Quantum propagators in a random metric

Abstract

We consider second order differential operators with coefficients which are Gaussian random fields. When the covariance becomes singular at short distances then the propagators of the Schr\"odinger equation as well as of the wave equation behave in an anomalous way. In particular, the Feynman propagator for the wave equation is less singular than the one with deterministic coefficients. We suggest some applications to quantum gravity.