CONSISTENCY OF SEMICLASSICAL GRAVITY

We discuss some subtleties that arise in the semiclassical approximati on to quantum gravity. We show that integrability conditions prevent t he existence of Tomonaga-Schwinger time functions on the space of thre e-metrics but admit them on superspace. The concept of semiclassical t ime is carefully examined. We point out that central charges in the ma tter sector spoil the consistency of the semiclassical approximation, unless the full quantum theory of gravity and matter is anomaly-free. We finally discuss consequences of these considerations for quantum fi eld theory in hat spacetime, but with arbitrary foliations.