stress energy tensor in [@WaldQFTCST] was an important starting point for the later development of local covariant quantum field theory, so the agreement between Wald’s result on the freedom in defining the stress energy tensor and ours, discussed above, are in no way coincidential.

How does this freedom come about? To see this, note that only the singularities of the Hadamard parametrix $\mbox{\sc h}_{\boldsymbol{M}}$ are completely fixed. One has the freedom of altering the smooth contributions to that Hadamard parametrix, and as long as this leads to an expected stress energy tensor which is still consistent with conditions [**LCSE-1...4**]{}, this yields an equally good definition of $\langle {\sf T}_{\boldsymbol{M}ab}(x) \rangle_{\omega_{\boldsymbol{M}}}$. Even requiring as in [@Wald78; @WaldQFTCST] that the expectation value of the stress energy tensor agrees in Minkowski spacetime with the expression obtained by normal ordering — implying that in Minkowski spacetime, the stress energy expectation value of the vacuum vanishes — doesn’t fully solve the problem. For if another definition is chosen so that the resulting difference term $C_{\boldsymbol{M}ab}$ is made of curvature terms, that difference vanishes on Minkowski spacetime. The problem could possibly be solved if there was a distinguished state $\boldsymbol{\omega} =
(\omega_{\boldsymbol{M}})$ for which the expectation 