Defining entanglement without tensor factoring: a Euclidean hourglass prescription

TL;DR

This paper introduces a gauge-invariant method to define entanglement entropy across a planar boundary using a Euclidean hourglass geometry, avoiding tensor factorization and applicable to scalar and Maxwell fields.

Contribution

It proposes a novel entanglement entropy prescription that does not rely on tensor factorization, providing a gauge-invariant and geometrically regulated approach.

Findings

Entropy for scalar fields is insensitive to non-minimal coupling.

Maxwell field entropy matches that of (d-2) scalars.

The method offers a manifest state-counting interpretation.

Abstract

We consider entanglement across a planar boundary in flat space. Entanglement entropy is usually thought of as the von Neumann entropy of a reduced density matrix, but it can also be thought of as half the von Neumann entropy of a product of reduced density matrices on the left and right. The latter form allows a natural regulator in which two cones are smoothed into a Euclidean hourglass geometry. Since there is no need to tensor-factor the Hilbert space, the regulated entropy is manifestly gauge-invariant and has a manifest state-counting interpretation. We explore this prescription for scalar fields, where the entropy is insensitive to a non-minimal coupling, and for Maxwell fields, which have the same entropy as $d-2$ scalars.