Entanglement, Soft Modes, and Celestial CFT

TL;DR

This paper investigates the vacuum entanglement entropy in free Maxwell theory using conformal transformations, soft modes, and edge states, revealing how soft charges contribute to entanglement across spacetime patches.

Contribution

It introduces a novel approach to analyze vacuum entanglement entropy via conformal inversions and soft modes, linking edge states to soft charge correlations in Maxwell theory.

Findings

Edge mode contributions arise from soft charge correlations.

Conformal inversion maps vacuum states to thermofield double states.

Soft modes at the entangling surface encode entanglement structure.

Abstract

We revisit the calculation of vacuum entanglement entropy in free Maxwell theory in four-dimensional Minkowski spacetime. Weyl invariance allows for this theory to be embedded as a patch inside the Einstein static universe. We use conformal inversions to extend the conformal primary solutions of the equations of motion labelled by a boost-weight $\Delta = 1 + i\lambda$ to an inverted Minkowski patch centered at spacelike infinity of the original patch. For $\lambda \neq 0$ these solutions admit an expansion in terms of wavefunctions supported in the (future) Milne wedges of the original and inverted Minkowski patches, that diagonalize the respective Milne times. The Minkowski vacuum can then be described as a thermofield double state on these two Milne wedges. We characterize the soft sectors of $\lambda = 0$ modes supported in the two Milne wedges. Upon reinterpreting the non-pure gauge $\lambda = 0$ wavefunctions as sourced by image charges in the inverse Minkowski patch, we construct an appropriate entangling constraint in the Minkowski theory, thereby characterizing the physical state space. We show that the edge mode contribution to the vacuum entanglement entropy is due to correlations between the soft charges of the two Milne patches, or equivalently non-trivial conformally soft mode configurations at the entangling surface.