4DN=4VMTM_onshell_final
On Shell Algebra and Holoraumy for the 4D N=4 vector tensor multiplet
Notebook Authors: Gabriel Hannon, Rui Xian Siew, and Kory Stiffler

$RecursionLimit = Infinity
<< Adinkra.wl

Function Rules And Shorthands

(*These rules ensure linearity of our derivative operator and give us some nice definitions for operators which will simplify code*)

Dia[tt_][ii_][aa_][Arg1_+Arg2_]:= Dia[tt][ii][aa][Arg1] + Dia[tt][ii][aa][Arg2]

Dia[tt_][ii_][aa_][Arg1_ Arg2_]/;NumberQ[Arg1]:=Arg1 Dia[tt][ii][aa][Arg2]

Dia[tt_][ii_][aa_][Derivative[nt_,nx_,ny_,nz_][Field_][t,x,y,z]]:=D[Dia[tt][ii][aa][Field[t,x,y,z]],{t,nt},{x,nx},{y,ny},{z,nz}]
Dia[tt_][ii_][aa_][Arg1_ Arg2_]/;NumberQ[Arg2]:=Arg2 Dia[tt][ii][aa][Arg1]
Dia[tt_][ii_][aa_][Arg1_]/;NumberQ[Arg1]:= Arg1 Dia[tt][ii][aa]

AlgebraTil0[Field_]:= Table[Dia[0][ii][aa][Dia[0][jj][bb][Field]] + Dia[0][jj][bb][Dia[0][ii][aa][Field]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm

AlgebraTil1[Field_]:= Table[Dia[1][ii][aa][Dia[1][jj][bb][Field]] + Dia[1][jj][bb][Dia[1][ii][aa][Field]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm

AlgebraCross[Field_]:=Table[Dia[0][ii][aa][Dia[1][jj][bb][Field]] + Dia[1][jj][bb][Dia[0][ii][aa][Field]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm

HoloraumyTil0[Field_]:= Table[Dia[0][ii][aa][Dia[0][jj][bb][Field]] -Dia[0][jj][bb][Dia[0][ii][aa][Field]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm

HoloraumyTil1[Field_]:= Table[Dia[1][ii][aa][Dia[1][jj][bb][Field]] -Dia[1][jj][bb][Dia[1][ii][aa][Field]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm

HoloraumyCross[Field_]:=Table[Dia[0][ii][aa][Dia[1][jj][bb][Field]] -Dia[1][jj][bb][Dia[0][ii][aa][Field]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm

UsualAlgebra[Field_]:= Table[2 I KroneckerDelta[ii,jj] Sum[γdown[mu][[aa,bb]]D[Field,Ϛ[mu]],{mu,0,3}]//Simplify,{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//MatrixForm

Transformation Laws, untilded is 0, tilded is 1 for instance:
Dia[0] is untilded D, Dia[1] is tilded (OverTilde[D])
Different numbering convention from the off-shell notebook
A[0] is untilded scalar A
A[1] is tilded scalar (OverTilde[A]), 
Ψ[0] is untilded Ψ
Ψ[1] is tilded (OverTilde[Ψ])

Vector, untilded transformation laws

A

Do[Dia[0][ii][aa][A[0][t,x,y,z]] = Ψ[0][ii][aa][t,x,y,z],{ii,1,2},{aa,1,4}]//Simplify

B

Do[Dia[0][ii][aa][B[0][t,x,y,z]] = I Sum[γ5[[aa,bb]]Ψ[0][ii][bb][t,x,y,z],{bb,1,4}]//Simplify,{ii,1,2},{aa,1,4}]//Simplify

Avec and Fmn

Do[Dia[0][ii][aa][Avec[0][mu][t,x,y,z]] = I Sum[sigma[2][[ii,jj]]γstdown[mu][[aa,bb]]Ψ[0][jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}]//Simplify,{ii,1,2},{aa,1,4},{mu,0,3}]//Simplify

Do[Fmn[mu,nu][t,x,y,z] = D[Avec[0][nu][t,x,y,z],Ϛ[mu]]-D[Avec[0][mu][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3}]

Psi

Do[Dia[0][ii][aa][Ψ[0][jj][bb][t,x,y,z]] = KroneckerDelta[ii,jj](I Sum[γdown[mu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]],{mu,0,3}]
 -Sum[γ5γdown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]],{mu,0,3}])
+(1/4)Sum[sigma[2][[ii,jj]]CommuteGammadown[mu,nu][[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}]//Simplify,{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify

Check that these laws run

(*Run supercovar derive to check*)

Dia[0][2][1][A[0][t,x,y,z]]

Dia[0][1][3][B[0][t,x,y,z]]

Dia[0][1][2][Avec[0][2][t,x,y,z]]

Dia[0][1][2][Ψ[0][1][2][t,x,y,z]]

Dia[0][1][2][Ψ[0][2][1][t,x,y,z]]

Tensor, untilded transformation laws

A

Do[Dia[0][ii][aa][A[1][t,x,y,z]] = Sum[sigma[3][[ii, jj]]Ψ[1][jj][aa][t,x,y,z], {jj, 1, 2}]//Simplify, {ii, 1,2}, {aa, 1,4}]

B

Do[Dia[0][ii][aa][B[1][t,x,y,z]] = I Sum[γ5[[aa,bb]]Ψ[1][ii][bb][t,x,y,z],{bb,1,4}], {ii, 1,2}, {aa, 1,4}]

cphi

Do[Dia[0][ii][aa][φ[1][t,x,y,z]] = Sum[sigma[1][[ii,jj]]Ψ[1][jj][aa][t,x,y,z],{jj,1,2}]//Simplify,{ii,1,2}, {aa, 1,4}]

Bmn

Do[Dia[0][ii][aa][Bmn[1][mu,nu][t,x,y,z]] = ((-I )/4)Sum[sigma[2][[ii,jj]]CommuteGammastdown[mu,nu][[aa,bb]]Ψ[1][jj][bb][t,x,y,z],{jj,1,2},{bb, 1,4}]//Simplify, {ii, 1,2}, {aa, 1,4}, {mu, 0, 3}, {nu, 0,3}]

Define Symmetries on Bmn

Bmn[1][1,0][t,x,y,z]=-Bmn[1][0,1][t,x,y,z];
Bmn[1][2,0][t,x,y,z]=-Bmn[1][0,2][t,x,y,z];
Bmn[1][3,0][t,x,y,z]=-Bmn[1][0,3][t,x,y,z];
Bmn[1][2,1][t,x,y,z]=-Bmn[1][1,2][t,x,y,z];
Bmn[1][1,3][t,x,y,z]=-Bmn[1][3,1][t,x,y,z];
Bmn[1][3,2][t,x,y,z] =-Bmn[1][2,3][t,x,y,z];
Do[Bmn[1][mi,mi][t,x,y,z] = 0,{mi,0,3}];

psi

Do[Dia[0][ii][aa][Ψ[1][jj][bb][t,x,y,z]] = KroneckerDelta[ii,jj](-Sum[γ5γdown[mu][[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]],{mu,0,3}])+I sigma[3][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[A[1][t,x,y,z],Ϛ[mu]],{mu,0,3}] + I sigma[1][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[mu]],{mu,0,3}] -I sigma[2][[ii,jj]] Sum[ϵdownupupup[mu,nu,ap,bt]γ5γdown[mu][[aa,bb]]D[Bmn[1][ap,bt][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

Hmna Field Strength

Do[Hmna[mu,nu,al][t,x,y,z]=D[Bmn[1][nu,al][t,x,y,z], Ϛ[mu]] + D[Bmn[1][al,mu][t,x,y,z], Ϛ[nu]]+D[Bmn[1][mu,nu][t,x,y,z], Ϛ[al]],{mu,0,3},{nu,0,3},{al,0,3}]

Check Laws Running

Dia[0][2][1][A[1][t,x,y,z]]

Dia[0][1][3][B[1][t,x,y,z]]

Dia[0][2][1][φ[1][t,x,y,z]]

Dia[0][1][2][Bmn[1][2,3][t,x,y,z]]

Dia[0][1][2][Ψ[1][2][1][t,x,y,z]]

Vector Tensor, tilded transformation laws

A

Do[Dia[1][ii][aa][A[0][t,x,y,z]] = I Sum[sigma[2][[ii,jj]]Ψ[1][jj][aa][t,x,y,z],{jj,1,2}],{ii,1,2},{aa,1,4}]//Simplify

B

Do[Dia[1][ii][aa][B[0][t,x,y,z]] = -Sum[sigma[2][[ii,jj]]γ5[[aa,bb]]Ψ[1][jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}]//Simplify,{ii,1,2},{aa,1,4}]//Simplify

Avec

Do[Dia[1][ii][aa][Avec[0][mu][t,x,y,z]] = Sum[KroneckerDelta[ii,jj]γstdown[mu][[aa,bb]]Ψ[1][jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}]//Simplify,{ii,1,2},{aa,1,4},{mu,0,3}]//Simplify

Psi

Do[Dia[1][ii][aa][Ψ[0][jj][bb][t,x,y,z]] =  I sigma[2][[ii,jj]](-Sum[γ5γdown[mu][[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]],{mu,0,3}])
- I sigma[1][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[A[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+  I sigma[3][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
- KroneckerDelta[ii,jj] Sum[ϵdownupupup[mu,nu,ap,bt]γ5γdown[mu][[aa,bb]]D[Bmn[1][ap,bt][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

Check that these laws run

(*Run supercovar derive to check*)

Dia[1][2][1][A[0][t,x,y,z]]

Dia[1][1][3][B[0][t,x,y,z]]

Dia[1][1][2][Avec[0][2][t,x,y,z]]

Dia[1][1][2][Ψ[0][1][2][t,x,y,z]]

Dia[1][1][2][Ψ[0][2][1][t,x,y,z]]

Tensor Vector

A

Do[Dia[1][ii][aa][A[1][t,x,y,z]] = - Sum[sigma[1][[ii, jj]]Ψ[0][jj][aa][t,x,y,z], {jj, 1, 2}]//Simplify, {ii, 1,2}, {aa, 1,4}]

B

Do[Dia[1][ii][aa][B[1][t,x,y,z]] = - 1 Sum[sigma[2][[ii,jj]]γ5[[aa,bb]]Ψ[0][jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}], {ii, 1,2}, {aa, 1,4}]

cphi

Do[Dia[1][ii][aa][φ[1][t,x,y,z]] = 1 Sum[sigma[3][[ii,jj]]Ψ[0][jj][aa][t,x,y,z],{jj,1,2}]//Simplify,{ii,1,2}, {aa, 1,4}]

Bmn

Do[Dia[1][ii][aa][Bmn[1][mu,nu][t,x,y,z]] =
-(1/4)Sum[KroneckerDelta[ii,jj]CommuteGammastdown[mu,nu][[aa,bb]]Ψ[0][jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}]//Simplify,{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4},{mu,0,3},{nu,0,3}]//Simplify

psi

Do[Dia[1][ii][aa][Ψ[1][jj][bb][t,x,y,z]] = 
 I sigma[2][[ii,jj]](I Sum[γdown[mu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]],{mu,0,3}]
 -Sum[γ5γdown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]],{mu,0,3}])
-(I/4)Sum[KroneckerDelta[ii,jj]CommuteGammadown[mu,nu][[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}]//Simplify,{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify

Check Laws Running

Dia[1][2][1][A[1][t,x,y,z]]

Dia[1][1][3][B[1][t,x,y,z]]

Dia[1][2][1][φ[1][t,x,y,z]]

Dia[1][1][3][Bmn[1][0,3][t,x,y,z]]

Dia[1][1][2][Ψ[1][2][1][t,x,y,z]]

Λ

Do[ΛTilTil[mu,aa,bb,ii,jj] = 2I Sum[KroneckerDelta[ii,jj]γdown[al][[aa,bb]]Bmn[1][al,mu][t,x,y,z],{al,0,3}]
+ I sigma[1][[ii,jj]]γstdowndown[mu][[aa,bb]]A[1][t,x,y,z]
-I sigma[3][[ii,jj]]γstdowndown[mu][[aa,bb]]φ[1][t,x,y,z]
+I sigma[2][[ii,jj]]γ5γstdowndown[mu][[aa,bb]]B[1][t,x,y,z],{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4},{mu,0,3}]

Do[ΛTil[mu,aa,bb,ii,jj] = (-1/2)sigma[2][[ii,jj]]Sum[CommuteGammastdownstupdown[mu,al][[aa,bb]]Avec[0][al][t,x,y,z],{al,0,3}] -I KroneckerDelta[ii,jj]γstdowndown[mu][[aa,bb]]A[0][t,x,y,z]-KroneckerDelta[ii,jj]γ5γstdowndown[mu][[aa,bb]]B[0][t,x,y,z],{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4},{mu,0,3}]

Do[λTil[aa,bb,ii,jj]=  sigma[2][[ii,jj]]Sum[CommuteGammadown[al,be][[aa,bb]]Bmn[1][al,be][t,x,y,z],{al,0,3},{be,0,3}]+  2I sigma[3][[ii,jj]]Cmetric[[aa,bb]]A[1][t,x,y,z]+  2I sigma[1][[ii,jj]]Cmetric[[aa,bb]]φ[1][t,x,y,z],{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]

Algebra Checks

Z, for {D,D} on untilded Psi

Z[ii_,jj_,kk_,ll_][aa_,bb_,cc_,dd_]:=(-3I/4)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,ll]γdown[al][[aa,bb]] (γstdown[al])[[cc,dd]],{al,0,3}]+(-I/32) Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,ll](CommuteGammadown[al,nu][[aa,bb]](CommuteGammastdown[al,nu])[[cc,dd]]),{al,0,3},{nu,0,3}]+(-I/4)Sum[sigma[1][[ii,jj]]sigma[1][[kk,ll]]γdown[al][[aa,bb]]((γstdown[al])[[cc,dd]]),{al,0,3}]+(I/32) Sum[sigma[1][[ii,jj]]sigma[1][[kk,ll]](CommuteGammadown[al,nu][[aa,bb]](CommuteGammastdown[al,nu])[[cc,dd]]),{al,0,3},{nu,0,3}]+(I/32) Sum[sigma[3][[ii,jj]]sigma[3][[kk,ll]](CommuteGammadown[al,nu][[aa,bb]](CommuteGammastdown[al,nu])[[cc,dd]]),{al,0,3},{nu,0,3}]+(-I/4)Sum[sigma[3][[ii,jj]]sigma[3][[kk,ll]]γdown[al][[aa,bb]]((γstdown[al])[[cc,dd]]),{al,0,3}]+(-I/4) Sum[sigma[2][[ii,jj]]sigma[2][[kk,ll]]γ5γdown[al][[aa,bb]]((γ5γstdown[al])[[cc,dd]]),{al,0,3}]+(-3I/4) sigma[2][[ii,jj]]sigma[2][[kk,ll]]Cmetric[[aa,bb]](KroneckerDelta[cc,dd])+(-3I/4) sigma[2][[ii,jj]]sigma[2][[kk,ll]]γ5down[[aa,bb]](γ5[[cc,dd]])

Vector 00: {D,D}

A

AlgebraTil0[A[0][t,x,y,z]] == UsualAlgebra[A[0][t,x,y,z]]//Simplify

B

AlgebraTil0[B[0][t,x,y,z]] == UsualAlgebra[B[0][t,x,y,z]]//Simplify

Psi

psiVMAlg00=Table[(2I)Sum[KroneckerDelta[ii,jj]γdown[mu][[aa,bb]]D[Ψ[0][kk][cc][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+(-3I/4)Sum[KroneckerDelta[ii,jj]γdown[mu][[aa,bb]]( (γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][kk][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4}]
+(-I/32) Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg](CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-I/4)Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(I/32) Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]](CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(I/32) Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]](CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-I/4)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-I/4) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-3I/4) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]Cmetric[[aa,bb]](γ[la][[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(-3I/4) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]γ5down[[aa,bb]](γ5γ[la][[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}],{aa,1,4},{bb,1,4},{cc,1,4},{ii,1,2},{jj,1,2},{kk,1,2}]//Simplify;

psiVMAlg00Paper = Table[(2I)Sum[KroneckerDelta[ii,jj]γdown[mu][[aa,bb]]D[Ψ[0][kk][cc][t,x,y,z],Ϛ[mu]],{mu,0,3}] +Sum[ Z[ii,jj,kk,ll][aa,bb,cc,dd]γ[mu][[dd,ee]]D[Ψ[0][ll][ee][t,x,y,z],Ϛ[mu]],{mu,0,3},{ee,1,4},{ll,1,2},{dd,1,4}],{aa,1,4},{bb,1,4},{cc,1,4},{ii,1,2},{jj,1,2},{kk,1,2}];

psiVMAlg00Math =Table[Dia[0][ii][aa][Dia[0][jj][bb][Ψ[0][kk][cc][t,x,y,z]]] + Dia[0][jj][bb][Dia[0][ii][aa][Ψ[0][kk][cc][t,x,y,z]]],{aa,1,4},{bb,1,4},{cc,1,4},{ii,1,2},{jj,1,2},{kk,1,2}]//Simplify;

psiVMAlg00==psiVMAlg00Math//Simplify

(psiVMAlg00Paper)==psiVMAlg00Math//Simplify

Field Strength

Table[AlgebraTil0[Fmn[mu,nu][t,x,y,z]]== UsualAlgebra[Fmn[mu,nu][t,x,y,z]],{mu,0,3},{nu,0,3}]//Simplify//MatrixForm

Vector

VecAlg00 = Table[ 2I KroneckerDelta[ii,jj]Sum[γdown[nu][[aa,bb]]Fmn[nu,mu][t,x,y,z],{nu,0,3}]+2 I sigma[2][[ii,jj]](I Cmetric[[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]]-γ5down[[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]]),{mu,0,3},{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify//MatrixForm;

VecAlg00Math = Table[Dia[0][ii][aa][Dia[0][jj][bb][Avec[0][mu][t,x,y,z]]]+Dia[0][jj][bb][Dia[0][ii][aa][Avec[0][mu][t,x,y,z]]],{mu,0,3},{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify//MatrixForm;

VecAlg00== VecAlg00Math//Simplify

Tensor 00: {D,D}

A

AlgebraTil0[A[1][t,x,y,z]]==UsualAlgebra[A[1][t,x,y,z]]//Simplify

B

AlgebraTil0[B[1][t,x,y,z]]==UsualAlgebra[B[1][t,x,y,z]]//Simplify

φ

AlgebraTil0[φ[1][t,x,y,z]]==UsualAlgebra[φ[1][t,x,y,z]]//Simplify

Psi, non off-shell closure zerms are referred to as Z-tilde in the paper

psiTMAlg00=Table[Sum[2I KroneckerDelta[ii,jj]γdown[mu][[aa,bb]]D[Ψ[1][kk][cc][t,x,y,z],Ϛ[mu]],{mu,0,3}]
-(I/2)Sum[KroneckerDelta[ii,jj]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][kk][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4}]
-(I/2)Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
-(I/2)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+ (I/2) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}],{aa,1,4},{bb,1,4},{cc,1,4},{ii,1,2},{jj,1,2},{kk,1,2}]//Simplify//MatrixForm;

psiTMAlg00Math=Table[Dia[0][ii][aa][Dia[0][jj][bb][Ψ[1][kk][cc][t,x,y,z]]] + Dia[0][jj][bb][Dia[0][ii][aa][Ψ[1][kk][cc][t,x,y,z]]],{aa,1,4},{bb,1,4},{cc,1,4},{ii,1,2},{jj,1,2},{kk,1,2}]//Simplify//MatrixForm;

psiTMAlg00==psiTMAlg00Math//Simplify

Field Strength

Table[AlgebraTil0[Hmna[mu,nu,al][t,x,y,z]]== UsualAlgebra[Hmna[mu,nu,al][t,x,y,z]],{mu,0,3},{nu,0,3},{al,0,3}]//Simplify//MatrixForm

Bmn

BmnAlg00 = Table[ 2I KroneckerDelta[ii,jj]Sum[γdown[al][[aa,bb]]Hmna[al,mu,nu][t,x,y,z],{al,0,3}]
+  I sigma[1][[ii,jj]]γstdowndown[mu][[aa,bb]]D[A[1][t,x,y,z],Ϛ[nu]]
-  I sigma[1][[ii,jj]]γstdowndown[nu][[aa,bb]]D[A[1][t,x,y,z],Ϛ[mu]]
-  I sigma[2][[ii,jj]]γ5γstdowndown[mu][[aa,bb]]D[B[1][t,x,y,z],Ϛ[nu]]
+  I sigma[2][[ii,jj]]γ5γstdowndown[nu][[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]]
-  I sigma[3][[ii,jj]]γstdowndown[mu][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[nu]]
+  I sigma[3][[ii,jj]]γstdowndown[nu][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{aa,1,4},{bb,1,4},{ii,1,2},{jj,1,2}]//Simplify//MatrixForm;

BmnAlg00Math = Table[Dia[0][ii][aa][Dia[0][jj][bb][Bmn[1][mu,nu][t,x,y,z]]]+ Dia[0][jj][bb][Dia[0][ii][aa][Bmn[1][mu,nu][t,x,y,z]]],{mu,0,3},{nu,0,3},{aa,1,4},{bb,1,4},{ii,1,2},{jj,1,2}]//Simplify//MatrixForm;

BmnAlg00Math== BmnAlg00//Simplify

Vector 11: {(OverTilde[D]),(OverTilde[D])}

A

AlgebraTil1[A[0][t,x,y,z]] == UsualAlgebra[A[0][t,x,y,z]]//Simplify

B

AlgebraTil1[B[0][t,x,y,z]] == UsualAlgebra[B[0][t,x,y,z]]//Simplify

Psi, non off-shell closure zerms are referred to as caligraphic Z in the paper

psiVMAlg11=Table[Sum[ 2I KroneckerDelta[ii,jj]γdown[mu][[aa,bb]]D[Ψ[0][kk][cc][t,x,y,z],Ϛ[mu]],{mu,0,3}]
-(I/2)Sum[KroneckerDelta[ii,jj]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][kk][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4}]
+(I/2)Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(I/2)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+ (I/2) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}],{aa,1,4},{bb,1,4},{cc,1,4},{ii,1,2},{jj,1,2},{kk,1,2}]//Simplify//MatrixForm;

psiVMAlg11Math=Table[Dia[1][ii][aa][Dia[1][jj][bb][Ψ[0][kk][cc][t,x,y,z]]] + Dia[1][jj][bb][Dia[1][ii][aa][Ψ[0][kk][cc][t,x,y,z]]],{aa,1,4},{bb,1,4},{cc,1,4},{ii,1,2},{jj,1,2},{kk,1,2}]//Simplify//MatrixForm;

psiVMAlg11==psiVMAlg11Math//Simplify

Field Strength

Table[AlgebraTil1[Fmn[mu,nu][t,x,y,z]]==  UsualAlgebra[Fmn[mu,nu][t,x,y,z]],{mu,0,3},{nu,0,3}]//Simplify//MatrixForm

Vector

VecAlg11 =Table[2I KroneckerDelta[ii,jj]Sum[γdown[nu][[aa,bb]]Fmn[nu,mu][t,x,y,z],{nu,0,3}]+- 2 I sigma[2][[ii,jj]](I Cmetric[[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]]- γ5down[[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]]),{mu,0,3},{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify//MatrixForm;

VecAlg11Math = Table[Dia[1][ii][aa][Dia[1][jj][bb][Avec[0][mu][t,x,y,z]]]+Dia[1][jj][bb][Dia[1][ii][aa][Avec[0][mu][t,x,y,z]]],{mu,0,3},{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify//MatrixForm;

VecAlg11Math == VecAlg11//Simplify

Tensor 11: {(OverTilde[D]),(OverTilde[D])}

A

AlgebraTil1[A[1][t,x,y,z]]== UsualAlgebra[A[1][t,x,y,z]]//Simplify

B

AlgebraTil1[B[1][t,x,y,z]]== UsualAlgebra[B[1][t,x,y,z]]//Simplify

cphi

AlgebraTil1[φ[1][t,x,y,z]]== UsualAlgebra[φ[1][t,x,y,z]]//Simplify

Psi, non off-shell closure zerms are referred to as caligraphic Z-tilde in the paper

psiTMAlg11Math=Table[Dia[1][ii][aa][Dia[1][jj][bb][Ψ[1][kk][cc][t,x,y,z]]] + Dia[1][jj][bb][Dia[1][ii][aa][Ψ[1][kk][cc][t,x,y,z]]],{aa,1,4},{bb,1,4},{cc,1,4},{ii,1,2},{jj,1,2},{kk,1,2}]//Simplify//MatrixForm;

psiTMAlg11=Table[(2I)Sum[KroneckerDelta[ii,jj]γdown[mu][[aa,bb]]D[Ψ[1][kk][cc][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+(-3I/4)Sum[KroneckerDelta[ii,jj]γdown[mu][[aa,bb]]( (γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][kk][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4}]
+(-I/32) Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg](CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(I/4)Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-I/32) Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]](CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-I/32) Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]](CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(I/4)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-I/4) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-3I/4) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]Cmetric[[aa,bb]](γ[la][[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(-3I/4) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]γ5down[[aa,bb]](γ5γ[la][[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}],{aa,1,4},{bb,1,4},{cc,1,4},{ii,1,2},{jj,1,2},{kk,1,2}]//Simplify//MatrixForm;

psiTMAlg11==psiTMAlg11Math//Simplify

Bmn

BmnAlg11 = Table[ 2 I KroneckerDelta[ii,jj]Sum[γdown[al][[aa,bb]]D[Bmn[1][mu,nu][t,x,y,z],Ϛ[al]],{al,0,3}]
-D[ΛTilTil[nu,aa,bb,ii,jj],Ϛ[mu]]+D[ΛTilTil[mu,aa,bb,ii,jj],Ϛ[nu]],{mu,0,3},{nu,0,3},{aa,1,4},{bb,1,4},{ii,1,2},{jj,1,2}]//Simplify//MatrixForm;

BmnAlg11Math = Table[Dia[1][ii][aa][Dia[1][jj][bb][Bmn[1][mu,nu][t,x,y,z]]]+ Dia[1][jj][bb][Dia[1][ii][aa][Bmn[1][mu,nu][t,x,y,z]]],{mu,0,3},{nu,0,3},{aa,1,4},{bb,1,4},{ii,1,2},{jj,1,2}]//Simplify//MatrixForm;

BmnAlg11Math== BmnAlg11//Simplify

Vector Cross: {D,(OverTilde[D])}

A

Table[-2I sigma[1][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[A[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+2I sigma[3][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
== Dia[0][ii][aa][Dia[1][jj][bb][A[0][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][A[0][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

AVMAlgCrossMath = Table[Dia[0][ii][aa][Dia[1][jj][bb][A[0][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][A[0][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

AVMAlgCross= Table[-2I sigma[1][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[A[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+2I sigma[3][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[mu]],{mu,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

AVMAlgCrossMath== AVMAlgCross//Simplify

B

Table[ 2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+(2/3)I KroneckerDelta[ii,jj]Sum[ϵup[mu,nu,al,be]γstdowndown[mu][[aa,bb]]Hmna[nu,al,be][t,x,y,z],{nu,0,3},{al,0,3},{be,0,3},{mu,0,3}]
== Dia[0][ii][aa][Dia[1][jj][bb][B[0][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][B[0][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BVMAlgCrossMath = Table[Dia[0][ii][aa][Dia[1][jj][bb][B[0][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][B[0][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BVMAlgCross= Table[ 2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+(2/3)I KroneckerDelta[ii,jj]Sum[ϵup[mu,nu,al,be]γstdowndown[mu][[aa,bb]]Hmna[nu,al,be][t,x,y,z],{nu,0,3},{al,0,3},{be,0,3},{mu,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BVMAlgCrossMath== BVMAlgCross//Simplify

Siew Psi with Better Gamma Basis

X: renamed 5-9 as 4-8 as there previously was no 4
Untilded X as in paper (hence the X1[0] for instance)

Do[X1[0][ii,jj,kk,ll] =(I/4) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-3/4)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(3/4) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-I/4) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X2[0][ii,jj,kk,ll] =(-I/8)  KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-1/8)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(1/8) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(I/8) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X3[0][ii,jj,kk,ll] =(-I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(3/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-3/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(5I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X4[0][ii,jj,kk,ll] =(-I/4)  KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-1/4)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(1/4) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(I/4) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X5[0][ii,jj,kk,ll] =(3I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-1/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(1/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X6[0][ii,jj,kk,ll] =(-9I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-1/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(1/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X7[0][ii,jj,kk,ll] =(3I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-5/16)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(5/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(5I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X8[0][ii,jj,kk,ll] =(-I/128) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-1/128)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(1/128) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(I/128) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Calc Psi

psiVMAlgCross=Table[ 2I Sum[X1[0][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψ[1][ll][cc][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3}]
+ 2I Sum[X2[0][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+  2I Sum[X3[0][ii,jj,kk,ll]γstdowndown[nu][[aa,bb]](γ[nu].γ[mu])[[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+  2I Sum[X4[0][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}]
+  2I Sum[X5[0][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]](γ5γ[nu].γ[mu])[[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+  2I Sum[X6[0][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}]
+  2I Sum[X7[0][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}]
+ 2I Sum[X8[0][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[la])[[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[la]],{ll,1,2},{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4}],{ii,1,2},{jj,1,2},{kk,1,2}, {aa, 1,4}, {bb, 1,4},{cc,1,4}]//Simplify//MatrixForm;

psiVMAlgCrossMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][Ψ[0][kk][cc][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][Ψ[0][kk][cc][t,x,y,z]]],{ii,1,2},{jj,1,2},{kk,1,2}, {aa, 1,4}, {bb, 1,4},{cc,1,4}]//Simplify//MatrixForm;

psiVMAlgCrossMath==psiVMAlgCross//Simplify

Vector

Table[-KroneckerDelta[ii,jj]Sum[(γ5.CommuteGammastdowndown[mu,nu])[[aa,bb]]UD[B[1][t,x,y,z],Ϛ[nu]],{nu,0,3}]
-2 sigma[2][[ii,jj]]Sum[CommuteGammadown[al,be][[aa,bb]]D[Bmn[1][be,mu][t,x,y,z],Ϛ[al]],{al,0,3},{be,0,3}]
-D[λTil[aa,bb,ii,jj],Ϛ[mu]]
== Dia[0][ii][aa][Dia[1][jj][bb][Avec[0][mu][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][Avec[0][mu][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3}]//Simplify//MatrixForm;

VecCrosAlg = Table[-KroneckerDelta[ii,jj]Sum[(γ5.CommuteGammastdowndown[mu,nu])[[aa,bb]]UD[B[1][t,x,y,z],Ϛ[nu]],{nu,0,3}]
-2 sigma[2][[ii,jj]]Sum[CommuteGammadown[al,be][[aa,bb]]D[Bmn[1][be,mu][t,x,y,z],Ϛ[al]],{al,0,3},{be,0,3}]
-D[λTil[aa,bb,ii,jj],Ϛ[mu]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3}]//Simplify//MatrixForm;

VecCrossAlgMath = Table[Dia[0][ii][aa][Dia[1][jj][bb][Avec[0][mu][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][Avec[0][mu][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3}]//Simplify//MatrixForm;

VecCrosAlg== VecCrossAlgMath//Simplify

Tensor Cross: {D,(OverTilde[D])}

A

Table[-2I sigma[1][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]],{mu,0,3}]
==  Dia[0][ii][aa][Dia[1][jj][bb][A[1][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][A[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

ATMAlgCross= Table[-2I sigma[1][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]],{mu,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

ATMAlgCrossMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][A[1][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][A[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

ATMAlgCrossMath== ATMAlgCross//Simplify

B

Table[2  sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+(1/2) KroneckerDelta[ii,jj]Sum[(γ5.CommuteGammadown[mu,nu])[[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}]
==  Dia[0][ii][aa][Dia[1][jj][bb][B[1][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][B[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BTMAlgCross=Table[2  sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+(1/2) KroneckerDelta[ii,jj]Sum[(γ5.CommuteGammadown[mu,nu])[[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BTMAlgCrossMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][B[1][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][B[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BTMAlgCross==BTMAlgCrossMath//Simplify

φ

Table[  2I sigma[3][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]],{mu,0,3}]
== Dia[0][ii][aa][Dia[1][jj][bb][φ[1][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][φ[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

cphiAlgCross=Table[  2I sigma[3][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]],{mu,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

cphiAlgCrossMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][φ[1][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][φ[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

cphiAlgCrossMath==cphiAlgCross//Simplify

Bmn

Table[(-1/2)KroneckerDelta[ii,jj]Sum[(γ5γ[al].CommuteGammastdowndown[mu,nu])[[aa,bb]]D[B[0][t,x,y,z],Ϛ[al]],{al,0,3}]
-(1/2)sigma[2][[ii,jj]]Sum[CommuteGammastdowndown[al,mu][[aa,bb]]UD[Avec[0][nu][t,x,y,z],Ϛ[al]],{al,0,3}]
+(1/2)sigma[2][[ii,jj]]Sum[CommuteGammastdowndown[al,nu][[aa,bb]]UD[Avec[0][mu][t,x,y,z],Ϛ[al]],{al,0,3}]
-D[ΛTil[nu,aa,bb,ii,jj],Ϛ[mu]]
+D[ΛTil[mu,aa,bb,ii,jj],Ϛ[nu]]
== Dia[0][ii][aa][Dia[1][jj][bb][Bmn[1][mu,nu][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][Bmn[1][mu,nu][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3},{nu,0,3}]//Simplify//MatrixForm;

BmnCrossAlg=Table[(-1/2)KroneckerDelta[ii,jj]Sum[(γ5γ[al].CommuteGammastdowndown[mu,nu])[[aa,bb]]D[B[0][t,x,y,z],Ϛ[al]],{al,0,3}]
-(1/2)sigma[2][[ii,jj]]Sum[CommuteGammastdowndown[al,mu][[aa,bb]]UD[Avec[0][nu][t,x,y,z],Ϛ[al]],{al,0,3}]
+(1/2)sigma[2][[ii,jj]]Sum[CommuteGammastdowndown[al,nu][[aa,bb]]UD[Avec[0][mu][t,x,y,z],Ϛ[al]],{al,0,3}]
-D[ΛTil[nu,aa,bb,ii,jj],Ϛ[mu]]
+D[ΛTil[mu,aa,bb,ii,jj],Ϛ[nu]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3},{nu,0,3}]//Simplify//MatrixForm;

BmnCrossAlgMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][Bmn[1][mu,nu][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][Bmn[1][mu,nu][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3},{nu,0,3}]//Simplify//MatrixForm;

BmnCrossAlgMath==BmnCrossAlg//Simplify

Siew Psi with Better Gamma Basis

Tilded X (hence the X1[1] for instance)

Do[X1[1][ii,jj,kk,ll] =(-I/4) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-3/4)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(3/4) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-I/4) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X2[1][ii,jj,kk,ll] =(I/8)  KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-1/8)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(1/8) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(I/8) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X3[1][ii,jj,kk,ll] =(I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(3/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-3/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(5I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X4[1][ii,jj,kk,ll] =(-I/4)  KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(1/4)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-1/4) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-I/4) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X5[1][ii,jj,kk,ll] =(3I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(1/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-1/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X6[1][ii,jj,kk,ll] =(-9I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(1/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-1/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X7[1][ii,jj,kk,ll] =(3I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(5/16)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-5/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-5I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X8[1][ii,jj,kk,ll] =(I/128) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-1/128)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(1/128) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(I/128) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Calc Psi

Table[ 2I Sum[X1[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψ[0][ll][cc][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3}]
+ 2I Sum[X2[1][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+  2I Sum[X3[1][ii,jj,kk,ll]γstdowndown[nu][[aa,bb]](γ[nu].γ[mu])[[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+  2I Sum[X4[1][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}]
+  2I Sum[X5[1][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]](γ5γ[nu].γ[mu])[[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+  2I Sum[X6[1][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}]
+  2I Sum[X7[1][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}]
+ 2I Sum[X8[1][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[la])[[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[la]],{ll,1,2},{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4}]
== Dia[0][ii][aa][Dia[1][jj][bb][Ψ[1][kk][cc][t,x,y,z]]] + Dia[1][jj][bb][Dia[0][ii][aa][Ψ[1][kk][cc][t,x,y,z]]],{ii,1,2},{jj,1,2},{kk,1,2}, {aa, 1,4}, {bb, 1,4},{cc,1,4}]//Simplify//MatrixForm

Holoraumy

Vector 00

A

AVMHol00 = Table[-2KroneckerDelta[ii,jj](Sum[γ5γdown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]],{mu,0,3}])
+((1/2))sigma[2][[ii,jj]]Sum[CommuteGammadown[mu,nu][[aa,bb]]
Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify//MatrixForm;

HoloraumyTil0[A[0][t,x,y,z]]== AVMHol00//Simplify

B

BVMHol00 = Table[2KroneckerDelta[ii,jj](Sum[γ5γdown[mu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]],{mu,0,3}])
+((I/2))sigma[2][[ii,jj]]Sum[(γ5.CommuteGammadown[mu,nu])[[aa,bb]]
Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify//MatrixForm;

HoloraumyTil0[B[0][t,x,y,z]]== BVMHol00//Simplify

Psi, in the body of the paper is quoted the results upon actually enforcing the dirac equation. Below are also given the results including these Dirac Equation terms

PsiVMHol00=Table[ (2I)Sum[ sigma[2][[ii,jj]]sigma[2][[kk,gg]]γdown[mu][[aa,bb]]D[Ψ[0][gg][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{gg,1,2}]
+(2I)Sum[ sigma[1][[ii,jj]]sigma[1][[kk,gg]]γ5γdown[mu][[aa,bb]](γ5[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{gg,1,2},{dd,1,4}]
+ (2I)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]γ5γdown[mu][[aa,bb]](γ5[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{gg,1,2},{dd,1,4}]
+(-3I/4) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]

+(-I/32)Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]CommuteGammadown[mu,nu][[aa,bb]]((CommuteGammastdown[mu,nu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]

+(-I/4)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]

+(-3I/4)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]

+(-3I/4)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]

+(-3I/4) Sum[sigma[1][[ii,jj]]sigma[1][[gg,kk]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+  (3I/4)Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(3I/4)Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(-3I/4) Sum[sigma[3][[ii,jj]]sigma[3][[gg,kk]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(3I/4)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(3I/4)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//Simplify;

cV1[VM,0,0][ii_,jj_,kk_,ll_]:=sigma[2][[ii,jj]]sigma[2][[kk,ll]]
cV2[VM,0,0][ii_,jj_,kk_,ll_]:=sigma[1][[ii,jj]]sigma[1][[kk,ll]] + sigma[3][[ii,jj]]sigma[3][[kk,ll]]

PsiVMHol00Paper =2 I Table[ Sum[(cV1[VM,0,0][ii,jj,kk,ll]γdown[mu][[aa,bb]]KroneckerDelta[cc,dd]  + cV2[VM,0,0][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]])D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] ,{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}];

PsiVMHol00Math = Table[Dia[0][ii][aa][Dia[0][jj][bb][Ψ[0][kk][cc][t,x,y,z]]]-Dia[0][jj][bb][Dia[0][ii][aa][Ψ[0][kk][cc][t,x,y,z]]],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//Simplify;

PsiVMHol00==PsiVMHol00Math//Simplify

PsiVMHol00Paper==PsiVMHol00Math//Simplify

DiracEquationΨ[ti_][ll_][aa_]:=Sum[γup[mu][[aa,bb]]D[Ψ[ti][ll][bb][t,x,y,z],Ϛ[mu]],{mu,0,3},{bb,1,4}]

EnforceDiracEquation = Table[Solve[DiracEquationΨ[ti][ll][aa]==0,D[Ψ[ti][ll][aa][t,x,y,z],t]],{ti,0,1},{ll,1,2},{aa,1,4}]//Flatten

Collect[(PsiVMHol00Paper==PsiVMHol00Math)//.EnforceDiracEquation,Derivative[_,_,_,_][Ψ[_][_][_]][t,x,y,z],Simplify]

Vector

AvecHol00 =Table[sigma[2][[ii,jj]](Sum[CommuteGammastdownstupdown[mu,nu][[aa,bb]]D[A[0][t,x,y,z], Ϛ[nu]],{nu,0,3}]+  I Sum[γ5[[aa,cc]]CommuteGammastdownstupdown[mu,nu][[cc,bb]]D[B[0][t,x,y,z],Ϛ[nu]],{nu,0,3},{cc,1,4}])+- KroneckerDelta[ii,jj]Sum[ϵdownupupup[mu,nu,al,be]γ5γstdowndown[nu][[aa,bb]]Fmn[al,be][t,x,y,z],{nu,0,3},{al,0,3},{be,0,3}],{mu,0,3},{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify//MatrixForm;

AvecHol00Math= Table[Dia[0][ii][aa][Dia[0][jj][bb][Avec[0][mu][t,x,y,z]]]-Dia[0][jj][bb][Dia[0][ii][aa][Avec[0][mu][t,x,y,z]]],{mu,0,3},{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify//MatrixForm;

AvecHol00Math== AvecHol00//Simplify

Tensor 00

A

ATMHol00 = Table[2sigma[3][[ii,jj]](-Sum[γ5γdown[mu][[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]],{mu,0,3}])
+2sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+ (( 2)/3)sigma[1][[ii,jj]]Sum[ϵdownupupup[mu,nu,al,be]γ5γdown[mu][[aa,bb]]Hmna[nu,al,be][t,x,y,z],{mu,0,3},{nu,0,3},{al,0,3},{be,0,3}],{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify//MatrixForm;

ATMHol00== HoloraumyTil0[A[1][t,x,y,z]]//Simplify

B

BTMHol00=Table[2Sum[γ5γdown[mu][[aa,bb]](sigma[3][[ii,jj]]D[A[1][t,x,y,z],Ϛ[mu]]+sigma[1][[ii,jj]]D[φ[1][t,x,y,z],Ϛ[mu]]),{mu,0,3}]
- (2/3)sigma[2][[ii,jj]]Sum[ϵdownupupup[mu,nu,al,be]γdown[mu][[aa,bb]]Hmna[nu,al,be][t,x,y,z],{mu,0,3},{nu,0,3},{al,0,3},{be,0,3}],{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify//MatrixForm;

BTMHol00== HoloraumyTil0[B[1][t,x,y,z]]//Simplify

φ

cphiHol00= Table[2sigma[1][[ii,jj]](-Sum[γ5γdown[mu][[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]],{mu,0,3}])
-2sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[A[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
-(( 2)/3)sigma[3][[ii,jj]]Sum[ϵdownupupup[mu,nu,al,be]γ5γdown[mu][[aa,bb]]Hmna[nu,al,be][t,x,y,z],{mu,0,3},{nu,0,3},{al,0,3},{be,0,3}],{ii,1,2},{jj,1,2},{aa,1,4},{bb,1,4}]//Simplify//MatrixForm;

cphiHol00== HoloraumyTil0[φ[1][t,x,y,z]]//Simplify

Psi, in the body of the paper is quoted the results upon actually enforcing the dirac equation. Below are also given the results including these Dirac Equation terms

PsiTMHol00 = Table[+(-I) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]CommuteGammadown[mu,nu][[aa,bb]](γstdown[mu])[[cc,dd]]D[Ψ[1][gg][dd][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3},{dd,1,4},{gg,1,2}]
+(2I)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]γ5γdown[mu][[aa,bb]]((γ5)[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{dd,1,4},{gg,1,2}]
+(I/2)Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(I/8) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]CommuteGammadown[mu,nu][[aa,bb]]((CommuteGammastdown[mu,nu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-I/2) Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(I)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(-I)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+  (-I/2) Sum[sigma[1][[ii,jj]]sigma[1][[gg,kk]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+ (-I) Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+ (I) Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(-I/2) Sum[sigma[3][[ii,jj]]sigma[3][[gg,kk]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-I)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(I)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//Simplify;

PsiTMHol00Math = Table[Dia[0][ii][aa][Dia[0][jj][bb][Ψ[1][kk][cc][t,x,y,z]]]-Dia[0][jj][bb][Dia[0][ii][aa][Ψ[1][kk][cc][t,x,y,z]]],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//Simplify;

PsiTMHol00== PsiTMHol00Math//Simplify

Check Paper

cV1[TM,0,0][ii_,jj_,kk_,ll_]:=KroneckerDelta[ii,jj]KroneckerDelta[kk,ll]
cV2[TM,0,0][ii_,jj_,kk_,ll_]:=sigma[2][[ii,jj]]sigma[2][[kk,ll]]

PsiTMHol00Paper = Table[ Sum[2 I cV1[TM,0,0][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}] +Sum[I cV2[TM,0,0][ii,jj,kk,ll]CommuteGammadown[mu,ld][[aa,bb]]γstdown[ld][[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{ld,0,3},{dd,1,4}],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//Simplify;

PsiTMHol00Paper==PsiTMHol00Math

(PsiTMHol00Paper==PsiTMHol00Math//.EnforceDiracEquation)//Simplify

Bmn

BmnHol00 = Table[sigma[2][[ii,jj]](Sum[ϵdowndownupup[mu,nu,de,la]γstdowndown[de][[aa,bb]]D[B[1][t,x,y,z],Ϛ[la]],{la,0,3},{de,0,3}])+ Sum[ϵdowndownupup[mu,nu,de,la]γ5γstdowndown[de][[aa,bb]](-sigma[1][[ii,jj]]D[A[1][t,x,y,z],Ϛ[la]]+sigma[3][[ii,jj]]D[φ[1][t,x,y,z],Ϛ[la]]),{de,0,3},{la,0,3}]
-(1/3) KroneckerDelta[ii,jj]Sum[ϵdownupupup[mu,la,al,be]γ5γstdowndown[nu][[aa,bb]]Hmna[la,al,be][t,x,y,z],{la,0,3},{al,0,3},{be,0,3}]
+(1/3)KroneckerDelta[ii,jj]Sum[ϵdownupupup[nu,la,al,be]γ5γstdowndown[mu][[aa,bb]]Hmna[la,al,be][t,x,y,z],{la,0,3},{al,0,3},{be,0,3}],{mu,0,3},{nu,0,3},{aa,1,4},{bb,1,4},{ii,1,2},{jj,1,2}]//Simplify//MatrixForm;

BmnHol00Math = Table[Dia[0][ii][aa][Dia[0][jj][bb][Bmn[1][mu,nu][t,x,y,z]]]-Dia[0][jj][bb][Dia[0][ii][aa][Bmn[1][mu,nu][t,x,y,z]]],{mu,0,3},{nu,0,3},{aa,1,4},{bb,1,4},{ii,1,2},{jj,1,2}]//Simplify//MatrixForm;

BmnHol00Math== BmnHol00//Simplify

Vector 11

A

AVMHolTil =Table[ KroneckerDelta[ii,jj](-2 Sum[γ5γdown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]],{mu,0,3}])
-(1/2) sigma[2][[ii,jj]]Sum[CommuteGammadown[mu,nu][[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

AVMHolTilMath=Table[Dia[1][ii][aa][Dia[1][jj][bb][A[0][t,x,y,z]]] -Dia[1][jj][bb][Dia[1][ii][aa][A[0][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

AVMHolTilMath==AVMHolTil//Simplify

B

BVMHolTil=Table[ KroneckerDelta[ii,jj](2 Sum[γ5γdown[mu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]],{mu,0,3}])
- I(1/2) sigma[2][[ii,jj]]Sum[(γ5.CommuteGammadown[mu,nu])[[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BVMHolTilMath=Table[Dia[1][ii][aa][Dia[1][jj][bb][B[0][t,x,y,z]]] -Dia[1][jj][bb][Dia[1][ii][aa][B[0][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BVMHolTil==BVMHolTilMath//Simplify

Psi, in the body of the paper is quoted the results upon actually enforcing the dirac equation. Below are also given the results including these Dirac Equation terms

psiVMHolTil=Table[
(-I) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]CommuteGammadown[mu,nu][[aa,bb]](γstdown[mu])[[cc,dd]]D[Ψ[0][gg][dd][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3},{dd,1,4},{gg,1,2}]
+(2I)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]γ5γdown[mu][[aa,bb]]((γ5)[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{dd,1,4},{gg,1,2}]
+(I/2)Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(I/8) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]CommuteGammadown[mu,nu][[aa,bb]]((CommuteGammastdown[mu,nu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-I/2) Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(I)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(-I)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+  (I/2) Sum[sigma[1][[ii,jj]]sigma[1][[gg,kk]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+ (I) Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+ (-I) Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(I/2) Sum[sigma[3][[ii,jj]]sigma[3][[gg,kk]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(I)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(-I)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[0][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//Simplify;

psiVMHolTilMath =Table[Dia[1][ii][aa][Dia[1][jj][bb][Ψ[0][kk][cc][t,x,y,z]]]-Dia[1][jj][bb][Dia[1][ii][aa][Ψ[0][kk][cc][t,x,y,z]]],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//Simplify;

psiVMHolTilMath==psiVMHolTil//Simplify

Check Paper

cV1[VM,1,1][ii_,jj_,kk_,ll_]:=KroneckerDelta[ii,jj]KroneckerDelta[kk,ll]
cV2[VM,1,1][ii_,jj_,kk_,ll_]:=sigma[2][[ii,jj]]sigma[2][[kk,ll]]

PsiVMHolTilPaper = I Table[Sum[(2cV1[VM,1,1][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]+ Sum[cV2[VM,1,1][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]],{nu,0,3}])D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{dd,1,4},{ll,1,2}],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//Simplify;

PsiVMHolTilPaper==psiVMHolTilMath

(PsiVMHolTilPaper==psiVMHolTilMath//.EnforceDiracEquation)//Simplify

Vector

Table[-sigma[2][[ii,jj]](Sum[CommuteGammastdownstupdown[mu,nu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[nu]],{nu,0,3}]
+( I) Sum[(γ5.CommuteGammastdownstupdown[mu,nu])[[aa,bb]]D[B[0][t,x,y,z],Ϛ[nu]],{nu,0,3}])
- KroneckerDelta[ii,jj]Sum[ϵdownupupup[mu,de,la,ka]γ5γstdowndown[de][[aa,bb]]Fmn[la,ka][t,x,y,z],{la,0,3},{ka,0,3},{de,0,3}]
==Dia[1][ii][aa][Dia[1][jj][bb][Avec[0][mu][t,x,y,z]]] -Dia[1][jj][bb][Dia[1][ii][aa][Avec[0][mu][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3}]//Simplify//MatrixForm;

AvecHolTil=Table[-sigma[2][[ii,jj]](Sum[CommuteGammastdownstupdown[mu,nu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[nu]],{nu,0,3}]
+( I) Sum[(γ5.CommuteGammastdownstupdown[mu,nu])[[aa,bb]]D[B[0][t,x,y,z],Ϛ[nu]],{nu,0,3}])
- KroneckerDelta[ii,jj]Sum[ϵdownupupup[mu,de,la,ka]γ5γstdowndown[de][[aa,bb]]Fmn[la,ka][t,x,y,z],{la,0,3},{ka,0,3},{de,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3}]//Simplify//MatrixForm;

AvecHolTilMath=Table[Dia[1][ii][aa][Dia[1][jj][bb][Avec[0][mu][t,x,y,z]]] -Dia[1][jj][bb][Dia[1][ii][aa][Avec[0][mu][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3}]//Simplify//MatrixForm;

AvecHolTilMath==AvecHolTil//Simplify

Tensor 11

A

ATMHolTil =Table[sigma[3][[ii,jj]](2 Sum[γ5γdown[mu][[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]],{mu,0,3}])
+2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+(2/3) sigma[1][[ii,jj]]Sum[ϵdownupupup[mu,nu,al,be]γ5γdown[mu][[aa,bb]]Hmna[nu,al,be][t,x,y,z],{mu,0,3},{nu,0,3},{al,0,3},{be,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

ATMHolTilMath=Table[Dia[1][ii][aa][Dia[1][jj][bb][A[1][t,x,y,z]]] -Dia[1][jj][bb][Dia[1][ii][aa][A[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

ATMHolTilMath==ATMHolTil//Simplify

B

BTMHolTil=Table[- 2 sigma[1][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
- 2 sigma[3][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]D[A[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+(2/3) sigma[2][[ii,jj]]Sum[ϵdownupupup[mu,nu,al,be]γdown[mu][[aa,bb]]Hmna[nu,al,be][t,x,y,z],{mu,0,3},{nu,0,3},{al,0,3},{be,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BTMHolTilMath=Table[Dia[1][ii][aa][Dia[1][jj][bb][B[1][t,x,y,z]]] -Dia[1][jj][bb][Dia[1][ii][aa][B[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BTMHolTilMath==BTMHolTil//Simplify

cphi

chpiHolTil=Table[sigma[1][[ii,jj]]( 2 Sum[γ5γdown[mu][[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]],{mu,0,3}])
- 2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[A[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
-(2/3) sigma[3][[ii,jj]]Sum[ϵdownupupup[mu,nu,al,be]γ5γdown[mu][[aa,bb]]Hmna[nu,al,be][t,x,y,z],{mu,0,3},{nu,0,3},{al,0,3},{be,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

chpiHolTilMath=Table[Dia[1][ii][aa][Dia[1][jj][bb][φ[1][t,x,y,z]]] -Dia[1][jj][bb][Dia[1][ii][aa][φ[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

chpiHolTilMath==chpiHolTil//Simplify

Psi, in the body of the paper is quoted the results upon actually enforcing the dirac equation. Below are also given the results including these Dirac Equation terms

PsiTMHolTil =Table[ (2I)Sum[ sigma[2][[ii,jj]]sigma[2][[kk,gg]]γdown[mu][[aa,bb]]D[Ψ[1][gg][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{gg,1,2}]
+(-2I)Sum[ sigma[1][[ii,jj]]sigma[1][[kk,gg]]γ5γdown[mu][[aa,bb]](γ5[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{gg,1,2},{dd,1,4}]
+ (-2I)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]γ5γdown[mu][[aa,bb]](γ5[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{gg,1,2},{dd,1,4}]
+(-3I/4) Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]γdown[mu][[aa,bb]]((γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-I/32)Sum[sigma[2][[ii,jj]]sigma[2][[kk,gg]]CommuteGammadown[mu,nu][[aa,bb]]((CommuteGammastdown[mu,nu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-I/4)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-3I/4)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(-3I/4)Sum[KroneckerDelta[ii,jj]KroneckerDelta[kk,gg]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(3I/4) Sum[sigma[1][[ii,jj]]sigma[1][[gg,kk]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+ (-3I/4)Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(-3I/4)Sum[sigma[1][[ii,jj]]sigma[1][[kk,gg]]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(3I/4) Sum[sigma[3][[ii,jj]]sigma[3][[gg,kk]]γ5γdown[mu][[aa,bb]]((γ5γstdown[mu].γ[la])[[cc,dd]])D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{mu,0,3},{la,0,3},{dd,1,4},{gg,1,2}]
+(-3I/4)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]Cmetric[[aa,bb]]((γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}]
+(-3I/4)Sum[sigma[3][[ii,jj]]sigma[3][[kk,gg]]γ5down[[aa,bb]]((γ5γ[la][[cc,dd]]))D[Ψ[1][gg][dd][t,x,y,z],Ϛ[la]],{la,0,3},{dd,1,4},{gg,1,2}],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//Simplify;

PsiTMHolTilMath =Table[Dia[1][ii][aa][Dia[1][jj][bb][Ψ[1][kk][cc][t,x,y,z]]] - Dia[1][jj][bb][Dia[1][ii][aa][Ψ[1][kk][cc][t,x,y,z]]],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//Simplify;

PsiTMHolTilMath==PsiTMHolTil//Simplify

Check Paper

cV1[TM,1,1][ii_,jj_,kk_,ll_]:=sigma[2][[ii,jj]]sigma[2][[kk,ll]]
cV2[TM,1,1][ii_,jj_,kk_,ll_]:=-(sigma[1][[ii,jj]]sigma[1][[kk,ll]] + sigma[3][[ii,jj]]sigma[3][[kk,ll]])

PsiTMHolTilPaper = 2 I Table[Sum[(cV1[TM,1,1][ii,jj,kk,ll]γdown[mu][[aa,bb]]KroneckerDelta[cc,dd]+cV2[TM,1,1][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]])D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{dd,1,4},{ll,1,2}],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//Simplify;

PsiTMHolTilPaper==PsiTMHolTilMath

(PsiTMHolTilPaper==PsiTMHolTilMath//.EnforceDiracEquation)//Simplify

Bmn

BmnHolTil=Table[-sigma[2][[ii,jj]]( Sum[ϵdowndownupup[mu,nu,de,la]γstdowndown[de][[aa,bb]]D[ B[1][t,x,y,z],Ϛ[la]],{la,0,3},{de,0,3}])
-  sigma[1][[ii,jj]]Sum[ϵdowndownupup[mu,nu,de,la]γ5γstdowndown[de][[aa,bb]]D[A[1][t,x,y,z],Ϛ[la]],{la,0,3},{de,0,3}]
+   sigma[3][[ii,jj]]Sum[ϵdowndownupup[mu,nu,de,la]γ5γstdowndown[de][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[la]],{la,0,3},{de,0,3}]
-(1/3)  KroneckerDelta[ii,jj]Sum[ϵdownupupup[mu,ka,al,be]γ5γstdowndown[nu][[aa,bb]]Hmna[ka,al,be][t,x,y,z],{ka,0,3},{al,0,3},{be,0,3}]
+(1/3)  KroneckerDelta[ii,jj]Sum[ϵdownupupup[nu,ka,al,be]γ5γstdowndown[mu][[aa,bb]]Hmna[ka,al,be][t,x,y,z],{ka,0,3},{al,0,3},{be,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3},{nu,0,3}]//Simplify//MatrixForm;

BmnHolTilMath=Table[Dia[1][ii][aa][Dia[1][jj][bb][Bmn[1][mu,nu][t,x,y,z]]] -Dia[1][jj][bb][Dia[1][ii][aa][Bmn[1][mu,nu][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3},{nu,0,3}]//Simplify//MatrixForm;

BmnHolTilMath==BmnHolTil//Simplify

Vector Cross

A

AVMHolCross=Table[sigma[2][[ii,jj]](2I Sum[γ5γdown[mu][[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]],{mu,0,3}])
-(2/3) KroneckerDelta[ii,jj]Sum[ϵdownupupup[mu,nu,al,be]γ5γdown[mu][[aa,bb]]Hmna[nu,al,be][t,x,y,z],{mu,0,3},{nu,0,3},{al,0,3},{be,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

AVMHolCrossMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][A[0][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][A[0][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

AVMHolCrossMath==AVMHolCross//Simplify

B

Table[-2 sigma[1][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]D[A[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+2 sigma[3][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]== Dia[0][ii][aa][Dia[1][jj][bb][B[0][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][B[0][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BVMHolCross= Table[-2 sigma[1][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]D[A[1][t,x,y,z],Ϛ[mu]],{mu,0,3}]
+2 sigma[3][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[mu]],{mu,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BVMHolCrossMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][B[0][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][B[0][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BVMHolCrossMath==BVMHolCross//Simplify

Vector

Table[2 KroneckerDelta[ii,jj](γ5down)[[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]]
+I sigma[3][[ii,jj]]Sum[CommuteGammastdownstupdown[mu,la][[aa,bb]]D[A[1][t,x,y,z],Ϛ[la]],{la,0,3}]
+I sigma[1][[ii,jj]]Sum[CommuteGammastdownstupdown[mu,la][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[la]],{la,0,3}]
+(2 I/3) sigma[2][[ii,jj]]Sum[ϵdownupupup[mu,nu,al,be](γ5down)[[aa,bb]]Hmna[nu,al,be][t,x,y,z],{nu,0,3},{al,0,3},{be,0,3}]
== Dia[0][ii][aa][Dia[1][jj][bb][Avec[0][mu][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][Avec[0][mu][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3}]//Simplify//MatrixForm;

AvecHolCross=Table[2 KroneckerDelta[ii,jj](γ5down)[[aa,bb]]D[B[1][t,x,y,z],Ϛ[mu]]
+I sigma[3][[ii,jj]]Sum[CommuteGammastdownstupdown[mu,la][[aa,bb]]D[A[1][t,x,y,z],Ϛ[la]],{la,0,3}]
+I sigma[1][[ii,jj]]Sum[CommuteGammastdownstupdown[mu,la][[aa,bb]]D[φ[1][t,x,y,z],Ϛ[la]],{la,0,3}]
+(2 I/3) sigma[2][[ii,jj]]Sum[ϵdownupupup[mu,nu,al,be](γ5down)[[aa,bb]]Hmna[nu,al,be][t,x,y,z],{nu,0,3},{al,0,3},{be,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3}]//Simplify//MatrixForm;

AvecHolCrossMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][Avec[0][mu][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][Avec[0][mu][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3}]//Simplify//MatrixForm;

AvecHolCrossMath==AvecHolCross//Simplify

Siew Psi with Better Gamma Basis, in the body of the paper is quoted the results upon actually enforcing the dirac equation. Below are also given the results including these Dirac Equation terms

X

Do[X1[6][ii,jj,kk,ll] =(I/4) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(1/4)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-1/4) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-I/4) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X2[6][ii,jj,kk,ll] =(I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-3/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(3/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(3I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X3[6][ii,jj,kk,ll] =(-I/8)  KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-1/8)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(1/8) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(I/8) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X4[6][ii,jj,kk,ll] =(-3I/128) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-3/128)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(3/128) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(3I/128) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X5[6][ii,jj,kk,ll] =(3I/4)  KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-1/4)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(1/4) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-3I/4) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X6[6][ii,jj,kk,ll] =(-3I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(1/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-1/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(7I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X7[6][ii,jj,kk,ll] =(-3I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(5/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-5/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-5I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X8[6][ii,jj,kk,ll] =(9I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(1/16)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-1/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Calc Psi

psiVMHolCross=Table[ 2I Sum[X1[6][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψ[1][ll][cc][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3}]
+  2I Sum[X2[6][ii,jj,kk,ll]γstdowndown[nu][[aa,bb]](γ[nu].γ[mu])[[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+ 2I Sum[X3[6][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+ 2I Sum[X4[6][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[la])[[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[la]],{ll,1,2},{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4}]
+  2I Sum[X5[6][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}]
+  2I Sum[X6[6][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]](γ5γ[nu].γ[mu])[[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+  2I Sum[X7[6][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}]
+  2I Sum[X8[6][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}]
,{ii,1,2},{jj,1,2},{kk,1,2}, {aa, 1,4}, {bb, 1,4},{cc,1,4}]//Simplify;

psiVMHolCrossMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][Ψ[0][kk][cc][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][Ψ[0][kk][cc][t,x,y,z]]],{ii,1,2},{jj,1,2},{kk,1,2}, {aa, 1,4}, {bb, 1,4},{cc,1,4}]//Simplify;

psiVMHolCross==psiVMHolCrossMath//Simplify

Check Paper

cV[xx_][ii_,jj_,kk_,ll_]:= ahat[xx]sigma[2][[ii,jj]]sigma[2][[kk,ll]] + bhat[xx](sigma[1][[ii,jj]]sigma[1][[kk,ll]] + sigma[3][[ii,jj]]sigma[3][[kk,ll]])+khat[xx]KroneckerDelta[ii,jj]KroneckerDelta[kk,ll]
straightV[xx_][ii_,jj_,kk_,ll_]:=Sum[I cV[xx][ii,rr,kk,ll]sigma[2][[rr,jj]],{rr,1,2}]

SetabkVM = {ahat[1]-> 1/4,bhat[1]->1/4,khat[1]->-1/4,ahat[2]->-1/8,bhat[2]->-1/8,khat[2]->1/8,ahat[3]->3/4,bhat[3]->-1/4,khat[3]->-3/4}

PsiVMHolCrossPaper =  Table[Sum[2 I straightV[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψ[1][ll][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2}]+2 I Sum[(Sum[straightV[2][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]],{nu,0,3}]+ straightV[3][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]])D[Ψ[1][ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//.SetabkVM//Simplify;

PsiVMHolCrossPaper==psiVMHolCrossMath

((PsiVMHolCrossPaper==psiVMHolCrossMath)//.EnforceDiracEquation)//Simplify

Tensor Cross

A

Table[sigma[1][[ii,jj]](2 Sum[γ5γdown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]],{mu,0,3}])
+ I(1/2) sigma[3][[ii,jj]]Sum[(CommuteGammadown[mu,nu])[[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}]
== Dia[0][ii][aa][Dia[1][jj][bb][A[1][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][A[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

ATMHolCross= Table[sigma[1][[ii,jj]](2 Sum[γ5γdown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]],{mu,0,3}])
+ I(1/2) sigma[3][[ii,jj]]Sum[(CommuteGammadown[mu,nu])[[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

ATMHolCrossMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][A[1][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][A[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

ATMHolCrossMath==ATMHolCross//Simplify

B

Table[sigma[2][[ii,jj]](-2I Sum[γ5γdown[mu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]],{mu,0,3}])
== Dia[0][ii][aa][Dia[1][jj][bb][B[1][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][B[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BTMHolCross=Table[sigma[2][[ii,jj]](-2I Sum[γ5γdown[mu][[aa,bb]]D[A[0][t,x,y,z],Ϛ[mu]],{mu,0,3}]),{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BTMHolCrossMath =Table[ Dia[0][ii][aa][Dia[1][jj][bb][B[1][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][B[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

BTMHolCrossMath==BTMHolCrossMath//Simplify

φ

Table[sigma[3][[ii,jj]](-  2 Sum[γ5γdown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]],{mu,0,3}])
+ (I/2) sigma[1][[ii,jj]]Sum[(CommuteGammadown[mu,nu])[[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}]
== Dia[0][ii][aa][Dia[1][jj][bb][φ[1][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][φ[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

cphiHolCross=Table[sigma[3][[ii,jj]](-  2 Sum[γ5γdown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]],{mu,0,3}])
+ (I/2) sigma[1][[ii,jj]]Sum[(CommuteGammadown[mu,nu])[[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

cphiHolCrossMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][φ[1][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][φ[1][t,x,y,z]]],{ii,1,2},{jj,1,2}, {aa, 1,4}, {bb, 1,4}]//Simplify//MatrixForm;

cphiHolCrossMath==cphiHolCross//Simplify

Bmn

Table[KroneckerDelta[ii,jj]Sum[ϵdowndownupup[mu,nu,de,la]γ5γstdowndown[de][[aa,bb]]D[A[0][t,x,y,z],Ϛ[la]],{la,0,3},{de,0,3}]
+KroneckerDelta[ii,jj](γ5γstdowndown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[nu]]-γ5γstdowndown[nu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]])
+(1/2) sigma[2][[ii,jj]] Sum[(γstdown[mu].γdown[de])[[aa,bb]]Fmn[nu,de][t,x,y,z],{de,0,3}]
- (1/2)  sigma[2][[ii,jj]]Sum[(γstdown[nu].γdown[de])[[aa,bb]]Fmn[mu,de][t,x,y,z],{de,0,3}]
+(I/2) sigma[2][[ii,jj]] Sum[ϵdowndownupup[mu,nu,ka,de]γ5down[[aa,bb]]Fmn[ka,de][t,x,y,z],{ka,0,3},{de,0,3}]

-(I/128) sigma[2][[ii,jj]]Sum[ϵdownupupup[nu,ka,de,la](γ5.CommuteGammastdowndown[la,mu])[[aa,bb]]Fmn[ka,de][t,x,y,z],{ka,0,3},{de,0,3},{la,0,3},{al,0,3},{be,0,3}]
+(I/128) sigma[2][[ii,jj]]Sum[ϵdownupupup[mu,ka,de,la](γ5.CommuteGammastdowndown[la,nu])[[aa,bb]]Fmn[ka,de][t,x,y,z],{ka,0,3},{de,0,3},{la,0,3},{al,0,3},{be,0,3}]
== Dia[0][ii][aa][Dia[1][jj][bb][Bmn[1][mu,nu][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][Bmn[1][mu,nu][t,x,y,z]]],{ii,1,1},{jj,2,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3},{nu,0,3}]//Simplify

BmnHolCross=Table[KroneckerDelta[ii,jj]Sum[ϵdowndownupup[mu,nu,de,la]γ5γstdowndown[de][[aa,bb]]D[A[0][t,x,y,z],Ϛ[la]],{la,0,3},{de,0,3}]
+KroneckerDelta[ii,jj](γ5γstdowndown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[nu]]-γ5γstdowndown[nu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]])
+(1/2) sigma[2][[ii,jj]] Sum[(γstdown[mu].γdown[de])[[aa,bb]]Fmn[nu,de][t,x,y,z],{de,0,3}]
- (1/2)  sigma[2][[ii,jj]]Sum[(γstdown[nu].γdown[de])[[aa,bb]]Fmn[mu,de][t,x,y,z],{de,0,3}]
+(I/2) sigma[2][[ii,jj]] Sum[ϵdowndownupup[mu,nu,ka,de]γ5down[[aa,bb]]Fmn[ka,de][t,x,y,z],{ka,0,3},{de,0,3}]
-(I/128) sigma[2][[ii,jj]]Sum[ϵdownupupup[nu,ka,de,la](γ5.CommuteGammastdowndown[la,mu])[[aa,bb]]Fmn[ka,de][t,x,y,z],{ka,0,3},{de,0,3},{la,0,3},{al,0,3},{be,0,3}]
+(I/128) sigma[2][[ii,jj]]Sum[ϵdownupupup[mu,ka,de,la](γ5.CommuteGammastdowndown[la,nu])[[aa,bb]]Fmn[ka,de][t,x,y,z],{ka,0,3},{de,0,3},{la,0,3},{al,0,3},{be,0,3}],{ii,1,1},{jj,2,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3},{nu,0,3}]//Simplify;

BmnHolCrossPaper =Table[KroneckerDelta[ii,jj]Sum[ϵdowndownupup[mu,nu,de,la]γ5γstdowndown[de][[aa,bb]]D[A[0][t,x,y,z],Ϛ[la]],{la,0,3},{de,0,3}]
+KroneckerDelta[ii,jj](γ5γstdowndown[mu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[nu]]-γ5γstdowndown[nu][[aa,bb]]D[B[0][t,x,y,z],Ϛ[mu]])
+ Cmetric[[aa,bb]]sigma[2][[ii,jj]] Sum[KroneckerDelta[mu,ro]D[Avec[0][ro][t,x,y,z],Ϛ[nu]]-KroneckerDelta[nu,ro]D[Avec[0][ro][t,x,y,z],Ϛ[mu]],{ro,0,3}]
+I sigma[2][[ii,jj]] Sum[ϵdowndownupup[mu,nu,la,ro]γ5down[[aa,bb]]D[Avec[0][ro][t,x,y,z],Ϛ[la]],{la,0,3},{ro,0,3}],{ii,1,1},{jj,2,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3},{nu,0,3}]//Simplify;

BmnHolCrossMath =Table[Dia[0][ii][aa][Dia[1][jj][bb][Bmn[1][mu,nu][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][Bmn[1][mu,nu][t,x,y,z]]],{ii,1,1},{jj,2,2}, {aa, 1,4}, {bb, 1,4},{mu,0,3},{nu,0,3}]//Simplify;

BmnHolCrossMath==BmnHolCross//Simplify

BmnHolCrossPaper ==BmnHolCrossMath//Simplify

BmnHolCrossPaper ==BmnHolCross//Simplify

Siew Psi with Better Gamma Basis, in the body of the paper is quoted the results upon actually enforcing the dirac equation. Below are also given the results including these Dirac Equation terms

X

Do[X1[5][ii,jj,kk,ll] =(I/4) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-1/4)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(1/4) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(I/4) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X2[5][ii,jj,kk,ll] =(I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(3/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-3/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-3I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X3[5][ii,jj,kk,ll] =(-I/8)  KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(1/8)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-1/8) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-I/8) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X4[5][ii,jj,kk,ll] =(-3I/128) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(3/128)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-3/128) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-3I/128) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X5[5][ii,jj,kk,ll] =(-3I/4)  KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(-1/4)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(1/4) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-3I/4) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X6[5][ii,jj,kk,ll] =(3I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(1/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-1/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(7I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X7[5][ii,jj,kk,ll] =(3I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(5/16) sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-5/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-5I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Do[X8[5][ii,jj,kk,ll] =(-9I/16) KroneckerDelta[ii,jj]sigma[2][[kk,ll]]
+(1/16)sigma[1][[ii,jj]]sigma[3][[kk,ll]]
+(-1/16) sigma[3][[ii,jj]]sigma[1][[kk,ll]]
+(-I/16) sigma[2][[ii,jj]]KroneckerDelta[kk,ll],{ii,1,2},{jj,1,2},{kk,1,2},{ll,1,2}]

Psi Calc

psiTMHolCross=Table[ 2I Sum[X1[5][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψ[0][ll][cc][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3}]
+  2I Sum[X2[5][ii,jj,kk,ll]γstdowndown[nu][[aa,bb]](γ[nu].γ[mu])[[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+ 2I Sum[X3[5][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+ 2I Sum[X4[5][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[la])[[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[la]],{ll,1,2},{mu,0,3},{nu,0,3},{la,0,3},{dd,1,4}]
+  2I Sum[X5[5][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}]
+  2I Sum[X6[5][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]](γ5γ[nu].γ[mu])[[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{nu,0,3},{dd,1,4}]
+  2I Sum[X7[5][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}]
+  2I Sum[X8[5][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{ll,1,2},{mu,0,3},{dd,1,4}],{ii,1,2},{jj,1,2},{kk,1,2}, {aa, 1,4}, {bb, 1,4},{cc,1,4}]//Simplify;

psiTMHolCrossMath=Table[Dia[0][ii][aa][Dia[1][jj][bb][Ψ[1][kk][cc][t,x,y,z]]] - Dia[1][jj][bb][Dia[0][ii][aa][Ψ[1][kk][cc][t,x,y,z]]],{ii,1,2},{jj,1,2},{kk,1,2}, {aa, 1,4}, {bb, 1,4},{cc,1,4}]//Simplify;

psiTMHolCrossMath==psiTMHolCross//Simplify

Check Paper

straightVt[xx_][ii_,jj_,kk_,ll_]:=Sum[I cV[xx][ii,rr,kk,ll]sigma[2][[rr,jj]],{rr,1,2}]

SetabkTM= {ahat[1]-> 1/4,bhat[1]->-1/4,khat[1]->1/4,ahat[3]->-1/8,bhat[3]->1/8,khat[3]->-1/8,ahat[5]->-3/4,bhat[5]->-1/4,khat[5]->-3/4}

PsiTMHolCrossPaper =  Table[Sum[2 I straightVt[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψ[0][ll][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2}]+2 I Sum[(Sum[straightVt[3][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]],{nu,0,3}]+ straightVt[5][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]])D[Ψ[0][ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}],{ii,1,2},{jj,1,2},{kk,1,2},{aa,1,4},{bb,1,4},{cc,1,4}]//.SetabkTM//Simplify;

PsiTMHolCrossPaper==psiTMHolCrossMath

((PsiTMHolCrossPaper==psiTMHolCrossMath)//.EnforceDiracEquation)//Simplify

Final Holoraumy Trues

HoloraumyTil0[A[0][t,x,y,z]]== AVMHol00//Simplify
AVMHolTilMath==AVMHolTil//Simplify
ATMHolCrossMath==ATMHolCross//Simplify

HoloraumyTil0[B[0][t,x,y,z]]== BVMHol00//Simplify
BVMHolTilMath==BVMHolTil//Simplify
BTMHolCrossMath==BTMHolCross//Simplify

PsiVMHol00==PsiVMHol00Math//Simplify
psiVMHolTilMath==psiVMHolTil//Simplify
psiVMHolCrossMath==psiVMHolCross//Simplify
(PsiVMHol00Paper==PsiVMHol00Math)//.EnforceDiracEquation//Simplify
(PsiVMHolTilPaper==psiVMHolTilMath//.EnforceDiracEquation)//Simplify
((PsiVMHolCrossPaper==psiVMHolCrossMath)//.EnforceDiracEquation)//Simplify

AvecHol00Math== AvecHol00//Simplify
AvecHolTilMath==AvecHolTil//Simplify
AvecHolCrossMath==AvecHolCross//Simplify

ATMHol00== HoloraumyTil0[A[1][t,x,y,z]]//Simplify
ATMHolTilMath==ATMHolTil//Simplify
ATMHolCrossMath==ATMHolCross//Simplify

BTMHol00== HoloraumyTil0[B[1][t,x,y,z]]//Simplify
BTMHolTilMath==BTMHolTil//Simplify
BTMHolCrossMath==BTMHolCross//Simplify

cphiHol00== HoloraumyTil0[φ[1][t,x,y,z]]//Simplify
chpiHolTilMath==chpiHolTil//Simplify
cphiHolCrossMath==cphiHolCross//Simplify

PsiTMHol00== PsiTMHol00Math//Simplify
PsiTMHolTilMath==PsiTMHolTil//Simplify
psiTMHolCrossMath==psiTMHolCross//Simplify
(PsiTMHol00Paper==PsiTMHol00Math)//.EnforceDiracEquation//Simplify
(PsiTMHolTilPaper==PsiTMHolTilMath//.EnforceDiracEquation)//Simplify
((PsiTMHolCrossPaper==psiTMHolCrossMath)//.EnforceDiracEquation)//Simplify

BmnHol00Math== BmnHol00//Simplify
BmnHolTilMath==BmnHolTil//Simplify
BmnHolCrossMath==BmnHolCross//Simplify

Adinkra reduce the Fermionic, on-shell holoraumy to 1D
The Paper versions go to zero upon enforcing the dirac equation (obviously). I see factors of 1 and 2 in the Holoraumy prior to enforcing DE, indicating a better place to find connections to SM-I through IV would be the “off-shell” holoraumy, which is really on shell anyway (just that the full off-shell multiplet that it is on shell from is not known).

AdinkraReduce = {Ψ[tt_][ii_][aa_][t,x,y,z]-> Ψ[tt][ii][aa][t],D[Ψ[tt_][ii_][aa_][t,x,y,z],t]->D[Ψ[tt][ii][aa][t],t] ,D[Ψ[tt_][ii_][aa_][t,x,y,z],x]->0 ,D[Ψ[tt_][ii_][aa_][t,x,y,z],y]->0,D[Ψ[tt_][ii_][aa_][t,x,y,z],z]->0  }

PsiTo16 = {Ψ[0][1][aa_]-> Function[{t},Ψ16[aa][t]],Ψ[0][2][aa_]-> Function[{t},Ψ16[4+aa][t]],Ψ[1][1][aa_]-> Function[{t},Ψ16[8+aa][t]],Ψ[1][2][aa_]-> Function[{t},Ψ16[12+aa][t]]}

PsiVMHol00Math//.AdinkraReduce//.PsiTo16

psiVMHolTilMath//.AdinkraReduce//.PsiTo16

psiVMHolCrossMath//.AdinkraReduce//.PsiTo16