# On Entanglement Entropy of Maxwell fields in 3+1 dimensions

## TL;DR

This paper investigates the entanglement entropy of free photon fields in 3+1 dimensions, revealing area proportionality and distinct entanglement behaviors of transverse and longitudinal magnetic modes due to gauge constraints.

## Contribution

It introduces a method to separate gauge-invariant quantities and analyzes the different entanglement structures of magnetic field modes in a gauge theory.

## Key findings

- Entanglement entropy is proportional to the transverse area.
- Transverse magnetic modes are entangled locally near the surface.
- Longitudinal modes exhibit long-range entanglement extending far from the surface.

## Abstract

We consider entanglement entropy between two halves of space separated by a plane, in the theory of free photon in 3+1 dimensions. We show how to separate local gauge invariant quantities that belong to the two spatial regions. We calculate the entanglement entropy by integrating over the degrees of freedom in one half space using an approximation that assumes slow variation of the magnetic fields in longitudinal direction. We find that the entropy is proportional to the transverse area as expected. Interestingly the entanglement properties of the 2D transverse and longitudinal modes of magnetic field are quite different. While the transverse fields are entangled mostly in the neighborhood of the separation surface as expected, the longitudinal fields are entangled through an infrared mode which extends to large distances from the entanglement surface. This long range entanglement arises due to necessity to solve the no-monopole constraint condition for magnetic field.