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 Teichmüller 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 Teichmüller space---and argue that the answer should not be sensitive to the simplifications made in my model.

Polymer quantum dynamics of the Taub universe

“…In particular, we choose a discretized x space, and solve the corresponding eigenvalue problem in the p polarization. Even though the bulk of the discussion of the relation of the polymer representation at different scales is based on the properties of the Hamiltonian as a quadratic form, we can nevertheless apply this paradigm to the Taub model, which is described by a linear Hamiltonian (8), after the well-known procedure, established in [21]. In fact, squaring the Hamiltonian leads to squared eigenvalues without affecting the corresponding eigenfunctions.…”

“…Considering the time evolution for the wave function Ψ as given by Ψ k (p, τ ) = e −ikτ ψ k (p) and the results of [21], we obtain the following eigenvalue problem…”

Quantum modular group in (2+1)-dimensional gravity

“…It may be checked that tracesR a satisfying (2.17) can be represented by [11,18,19] 20) where the operatorsr…”

“…In ADM quantization, the natural configuration space is Teichmüller space, and the group of large diffeomorphisms-the modular group-has a well-understood and well-behaved action on this space. As a consequence, standard mathematical results allow us to construct invariant (or more general "covariant") wave functions [9,20]. In the holonomy representation, on the other hand, the modular group does not act nicely (i.e., properly discontinuously) on the natural configuration space, and the construction of invariant wave functions is much more problematic [21][22][23].…”

Canonical quantization of (2+1)-dimensional gravity

“…Both the dynamical and the mapping class problems have been solved for the scattering of two particles on IR 2 [13], and the Hamiltonian is known for the quantum torus [7], with work under way on the mapping class problem [33].…”