Lattice fermions with gauge noninvariant measure

We define Weyl fermions on a finite lattice in such a way that in the path integral the action is gauge invariant but the functional measure is not. Two variants of such a formulation are tested in the perturbative calculation of the fermion determinant in the chiral Schwinger model. We find that one of these variants ensures restoring the gauge invariance of the nonanomalous part of the determinant in the continuum limit. A ‘perfect’ perturbative regularization of the chiral fermions is briefly discussed.