# The Potential in General Linear Electrodynamics: Causal Structure, Propagators and Quantization

February 02, 2016

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 general linear electrodynamics whose
best known application are the effective description of electrodynamics inside
media including, e.g., birefringence. We will 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.

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