Towards a covariant formulation of electromagnetic wave polarization

2.4. GROUP THEORY OF TRANSFORMATION MATRICES 23
This is the group of linear unitary transformations that leave a metric, δ α β ,
invariant. Here we also have the possibility of different signature and it it is
written in the the same way as for O(n + , n − ). The special unitary group is a
subgroup of U(n), given by
SU(n) = SL n (C) ∩ U(n). (2.111)
This is a subgroup of U(n), which admits a totally anti-symmetric tensor of
rank n as an invariant tensor. As in the real case, the orthogonal groups,
the unitary group, U(n), includes reflections, while the special unitary group,
SU(n), do not have reflections, but allows the totally anti-symmetric tensor as
as invariant. Similar to U(n + , n − ), we can have a signature of the metric tensor,
with the same notation, SU(n + , n − ).
2.4.2 Physical use of the groups
The groups considered in this work are SO(2), SU(2), SO(3), SU(3), SO(3, 1)
and SU(3, 1).
The generators of the first group, SU(2), are known as the Pauli spin matrices,
σ ν , given by Equation (2.54). The group SU(2) is isomorphic to the
next group, SO(3), which is the rotation group in three-dimensional Euclidean
space. The SU(3) group is similar to SO(3), but is a complex group, consisting
of complex rotations in C 3 . The generators of SU(3) are found in Appendix A
and are used to extract the generalized polarization parameters.
The proper homogeneous Lorentz group, with the flat metric g µν , given
by Equation (2.6), is the group SO(3, 1) and the generalization to a complex
vector space is the group SU(3, 1). Note that the signature of the metric is
(1, −1, −1, −1), which really is the same as (−1, 1, 1, 1) considering Lorentz
transformations, i.e. SU(3, 1) = SU(1, 3).
If we enlarge the groups SO(3, 1) and SU(3, 1) to include reflections, we
have the groups O(3, 1) and U(3, 1), respectively. These groups includes the
improper Lorentz transformations, which have det A = −1 . The proper Lorentz
transformations, together with translations, forms the Poincare group (proper
inhomogeneous Lorentz group), a group often encountered in the subject of the
Lorentz group.
The groups we are considering are SU(2), SO(3), SU(3), SO(3, 1) and SU(3, 1).
The special unitary and special orthogonal groups have their respective metric
tensor and the totally anti-symmetric tenor as invariant tensors. Using these invariant
tensors we can decompose tensors of the groups into lower rank tensors.
The unitary groups will include complex rotations, given by spherical tensors
(see Sakurai [20]), which can be interpreted as a change in the phase of the wave.
Restricting the rotations to only be real rotations of the Cartesian coordinate
axes, we can use the groups SO(3) and SO(3, 1) on complex vector spaces.
2.4.3 Irreducible tensor representation of Lie groups
In Section 2.3 we made a decomposition of a tensor product, ⃗ A ⊗ B, which is a
rank two tensor, into a scalar (rank zero tensor), a vector (rank one tensor) and
a tensor (rank two tensor). The tensors produced in this process are irreducible,
which means that they can not be written as a product of lower rank tensors