Towards a covariant formulation of electromagnetic wave polarization

Contents
1 Introduction 1
2 Background 3
2.1 Covariant formulation of Maxwell equations . . . . . . . . . . . . 3
2.1.1 Basic relativity and tensor operations . . . . . . . . . . . 3
2.1.2 Microscopic Maxwell equations . . . . . . . . . . . . . . . 6
2.1.3 Construction of the electromagnetic field strength tensor . 8
2.1.4 Electromagnetic energy-momentum tensor . . . . . . . . . 9
2.2 Generalized Stokes parameters . . . . . . . . . . . . . . . . . . . 11
2.2.1 Spectral density matrix . . . . . . . . . . . . . . . . . . . 11
2.2.2 Instantaneous Stokes’ parameters . . . . . . . . . . . . . . 11
2.2.3 Three-dimensional polarization parameters . . . . . . . . 13
2.2.4 Interpretation of the 3D polarization parameters . . . . . 13
2.2.5 The Storey and Lefeuvre six-quasivector . . . . . . . . . . 15
2.3 Irreducible representation . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Complex vector algebra . . . . . . . . . . . . . . . . . . . 17
2.3.2 The electromagnetic field . . . . . . . . . . . . . . . . . . 18
2.3.3 The product E ⃗ ⊗ B ⃗ . . . . . . . . . . . . . . . . . . . . . 20
2.4 Group theory of transformation matrices . . . . . . . . . . . . . . 22
2.4.1 The general linear group and subgroups . . . . . . . . . . 22
2.4.2 Physical use of the groups . . . . . . . . . . . . . . . . . . 23
2.4.3 Irreducible tensor representation of Lie groups . . . . . . 23
3 Covariant spectral density tensor 27
3.1 Covariant spectral density tensor . . . . . . . . . . . . . . . . . . 27
3.2 Pauli spin matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Dirac gamma matrices . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Decomposition of τ αβγδ . . . . . . . . . . . . . . . . . . . . . . . 29
4 Irreducible representation of polarization 31
4.1 The irreducible parameters under SO(3) . . . . . . . . . . . . . . 31
4.1.1 Σ, the sum . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.2 ∆, the difference . . . . . . . . . . . . . . . . . . . . . . . 33
4.1.3 χ, the real part . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1.4 ψ, the imaginary part . . . . . . . . . . . . . . . . . . . . 34
4.2 Table of the parameters . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Normalization of the S, V, T parameters . . . . . . . . . . . . . . 35
4.3.1 Example: the refractive index . . . . . . . . . . . . . . . . 35
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