The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold M, to each spin structure σ and Riemannian metric g there is associated a space S_(σ,g) of spinor fields on M and a Hilbert space of L~2-spinors of S _(σ,g). The group Diff~+(M) of orientation-preserving diffeomorphisms of M acts both on g (by pullback) and on [σ] (by a suitably defined pullback f~*σ). Any f ∈ Diff~+(M) lifts in exactly two ways to a unitary operator U from H_(σ, g) to H_(f*σ, 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.
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