Causal diamonds in 2+1 dimensional quantum gravity

TL;DR

This paper develops a quantum theory of causal diamonds in 2+1 dimensional gravity, revealing a phase space linked to diffeomorphisms and quantization conditions on boundary twists, advancing understanding of quantum geometry in lower dimensions.

Contribution

It introduces a novel phase space description and applies group-theoretic quantization to causal diamonds in 2+1D gravity, connecting boundary geometry to Virasoro and BMS_3 groups.

Findings

Quantization of boundary twist in integer or half-integer multiples of Planck length over boundary length.

Phase space identified as cotangent bundle of Diff^+(S^1)/PSL(2,R).

Quantum states described by wavefunctions on Virasoro coadjoint orbits.

Abstract

We develop the reduced phase space quantization of causal diamonds in pure 2+1 dimensional gravity with a non-positive cosmological constant. The system is defined as the domain of dependence of a topological disc with fixed boundary metric. By solving the initial value constraints in a constant-mean-curvature time gauge and removing all the spatial gauge redundancy, we find that the phase space is the cotangent bundle of Diff^+(S^1)/PSL(2,R). To quantize this phase space we apply Isham's group-theoretic quantization scheme, with respect to a BMS_3 group, and find that the quantum theory can be realized by wavefunctions on some coadjoint orbit of the Virasoro group, with labels in irreducible unitary representations of the corresponding little group. We find that the twist of the diamond boundary loop is quantized in integer or half-integer multiples of the ratio of the Planck length to the boundary length.