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Quantum gravity of the Heisenberg algebra

Ahmed Almheiri, Akash Goel, Xu-Yao Hu

TL;DR

This work analyzes a tractable toy model for the high-temperature limit of double-scaled SYK, where the Hamiltonian becomes the harmonic-oscillator position operator and the length of a wormhole is the central dynamical degree of freedom. By computing the full generating functional for length observables and their coupling to matter, it reveals that the effective action for the length is non-local and that the length dynamics exhibit de Sitter-like features, including slowed growth under perturbations. The authors reconstruct an emergent bulk metric from on-shell geodesic lengths and reverse-engineer a dilaton-gravity theory whose potential depends on temperature, highlighting nonlocal bulk behavior. The results illuminate how simple quantum-mechanical models can encode bulk geometry and offer a controlled setting to study wormhole dynamics, matter interactions, and potential connections to traversable-wormhole protocols, while noting discrepancies with standard scrambling in local gravity.

Abstract

We consider a simplified model of double scaled SYK (DSSYK) in which the Hamiltonian is the position operator of the Harmonic oscillator. This model captures the high temperature limit of DSSYK but could also be defined as a quantum theory in its own right. We study properties of the emergent geometry including its dynamics in response to inserting matter particles. In particular, we find that the model displays de Sitter-like properties such as that infalling matter reduces the rate of growth of geodesic slices between the two boundaries. The simplicity of the model allows us to compute the full generating functional for correlation functions of the length mode or any number of matter operators. We provide evidence that the effective action of the geodesic length between boundary points is non-local. Furthermore, we use the on-shell solution for the geodesic lengths between any two boundary points to reconstruct an effective bulk metric and reverse engineer the dilaton gravity theory that generates this metric as a solution.

Quantum gravity of the Heisenberg algebra

TL;DR

This work analyzes a tractable toy model for the high-temperature limit of double-scaled SYK, where the Hamiltonian becomes the harmonic-oscillator position operator and the length of a wormhole is the central dynamical degree of freedom. By computing the full generating functional for length observables and their coupling to matter, it reveals that the effective action for the length is non-local and that the length dynamics exhibit de Sitter-like features, including slowed growth under perturbations. The authors reconstruct an emergent bulk metric from on-shell geodesic lengths and reverse-engineer a dilaton-gravity theory whose potential depends on temperature, highlighting nonlocal bulk behavior. The results illuminate how simple quantum-mechanical models can encode bulk geometry and offer a controlled setting to study wormhole dynamics, matter interactions, and potential connections to traversable-wormhole protocols, while noting discrepancies with standard scrambling in local gravity.

Abstract

We consider a simplified model of double scaled SYK (DSSYK) in which the Hamiltonian is the position operator of the Harmonic oscillator. This model captures the high temperature limit of DSSYK but could also be defined as a quantum theory in its own right. We study properties of the emergent geometry including its dynamics in response to inserting matter particles. In particular, we find that the model displays de Sitter-like properties such as that infalling matter reduces the rate of growth of geodesic slices between the two boundaries. The simplicity of the model allows us to compute the full generating functional for correlation functions of the length mode or any number of matter operators. We provide evidence that the effective action of the geodesic length between boundary points is non-local. Furthermore, we use the on-shell solution for the geodesic lengths between any two boundary points to reconstruct an effective bulk metric and reverse engineer the dilaton gravity theory that generates this metric as a solution.
Paper Structure (27 sections, 141 equations, 2 figures)

This paper contains 27 sections, 141 equations, 2 figures.

Table of Contents

  1. Introduction
  2. High temperature approximation and the Heisenberg algebra
  3. Analysis of the Heisenberg model
  4. Thermodynamics
  5. Wormhole length
  6. Generating functional for lengths
  7. Path integral
  8. Lagrangian at Large $\beta$
  9. The Heisenberg model with particles
  10. Correlation functions
  11. Dynamics of the length
  12. Individual lengths and folded space
  13. Colliding shockwaves inside the wormhole
  14. Single shockwave lengths
  15. Overlap and bulk correlations
  16. ...and 12 more sections

Figures (2)

  • Figure 1: A geodesic of length $l$ spans a central angle $\Theta$. $r$ ($\rho$) and $\theta$ ($\tau$) are the radial and angular coordinates, respectively, in the rotationally invariant metric (\ref{['eq:rotationally invariant metric ansatz']}) and the black hole metric (\ref{['rhometric']}).
  • Figure 2: The two matter cords split the diagram into four regions.