4D N = 1 CS On Shell 4DCalcs.nb

Setting Rep and Importing Adinkra package

$RecursionLimit=Infinity

Rep = CSOnShell;

<<Adinkra`

BuildDate[Adinkra]

fields: Afield, Bfield,Ffield, Gfield, ψ[a], all free indices covariant
AdinkraReduce
(Subscript[Φ, i]) = {(OverDot[A]),(OverDot[B])}
(Subscript[Ψ, (OverHat[i])])= {(Subscript[(OverDot[ψ]), a])}

AdinkraReduce Encodes reducing the dependence on the fields from 4D space-time to only time, doesn’t matter if this stays off shell

AdinkraReduce={Afield-> Function[{t,x,y,z},Afield[t]],Bfield-> Function[{t,x,y,z},Bfield[t]],Ffield-> Function[{t,x,y,z},Ffield[t]],Gfield-> Function[{t,x,y,z},Gfield[t]],ψ[a_]->Function[{t,x,y,z},ψ[a][t]]}

fields := prints out the adinkra reduced fields in subscript form, doesn’t matter if this stays off shell

fields={ψ[a_][t]->(Subscript[ψ, a]),ψ[a_]'[t]->(Subscript[(OverDot[ψ]), a]),ψ[a_]''[t]->(Subscript[ψ, a])'',
 Afield[t]->A,  Afield'[t]->(OverDot[A]),  Afield''[t]->A'',Bfield[t]->B,  Bfield'[t]->(OverDot[B]),  Bfield''[t]->B'', Ffield[t]->F,  Ffield'[t]->(OverDot[F]), Gfield[t]->G, Gfield'[t]->(OverDot[G])}

ψ[2]'[t]/.fields

ψ[1][t,x,y,z]/.AdinkraReduce/.fields

D[ψ[1][t,x,y,z],t]/.AdinkraReduce/.fields

Phi[t][[i]] = (Subscript[Φ, i]) = {(OverDot[A]),(OverDot[B])}
Psi[t][[ihat]] = (Subscript[Ψ, (OverHat[i])])= {(Subscript[(OverDot[ψ]), a])}

Phi[t] = {Afield'[t],Bfield'[t]}

Psi[t] = -I {ψ[1]'[t],ψ[2]'[t],ψ[3]'[t],ψ[4]'[t]}

Phi[t]/.fields

I Psi[t]/.fields

FieldList:= list of the names of the fields

FieldList = {Afield[t,x,y,z],Bfield[t,x,y,z],ψ[1][t,x,y,z],ψ[2][t,x,y,z],ψ[3][t,x,y,z],ψ[4][t,x,y,z]}

Symmetries, Constraints, and raised indices such as Vfieldup, etc.

ψup[a] := (ψ^a)

Do[ψup[ai][t,x,y,z] = RaiseFermionIndex[ψ[ai][t,x,y,z]],{ai,1,4}]

InverseCmetric//MatrixForm

Table[ψup[ai][t,x,y,z] ,{ai,1,4}]

Check UD[Field_,Ϛ[μ]] := (∂^μ)Field   ,   Lap[Field_] := (∂^μ)(∂_μ)Field defined in SpaceTime2.nb. Doesn’t matter in which order these are taken when strung together.

D[UD[Afield[t,x,y,z],Ϛ[0]],Ϛ[0]]

Lap[D[Afield[t,x,y,z],Ϛ[0]]]

UD[D[Lap[Afield[t,x,y,z]],Ϛ[0]],Ϛ[0]]

Lap[Lap[Bfield[t,x,y,z]]]

Lap[ψ[1][t,x,y,z]]

Transformation Rules

Do[Da[a][Afield[t,x,y,z]]=  ψ[a][t,x,y,z] ,{a,1,4}]

DColor[Afield'[t]] = Table[D[Da[Color][Afield[t,x,y,z]]/.AdinkraReduce,t],{Color,1,4}]

PrintDColor[Afield'[t]]=Table[DColor[Afield'[t]][[Color]],{Color,1,4}]/.fields//MatrixForm

Da[1][Afield[t,x,y,z]]

Da[2][Afield[t,x,y,z]]

Do[Da[a][Bfield[t,x,y,z]]= I Sum[γ5[[a,d]]ψ[d][t,x,y,z],{d,1,4}] ,{a,1,4}]

DColor[Bfield'[t]] = Table[D[Da[Color][Bfield[t,x,y,z]]/.AdinkraReduce,t],{Color,1,4}]

PrintDColor[Bfield'[t]]=Table[DColor[Bfield'[t]][[Color]],{Color,1,4}]/.fields//MatrixForm

Da[1][Bfield[t,x,y,z]]

Do[Da[a][ψ[b][t,x,y,z]] = Sum[I γdown[mu][[a,b]]D[Afield[t,x,y,z],Ϛ[mu]] -γ5γdown[mu][[a,b]]D[Bfield[t,x,y,z],Ϛ[mu]],{mu,0,3}]  ,{a,1,4},{b,1,4}]

Da[3][ψ[1][t,x,y,z]]

Do[DColor[-I ψ[bi]'[t]] = Table[D[-I Da[Color][ψ[bi][t,x,y,z]]/.AdinkraReduce,t],{Color,1,4}],{bi,1,4}]

PrintDColor[-I ψ'[t]]=Table[Collect[DColor[-I ψ[bi]'[t]][[Color]],{Afield'[t],Bfield'[t],Ffield[t],Gfield[t]} ,Simplify],{bi,1,4},{Color,1,4}]/.fields//MatrixForm

DD

SUSYTransformFields[Operator_]:=Table[FieldList[[FieldIndex]]->Operator[FieldList[[FieldIndex]]],{FieldIndex,1,Length[FieldList]}]

SUSYTransformJustB[Operator_]:=Table[FieldList[[FieldIndex]]->Operator[FieldList[[FieldIndex]]],{FieldIndex,1,4}]

Do[SUSYTransform[Da[ai]] = Join[SUSYTransformFields[Da[ai]],D[SUSYTransformFields[Da[ai]],t],D[SUSYTransformFields[Da[ai]],x],D[SUSYTransformFields[Da[ai]],y],D[SUSYTransformFields[Da[ai]],z],D[SUSYTransformJustB[Da[ai]],t,t],D[SUSYTransformJustB[Da[ai]],t,x],D[SUSYTransformJustB[Da[ai]],t,y],D[SUSYTransformJustB[Da[ai]],t,z],D[SUSYTransformJustB[Da[ai]],x,x],D[SUSYTransformJustB[Da[ai]],x,y],D[SUSYTransformJustB[Da[ai]],x,z],D[SUSYTransformJustB[Da[ai]],z,z],D[SUSYTransformJustB[Da[ai]],y,y],D[SUSYTransformJustB[Da[ai]],y,z]],{ai,1,4}]

DD[LeftOperator_,RightOperator_][Field_]:=RightOperator[Field]/.SUSYTransform[LeftOperator]

AlgebraDD[LeftOperator_,RightOperator_][Field_]:= DD[LeftOperator,RightOperator][Field] + DD[RightOperator,LeftOperator][Field]

AlgebraDDTable[Field_] := Table[AlgebraDD[Da[ai],Da[bi]][Field],{ai,1,4},{bi,1,4}]

HoloraumyDD[LeftOperator_,RightOperator_][Field_]:= DD[LeftOperator,RightOperator][Field] - DD[RightOperator,LeftOperator][Field]

HoloraumyDDTable[Field_]:=Table[HoloraumyDD[Da[ai],Da[bi]][Field],{ai,1,4},{bi,1,4}]

Closure[ai_,bi_][Field_]:=Sum[2 I γdown[m][[ai,bi]]D[Field,Ϛ[m]],{m,0,3}]

ClosureTable[Field_]:=Table[Closure[ai,bi][Field],{ai,1,4},{bi,1,4}]

Table[Closure[ii,ji][ψ[ki][t,x,y,z]],{ii,1,4},{ji,1,4},{ki,1,4}]

Algebra

{(Subscript[D, a]),(Subscript[D, b])}A= 2 i (Subscript[((γ^μ)), ab])(∂_μ)A

(AlgebraDDTable[Afield[t,x,y,z]] ==ClosureTable[Afield[t,x,y,z]])//Simplify

{(Subscript[D, a]),(Subscript[D, b])}B= 2 i (Subscript[((γ^μ)), ab])(∂_μ)B

(AlgebraDDTable[Bfield[t,x,y,z]] ==ClosureTable[Bfield[t,x,y,z]])//Simplify

where

Z0[Rep] = Table[Sum[I γdown[ro][[aa,bb]]γγstdownstup[ro,mu][[cc,dd]]D[ψ[dd][t,x,y,z],Ϛ[mu]],{ro,0,3},{mu,0,3},{dd,1,4}],{aa,1,4},{bb,1,4},{cc,1,4}];

(Table[AlgebraDD[Da[aa],Da[bb]][ψ[cc][t,x,y,z]],{aa,1,4},{bb,1,4},{cc,1,4}] ==Table[Closure[aa,bb][ψ[cc][t,x,y,z]] -Z0[Rep][[aa,bb,cc]],{aa,1,4},{bb,1,4},{cc,1,4}])//Simplify

Test that 2 (Subscript[((γ^μ)), ab])((Subscript[δ, c])^d) - (Subscript[((γ^ρ)), ab])((Subscript[((Subscript[γ, ρ])(γ^μ)), c])^d)= (Subscript[((γ^μ)), ab])((Subscript[δ, c])^d)-(1/2)(Subscript[((γ^ρ)), ab])((Subscript[([(Subscript[γ, ρ]),(γ^μ)]), c])^d)

Table[2 γdown[mu][[aa,bb]]KroneckerDelta[cc,dd]  - Sum[γdown[ro][[aa,bb]]γγstdownstup[ro,mu][[cc,dd]],{ro,0,3}],{mu,0,3},{aa,1,4},{bb,1,4},{cc,1,4},{dd,1,4}]== Table[γdown[mu][[aa,bb]]KroneckerDelta[cc,dd]  -(1/2)Sum[γdown[ro][[aa,bb]]CommuteGammastdownstup[ro,mu][[cc,dd]],{ro,0,3}],{mu,0,3},{aa,1,4},{bb,1,4},{cc,1,4},{dd,1,4}]

Holoraumy

[(Subscript[D, a]),(Subscript[D, b])]A = -2(Subscript[((γ^5)(γ^μ)), ab])(∂_μ)B

HoloraumyDDTable[Afield[t,x,y,z]] ==  Table[Sum[-2γ5γdown[mu][[ai,bi]]D[Bfield[t,x,y,z],Ϛ[mu]],{mu,0,3}],{ai,1,4},{bi,1,4}]

[(Subscript[D, a]),(Subscript[D, b])]B= 2(Subscript[((γ^5)(γ^μ)), ab])(∂_μ)A

HoloraumyDDTable[ Bfield[t,x,y,z]] ==  Table[Sum[2γ5γdown[mu][[ai,bi]]D[Afield[t,x,y,z],Ϛ[mu]],{mu,0,3}],{ai,1,4},{bi,1,4}]

Bosonic Holoraumy in su(2) structure:

BHoloTerms[mu_]:=Outer[Times,sigma[2],γ5γdown[mu]]

Dimensions[BHoloTerms[0]]

BHoloTerms[0][[1,2]]== -I γ5γdown[0]

Do[BHolo[Rep][mu] = BHoloTerms[mu],{mu,0,3}];

(({{HoloraumyDDTable[Afield[t,x,y,z]] }, {HoloraumyDDTable[Bfield[t,x,y,z]]}}))==-2I Table[(Sum[Table[BHolo[Rep][mu][[su2a,su2b,aa,bb]],{su2a,1,2},{su2b,1,2}].(({{D[Afield[t,x,y,z],Ϛ[mu]]}, {D[Bfield[t,x,y,z],Ϛ[mu]]}})),{mu,0,3}])[[su2c]],{su2c,1,2},{aa,1,4},{bb,1,4}]//Simplify

with and with

FHoloTerms[mu_][aa_,bb_,cc_,dd_]:=-γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]

Do[FHolo[Rep][mu] = Table[ FHoloTerms[mu][aa,bb,cc,dd],{aa,1,4},{bb,1,4},{cc,1,4},{dd,1,4}],{mu,0,3}]

DEqm[Rep] = Table[Sum[γ[mu][[dd,ee]]D[ψ[ee][t,x,y,z],Ϛ[mu]],{mu,0,3},{ee,1,4}],{dd,1,4}];

ZHoloTerms[aa_,bb_,cc_]:= I Sum[γ5γdown[kp][[aa,bb]]γ5γstdown[kp][[cc,dd]]DEqm[Rep][[dd]],{kp,0,3},{dd,1,4}]

ZHolo[Rep] = Table[ZHoloTerms[aa,bb,cc],{aa,1,4},{bb,1,4},{cc,1,4}];

Table[HoloraumyDD[Da[aa],Da[bb]][ψ[cc][t,x,y,z]],{aa,1,4},{bb,1,4},{cc,1,4}]==-2I Table[Sum[FHolo[Rep][mu][[aa,bb,cc,dd]]D[ψ[dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{dd,1,4}],{aa,1,4},{bb,1,4},{cc,1,4}]-ZHolo[Rep]//Simplify

Diagonal Lagrangian
ℒ = (1/2) A□A+(1/2) B□B+ i (1/2) (((Subscript[ψ, a])((γ^μ)))^ab)(∂_μ)(Subscript[ψ, b])  
(Subscript[D, c])ℒ = 0  = ((Subscript[D, c])A) □A + ((Subscript[D, c])B) □A  - i (Subscript[ψ, a])((((γ^μ)))^ab)(∂_μ)((Subscript[D, c])(Subscript[ψ, b]))

Integration by parts commands: Put all derivatives on the bosons

ByParts={  Derivative[0,0,0,1][ψ[xp__]][t,x,y,z]efield__[t,x,y,z]-> -  ψ[xp][t,x,y,z]D[efield[t,x,y,z],z],
Derivative[0,0,1,0][ψ[xp__]][t,x,y,z]efield__[t,x,y,z]-> -  ψ[xp][t,x,y,z]D[ efield[t,x,y,z],y],Derivative[0,1,0,0][ψ[xp__]][t,x,y,z]efield__[t,x,y,z]-> -   ψ[xp][t,x,y,z]D[ efield[t,x,y,z],x],Derivative[1,0,0,0][ψ[xp__]][t,x,y,z]efield__[t,x,y,z]-> -   ψ[xp][t,x,y,z]D[ efield[t,x,y,z],t]}

DLA = ((Subscript[D, c])A) □A

DLA = Table[Da[ci][Afield[t,x,y,z]]Lap[Afield[t,x,y,z]],{ci,1,4}]

DLB = ((Subscript[D, c])B) □B

DLB = Table[Da[ci][Bfield[t,x,y,z]]Lap[Bfield[t,x,y,z]],{ci,1,4}]

DLPsi =  - i (Subscript[ψ, a])((((γ^μ)))^ab)(∂_μ)((Subscript[D, c])(Subscript[ψ, b]))

DLPsi = Table[ - I Sum[ψ[ai][t,x,y,z]γup[mu][[ai,bi]]D[Da[ci][ψ[bi][t,x,y,z]],Ϛ[mu]],{ai,1,4},{bi,1,4},{mu,0,3}],{ci,1,4}]

Lagrangian invariance:

DLTotal=(DLA +DLB + DLPsi );

DLFinal = Collect[DLTotal,{ψ[_][t,x,y,z],Derivative[_,_,_,_][Afield][t,x,y,z],Afield[t,x,y,z],Derivative[_,_,_,_][Bfield][t,x,y,z],Bfield[t,x,y,z],Derivative[_,_,_,_][Ffield][t,x,y,z],Ffield[t,x,y,z],Derivative[_,_,_,_][Gfield][t,x,y,z],Gfield[t,x,y,z]},Simplify]