Loop Quantum Cosmology on a Torus

TL;DR

This paper explores how a torus topology influences Loop Quantum Cosmology by deriving the Teichmueller space, constructing a Hamiltonian, and analyzing the quantum properties, revealing a surprising continuous spectrum in the quantization.

Contribution

It introduces a novel analysis of torus topologies in Loop Quantum Cosmology, including the derivation of Teichmueller space and new quantization methods for the triad components.

Findings

Derived Teichmueller space for torus universes

Constructed a Hamiltonian for torus cosmologies

Found a continuous spectrum in the quantization of triad components

Abstract

In this paper we study the effect of a torus topology on Loop Quantum Cosmology. We first derive the Teichmueller space parametrizing all possible tori using Thurston's theorem and construct a Hamiltonian describing the dynamics of these torus universes. We then compute the Ashtekar variables for a slightly simplified torus such that the Gauss constraint can be solved easily. We perform a canonical transformation so that the holomies along the edges of the torus reduce to a product between almost and strictly periodic functions of the new variables. The drawback of this transformation is that the components of the densitized triad become complicated functions of these variables. Nevertheless we find two ways of quantizing these components, which in both cases leads surprisingly to a continuous spectrum.