Quantization of causal diamonds in (2+1)-dimensional gravity -- Part II: Group-theoretic quantization

TL;DR

This paper develops a non-perturbative, group-theoretic quantization of causal diamonds in (2+1)-dimensional gravity, revealing a novel quantization of the corner loop twist related to fundamental length scales.

Contribution

It introduces a new quantization scheme based on the BMS3 group and coadjoint orbits, extending Isham's method to a non-linear phase space in (2+1)-dimensional gravity.

Findings

Quantization of the corner loop twist in terms of Planck length and perimeter.

Proposal of a class of quantum theories labeled by Virasoro coadjoint orbits.

Hilbert space constructed from wavefunctions on Diff^+(S^1)/PSL(2,R) with unitary irreducible representations.

Abstract

We develop the non-perturbative reduced phase space quantization of causal diamonds in (2+1)-dimensional gravity with a nonpositive cosmological constant. In Part I we described the classical reduction process and the reduced phase space, $\widetilde{\mathcal P} = T^*(\text{Diff}^+\!(S^1)/\text{PSL}(2, \mathbb R))$, while in Part II we discuss the quantization of the phase space and quantum aspects of the causal diamonds. Because the phase space does not have a natural linear structure, a generalization of the standard canonical (coordinate) quantization is required. In particular, as the configuration space is a homogeneous space for the $\text{Diff}^+\!(S^1)$ group, we apply Isham's group-theoretic quantization scheme. We propose a quantization based on (projective) unitary irreducible representations of the $\text{BMS}_3$ group, which is obtained from a natural prescription for extending $\text{Diff}^+\!(S^1)$ into a transitive group of symplectic symmetries of the phase space. We find a class of suitable quantum theories labelled by a choice of a coadjoint orbit of the Virasoro group and an irreducible unitary representation of the corresponding little group. The most natural choice, justified by a Casimir matching principle, corresponds to a Hilbert space realized by wavefunctions on $\text{Diff}^+\!(S^1)/\text{PSL}(2, \mathbb R)$ valued in some unitary irreducible representation of $\text{SL}(2, \mathbb R)$. A surprising result is that the twist of the diamond corner loop is quantized in terms of the ratio of the Planck length to the corner perimeter.