GordonSimplify
GordonSimplify[exp]
rewrites spinor chains describing a vector or an axial-vector current using Gordon identities.
See also
Overview, DiracGamma, Spinor, SpinorChainTrick.
Examples
SpinorUBar[p1, m1] . GA[\[Mu]] . SpinorU[p2, m2]
GordonSimplify[%]
uˉ(p1,m1).γˉμ.u(p2,m2)
m1+m2(p1+p2)μ(φ(p1,m1)).(φ(p2,m2))+m1+m2i(φ(p1,m1)).σμp1−p2.(φ(p2,m2))
SpinorUBar[p1, m1] . GA[\[Mu], 5] . SpinorV[p2, m2]
GordonSimplify[%]
uˉ(p1,m1).γˉμ.γˉ5.v(p2,m2)
m1+m2(p1+p2)μ(φ(p1,m1)).γˉ5.(φ(−p2,m2))+m1+m2i(φ(p1,m1)).σμp1−p2.γˉ5.(φ(−p2,m2))
Relations involving projectors can be used to trade the right projector for a left one
SpinorVBar[p1, m1] . GA[\[Mu], 6] . SpinorV[p2, m2]
GordonSimplify[%]
vˉ(p1,m1).γˉμ.γˉ6.v(p2,m2)
−m1i(φ(−p1,m1)).σμp1−p2.γˉ6.(φ(−p2,m2))−m1m2(φ(−p1,m1)).γˉμ.γˉ7.(φ(−p2,m2))−m1(p1+p2)μ(φ(−p1,m1)).γˉ6.(φ(−p2,m2))
Use the Select
option to achieve the opposite
ex = SpinorVBar[p1, m1] . GA[\[Mu], 7] . SpinorV[p2, m2]
GordonSimplify[ex]
vˉ(p1,m1).γˉμ.γˉ7.v(p2,m2)
(φ(−p1,m1)).γˉμ.γˉ7.(φ(−p2,m2))
GordonSimplify[ex, Select -> {{Spinor[__], DiracGamma[__], GA[7], Spinor[__]}}]
−m1i(φ(−p1,m1)).σμp1−p2.γˉ7.(φ(−p2,m2))−m1m2(φ(−p1,m1)).γˉμ.γˉ6.(φ(−p2,m2))−m1(p1+p2)μ(φ(−p1,m1)).γˉ7.(φ(−p2,m2))
We can choose between having expressions proportional to 1/m1 (mass of the first spinor) or 1/m2 (mass of the second spinor)
GordonSimplify[SpinorVBar[p1, m1] . GA[\[Mu], 6] . SpinorV[p2, m2], Inverse -> First]
−m1i(φ(−p1,m1)).σμp1−p2.γˉ6.(φ(−p2,m2))−m1m2(φ(−p1,m1)).γˉμ.γˉ7.(φ(−p2,m2))−m1(p1+p2)μ(φ(−p1,m1)).γˉ6.(φ(−p2,m2))
GordonSimplify[SpinorVBar[p1, m1] . GA[\[Mu], 6] . SpinorV[p2, m2], Inverse -> Last]
−m2i(φ(−p1,m1)).σμp1−p2.γˉ7.(φ(−p2,m2))−m2m1(φ(−p1,m1)).γˉμ.γˉ7.(φ(−p2,m2))−m2(p1+p2)μ(φ(−p1,m1)).γˉ7.(φ(−p2,m2))
In D-dimensions chiral Gordon identities are scheme dependent!
ex = SpinorVBarD[p1, m1] . GAD[\[Mu], 5] . SpinorVD[p2, m2]
vˉ(p1,m1).γμ.γˉ5.v(p2,m2)
FCGetDiracGammaScheme[]
GordonSimplify[ex]
NDR
−m1−m2(p1+p2)μ(φ(−p1,m1)).γˉ5.(φ(−p2,m2))−m1−m2i(φ(−p1,m1)).σμp1−p2.γˉ5.(φ(−p2,m2))
FCSetDiracGammaScheme["BMHV"]
GordonSimplify[ex]
BMHV
−m1−m2i(φ(−p1,m1)).σμp1−p2.γˉ5.(φ(−p2,m2))−m1−m2(p1+p2)μ(φ(−p1,m1)).γˉ5.(φ(−p2,m2))+m1−m22(φ(−p1,m1)).γμ.(γ^⋅p2^).γˉ5.(φ(−p2,m2))
FCSetDiracGammaScheme["NDR"]
NDR