${\mathcal{W}}_\omega$ is of the form ${\mathcal{W}}_\omega = (i{\nabla\hspace{-3mm}{/}}+\mu)\otimes\mathbbm{1}u$ where $u$ is an auxiliary two-point function. Consequently, we have, within a regular domain $X$, that $${\mathcal{W}}_\omega = {\,}^\psi H_k^{(+)} \quad \textrm{modulo}  \quad C^k(X)$$ where ${\,}^\psi H^{(+)}_k =
(i{\nabla\hspace{-3mm}{/}}+\mu)\otimes\mathbbm{1}H^{(+)}_{k+1}$. We also define ${\,}^\psi H^{(-)}_k = (i{\nabla\hspace{-3mm}{/}}+\mu) \otimes\mathbbm{1}H^{(-)}_{k+1}$. Hence, within a regular domain $X$, for any Hadamard state $\omega$ we have the following identities: $$\begin{aligned}
\label{dirac hadamard relations}
{\mathcal{W}}_\omega &=& {\,}^\psi H_k^{(+)} \quad \textrm{modulo } C^k(X) \, , \label{dirac hadamard relations 1} \\
{\mathcal{W}}^\Gamma_\omega &=& -{\,}^\psi H^{(-)}_k \quad \textrm{modulo } C^k(X) \label{dirac hadamard relations 2}\, ,
\label{sign note}\\
iS_{\mathrm{sp}}&=& {\,}^\psi H^{(+)}_k - {\,}^\psi H^{(-)}_k \quad \textrm{modulo } C^k(X)  \, . \label{dirac hadamard relations 3}\end{aligned}$$ Two remarks are in order: First, note that we require the sign in (\[sign note\]) so as to ensure that the anticommutation relation holds. Second, observe that as $k$ increases ${\mathcal{W}}_\omega - {\,}^\psi H_k^{(+)}$ becomes more regular and that, for sufficiently high $k$, ${\mathcal{W}}_\omega - {\,}^\psi H_k^{(+)}$ has a well defined coincidence limit.

The stress energy tensor {#energy section}
------------------------

We open this section with a few remarks about obtaining scalar distributions from spinorial ones as this will be the basis of our analysis of the energy density for 