The Fermionic Entanglement Entropy and Area Law for the Relativistic Dirac Vacuum State

TL;DR

This paper proves an area law for fermionic entanglement entropy in the free Dirac field, demonstrating how it scales with boundary area under certain limits, and extends Widom's theorem to specific pseudo-differential operators.

Contribution

It introduces a regularization method for the Dirac field and generalizes Widom's theorem to handle discontinuities in principal symbols.

Findings

Proves an area law for fermionic entanglement entropy in the Dirac vacuum.

Establishes the behavior of entropy as volume tends to infinity or regularization length tends to zero.

Extends mathematical theorems to new classes of pseudo-differential operators.

Abstract

We consider the fermionic entanglement entropy for the free Dirac field in a bounded spatial region of Minkowski spacetime. In order to make the system ultraviolet finite, a regularization is introduced. An area law is proven in the limiting cases where the volume tends to infinity and/or the regularization length tends to zero. The technical core of the paper is to generalize a theorem of Harold Widom to pseudo-differential operators whose principal symbols develop a specific discontinuity at a single point.