  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.
