•Splitting in components

We now want to express the result as

ξ^k _ 1 (-d _ μ^(k _ 1 j) d _ μ^(j k _ 2) + Y^(k _ 1 k _ 2)) ξ^k _ 2

with X anti-hermitian and Y hermitian and

d _ μ^(k _ 1 k _ 2) = δ^(k _ 1 k _ 2)∂ _ μ + X _ μ^(k _ 1 k _ 2).

That is, as

ξ^k _ 1 (-δ^(k _ 1 k _ 2)∂ _ μ∂ _ μ - X _ μ^(k _ 1 j)X _ μ^(j k _ 2) - δ^(k _ 1 j)(∂ _ μ X _ μ^(j k _ 2)) - 2 δ^(k _ 1 j) X _ μ^(j k _ 2)∂ _ μ + Y^(k _ 1 k _ 2)) ξ^k _ 2.

The third term doesn't contribute because of the anti-hermiticity of X.

dd = Select[sss // Expand, (Count[#, PartialD, Infinity, Heads -> True] === 2) &]

-1/2 < σ^k _ 1 '6 σ^k _ 2 > ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 1 ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 2

dd // ExpandUGenerators

-δ _ (k _ 1 k _ 2)^(3) ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 1 ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 2

The non-derivative part of the Y and X contribution.

xxy = sss /. QuantumField[PartialD[LorentzIndex[_]], __][x] -> 0 /. _FieldDerivative -> 0 /. QuantumField[Particle[(Vector | AxialVector)[0], ___], ___] -> 0 /. QuantumField[Particle[l | r, ___], ___][_] -> 0 // Simplify

1/4 (< χ _ + '6 σ^k _ 1 '6 σ^k _ 2 > + 4 < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 '6 Γ _ ρ _ 1 > + < σ^k _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 '6 u _ ρ _ 1 > - 4 < σ^k _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 > - < σ^k _ 1 '6 u _ ρ _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 >) ξ^( )^k _ 1 ξ^( )^k _ 2


We may split this in a symmetric and an anti-symmetric part of which only the
symmetric part contributes:

xxyS = Symmetrize[xxy, Union[Cases[xxy, SUNIndex[__], Infinity, Heads -> True]]] // CycleUTraces // Simplify

1/8 (< χ _ + '6 σ^k _ 1 '6 σ^k _ 2 > + < χ _ + '6 σ^k _ 2 '6 σ^k _ 1 > + 4 < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 '6 Γ _ ρ _ 1 > + < σ^k _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 '6 u _ ρ _ 1 > - 8 < σ^k _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 > + 4 < σ^k _ 1 '6 Γ _ ρ _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 > - 2 < σ^k _ 1 '6 u _ ρ _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 > + < σ^k _ 1 '6 u _ ρ _ 1 '6 u _ ρ _ 1 '6 σ^k _ 2 >) ξ^( )^k _ 1 ξ^( )^k _ 2

The part 2X _ μ ∂^μ:

xd = sss - xxy - dd // IndicesCleanup // CycleUTraces // Simplify

(< σ^k _ 1 '6 Γ _ τ _ 1 '6 σ^k _ 2 > - < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ τ _ 1 >) ξ^( )^k _ 1 ∂ _ τ _ 1 ξ^( ) _ ó ^k _ 2

X:

X[Sequence @@ ((Pattern1[#[[1]], Blank[]] & /@ Union[Cases[xd, LorentzIndex[__], Infinity, Heads -> True]]) /. Pattern1 -> Pattern)] = xd/2 /. QuantumField[___, Particle[PseudoScalar[12]], SUNIndex[_]][x] -> 1 // Simplify

1/2 (< σ^k _ 1 '6 Γ _ τ _ 1 '6 σ^k _ 2 > - < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ τ _ 1 >)

Componentized version:

X[Sequence @@ ((Pattern1[#[[1]], Blank[]] & /@ Union[Cases[X[μ1], SUNIndex[__], Infinity, Heads -> True]]) /. Pattern1 -> Pattern), μ1_] = X[μ1] ;

X is anti-symmetric:

Symmetrize[X[k1, k2, μ], {k1, k2}] // CycleUTraces // Simplify

0

The square of X.

XX = NM[X[k1, j, μ1], X[j, k2, μ1]] // NMExpand // Expand // SUNReduce // IndicesCleanup // CycleUTraces // Simplify

1/2 (< σ^k _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 '6 Γ _ ρ _ 1 > - 2 < σ^k _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 > + < σ^k _ 1 '6 Γ _ ρ _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 >)

The square of X is symmetric:

xx = QuantumField[Particle[PseudoScalar[12]], SUNIndex[k1]][x] XX QuantumField[Particle[PseudoScalar[12]], SUNIndex[k2]][x]

1/2 (< σ^k _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 '6 Γ _ ρ _ 1 > - 2 < σ^k _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 > + < σ^k _ 1 '6 Γ _ ρ _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 >) ξ^( )^k _ 1 ξ^( )^k _ 2

AntiSymmetrize[XX, {k1, k2}] // CycleUTraces // Simplify

0

The field strength associated with X is [σ^k _ 1,σ^k _ 2] times the field strength associated with Γ:

XFST[k1_, k2_, μ1_, μ2_] = FieldDerivative[X[k1, k2, μ2], x, {μ1}] - FieldDerivative[X[k1, k2, μ1], x, {μ2}] + NM[X[k1, j, μ1], X[j, k2, μ2]] - NM[X[k1, j, μ2], X[j, k2, μ1]] // NMExpand // Expand // SUNReduce // CycleUTraces // Simplify

1/2 (< ∂ _ μ _ 2(Γ _ μ _ 1) '6 σ^k _ 1 '6 σ^k _ 2 > - < ∂ _ μ _ 2(Γ _ μ _ 1) '6 σ^k _ 2 '6 σ^k _ 1 > - < ∂ _ μ _ 1(Γ _ μ _ 2) '6 σ^k _ 1 '6 σ^k _ 2 > + < ∂ _ μ _ 1(Γ _ μ _ 2) '6 σ^k _ 2 '6 σ^k _ 1 > - < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ μ _ 1 '6 Γ _ μ _ 2 > + < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ μ _ 2 '6 Γ _ μ _ 1 > + < σ^k _ 1 '6 Γ _ μ _ 1 '6 Γ _ μ _ 2 '6 σ^k _ 2 > - < σ^k _ 1 '6 Γ _ μ _ 2 '6 Γ _ μ _ 1 '6 σ^k _ 2 >)

The part ∂^μ Xwhich doesn't contribute because of the afore mentioned anti-symmetry:

dx = QuantumField[Particle[PseudoScalar[12]], SUNIndex[k1]][x] QuantumField[Particle[PseudoScalar[12]], SUNIndex[k2]][x] UTrace[FieldDerivative[X[μ1], x, {μ1}] /. FieldDerivative[x, LorentzIndex[_]] -> 0 /. UTrace1 -> Identity] // NMExpand // Simplify

1/2 (< σ^k _ 1 '6 ∂ _ μ _ 1(Γ _ μ _ 1) '6 σ^k _ 2 > - < σ^k _ 1 '6 σ^k _ 2 '6 ∂ _ μ _ 1(Γ _ μ _ 1) >) ξ^( )^k _ 1 ξ^( )^k _ 2

Symmetrize[dx, Union[Cases[dx, SUNIndex[__], Infinity, Heads -> True]]] // CycleUTraces // Simplify

0

The parts coming from Y:

y = Expand[NMExpand[xxyS - xx]] // CycleUTraces // Simplify

1/8 (< χ _ + '6 σ^k _ 1 '6 σ^k _ 2 > + < χ _ + '6 σ^k _ 2 '6 σ^k _ 1 > + < σ^k _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 '6 u _ ρ _ 1 > - 2 < σ^k _ 1 '6 u _ ρ _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 > + < σ^k _ 1 '6 u _ ρ _ 1 '6 u _ ρ _ 1 '6 σ^k _ 2 >) ξ^( )^k _ 1 ξ^( )^k _ 2

Componentized version:

Y[Sequence @@ ((Pattern1[#[[1]], Blank[]] & /@ Union[Cases[y, SUNIndex[__], Infinity, Heads -> True]]) /. Pattern1 -> Pattern)] = y /. QuantumField[___, Particle[PseudoScalar[12]], SUNIndex[_]][x] -> 1 ;

Y is symmetric:

AntiSymmetrize[y, Union[Cases[y, SUNIndex[__], Infinity, Heads -> True]]] // CycleUTraces // Simplify

0

The result can then be written:

res = dd + xx + xd + y // Simplify

1/8 (4 (< σ^k _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 '6 Γ _ ρ _ 1 > - 2 < σ^k _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 > + < σ^k _ 1 '6 Γ _ ρ _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 >) ξ^( )^k _ 1 ξ^( )^k _ 2 + (< χ _ + '6 σ^k _ 1 '6 σ^k _ 2 > + < χ _ + '6 σ^k _ 2 '6 σ^k _ 1 > + < σ^k _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 '6 u _ ρ _ 1 > - 2 < σ^k _ 1 '6 u _ ρ _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 > + < σ^k _ 1 '6 u _ ρ _ 1 '6 u _ ρ _ 1 '6 σ^k _ 2 >) ξ^( )^k _ 1 ξ^( )^k _ 2 - 4 < σ^k _ 1 '6 σ^k _ 2 > ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 1 ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 2 + 8 (< σ^k _ 1 '6 Γ _ τ _ 1 '6 σ^k _ 2 > - < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ τ _ 1 >) ξ^( )^k _ 1 ∂ _ τ _ 1 ξ^( ) _ ó ^k _ 2)

sss

1/4 (< χ _ + '6 σ^k _ 1 '6 σ^k _ 2 > ξ^( )^k _ 1 ξ^( )^k _ 2 + 4 < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 '6 Γ _ ρ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 + < σ^k _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 '6 u _ ρ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 - 4 < σ^k _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 - < σ^k _ 1 '6 u _ ρ _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 - 4 < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 > ξ^( )^k _ 1 ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 2 + 4 < σ^k _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 > ξ^( )^k _ 1 ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 2 - 2 < σ^k _ 1 '6 σ^k _ 2 > ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 1 ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 2)

The difference is anti-symmetric in k _ 1 and k _ 2 and thus vanishes:

Symmetrize[sss - res, Union[Cases[sss - res, SUNIndex[__], Infinity, Heads -> True]]] // CycleUTraces // Expand // IndicesCleanup

0

Converted by Mathematica  (July 10, 2003)