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Covariant phase space and the semi-classical Einstein equation

Abhirup Bhattacharya, Onkar Parrikar

TL;DR

This work advances a covariant phase space framework to semi-classical gravity by introducing the semi-classical symplectic form $\boldsymbol{F}=\Omega_{\text{grav.}}+\mathsf{f}$, where $\mathsf{f}$ is the Berry curvature of quantum matter and $\mathsf{a}=-i\langle\psi|\delta\psi\rangle$. It shows $\boldsymbol{F}$ is slice-independent and satisfies a quantum Hollands–Iyer–Wald identity, with a gauge-invariant extension to subregions via Connes cocycles. The formalism yields semi-classical charges generated by diffeomorphisms and connects bulk and boundary Berry curvatures in AdS/CFT, producing quantum-corrected relations between bulk canonical energy and boundary modular quantities (e.g., FLM and quantum extremal surface prescriptions). The results provide a covariant, quantum-corrected phase-space perspective on holographic entanglement and black hole entropy, offering a foundation for incorporating quantum matter effects into gravitational phase-space geometry and their CFT duals.

Abstract

The covariant phase space formalism in general relativity is a covariant method for constructing the symplectic two-form, Hamiltonian and other conserved charges on the phase space of solutions to the Einstein equation with classical matter. In this note, we consider a generalization of this formalism to the semi-classical Einstein equation coupled to quantum matter. Given a family of solutions in semi-classical gravity, we define the semi-classical symplectic two-form -- a natural generalization of the classical sympelctic two-form -- as the sum of the gravitational symplectic form and the Berry curvature associated to the quantum state of matter. We show that the semi-classical symplectic two-form is independent of the Cauchy slice, and satisfies the quantum generalization of the classical Hollands-Iyer-Wald identity. For small perturbations, we also extend our discussion to gauge-invariantly defined subregions of spacetime, where the quantum contribution is replaced by the Berry curvature of certain special purifications involving the Connes cocycle. In the AdS/CFT context, the semi-classical symplectic form defined here is naturally dual to the Berry curvature in the boundary CFT.

Covariant phase space and the semi-classical Einstein equation

TL;DR

This work advances a covariant phase space framework to semi-classical gravity by introducing the semi-classical symplectic form , where is the Berry curvature of quantum matter and . It shows is slice-independent and satisfies a quantum Hollands–Iyer–Wald identity, with a gauge-invariant extension to subregions via Connes cocycles. The formalism yields semi-classical charges generated by diffeomorphisms and connects bulk and boundary Berry curvatures in AdS/CFT, producing quantum-corrected relations between bulk canonical energy and boundary modular quantities (e.g., FLM and quantum extremal surface prescriptions). The results provide a covariant, quantum-corrected phase-space perspective on holographic entanglement and black hole entropy, offering a foundation for incorporating quantum matter effects into gravitational phase-space geometry and their CFT duals.

Abstract

The covariant phase space formalism in general relativity is a covariant method for constructing the symplectic two-form, Hamiltonian and other conserved charges on the phase space of solutions to the Einstein equation with classical matter. In this note, we consider a generalization of this formalism to the semi-classical Einstein equation coupled to quantum matter. Given a family of solutions in semi-classical gravity, we define the semi-classical symplectic two-form -- a natural generalization of the classical sympelctic two-form -- as the sum of the gravitational symplectic form and the Berry curvature associated to the quantum state of matter. We show that the semi-classical symplectic two-form is independent of the Cauchy slice, and satisfies the quantum generalization of the classical Hollands-Iyer-Wald identity. For small perturbations, we also extend our discussion to gauge-invariantly defined subregions of spacetime, where the quantum contribution is replaced by the Berry curvature of certain special purifications involving the Connes cocycle. In the AdS/CFT context, the semi-classical symplectic form defined here is naturally dual to the Berry curvature in the boundary CFT.
Paper Structure (20 sections, 134 equations, 2 figures)

This paper contains 20 sections, 134 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A primer on field spaces and infinite dimensional manifolds
  4. The covariant symplectic form
  5. Gravitational phase space with classical matter
  6. Boundary charges for diffeomorphisms
  7. The classical Hollands-Wald identity
  8. Covariant phase space with quantum matter
  9. Berry curvature
  10. Semi-classical generalization of the symplectic form
  11. Boundary charges
  12. The semi-classical Hollands-Wald identity
  13. Covariant phase space formalism for subregions
  14. Purifications and the subregion Berry connection in QFT
  15. The subregion Berry connection in semi-classical gravity
  16. ...and 5 more sections

Figures (2)

  • Figure 1: We consider spacetimes obtained by specifying asymptotic boundary conditions on the boundary $Y^-$, which involves a Euclidean part and a Lorentzian part. This data completely specifies the semi-classical geometry within the diamond $D(\Sigma_t)$ defined at asymptotic time $t$.
  • Figure 2: The bulk matter states are obtained by adding sources for single/multi-trace operators on the past asymptotic boundary $Y^-$.