Quantization of free electromagnetic field: A different approach to the time-like polarization

Abstract

A four-dimensional photon polarization space, such that gives a different interpretation of the ladder operators for the time-like degree comparing to the Gupta-Bleuler formulation is presented. This interpretation, coming from the construction of a covariant Hamiltonian, gives positive defined norms for all the four polarization degrees of freedom. This paper is written in the background of reducible representation algebras introduced by Czachor, although the interpretation of the ladder operators as presented here could be done independently. States that reproduce standard electromagnetic fields (i.e. photons with two transverse polarizations) from the four-dimensional covariant formalism (i.e. with two additional longitudinal and time-like polarizations) are shown explicitly. Lastly, Lorentz and gauge transformations are discussed in detail.