Isometric evolution in de Sitter quantum gravity

TL;DR

This work demonstrates that in simple de Sitter quantum-gravity models, summing over smooth geometries yields isometric rather than unitary time evolution, effectively projecting crunching initial conditions out of the bulk while preserving a non-crunching code subspace. It establishes a matrix-model dual for de Sitter JT gravity with a varying boundary dilaton and shows how bulk-to-boundary evolution becomes an isometry, with the full evolution operator a projector $\widehat{\mathcal{U}}=\widehat{V}\widehat{V}^{\dagger}$ and $\widehat{V}^{\dagger}\widehat{V}=I_{\rm bulk}$. A minisuperspace analysis in Einstein gravity with positive $\Lambda$ finds a similar pruning: smooth geometries correspond to bouncing cosmologies and the path integral eliminates crunch-like histories, suggesting the mechanism persists beyond JT. Together, these results imply a holographically limited, non-perturbative bulk Hilbert space in de Sitter gravity and motivate further study of more realistic models and UV completions where the true state space may be finite-dimensional.

Abstract

We study time evolution in two simple models of de Sitter quantum gravity, Jackiw-Teitelboim gravity and a minisuperspace approximation to Einstein gravity with a positive cosmological constant. In the former we find that time evolution is isometric rather than unitary, and find suggestions that this is true in Einstein gravity as well. The states that are projected out under time evolution are initial conditions that crunch. Along the way we establish a matrix model dual for Jackiw-Teitelboim gravity where the dilaton varies on the boundary.

Figures (4)

• Figure 1: A depiction of the inner product $\langle e^{\varphi_1}|e^{\varphi_2}\rangle$. Following Cotler:2019dcj, we consider boundary conditions in the future asymptotic region corresponding to a bra and a ket, and perform the path integral over those metrics that interpolate between the boundary conditions in the limit that the corresponding boundaries approach one another.
• Figure 2: The no-boundary state evolved to the infinite future to give the Hartle-Hawking state. The state $|\varnothing\rangle$ corresponding to the Euclidean cap is prepared at a finite time, and is then evolved in Lorentzian time by $\widehat{V}$ to the infinite future. The wavefunction is naturally computed in the $\Phi$-basis by projecting onto $\langle \Phi|$ in the far future.
• Figure 3: The JT de Sitter S-matrix, starting in the state $|p_2\rangle$ and ending in the state $\langle p_1|$. Time evolution from past infinity to the bottleneck is given by $\widehat{V}^\dagger$, and time evolution from the bottleneck to future infinity is given by $\widehat{V}$. Since $\widehat{V}$ is an isometry, the total time evolution $\widehat{\mathcal{U}} = \widehat{V}\widehat{V}^\dagger$ is a projector.
• Figure 4: Plot of potential for the minisuperspace equations. For energies $E < V_{\max}$ the scale factor will reach a minimum size and bounce, whereas for $E > V_{\max}$ the warp factor will crunch to zero size.