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 [tex]S = P_0S_{FP}P_0,\tag{9.59}[/tex] 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 [tex]\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}[/tex] 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 [tex]SS^\dagger = P_0S_{FP}P_0S_{FP}^\dagger P_0 = P_0S_{FP}S_{FP}^\dagger P_0.\tag{9.61}[/tex] 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!(adsbygoogle = window.adsbygoogle || []).push({});

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# S matrix Unitarity Proof, pg 298 Peskin Schroeder

Loading...

Similar Threads - matrix Unitarity Proof | Date |
---|---|

A Question about the matrix of a pseudoscalar meson | Jan 20, 2018 |

A Reduced matrix element for 0_ --> 0+ forbidden beta decay | Aug 9, 2017 |

A Tree-level unitarity constraints in Two-Higgs Doublet Model | Oct 27, 2016 |

Unitarity of the PMNS matrix | Jul 31, 2011 |

Unitarity of CKM matrix | May 24, 2008 |

**Physics Forums - The Fusion of Science and Community**