would appear in the variance of the stress energy, need extra definition, but we may regard this as an additional issue. Following the idea above for defining the quantized stress energy tensor, one can see that the starting point for the expectation value of the stress energy tensor in a state $\omega_{\boldsymbol{M}}$ on $\mathscr{A}(\boldsymbol{M})$ is the corresponding two-point function $$w_{\boldsymbol{M}}(x,y) = \omega_{\boldsymbol{M}}(\Phi_{\boldsymbol{M}}(x)
\Phi_{\boldsymbol{M}}(y)) \,,$$ written here symbolically as a function, although properly it is a distribution on $C_0^\infty(\boldsymbol{M} \times \boldsymbol{M})$. One now considers a particular set $\mathcal{S}_{\mu sc}(\boldsymbol{M})$, the states on $\mathscr{A}(\boldsymbol{M})$ whose two-point functions fulfill the [*microlocal spectrum condition*]{} [@Rad; @BFK; @SVW]. The microlocal spectrum condition specifies the wavefront set of the distribution $w_{\boldsymbol{M}}$ in a particular, asymmetric way, which is reminiscent of the spectrum condition for the vacuum state in Minkowski spacetime. Using the transformation properties of the wavefront set under diffeomorphisms, one can show (1) the sets $\mathcal{S}_{\mu sc}(\boldsymbol{M})$ transform contravariantly under arrows $\boldsymbol{M} \overset{\psi}{\longrightarrow} \boldsymbol{N}$ in [Loc]{}, meaning that every state in $\mathcal{S}_{\mu sc}(\boldsymbol{N})$ restricts to a state of $\mathcal{S}_{\mu sc}(\psi(\boldsymbol{M}))$ and $\mathcal{S}_{\mu sc}(\psi(\boldsymbol{M})) \circ \mathscr{A}(\psi) =
\mathcal{S}_{\mu sc}(\boldsymbol{M})$, and moreover (2) that the microlocal spectrum condition is equivalent to the [*Hadamard condition*]{} on $w_{\boldsymbol{M}}$ [@Rad; 