argument against an invariant state with the property is given in Fewster’s contribution [@cjfgm].

The Quantized Stress Energy Tensor
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A general property of quantum field theories is the existence of a causal dynamical law, or time-slice axiom. In a local covariant quantum field theory $\boldsymbol{\Phi}$, this leads to a covariant dynamics [@BFV] termed “relative Cauchy evolution” in Fewster’s contribution [@cjfgm], to which we refer again for further details. The relative Cauchy evolution consists, for each $\boldsymbol{M}$ in [Loc]{}, of an isomorphism $${\rm rce}_{\boldsymbol{M}}(h) : \mathscr{A}(\boldsymbol{M}) \to
 \mathscr{A}(\boldsymbol{M})$$ which describes the effect of an additive perturbation of the metric of $\boldsymbol{M}$ by a symmetric 2-tensor field $h$ on the propagation of the quantum field, akin to a scattering transformation brought about by perturbation of the background metric. One may consider the derivation – assuming it exists – which is obtained as $\left. d/ds \right|_{s=0}{\rm rce}_{\boldsymbol{M}}(h(s))$ where $h(s)$ is any smooth family of metric perturbations with $h(s=0) = 0$. Following the spirit of Bogoliubov’s formula [@Bogo], this gives rise (under fairly general assumptions) to a local covariant quantum field $({\sf T}_M)$ which generates the derivation upon taking commutators, and can be identified (up to some multiplicative constant) with the quantized stress 