vanishing trace property for the expectation value of the stress energy tensor in the presence of curvature if one insists on the stress energy tensor having vanishing divergence: Under these circumstances, one necessarily finds $$\langle {\sf T}_{\boldsymbol{M}}^a{}_a \rangle_{\omega_{\boldsymbol{M}}} = -4 Q_{\boldsymbol{M }}\,.$$ Note that this is independent of the choice of Hadamard state $\omega_{\boldsymbol{M}}$. The origin of this trace-anomaly lies in having subtracted the divergence-causing term $Q g_{ab}$ in the definition of the renormalized stress energy tensor. In a sense, non-vanishing divergence of the renormalized stress energy tensor could also be viewed as an anomaly, so in the case of the conformally coupled, massless quantized linear scalar, one can trade the non-vanishing “divergence anomaly” of the renormalized stress energy tensor for the trace anomaly. The condition [**LCSE-4**]{} assigns higher priority to vanishing divergence, whence one has to put up with the trace anomaly.

One may wonder why the features of the quantized/renormalized stress energy tensor have been discussed here at such an extent while our main topic are local thermal thermal equilibrium states. The reason is that the ambiguities and anomalies by which the definition of the stress energy tensor for quantum fields in curved spacetime is plagued will also 