Trace anomaly effective actions - a critique

Abstract: The existence of trace anomalies for conformally coupled quantum fields in curved spacetimes, despite the vanishing of the trace of the conformally invariant classical stress-energy tensor, was firmly established in the 1970's, and expressions for the anomalous trace of the semi-classical quantum stress-energy tensor constructed from the curvature tensors of the background spacetime were proven to hold independent of the quantum state of the field. Subsequently, effective actions were devised whose variation with respect to the metric gives a conserved "anomalous stress-energy tensor" or ASET. These ASETs, sometimes together with a contribution from a conformally invariant contribution to the effective action, were used to derive approximations to the full semi-classical stress-energy tensor (SCSET), which is state dependent. I argue that the ASET should be defined as state independent, and for a Schwarzschild black hole spacetime should be regular on the horizon and should fall off at infinity fast enough to be consistent with asymptotic flatness. Proposals for anomaly effective actions discussed in the literature are unsatisfactory in this regard, but I show that it is possible to meet these criteria. The ASET is evaluated and compared with numerical results for the full SCSET.