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Renormalization of Entanglement Entropy and the Gravitational Effective Action

Joshua H. Cooperman, Markus A. Luty

TL;DR

Problem addressed: UV divergences in entanglement entropy and its relation to gravitational physics. Approach: formulate entanglement entropy via the Callan-Wilczek formula on conical spaces and renormalize using the gravitational effective action. Key results: leading term is the renormalized Bekenstein-Hawking entropy; subleading, state-dependent UV terms match Wald entropy for higher-derivative gravity. Implications: supports interpreting entanglement entropy as a finite observable in quantum gravity for a broad class of setups, while noting limitations related to gravitational fluctuations and restricted entangling surfaces.

Abstract

The entanglement entropy associated with a spatial boundary in quantum field theory is UV divergent, with the leading term proportional to the area of the boundary. For a class of quantum states defined by a path integral, the Callan-Wilczek formula gives a geometrical definition of the entanglement entropy. We show that, for this class of quantum states, the entanglement entropy is rendered UV-finite by precisely the counterterms required to cancel the UV divergences in the gravitational effective action. In particular, the leading contribution to the entanglement entropy is given by the renormalized Bekenstein-Hawking formula, in accordance with a proposal of Susskind and Uglum. We show that the subleading UV-divergent terms in the entanglement entropy depend nontrivially on the quantum state. We compute new subleading terms in the entanglement entropy and find agreement with the Wald entropy formula for black hole spacetimes with bifurcate Killing horizons. We speculate that the entanglement entropy of an arbitrary spatial boundary may be a well-defined observable in quantum gravity.

Renormalization of Entanglement Entropy and the Gravitational Effective Action

TL;DR

Problem addressed: UV divergences in entanglement entropy and its relation to gravitational physics. Approach: formulate entanglement entropy via the Callan-Wilczek formula on conical spaces and renormalize using the gravitational effective action. Key results: leading term is the renormalized Bekenstein-Hawking entropy; subleading, state-dependent UV terms match Wald entropy for higher-derivative gravity. Implications: supports interpreting entanglement entropy as a finite observable in quantum gravity for a broad class of setups, while noting limitations related to gravitational fluctuations and restricted entangling surfaces.

Abstract

The entanglement entropy associated with a spatial boundary in quantum field theory is UV divergent, with the leading term proportional to the area of the boundary. For a class of quantum states defined by a path integral, the Callan-Wilczek formula gives a geometrical definition of the entanglement entropy. We show that, for this class of quantum states, the entanglement entropy is rendered UV-finite by precisely the counterterms required to cancel the UV divergences in the gravitational effective action. In particular, the leading contribution to the entanglement entropy is given by the renormalized Bekenstein-Hawking formula, in accordance with a proposal of Susskind and Uglum. We show that the subleading UV-divergent terms in the entanglement entropy depend nontrivially on the quantum state. We compute new subleading terms in the entanglement entropy and find agreement with the Wald entropy formula for black hole spacetimes with bifurcate Killing horizons. We speculate that the entanglement entropy of an arbitrary spatial boundary may be a well-defined observable in quantum gravity.

Paper Structure

This paper contains 13 sections, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Entanglement Entropy and Conical Spaces
  3. Geometrical Formulation
  4. Entanglement Entropy from Conical Geometry
  5. Flat Spacetime
  6. Global Schwarzschild Spacetime
  7. Entanglement Entropy and the Gravitational Action
  8. Regulating the Cone
  9. The Conical Limit
  10. Calculations
  11. Wald Entropy
  12. Gravitational Fluctuations
  13. Conclusions

Figures (2)

  • Figure 1: Definition of fields for the Euclidean path integral defining the density matrix Eq. (\ref{['eq:rhoreduced']}) and the quantum states Eqs. (\ref{['eq:statepathintket']}) and (\ref{['eq:statepathintbra']}).
  • Figure 2: Coordinates for boost invariant spacetime. The arrows show the orbits of the boost symmetry, and the shaded region corresponds to $\kappa = U^2 - V^2 > 0$.