A Fermion Primer

Abstract

This is a review paper intended to illustrate a few critical issues concerning Dirac, Weyl, and Majorana fermions and their differences. The first part consists of a basic introduction to fermions, with further detail on fermions in 4d, focusing in particular on what differentiates the three types of fermions: chirality, helicity, mass, field equations, properties under discrete symmetry transformations, actions, and observables. On this basis, we tackle a series of challenging and sometimes controversial problems: the non-existence of Weyl fermion propagators, and ways to circumvent it; the regularizations for Weyl fermion amplitudes, in particular the appropriateness of using PV regularization; the definition of a functional integral for a Weyl fermion; the difference between Weyl and massless Majorana fermions; and the difference between Dirac and Majorana mass terms. Most notably, we come to the conclusion that the method of replacing a non-existing Weyl fermion propagator with a massless Dirac one, although it may yield, in some cases, correct results, is flawed at the very origin by a logical loophole. We then show what is the correct way to proceed in this case. We also consider the topic of applying a Wick rotation at the classical action level for Weyl fermions and conclude that this procedure leads to a nonequivalent theory. Finally, although anomalies are not the central focus here, we have deemed it useful to summarily review the relation, and its cohomological basis, that exists between the lack of a Weyl fermion propagator (non-invertibility of the Weyl–Dirac operator) and the appearance of dangerous anomalies in the theory.