Anomalies and symmetries of the regularized action

We show that the Pauli–Villars regularized action for a scalar field in a gravitational background in 1+1 dimensions has, for any value of the cutoff M, a symmetry which involves non-local transformations of the regulator field plus (local) Weyl transformations of the metric tensor. These transformations, an extension to the regularized action of the usual Weyl symmetry transformations of the classical action, lead to a new interpretation of the conformal anomaly in terms of the (non-anomalous) Jacobian for this symmetry. Moreover, the Jacobian is automatically regularized, and yields the correct result when the masses of the regulators tend to infinity. In this limit the transformations, which are non-local on a scale of 1M, become the usual Weyl transformations of the metric. We also present the example of the chiral anomaly in 1+1 dimensions, showing that the Pauli–Villars regularized action has a non-local symmetry. This symmetry is similar to the one of (lattice) Ginsparg–Wilson fermions, with the ultraviolet cutoff playing the role of the inverse of the lattice spacing.