energy tensor (and has been shown in examples to agree indeed with the derivation induced by forming the commutator with the quantized stress energy tensor [@BFV]). Assuming for the moment that ${\sf T}_{\boldsymbol{M}}(f)$ takes values in $\mathscr{A}(\boldsymbol{M})$, we see that in a local covariant quantum field theory which fulfills the time-slice axiom the stress energy tensor is characterized by the following properties:\
[**LCSE-1**]{} : $$2i [{\sl T}_{\boldsymbol{M}}(f),A] = \left. \frac{d}{ds}\right|_{s = 0} {\rm rce}_{\boldsymbol{M}}(h(s))
(A) \,, \quad A \in \mathscr{A}(\boldsymbol{M})\,, \quad f = \left.\frac{d}{ds}\right|_{s=0} h(s)$$ Note that $f = f^{ab}$ is a smooth, symmetric $C_0^\infty$ two-tensor field on $\boldsymbol{M}$. Using the more ornamental abstract index notation, one would therefore write $${\sf T}_{\boldsymbol{M}ab}(f^{ab}) = {\sf T}_{\boldsymbol{M}}(f) \,.$$ [**LCSE-2**]{} : $$\mathscr{A}(\psi) \left({\sf T}_{\boldsymbol{M}}(f) \right) = 
 {\sf T}_{\boldsymbol{N}}(\psi_*f)$$ whenever $\boldsymbol{M} \overset{\psi}{\longrightarrow} \boldsymbol{N}$ is an arrow in [Loc]{}.\
[**LCSE-3**]{} : $${\sf T}_{\boldsymbol{M}ab} = {\sf T}_{\boldsymbol{M}ba} \,.$$ [**LCSE-4**]{} : $$\nabla^a{\sf T}_{\boldsymbol{M}ab} = 0\,,$$ where $\nabla$ is the covariant derivative of the metric of $\boldsymbol{M}$.\
Conditions [**LCSE-3,4**]{} are (not strictly, but morally) consequences of the previous two conditions, see [@BFV] for discussion.

The conditions [**LCSE-1...4**]{} ought to be taken as characterizing for any stress energy tensor of a given local covariant quantum field theory $\boldsymbol{\Phi}$. However, 