relation, $\{\Psi(\mathcal{F}_1),\Psi(\mathcal{F}_2) \}=iS(\mathcal{F}_1,\mathcal{F}_2)\mathbbm{1}$.
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Here $\mathcal{F},
\mathcal{F}_1,\mathcal{F}_2\in\mathcal{D}(D^*{\mathcal{M}}\oplus D{\mathcal{M}})$ and $\alpha_1,\alpha_2\in{\mathbb{C}}$. It is relation (iv) that quantises the theory.

The usual Dirac field $\psi$ and its adjoint $\psi^+$ are special cases of the above construction. For $h\in C^\infty_0(D^*{\mathcal{M}})$, $f\in C^\infty_0(D{\mathcal{M}})$ we define $$\psi(h) := \Psi\left( \begin{array}{c} h \\ 0 \end{array}  \right)  \quad
\textrm{and} \quad \psi^+(f) = \Psi \left( \begin{array}{c} 0 \\ f \end{array}  \right)$$ which we interpret as smeared fields.

A *state* $\omega : \mathfrak{F}({\mathcal{M}},g)\mapsto {\mathbb{C}}$ is a linear functional which is positive, i.e. $\omega(A^*A)\geq 0$ $\forall A\in\mathfrak{F}({\mathcal{M}},g)$, and normalised such that $\omega(\mathbbm{1})=1$. We shall restrict our attention to states for which the two-point function, defined by $$\omega(\Psi(\mathcal{F}_1)\Psi(\mathcal{F}_2)) \quad \forall \mathcal{F}_1,\mathcal{F}_2 \in {\mathcal{D}}(D^*{\mathcal{M}}\oplus D{\mathcal{M}}) \, ,$$ is a distribution on $\mathcal{D}(D^*{\mathcal{M}}\oplus D{\mathcal{M}})\otimes \mathcal{D}(D^*{\mathcal{M}}\oplus D{\mathcal{M}})$. Associated to each state $\omega$ we define what we shall call the Dirac and Dirac adjoint two-point functions ${\mathcal{W}}_\omega, {\mathcal{W}}^\Gamma_\omega \in
{\mathcal{D}}'(D{\mathcal{M}}\times D^*{\mathcal{M}})$ respectively by $${\mathcal{W}}_\omega(f,h)=\omega
(\psi^+(f) \psi(h)) \quad \textrm{and} \quad
{\mathcal{W}}^\Gamma_\omega(f,h)=\omega(\psi(h)\psi^+(f)) \, .$$ As a consquence of the positivity of states we immediately have that $${\mathcal{W}}(f,f^+) \geq 0 \quad \forall f \in C^\infty_0(D{\mathcal{M}}) \quad \textrm{and} \quad {\mathcal{W}}^\Gamma(h^+,h)\geq 0 \quad \forall h\in C^\infty(D^*{\mathcal{M}}) \, .$$ The covariant anticommutation relation in terms of ${\mathcal{W}}_\omega$ and ${\mathcal{W}}^\Gamma_\omega$ 