Title: Quantization of a causal diamond in 2+1 dimensional gravity
Abstract: We develop the reduced phase space quantization of causal diamonds in pure 2+1 dimensional gravity with a negative cosmological constant. The system is defined as the domain of dependence of a spacelike topological disk with fixed boundary length. 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). To quantize this nonlinear phase space we apply Isham's group-theoretic quantization scheme, 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 terms of the ratio of the Planck length to the boundary length.