Quantum gravity of the Heisenberg algebra

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.

30 pages + appendices; v2: typos corrected, references added

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Cited by in corpus (6)

• Krylov spread complexity as holographic complexity beyond JT gravity
• The dilaton gravity hologram of double-scaled SYK
• Tridiagonal Hamiltonians modeling the density of states of the Double-Scaled SYK model
• Quantum group origins of edge states in double-scaled SYK
• Gravitational wavefunctions in JT supergravity
• Composite AdS geodesics for CFT correlators and timelike entanglement entropy