@San]. This condition says [@KayWald] that the two-point function $w_{\boldsymbol{M}}$ splits in the form $$w_{\boldsymbol{M}}(x,y) = \mbox{\sc h}_{\boldsymbol{M}}(x,y)  + u(x,y)$$ where $\mbox{\sc h}_{\boldsymbol{M}}$ is a Hadamard parametrix for the wave-equation , and $u$ is a $C^\infty$ integral kernel. Consequently, the singular behaviour of $\mbox{\sc h}_{\boldsymbol{M}}$ and hence of $w_{\boldsymbol{M}}$ is determined by the spacetime geometry of $\boldsymbol{M}$ and is state-independent within the set $\mathcal{S}_{\mu sc}(\boldsymbol{M})$ whereas different states are distinguished by different smooth terms $u(x,y)$. With respect to such a splitting of a two-point function, one can then define the expectation value of the stress energy tensor in a state $\omega_{\boldsymbol{M}}$ in $\mathcal{S}(\boldsymbol{M})$ as $$\begin{aligned}
 \label{Ttilde}
\langle \tilde{\sf T}_{\boldsymbol{M}ab}(x) \rangle_{\omega_{\boldsymbol{M}}}
 & = \lim_{y \to x} P_{ab}(x,y;\nabla_{(x)},\nabla_{(y)}) \left( w_{\boldsymbol{M}} -
 \mbox{\sc h}_{\boldsymbol{M}} \right) \,.\end{aligned}$$ Note that $\langle \tilde{\sf T}_{\boldsymbol{M}ab}(x) \rangle_{\omega_{\boldsymbol{M}}}$ is, in fact, smooth in $x$ and therefore is a $C^\infty$ tensor field on $\boldsymbol{M}$. The reason for the appearance of the twiddle on top of the symbol for the just defined stress energy tensor is due to the circumstance that with this definition, te stress energy tensor is (apart from exceptional cases) not divergence-free. However, Wald [@Wald78; @WaldQFTCST] has shown that this can be repaired by subtracting the divergence-causing term. 