Energy-momentum conservation in pre-metric electrodynamics with magnetic charges

TL;DR

This paper establishes the precise conditions for energy-momentum conservation in a topological, pre-metric framework of electrodynamics that includes magnetic charges, highlighting the role of constitutive relations and electric-magnetic reciprocity.

Contribution

It provides a necessary and sufficient condition for energy-momentum conservation in pre-metric electrodynamics with magnetic charges, clarifying the role of reciprocity and complex structures.

Findings

Energy-momentum conservation linked to constitutive relation G=G(F)

Reciprocity defines a complex structure on tensor pairs (F,G)

Contradicts recent claims about complex structure on 2-forms

Abstract

A necessary and sufficient condition for energy-momentum conservation is proved within a topological, pre-metric approach to classical electrodynamics including magnetic as well as electric charges. The extended Lorentz force, consisting of mutual actions by F=(E, B) on the electric current and G=(H, D) on the magnetic current, can be derived from an energy-momentum "potential" if and only if the constitutive relation G=G(F) satisfies a certain vanishing condition. The electric-magnetic reciprocity introduced by Hehl and Obukhov is seen to define a complex structure on the tensor product of 2-form pairs (F,G) which is independent of but consistent with the Hodge star operator defined by any Lorentzian metric. Contrary to a recent claim in the literature, it does not define a complex structure on the space of 2-forms itself.