Discrete symmetries and the propagator approach to coupled fermions in Quantum Field Theory. Generalities. The case of a single fermion-antifermion pair

TL;DR

This paper explores discrete symmetries in quantum field theory, detailing their effects on fermions and their propagators, especially focusing on Majorana fermions and the constraints imposed by C and CP invariance.

Contribution

It provides a comprehensive account of discrete symmetries for fermions, including transformation rules and constraints on propagators, with a focus on coupled fermion-antifermion systems.

Findings

Derived transformation rules for Weyl spinors at classical and quantum levels.

Outlined ambiguities in classical fermionic Lagrangians related to symmetry invariance.

Established criteria on propagators for C and CP invariance, especially for Majorana fermions.

Abstract

Starting from Wigner's symmetry representation theorem, we give a general account of discrete symmetries (parity P, charge conjugation C, time-reversal T), focusing on fermions in Quantum Field Theory. We provide the rules of transformation of Weyl spinors, both at the classical level (grassmanian wave functions) and quantum level (operators). Making use of Wightman's definition of invariance, we outline ambiguities linked to the notion of classical fermionic Lagrangian. We then present the general constraints cast by these transformations and their products on the propagator of the simplest among coupled fermionic system, the one made with one fermion and its antifermion. Last, we put in correspondence the propagation of C eigenstates (Majorana fermions) and the criteria cast on their propagator by C and CP invariance.