The classical and quantum photon field for non-compact manifolds with boundary and in possibly inhomogeneous media

Abstract

In this article I give a rigorous construction of the classical and quantum photon field on non-compact manifolds with boundary and in possibly inhomogeneous media. Such a construction is complicated by the presence of zero-modes that may appear in the presence of non-trivial topology of the manifold or the boundary. An important special case is $\mathbb{R}^3$ with obstacles. In this case the zero modes have a direct interpretation in terms of the topology of the obstacle. I give a formula for the renormalised stress energy tensor in terms of an integral kernel of an operator defined by spectral calculus of the Laplace Beltrami operator on differential forms with relative boundary conditions.