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Wormholes in Quantum Mechanics

Herman Verlinde

TL;DR

The work extends wormhole physics beyond gravity by defining a geometric path integral for wormhole partition functions in a broad class of quantum systems derived from phase-space quantization. It shows that the $n$-fold wormhole partition function is the $n$-th Rényi entropy of a thermo-mixed double state, realized via a replica construction that isolates contributions from different quantum numbers and their correlations. The approach yields exact results for a particle on a group manifold and connects to 2D CFTs with Virasoro symmetry, reproducing holographic AdS${}_3$ gravity in the appropriate limit. Overall, the paper provides a microscopic, non-gravitational realization of replica wormholes and a novel thermal mixed state that clarifies how classical correlations, quantum entanglement, and degeneracies contribute to entropy and information flow in wormhole geometries.

Abstract

We introduce a geometric path integral definition of wormhole partition functions in a general class of 1D quantum systems obtained by quantizing a phase space. We compute the wormhole partition function in a semi-classical limit and in some simple examples. The partition function of the n-fold wormhole is found to be identical to the n-th Renyi entropy of a thermal mixed state of the doubled system. This mixed state incorporates three types of quantum statistical behavior: classically correlated, quantum entangled, and classically uncorrelated. We apply our prescription to 2D CFTs with Virasoro symmetry and recover the holographic dual formulation in terms of AdS3 gravity.

Wormholes in Quantum Mechanics

TL;DR

The work extends wormhole physics beyond gravity by defining a geometric path integral for wormhole partition functions in a broad class of quantum systems derived from phase-space quantization. It shows that the -fold wormhole partition function is the -th Rényi entropy of a thermo-mixed double state, realized via a replica construction that isolates contributions from different quantum numbers and their correlations. The approach yields exact results for a particle on a group manifold and connects to 2D CFTs with Virasoro symmetry, reproducing holographic AdS gravity in the appropriate limit. Overall, the paper provides a microscopic, non-gravitational realization of replica wormholes and a novel thermal mixed state that clarifies how classical correlations, quantum entanglement, and degeneracies contribute to entropy and information flow in wormhole geometries.

Abstract

We introduce a geometric path integral definition of wormhole partition functions in a general class of 1D quantum systems obtained by quantizing a phase space. We compute the wormhole partition function in a semi-classical limit and in some simple examples. The partition function of the n-fold wormhole is found to be identical to the n-th Renyi entropy of a thermal mixed state of the doubled system. This mixed state incorporates three types of quantum statistical behavior: classically correlated, quantum entangled, and classically uncorrelated. We apply our prescription to 2D CFTs with Virasoro symmetry and recover the holographic dual formulation in terms of AdS3 gravity.

Paper Structure

This paper contains 17 sections, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Wormholes in Quantum Mechanics
  3. Wormhole Partition Function
  4. Semi-classical computation
  5. Particle on a group I
  6. A New Thermal Mixed State
  7. Three thermal states
  8. Entropy and mutual information
  9. Conclusion
  10. Wormholes in QM: Three examples
  11. Coupled oscillator
  12. Particle on a group II
  13. Schwarzian Quantum Mechanics
  14. Wormholes in 2D CFT
  15. Geometric phase space of 2D CFT
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Replica wormhole geometry for the third Rényi entropy represented as a quotient of the Poincaré disc. The side edges are all labeled by $SL(2,\mathbb{R})$ group elements.
  • Figure 2: The two entanglement cuts surrounding the near horizon region of the two-sided black hole introduce edge states on each side of the cut. The combined state of edges across each cut are described by a boundary state of the holographic CFT.