a notion of Hadamard states. The essential feature of Hadamard states is that they all share a common singularity structure; in particular their two-point functions, and their Dirac adjoints, have a local and covariantly determined singular expansion. We denote the Hadamard series corresponding to the singularity structure of the Dirac two-point function ${\mathcal{W}}_\omega$ by ${\,}^\psi H_k^{(+)}$ and that corresponding to the singular structure of the adjoint Dirac two-point function ${\mathcal{W}}^\Gamma_\omega$ by ${\,}^\psi H^{(-)}_k$. The salient feature of such states is that one may define a finite stress energy density $\langle \rho^{\mathrm{fin}}\rangle_\omega(x)$ by using ${\mathcal{W}}_\omega-{\,}^\psi H_k^{(+)}$ and a point-splitting prescription. Indeed, define an operator $\rho^\mathrm{split}$ such that the finite contribution to the energy density is given by $\rho^\textrm{split}$ acting on the regularised two-point function of the Dirac field ${\mathcal{W}}_\omega - {\,}^\psi H_k^{(+)}$; i.e. $$\langle \rho^{\mathrm{fin}}\rangle_\omega (x) := \lim_{x'\rightarrow x} [\rho^\mathrm{split} ( {\mathcal{W}}_\omega -{\,}^\psi H_1^{(+)} ) ](x,x') \, .$$ The precise form of $\rho^\textrm{split}$ is given in §\[energy section\]. The quantity $\langle \rho^{\mathrm{fin}}\rangle_\omega$ is equal to the renormalised energy density modulo a local curvature term; we shall return to this issue later after proving our main result in theorem \[main result\]. Our result then reads for any real valued $f\in C^\infty_0({\mathbb{R}})$ and 