S matrix Unitarity Proof, pg 298 Peskin Schroeder

[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!

 

[AndyW]

Thank you for your inquiry regarding the derivation in Peskin and Schroeder's QFT book. I can certainly understand your confusion and will do my best to clarify the steps involved in the proof.

Firstly, I would like to point out that the key idea behind this derivation is to show that the gauge invariant S matrix, defined as S = P_0S_{FP}P_0, is unitary. This can be done by proving that SS^\dagger = I, where I is the identity matrix.

Now, let's break down the steps involved in the proof:

Step 1: Using the LHZ formalism, we can write the correlation function as \langle \psi_1 \psi_2 \rangle = \sum_{i=1,2}\epsilon^\ast_{i\mu}\epsilon_{i\nu}\mathcal{M}^\mu\mathcal{M}^{\ast\nu}, where \psi_1 and \psi_2 are general particle states and \epsilon_{i\mu} are the polarization vectors for the two particles. Note that the sum runs over only transverse polarization states, as mentioned in equation (9.60).

Step 2: Using equation (9.60), we can rewrite the correlation function as \langle \psi_1 \psi_2 \rangle = -g_{\mu\nu}\mathcal{M}^\mu\mathcal{M}^{\ast\nu}. This is a key step in the proof, as it shows that the correlation function is independent of the polarization vectors \epsilon_{i\mu}.

Step 3: Now, let's consider the S matrix element between two gauge invariant states, \langle \psi_1^{GI} \psi_2^{GI} \rangle. Using the definition of the gauge invariant S matrix, we can write this as \langle \psi_1^{GI} \psi_2^{GI} \rangle = P_0\langle \psi_1 \psi_2 \rangle P_0. This is because the projection operator P_0 projects general particle states into gauge invariant states.

Step 4: Using the result from step 2, we can rewrite the S matrix element as \langle \psi_1^{GI} \psi_2^{GI} \rangle = P_0(-g_{\mu\nu}\mathcal{M}