Assuming that a massive spin-1 particle has momentum only in the z-direction, the polarization vectors are given by
εμ(Jz=+1)=(0,−1√2,−i√2,0)
εμ(Jz=0)=(pm,0,0,Em)
εμ(Jz=−1)=(0,1√2,−i√2,0)
The so-called spin-sum is the claimed to be
∑Jz=−1,0,+1εμε∗ν=gμν+pμpνm2
I absolutely dont understand how this spin-sum is evaluated.
What does εμε∗ν even exactly mean? Is it a scalar product between two of the three above polarization vectors, or is it a "tensor-product" between the components of a single polarization vector which results in a 4x4 matrix and one has finally to sum all such matrices for the three possible values of Jz?
I would really appreciate it if somebody can explain to me what this spin-sum exactly means and how it is evaluated step-by-step.869