QED symmetries in real-time thermal field theory

TL;DR

This paper investigates the discrete and gauge symmetries of Quantum Electrodynamics at finite temperature using the real-time formalism, deriving Ward identities and analyzing vertex functions and their singularities.

Contribution

It derives finite temperature Ward identities in QED and reveals the independence of vertex functions and their singular behavior at zero photon momentum.

Findings

Only one longitudinal vertex function is independent.

Certain vertex function elements become singular as photon momentum approaches zero.

Ward identities relate vertex functions to self-energy components.

Abstract

We study the discrete and gauge symmetries of Quantum Electrodynamics at finite temperature within the real-time formalism. The gauge invariance of the complete generating functional leads to the finite temperature Ward identities. These Ward identities relate the eight vertex functions to the elements of the self-energy matrix. Combining the relations obtained from the $Z_2$ and the gauge symmetries of the theory we find that only one out of eight longitudinal vertex functions is independent. As a consequence of the Ward identities it is shown that some elements of the vertex function are singular when the photon momentum goes to zero.