Towards a covariant formulation of electromagnetic wave polarization

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Abstract
A covariant formulation of electromagnetic wave polarization is introduced. It
is shown that 36 parameters are needed to fully describe the polarization of a
partially coherent wave. The 36 parameters becomes more understandable when
they are organized according to their tensor nature. An attempt is made to find
the largest set of tensor constituents, which uniquely determines the 36 polarization
parameters. The formulation is based on an irreducible representation
of electromagnetic wave polarization, using geometrical quantities with known
transformation properties under certain groups. The groups we concentrate on
are the Cartesian rotation group, SO(3), and the proper homogeneous Lorentz
group, SO(3, 1). Specifically, the parameters are obtained by decomposing tensor
products, of the electric and magnetic vector fields, ⃗ E and ⃗ B respectively,
into scalars, vectors and tensors. A set of Lorentz covariant polarization tensors,
the group SO(3, 1), are constructed from the electromagnetic field strength
tensor and its dual tensor. These covariant tensors have the SO(3) parameters
as components. The geometric quantities under SO(3) have a known physical
interpretation, only two are not previously found in the literature. One application
of these polarization parameters is to characterize the polarization of waves
in plasma, e.g., by determining the refractive index vector, ⃗n. The covariant
spectral density tensor is introduced and defined as the tensor product of the
field strength tensor with it-self. The covariant spectral density tensor is expanded
in a basis (the Dirac gamma matrices) and decomposed into a reducible
tensor representation. It is found that this reducible representation consists of
known covariant quantities. Since waves are considered, it is convenient to use
the frequency domain which implies the use of complex spaces.