de Sitter Hilbert Space in Quantum Gravity

Updated 19 November 2025

de Sitter Hilbert space is the mathematical state space in quantum field theory and quantum gravity on de Sitter backgrounds, incorporating observer-centric descriptions and finite entropy bounds.
Its construction employs canonical quantization methods, Wheeler–DeWitt equation solutions, and a group-averaging procedure to yield a positive-definite, gauge-invariant inner product.
The sector structure, dividing bounce, bang, and crunch phases, provides insights into cosmological dynamics and supports holographic as well as observer-dependent analyses.

A de Sitter Hilbert space refers to the precise mathematical structure of the state space underlying quantum field theory (QFT), quantum gravity, and related models formulated on backgrounds or solutions with de Sitter (dS) symmetry. The rigorous construction and interpretation of this Hilbert space are essential for understanding fundamental issues in quantum cosmology, quantum gravity with a positive cosmological constant, observer-dependent entropy, and the possibility of a finite, observer-centric quantum theory for cosmological spacetimes. The following sections survey methodologies, foundational choices, and implications of the de Sitter Hilbert space, with an emphasis on both canonical quantum gravity exemplified by Jackiw–Teitelboim (JT) gravity and representation-theoretic QFT perspectives.

1. Canonical Construction and Wheeler–DeWitt Quantization

The canonical approach to the de Sitter Hilbert space is exemplified in the context of two-dimensional de Sitter JT gravity. The starting point is the action

$I_\mathrm{JT} = \frac{1}{4\pi}\int d^2x\sqrt{-g}\,\Phi(R-2)\,,$

where $\Phi$ is the dilaton. The phase space is coordinatized by spatial metrics $a(\theta)$ and dilaton profiles $\Phi(\theta)$ , with canonical momenta $p(\theta)$ and $k(\theta)$ . The dynamics is constrained by both Hamiltonian $\mathcal{H}$ and momentum $\mathcal{P}$ densities; after suitable gauge fixing (e.g., constant-extrinsic-curvature gauge $a'=k'=0$ ), only zero modes remain, and the entire theory reduces to quantum mechanics with a single Hamiltonian constraint:

$H = \frac{\partial^2}{\partial\Phi\,\partial a} + a\Phi \,, \qquad H|\psi\rangle = 0\,.$

Physical (Dirac) states—called “invariants”—are solutions to the Wheeler–DeWitt (WdW) equation, $H|\psi\rangle_\text{phys} = 0$ . Dual to this is the space of “co-invariants,” neither strictly constrained nor entirely unconstrained, but equivalence classes under $|\psi\rangle \sim |\psi\rangle + H|\phi\rangle$ . This distinction is crucial, both for explicit construction and for defining inner products in quantum gravity, where constraints are operator-valued distributions (Held et al., 18 Oct 2024).

2. Inner Products and the Group-Averaging Procedure

The central challenge in constructing a Hilbert space subject to Hamiltonian constraints is the definition of a positive-definite, gauge-invariant inner product. The group-averaging (or “rigging map”) procedure defines a noncompact projector onto physical states using the operator $\eta=2\pi\delta(H) = \int_{-\infty}^\infty dt\,e^{-iHt}$ . For co-invariant states $|\psi)$ , $|\psi')$ this yields the physical inner product

$(\psi'|\psi) = \langle \psi' | \eta | \psi \rangle_{H_0} \,.$

Explicitly in the variable $(a, k)$ representation, one finds that the “boost” generator $H = i(k\partial_a + a\partial_k)$ allows for “geodesic” sectors $\delta(k)\delta(a-b)$ , with the group-averaging mapping to

$\eta\,\delta(k)\delta(a-b) = (1/b)\,\delta(\sqrt{a^2 - k^2} - b)\,,$

and a positive measure given by

$\| \psi \|^2 = \int_0^\infty b\, db\, |\psi_0(b)|^2 + \int_0^\infty \beta\, d\beta\, [\,|\psi_+(\beta)|^2 + |\psi_-(\beta)|^2\,]\,.$

This group-averaging ensures that the physical Hilbert space is positive-definite and that key geometric observables (forming an $\mathfrak{sl}(2,\mathbb{R})$ algebra) are self-adjoint (Held et al., 18 Oct 2024).

3. Sector Structure and Geometric Interpretation

The full de Sitter JT Hilbert space decomposes into three distinct sectors, corresponding to classical branches of solutions:

Bounce: $a^2 > k^2$ (contracting to expanding closed universes); Hilbert space sector parameterized by minimum radius $b$ .
Bang: $k > a$ (big-bang cosmologies expanding from $a=0$ ); sector labeled by a Milne parameter $\beta$ .
Crunch: $-k > a$ (time-reversed bangs).

The WdW equation and group-averaging procedure enforce that physical wavefunctions are functions only of the invariant radii ( $b$ or $\beta$ ), corresponding to boost-invariant “radial” operators. This mirrors the classical “fishbone” phase-space structure, manifesting at the quantum level as disjoint Hilbert space sectors (Held et al., 18 Oct 2024).

4. Relation to Ambient Space and Group Representation Formalism

The representation-theoretic approach leverages the full de Sitter symmetry group. In higher dimensions (as in dS $_4$ ), the state space of single fields consists of unitary irreducible representations (UIRs) of SO(1,4), with the Hilbert space built from solutions to the Klein–Gordon equation on the hyperboloid,

$(Q_{dS} - \nu^2 - 9/4) \,\phi(x) = 0\,.$

Two natural bases arise:

Configuration basis: $|x;\pm\rangle$ states on the de Sitter manifold, orthonormal under the SO(1,4)-invariant measure;
Plane-wave basis: $|\nu,\xi\rangle$ , with $\xi$ a future-pointing null vector parameterizing the celestial sphere $S^3$ .

The Klein–Gordon inner product is manifestly invariant and positive-definite for principal series UIRs. The Fock space constructed from these one-particle states has precisely defined completeness properties—crucially, asymptotic “in” and “out” states can be defined observer-independently under an extended adiabatic assumption, yielding a well-defined unitary $S$ -matrix operator (Takook et al., 2023).

5. Inner Product and Observables: Uniqueness and Extension

At the quantum gravitational level, especially in “mini-superspace” truncations or large-volume limits (as in the analysis of the full de Sitter Wheeler–DeWitt equation), the Hilbert space reduces to a set of diffeomorphism- and Weyl-invariant functionals $Z[g,\chi]$ , subject to precise Ward identities. The space of all such $Z$ defines a “theory space,” and the positive-definite, Hermitian inner product is inherited from $L^2$ on functional space, gauge-fixed appropriately. This structure extends Higuchi’s group-averaged basis to finite gravitational coupling, linking the inner product to boundary conformal data (Chakraborty et al., 2023).

In the setting of observer-centric algebras, as in the construction of von Neumann algebras of observables for a static patch, the inclusion of the observer (“clock” degrees of freedom) converts the naive, ill-behaved Type III factor available in QFT into a well-defined Type II $_1$ algebra with a finite trace. The entropy of a physical state then matches the generalized entropy $A/4G_N + S_\text{out}$ , with the maximally entropic (trace) state corresponding to the Bunch–Davies vacuum (Chandrasekaran et al., 2022).

6. Implications for Quantum Gravity and Complementarity

The sector structure, group-averaged inner product, and the dual roles of invariants and co-invariants provide a canonical model for building physical Hilbert spaces in quantum gravity, characterized by genuinely gauge-invariant observables and equipped with positive-definite norms. In cases including gravitating observers or black holes, the algebra of observables can become Type II $_1$ , enforcing a finite entropy bound and encoding the peculiarities of cosmological complementarity—only a finite number of independent degrees of freedom are accessible to any single observer.

In JT gravity, these constructions are exact at the quantum level; in higher dimensions, similar structures are conjectured but require further understanding of nonperturbative quantum gravitational effects. The de Sitter Hilbert space remains a critical testing ground for foundational quantum gravity—both for canonical quantization and for holographic (dS/CFT) approaches (Held et al., 18 Oct 2024, Takook et al., 2023, Chakraborty et al., 2023).

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