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

TL;DR

This work provides a concrete canonical framework for de Sitter JT gravity, constructing a positive-definite physical Hilbert space by pairing invariant Wheeler–DeWitt states with co-invariant representations through group averaging. It reveals a rich phase-space structure (the fishbone) that splits into bang, bounce, and crunch sectors, with a tractable mini-superspace truncation that is exact for these sectors. The authors show how Klein–Gordon inner products, various gauge-fixing schemes, and a non-self-adjoint constraint can be reconciled with a well-defined inner product, and they connect the full 2D theory to a Schwarzian description in appropriate limits. The framework naturally extends to matter couplings, yielding a direct-integral Hilbert space and explicit examples with conformal matter, while also addressing asymptotic scattering states and potential topology-changing effects. Overall, the paper provides a robust, gauge-aware route to canonical quantum gravity in a cosmological setting and clarifies how bulk observables and inner products can be consistently defined in de Sitter JT gravity."

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.

Figures (6)

• Figure 1: The 'fishbone' is the space of connected closed universe spacetimes with constant positive curvature $\mathcal{R}=2$. This divides into sectors, each of which has a single real parameter labelling different solutions. On the left are big bang and big crunch solutions with an initial or final Milne singularity, which we call the bang and crunch sectors. The 'spine' of the fishbone gives bounce solutions which expand exponentially in both past and future, parameterised by the minimum radius of the universe. Finally, the 'ribs' are an infinite sequence of $\mathbf{n\times\text{dS}_2}$ sectors labelled by positive integers $n$ (joining the 'bounce' branch at radius $n$). These spacetimes are $n$-fold covers of dS$_2$, except that the identification of the angular periodic spatial coordinate comes along with a boost (or static patch time translation). The solutions of JT are described by one more real parameter which gives the magnitude of the dilaton solution, which is the conjugate momentum to the parameter describing the geometry. The phase space is thus the cotangent space of this fishbone (at least away from degenerate geometries, which are more subtle).
• Figure 2: Each spacetime is a piece of $\widetilde{\mathrm{dS}_2}$, with some identifications. These are illustrated here, along with integral curves of the one-parameter isometry group (which are also the curves of constant dilaton) in each case. a) Identifying two curves related by a boost in the future Milne wedge of one copy of global $\mathrm{dS}_2$ lead to a big bang geometry. A similar identification in the past Milne wedge would yield a big crunch geometry. b) Identifying the worldlines of two static observers at different spatial locations. These result in bounce geometries with the minimal spatial volume depending on the minimal proper separation between the two worldliness. c) Finally, if the worldines in the previous case are separated by exactly an integer multiple of $2\pi$ in $\theta$, we can perform the identifications with a relative boost between the two worldines. This results in what we call the $n\times\mathrm{dS}_2$ spacetimes.
• Figure 3: The gauge condition \ref{['eq:ksqGF']} is the equation of motion for a particle moving in a quartic potential $V(k)$ depicted here, where we require solutions to be $2\pi$ periodic. Solutions with $k(\theta)=\mathrm{constant}$ correspond to a particle sitting at the extrema of this potential with $k(\theta)= 0,\pm\sqrt{2a}$ corresponding to bouncing, crunching, and banging geometries respectively. Non-constant $k(\theta)$ satisfying the gauge condition have solutions which oscillate about the minima of the potential with minimal period $\frac{2\pi}{n}$, and correspond to $n\times\mathrm{dS}_2$ geometries not captured using the $k'=0$ gauge.
• Figure 4: When we impose the $a'=k'=0$ gauge fixing and pass to the mini-superspace theory, we do not pick up the $n\times\mathrm{dS}_2$ geometries. The space of remaining $\mathcal{R}=2$ geometries is depicted above. There are three remaining sectors --- bouncing, banging, and crunching solutions, each labelled by a single parameter --- meeting at a vertex (which encompasses two distinct geometries).
• Figure 5: When expressed in the basis of $a$ and $k$ eigenstates the Hamiltonian becomes a first order differential operator that we can interpret as generating boosts on the right half of a two-dimensional Minkowski space with $a$ and $k$ acting as space and time respectively. Each orbit of the boost corresponds to a definite spacetime geometry (a point on the configuration space in figure \ref{['fig:miniPhaseSpace']}). 'Timelike' orbits in the 'Rindler wedge' $a>|k|$ correspond to bounce states while orbits in the 'Milne' wedges correspond to bang and crunch states.