they don’t fix ${\sf T}_{\boldsymbol{M}}$ completely. Suppose that we have made some choice, $({\sf T}^{[1]}_{\boldsymbol{M}})$, of a local covariant stress energy tensor for our local covariant quantum field theory $\boldsymbol{\Phi}$. Now choose a local covariant family $(C_{\boldsymbol{M}})$ of (number-valued) smooth tensor fields $C_{\boldsymbol{M}} = C_{\boldsymbol{M}ab}$ on $\boldsymbol{M}$, and write $C_{\boldsymbol{M}}(f) = \int_{\boldsymbol{M}} C_{\boldsymbol{M}ab} f^{ab}
\,d{\rm vol}_{\boldsymbol{M}}$ where $d{\rm vol}_{\boldsymbol{M}}$ denotes the metric-induced volume form on $\boldsymbol{M}$. If $(C_{\boldsymbol{M}})$ is not only local covariant, but also symmetric and has vanishing divergence, then we may set $${\sf T}^{[2]}_{\boldsymbol{M}}(f) =  {\sf T}^{[1]}_{\boldsymbol{M}}(f) + C_{\boldsymbol{M}}(f)
{\boldsymbol{1}}$$ (where $\boldsymbol{1}$ is the unit of $\mathscr{A}(\boldsymbol{M})$) to obtain in this way another choice, $({\sf T}^{[2]}_{\boldsymbol{M}})$, of a stress energy tensor for $\boldsymbol{\Phi}$ which is as good as the previous one since it satisfies the conditions [**LCSE-1...4**]{} equally well. Again under quite general conditions, we may expect that this is actually the complete freedom for the stress energy tensor that is left by the above conditions, in particular the freedom should in fact be given by a multiple of unity (be state-independent). Since the $C_{\boldsymbol{M}}$ must be local covariant and divergence-free, they should be local functionals of the spacetime metric of $\boldsymbol{M}$, and one can prove this upon 