Reduction of complex electromagnetic fields to the irreducible representations of the inhomogeneous Lorentz group

Abstract

In the present paper we study the reduction of the wave function which transforms as electromagnetic fields to the irreducible representation of proper, inhomogeneous, orthochronous Lorentz group. The electromagnetic fields are expanded into the modes with the amplitudes as wave functions for massless particles of spin 0 and 1. The values of electromagnetic fields in terms of these expansions are calculated as the solution of Maxwell's equations in free space with and without the source. In the presence of source, we derive the expansions for longitudinal and transverse electric and magnetic fields and prove that in electromagnetic waves the horizontal component of the magnetic field is always zero, while in the absence of source we discuss the two types of circular polarizations of electromagnetic waves by proving that only spin-1 wave functions contribute to the electromagnetic fields in this case. In the end we study the second quantization of the electromagnetic fields on replacing the wave function and their complex conjugates by destruction and the construction operators respectively.