The Hilbert space of gauge theories: group averaging and the quantization of Jackiw-Teitelboim gravity

TL;DR

The paper addresses the challenge of constructing a gravitational Hilbert space when the gauge group is non-compact, where standard group averaging can diverge due to infinite-volume stabilizers. It introduces a renormalized group-averaging framework that partitions the kinematic space into orbit-type sectors and defines sector-specific rigging maps, yielding a direct-sum physical Hilbert space with a positive-definite inner product. Applied to Jackiw-Teitelboim gravity in closed universes with positive cosmological constant, the method provides a complete quantization capturing expand/crunch, singular, and black-hole sectors, unifying Dirac quantization with gravitational path-integral intuition. These results offer a robust approach to quantizing gravity in settings with isometries, clarify the role of superselection sectors, and suggest pathways to generalize the framework beyond finite-dimensional gauge groups and into broader gravitational contexts.

Abstract

When the gauge group of a theory has infinite volume, defining the inner product on physical states becomes subtle. This is the case for gravity, even in exactly solvable models such as minisuperspace or low-dimensional theories: the physical states do not inherit an inner product in a straightforward manner, and different quantization procedures yield a priori inequivalent prescriptions. This is one of the main challenges when constructing gravitational Hilbert spaces. In this paper we study a quantization procedure known as group averaging, which is a special case of the BRST/BV formalism and has gained popularity as a promising connection between Dirac quantization and gravitational path integrals. We identify a large class of theories for which group averaging is ill-defined due to isometry groups with infinite volume, which includes Jackiw-Teitelboim gravity. We propose a modification of group averaging to renormalize these infinite volumes and use it to quantize Jackiw-Teitelboim gravity with a positive cosmological constant in closed universes. The resulting Hilbert space naturally splits into infinite-dimensional superselection sectors and has a positive-definite inner product. This is the first complete Dirac quantization of this theory, as we are able to capture all the physical states for the first time.

Figures (4)

• Figure 1: The infinite strip, with $\tau\in(-\pi/2,\pi/2)$ and $\sigma\in\mathbb{R}$, is the universal cover of the de Sitter hyperboloid, (\ref{['hyperboloid']}). Figure borrowed from Alonso-Monsalve:2024oii.
• Figure 2: The region $\mathcal{R}$ (\ref{['R']}) in three-dimensional Minkowski space is in one-to-one correspondence with elements of $\mathcal{G}$. The blue points correspond to $Q$ which generate rotations, the yellow points boosts, and the green points shears/null rotations. $\mathcal{R}$ lies between the sheets of the hyperboloid $Q^iQ_i=-\pi^2$ (dark blue), with the upper sheet included but not the lower one. Figure borrowed from Alonso-Monsalve:2024oii.
• Figure 3: Representative actions on the infinite strip by elements of the various orbit types of $\widetilde{\mathcal{G}}$. The colorful dashed line is the image of the black dashed line $(\sigma=0)$ under each isometry. The Killing vector fields for each isometry are shown as gray arrows. The vectors of Lie-algebra charges that generate each transformation (aside from $2\pi n$ translations) are (a) $Q^i=(a,0,0)$, timelike; (b) $Q^i=(b,-b,0)$, null; and (c) $Q^i=(0,-c,0)$, spacelike.
• Figure 4: Dilaton solutions on the geometries from Figure \ref{['fig:quotients']}. The dashed lines are identified under each quotient isometry. We interpret the dilaton as a proxy for volume in 2 spacetime dimensions, as suggested by its connection with near-extremal black hole limits in higher dimensions (see a discussion for e.g. in Alonso-Monsalve:2024oii). The regions where $\Phi\rightarrow +\infty$ are expanding dS regions, while the regions where $\Phi\rightarrow -\infty$ are curvature singularities. Then (a) and (b) represent expanding or crunching solutions, while (c) represents $n$ black holes which alternate with inflating regions.