for the cospinor field resulting in $S^\pm_\textrm{cosp}$. The advanced-minus-retarded fundamental solution for spinors is $S_\mathrm{sp}=S^-_\mathrm{sp}-S^+_\mathrm{sp}$ and for cospinors $S_\mathrm{cosp}=S^-_\mathrm{cosp}-S^+_\mathrm{cosp}$. An additional antilinear map $\Gamma :D^*{\mathcal{M}}\oplus D{\mathcal{M}}\mapsto D^*{\mathcal{M}}\oplus D{\mathcal{M}}$ acts by $$\Gamma\left( \begin{array}{c} h \\ f \end{array} \right)
= \left( \begin{array}{c} f^+ \\ h^+\end{array} \right) \,  ,$$ where $f\in C^\infty_0(D{\mathcal{M}})$ and $h\in C^\infty_0(D^*{\mathcal{M}})$. The map $\Gamma$ makes explicit the symmetry between spinors and cospinors in this dual setting.

The quantum Dirac field and Hadamard states {#quantum dirac hadamard}
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To define the field algebra we denote by $\mathcal{D}(D^*{\mathcal{M}}\oplus D{\mathcal{M}}) =
C^\infty_0(D^*{\mathcal{M}})\oplus C^\infty_0(D{\mathcal{M}})$ the space of all test cospinors and test spinors on which the operators $$D := i\left(   \begin{array}{cc} i{\nabla\hspace{-3mm}{/}}+\mu & 0 \\ 0 & -i{\nabla\hspace{-3mm}{/}}+ \mu \end{array}\right) \, \quad S := i\left(   \begin{array}{cc}
S_\mathrm{cosp} & 0 \\ 0 & S_\mathrm{sp}  \end{array}\right)$$ act. The elements $\mathcal{F}\in\mathcal{D}(D^*{\mathcal{M}}\oplus D{\mathcal{M}})$ may be used to label a set of abstract objects $\{ \Psi(\mathcal{F}) \mid \mathcal{F} \in
\mathcal{D}(D^*{\mathcal{M}}\oplus D{\mathcal{M}})\}$ which, when equipped with $\mathbbm{1}$, generates a unital \*-algebra $\mathfrak{F}$. We define the *algebra of smeared fields* $\mathfrak{F}({\mathcal{M}},g)$ to be $\mathfrak{F}$ quotiented by\

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      i) Adjoint, $\Psi (\mathcal{F})^* = \Psi(\Gamma\mathcal{F})$;

     ii) Linearity, $\Psi(\alpha_1 \mathcal{F}_1 + \alpha_2\mathcal{F}_2) =
         \alpha_1\Psi(\mathcal{F}_1)+\alpha_2\Psi(\mathcal{F}_2)$;

    iii) The field equation, $\Psi(D\mathcal{F})=0$;

     iv) Canonical anticommutation 