Nonlocal parity transformations and anomalies

We present an alternative derivation of the parity anomaly for a massless Dirac field in 2+1 dimensions coupled to a gauge field. The anomaly functional, a Chern–Simons action for the gauge field, is obtained from the nontrivial Jacobian corresponding to a nonlocal symmetry of the Pauli–Villars regularized action. That Jacobian is well-defined, finite, and yields the standard Chern–Simons term when the cutoff tends to infinity.