Towards a covariant formulation of electromagnetic wave polarization

Chapter 4
Irreducible representation
of SO(3) covariant
polarization
In Section 2.3 we defined the tensor product between two fields and represented
it as a sum of a scalar, a vector and a tensor, using SO(3) symmetry. In this
chapter we use this to define a new set of parameters describing the electromagnetic
field. The number of linearly independent components for a total
description of the fields are 36. Comparing with Storey and Lefeuvre [23], this
is the number of quadratic terms in the product ( E, ⃗ B) ⃗ ⊗ ( E ⃗ † , B ⃗ † ).
4.1 The irreducible parameters under SO(3)
The three-dimensional rotation group, SO(3) (see Section 2.4), is just rotation
of coordinates in three-dimensional Euclidean space. The generators of SO(3)
are rotation around each of the coordinate axis. The geometrical objects of this
group, SO(3), and the ordinary Cartesian three-dimensional space are scalars,
vectors and tensors. 1 To label the parameters in a convenient way we use the
letters S, V, T , which are mnemonics for Scalar, V ector and T ensor. The general
decomposition scheme can be written
F ⊗ F † = S ⊕ V ⊕ T. (4.1)
This is a mathematical way of displaying the decomposition into different subspaces
of geometrical objects and that the subspaces are closed under SO(3)
transformations.
The superscripts used to label the parameters are Σ, ∆, χ, ψ. The labels Σ
and ∆ denotes the autocorrelated parameters involving only multiplication of ⃗ E
with itself and ⃗ B with itself. Σ stands for the sum between electric and magnetic
components and ∆ is the difference. The last two labels, χ and ψ, denotes the
cross-correlated parameters, where the components only contain multiplication
1 For more information on SO(3) and similar groups, good sources are Arfken & Weber [10],
Goldstein [11] and Tung [24].
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