Renormalization of Entanglement Entropy and the Gravitational Effective Action

TL;DR

This paper demonstrates that the UV divergences in entanglement entropy can be canceled by gravitational counterterms, linking entanglement entropy to the renormalized gravitational action and black hole entropy formulas.

Contribution

It establishes a direct connection between entanglement entropy and the renormalized gravitational effective action, including new subleading terms and their relation to Wald entropy.

Findings

Entanglement entropy is UV finite after renormalization with gravitational counterterms.

Leading entanglement entropy matches the renormalized Bekenstein-Hawking formula.

Subleading terms agree with Wald entropy for black holes.

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.