Causal diamonds in ($2+1$)-dimensional quantum gravity

We develop the reduced phase space quantization of causal diamonds in pure ($2+1$)-dimensional gravity with a nonpositive 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 ${\text{Diff}}^{+}(S^1)/\mathrm{PSL}(2,\mathbb{R})$. To quantize this phase space we apply Isham’s group-theoretic quantization scheme, with respect to a ${\mathrm{BMS}}_3$ group, and find that the quantum theory can be realized by wave functions 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.

Figure 1

A generic classical state corresponds to a causal diamond in ${\mathrm{AdS}}_3$ (or in ${\mathrm{Mink}}_3$ if $\mathrm{\Lambda }=0$) with boundary length $\ell$. Note that in general the Cauchy horizon is not smooth since the null generators exit at caustics.