# S matrix Unitarity Proof, pg 298 Peskin Schroeder

1. Sep 16, 2015

### nickodel4

I have a question regarding a derivation in Peskin and Schroeder's QFT book. On page 298, he is discussing a method for defining a gauge invariant S matrix. He does this by defining projection operators $P_0$ that project general particle states into gauge invariant states, and then defining an S matrix as $$S = P_0S_{FP}P_0,\tag{9.59}$$ where $S_{FP}$ is the general S matrix between general states. This S matrix is therefore by definition gauge invariant, but now its unitarity can be questioned. He "proves" that this new $S$ matrix is unitary first by stating that $$\sum_{i=1,2}\epsilon^\ast_{i\mu}\epsilon_{i\nu}\mathcal{M}^\mu\mathcal{M}^{\ast\nu} = -g_{\mu\nu}\mathcal{M}^\mu\mathcal{M}^{\ast\nu},\tag{9.60}$$ where the sum runs over only transverse polarization states. He also points out that this identity holds true even if the two amplitudes are distinct. He claims that this is the exact information we need to prove the following $$SS^\dagger = P_0S_{FP}P_0S_{FP}^\dagger P_0 = P_0S_{FP}S_{FP}^\dagger P_0.\tag{9.61}$$ From here it is easy to see that, on the subspace of gauge invariant states, this is equal to the identity, because the $S_{FP}$ matix is unitary. However, I do not see the relation between equations 9.60 and 9.61, or how he uses 9.60 to justify the second step in the 9.61. The only clue I can see is that the sum in 9.60 is a result of using the LHZ formalism to go from correlation functions to S matrix elements, where polarizations must be summed over. But I am at a loss from there. Any help would be much appreciated!

2. Sep 21, 2015