•SU(2) matrix relations for χ _ + and χ _ -.

Independent combinations: < χ _ +^2 - χ _ -^2> = 4 <χ^†χ>,
                                              < χ _ +^2 + χ _ -^2> = < χ _ + > ^2+< χ _ - > ^2- 2 (det(χ)+det(χ^†))

The second relation follows from the Newton formula (CharacteristicCoefficient[UMatrix[a],UDimension->2][0]) for 2x2 matrices:
det (a) = 1/2 (< a >^2 - < a a >) and multiplicability of the determinant.

CharacteristicCoefficient[UMatrix[a], UDimension -> 2][0]

1/2 (< a >^2 - < a '6 a >)

Det[(a11 a12)] + 1/2 (UTrace[NM[(a11 a12), (a11 a12)]] - UTrace[(a11 a12)]^2) // CommutatorReduce // Expand a21 a22 a21 a22 a21 a22 a21 a22

0

UTrace[NM[UChiPlus[x], UChiPlus[x]] - NM[UChiMinus[x], UChiMinus[x]]] /. $Substitutions // NMExpand // Expand // CommutatorReduce // UReduce[#, SMMToMM -> True] &

4 < χ^† '6 χ >

UTrace[NM[UChiPlus[x], UChiPlus[x]] + NM[UChiMinus[x], UChiMinus[x]]] /. $Substitutions // NMExpand // Expand // CommutatorReduce // UReduce[#, SMMToMM -> True] &

2 < ÷„^† '6 χ '6 ÷„^† '6 χ > + 2 < χ^† '6 ÷„ '6 χ^† '6 ÷„ >

UTrace1[NM[Adjoint[MM[x_]], UMatrix[UChi[chopts___]][x_], Adjoint[MM[x_]], UMatrix[UChi[___]][x_]]] -> -UTrace1[NM[Adjoint[UMatrix[UChi[chopts]][x]], MM[x], Adjoint[UMatrix[UChi[chopts]][x]], MM[x]]] + 1/2 UTrace[UChiPlus[x]]^2 + 1/2 UTrace[UChiMinus[x]]^2 - Det[UMatrix[UChi[chopts]][x]] - Det[Adjoint[UMatrix[UChi[chopts]][x]]]

< ÷„^† '6 χ '6 ÷„^† '6 χ > -> < χ _ - >^2/2 + < χ _ + >^2/2 - {χ^†} - {χ} - < χ^† '6 ÷„ '6 χ^† '6 ÷„ >

detChiRule = UTrace1[NM[Adjoint[MM[x_]], UMatrix[UChi[chopts___]][x_], Adjoint[MM[x_]], UMatrix[UChi[___]][x_]]] -> -UTrace1[NM[Adjoint[UMatrix[UChi[chopts]][x]], MM[x], Adjoint[UMatrix[UChi[chopts]][x]], MM[x]]] + 1/2 UTrace[UChiPlus[x]]^2 + 1/2 UTrace[UChiMinus[x]]^2 - Det[UMatrix[UChi[chopts]][x]] - Det[Adjoint[UMatrix[UChi[chopts]][x]]]

< ÷„^† '6 χ '6 ÷„^† '6 χ > -> < χ _ - >^2/2 + < χ _ + >^2/2 - {χ^†} - {χ} - < χ^† '6 ÷„ '6 χ^† '6 ÷„ >

Converted by Mathematica  (July 10, 2003)