Symmetry Properties of Electromagnetic Field in the Matter

TL;DR

This paper explores the symmetry properties of electromagnetic fields in matter, revealing a complex structure with multiple potential vectors, dual spaces, and gauge symmetries, and suggests that electromagnetic fields in matter are inherently complex.

Contribution

It introduces a novel partition of the electromagnetic field space into subspaces with different symmetry properties and links these to multiple potential vectors and gauge groups.

Findings

Electromagnetic field in matter is a complex field with multiple potential vectors.

The space of electromagnetic fields is partitioned into four subspaces with distinct symmetry properties.

The gauge symmetry of electromagnetic fields in matter is governed by a two-parametric group.

Abstract

The sets ${\Phi({F}^{\mu \nu})}, {\Phi(\tilde {F}^{\mu \nu})}$ of linear functionals on the space $< F,+,\cdot >$ represent themself linear space $< \Phi,+,\cdot >$ over the field of \textit{scalars} $P$, which is dual to space $< F,+,\cdot >$, but it is substantial, that given linear space is not self-dual. It has been found, that the partition of linear space $< F,+,\cdot >$ over the field of genuine scalars and pseudoscalars, the vectors in which are sets of contravariant and covariant electromagnetic field tensors and pseudotensors ${{F}^{\mu\nu}}$, ${\tilde{F}^{\mu\nu}}$, ${{F}_{\mu\nu}}$, ${\tilde{F}_{\mu\nu}}$, on 4 subspaces takes place. It corresponds to appearance of 4 kinds of electromagnetic field potential 4-vectors $A_{\mu}$, which are transformed according to the representations of general Lorentz group with various symmetry relatively improper rotations. It has been found, that conserving quantity, corresponding to complex fields is complex charge. It is argued, that electromagnetic field in the matter is complex field and that two-parametric group $\Gamma(\alpha,\beta) = U_{1}(\alpha) \otimes \mathfrak R(\beta)$, where $\mathfrak R(\beta)$ is abelian multiplicative group of real numbers (excluding zero), determines the gauge symmetry of electromagnetic field. It is also argued, that free electromagnetic field is characterized by pure imagine charge.