The Hilbert space of de Sitter JT: a case study for canonical methods in quantum gravity

TL;DR

This paper explores the Hilbert space structure of de Sitter Jackiw-Teitelboim gravity using canonical methods, addressing the challenge of Hamiltonian constraints and revealing a rich, sector-divided quantum structure.

Contribution

It introduces a canonical framework for de Sitter JT gravity, constructing the Hilbert space via invariants and gauge invariants, and relates it to mini-superspace models.

Findings

Identifies a positive-definite Hilbert space with multiple sectors.

Relates the quantum structure to classical phase space features.

Shows some sectors are exactly solvable via mini-superspace models.

Abstract

We study de Sitter JT gravity in the canonical formulation to illustrate constructions of Hilbert spaces in quantum gravity, which is challenging due to the Hamiltonian constraints. The key ideas include representing states as "invariants" (solutions to the Wheeler-DeWitt equation) or dual "co-invariants" (equivalence classes under gauge transformations), defining a physical inner product by group averaging, and relating this to Klein-Gordon inner products via gauge-fixing conditions. We identify a rich Hilbert space with positive-definite inner product which splits into distinct sectors, mirroring a similar structure in the classical phase space. Many (but not all) of these sectors are described exactly (in a constant extrinsic curvature gauge) by a mini-superspace theory, a quantum mechanical theory with a single constraint.