Dirac operator on spinors and diffeomorphisms

The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold M, to each spin structure sigma and Riemannian metric g there is associated a space S-sigma,S-g of spinor fields on M and a Hilbert space H-sigma,H-g = L-2(S-sigma,S-g, vol(g)(M)) of L-2-spinors of S-sigma,S-g. The group Diff(+)(M) of orientation-preserving diffeomorphisms of M acts both on g (by pullback) and on [sigma] (by a suitably defined pullback f*sigma). Any f is an element of Diff(+)(M) lifts in exactly two ways to a unitary operator U from H-sigma,H-g to H-f*sigma,H-,H-f*g. The canonically defined Dirac operator is shown to be equivariant with respect to the action of U, so in particular its spectrum is invariant under the diffeomorphisms.