On the square root of the Laplace--Beltrami operator as a Hamiltonian

Abstract

The Einstein equations for a spacetime of the form can be reduced to a Hamiltonian system on the Teichmuller space. However, the resulting Hamiltonian is the square root of a quadratic form in the momenta. Trying to make sense of this as a quantum operator is problematic since the Hamiltonian operator would be non-polynomial and non-local. In the first half of this article, I will examine the eigenfunctions of this operator. These go under the name of Maass functions and have been studied extensively by number theorists. In the second half, I will show that the quantum evolution due to this Hamiltonian does not cause the spacetime to collapse in a finite lapse of mean curvature time. Because of the difficult number theory involved with the Maass functions, I will actually perform the calculation for a simplified problem---a model of Teichmuller space---and argue that the answer should not be sensitive to the simplifications made in my model.