Quantization of causal diamonds in (2+1)-dimensional gravity -- Part I: Classical reduction

TL;DR

The paper tackles quantum gravity in a quasi-local setting by performing a non-perturbative reduced phase space analysis of causal diamonds in (2+1)-dimensional gravity with Λ≤0. It implements a CMC-time gauge with Dirichlet boundary conditions to show that the classical reduced phase space is the cotangent bundle $T^*[ ext{Diff}^+(S^1)/ ext{PSL}(2, ext{R})]$, corresponding to causal diamonds embedded in AdS$_3$ (or Mink$_3$ for Λ=0) with fixed corner length. The reduction proceeds via solving the Lichnerowicz equation, quotienting out boundary-trivial conformal and diffeomorphism symmetries, and expressing the reduced space either through conformal data or conformal coordinates, with an explicit projection map $J$ that removes PSL(2,R) redundancies. This Part I sets the stage for Part II, where Isham-type group quantization will be employed on the reduced phase space to address quantum causal diamonds, highlighting the role of boundary gravitons and the York time Hamiltonian. The work advances a non-perturbative, gauge-invariant description of gravitational subregions, with potential implications for locality and subsystem structure in quantum gravity.

Abstract

We develop the non-perturbative reduced phase space quantization of causal diamonds in (2+1)-dimensional gravity with a nonpositive cosmological constant. In this Part I we focus on the classical reduction process, and the description of the reduced phase space, while in Part II we discuss the quantization of the phase space and quantum aspects of the causal diamonds. 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^+(S^1)/PSL(2,R), i.e., the group of orientation-preserving diffeomorphisms of the circle modulo the projective special linear subgroup. Classically, the states correspond to causal diamonds embedded in AdS_3 (or Mink_3 if $Λ= 0$), with fixed corner length, and whose Cauchy surfaces have the topology of a disc.

Figures (14)

• Figure 1: A typical causal diamond, obtained by maximally developing geometric data on a Cauchy slice with disc topology.
• Figure 2: Starting with ADM data, satisfying the Dirichlet condition on the (induced) boundary metric, we gauge-fix "time" by imposing the CMC condition. Then we impose the momentum constraint (M. C.) to define "seed data". Further imposing the Hamiltonian constraint (H. C.) leads to "valid data" (i.e., data satisfying all the constraints); from arguments involving the Lichnerowicz equation, the space of valid data can be identified with the space of seed data modulo (boundary-trivial) Weyl transformations. The reduced phase space is then obtained by further quotienting the space of valid data by (boundary-trivial) diffeomorphisms; this identifies the reduced phase space with the space of seed data modulo (boundary-trivial) conformal transformations.
• Figure 3: The phase space consists of causal diamonds in $\text{\sl AdS}_3$ (or $\text{\sl Mink}_3$ if $\Lambda = 0$) with topologically trivial Cauchy slices (discs) whose corner loops have fixed length $\ell$.
• Figure 4: A diagrammatic summary of the phase space reduction process for the diamond. From $\mathcal{P}$ to $\bar{\mathcal{P}}$ the phase space is enlarged by introducing "conformal coordinates" $(\Psi, \Omega, \bar{\sigma}^{ab})$, and the constraints define a submanifold $\bar{\mathcal{S}}$. The order of these two steps can be interchanged, first going to $\mathcal{S}$ by imposing the constraints and then enlarging to $\bar{\mathcal{S}}$ by introducing conformal coordinates $(\Psi, \bar{\sigma}^{ab})$. From $\bar{\mathcal{S}}$ to $\hat{\mathcal{S}}$ the bulk diffeomorphisms are removed, and to $\widetilde{\mathcal{P}}$ the remaining $\text{\sl PSL}(2, \mathbb{R})$ action is quotiented out.
• Figure 5: Illustration of the uniformization algorithm. The space $(h, D)$ is flattened with a Weyl transformation, then isometrically embedded in the complex plane, then analytically deformed into the unit disc, and naturally identified with $(\bar{h}, D)$.