In Search of Fundamental Discreteness in 2+1 Dimensional Quantum Gravity

TL;DR

This paper investigates whether fundamental discreteness exists in 2+1 dimensional Lorentzian quantum gravity by analyzing length operators, finding that their spectra are continuous, challenging the notion of inherent discretization.

Contribution

It reexamines the discretization of time in 2+1D quantum gravity using the Chern-Simons formulation and identifies continuous spectra for length operators.

Findings

Length operators have continuous spectra.

Spectra are not bounded away from zero.

Discreteness of time is not supported by these results.

Abstract

Inspired by previous work in 2+1 dimensional quantum gravity, which found evidence for a discretization of time in the quantum theory, we reexamine the issue for the case of pure Lorentzian gravity with vanishing cosmological constant and spatially compact universes of genus larger than 1. Taking as our starting point the Chern-Simons formulation with Poincare gauge group, we identify a set of length variables corresponding to space- and timelike distances along geodesics in three-dimensional Minkowski space. These are Dirac observables, that is, functions on the reduced phase space, whose quantization is essentially unique. For both space- and timelike distance operators, the spectrum is continuous and not bounded away from zero.