Complex representation theory of the electromagnetic field

TL;DR

This paper explores the complex representation theory of the electromagnetic field, analyzing its Lorentz group representations and highlighting differences with Dirac equations, especially regarding mass terms and current coupling.

Contribution

It provides a detailed analysis of the irreducible Lorentz group representations of the electromagnetic field and compares them with Dirac and Majorana formalisms.

Findings

Mass term conflicts with relativistic invariance in electric-magnetic formalism

Differences between Dirac and Maxwell equations in coupling to electric current

Distinct representation structures for electromagnetic fields and fermions

Abstract

A concise discussion of the 3-dimensional irreducible (1,0) and (0,1) representations of the restricted Lorentz group and their application to the description of the electromagnetic field is given. It is shown that a mass term is in conflict with relativistic invariance of a formalism using electric and magnetic fields only, contrasting the case of the two-component Majorana field equations. An important difference between the Dirac equation and the Dirac form of Maxwell's equations is highlighted by considering the coupling of the electromagnetic field to the electric current.