Edge State Quantization: Vector Fields in Rindler

TL;DR

This paper explores the entanglement and edge state structure of vector fields, particularly Maxwell and Proca theories, in Rindler space, revealing their thermodynamic properties and connections to black hole microstates.

Contribution

It introduces a detailed analysis of edge sectors and gauge transformations in vector field theories in Rindler space, linking them to black hole microstates and extending to tensor fields and Chern-Simons theory.

Findings

Maxwell theory in Rindler space is thermodynamically trivial in 1+1 dimensions.

Edge states are characterized as eigenstates of horizon electric flux and large gauge transformations.

The edge Hilbert space is generated by Wilson line punctures, representing black hole microstates.

Abstract

We present a detailed discussion of the entanglement structure of vector fields through canonical quantization. We quantize Maxwell theory in Rindler space in Lorenz gauge, discuss the Hilbert space structure and analyze the Unruh effect. As a warm-up, in 1+1 dimensions, we compute the spectrum and prove that the theory is thermodynamically trivial. In d+1 dimensions, we identify the edge sector as eigenstates of horizon electric flux or equivalently as states representing large gauge transformations, localized on the horizon. The edge Hilbert space is generated by inserting a generic combination of Wilson line punctures in the edge vacuum, and the edge states are identified as Maxwell microstates of the black hole. This construction is repeated for Proca theory. Extensions to tensor field theories, and the link with Chern-Simons are discussed.