Towards a covariant formulation of electromagnetic wave polarization

Chapter 6
Reducible representation of
the covariant spectral
density tensor
In Chapter 3 we introduced the covariant spectral density tensor, τ αβγδ , as
the tensor product of the field strength tensor with its complex conjugate,
F αβ (F γδ ) ∗ . This tensor contains all correlation components of the electromagnetic
field and it is manifestly Lorentz covariant. One would like to decompose
the covariant spectral density tensor in an irreducible tensor representation (see
Sections 2.3 and 2.4). To keep it simple one can begin with a reducible representation,
which can be expanded into an irreducible representation. In this
Chapter we decompose the covariant spectral density tensor, into a reducible representation
consisting of irreducible tensors The irreducible tensors are identified
as known scalars and tensors. The covariant polarization tensors in Chapter 5
are reconstructed directly from the covariant spectral density tensor.
6.1 The covariant spectral density tensor
As shown in Chapter 3, it is the possible to construct a rank four tensor, τ αβγδ =
F αβ (F γδ ) ∗ , which consists of quadratic components. This tensor is the covariant
spectral density tensor, and it contains all quadratic correlation components of
the electromagnetic field.
Since τ αβγδ is the tensor product of of the field strength tensor with itself,
it has some convenient symmetries and zeroes:
τ αβγδ = (τ γδαβ ) ∗ , (6.1)
τ αβγδ = −τ βαγδ = −τ αβδγ = τ βαδγ , (6.2)
τ αβγδ = 0 , if α = β or(and) γ = β. (6.3)
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