JT Gravity in de Sitter Space and Its Extensions

We discuss and extend some aspects pertaining to the canonical quantisation of JT gravity in de Sitter space, including the problem of time and the construction of a Hilbert space. We then extend this discussion to other two dimensional models obtained by changing the dilaton potential and show that the canonical quantisation procedure can be carried out for a large class of such models. Some discussion leading towards a path integral understanding for states, other than the Hartle Hawking state, is also included here, along with comments pertaining to Holography and the entropy of de Sitter space.

💡 Research Summary

This paper provides a comprehensive study of Jackiw‑Teitelboim (JT) gravity in two‑dimensional de Sitter (dS) space, focusing on its canonical quantisation, the associated problem of time, and the construction of a bulk Hilbert space. The authors begin by reviewing the JT action with a dilaton ϕ, treating ϕ as the physical clock. By fixing ϕ on constant‑dilaton hypersurfaces they define a conserved inner product, which allows the Wheeler‑DeWitt (WdW) equation to be solved and a Hilbert space of single‑universe states to be built. Two bases are introduced: a Rindler‑type basis and an “M‑basis” built from the conserved quantity M = (∇ϕ)² + ϕ², which plays the role of a mass parameter distinguishing expanding (M > 0) and contracting (M < 0) branches. The paper analyses singularities, orbifold points, and the bounce sector obtained by analytically continuing ϕ to negative values, showing how time‑reversal symmetry and the resolution of classical singularities are encoded in the quantum theory.

The authors then connect JT dS to the supersymmetric Sachdev‑Ye‑Kitaev (SSS) double‑scaled matrix model. They demonstrate that the Hartle‑Hawking (HH) wavefunction of JT dS can be reproduced by the matrix model with a coefficient function ρ(M) that coincides with the entropy of a two‑dimensional de Sitter black hole of mass M. By introducing a second coefficient ˜ρ(M) they map states on the contracting branch as well, establishing a holographic dictionary between bulk wavefunctions and matrix‑model amplitudes. The matrix model naturally incorporates topology‑changing processes (creation/annihilation of universes, tunnelling between expanding and contracting branches), which explains why the Hilbert space obtained from the matrix model differs from the single‑universe Hilbert space derived via canonical quantisation.

A major part of the work is devoted to generalising the dilaton potential U(ϕ). The authors consider any potential that asymptotically behaves as U(ϕ) → ϕ² for ϕ → ∞, ensuring that the large‑ϕ sector matches JT gravity. They analyse classical solutions for polynomial potentials, exponential corrections, and mixed forms such as U(ϕ)=ϕ²+ae^{−αϕ}. For each class they solve the classical equations, identify the allowed regions (oscillatory vs. exponentially damped wavefunctions), and perform canonical quantisation using the same conserved inner product. In the exponential‑correction case U(ϕ)=ϕ²−2∑iεiαi e^{−αiϕ}, they explicitly evaluate the path integral, obtaining a wavefunction that solves the JT WdW equation at large ϕ but differs from the JT HH state, thereby providing a concrete example of non‑HH states in JT dS.

Section 8 extends the no‑boundary (Hartle‑Hawking) construction to a broad class of potentials, showing that the semiclassical density of states and the relation between the coefficient function and horizon entropy persist beyond the pure JT case. The authors discuss the role of the parameter S₀, which in higher‑dimensional origins equals the entropy of the near‑extremal Nariai black hole; taking S₀ → ∞ corresponds to the single‑universe sector used in the canonical quantisation.

Finally, the paper explores the possibility of a non‑perturbative completion via a finite‑rank double‑scaled matrix model. When the matrix rank L is large but finite (L ≈ e^{S₀}), the model captures subleading topology contributions and may provide a microscopic account of de Sitter entropy in terms of matrix degrees of freedom. The authors also comment on recent works that uncover additional classical branches not included in the original analysis and on alternative choices of physical clocks (e.g., constant extrinsic curvature slices), suggesting further avenues for research.

Overall, the work unifies canonical quantisation and matrix‑model holography for JT gravity in de Sitter space, demonstrates that a wide class of dilaton potentials share the same quantum structure, and offers concrete pathways to construct and interpret states beyond the Hartle‑Hawking wavefunction.