is stated in section \[punch line\].

The Dirac field in curved spacetime {#dirac background}
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The reader who is familiar with the formalism necessary to describe the classical Dirac field on a curved background is encouraged to skip ahead to section §\[quantum dirac hadamard\].

Spin structures and spinors on curved spacetimes {#dirac background 1}
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We begin by reviewing the geometry necessary to discuss the Dirac field in a curved spacetime. We shall employ the algebraic framework for describing the Dirac quantum field in a classical curved four-dimensional spacetime $(\mathcal{M},g)$. Here $\mathcal{M}$ is a four-dimensional smooth manifold (assumed Hausdorff, paracompact and without boundary) with a Lorentz metric $g_{ab}$ of signature ($+---$). Furthermore, we require $({\mathcal{M}},g)$ to be *globally hyperbolic*, that is ${\mathcal{M}}$ contains a Cauchy surface. Where index notation is used, Latin indices will run over the range $0,1,2,3$ unless explicitly stated otherwise, while Greek characters will denote frame indices and also run over $0,1,2,3$ unless explicitly stated otherwise. We employ units in which $c=\hbar=1$.

In Minkowski spacetime the spinors are nothing more than the spin-half representation of the Poincaré group, however, a general manifold does not exhibit this symmetry globally: Therefore, the usual (i.e. Minkowski spacetime) interpretation of a 