
γ
5
γ
μ
-1
Check Weinberg Vol 1, page 218, (5.4); = + : True
γ
5
γ
μ
T
)
γ
5
γ
μ
-1
Bibliography
S. Weinberg, The Quantum Theory of Fields, Vol. 1, Cambridge University Press, 1995.
S. Weinberg, The Quantum Theory of Fields, Vol. 1, Cambridge University Press, 1995.
8. Appendix: Calculations related to the paper
Intrinsic vector potential and electromagnetic mass
Richard Shurtleff, Department of Sciences,
Wentworth Institute of Technology, 550 Huntington Avenue,
Boston, MA, USA, 02115,
shurtleffr@wit.edu
Abstract
Electric charges may have mass in part or in full because they charged. Supplying details is the electromagnetic mass problem. Here, the charge’s mass is associated with intrinsic quantum mechanical quantities so that the classical problems with extended charge distributions, for example, are irrelevant. An intrinsic vector potential is defined, based on intrinsic linear momentum defined as a matrix generator in an 8-spinor Poincare algebra. The charge-electromagnetic field interaction energy is gauge-dependent and the needed mass term is placed with the interaction energy in the intrinsic gauge. Traditional electromagnetism retains its gauge invariance. The field equations for the 8-spinor field describe a massive, charged 4-spinor Dirac electron-like particle and, independent of the charged 4-spinor, an uncharged, massless neutrino-like particle, formulas that have been a part of physics for nearly a century.
Richard Shurtleff, Department of Sciences,
Wentworth Institute of Technology, 550 Huntington Avenue,
Boston, MA, USA, 02115,
shurtleffr@wit.edu
Abstract
Electric charges may have mass in part or in full because they charged. Supplying details is the electromagnetic mass problem. Here, the charge’s mass is associated with intrinsic quantum mechanical quantities so that the classical problems with extended charge distributions, for example, are irrelevant. An intrinsic vector potential is defined, based on intrinsic linear momentum defined as a matrix generator in an 8-spinor Poincare algebra. The charge-electromagnetic field interaction energy is gauge-dependent and the needed mass term is placed with the interaction energy in the intrinsic gauge. Traditional electromagnetism retains its gauge invariance. The field equations for the 8-spinor field describe a massive, charged 4-spinor Dirac electron-like particle and, independent of the charged 4-spinor, an uncharged, massless neutrino-like particle, formulas that have been a part of physics for nearly a century.
Appendix More Symbols: The table has the symbol from the paper, its Mathematica notation, and its definition y[[μ]] intrinsic Minkowski coordinates, independent of αμ[[μ]] the 8x8 matrix intrinsic vector potentiala ca a constant associated with χ χ gauge functionm m massq q charge
μ
y
μ
x
μ
α
μ
α
Appendix. More Definitions:
Appendix. More Definitions:
In[]:=
αμ[y_]:=αμ[y]=Table[FullSimplify[-caγμ[[4]].D1bDagger[y].γμ[[4]].γμ[[μ]].D1b[y],{kc,y}∈Reals],{μ,4}];χ[y_]:=-caSumημν[[λ1,λ2]]1-y[[λ1]]γμ[[λ2]]+y[[λ1]]γμ11[[λ2]]-y[[λ1]]πμ[[λ2]],{λ1,4},{λ2,4}+kcxSQUARED[y](one21+one12)
m
4caq
2
kc
xSQUARED[y]
3
1
qca
In[]:=
αμT[y_]:=Table[Transpose[αμ[y][[μ]]],{μ,4}]cαμcd[y_]:=Table[(Cd.αμ[y][[μ]].(-Cd)),{μ,4}]
Intrinsic vector potential and electromagnetic mass, Section 4. Currents and the intrinsic E-M vector potential
In the Dirac formalism of quantum mechanics, the probability current of a 4-spinor ψ is defined to be = ψ The same formula works here for 8-spinors ψ .In the paper, there are two 8-spinors ψ and Φ .For a Poincare transformation (Λ,b) of spacetime, the Lorentz 8-spinor ψ transforms by = D(Λ,0) ψ,which means ψ does not change with translations along , but does respond to spacetime rotations (hence Lorentz). This is the traditional convention, widely assumed. See, for example, Weinberg early in Chapter 5 where a Poincare transformation, including translations, is applied to annihilation/creation operators resulting in a Lorentz (no translation at all) transformation of the field ψ. The 8-spinor Φ is defined to be Φ = D(1,y)ψ,which is a translation along some arbitrary applied to ψ. The 8-spinor Φ transforms with translations and rotations, = D(Λ,b) Φ .The probability current of the 8-spinor Φ is given by = Φ = (y)ψ .Therefore, and differ because α replaces the γ in = ψ . There is a parasitic constant a put in the paper and dropped here. Also, the normalization constants have been dropped in this summary.
μ
j
μ
j
ψ
μ
γ
ψ'
μ
b
μ
y
Φ'
μ
J
μ
J
Φ
μ
γ
ψ
μ
α
μ
j
μ
J
μ
j
ψ
μ
γ
In[]:=
Print"check = + - ( + ) + - 2 : ",{0}UnionFlattenTableFullSimplify-(+γμ[[μ]]-kcSum[ημν[[ρ1,ρ2]]y[[ρ1]](γμ12[[ρ2]].γμ22[[μ]]+γμ22[[μ]].γμ21[[ρ2]]),{ρ1,4},{ρ2,4}]+xSQUARED[y]γμ11[[μ]]-2y[[μ]]Sum[ημν[[ρ1,ρ2]]y[[ρ1]]γμ11[[ρ2]],{ρ1,4},{ρ2,4}]),{kc,b}∈Reals,{μ,4}
μ
α
a
μ
γ
ky
ρ
ρ
γ
12
μ
γ
22
μ
γ
22
ρ
γ
21
2
k
2
y
μ
γ
11
2
k
μ
y
y
ρ
ρ
γ
11
αμ[y][[μ]]
ca
2
kc
2
kc
check = + - ( + ) + - 2 : True
μ
α
a
μ
γ
ky
ρ
ρ
γ
12
μ
γ
22
μ
γ
22
ρ
γ
21
2
k
2
y
μ
γ
11
2
k
μ
y
y
ρ
ρ
γ
11
In[]:=
Print["Check () = -(-) : ",{0}Union[Flatten[Simplify[Table[βd.Simplify[αμ[y][[μ]],{ca,kc,m,y}∈Reals].βd-(-Simplify[Sum[ημν[[μ,μ1]]αμ[{-y1,-y2,-y3,y4}][[μ1]],{μ1,4}],{ca,kc,m,y}∈Reals]),{μ,4}]]]]]
μ
βα
ν
y
-1
β
α
μ
y
ν
Check () = -(-) : True
μ
βα
ν
y
-1
β
α
μ
y
ν
In[]:=
Print["Check ((y) = -[(-y)(-C)] : ",{0}Union[Flatten[Table[Simplify[αμT[y][[μ]]-(-cαμcd[-y][[μ]])],{μ,4}]]]]
μ
α
T
)
μ
Cα
Check ((y) = -[(-y)(-C)] : True
μ
α
T
)
μ
Cα
In[]:=
Print"check - = 12 : ",{0}Union[Flatten[Table[(Sum[ημν[[ρ1,ρ2]]D[αμ[y][[μ]],y[[ρ1]],y[[ρ2]]],{ρ1,4},{ρ2,4}]-Sum[ημν[[μ,ρ2]]D[αμ[y][[ρ1]],y[[ρ1]],y[[ρ2]]],{ρ1,4},{ρ2,4}])-(12caγμ11[[μ]]),{μ,4}]]]
′
ρ
∂
′
∂
ρ
μ
α
′
μ
∂
′
∂
ρ
ρ
α
2
ak
μ
γ
11
2
kc
check - = 12 : True
′
ρ
∂
′
∂
ρ
μ
α
′
μ
∂
′
∂
ρ
ρ
α
2
ak
μ
γ
11
In[]:=
Print["Check () = -(-) : ",{0}Union[Flatten[Simplify[Table[βd.Simplify[αμ[y][[μ]],{ca,kc,m,y}∈Reals].βd-(-Simplify[Sum[ημν[[μ,μ1]]αμ[{-y1,-y2,-y3,y4}][[μ1]],{μ1,4}],{ca,kc,m,y}∈Reals]),{μ,4}]]]]]
μ
βα
ν
y
-1
β
α
μ
y
ν
Check () = -(-) : True
μ
βα
ν
y
-1
β
α
μ
y
ν
Intrinsic vector potential and electromagnetic mass, Section 5. Gauge, mass term, field equations
In[]:=
Print"check χ = -a1 - + ( + ) + + : ",{0}UnionFlattenχ[y]--caSumημν[[λ1,λ2]]1-y[[λ1]]γμ[[λ2]]+xSQUARED[y]y[[λ1]]γμ11[[λ2]]-y[[λ1]]πμ[[λ2]],{λ1,4},{λ2,4}+kcxSQUARED[y](one12+one21)
m
4aq
y
λ
λ
γ
2
ky
1
12
1
21
2
k
3
2
y
y
λ
λ
γ
11
1
q
y
λ
λ
π
m
4caq
2
kc
3
1
qca
check χ = -a1 - + ( + ) + + : True
m
4aq
y
λ
λ
γ
2
ky
1
12
1
21
2
k
3
2
y
y
λ
λ
γ
11
1
q
y
λ
λ
π
In[]:=
Print"Check + χ = -a- + + + 2( + ) - + + : ",{0}UnionFlattenTableFullSimplify(+αμ[y][[μ]]+Sum[ημν[[μ,λ]]D[χ[y],y[[λ]]],{λ,4}])--ca-γμ[[μ]]+kcSum[ημν[[ρ1,ρ2]]y[[ρ1]](γμ12[[ρ2]].γμ22[[μ]]+γμ22[[μ]].γμ21[[ρ2]]),{ρ1,4},{ρ2,4}]+2kcy[[μ]](one12+one21)-xSQUARED[y]γμ11[[μ]]+y[[μ]]Sum[ημν[[λ1,λ2]]y[[λ1]]γμ11[[λ2]],{λ1,4},{λ2,4}]+πμ[[μ]],{ca,kc,m,y}∈Reals,{μ,4}
μ
α
μλ
η
′
∂
λ
m
4aq
μ
γ
ky
λ
λ
γ
12
μ
γ
22
μ
γ
22
λ
γ
21
μ
ky
1
12
1
21
2
3
2
k
2
y
μ
γ
11
8
3
2
k
μ
y
y
λ
λ
γ
11
1
q
μ
π
m
4caq
2
3
2
kc
8
3
2
kc
1
q
Check + χ = -a- + + + 2( + ) - + + : True
μ
α
μλ
η
′
∂
λ
m
4aq
μ
γ
ky
λ
λ
γ
12
μ
γ
22
μ
γ
22
λ
γ
21
μ
ky
1
12
1
21
2
3
2
k
2
y
μ
γ
11
8
3
2
k
μ
y
y
λ
λ
γ
11
1
q
μ
π
In[]:=
Print["Check - - qχ = -m : ",{0}Union[Flatten[FullSimplify[Sum[ημν[[κ1,κ2]](+πμ[[κ1]].γμ[[κ2]]-qαμ[y][[κ1]].γμ11[[κ2]]),{κ1,4},{κ2,4}]-Sum[qD[χ[y],y[[κ]]].γμ11[[κ]],{κ,4}]-(-mone11),{ca,kc,m,y}∈Reals]]]]
π
λ
λ
γ
qα
λ
λ
γ
11
′
∂
λ
λ
γ
11
1
11
Check - - qχ = -m : True
π
λ
λ
γ
qα
λ
λ
γ
11
′
∂
λ
λ
γ
11
1
11
In[]:=
Print["Check parity, () = -(-) : ",{0}Union[Flatten[Simplify[Table[βd.Simplify[αμ[y][[μ]],{ca,kc,m,y}∈Reals].βd-(-Simplify[Sum[ημν[[μ,μ1]]αμ[{-y1,-y2,-y3,y4}][[μ1]],{μ1,4}],{ca,kc,m,y}∈Reals]),{μ,4}]]]]]
μ
βα
ν
y
-1
β
α
μ
y
ν
Check parity, () = -(-) : True
μ
βα
ν
y
-1
β
α
μ
y
ν
In[]:=
MaxMemoryUsed[]
Out[]=
124069744
Cite this as: Richard Shurtleff, "An 8x8 Matrix Rep Of Spacetime Symmetries" from the Notebook Archive (2020), https://notebookarchive.org/2020-02-396rs6i
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