Quantum gravitational corrections to the functional Schrödinger equation

Abstract

We derive corrections to the Schr\"odinger equation which arise from the quantization of the gravitational field. This is achieved through an expansion of the full functional Wheeler-DeWitt equation with respect to powers of the gravitational constant. The correction terms fall into two classes: One describes the breakdown of the classical background picture while the other corresponds to quantum gravitational corrections for the matter fields themselves. The latter are independent of the factor ordering which is chosen for the gravitational kinetic term. If the total state evolves adiabatically, the only correction term that survives contains the square of the matter Hamiltonian. In the general case there are also smaller terms which describe a gravitationally induced violation of unitarity. The corrections are numerically extremely tiny except near the big bang and the final stages of a black hole. They are also of principle significance for quantum field theories near the Planck scale.