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

Setting Rep and Importing Adinkra package

$RecursionLimit=Infinity

Rep = VSOnShell;

<<Adinkra`

BuildDate[Adinkra]

fields: Afield[mu],dfield, λ[a], all free indices covariant
AdinkraReduce
(Subscript[Φ, i]) = {(Subscript[(OverDot[A]), 1]),(OverDot[(Subscript[A, 2])]),(OverDot[(Subscript[A, 3])])}
(Subscript[Ψ, (OverHat[i])])= {(Subscript[(OverDot[λ]), a])}

AdinkraReduce Encodes reducing the dependence on the fields from 4D space-time to only time

AdinkraReduce={Afield[mu_]-> Function[{t,x,y,z},Afield[mu][t]],dfield-> Function[{t,x,y,z},dfield[t]],λ[a_]->Function[{t,x,y,z},λ[a][t]]}

fields := prints out the adinkra reduced fields in subscript form

fields={λ[a_][t]->(Subscript[λ, a]),λ[a_]'[t]->(Subscript[(OverDot[λ]), a]),λ[a_]''[t]->(Subscript[λ, a])'',
 Afield[mu_][t]->(Subscript[A, mu]),  Afield[mu_]'[t]->(Subscript[(OverDot[A]), mu]),  Afield[mu]''[t]->(Subscript[A, mu])'',dfield[t]->d, dfield'[t]->(OverDot[d])}

λ[2]'[t]/.fields

TeXForm[%]

λ[1][t,x,y,z]/.AdinkraReduce/.fields

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

Phi[t][[i]] = (Subscript[Φ, i]) = {(Subscript[(OverDot[A]), 1]),(OverDot[(Subscript[A, 2])]),(OverDot[(Subscript[A, 3])])}
Psi[t][[ihat]] = (Subscript[Ψ, (OverHat[i])])= {(Subscript[(OverDot[λ]), a])}

Phi[t] = {Afield[1]'[t],Afield[2]'[t],Afield[3]'[t]}

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

Phi[t]/.fields

I Psi[t]/.fields

FieldList:= list of the names of the fields

FieldList = {Afield[0][t,x,y,z],Afield[1][t,x,y,z],Afield[2][t,x,y,z],Afield[3][t,x,y,z],λ[1][t,x,y,z],λ[2][t,x,y,z],λ[3][t,x,y,z],λ[4][t,x,y,z]}

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

λup[a] := (λ^a)

Do[λup[ai][t,x,y,z] = RaiseFermionIndex[λ[ai][t,x,y,z]],{ai,1,4}]

InverseCmetric//MatrixForm

Table[λup[ai][t,x,y,z] ,{ai,1,4}]

Afieldup

Do[Afieldup[mu][t,x,y,z] = RaiseSTIndex[Afield[mu][t,x,y,z]],{mu,0,3}]

Table[Afieldup[mu][t,x,y,z],{mu,0,3}]

The Field strength, covariant and contravariant:  Fmn[μ,ν]:=(Subscript[F, μν]) = (∂_μ)(Subscript[A, ν]) - (∂_ν)(Subscript[A, μ])  ,  Fmnup[μ,ν][t,x,y,z] := (F^(μν)) = (∂^μ)(A^ν)-(∂^ν)(A^μ)

Do[Fmn[μ,ν][t,x,y,z] = D[Afield[ν][t,x,y,z],Ϛ[μ]] - D[Afield[μ][t,x,y,z],Ϛ[ν]], {μ,0,3},{ν,0,3}]

Do[Fmnup[μ,ν][t,x,y,z] =  RaiseSTIndex[FmnHolder[μ,ν][t,x,y,z],UP,UP]/.FmnHolder->Fmn,{μ,0,3},{ν,0,3}]

Table[Fmn[μ,ν][t,x,y,z],{μ,0,3},{ν,0,3}]//MatrixForm

Table[Fmnup[μ,ν][t,x,y,z],{μ,0,3},{ν,0,3}]//MatrixForm

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.

Ϛ[3]

D[UD[dfield[t,x,y,z],Ϛ[0]],Ϛ[0]]

Lap[D[dfield[t,x,y,z],Ϛ[0]]]

UD[D[Lap[dfield[t,x,y,z]],Ϛ[0]],Ϛ[0]]

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

Lap[λ[1][t,x,y,z]]

TemporalGauge

TemporalGauge = Afield[0]-> Function[{t},0]

Table[Afield[mu][t,x,y,z],{mu,0,3}]/.AdinkraReduce/.TemporalGauge

Transformation Rules

Do[Da[a][Afield[mu][t,x,y,z]]=  Sum[γstdown[mu][[a,b]] λ[b][t,x,y,z],{b,1,4}],{mu,0,3},{a,1,4}]

Do[DColor[Afield[mu]'[t]] = Table[D[Da[Color][Afield[mu][t,x,y,z]]/.AdinkraReduce,t],{Color,1,4}],{mu,0,3}]

PrintDColor[Afield'[t]]=Table[DColor[Afield[mu]'[t]][[Color]],{mu,0,3},{Color,1,4}]/.fields//MatrixForm

PrintDColor[Afield'[t]]

Do[Da[a][λ[b][t,x,y,z]] = -I        (1/4) Sum[CommuteGammadown[mu,nu][[a,b]](D[Afield[nu][t,x,y,z],Ϛ[mu]]-D[Afield[mu][t,x,y,z],Ϛ[nu]]),{mu,0,3},{nu,0,3}],{a,1,4},{b,1,4}]

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

PrintDColor[-I λ'[t]]=Table[Collect[DColor[-I λ[bi]'[t]][[Color]],{Afield[_]'[t],dfield[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][λ[ki][t,x,y,z]],{ii,1,4},{ji,1,4},{ki,1,4}]

Algebra

{(Subscript[D, a]),(Subscript[D, b])}(Subscript[A, ν])= 2 i (Subscript[((γ^μ)), ab])(∂_μ)(Subscript[A, ν]) -(∂_ν)(Subscript[Λ, ab]^((VS)))  where (Subscript[Λ, ab]^((VS)))= 2i (Subscript[((γ^μ)), ab])(Subscript[A, μ])

ΛGauge[Rep]=Table[Sum[2I γdown[mu][[aa,bb]]Afield[mu][t,x,y,z],{mu,0,3}],{aa,1,4},{bb,1,4}];

(Table[AlgebraDD[Da[aa],Da[bb]][Afield[mu][t,x,y,z]],{aa,1,4},{bb,1,4},{mu,0,3}]== Table[Closure[aa,bb][Afield[mu][t,x,y,z]]-D[ΛGauge[Rep][[aa,bb]],Ϛ[mu]],{aa,1,4},{bb,1,4},{mu,0,3}])//Simplify

Z0[Rep]=Table[Sum[(I (1/2)Sum[γdown[nu][[aa,bb]]γγstdownstup[nu,mu][[cc,dd]],{nu,0,3}] + I (1/16)Sum[(CommuteGammastdowndown[ap,bt][[aa,bb]]CommuteGamma[ap,bt].γ[mu])[[cc,dd]],{ap,0,3},{bt,0,3}])D[λ[dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{dd,1,4}],{aa,1,4},{bb,1,4},{cc,1,4}];

(Table[AlgebraDD[Da[ai],Da[bi]][λ[ci][t,x,y,z]],{ai,1,4},{bi,1,4},{ci,1,4}] ==Table[Closure[ai,bi][λ[ci][t,x,y,z]] ,{ai,1,4},{bi,1,4},{ci,1,4}] - Z0[Rep])//Simplify

Holoraumy

with

BHoloTerms[mu_,ap_,bt_][aa_,bb_]:= -I Sum[ϵdownupupup[mu,nu,ap,bt]γ5γstdowndown[nu][[aa,bb]],{nu,0,3}]

Do[BHolo[Rep][mu,ap,bt]= Table[BHoloTerms[mu,ap,bt][aa,bb],{aa,1,4},{bb,1,4}],{mu,0,3},{ap,0,3},{bt,0,3}]

Table[HoloraumyDDTable[Afield[mu][t,x,y,z]],{mu,0,3}] ==  Table[-2 I Sum[BHolo[Rep][mu,ap,bt]D[Afield[bt][t,x,y,z],Ϛ[ap]],{ap,0,3},{bt,0,3}],{mu,0,3}]//Simplify

with and 
and

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[λ[ee][t,x,y,z],Ϛ[mu]],{mu,0,3},{ee,1,4}],{dd,1,4}];

ZHoloTerms[aa_,bb_,cc_]:=Sum[I((3/2)Cmetric[[aa,bb]]KroneckerDelta[cc,dd]+(3/2) γ5down[[aa,bb]]γ5[[cc,dd]] -(1/2)Sum[γ5γdown[kp][[aa,bb]](γ5γstdown[kp])[[cc,dd]],{kp,0,3}])DEqm[Rep][[dd]],{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]][λ[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[λ[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 = -(1/4) (Subscript[F, μν])(F^(μν))+ i (1/2) (((Subscript[λ, a])((γ^μ)))^ab)(∂_μ)(Subscript[λ, b])   
(Subscript[D, c])ℒ = 0  =  (∂_μ)(F^(μν))((Subscript[D, c])(Subscript[A, ν]))   - i (Subscript[λ, a])((((γ^μ)))^ab)(∂_μ)((Subscript[D, c])(Subscript[λ, b]))

Integration by parts commands: Put all derivatives on the bosons

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

DLA =  (∂_μ)(F^(μν))((Subscript[D, c])(Subscript[A, ν]))

DLA = Table[Sum[ Da[ci][Afield[nu][t,x,y,z]]D[Fmnup[mu,nu][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3}],{ci,1,4}]

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

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

Lagrangian invariance:

DLTotal=(DLA +DLlambda);

DLFinal = Collect[DLTotal,{λ[_][t,x,y,z],Derivative[_,_,_,_][Afield[_]][t,x,y,z],Afield[_][t,x,y,z],Derivative[_,_,_,_][dfield][t,x,y,z],dfield[t,x,y,z]},Simplify]

LagrangianClosure = DLFinal == {0,0,0,0}