Wave function of the quantum black hole

Abstract

We show that the Wald Noether charge entropy is canonically conjugate to the opening angle at the horizon. Using this canonical relation we extend the Wheeler-DeWitt equation to a Schroedinger equation in the opening angle, following Carlip and Teitelboim. We solve the equation in the semiclassical approximation by using the correspondence principle and find that the solutions are minimal uncertainty wavefunctions with a continuous spectrum for the entropy and therefore also of the area of the black hole horizon. The fact that the opening angle fluctuates away from its classical value of 2 pi indicates that the quantum black hole is a superposition of horizonless states. The classical geometry with a horizon serves only to evaluate quantum expectation values in the strict classical limit.