On Maxwell's Equations on Globally Hyperbolic Spacetimes with Timelike Boundary

TL;DR

This paper investigates Maxwell's equations on globally hyperbolic spacetimes with timelike boundary, characterizing solutions, boundary conditions, gauge equivalences, and constructing associated algebraic structures for quantum field theory.

Contribution

It introduces boundary conditions for Maxwell's equations on such spacetimes, classifies gauge equivalence classes, and constructs algebraic observables, extending previous theories to spacetimes with boundary.

Findings

Boundary conditions yielding Green's formulas are identified.

Gauge equivalence classes of solutions are classified under specific boundary conditions.

The algebra of observables has a non-trivial center, similar to boundary-less cases.

Abstract

We study Maxwell's equation as a theory for smooth $k$-forms on globally hyperbolic spacetimes with timelike boundary as defined by Ak\'e, Flores and Sanchez. In particular we start by investigating on these backgrounds the D'Alembert - de Rham wave operator $\Box_k$ and we highlight the boundary conditions which yield a Green's formula for $\Box_k$. Subsequently, we characterize the space of solutions of the associated initial and boundary value problem under the assumption that advanced and retarded Green operators do exist. This hypothesis is proven to be verified by a large class of boundary conditions using the method of boundary triples and under the additional assumption that the underlying spacetime is ultrastatic. Subsequently we focus on the Maxwell operator. First we construct the boundary conditions which entail a Green's formula for such operator and then we highlight two distinguished cases, dubbed $\delta\mathrm{d}$-tangential and $\delta\mathrm{d}$-normal boundary conditions. Associated to these we introduce two different notions of gauge equivalence and we prove that in both cases, every equivalence class admits a representative abiding to the Lorenz gauge. We use this property and the analysis of the operator $\Box_k$ to construct and to classify the space of gauge equivalence classes of solutions of the Maxwell's equations with the prescribed boundary conditions. As a last step and in the spirit of future applications in the framework of algebraic quantum field theory, we construct the associated unital $*$-algebras of observables proving in particular that, as in the case of the Maxwell operator on globally hyperbolic spacetimes with empty boundary, they possess a non-trivial center.