Electromagnetic Potential in Pre-Metric Electrodynamics: Causal Structure, Propagators and Quantization

TL;DR

This paper develops a framework for pre-metric electrodynamics, analyzing the classical electromagnetic potential, its causal structure, and propagators, to facilitate covariant quantization beyond metric-dependent theories.

Contribution

It introduces a classical analysis of pre-metric electrodynamics, deriving propagators and causal structure without relying on a metric, and sets the foundation for covariant quantization.

Findings

Derived retarded and advanced propagators.

Analyzed causal structure based on constitutive law.

Outlined classical phase space and quantum field algebra.

Abstract

An axiomatic approach to electrodynamics reveals that Maxwell electrodynamics is just one instance of a variety of theories for which the name electrodynamics is justified. They all have in common that their fundamental input are Maxwell's equations $\textrm{d} F = 0$ (or $F = \textrm{d} A$) and $\textrm{d} H = J$ and a constitutive law $H = # F$ which relates the field strength two-form $F$ and the excitation two-form $H$. A local and linear constitutive law defines what is called local and linear pre-metric electrodynamics whose best known application are the effective description of electrodynamics inside media including, e.g., birefringence. We analyze the classical theory of the electromagnetic potential $A$ before we use methods familiar from mathematical quantum field theory in curved spacetimes to quantize it in a locally covariant way. Our analysis of the classical theory contains the derivation of retarded and advanced propagators, the analysis of the causal structure on the basis of the constitutive law (instead of a metric) and a discussion of the classical phase space. This classical analysis sets the stage for the construction of the quantum field algebra and quantum states. Here one sees, among other things, that a microlocal spectrum condition can be formulated in this more general setting.