We study Maxwell's equations as a theory for smooth $k$-forms on globally hyperbolic spacetimes with a timelike boundary. For that, we investigate the wave operator $\Box_k$ with appropriate boundary conditions and characterize the space of solutions of the associated initial and boundary value problem under reasonable assumptions. Subsequently we focus on the Maxwell operator $\delta\mathrm{d}$. First we introduce two distinguished boundary conditions, dubbed $\delta\mathrm{d}$ -tangential and $\delta\mathrm{d}$ -normal boundary conditions. Associated to these we introduce two different notions of gauge equivalence for the solutions of the Maxwell's operator $\delta\mathrm{d}$ and we prove that in both cases, every equivalence class admits a representative abiding to the Lorentz gauge. We then construct a unital -algebra $\mathcal A$ of observables for the system described by the Maxwell's operator. Finally we prove that, as in the case of the Maxwell operator on globally hyperbolic spacetimes with empty boundary, $\mathcal A$ possesses a non-trivial center.
This talk is part of series of talks affiliated with the virtual Mittag-Leffler workshop "Scattering, microlocal analysis and renormalization", organized by Claudio Dappiaggi, Jacob Schach Møller and Michal Wrochna. The full schedule can be found at: