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 non-perturbative 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 re-expressed in terms of $2N$ Majorana fermions. We discuss how all this is reflected in chiral perturbation theory.