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

TL;DR

This paper rigorously constructs classical and quantum photon fields on complex geometries with boundaries and inhomogeneous media, addressing zero-modes and providing a formula for the renormalised stress energy tensor.

Contribution

It introduces a rigorous framework for photon fields on non-compact manifolds with boundary, including zero-mode analysis and spectral calculus techniques.

Findings

Explicit formula for the renormalised stress energy tensor.

Analysis of zero-modes related to topology and obstacles.

Application to $ ext{R}^3$ with obstacles.

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.