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Starting from Wigner's symmetry representation theorem, we give a generalaccount of discrete symmetries (parity P, charge conjugation C, time-reversalT), focusing on fermions in Quantum Field Theory. We provide the rules oftransformation of Weyl spinors, both at the classical level (grassmanian wavefunctions) and quantum level (operators). Making use of Wightman's definitionof invariance, we outline ambiguities linked to the notion of classicalfermionic Lagrangian. We then present the general constraints cast by thesetransformations and their products on the propagator of the simplest amongcoupled fermionic system, the one made with one fermion and its antifermion.Last, we put in correspondence the propagation of C eigenstates (Majoranafermions) and the criteria cast on their propagator by C and CP invariance.

annals of physics

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