Phase ambiguity of the measure for continuum Majorana fermions

Integrating over a continuum Majorana fermion formally yields a functional Pfaffian. We show that the phase of this Pfaffian is ambiguous, as it depends on the choice of basis. This ambiguity is naturally resolved within a nonperturbative lattice definition, allowing us to discuss the relation between the phase of the lattice Pfaffian and the effective theta angle of the theory. We also resolve an apparent paradox regarding the induced theta angle when a theory of N Dirac fermions in a real representation of the gauge group is reexpressed in terms of 2N Majorana fermions. We discuss how all this is reflected in chiral perturbation theory.