in order to obtain a well-defined quantity. In Minkowski spacetime, this is usually achieved by normal ordering with respect to the vacuum. This makes explicit reference to the vacuum state in Minkowski spacetime, the counterpart of which is not available in case of generic spacetimes. Therefore, one proceeds in a different manner on generic curved spacetimes (but in Minkowski spacetime, the result coincides with what one obtains from normal ordering, up to a renormalization freedom).

Since one is mainly interested in expectation values of the quantized stress energy tensor — as these are the quantities entering on the right hand side of the semiclassical Einstein equations, discussed in the next section — one may concentrate on a class of states $\mathcal{S}(\boldsymbol{M})$ on $\mathscr{A}(\boldsymbol{M})$ for which the expectation value of the stress energy tensor can be defined as unambiguously as possible, and in a manner consistent with the conditions [**LCSE-1...4**]{} above. Defining only the expectation value of the stress energy tensor has the additional benefit that one need not consider the issue if, or in which sense, ${\sf T}_{\boldsymbol{M}}(f)$ is contained in $\mathscr{A}(\boldsymbol{M})$ (or its suitable extensions). Of course, it has the drawback that non-linear expressions in ${\sf T}_{\boldsymbol{M}}(f)$, as they 