Quantization of causal diamonds in (2+1)-dimensional gravity -- Part II: Group-theoretic quantization

Abstract

We develop the non-perturbative reduced phase space quantization of causal diamonds in (2+1)-dimensional gravity with a nonpositive cosmological constant. In Part I we described the classical reduction process and the reduced phase space, $\tilde{\mathcal{P}}=T^{\ast }({\text{Diff}}^{+}(S^1)/\text{PSL}(2,\mathbb{R}))$, while in Part II we discuss the quantization of the phase space and quantum aspects of the causal diamonds. Because the phase space does not have a natural linear structure, a generalization of the standard canonical (coordinate) quantization is required. In particular, as the configuration space is a homogeneous space for the ${\text{Diff}}^{+}(S^1)$ group, we apply Isham's group-theoretic quantization scheme. We propose a quantization based on (projective) unitary irreducible representations of the ${\text{BMS}}_3$ group, which is obtained from a natural prescription for extending ${\text{Diff}}^{+}(S^1)$ into a transitive group of symplectic symmetries of the phase space. 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 ${\text{Diff}}^{+}(S^1)/\text{PSL}(2,\mathbb{R})$ valued in some unitary irreducible representation of $\text{SL}(2,\mathbb{R})$. 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.

82 pages (127 including appendices), 0 figures. See also Part I: Classical reduction (arxiv.org/abs/2308.11741). [In this revised version, we have relocated some of the material to the appendices in order to streamline the main text and improve readability.]