Quantization of causal diamonds in (2+1)-gravity

Abstract

We develop the non-perturbative reduced phase space quantization of causal diamondsin (2+1)-dimensional gravity with a nonpositive cosmological constant. 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+(S1)/PSL(2,R). Classically, the states correspond to causal diamonds embedded in AdS3 (or Mink3 if Λ = 0), with fixed corner length, and whose Cauchy surfaces have the topology of a disc. Because the phase space does not have a natural linear structure, a generalization of the standard canonical (coordinate) quantization is required. As the configuration space is a homogeneous space for the Diff+(S1) group, we apply Isham’s group-theoretic quantization scheme. We propose a quantization based on (projective) unitary irreducible representations of the BMS3 group. 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 Diff+(S1)/PSL(2,R) valued in some unitary irreducible representation of SL(2,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.