Quantization of 2+1 Gravity on the Torus

TL;DR

This paper performs the canonical quantization of 2+1-dimensional gravity on a torus using polygon representation, revealing quantum behaviors like a big bounce and interference effects, and discusses the problem of time in quantum gravity.

Contribution

It provides an explicit quantization method for 2+1 gravity on a torus and explores the implications of different internal time choices on quantum cosmology.

Findings

Wave function exhibits a big bounce behavior.

Interference patterns resemble linear gratings.

Different internal times lead to distinct quantum interpretations.

Abstract

We use the polygon representation of 2+1--dimensional gravity to explicitly carry out the canonical quantization of a universe with the topology of a torus. The mapping-class-invariant wave function for a quantum ''big bounce'', is reminiscent of the interference patterns of linear gratings. We consider the ``problem of time'' of quantum gravity: for one choice of internal time the universe recovers a semiclassical interpretation after the bounce, with a wave packet centered at a single geometry; for another choice of internal time, the quantum solutions involve interference between macroscopically distinct universes.