Quantization of causal diamonds in (2+1)-dimensional gravity -- Part I: Classical reduction

TL;DR

This paper develops the classical phase space reduction for causal diamonds in (2+1)-dimensional gravity with nonpositive cosmological constant, revealing a geometric structure related to circle diffeomorphisms.

Contribution

It provides a detailed classical reduction of the phase space for causal diamonds, identifying it as the cotangent bundle of Diff^+(S^1)/PSL(2,R), setting the stage for quantum analysis.

Findings

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

States correspond to causal diamonds in AdS_3 or Mink_3

Classical reduction clarifies geometric structure of the system

Abstract

We develop the non-perturbative reduced phase space quantization of causal diamonds in (2+1)-dimensional gravity with a nonpositive cosmological constant. In this Part I we focus on the classical reduction process, and the description of the reduced phase space, while in Part II we discuss the quantization of the phase space and quantum aspects of the causal diamonds. The system is defined as the domain of dependence of a spacelike topological disk with fixed boundary metric. By solving the 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), i.e., the group of orientation-preserving diffeomorphisms of the circle modulo the projective special linear subgroup. Classically, the states correspond to causal diamonds embedded in AdS_3 (or Mink_3 if $\Lambda = 0$), with fixed corner length, and whose Cauchy surfaces have the topology of a disc.