the remainder of our discussion. Let $E_A$ be the spinor field derived from a local section $E$ of the spin bundle ${S{\mathcal{M}}}$. Then one can derive matrices ${\mathcal{W}}_{\omega AB}$ and ${\mathcal{W}}^\Gamma_{\omega AB}$ from ${\mathcal{W}}_\omega$ and ${\mathcal{W}}^\Gamma_\omega$ via $${\mathcal{W}}_{\omega AB}(f,f') = {\mathcal{W}}_\omega(fE_A,f'E_B^+) \quad {\mathcal{W}}_{\omega AB}^\Gamma (f,f') = {\mathcal{W}}^\Gamma_\omega (fE_A,f'E^+_B)$$ for all $f,f'\in C^\infty_0({\mathcal{M}})$. Scalar bi-distributions ${\mathrm{W}}_\omega$ and ${\mathrm{W}}^\Gamma_\omega$ (in ${\mathcal{D}}'({\mathcal{M}}\times{\mathcal{M}})$) may be constructed by taking the traces of the matrices, i.e. ${\mathrm{W}}_\omega=\delta^{AB}{\mathcal{W}}_{\omega AB}$ and ${\mathrm{W}}^\Gamma_\omega
= \delta^{AB}{\mathcal{W}}^\Gamma_{\omega AB}$. Similarly, one may define a scalar version ${\,}^\psi{\mathrm{H}}_k^{(\pm)}$ of the Hadamard series ${\,}^\psi H_k^{(\pm)}$.

The stress energy tensor of the classical spin-1/2 field $\psi$ is given by $$T_{ab} = \frac{i}{2}\bigg( \psi^+\gamma_{(a}\nabla_{b)}\psi - (\nabla_{(a}\psi^+)\gamma_{b)}\psi 
\bigg) \,$$ where the subscript parentheses denote symmetrisation, i.e $\tau_{(ab)}=(\tau_{ab}+\tau_{ba})/2$. As advertised, we shall concentrate exclusively on proving an absolute QWEI along a timelike worldline. Therefore, we pick a properly parametrised smooth timelike worldline $\gamma:{\mathbb{R}}\mapsto {\mathcal{M}}$ and consider a spacetime tube $\tau_\gamma\subset {\mathcal{M}}$ centred about it, a precise construction of this tube will be given shortly. In $\tau_\gamma$ we may construct a tetrad $\{e^a_\alpha
\}_{\alpha=0,1,2,3}$ such that $e^a_0 = \dot{\gamma}^a$ and the remaining $\{ e^a_\alpha\}_{\alpha=1,2,3}$ are orthonormal to this vector. Then, in the (dual) frame $e^\alpha_a\otimes e^\beta_b$, the tensor $T_{ab}$ 