is equally expressed $${\mathcal{W}}_\omega + {\mathcal{W}}^\Gamma_\omega = iS_{\mathrm{sp}}\, .$$

It is clear from our discussion concerning the advanced and retarded fundamental solutions of the Dirac field that one may use the Lichnérowicz identity to define a notion of Hadamard state for the Dirac field. In order to give the precise formulation of the Hadamard series construction we must first discuss some geometry and here we follow [@sahlmann2]. We denote by $\mathfrak{X}\subset {\mathcal{M}}\times{\mathcal{M}}$ the set $$\begin{aligned}
\mathfrak{X} &=& \{ (x,x^\prime)\in{\mathcal{M}}\times{\mathcal{M}}\mid  x,x^\prime \text{
are causally
related and} {\nonumber \\}&\,& \qquad J^+(x)\cap J^-(x^\prime) \,\textrm{and}\, J^-(x)\cap J^+(x^\prime)
\text{ are
contained} {\nonumber \\}&\,& \qquad \text{within a convex normal neighbourhood} 
\} \, .\end{aligned}$$ Let $X\subset \mathfrak{X}$ be an open subset of $\mathfrak{X}$ such that between any pair $(x,x^\prime)\in X$ there exists a unique geodesic connecting them such that the (signed) geodesic separation of points defines a smooth function $\sigma$ on $X$. We make the additional requirement that the Hadamard construction (to be described shortly) can be carried out on $X$. Subject to all these requirements, we call $X$ \[notation reg dom\]a *regular domain*. We define two sequences of distributions $\{ H_k^{(\pm)}\}_{k=0,1,2,\dots} \in
{\mathcal{D}}^\prime (X)$ by $$\begin{aligned}
\label{parametrix}
H_k^{(\pm)}(x,x^\prime) &=&  \frac{1}{4\pi^2} \bigg\{ \frac{\Delta^\frac{1}{2}(x,x^\prime)}{\sigma_\pm (x,x^\prime)} + \sum^k_{j=0}v_j
(x,x^\prime)\frac{\sigma^j(x,x^\prime)}{\ell 