Probabilistic deconstruction of a theory of gravity, Part II: curved space

TL;DR

This work advances a bottom-up view in which spacetime is the target space of a quantum observable and its volume measure is a probability density evolving under a quantum stochastic process. In AdS JT gravity, Einstein's equations emerge as the semi-classical generator equation governing joint quantum distributions of the boundary observable, with the dilaton/area playing the role of the evolving probability density. The analysis hinges on the Schwarzian regime, where long-time, near-boundary dynamics yield tractable two- and three-point correlators whose saddle-point evaluation reproduces the JT gravitational dynamics, including a negative cosmological constant. The authors conjecture that general relativity arises in the semi-classical limit of evolving probability under quantum stochastic processes, and outline future directions to extend the framework beyond AdS, incorporate matter, connect to SYK, and potentially derive holographic relations like the Ryu-Takayanagi formula from this probabilistic viewpoint.

Abstract

We propose that the underlying context of holographic duality and the Ryu-Takayanagi formula is that the volume measure of spacetime is a probability measure constrained by quantum dynamics. We define quantum stochastic processes using joint quantum distributions which are realized in a quantum system as expectation values of products of projectors. In anti-de Sitter JT gravity, we show that Einstein's equations arise from the evolution of probability under the quantum stochastic process induced by the boundary, with the area of compactified space in the gravitational theory identified as a probability density evolving under the quantum process. Extrapolating these and related results in flat JT gravity found in arXiv:2108.10916, we conjecture that general relativity arises in the semi-classical limit of the evolution of probability with respect to quantum stochastic processes.

Figures (6)

• Figure 1: Taking the holographic limit \ref{['hollim']}, two-point functions of the boundary particle have a closed-form expansion in large $\gamma$ only at i) short time scales, when it sees flat space, ii) long time scales, when it sees the asymptotic near-boundary geometry of $\mathop{\mathrm{\widetilde{AdS}}}\nolimits_2$.
• Figure 2: In the asymptotic near-boundary geometry of $\mathop{\mathrm{\widetilde{AdS}}}\nolimits_2$ given by \ref{['Mmetric']} with $\gamma \to \infty$, light cones are flattened so that only relative regions $6$ and its copies remain. In particular, the long-time dynamics in $\mathop{\mathrm{\widetilde{AdS}}}\nolimits_2$ corresponding to regions $6$ and $6'$ extends to vanishing times, or arbitrarily close to reference point $x'$.
• Figure 3: Depiction of a) level curves of relative coordinate $u$ with respect to reference point, and b) points on the classical trajectory of a spin-$\nu$ particle, ordered in increasing proper time.
• Figure 4: Semi-classically, the probability at a point of the black hole solution can only flow to inside of light-cones (shown in green). Thus in the two-sided black hole solution, the effective area at a point outside the left (right) horizon consists of probability for the left (right) boundary particle to be there, whereas the effective area inside the horizon is probability for either the left or right particle to be there.
• Figure 5: Depiction of a) a Poincare patch of the asymptotic space $\mathcal{M}$, and level curves of its spatial coordinate $q'$, and b) the map fixing $u_{12}$, which maps $x_1$ to a point with same spatial Poincaré coordinate as the base point $(\phi=0, \phi'=1)$.