To Prove:
ˉU(P2)γμU(P1)=12MˉU(P2)[(P1+P2)μ+ισμν(P2−P1)ν]U(P1)
We use 2gμν=[γμ,γν]+ and σμν=ι2[γμ,γν]− to get: ˉU(P2)γμU(P1)=12MˉU(P2)[(P1+P2)μ−12γμγν(P2−P1)ν+12γνγμ(P2−P1)ν]U(P1)
ˉU(P2)γμU(P1)=12MˉU(P2)[(P1+P2)μ−12γμγν(P2−P1)ν+12(2gμν−γμγν)(P2−P1)ν]U(P1)
contraction of indices gives:
ˉU(P2)γμU(P1)=12MˉU(P2)[(P1+P2)μ−12γμγν(P2−P1)ν+(P2−P1)μ−γμγν2(P2−P1)ν]U(P1)
rearranging:
ˉU(P2)γμU(P1)=12MˉU(P2)[2Pμ2−γμγν(P2−P1)ν]U(P1)
now using Dirac's equation
(ιγμ∂μ−M)Ψ=0,ˉΨ(ιγμ∂μ+M)=0,gives
ˉU(P2)γμU(P1)=12MˉU(P2)[2Pμ2+2γμM]U(P1)
ˉU(P2)γμU(P1)=ˉU(P2)Pμ2MU(P1)+ˉU(P2)γμU(P1)
Please tell me how to proceed further
This post imported from StackExchange Physics at 2014-12-06 09:31 (UTC), posted by SE-user Akshansh Singh