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 $\ensuremath{\theta}$ angle of the theory. We also resolve an apparent paradox regarding the induced $\ensuremath{\theta}$ 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.