Towards a covariant formulation of electromagnetic wave polarization

Towards a 

covariant formulation of 

electromagnetic wave 

polarization 

by 

Mikael Olofsson 

Swedish Institute of Space Physics 

Uppsala-division 

Box 537 

751 21 Uppsala 

IRF Scientific Report 273 

January 2001

i 

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.

ii 

Preface 

This report is a Master thesis in physics at the University of Uppsala. The thesis 

was performed at the Swedish Institute of Space Physics, Uppsala division (IRF- 

U). Supervisors for this work have been Jan Bergman and Tobia Carozzi and 

examinator was professor Bo Thidé. 

I would like to thank my supervisors Jan Bergman and Tobia Carozzi for 

suggesting this problem to me and for their help and tutoring in completing 

this thesis. I would also like to thank professor Bo Thidé for giving me the 

opportunity to do my thesis at the Swedish Institute of Space Physics, Uppsala 

division (IRF-U) and for his help with correcting and completing this report. 

Thanks to all the personnel at IRF-U for a pleasant time.

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 

iii

iv 

CONTENTS 

4.4 Stokes parameters from the S, V, T parameters . . . . . . . . . . 36 

D Covariant polarization tensors 61 

D.1 The X tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 

D.2 The Y tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 

D.3 The K and K tensors . . . . . . . . . . . . . . . . . . . . . . . . 64 

6 Reducible representation of τ αβγδ 45 

6.1 The covariant spectral density tensor . . . . . . . . . . . . . . . . 45 

6.2 The irreducible tensors . . . . . . . . . . . . . . . . . . . . . . . . 46 

6.3 Reducible representation . . . . . . . . . . . . . . . . . . . . . . . 47 

7 Conclusions 49 

A Generalized polarization parameters 51 

B Parameters from Dirac gamma matrices 53 

C E, ⃗ B ⃗ component form of S, V, T parameters 57 

C.1 Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 

C.2 ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 

C.3 χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 

C.4 ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 

D Covariant polarization tensors 61 

D.1 The X tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 

D.2 The Y tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 

D.3 The K and K tensors . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 1 

Introduction 

A fundamental principle in constructing a physical law is to define it with equations 

that are independent of the choice of coordinates. This way of formulating 

physics is known as a covariant formulation. The word covariant relates to tensors 

which are geometrical objects defined by their transformation properties. 

The simplest example is a scalar, being a rank zero tensor, which is known to 

be invariant for any choice of coordinate system. 

The purpose of this theoretical work is to find a manifestly covariant set of 

parameters describing classical electromagnetic wave polarization. Traditionally, 

electromagnetic wave polarization is characterized using Stokes’ parameters 

(see e.g. Stokes [22] and Born & Wolf [3]). Considering Stokes’ parameters, 

which are manifestly covariant in the two-dimensional representation (this is 

shown in Section 2.4, using the SO(2) group), it is not immediately apparent 

how they transform under the Lorentz group. Furthermore it is not clear how 

Stokes’ parameters handle waves which are not simply quasi-monochromatic 

transverse waves. 

The approach used in this thesis is that of classical macroscopic electromagnetic 

field waves and not the photonic or quantum mechanical picture. In 

this context, we define polarization as the second order statistical correlation 

between electric and magnetic field components. Therefore we compare the 

components of the field pairwise to see how they are related to each other, i.e., 

one has quadratic components. Invariants are generally quadratic quantities in 

some way, for example the length of a vector (using the Euclidean three dimensional 

space with the ordinary metric) is invariant under rotation but the 

three different components of the vector are changed under a rotation of coordinates. 

Many of the calculations presented in this work are not restricted to 

electromagnetic fields but can be applied to a general field. 

As we consider the wave properties of electric and magnetic fields, it is convenient 

to use the frequency domain representation, i.e., the Fourier transform 

of the wave. When using the frequency domain representation of the wave it 

becomes necessary to use complex spaces. In a complex space, one ends up with 

36 correlation parameters with two components combined into the form AB † , 

where A and B are two of the six field components (A and B can be the same 

component). 

After formulating polarization using the covariant formalism, the main emphasis 

is on combining these 36 quantities to obtain parameters which transform 

1

2 CHAPTER 1. INTRODUCTION 

according to certain transformation rules (the number of degrees of freedom will 

not change). These parameters are obtained by means of an irreducible tensor 

representation of the spectral density matrix. 

Chapter 2 gives an introduction and a background where some useful concepts 

are introduced: Stokes’ parameters and their three dimensional generalization 

(see Barut [5] and Carozzi, Karlsson & Bergman [6]) are introduced and 

the Storey and Lefeuvre [23] representation of the electromagnetic field is discussed. 

A general decomposition of second rank tensors, in three dimensions, is 

introduced and applied to a general field. The relativistic meaning of covariance 

and the tensor algebra of relativity is presented, together with the construction 

of the electromagnetic field strength tensor (often referred to as just the field 

tensor). In Chapter 3 we introduce and define the covariant spectral density 

tensor. This tensor is expanded in a basis consisting of the Dirac gamma matrices. 

In Chapter 4 a new set of parameters giving a covariant representation of 

the electromagnetic field with geometrical quantities is presented. The covariance 

here refers to real rotations, SO(3), and the space is the three dimensional 

complex Euclidean space. Chapter 5 contains a construction of relativistically 

covariant tensors in which the new parameters are found as entries. The group 

we use here is the real proper homogeneous Lorentz group, SO(3, 1). In Chapter 

6 the covariant spectral density tensor is decomposed into a reducible tensor 

representation, consisting of irreducible tensors and a reducible rest tensor. All 

tensors in this representation are manifestly covariant and the irreducible tensors 

are identified with known physical quantities. In Chapter 7 the thesis is 

summarized and concluded.

Chapter 2 

Background 

2.1 Covariant formulation of Maxwell equations 

In this Section we introduce some basic concepts from the theory of relativity 

and a covariant formulation of Maxwell equations. In the process we derive 

the field strength tensor, which is a fundamental object in electromagnetism, 

that will be used later in this paper. With the field strength tensor we express 

Maxwell’s equations as covariant 1 tensor equations. We will use Lorentz 

notation with the proper four-vector x µ = (ct, x, y, z). This notation together 

with the Lorentz scalar product, Equation (2.11), is often called Lorentz space 

(Barut [1]). 

2.1.1 Basic relativity and tensor operations 

In relativity one considers the concept of Lorentz invariance. The Lorentz transformations 

are the transformations that keep the expression 

c 2 ∆t 2 − ∆x 2 − ∆y 2 − ∆z 2 , (2.1) 

invariant. The ∆ refers to a difference between two vectors, which gives the 

invariance under translations. These transformations form a group, which is 

usually divided into different parts. 

The transformations satisfying Equation (2.1) include translations and form 

the inhomogeneous Lorentz group, which is also called the Poincaré group (see 

Barut [1] or Tung [24]). The transformations which keep the expression 

c 2 t 2 − x 2 − y 2 − z 2 , (2.2) 

invariant form the inhomogeneous Lorentz group and does not include translations. 

All homogeneous Lorentz transformations, denoted by L, have det L = 

±1, see Barut [1]. Transformations with det L = 1 are the proper Lorentz transformations 

and the transformations with det L = −1 are the improper Lorentz 

transformations, which include reflections. The proper Lorentz transformations 

1 The covariance refers to a specific group, in this case the Lorentz group (see Section 

2.4). Lorentz covariant quantities and equations are often referred to as proper quantities and 

equations. 

3

4 CHAPTER 2. BACKGROUND 

form a subgroup, the improper transformations does not. 2 In the literature, 

the proper homogeneous Lorentz group is often referred to as just the Lorentz 

group. A proper four-vector is a vector that transforms like x µ = (ct, x, y, z), µ = 

0, 1, 2, 3, under Lorentz transformations. 

A space similar to Lorentz space is the Minkowski space. In Minkowski 

space, the time coordinate is imaginary and a four-vector is written as x µ = 

(x, y, z, ict), where µ = 1, 2, 3, 4. The Lorentz transformations are in this space 

orthogonal transformations that preserves the ordinary Cartesian scalarproduct. 

The reason for not using the Minkowski space is that it is very complicated to 

add a curvature to this space. 

Considering Lorentz space, every object is a tensor. A scalar is a rank zero 

tensor, a vector is a tensor of rank one and so on. When dealing with tensors 

there are a number of tools and operations at our disposal. These are called 

permissible tensor operation since the results are new tensor components. The 

operations are (Schutz [21]) 3 : 

1. Linear combinations of tensors of the same type, meaning that the tensors 

have the same number of covariant (subscripts) indices and same number 

of contravariant (superscripts) indices. 

2. Multiplication of components of two tensors (the tensor product 4 ) gives 

a new tensor of type equal to the sum of the types. 

3. Contraction on a pair of indices. (only between covariant (lower) and 

contravariant (upper) indices) 

4. Covariant differentiation, see Dirac [8] or Schutz [21], defined for a covariant 

vector as (using Einstein sum convention 5 ) 

and for a contravariant vector as 

A µ;ν ≡ ∂A µ 

∂x ν − Γα µνA α , (2.3) 

A µ ;ν ≡ ∂Aµ 

∂x ν + Γµ ανA α . (2.4) 

For tensors of higher rank, a Γ term appears for each index, with a minus 

sign for a covariant index and a plus sign for a contravariant index. The 

covariant derivative of a rank two tensor will be 

T µ ν;σ ≡ ∂T µ ν 

∂x σ + Γµ ασT α ν − Γ α νσT µ α . (2.5) 

The Γ’s are called the Christoffel symbols and are defined later in this 

section. 

2 The group theory of the Lorentz groups are further discussed in Section 2.4 

3 For further information on tensors and tensor algebra see e.g. Schutz [21], Arfken & 

Weber [10], Barut [1] [18] or Misner, Thorne & Wheeler [7]. 

4 Also called the outer product or the direct product, but these terms are sometimes used 

for other types of products. 

5 The Einstein sum convention is that in a term you sum over repeated indices. Note that 

in general relativity you can only sum over a covariant index and a contravariant index, i.e., 

contraction.

2.1. COVARIANT FORMULATION OF MAXWELL EQUATIONS 5 

The metric, g µν , is a symmetric tensor, which describes the curvature of space, 

and is used to ’raise’ and ’lower’ indices. This metric is constant under covariant 

differentiation. The metric we use is the flat metric 6 

⎛ 

⎞ 

1 0 0 0 

g µν = g µν = ⎜0 −1 0 0 

⎟ 

⎝0 0 −1 0 ⎠ . (2.6) 

0 0 0 −1 

We are also allowed to use the four dimensional Levi-Civita symbol or the 

totally anti-symmetric tensor 7 (see e.g. Goldstein [11] or Misner, Thorne & 

Wheeler [7]). With the flat metric, g µν , it is defined by 

ɛ αβγδ = −ɛ αβγδ 

ɛ 0123 = 1 

ɛ αβγδ = [αβγδ] = 

{ 

1 αβγδ even permutation of 0,1,2,3, 

−1 αβγδ odd permutation of 0,1,2,3, 

(2.7) 

where [αβγδ] is the permutation symbol. Generalizing the Levi-Civita tensor 

to an arbitrary metric it is defined as 

ɛ αβγδ = √ −g[αβγδ], (2.8) 

where 

g = det g µν . (2.9) 

The contravariant and mixed Levi-Civita tensors are then formed by raising the 

indices. Note that the sign convention of ɛ 0123 varies and some authors define 

it as -1, which just changes the sign of ɛ αβγδ and ɛ αβγδ . 

The differential invariant distance can, in our flat space be written 

ds 2 = g µν dx µ dx ν = c 2 dt 2 − dx 2 − dy 2 − dz 2 = c 2 dτ 2 . (2.10) 

In the last step here we have introduced the proper time, τ, a term used in 

relativity and it is the time experienced by the particle in question and not 

by an observer of the particle. Note that the coordinate time, t, is the time 

experienced by the observer defining the coordinate system and also defining 

the metric. 

The scalar product in Lorentz space is defined according to contraction. The 

procedure is the following: a vector a µ , multiplied with a vector b µ , gives a rank 

two tensor, which via contraction gives a scalar. This is also the covariant trace 

of a tensor, the formula is 

a µ b ν = c µν , g µν c µν = c µ µ = a 0 b 0 − a 1 b 1 − a 2 b 2 − a 3 b 3 . (2.11) 

6 The strong equivalence principle (see Schutz [21]) states that ’any physical law which can 

be expressed in tensor notation in special relativity has exactly the same form in a locally 

inertial frame of curved space-time’, i.e. if we find a tensor equation using the flat metric, it 

will be the same in a curved space time but with another metric. 

7 The totally anti-symmetric tensor will be further discussed in Section 2.4, when the group 

theory of the Lorentz group is considered.


For a tensor of rank two, the trace is defined as contraction of the indices, which 

is a permissible tensor operation. In contrast, the ordinary trace is not a permissible 

operation. The four-gradients, ∂ µ (contravariant) and ∂ µ (covariant), 

are defined as 

∂ µ ≡ 

∂ 

∂x µ = ( ∂ 

∂ct , ∂ 

∂x , ∂ ∂y , ∂ ). (2.12) 

∂z 

An often used notation is ∂ µ A = A, µ (compare with the covariant derivative, 

A ;µ ). 

In covariant differentiation, Equation (2.3), the Γ α µν comes in to assure covariance. 

These Γ’s are called Christoffel symbols and are defined as 

Γ γ βµ = 1 2 gαγ (g αβ,µ + g αµ,β − g µβ,α ). (2.13) 

In flat space, these Christoffel symbols vanish everywhere and from Equation 

(2.3) we have 

A ,µ = A ;µ . (2.14) 

To find a covariant form of an equation one needs to find proper four-vectors 

and (or) proper tensors that, by performing the permissible operations on them, 

leads to the equation in question. This is what we will do with the Maxwell 

equations in the next section. 

2.1.2 Microscopic Maxwell equations 

In vacuum the Maxwell equations in the general differential form do not include 

the ⃗ H and ⃗ D fields. These are the so-called microscopic Maxwell equations (see 

Arfken & Weber [10], Wangsness [25] or Jackson [12]) and have the following 

form 

∇ · ⃗E = ρ ε 0 

, (2.15) 

∇ · ⃗B = 0, (2.16) 

∇ × ⃗ E = − ∂ ∂t ⃗ B, (2.17) 

∇ × ⃗ B = µ 0 

⃗j + 1 c 2 ∂ 

∂t ⃗ E, (2.18) 

where ⃗j is the current density and ρ is the charge density. It is now convenient 

to use Lorentz gauge, which is the condition 

∇ · ⃗A + ɛµ ∂φ 

∂t 

+ µσφ = 0, (2.19) 

where σ is the conductivity, ɛ is the permittivity and µ is the permeability. 

In vacuum (σ = 0, ɛ = ɛ 0 , µ = µ 0 ), using ɛ 0 µ 0 = 1/c 2 , the Lorentz condition 

reduces to 

∇ · ⃗A + 1 ∂φ 

c 2 = 0. (2.20) 

∂t 

Here we have introduced the electric potential, φ, and the magnetic vector potential, 

A. ⃗ The electric and magnetic field can be calculated from the potentials 

as 

⃗B = ∇ × A, ⃗ (2.21)


⃗E = −∇ · φ − 1 ∂A 

⃗ 

c ∂t . (2.22) 

By inserting Equation (2.21) and Equation (2.22) into the inhomogeneous Maxwell 

equations (Eq. (2.15) and Eq. (2.18)) and using the Lorentz condition for vacuum 

(Eq. (2.20)) we get two separated equations for the potentials, 

for the magnetic vector potential, and 

∇ 2 ⃗ A − 

1 

c 2 ∂ 2 ⃗ A 

∂ 2 t = −µ 0 ⃗ j, (2.23) 

∇ 2 φ − 1 c 2 ∂ 2 φ 

∂ 2 t = − ρ ɛ 0 

, (2.24) 

for the electric scalar potential. The differential operator in the above equations 

is known as the d’Alembertian (with a minus sign) and is a fundamental operator 

in electrodynamics and is known to be invariant (see e.g. Wagnsness [25], Arfken 

& Weber [10] or Jackson [12]). Using four-vector notation it it can be written: 

□ 2 = 1 c 2 ∂ 2 

∂ 2 t − ∇2 = ∂ 2 = ∂ µ ∂ µ . (2.25) 

From Equation (2.23) and Equation (2.24) we get the following definitions of 

four-vectors, 

A µ = (φ, c ⃗ A), (2.26) 

J µ = (ρ,⃗j/c), (2.27) 

which are the four-potential and the four-current, respectively. We insert the c 

to get the correct dimension. 8 Then we can write Equations (2.23) and (2.24) 

together as a four-vector differential equation 

∂ 2 A µ = µ 0 cJ µ . (2.28) 

To see that this is a tensor equation, i.e. showing that A µ and J µ are proper 

four-vectors, it suffices to show that J µ is a proper four-vector. This is because 

the d’Alembertian is an invariant scalar (see e.g. Jackson [12] or Barut [1]). 

Then, by the quotient rule (see Schutz [21], Arfken & Weber [10]), A µ is a 

four-vector if J µ is a four-vector. Since an electric charge element de = ρd 3 x is 

invariant we compare this with the four-dimensional volume element d 4 x. We 

see that ρ must transform the same way as dx 0 . Consider J 1 , we have 

J 1 = ρv x = ρ dx1 

dt 

= cρ dx1 

d(ct) = J 0 dx1 

dx 0 . (2.29) 

Since J 0 transforms as dx 0 , the equation above shows that J 1 transforms as dx 1 

and thus J 2 and J 3 transform as dx 2 and dx 3 , respectively. So, Equation (2.28) 

is a proper tensor equation and A µ and J µ are proper four-vectors. However, A µ 

is not a measurable quantity and it also depends on the gauge we have chosen. 

What we would like to have is a covariant expression that involves ⃗ E and ⃗ B 

since these are the quantities we measure and they are gauge invariant, which 

is not the case with the potential. 

8 The potentials can be defined differently and different units can be used, and thus gives 

a different field strength tensor, see e.g. Jackson [12] or Wangsness [25].


2.1.3 Construction of the electromagnetic field strength 

tensor 

The relation between the potentials and the electric and magnetic fields are 

given by in Equations (2.21) and (2.22), so we first rewrite these equations 

using the four-vector potential, A µ . Considering the i:th component of the field 

we have 

⃗B = ∇ × ⃗ A ⇒ cB i = ∂Ak 

∂x j − ∂Aj 

∂x k = ∂j A k − ∂ k A j (i, j, k) = (1, 2, 3), (2.30) 

and 

⃗E = −∇φ − ∂ A ⃗ 

∂t ⇒ E i = ∂A0 

∂x i − ∂Ai 

∂x 0 = ∂i A 0 − ∂ 0 A i (i = 1, 2, 3). (2.31) 

Equations (2.30) and (2.31), are proper tensor equations since ∂ µ and A µ are 

proper four-vectors. Thus we have the equations on the form we want and 

proceed by defining an antisymmetric rank two tensor, F µν , with elements given 

by the equations above (Eqs. (2.30) and (2.31)). The tensor is found to be 

⎛ 

⎞ 

0 −E 1 −E 2 −E 3 

F µν ≡ ∂ µ A ν − ∂ ν A µ = ⎜ E 1 0 −cB 3 cB 2 

⎟ 

⎝ E 2 cB 3 0 −cB 1 

⎠ . (2.32) 

E 3 −cB 2 cB 1 0 

This is the field strength tensor which is not uniquely defined. The form of 

the components depends on the choice of space (Lorentz or Minkowski space) 

and the choice of units. 9 Note that an invariant four-vector multiplied with a 

constant is again an invariant four-vector. Next step is to rewrite the Maxwell 

equations in terms of the field strength tensor, but first we define the dual tensor 

of F µν , in the following way 

⎛ 

⎞ 

0 −cB 1 −cB 2 −cB 3 

F αβ = 1 2 ɛαβγδ F γδ = 1 2 ɛαβγδ g γµ g δν F µν = ⎜ cB 1 0 E 3 −E 2 

⎟ 

⎝ cB 2 −E 3 0 E 1 

⎠ . 

cB 3 E 2 −E 1 0 

(2.33) 

Now using the field strength tensor and its dual we can rewrite the Maxwell 

equations in the covariant form for the flat metric, g µν from Equation (2.6), 

which is valid in special relativity. We have the continuity equation 

∂ µ J µ = ∂J µ 

= 0, (2.34) 

∂x 

µ 

the inhomogeneous Maxwell equations (Eqs. (2.15) and (2.18)) 

∂ µ F µν = µ 0 cJ ν , (2.35) 

and the homogeneous Maxwell equations (Eqs. (2.16) and (2.17)) 

∂ µ F µν = 0. (2.36) 

9 For some other variants of the field strength tensor see Barut [1] or Wangsness [25].


To generalize these equations to an arbitrary metric we need to replace the 

derivative with the covariant derivative (see Eq. (2.3)). We the obtain a manifestly 

covariant form of Maxwell’s equations. The continuity equation 

the inhomogeneous Maxwell equations 

and the homogeneous Maxwell equations 

J µ ;µ = 0, (2.37) 

F µν 

;µ = µ 0 cJ ν , (2.38) 

F µν 

;µ = 0. (2.39) 

According to the strong equivalence principle (SEP), see Schutz [21], we can 

replace the four-gradient, ∂ µ , with the covariant derivative if we have the equations 

as proper tensor equations in special relativity, i.e. using the flat metric 

g µν . In the form above (Eqs. (2.34), (2.35) and (2.36)), Maxwell equations are 

proper tensor equations and thus we can just replace the ordinary derivative 

with the covariant derivative. 

Invariant quadratic quantities from the field strength tensor 

Using the permissible tensor operations on the field strength tensor tensor and 

its dual, it is possible to construct invariant scalars. This is done by contracting 

the tensor product of the field strength tensor with itself or its dual tensor. 

From this we get the invariant scalars 

and 

F µν F ∗ µν = −2c ⃗ E · ⃗B † − 2c ⃗ B · ⃗E † = −4cRe[ ⃗ E · ⃗B † ] (2.40) 

F µν F ∗ µν = 2(c 2 ⃗ B 2 − ⃗ E 2 ). (2.41) 

The first invariant is the real part of the scalar product of ⃗ E and ⃗ B, and the 

second is recognised as the Lagrangian, which corresponds to the free energy of 

an electromagnetic wave in vacuum. 

2.1.4 Electromagnetic energy-momentum tensor 

This is a good time to introduce another proper rank two tensor , the electromagnetic 

energy-momentum tensor 10 . The fields are real valued and we first 

define the energy density E and the Poynting vector ⃗ S to be 

E = 1 2 ( ⃗ E 2 + c 2 ⃗ B 2 ), (2.42) 

⃗S = ⃗ E × c ⃗ B. (2.43) 

In terms of the field strength tensor, the electromagnetic energy-momentum 

tensor can be defined as (Dirac [8]) 

T µν = −F µ α F να + 1 4 gµν F αβ F αβ . (2.44) 

10 Also called the stress-energy tensor or the canonical stress tensor, and is a fundamental 

object in general relativity (see Schutz [21], Dirac [8], Barut [1], Jackson [12] or Misner, Thorne 

& Wheeler [7]).


With the energy density, E, and the Poynting vector, S, the energy-momentum 

tensor can be written as 

⎛ 

⎞ 

E S 1 S 2 S 3 

T µν = ⎜S 1 −E1 2 − c 2 B1 2 + E −E 1 E 2 − c 2 B 1 B 2 −E 1 E 3 − c 2 B 1 B 3 

⎟ 

⎝S 2 −E 2 E 1 − c 2 B 2 B 1 −E2 2 − c 2 B2 2 + E −E 2 E 3 − c 2 B 2 B 3 

⎠ . 

S 3 −E 3 E 1 − c 2 B 3 B 1 E 3 E 1 − c 2 B 3 B 1 −E3 2 − c 2 B3 2 + E 

(2.45) 

The components T mn (m, n = 1, 2, 3) together form the Maxwell stress tensor 

(see Section 2.3 Equation (2.95)).

2.2. GENERALIZED STOKES PARAMETERS 11 

2.2 Generalized Stokes parameters 

In this Section we give the definitions of the two-dimensional Stokes parameters 

and the generalized polarization parameters introduced by T. Carozzi, R. Karlsson 

and J. Bergman [6]. These polarization parameters are a generalization of 

the Stokes parameters into three-dimensions. A similar set of parameters were 

introduced by Brosseau [5]. The Poynting vector and the Maxwell stress tensor 

are introduced and defined. Storey and Lefeuvre’s [23] six dimensional electromagnetic 

quasi-vector and their representation of the quadratic quantities of an 

electromagnetic field are also discussed. 

2.2.1 Spectral density matrix 

Consider an arbitrary vector field, ⃗ f(⃗r, t). The field is a superposition of all 

waves and stationary fields at the point ⃗r. The spectral components of the field 

⃗f(⃗r, t), one obtains by taking the Fourier transform, ⃗ F (⃗r, ω), of the field, defined 

as 

⃗F (⃗r, ω) = 

∫ ∞ 

−∞ 

⃗f(⃗r, t)e iωt dt. (2.46) 

The vector, ⃗ F , represents a complex valued vector field which can be separated 

into three complex amplitudes (the x, y, z components), or into three real 

amplitudes (F 1 , F 2 , F 3 ) and their real phases (δ 1 , δ 2 , δ 3 ), in the following way 

⎛ 

⃗F (⃗r, ω) = ⎝ F ⎞ ⎛ 

⎞ 

x(⃗r, ω) F 1 (⃗r, ω)e δ1(⃗r,ω) 

F y (⃗r, ω) ⎠ = ⎝F 2 (⃗r, ω)e δ2(⃗r,ω) ⎠ . (2.47) 

F z (⃗r, ω) F 3 (⃗r, ω)e δ3(⃗r,ω) 

The spectral density matrix, S d (⃗r, ω), is constructed by taking the tensor product 

between F ⃗ = F ⃗ (⃗r, ω), and its Hermitian conjugate, F ⃗ † = F ⃗ † (⃗r, ω). This 

gives 

⎛ 

⎞ 

S d (⃗r, ω) = F ⃗ ⊗ F ⃗ F x F † x ∗ F x Fy ∗ F x Fz 

∗ 

= ⎝F y Fx ∗ F y Fy ∗ F y Fz 

∗ ⎠ , (2.48) 

F z Fx ∗ F z Fy ∗ F z Fz 

∗ 

where F ∗ denotes complex conjugate. We will assume (⃗r, ω)-dependence and 

the notation S d = S d (⃗r, ω) will be used in the following. 

2.2.2 Instantaneous Stokes’ parameters 

Stokes’ parameters describe the polarization state of a transverse electromagnetic 

wave in the polarization plane. One usually assumes that the wave propagates 

in the z-direction and the polarization ellipse lies in the xy-plane (see 

Brosseau [4]). Stokes parameters can be written (using the notation 11 of R. 

11 The notation for Stokes’ parameters is not universal. The notation of Stokes [22] was 

(A,B,C,D), another notation is (S 0 , S 1 , S 2 , S 3 ) used by e.g. Jackson [12] and Brosseau [4].


Karlsson [13], Born and Wolf [3]) 

I = F 2 x + F 2 y , (2.49) 

Q = F 2 x − F 2 y , (2.50) 

U = 2Re[F y F ∗ x ] = F x F ∗ y + F y F ∗ x , (2.51) 

V = 2Im[F y F ∗ x ] = i ( F x F ∗ y − F y F ∗ x 

) 

, (2.52) 

where I denotes the intensity of the field, Q and U describe the linear polarization 

and V is related to circular polarization. 12 In the two-dimensional case the 

spectral density tensor, Equation (2.48), is reduced to 

( ) 

Fx Fx S d = ∗ F x Fy 

∗ 

F y Fx 

∗ F y Fy 

∗ . (2.53) 

The two-dimensional spectral density matrix can be decomposed using the unit 

matrix, I 2 , and the generators of the special unitary symmetry group, SU(2), 

which are recognized as the Pauli spin matrices, σ i , i = 1, 2, 3. These can be 

written 

σ 1 = 

( 0 1 

1 0 

) ( 0 −i 

, σ 2 = 

i 0 

) ( 1 0 

, σ 3 = 

0 −1 

) ( 1 0 

, I 2 = 

0 1 

(2.54) 

Writing the spectral density matrix, S d , as a linear combination of I 2 and the 

Pauli spin matrices, Equation (2.54), one identifies the coefficients as the Stokes 

parameters, I, Q, U and V. The spectral density matrix, S d , can then be written 

in the following form 

S d = 1 ) 

(II 2 + Uσ 1 + Vσ 2 + Qσ 3 = 1 ( ) 

I + Q U − iV 

. (2.55) 

2 

2 U + iV I − Q 

This is a convenient way to express the spectral density matrix, S d , since the 

Stokes parameters are just scalars and the Pauli spin matrices are unitary with 

determinant equal to ±1. This procedure can be generalized into three dimensions, 

and leads to a similar relation. 

For a deterministic wave we have 

) 

. 

I 2 = U 2 + Q 2 + V 2 , (2.56) 

which means that the wave is totally polarized and hence one of the parameters 

is abundant for a complete description of the state of polarization. In reality 

we can not have a deterministic wave and must therefore integrate over a small 

interval (bandwidth). The relation above then becomes 

I 2 ≥ U 2 + Q 2 + V 2 . (2.57) 

This means that for a deterministic (or a fully polarized wave) wave we do 

not need all the parameters to describe the polarization of the wave, but for a 

partially polarized wave all four parameters are needed. 

12 For a longer discussion on Stokes parameters see Brosseau [4] or Born & Wolf [3].


2.2.3 Three-dimensional polarization parameters 

We want to obtain a more general form of Equation (2.53) applicable to an 

arbitrary wave in three dimensions (to use Stokes parameters it is necessary to 

know the direction of propagation). To achieve this we expand the full spectral 

density matrix, S d , using the unit matrix, I 3 , and the generators of the SU(3) 

symmetry group, λ i , (i = 1, . . . , 8), as given by Gell-Mann and Ne’eman [17] 

(see Appendix A Equation (A.2)). The spectral density matrix is then formed 

as a linear combination of these matrices in the following way 

S d = 1 3 Λ 0I 3 + 1 8∑ 

Λ i λ i = 

2 

i=1 

⎛ 

1 

3 

⎜ 

Λ 0 + 1 2 Λ 3 + 1 

1 

⎝ 2 Λ 1 + i 2 Λ 2 

1 

2 Λ 4 + i 2 Λ 5 

2 √ 3 Λ 8 

1 

2 Λ 1 − i 2 Λ 2 

1 

3 Λ 0 − 1 2 Λ 3 + 1 

1 

2 Λ 6 + i 2 Λ 7 

2 √ 3 Λ 8 

1 

2 Λ 4 − i 2 Λ ⎞ 

5 

1 

2 Λ 6 − i 2 Λ ⎟ 

7 ⎠ , (2.58) 

1 

3 Λ 0 − √ 1 

3 

Λ 8 

where Λ i , i = 0, 1, . . . 8, are the generalized Stokes parameters. The trace of 

this matrix, S d , is Λ 0 which is the energy density of the field. For the two 

dimensional case this is the Stokes parameter I. The Λ i are found by comparing 

Equation (2.48) with Equation (2.58), the explicit expressions for the Λ i are 

found in Appendix A. In the case of a transverse wave propagating in the z- 

direction, i.e. F z = 0, the spectral density matrix, Equation (2.58), reduces to 

two-dimensional form 

⎛ 

⎞ 

S d = 1 Λ 0 + Λ 3 Λ 1 − iΛ 2 0 

⎝Λ 1 + iΛ 2 Λ 0 − Λ 3 0⎠ , (2.59) 

2 

0 0 0 

where the Λ i have reduced to the Stokes parameters as in Equation (2.53). This 

is just a rotation of the coordinate system into the one of the wave with the 

z-axis being the direction of propagation. 

There are different normalizations of the generalized polarization parameters. 

Brosseau [5] uses 1 3 instead of 1 2 

, which we have used, in front of the SU(3) 

generators in Equation (2.58). If one uses 1 2 

, Equation (2.59) (transverse wave 

propagating in the z-direction) reduces to Stokes parameters, Equation (2.55), 

which is not the case if one uses 1 3 . 

2.2.4 Interpretation of the 3D polarization parameters 

Every tensor can be decomposed into its symmetric and anti-symmetric parts, 

Arfken & Weber [10]. Thus, Equation (2.58) can be written 

S d = S S d + S A d = 1 2( 

Sd + S T d 

) 1( + Sd − S T ) 

d . (2.60) 

2 

The superscript T denotes a transposed tensor. From Equation (2.58) it is easily 

seen that the symmetric part, Sd S, and anti-symmetric part, SA d 

, are given by 

⎛ 

1 

3 

Sd S ⎜ 

Λ 0 + 1 2 Λ 3 + 1 

2 √ Λ 1 

3 

8 

2 Λ 1 

1 

2 Λ ⎞ 

4 

1 

= ⎝ 

2 Λ 1 

1 

3 Λ 0 − 1 2 Λ 3 + 1 

2 √ Λ 1 

3 

8 2 Λ ⎟ 

6 ⎠ (2.61) 

1 

2 Λ 1 

4 

2 Λ 1 

6 

3 Λ 0 − √ 1 

3 

Λ 8


and 

⎛ 

0 − i 

Sd A 2 Λ 2 − i 2 Λ ⎞ 

5 

= ⎝ i 

2 Λ 2 0 − i 2 Λ ⎠ 

7 , (2.62) 

i 

2 Λ i 

5 2 Λ 7 0 

respectively. To any anti-symmetric rank two tensor, in three dimensions 13 a 

dual pseudo-vector is associated (see Arfken & Weber [10]). This dual pseudovector 

is defined by 

V i = 1 2 ɛ ijkT jk , (2.63) 

where ɛ ijk is the totally anti-symmetric tensor, also known as the Levi-Civita 

tensor. 14 The dual pseudo vector to Sd 

A is then 

⃗V ′ = i(−Λ 7 , Λ 5 , −Λ 2 ). (2.64) 

The totally anti-symmetric tensor, ɛ ijk , contracted with a vector of ones, (1,1,1) 

becomes 

⎛ 

0 1 

⎞ 

−1 

ɛ ijk (1, 1, 1) k = ⎝−1 0 1 ⎠ . (2.65) 

1 −1 0 

We see that it has a different sign convention compared to λ 7 and λ 2 and 

therefore we get the minus sign on them (see Appendix A). This vector ⃗ V ′ is 

an imaginary vector and for later use we would like to have a real vector so 

following Carozzi, Karlsson and Bergman [6] we therefore define a vector ⃗ V by 

which is real. The components of ⃗ V are 

⃗V ≡ (Λ 7 , −Λ 5 , Λ 2 ). (2.66) 

Λ 7 = i(F y F ∗ z − F z F ∗ y ) = 2Im(F z F ∗ y ) = −2Im(F y F ∗ z ), 

Λ 5 = i(F x F ∗ z − F z F ∗ x ) = 2Im(F z F ∗ x ) = −2Im(F x F ∗ z ), 

Λ 2 = i(F x F ∗ y − F y F ∗ x ) = 2Im(F y F ∗ x ) = −2Im(F x F ∗ y ). 

(2.67a) 

(2.67b) 

(2.67c) 

Recalling that ⃗ V is a pseudo-vector and that a crossproduct between two polar 

(ordinary) vectors or two axial (pseudo) vectors is a pseudo-vector, we can from 

the form of the components conclude that ⃗ V can be defined by 

⃗V = i ⃗ F × ⃗ F † . (2.68) 

The complex vector F ⃗ and its complex conjugate F ⃗ † form a polarization plane 

in space, see e.g. Lindell [14] or Karlsson [13], and because V ⃗ = iF ⃗ × F ⃗ † it is 

perpendicular to both F ⃗ and F ⃗ † and thus parallel or anti-parallel to the normal 

of the polarization plane (see Karlsson [13] or Lindell [14]). Λ 2 , Λ 5 and Λ 7 are 

the three-dimensional generalization of circular polarization. Comparing the 

13 In four dimensions a dual second rank tensor is associated to an anti-symmetric second 

rank tensor (see Sections 2.1 and 2.4). 

14 The sign of the metric in the three-dimensional Euclidean space (the ordinary threedimensional 

space) is (1, 1, 1). This gives that the totally anti-symmetric tensor of this space 

is equal to the permutation symbol and contraction between indices of the same kind is allowed 

(compare with Sections 2.1 and 2.4)


expressions for the Λ i ’s, Equation (2.67), with Stokes’ parameter for circular 

polarization, V, we have the relation (see Carozzi, Karlsson and Bergman [6]) 

|V| = 

√ 

Λ 2 7 + Λ2 5 + Λ2 2 = |⃗ V |. (2.69) 

We define left- and right-handed polarization as seen by the wave (opposite to 

an observer looking into the wave). For a right-handed polarized wave, V ⃗ will 

be parallel to the direction of propagation and anti-parallel for a left-handed 

polarized wave. 15 Because measurements is generally made in one point, i.e. 

we have the field components in one point, we really have two possibilities with 

opposite signs (opposite direction of propagation). To get a correct sign on V ⃗ 

one needs further information about the direction of propagation of the wave. 

2.2.5 The Storey and Lefeuvre six-quasivector 

In order to obtain a straightforward description of the electromagnetic field with 

all possible combinations of the electromagnetic field components, Storey and 

Lefeuvre introduced a six dimensional quasivector, for the field components, 

defined as 

⃗E = (E 1 , E 2 , E 3 , cB 1 , cB 2 , cB 3 ). (2.70) 

For a continuum of plane waves Storey and Lefeuvre introduces a generalized 

electric field, e m i (⃗r, t), where the i-th component is the real part of the evolution 

e m i (⃗r, t) = Re[e i exp i(ωt − ⃗ k · ⃗r)]. (2.71) 

Here e i is the complex amplitude. The 6 × 6 matrix they define is constructed 

as 

a ij = e ie ∗ j 

δ . (2.72) 

These are constant since the complex amplitude is constant for the plane wave 

in Equation (2.71). The six-vector, Equation (2.70), is not a vector, in the 

geometrical meaning, more than in the sense that it has six entries. The first 

three components, the electric part, together form the E ⃗ vector (polar vector) 

and the last three form the B ⃗ vector which is a pseudo-vector (axial vector). So, 

the six-vector defined here has three components transforming as a polar vector 

and three components transforming as a pseudo-vector. 16 Anyhow, using this 

six-vector we obtain a spectral density matrix 

⎛ 

⎞ 

E1 2 E 1 E2 ∗ E 1 E3 ∗ E 1 B1 ∗ E 1 B2 ∗ E 1 B3 

∗ E 2 E1 ∗ E2 2 E 2 E3 ∗ E 2 B1 ∗ E 2 B2 ∗ E 2 B ∗ 3 

S d = 

E 3 E1 ∗ E 3 E2 ∗ E3 2 E 3 B1 ∗ E 3 B2 ∗ E 3 B3 

∗ ⎜B 1 E1 ∗ B 1 E2 ∗ B 1 E3 ∗ B1 2 B 1 B2 ∗ B 1 B3 

∗ . (2.73) 

⎟ 

⎝B 2 E1 ∗ B 2 E2 ∗ B 2 E3 ∗ B 2 B1 ∗ B2 2 B 2 B3 

∗ ⎠ 

B 3 E1 ∗ B 3 E2 ∗ B 3 E3 ∗ B 3 B1 ∗ B 3 B2 ∗ B3 

2 

As seen from its structure, this matrix is symmetric in the real part and antisymmetric 

in the imaginary part. It has 36 different entries of which 21 are 

15 For more information on the ⃗ V and complex vector analysis see Lindell [14] [15]. 

16 More information on polar and axial vectors are found in Arfken & Weber [10]


real and 15 imaginary. Since the six-vector, ⃗ E, is not a geometrical quantity 

and do not have the convenient transformation properties of a rank one tensor, 

this matrix is no more than a table of the 36 correlation components. Even so, 

these 36 correlation components are those components we want to combine into 

geometrical quantities to get a more physical representation of polarization.

2.3. IRREDUCIBLE REPRESENTATION 17 

2.3 Irreducible representation in terms of geometrical 

objects 

A quite natural way of representing a complex vector field in quadratic components 

is to take the tensor product of the field and its hermitian conjugate. The 

tensor product between any two three-dimensional vectors is defined as 

⎛ 

⃗A ⊗ B ⃗ † = ⎝ A ⎞ 

1B1 ∗ A 1 B2 ∗ A 1 B3 

∗ 

A 2 B1 ∗ A 2 B2 ∗ A 2 B3 

∗ ⎠ , (2.74) 

A 3 B1 ∗ A 3 B2 ∗ A 3 B3 

∗ 

where † denotes Hermitian conjugate and ∗ denotes complex conjugate. This 

tensor product is the fundamental object in this chapter and we will decompose 

it into different geometrical objects. Since it is constructed from the tensor 

product of two vectors it is also clear that it is a tensor. This matrix is often 

called the correlation matrix. The symmetry used here is that of the rotation 

group, SO(3) (see Arfken & Weber [10], Tung [24] or Barut & Raczka [18]), since 

we are in the complex Euclidean three dimensional space and we are restricting 

the rotations to real rotations. 

2.3.1 Complex vector algebra 

In the spectral domain, the Fourier transform of the electromagnetic field, we 

have a complex vector field. For a good introduction to complex algebra, the 

paper[14] and book [15] by Lindell is recommended. The complex vector algebra, 

with notations, used in this work is presented here. 

The Hermitian conjugate of a scalar is the same as the complex conjugate, 

and for a vector 

⃗F = ⃗a + i ⃗ b ⇒ ⃗ F † = ⃗a − i ⃗ b, (2.75) 

it is also the same. The main rule is that in in a multiplication of components, 

we have the conjugation on the second factor. This gives the scalar product of 

⃗A and ⃗ B, to be 

scalar( ⃗ A, ⃗ B) = ⃗ A · ⃗B † , (2.76) 

and the cross product 

cross( ⃗ A, ⃗ B) = ⃗ A × ⃗ B † . (2.77) 

For a matrix or tensor, the Hermitian conjugate is conjugation and transponation, 

A † = (A T ) ∗ . In this work we use the products defined in a real space and 

write out the conjugate, ⃗ A · ⃗B † . 

Following Lindell [14], we have some useful relations for complex vectors. 

For determining if a vector is zero, we have 

⃗A · ⃗A † = 0 ⇒ ⃗ A = 0, (2.78) 

⃗A × ⃗ B = 0, ⃗ A · ⃗ B † = 0 ⇒ ⃗ A = 0 or ⃗ B = 0. (2.79) 

If the cross product of two vectors is zero, we have the relation 

⃗A × ⃗ B = 0, ⃗ A ≠ 0 ⇒ ∃α ∈ C, ⃗ B = α ⃗ A, (2.80)


and if the scalar product is zero, we have 

⃗A · ⃗B = 0, A ≠ 0 ⇒ ∃ ⃗ C, ⃗ B = ⃗ C × ⃗ A. (2.81) 

With complex field ⃗ A, the condition for linear polarization is 

⃗A × ⃗ A † = 0, (2.82) 

having ⃗ A = ⃗a+i ⃗ b, this becomes ⃗a× ⃗ b = 0. The condition for circular polarization 

is 

⃗A · ⃗A = 0, (2.83) 

which means that |Im[A]| = |Re[A]|, i.e. the complex and real part are of equal 

size. This section gives the main features of complex vector algebra, which 

we are using in this work. Complex vector algebra is used extensively when 

considering electromagnetic waves, but is generally not thoroughly explained in 

the literature. 

2.3.2 The electromagnetic field 

For the complex electric or the magnetic field separately ( F ⃗ = F ⃗ (⃗r, ω) is either 

⃗E or B) ⃗ the tensor product is usually called the spectral density matrix. We 

have 

⎛ 

⎞ 

⃗F ⊗ F ⃗ F 1 F † 1 ∗ F 1 F2 ∗ F 1 F3 

∗ 

= ⎝F 2 F1 ∗ F 2 F2 ∗ F 2 F3 

∗ ⎠ . (2.84) 

F 3 F1 ∗ F 3 F2 ∗ F 3 F3 

∗ 

This matrix contains 9 parameters which is the number of degrees of freedom 

needed to completely describe the wave field. Since ⃗ F is a complex vector we 

can separate its real and imaginary parts. With this in mind one can easily 

show that Equation (2.84) is symmetric in the real part and anti-symmetric in 

the imaginary part. We have that the spectral density matrix consists of six real 

(the components on the diagonal are real) and three imaginary components, a 

total of 9 different degrees of freedom. 

This way of describing the wave is not very convenient since the parameters 

are in general not very physical and are not invariant under coordinate transformations. 

A better way is to write the spectral density matrix as a sum of 

geometric quantities. Recall from the previous chapter, with ⃗ F = ⃗a + i ⃗ b, the 

quantities 

I F = ⃗ F · ⃗F † = a 2 + b 2 = | ⃗ F | 2 (2.85) 

and 

⃗V = i ⃗ F × ⃗ F † = i(⃗a + i ⃗ b) × (⃗a − i ⃗ b) = 2⃗a × ⃗ b. (2.86) 

I F is the energy density of the field in question. This V ⃗ is parallel (or antiparallel 

depending on whether the wave is left or rigthand polarized) to the 

direction of propagation of the wave. Note that E ⃗ is a polar vector and B ⃗ is an 

axial vector, but E ⃗ × E ⃗ † and B ⃗ × B ⃗ † are both axial vectors, since B ⃗ × B ⃗ † is a 

product of two axial vectors. 

The trace of the spectral density matrix is equal to the energy density of the 

field, which is an invariant scalar. From Equation (2.63) we know that we can


associate a vector to the anti-symmetric part of the matrix. This means that 

we can write the spectral density matrix in the following way 

⃗F ⊗ ⃗ F † = 1 3 Iδ ij + iɛ ijk v k + Q ij , (2.87) 

where I is a real scalar, ⃗v is the real dual pseudo-vector (associated to the antisymmetric 

imaginary part, see Eq. (2.63) ) and Q is a real symmetric tensor (the 

anti-symmetric part is contained in ɛ ijk v k ). This way of representing a tenor is 

termed an irreducible tensor representation (see e.g. Sakurai [20] and Arfken 

& Weber [10]). Since I and Q are real valued we know that ⃗v must contain the 

imaginary part. Hence, 

⎛ 

⎞ 

0 Im[F 1 F2 ∗ ] Im[F 1 F3 ∗ ] 

ɛ ijk v k = ⎝Im[F 2 F1 ∗ ] 0 Im[F 2 F3 ∗ ] ⎠ , (2.88) 

Im[F 3 F1 ∗ ] Im[F 3 F2 ∗ ] 0 

which gives 

⃗v = ( Im[F 2 F ∗ 3 ], Im[F 3 F ∗ 1 ], Im[F 1 F ∗ 2 ] ) . (2.89) 

To get ⃗v in terms of ⃗ F , we look at one component of ⃗ V , say V 3 (see Eq. (2.66)) 

V 3 = i(F 1 F ∗ 2 − F 2 F ∗ 1 ) = i2i(b 1 a 2 − a 1 b 2 ) = −2Im(F 1 F ∗ 2 ) = 2Im(F 2 F ∗ 1 ). (2.90) 

And similarly for the other two components. We want ⃗v to be the imaginary 

part and thus we get 

⃗V = 2 ( Im(F 3 F ∗ 2 ), Im(F 1 F ∗ 3 ), Im(F 2 F ∗ 1 ) ) = −2⃗v. (2.91) 

Rewriting ⃗v in terms of ⃗ F we have the following expression for ⃗v 

⃗v = i [ 

(F3 F2 ∗ − F 2 F3 ∗ ), (F 1 F3 ∗ − F 3 F1 ∗ ), (F 2 F1 ∗ − F 1 F2 ∗ ) ] 

2 

= ( − Im(F 3 F ∗ 2 ), −Im(F 1 F ∗ 3 ), −Im(F 2 F ∗ 1 ) ) = − i 2 ⃗ F × ⃗ F † . (2.92) 

Comparing ⃗v with V ⃗ , we have that ⃗v defines left-handed and right-handed circular 

polarization as seen by an observer receiving the wave (standing in the 

direction of propagation), the size of this vector, ⃗v is 1 π 

times the area of the polarization 

ellipse (see Lindell [14]). The relation between ⃗v and Stokes parameter 

V is (compare with Equation (2.69)) 

√ 

|V| = 2 v1 2 + v2 2 + v2 3 = |2⃗v|. (2.93) 

By subtraction we get Q from Equation (2.87), and we have 

⎛ 

F1 2 − 1 3 I 1 

2 (F 1F2 ∗ + F 2 F ∗ 1 

1 ) 

2 (F ⎞ 

3F1 ∗ + F 1 F3 ∗ ) 

Q = ⎝ 1 

2 (F 1F2 ∗ + F 2 F1 ∗ ) F2 2 − 1 3 I 1 

2 (F 3F2 ∗ + F 2 F3 ∗ ) ⎠ . (2.94) 

1 

2 (F 3F1 ∗ + F 1 F3 ∗ 1 

) 

2 (F 3F2 ∗ + F 2 F3 ∗ ) F3 2 − 1 3 I 

We see that ⃗v contains the imaginary part of the matrix ⃗ F ⊗ ⃗ F † and that Q is 

real valued and symmetric.


Maxwell stress tensor 

A tensor similar to Q is the Maxwell stress tensor (see Wangsness [25], Jackson 

[12] and Marion & Heald [16]), for real fields defined as 

T ij = ɛ 0 

( 

Ei E j − 1 2 ⃗ E 2 δ ij 

) 

+ 

1 

µ 0 

( 

Bi B j − 1 2 ⃗ B 2 δ ij 

) 

= 

⎛ 

E1 2 − 1 2E ⃗ 2 ⎞ 

E 1 E 2 E 1 E 3 

ɛ 0 

⎝ E 2 E 1 E2 2 − 1 ⃗ 2E 2 E 2 E 3 

⎠ + 

E 3 E 1 E 3 E 2 E3 2 − 1 ⃗ 2E 2 

The trace of this symmetric tensor is 

⎛ 

B 

1 

1 2 − 1 2B ⃗ 2 ⎞ 

B 1 B 2 B 1 B 3 

⎝ B 

µ 2 B 1 B2 2 − 1 ⃗ 2B 2 B 2 B 3 

⎠ . (2.95) 

0 

B 3 B 1 B 3 B 2 B3 2 − 1 ⃗ 2B 2 

T r(T ij ) = − ɛ 0 

2 ⃗ E 2 − 1 

2µ 0 

⃗ B 2 , (2.96) 

which is minus the energy density of the electromagnetic field. 

2.3.3 The product ⃗ E ⊗ ⃗ B 

We now form the tensor product of E ⃗ and B ⃗ † , which is 

⎛ 

⃗E ⊗ B ⃗ † = ⎝ E ⎞ 

1B1 ∗ E 1 B2 ∗ E 1 B3 

∗ 

E 2 B1 ∗ E 2 B2 ∗ E 2 B3 

∗ ⎠ . (2.97) 

E 3 B1 ∗ E 3 B2 ∗ E 3 B3 

∗ 

This matrix has a total of 18 components and is a tensor since it is constructed 

from two vectors. Since it is a tensor and we can write any tensor as a sum of 

its symmetric and anti-symmetric parts we follow the same procedure as we did 

with the field ⃗ F in the previous section, Equation (2.87), and we get 

I EB = E 1 B ∗ 1 + E 2 B ∗ 2 + E 3 B ∗ 3, 

(2.98a) 

⃗v EB = i 2[ 

(E3 B ∗ 2 − E 2 B ∗ 3), (E 1 B ∗ 3 − E 3 B ∗ 1), (E 2 B ∗ 1 − E 1 B ∗ 2) ] , (2.98b) 

⎛ 

E 1 B1 ∗ − 1 3 I EB 

Q EB = ⎝ 1 

2 (E 1B2 ∗ + E 2 B1) ∗ E 2 B2 ∗ − 1 3 I EB 

1 

2 (E 1B2 ∗ + E 2 B1) ∗ 1 

2 (E ⎞ 

3B1 ∗ + E 1 B3) 

∗ 

1 

2 (E 3B2 ∗ + E 2 B3) 

∗ ⎠ . (2.98c) 

1 

2 (E 3B1 ∗ + E 1 B3) ∗ 1 

2 (E 3B2 ∗ + E 2 B3) ∗ E 3 B3 ∗ − 1 3 I EB 

When considering reflections, these quantities are different from the decomposition 

of the tensor product of the fields with themselves (see previous Section, 

decomposition of Equation (2.84)). In Equation (2.98), we have that I EB is a 

pseudo-scalar (odd under spatial inversion) and Q EB is a pseudo-tensor (odd 

under spatial inversion). The vector, ⃗v EB , is a polar vector which is odd under 

spatial inversion. 17 Spatial inversion, or reflection, is an example of an improper 

17 For definitions and transformation properties of pseudo-quantities, see Goldstein [11], 

Arfken & Weber [10] or Jackson [12].


rotation, which changes the coordinate system from right-handed to left-handed 

or vice versa. 

All the components in the equations above (Eq. (2.98)) are complex so we 

separate the real and imaginary parts according to 

⃗E ⊗ ⃗ B † = Re[ ⃗ E ⊗ ⃗ B † ] + iIm[ ⃗ E ⊗ ⃗ B † ]. (2.99) 

These two matrices, the real and imaginary parts, are two real matrices and 

can be divided in the same way just taking real and imaginary parts of the 

EB-components separately, and we have 

This is a separation we will use later on. 

The Poynting vector 

Re[ ⃗ E ⊗ ⃗ B † ] = 1 2( ⃗E ⊗ ⃗ B † + ⃗ B ⊗ ⃗ E †) , (2.100) 

iIm[ ⃗ E ⊗ ⃗ B † ] = 1 2( ⃗E ⊗ ⃗ B † − ⃗ B ⊗ ⃗ E †) . (2.101) 

The vector ⃗v EB (Eq. (2.98b)) is very similar to the Poynting vector. For real 

valued vector fields, the Poynting vector is defined as 

⃗S = ⃗ E × ⃗ H, (2.102) 

where H ⃗ is the magnetic H-field (see Wagnsness [25] or Jackson [12]). Compared 

to the vector ⃗v EB , the Poynting vector is recognised as a polar vector. The 

Poynting vector, S, is the instantaneous energy flow and points in the direction 

of energy flow. In the complex case, using our notation with E ⃗ and B, ⃗ it becomes 

( 

) 

⃗S = cE ⃗ × B ⃗ † = c E 2 B3 ∗ − E 3 B2, ∗ E 3 B1 ∗ − E 1 B3, ∗ E 1 B2 ∗ − E 2 B1 

∗ . (2.103) 

The real part of Equation (2.103) is the instantaneous energy flow of the field 

and is varying with time. The time average of the energy flow, 〈 ⃗ S〉, is often 

more useful. Using the complex notation for a plane wave where ⃗ E and ⃗ B are 

the complex vector amplitudes this time average is very easy to calculate, we 

get 

〈 ⃗ S〉 = 1 2 cRe[ ⃗ E × ⃗ B † ]. (2.104) 

Comparing Equa- 

Which is just half of the real part of Equation (2.103). 

tion (2.103) with Equation (2.98b) we see that 

⃗S = 2i⃗v EB . (2.105) 

Thus, we have a simple relation between the complex Poynting vector and the 

⃗v EB vector.


2.4 Group theory of transformation matrices 

In this section we give an introduction to the group theory of the Lie groups 

used in this thesis. The definitions of the groups are given together with physical 

interpretations. We also introduce the mathematical concept of irreducible 

tensor representation and apply it to the groups SO(2), SO(3) and SO(3, 1). 

2.4.1 The general linear group and subgroups 

In this thesis there are a number of different symmetry groups consisting of operations, 

such as rotation or reflection, on a real or complex vector space. These 

groups can be thought of as matrices (see Fieseler [9] or Arfken & Weber [10]), 

i.e. transformation matrices. These groups are continuous groups or Lie groups. 

For more information on these groups, see Barut & Raczka [18] or Tung [24]. 

The first group, the general linear group, is the group of all invertible matrices, 

and can be expressed as 

GL n (K) = {A ∈ K n,n ; det A ≠ 0}, (2.106) 

where K, in our case, is either the real numbers, R, or the complex numbers, 

C. 

A subgroup to GL n (K) is the the Special linear group, and is defined as 

SL n (K) = {A ∈ GL n (K); det A = 1} = {A ∈ K n,n ; det A = 1}. (2.107) 

This group leaves any totally anti-symmetric tensor of rank n invariant. 

The orthogonal group is the subgroup of GL n (K) given by 

O(n) = {A ∈ GL n (R); A⃗x · A⃗y = ⃗x · ⃗y ∀⃗x, ⃗y ∈ R n } 

= {A ∈ GL n (R); A T A = I n }. (2.108) 

This is the set of all linear transformations that leaves the Euclidean metric in n 

dimensions, δ µν (n) , invariant. This notation can be extended to any metric tensor 

with signature (n + , n − ), and the group is O(n + , n − ). For the flat metric, g µν 

(Eq. (2.6)), with signature (1, −1, −1, −1), we have the group O(1, 3), which is 

called the orthogonal group of signature (1, −1, −1, −1). 

The special orthogonal group, SO(n), or SO(n + , n − ), is the subgroup of 

O(n) which admits the totally anti-symmetric tensor, ɛ [n] (the [n] on ɛ [n] is the 

number of suffixes, the rank, of the tensor), as an additional invariant tensor. 

This group is the intersection 

SO(n) = SL n (C) ∩ O(n), (2.109) 

of the special linear group and the orthogonal group. An important difference 

between the O(n) and SO(n) groups is that the O(n) groups include improper 

rotations (reflection, inversion), which is not included in the SO(n) groups because 

of the detA = 1 condition. This absence of reflections gives the SO(n) 

groups the extra invariant tensor ɛ [n] , the totally anti-symmetric tensor. 

Similar to the orthogonal group, O(n), which is real, we have in a complex 

vector space the unitary group, defined as 

U(n) = {A ∈ GL n (C); A⃗x · (A⃗y) † = ⃗x · ⃗y † ∀⃗x, ⃗y ∈ C n } 

= {A ∈ GL n (C); A † A = AA † = I n }. (2.110)

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


using the metric tensor, g µν , and the totally anti-symmetric tensor, ɛ [n] . In 

the rotation group, SO(3), the irreducible tensor representation of a general 

second rank tensor is just the decomposition done in Section 2.3, with the metric 

g ab = δ ab , the Kronecker delta. The decomposition is given by 

where 

and 

A ij = 1 3 Θδ ij + 1 2 ɛ ijkω k + σ ij , (2.112) 

θ = A i i, (2.113) 

ω i = ɛ ijk A jk (2.114) 

σ ij = 1 2 (A ij + A ji ) − 1 3 g ijA k k (2.115) 

are the trace, twice the dual vector and the symmetric part of A ij , respectively. 

Here we considered SO(3), but this decomposition is the same for SU(3) in the 

complex vector space, which we have done in Section 2.3 (where we use a vector 

algebra approach), but under SU(3) this representation is not irreducible (see 

Section 2.4.2). The constant, 1 2 , in the anti-symmetric term, ωk , is dependent 

on how one defines the dual vector (compare with Equation (2.87)). 

Considering the groups SO(3, 1), real vector space, and SU(3, 1), complex 

vector space, the representations are the equivalent (compare with SU(3) ↔ 

SO(3) above) but SO(3, 1) is real and SU(3, 1) is complex. In the four-dimensional 

groups, SO(3, 1) and SU(3, 1), the invariant tensors are a rank two tensor, the 

metric tensor, and a rank four tensor, the totally anti-symmetric tensor. For the 

flat metric, g µν , with signature (1, −1, −1, −1), and the definition of complex 

scalar product, ⃗a · ⃗b † , SO(3, 1) and SU(3, 1) are the proper Lorents groups for 

real and complex vector spaces, respectively. The decomposition for a general 

rank two tensor, A µν , is given by 

A µν = 1 4 Sg µν + 1 2 ɛ µναβT (a)αβ + T µν , (2.116) 

where S is a scalar, recognised as the trace of A. T (a) is a dual tensor and 

contains the anti-symmetric part of A, ɛ is antisymmetric. T is a tensor and 

is the symmetric part of A. Comparing with the three-dimensional case, Equation 

(2.112), S, T (a) and T are given by 

and 

S = A µν g µν , (2.117) 

T (a) 

µν = ɛ µναβ A αβ (2.118) 

T µν = 1 2 (A µν + A νµ ) − 1 4 g µνA α α. (2.119) 

Comparing with Equation (2.112), there are no vectors, i.e. rank one tensors, 

in Equation (2.116). This is due to the fact that we have an even number of 

dimensions and the invariant tensors has even rank. This gives that a traceless

2.4. GROUP THEORY OF TRANSFORMATION MATRICES 25 

rank two tensor is irreducible under SO(3, 1) but can be split into its symmetric 

and anti-symmetric parts, T and T (a) in Equation (2.116) respectively. Since 

contraction is over two indices we do not get any odd rank tensors. Note, this 

does not mean that proper four-vectors are forbidden but we can not construct 

them directly using the invariant tensors of the group. To achieve the odd rank 

tensors we have to consider the pertinent physical equations. 

Before moving on, there are some remarks concerning index notation of 

tensors under different groups. In the group SO(3) we have a diagonal metric 

with signature (1, 1, 1), i.e. the metric is the Kronecker delta. With this metric 

there is no difference between a covariant (lower) and contravariant (upper) 

index. I.e. the relation A ij... = A ij... is valid for every tensor and the relation 

holds for randomly mixed indices. In contrast to SO(3) we have SO(3, 1) with a 

metric signature (1, −1, −1, −1). For a general second rank tensor we have that 

A µν ≠ A µν and A µ ν ≠ Aµ ν , where the last relation is the difference between 

raising (or lowering) the first or the second index. With mixed indices the 

metric tensor is just the Kronecker delta, which can be convenient to use in 

certain calculations. In decomposition and construction of lower rank tensors it 

is favourable to use non-mixed tensors since it allows direct permutation of the 

indices. 

This decomposition can be generalized for tensors of higher rank and will 

include pseudo-scalars and higher order pseudo-tensors, but to obtain an irreducible 

form we must form all possible invariant tensors, using products of the 

invariant tensors, g µν and ɛ αβγδ , and lower rank tensors. The number of terms 

will increase dramatically with higher rank and it is very difficult to determine 

what each tensor has as its components. To make it simple we can do a reducible 

representation with fewer terms, where it is much easier to identify each tensor. 

Considering a general rank four tensor, A αβγδ , we can get it in a representation 

similar to a tensor product of a rank two representation, Equation (2.116). This 

gives the following decomposition 

A αβγδ = Sg αβ g γδ + (1) T µν g αβ ɛ γδµν + (2) T γδ g αβ 

+ (3) T µν ɛ αβµν g γδ + (4) T µν (5) T ψχ ɛ αβµν ɛ γδψχ 

+ (6) T µν (7) T γδ ɛ αβµν + (8) T αβ g γδ + (9) T αβ (10) T µν ɛ γδµν + U αβγδ , (2.120) 

where S is a scalar, the n T ’s are rank two tensors that are either symmetric 

or anti-symmetric and U is a rank four tensor. This representation is generally 

not an irreducible representation. For this to be a proper tensor representation 

the S, T and U tensors have to be proper tensors, and for a general rank four 

tensor, they are difficult to determine. The technique we use is to construct 

lower rank tensors from the original tensor, A, and use these lower rank tensors 

in the decomposition. We construct the tensors 

and 

S = A αβγδ g αβ g γδ , 

S (p) = A αβγδ ɛ αβγδ 

(2.121) 

T αβ = A αβγδ g γδ − 1 S. (2.122) 

4


Using these tensors to decompose A αβγδ , we obtain 

A αβγδ = 1 16 Sgαβ g γδ + 1 4! S(p) ɛ αβγδ + 1 2 T αβ g γδ + Ψ αβγδ , (2.123) 

where Ψ is a reducible rank four tensor. The tensor T αβ is a rank two tensor and 

thus can be reduced according to Equation (2.116). The irreducible tensors are 

scalar, pseudo-scalar, symmetric traceless rank two tensor and anti-symmetric 

rank two tensor. 

To do a decomposition of higher rank tensors, one makes a representation 

consisting of these irreducible tensors and products thereof, iterating the symmetry 

and anti-symmetry between indices. Comparing this with the symmetries 

of the tensor we want to decompose, it will be easier to construct the proper 

tensors in the representation. 

Example: Irreducible tensor representation of the two-dimensional 

spectral density matrix 

A simple example of irreducible tensor representation is to take the two dimensional 

spectral density tensor, Equation (2.53), which is a tensor under the 

group SO(2), and decompose it into an irreducible representation. In SO(2) 

we have the invariant metric tensor, δ αβ , and the totally anti-symmetric tensor, 

ɛ αβ . For a general rank two tensor, A αβ , the irreducible representation will be 

A αβ = 1 2 Sδ αβ + 1 2 S(p) ɛ αβ + T αβ , (2.124) 

where we have S as the trace, S = A αβ δ αβ , and the anti-symmetric part, S (p) = 

A αβ ɛ αβ , which is a pseudo-scalar. Performing this on the two dimensional 

spectral density matrix, S d (Eq. (2.53)), we have 

( ) 

F1 F1 S d = ∗ F 1 F2 

∗ 

F 2 F1 ∗ F 2 F2 

∗ = 1 2 (F 1 2 + F2 2 )δ αβ + 1 2 (F 1F † 2 − F 2F † 1 )ɛ αβ+ 

( F 

2 

1 − F2 2 F 1 F2 ∗ + F 2 F1 

∗ 

1 

2 

F 1 F † 2 + F 2F ∗ 1 F 2 2 − F 2 1 

) 

. (2.125) 

Comparing this representation with Stokes’ parameters, Equation (2.49), we see 

that it can be written 

( 

F1 F † 

S d = 

1 F 1 F † ) 

2 

F 2 F † 1 F 2 F † = 1 

2 

2 Iδ αβ − i 2 Vɛ αβ + 1 ( ) Q U 

, (2.126) 

2 U −Q 

where the energy density, I, is the invariant scalar and the degree of circular 

polarization, V, is the invariant pseudo scalar. The last two Stokes parameters, 

corresponding to linear polarization along the axis (Q) and linear polarization in 

45 ◦ angle to the axis (U), together form a traceless symmetric rank two tensor. 

This is the irreducible tensor representation of the two-dimensional spectral 

density tensor under SO(2).

Chapter 3 

The covariant spectral 

density tensor and a generic 

decomposition scheme 

In this chapter we introduce the covariant spectral density tensor, τ αβγδ ≡ 

F αβ (F γδ ) ∗ , which is a manifestly Lorentz covariant rank four tensor, containing 

all quadratic components of the electromagnetic field. An alternative way 

of constructing a covariant spectral density tensor is to consider A µ A ν (see 

Barut [1]), where A µ is the electromagnetic four-potential (see Section 2.1.2), 

The four-potential is however not a measurable physical quantity and it also 

depends on the choice of gauge. Since our approach is to consider measurable 

parameters of polarization (just as Stokes’ parameters) the seemingly more complicated 

rank four tensor τ αβγδ is preferable. One way to expand the covariant 

spectral density tensor is to use the Dirac gamma matrices. In this chapter 

two different definitions of the Dirac gamma matrices is presented and we use 

the one that comes from the flat metric g µν (see Section 2.1, Equation (2.6)) to 

extract the 36 parameters needed for a complete representation of the field. The 

technique here is similar to that in Section 2.2, where a three-dimensional tensor 

of rank two is decomposed in terms of a basis of matrices. In this Chapter we 

use the original four Dirac gamma matrices and expand them into a complete 

16 dimensional matrix basis. The covariant spectral density tensor is expanded 

in this basis. 

3.1 Covariant spectral density tensor 

To have all quadratic electromagnetic field components collected into one object, 

we introduce and define the covariant spectral density tensor, τ αβγδ , by 

τ αβγδ ≡ F αβ (F γβ ) ∗ . (3.1) 

The definition for real fields is the same but the complex conjugate is then 

abundant. This is the deterministic version where we know the pertinent field 

completely. For a physical field, where one considers a certain bandwidth, this 

27

28 CHAPTER 3. COVARIANT SPECTRAL DENSITY TENSOR 

tensor becomes 

τ αβγδ ≡ 〈F αβ (F γβ ) ∗ 〉, (3.2) 

where the average is taken separately on each component. When performing calculations 

we can ignore the average, 〈〉, and do the analysis for a deterministic 

wave. 

The deterministic approach, in contrast to consider a stochastical wave, is 

used throughout this thesis due to the fact that the average can be defined 

differently. To keep the generality we therefore do not explicitly define the 

〈〉-operator. 1 

3.2 Pauli spin matrices 

The gamma matrices are usually written in terms of the Pauli spin matrices 

(σ 1 , σ 2 , σ 3 ) and the 2 × 2 unit matrix (I 2 ). The Pauli matrices comes from the 

non-relativistic treatment of spin- 1 2 

particles and they are defined as 

σ 1 = 

( 0 1 

1 0 

) ( 0 −i 

, σ 2 = 

i 0 

) ( 1 0 

, σ 3 = 

0 −1 

) ( 1 0 

, I 2 = 

0 1 

These are the well known Pauli spin matrices from quantum mechanics. 

3.3 Dirac gamma matrices 

) 

. 

(3.3) 

The gamma matrices arise in the relativistic treatment of spin- 1 2 

particles and is 

the four dimensional generalization of the Pauli matrices. There are two different 

ways of defining the gamma matrices, one comes from using the Minkowski 

notation, (x, y, z, ict), and the second one from using Lorentz space with the 

flat metric, g µν (Eq. (2.6)). 

The first notation (used by Sakurai [19]) is the one originally used by Dirac, 

who defined the matrices as 

( ) 

( ) 

0 −iσµ 

I2 0 

γ µ = 

, µ = 1, 2, 3, γ 

iσ µ 0 

4 = 

. (3.4) 

0 −I 2 

One also defines a fifth matrix as 

γ 5 = γ 1 γ 2 γ 3 γ 4 (3.5) 

Here the γ µ ’s are Hermitian for µ=1,2,3,4,5. The second notation (used by for 

example Bjorken & Drell [2], Arfken & Weber [10]) one gets when using the 

metric g µν (see Eq. (2.6)), Lorentz notation, and has the following form 

( 

γ µ 0 σ 

µ 

= 

−σ µ 0 

The fifth matrix in this case looks as 

) 

, µ = 1, 2, 3 , γ 0 = 

( ) 

I2 0 

. (3.6) 

0 −I 2 

γ 5 = γ 5 = iγ 0 γ 1 γ 2 γ 3 . (3.7) 

1 Compare with quantum mechanics, where the 〈〉-operator is completely different from 

the classical operator (see e.g. Sakurai [19] [20]).

3.4. DECOMPOSITION OF τ αβγδ 29 

These matrices have the properties 

γ µ γ ν + γ ν γ µ = 2g µν µ, ν = 0, 1, 2, 3, (3.8) 

γ 0† = γ 0 , γ k† = −γ k , k = 1, 2, 3. (3.9) 

With these matrices we can form a 16-dimensional algebra, using the unit matrix 

and the symmetric combinations of the matrices. It is a complete basis 

with convenient Lorentz transformation properties. These 16 matrices are constructed 

in the following way 2 

I 4 , γ 5 = iγ 0 γ 1 γ 2 γ 3 , γ µ , γ 5 γ µ , σ µν = i(γ µ γ ν − γ ν γ µ )/2. (3.10) 

Considering Lorentz transformations, the first four tensors, γ µ , µ = 0, 1, 2, 3, 

transforms as a vector and γ 5 γ µ transforms as an axial vector (pseudo-vector). 

The six anti-symmetric combinations, σ µν transforms as a tensor. 3 

3.4 Decomposition of τ αβγδ 

We want to extract the quadratic forms of the components in the covariant 

spectral density tensor, τ αβγδ = F αβ (F γδ ) ∗ . This tensor contains all correlation 

components and is Lorentz covariant. 4 

Here we use the notation that γ 4 is the unit 4 × 4 matrix and perform the 

operations 

F µν (F αβ ) ∗ γ n µνγ n αβ , n = 0, 1, . . . , 15. (3.11) 

This gives a total of 256 scalars (16 × 16) but only 36 are non-zero (all 36 are 

found in appendix B). These 36 nonzero are the combinations of γ 1 , γ 3 , γ 6 , γ 8 , γ 11 

and γ 14 , which are the antisymmetric gamma matrices. We make the contractions 

F µν γ n µν, which yields 

F µν γ 1 µν = −2(E 3 + cB 3 ), 

F µν γ 3 µν = −2(E 2 + cB 2 ), 

F µν γ 6 µν = 2(E 2 − cB 2 ), 

F µν γ 8 µν = 2i(−E 1 + cB 1 ), 

F µν γ 11 

µν = 2(−E 3 + cB 3 ), 

F µν γ 14 

µν = −2i(E 1 + cB 1 ). 

(3.12) 

Performing this operation with the symmetric gamma matrices gives zero since 

the field strength tensor is antisymmetric and has zeroes as its diagonal elements. 

Our 36 quadratic parameters are then the combinations of any two of the linear 

expressions in Equation (3.12), remembering that in general, ab ∗ ≠ a ∗ b. 

Rewriting the field strength tensor as a sum, using the Dirac gamma matrices, 

we obtain 

2F µν = −(E 3 + cB 3 )γ 1 − (E 2 + cB 2 )γ 3 + (E 2 − cB 2 )γ 6 

− i(−E 1 + cB 1 )γ 8 + (−E 3 + cB 3 )γ 11 + i(E 1 + cB 1 )γ 14 . (3.13) 

2 All 16 gamma matrices are found in appendix B where they are written out in matrix 

form. 

3 More information on the transformation properties of Dirac gamma matrices is found in 

Bjorken & Drell [2], Sakurai [19] and Arfken & Weber [10]. 

4 The covariant spectral density tensor is explored and used more extensively in Chapter 6.

30 CHAPTER 3. COVARIANT SPECTRAL DENSITY TENSOR 

We have here a method of constructing 36 linearly independent parameters 

of the electromagnetic field but it is not an easy and direct method, since the 

parameters we gain are not collected into different geometrical quantities. On 

the good side we are just using the field strength tensor and the Dirac gamma 

matrices, which have known transformation properties under the Lorentz group.

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]. 

31

32CHAPTER 4. IRREDUCIBLE REPRESENTATION OF POLARIZATION 

between an electric and a magnetic quantity. The label χ corresponds to the sum 

which means that it is the real part. The label ψ corresponds to the difference 

and thus the imaginary part. The parameters are defined as 

Σ : ⃗ E ⊗ ⃗ E † + ⃗ B ⊗ ⃗ B † = 1 3 δ ijS Σ + iɛ ijk V Σ 

k + T Σ ij , (4.2) 

∆ : ⃗ E ⊗ ⃗ E † − ⃗ B ⊗ ⃗ B † = 1 3 δ ijS ∆ + iɛ ijk V ∆ 

k + T ∆ ij , (4.3) 

χ : ⃗ E ⊗ ⃗ B † + ⃗ B ⊗ ⃗ E † = 1 3 δ ijS χ + iɛ ijk V χ 

k + T χ ij , (4.4) 

ψ : ⃗ E ⊗ ⃗ B † − ⃗ B ⊗ ⃗ E † = 1 3 δ ijS ψ + iɛ ijk V ψ 

k + T ψ ij . (4.5) 

The number of components and degrees of freedom are distributed as 

Adding up to a total of 36 components. 

4.1.1 Σ, the sum 

parameter label S V i T ij Sum 

Σ 1 3 5 9 

∆ 1 3 5 9 

χ 1 3 5 9 

ψ 1 3 5 9 

total 4 12 20 36 

The first three parameters S Σ , V Σ and T Σ are labelled with Σ to denote the 

sum. The first parameter, S Σ , given by 

S Σ = ⃗ E 2 + c 2 ⃗ B 2 = E 2 1 + E 2 2 + E 2 3 + c 2 B 2 1 + c 2 B 2 2 + c 2 B 2 3, (4.6) 

is a real scalar and corresponds to the total energy density of the fields. The 

second parameter, ⃗ V Σ , is a real pseudo-vector that holds the imaginary part of 

the sum. This is easy to see since it is a crossproduct (pseudo-vector) and the 

components have the form A − A † . It has the following form 

⃗V Σ = ( Im[E 2 E ∗ 3 + c 2 B 2 B ∗ 3], Im[E 3 E ∗ 1 + c 2 B 3 B ∗ 1], Im[E 1 E ∗ 2 + c 2 B 1 B ∗ 2] ) 

= − i 2( ⃗E × ⃗ E † + c 2 ⃗ B × ⃗ B 

† ) . (4.7) 

In the case of transverse waves, V ⃗ is parallel or anti-parallel to the direction of 

propagation. The third parameter is 

⎛ 

⎞ 

E1 2 + c 2 B1 2 − 1 3 SΣ Re[E 1 E2 ∗ + c 2 B 1 B2] ∗ Re[E 1 E3 ∗ + c 2 B 1 B3] 

∗ Tij Σ = 

⎜Re[E 1 E2 ∗ + c 2 B 1 B2] ∗ E2 2 + c 2 B2 2 − 1 3 

⎝ 

SΣ Re[E 2 E3 ∗ + c 2 B 2 B3] 

∗ ⎟ 

⎠ . 

Re[E 1 E3 ∗ + c 2 B 1 B3] ∗ Re[E 2 E3 ∗ + c 2 B 2 B3] ∗ E3 2 + c 2 B3 2 − 1 3 SΣ (4.8) 

This parameter, Tij Σ , is similar to the Maxwell stress tensor, see Equation (2.95), 

the only difference being the 1 3 on the SΣ ’s on the diagonal. This tensor is 

traceless (it is made traceless in the definition), this is not the case with the 

Maxwell stress tensor, which has the trace equal to the energy density.

4.1. THE IRREDUCIBLE PARAMETERS UNDER SO(3) 33 

4.1.2 ∆, the difference 

The next label, ∆, corresponds to the difference between the tensor products. 

The first one, S ∆ , is the difference in energy 

S ∆ = ⃗ E 2 − c 2 ⃗ B 2 = E 2 1 + E 2 2 + E 2 3 − c 2 B 2 1 − c 2 B 2 2 − c 2 B 2 3, (4.9) 

and is recognised as the Lagrangian for a free electromagnetic field, which is a 

known invariant scalar. 2 The vector, ⃗ V ∆ , is given by 

⃗V ∆ = ( Im[E 2 E ∗ 3 − c 2 B 2 B ∗ 3], Im[E 3 E ∗ 1 − c 2 B 3 B ∗ 1], Im[E 1 E ∗ 2 − c 2 B 1 B ∗ 2] ) 

= − i 2( ⃗E × ⃗ E † − c 2 ⃗ B × ⃗ B 

† ) . (4.10) 

Recalling the V ⃗ defined in Section 2.4, V ⃗ Σ is the difference V ⃗ E − V ⃗ B , i.e. a 

measure of the difference of the polarization ellipses. Here we get the tensor to 

be 

⎛ 

⎞ 

E1 2 − c 2 B1 2 − 1 3 S∆ Re[E 1 E2 ∗ − c 2 B 1 B2] ∗ Re[E 1 E3 ∗ − c 2 B 1 B3] 

∗ Tij ∆ = 

⎜Re[E 1 E2 ∗ − c 2 B 1 B2] ∗ E2 2 − c 2 B2 2 − 1 3 

⎝ 

S∆ Re[E 2 E3 ∗ − c 2 B 2 B3] 

∗ ⎟ 

⎠ . 

Re[E 1 E3 ∗ − c 2 B 1 B3] ∗ Re[E 2 E3 ∗ − c 2 B 2 B3] ∗ E3 2 − c 2 B3 2 − 1 3 S∆ (4.11) 

It is real, symmetric and traceless as it should be. Again all parameters are real 

and V ⃗ ∆ is the imaginary part. Rewriting Tij Σ and T ij ∆ in the same manner as 

the Maxwell stress tensor (Eq. (2.95)) we have 

T Σ ∆ 

ij = ( E i E j − 1 3 ⃗ E 2 δ i j ) ± c 2( B i B j − 1 3 ⃗ B 2 δ i j ) , (4.12) 

where the plus-sign corresponds to Tij Σ and the minus to T ij ∆ , and it is easy to 

see the similarity with the Maxwell stress tensor. 

4.1.3 χ, the real part 

The label χ corresponds to the real part of the cross-correlated components, 

i.e. the components containing EB ∗ and BE ∗ parts. The scalar S χ is just two 

times the real part of the trace of Equation (2.97), the correlation matrix for ⃗ E 

and ⃗ B, and is given by 

S χ = c ⃗ E · ⃗B † + c ⃗ B · ⃗E † = 2Re[c ⃗ E · ⃗B † ], (4.13) 

The vector ⃗ V χ corresponds to the antisymmetric imaginary part of the correlation 

matrix and the imaginary part of the Poynting vector and is given by 

( 

) 

⃗V χ = c Im[E 2 B3 ∗ + B 2 E3], ∗ Im[E 3 B1 ∗ + B 3 E1], ∗ Im[E 1 B2 ∗ + B 1 E2] 

∗ 

= −c i 2( ⃗E × ⃗ B † + ⃗ B × ⃗ E †) . (4.14) 

2 The Lagrangian is really half of this (see Barut [1]) but the 1 is just a constant that can 

2 

be ommited since we are just interested in covariance of the parameters.


The tensor, T χ ij , is the symmetric real part of Equation (2.97) made traceless 

with S χ , and is written 

⎛ 

⎞ 

2cRe[E 1 B1] ∗ − 1 3 Sχ cRe[E 1 B2 ∗ + E 2 B1] ∗ cRe[E 1 B3 ∗ + E 3 B1] 

∗ T χ ij = ⎜cRe[E 1 B2 ∗ + E 2 B1] ∗ 2cRe[E 2 B2] ∗ − 1 3 

⎝ 

Sχ cRe[E 2 B3 ∗ + E 3 B2] 

∗ ⎟ 

⎠ . 

cRe[E 1 B3 ∗ + E 3 B1] ∗ cRe[E 2 B3 ∗ + E 3 B2] ∗ 2cRe[E 3 B3] ∗ − 1 3 Sχ 

(4.15) 

4.1.4 ψ, the imaginary part 

The last label, ψ, is a bit different. The scalar S ψ is two times the imaginary 

part of the trace of the correlation matrix ( ⃗ E ⊗ ⃗ B † ), hence 

S ψ = c ⃗ E · ⃗B † − c ⃗ B · ⃗E † = 2icIm[ ⃗ E · ⃗B † ]. (4.16) 

The vector, V ψ , contains the antisymmetric real part and is given by 

( 

) 

⃗V ψ = −ic Re[E 2 B3 ∗ − E 3 B2], ∗ Re[E 3 B1 ∗ − E 1 B3], ∗ Re[E 1 B2 ∗ − E 2 B1] 

∗ 

= −c i 2( ⃗E × ⃗ B † − ⃗ B × ⃗ E †) . (4.17) 

This is also the real part of the complex Poynting vector, Equation (2.103). The 

tensor T ψ ij holds the symmetric imaginary part and is written 

⎛ 

⎞ 

2icIm[E 1 B1] ∗ − 1 3 Sψ icIm[E 1 B2 ∗ + E 2 B1] ∗ icIm[E 1 B3 ∗ + E 3 B1] 

∗ T ψ ij = ⎜icIm[E 1 B2 ∗ + E 2 B1] ∗ 2icIm[E 2 B2] ∗ − 1 3 

⎝ 

Sψ icIm[E 2 B3 ∗ + E 3 B2] 

∗ ⎟ 

⎠ . 

icIm[E 1 B3 ∗ + E 3 B1] ∗ icIm[E 2 B3 ∗ + E 3 B2] ∗ 2icIm[E 3 B3] ∗ − 1 3 Sψ 

The total complex Poynting vector is then given by 

(4.18) 

⃗S = i ⃗ V ψ + i ⃗ V χ . (4.19) 

4.2 Table of the parameters 

There is a total of 12 parameters in the form of scalars, vectors and tensors. 

Here is a table of the parameters for a better overview:

4.3. NORMALIZATION OF THE S, V, T PARAMETERS 35 

S Σ 

S ∆ 

Total energy density 

Lagrangian for a free electromagnetic field 

S χ Re[cE ⃗ · ⃗B † ] 

S ψ Im[cE ⃗ · ⃗B † ] 

⃗V Σ 

⃗V ∆ 

⃗V χ 

⃗V ψ 

Tij 

Σ 

Tij 

∆ 

T χ ij 

Sum of the ⃗v vectors (see Section 2.3) for E ⃗ and B. ⃗ Sum of the 

polarization ellipses. 

Difference of the ⃗v vectors for E ⃗ and B. ⃗ Difference 

of the polarization ellipses 

Imaginary part of the complex Poynting vector. 

Real part of the complex Poynting vector. 

Similar to Maxwell stress tensor but traceless. 

Similar to Tij Σ but corresponds to the difference of the tensors for E ⃗ and B ⃗ 

Not found in the literature. Real part of a cross-correlated 

version of Maxwell stress tensor. 

T ψ ij 

Not found in the literature. The imaginary part to T χ ij 

The last two are of special interesting since to the best of our knowledge they 

have not previously been defined or used in the literature. 

4.3 Normalization of the S, V, T parameters 

The parameter S Σ , the total energy density, is the only parameter that is a 

positive real number for any electromagnetic field. Using s, v and t as mnemonics 

for the normalized parameters, we have 

s Σ = SΣ 

S Σ = 1, 

⃗v Σ = ⃗ V Σ 

S Σ , 

t Σ ij = T Σ ij 

S Σ , 

s∆ = S∆ 

S Σ , 

⃗v∆ = ⃗ V ∆ 

S Σ , 

t∆ ij = T ∆ ij 

S Σ , 

sχ = Sχ 

S Σ , 

⃗vχ = ⃗ V χ 

S Σ , 

tχ ij = T χ ij 

S Σ , 

sψ = Sψ 

S Σ , (4.20) 

⃗vψ = ⃗ V ψ 

S Σ , (4.21) 

tψ ij = T ψ ij 

S Σ . (4.22) 

These are normalized versions where the components of the parameters have a 

maximum absolute value of one. 

4.3.1 Example: the refractive index 

In this section we use some complex vector algebra. A good source of information 

on this subject is Lindell [14]. This example illustrates how to determine 

the refractive index vector ⃗n which is then expressed in terms of the parameters 

S, V and T . The refractive index is defined by 

⃗n ≡ c ω ⃗ k. (4.23) 

The Fourier transformed version of Faraday’s law is given by 

⃗ k × ⃗ E = ω ⃗ B ⇔ ⃗n × ⃗ E = c ⃗ B. (4.24)


The condition of a divergence free magnetic field is easily seen from this equation. 

By taking the scalar product with ⃗n, from the left we obtain 

⃗n · (⃗n × ⃗ E) = c⃗n · ⃗B ⇒ ⃗n · ⃗B = 0, (4.25) 

where we have used the vector relations ⃗a · ( ⃗ b × ⃗c) = (⃗a × b) · ⃗c and ⃗n × ⃗n = 0. 

We also multiply with ⃗n † , from the left, hence 

⃗n † · (⃗n × ⃗ E) = c⃗n † · ⃗B ⇒ ⃗n † · ⃗B = 0. (4.26) 

The condition for the last equation to be zero is that ⃗n † ×⃗n = 0. This is fulfilled 

if ⃗n can be written ⃗n = n⃗r, where n is a complex scalar and ⃗r is a real vector. 

This means that the refractive index is the same in all direction, which is true 

for a homogeneous medium. To get an expression for ⃗n we use Faraday’s law 

cross multiplied from the right with ⃗ B † 

(⃗n × ⃗ E) × ⃗ B † = c ⃗ B × ⃗ B † , (4.27) 

which leads to 

−⃗n( ⃗ B † · ⃗E) + ⃗ E( ⃗ B † · ⃗n) = c ⃗ B × ⃗ B † . (4.28) 

If we restrict ourself to the case ⃗n × ⃗n † = 0 (homogeneous medium), the second 

term on the left hand side vanishes and we are left with 

which for ⃗ E · ⃗B † ≠ 0 can be rewritten as 

−⃗n( ⃗ E · ⃗B † ) = c ⃗ B × ⃗ B † , (4.29) 

B 

⃗n = −c ⃗ × B ⃗ † 

. (4.30) 

⃗E · ⃗B 

† 

If we compare this expression with the parameters defined earlier in this chapter 

we have 

⃗B × ⃗ B † = i 

c 2 ( ⃗V Σ − ⃗ V ∆) , (4.31) 

and 

Inserting this in the expression for ⃗n we get 

⃗E · ⃗B † = 1 2c( 

S χ + S ψ) . (4.32) 

⃗n = −2 i( ⃗ V Σ − ⃗ V ∆) 

S χ + S ψ . (4.33) 

Given the S, V and T parameters this is a simple calculation. 

4.4 Stokes parameters from the S, V, T parameters 

In the two dimensional case, with E 3 = 0 and only considering the electric field 

(putting ⃗ B = 0), we have Stokes parameters. In terms of the S, V, T parameters

4.4. STOKES PARAMETERS FROM THE S, V, T PARAMETERS 37 

they can be written 

I = F 2 x + F 2 y = S Σ , (4.34) 

Q = F 2 x − F 2 y = T Σ 11 − T Σ 22, (4.35) 

U = 2Re[F y F ∗ x ] = 2T Σ 12, (4.36) 

V = 2Im[F y F ∗ x ] = −2V Σ 

3 , (4.37) 

where Σ can be replaced by ∆, since the magnetic field, ⃗ B , is zero. It is similar 

for the polarization of the magnetic field. The two dimensional spectral density 

matrix (Eq. (2.53)) becomes 

( ) 

Fx Fx S d = ∗ F x Fy 

∗ 

F y Fx 

∗ F y Fy 

∗ = 1 2 

( ) 

S Σ + T11 Σ − T22 Σ 2T12 Σ + i2V3 

Σ 

2T12 Σ − i2V3 Σ S Σ − T11 Σ + T22 

Σ , (4.38) 

where we have put in the Stokes parameters in the S, V, T parameter form. We 

can simplify this to 

( ) 

T 

Σ 

S d = 11 T12 Σ + iV3 

Σ 

T12 Σ − iV3 Σ T22 

Σ . (4.39)

38CHAPTER 4. IRREDUCIBLE REPRESENTATION OF POLARIZATION

Chapter 5 

Covariant polarization 

tensors in E,B-component 

form 

In Chapter 5 we saw that the S, V, T parameters can be found in Lorentz covariant 

tensors. These tensors were constructed according to 

where 

is the metric, 

and 

1 X µν = F αµ (F αδ ) ∗ g δν , (5.1) 

2 X µν = F αµ (F αδ ) ∗ g δν , (5.2) 

3 X µν = F αµ (F αδ ) ∗ g δν , (5.3) 

4 X µν = F αµ (F αδ ) ∗ g δν , (5.4) 

⎛ 

⎞ 

1 0 0 0 

g µν = g µν = ⎜0 −1 0 0 

⎟ 

⎝0 0 −1 0 ⎠ , (5.5) 

0 0 0 −1 

⎛ 

⎞ 

0 −E 1 −E 2 −E3 

F µν = ⎜E 1 0 −cB 3 cB 2 

⎟ 

⎝E 2 cB 3 0 −cB 1 

⎠ (5.6) 

E 3 −cB 2 cB 1 0 

⎛ 

⎞ 

0 E 1 E 2 E 3 

F µν = ⎜−E 1 0 −cB 3 cB 2 

⎟ 

⎝−E 2 cB 3 0 −cB 1 

⎠ (5.7) 

−E 3 −cB 2 cB 1 0 

are the contravariant and covariant field strength tensors, respectively, and 

⎛ 

⎞ 

0 −cB 1 −cB 2 −cB 3 

F µν = ⎜cB 1 0 E 3 −E 2 

⎟ 

⎝cB 2 −E 3 0 E 1 

⎠ (5.8) 

cB 3 E 2 −E 1 0 

39

40 CHAPTER 5. COVARIANT POLARIZATION TENSORS 

and 

⎛ 

⎞ 

0 cB 1 cB 2 cB 3 

F µν = ⎜−cB 1 0 E 3 −E 2 

⎟ 

⎝−cB 2 −E 3 0 E 1 

⎠ (5.9) 

−cB 3 E 2 −E 1 0 

are the corresponding dual tensors. 

5.1 The X tensors 

In E, B-component form the n X µν tensors are given by 

1 X µν = 

[−E 1 E ∗ 1 − E 2 E ∗ 2 − E 3 E ∗ 3, −E 2 cB ∗ 3 + E 3 cB ∗ 2, E 1 cB ∗ 3 − E 3 cB ∗ 1, −E 1 cB ∗ 2 + E 2 cB ∗ 1] 

[−cB 3 E ∗ 2 + cB 2 E ∗ 3, E 1 E ∗ 1 − c 2 B 3 B ∗ 3 − c 2 B 2 B ∗ 2, E 1 E ∗ 2 + c 2 B 2 B ∗ 1, E 1 E ∗ 3 + c 2 B 3 B ∗ 1] 

[cB 3 E ∗ 1 − cB 1 E ∗ 3, E 2 E ∗ 1 + c 2 B 1 B ∗ 2, E 2 E ∗ 2 − c 2 B 3 B ∗ 3 − c 2 B 1 B ∗ 1, E 2 E ∗ 3 + c 2 B 3 B ∗ 2] 

[−cB 2 E ∗ 1 +cB 1 E ∗ 2, E 3 E ∗ 1 +c 2 B 1 B ∗ 3, E 3 E ∗ 2 +c 2 B 2 B ∗ 3, E 3 E ∗ 3 −c 2 B 2 B ∗ 2 −c 2 B 1 B ∗ 1] 

(5.10) 

2 X µν = 

[−c 2 B 1 B ∗ 1−c 2 B 2 B ∗ 2−c 2 B 3 B ∗ 3, −cB 3 E ∗ 2+cB 2 E ∗ 3, cB 3 E ∗ 1−cB 1 E ∗ 3, −cB 2 E ∗ 1+cB 1 E ∗ 2] 

[−E 2 cB ∗ 3 + E 3 cB ∗ 2, c 2 B 1 B ∗ 1 − E 3 E ∗ 3 − E 2 E ∗ 2, E 2 E ∗ 1 + c 2 B 1 B ∗ 2, E 3 E ∗ 1 + c 2 B 1 B ∗ 3] 

[E 1 cB ∗ 3 − E 3 cB ∗ 1, E 1 E ∗ 2 + c 2 B 2 B ∗ 1, c 2 B 2 B ∗ 2 − E 3 E ∗ 3 − E 1 E ∗ 1, E 3 E ∗ 2 + c 2 B 2 B ∗ 3] 

[−E 1 cB ∗ 2 + E 2 cB ∗ 1, E 1 E ∗ 3 + c 2 B 3 B ∗ 1, E 2 E ∗ 3 + c 2 B 3 B ∗ 2, c 2 B 3 B ∗ 3 − E 2 E ∗ 2 − E 1 E ∗ 1] 

(5.11) 

3 X µν = 

[−cB 1 E ∗ 1−cB 2 E ∗ 2−cB 3 E ∗ 3, −c 2 B 2 B ∗ 3+c 2 B 3 B ∗ 2, c 2 B 1 B ∗ 3−c 2 B 3 B ∗ 1, −c 2 B 1 B ∗ 2+c 2 B 2 B ∗ 1] 

[E 3 E ∗ 2 − E 2 E ∗ 3, cB 1 E ∗ 1 + E 3 cB ∗ 3 + E 2 cB ∗ 2, cB 1 E ∗ 2 − E 2 cB ∗ 1, cB 1 E ∗ 3 − E 3 cB ∗ 1] 

[−E 3 E ∗ 1 + E 1 E ∗ 3, cB 2 E ∗ 1 − E 1 cB ∗ 2, cB 2 E ∗ 2 + E 3 cB ∗ 3 + E 1 cB ∗ 1, cB 2 E ∗ 3 − E 3 cB ∗ 2] 

[E 2 E ∗ 1 − E 1 E ∗ 2, cB 3 E ∗ 1 − E 1 cB ∗ 3, cB 3 E ∗ 2 − E 2 cB ∗ 3, cB 3 E ∗ 3 + E 2 cB ∗ 2 + E 1 cB ∗ 1] 

(5.12) 

4 X µν = 

[−E 1 cB ∗ 1 − E 2 cB ∗ 2 − E 3 cB ∗ 3, −E 3 E ∗ 2 + E 2 E ∗ 3, E 3 E ∗ 1 − E 1 E ∗ 3, −E 2 E ∗ 1 + E 1 E ∗ 2] 

[c 2 B 2 B ∗ 3 −c 2 B 3 B ∗ 2, E 1 cB ∗ 1 +cB 3 E ∗ 3 +cB 2 E ∗ 2, −cB 2 E ∗ 1 +E 1 cB ∗ 2, −cB 3 E ∗ 1 +E 1 cB ∗ 3] 

[−c 2 B 1 B ∗ 3+c 2 B 3 B ∗ 1, −cB 1 E ∗ 2+E 2 cB ∗ 1, cB 3 E ∗ 3+E 2 cB ∗ 2+cB 1 E ∗ 1, −cB 3 E ∗ 2+E 2 cB ∗ 3] 

[c 2 B 1 B ∗ 2−c 2 B 2 B ∗ 1, −cB 1 E ∗ 3+E 3 cB ∗ 1, −cB 2 E ∗ 3+E 3 cB ∗ 2, cB 2 E ∗ 2+E 3 cB ∗ 3+cB 1 E ∗ 1] 

(5.13)

5.2. THE Y TENSORS 41 

5.2 The Y tensors 

The Y tensors, constructed by the combinations 

Σ Y µν = 1 2 

( 1 

X µν + 2 X µν ) 

, 

χ Y µν = 1 ( 3 

X µν + 4 X µν ) 

, 

2 

∆ Y µν = 1 ( 1 

X µν − 2 X µν ) 

(5.14) 

, 

2 

ψ Y µν = 1 ( 3 

X µν − 4 X µν ) 

, 

2 

and are then found to be 

Σ Y µν = 1 2 

[−E 1 E ∗ 1−E 2 E ∗ 2−E 3 E ∗ 3−cB 1 cB ∗ 1−cB 2 cB ∗ 2−cB 3 cB ∗ 3, −cB 3 E ∗ 2+cB 2 E ∗ 3−E 2 cB ∗ 3+E 3 cB ∗ 2, 

E 1 cB ∗ 3 − E 3 cB ∗ 1 + cB 3 E ∗ 1 − cB 1 E ∗ 3, −E 1 cB ∗ 2 + E 2 cB ∗ 1 − cB 2 E ∗ 1 + cB 1 E ∗ 2] 

[−cB 3 E ∗ 2+cB 2 E ∗ 3−E 2 cB ∗ 3+E 3 cB ∗ 2, E 1 E ∗ 1−cB 3 cB ∗ 3−cB 2 cB ∗ 2+cB 1 cB ∗ 1−E 3 E ∗ 3−E 2 E ∗ 2, 

E 2 E ∗ 1 + cB 1 cB ∗ 2 + E 1 E ∗ 2 + cB 2 cB ∗ 1, E 3 E ∗ 1 + cB 1 cB ∗ 3 + E 1 E ∗ 3 + cB 3 cB ∗ 1] 

[E 1 cB ∗ 3 − E 3 cB ∗ 1 + cB 3 E ∗ 1 − cB 1 E ∗ 3, E 2 E ∗ 1 + cB 1 cB ∗ 2 + E 1 E ∗ 2 + cB 2 cB ∗ 1, 

E 2 E ∗ 2−cB 3 cB ∗ 3−cB 1 cB ∗ 1+cB 2 cB ∗ 2−E 3 E ∗ 3−E 1 EC 1 , E 3 E ∗ 2+cB 2 cB ∗ 3+E 2 E ∗ 3+cB 3 cB ∗ 2] 

[−E 1 cB ∗ 2 + E 2 cB ∗ 1 − cB 2 E ∗ 1 + cB 1 E ∗ 2, E 3 E ∗ 1 + cB 1 cB ∗ 3 + E 1 E ∗ 3 + cB 3 cB ∗ 1, 

E 3 E ∗ 2+cB 2 cB ∗ 3+E 2 E ∗ 3+cB 3 cB ∗ 2, E 3 E ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+cB 3 cB ∗ 3−E 2 E ∗ 2−E 1 E ∗ 1] 

(5.15) 

χ Y µν = 1 2 

[−cB 1 E ∗ 1−cB 2 E ∗ 2−cB 3 E ∗ 3−E 1 cB ∗ 1−E 2 cB ∗ 2−E 3 cB ∗ 3, −cB 2 cB ∗ 3+cB 3 cB ∗ 2−E 3 E ∗ 2+E 2 E ∗ 3, 

cB 1 cB ∗ 3 − cB 3 cB ∗ 1 + E 3 E ∗ 1 − E 1 E ∗ 3, −cB 1 cB ∗ 2 + cB 2 cB ∗ 1 − E 2 E ∗ 1 + E 1 E ∗ 2] 

[E 3 E ∗ 2−E 2 E ∗ 3+cB 2 cB ∗ 3−cB 3 cB ∗ 2, cB 1 E ∗ 1+E 3 cB ∗ 3+E 2 cB ∗ 2+E 1 cB ∗ 1+cB 3 E ∗ 3+cB 2 E ∗ 2, 

cB 1 E ∗ 2 − E 2 cB ∗ 1 − cB 2 E ∗ 1 + E 1 cB ∗ 2, cB 1 E ∗ 3 − E 3 cB ∗ 1 − cB 3 E ∗ 1 + E 1 cB ∗ 3] 

[−E 3 E ∗ 1 + E 1 E ∗ 3 − cB 1 cB ∗ 3 + cB 3 cB ∗ 1, cB 2 E ∗ 1 − E 1 cB ∗ 2 − cB 1 E ∗ 2 + E 2 cB ∗ 1, 

cB 1 E ∗ 1+E 3 cB ∗ 3+E 2 cB ∗ 2+E 1 cB ∗ 1+cB 3 E ∗ 3+cB 2 E ∗ 2, cB 2 E ∗ 3−E 3 cB ∗ 2−cB 3 E ∗ 2+E 2 cB ∗ 3] 

[E 2 E ∗ 1 − E 1 E ∗ 2 + cB 1 cB ∗ 2 − cB 2 cB ∗ 1, cB 3 E ∗ 1 − E 1 cB ∗ 3 − cB 1 E ∗ 3 + E 3 cB ∗ 1, 

cB 3 E ∗ 2−E 2 cB ∗ 3−cB 2 E ∗ 3+E 3 cB ∗ 2, cB 1 E ∗ 1+E 3 cB ∗ 3+E 2 cB ∗ 2+E 1 cB ∗ 1+cB 3 E ∗ 3+cB 2 E ∗ 2] 

(5.16)

42 CHAPTER 5. COVARIANT POLARIZATION TENSORS 

∆ Y µν = 1 2 

[−E 1 E ∗ 1−E 2 E ∗ 2−E 3 E ∗ 3+cB 1 cB ∗ 1+cB 2 cB ∗ 2+cB 3 cB ∗ 3, cB 3 E ∗ 2−E 2 cB ∗ 3−cB 2 E ∗ 3+E 3 cB ∗ 2, 


[cB 2 E ∗ 3−E 3 cB ∗ 2−cB 3 E ∗ 2+E 2 cB ∗ 3, E 1 E ∗ 1−cB 3 cB ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+E 3 E ∗ 3+E 2 E ∗ 2, 

− cB 1 cB ∗ 2 + cB 2 cB ∗ 1 − E 2 E ∗ 1 + E 1 E ∗ 2, −E 3 E ∗ 1 + E 1 E ∗ 3 − cB 1 cB ∗ 3 + cB 3 cB ∗ 1] 

[cB 3 E ∗ 1 − E 1 cB ∗ 3 − cB 1 E ∗ 3 + E 3 cB ∗ 1, E 2 E ∗ 1 − E 1 E ∗ 2 + cB 1 cB ∗ 2 − cB 2 cB ∗ 1, 

E 1 E ∗ 1−cB 3 cB ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+E 3 E ∗ 3+E 2 E ∗ 2, −cB 2 cB ∗ 3+cB 3 cB ∗ 2−E 3 E ∗ 2+E 2 E ∗ 3] 

[cB 1 E ∗ 2 − E 2 cB ∗ 1 − cB 2 E ∗ 1 + E 1 cB ∗ 2, cB 1 cB ∗ 3 − cB 3 cB ∗ 1 + E 3 E ∗ 1 − E 1 E ∗ 3, 

E 3 E ∗ 2−E 2 E ∗ 3+cB 2 cB ∗ 3−cB 3 cB ∗ 2, E 1 E ∗ 1−cB 3 cB ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+E 3 E ∗ 3+E 2 E ∗ 2] 

(5.17) 

ψ Y µν = 1 2 

[−cB 1 E ∗ 1−cB 2 E ∗ 2−cB 3 E ∗ 3+E 1 cB ∗ 1+E 2 cB ∗ 2+E 3 cB ∗ 3, E 3 E ∗ 2−E 2 E ∗ 3−cB 2 cB ∗ 3+cB 3 cB ∗ 2, 

cB 1 cB ∗ 3 − cB 3 cB ∗ 1 − E 3 E ∗ 1 + E 1 E ∗ 3, −cB 1 cB ∗ 2 + cB 2 cB ∗ 1 + E 2 E ∗ 1 − E 1 E ∗ 2] 

[E 3 E ∗ 2−E 2 E ∗ 3−cB 2 cB ∗ 3+cB 3 cB ∗ 2, cB 1 E ∗ 1+E 3 cB ∗ 3+E 2 cB ∗ 2−E 1 cB ∗ 1−cB 3 E ∗ 3−cB 2 E ∗ 2, 

cB 2 E ∗ 1 − E 1 cB ∗ 2 + cB 1 E ∗ 2 − E 2 cB ∗ 1, cB 3 E ∗ 1 − E 1 cB ∗ 3 + cB 1 E ∗ 3 − E 3 cB ∗ 1] 

[cB 1 cB ∗ 3 − cB 3 cB ∗ 1 − E 3 E ∗ 1 + E 1 E ∗ 3, cB 2 E ∗ 1 − E 1 cB ∗ 2 + cB 1 E ∗ 2 − E 2 cB ∗ 1, 

cB 2 E ∗ 2+E 3 cB ∗ 3+E 1 cB ∗ 1−cB 3 E ∗ 3−E 2 cB ∗ 2−cB 1 E ∗ 1, cB 3 E ∗ 2−E 2 cB ∗ 3+cB 2 E ∗ 3−E 3 cB ∗ 2] 

[−cB 1 cB ∗ 2 + cB 2 cB ∗ 1 + E 2 E ∗ 1 − E 1 E ∗ 2, cB 3 E ∗ 1 − E 1 cB ∗ 3 + cB 1 E ∗ 3 − E 3 cB ∗ 1, 

cB 3 E ∗ 2−E 2 cB ∗ 3+cB 2 E ∗ 3−E 3 cB ∗ 2, cB 3 E ∗ 3+E 2 cB ∗ 2+E 1 cB ∗ 1−cB 2 E ∗ 2−E 3 cB ∗ 3−cB 1 E ∗ 1] 

(5.18) 

5.3 The K and K tensors 

The K and K tensors are formed by removing the trace from ∆ Y µν and χ Y µν , 

respectively, and are given by 

K µν = 1 2 

[0, cB 3 E ∗ 2 − E 2 cB ∗ 3 − cB 2 E ∗ 3 + E 3 cB ∗ 2, cB 1 E ∗ 3 − E 3 cB ∗ 1 − cB 3 E ∗ 1 + E 1 cB ∗ 3, 

cB 2 E ∗ 1 − E 1 cB ∗ 2 − cB 1 E ∗ 2 + E 2 cB ∗ 1] 

[cB 2 E ∗ 3 − E 3 cB ∗ 2 − cB 3 E ∗ 2 + E 2 cB ∗ 3, 0, −cB 1 cB ∗ 2 + cB 2 cB ∗ 1 − E 2 E ∗ 1 + E 1 E ∗ 2, 

− E 3 E ∗ 1 + E 1 E ∗ 3 − cB 1 cB ∗ 3 + cB 3 cB ∗ 1] 

[cB 3 E ∗ 1 − E 1 cB ∗ 3 − cB 1 E ∗ 3 + E 3 cB ∗ 1, E 2 E ∗ 1 − E 1 E ∗ 2 + cB 1 cB ∗ 2 − cB 2 cB ∗ 1, 0, 

− cB 2 cB ∗ 3 + cB 3 cB ∗ 2 − E 3 E ∗ 2 + E 2 E ∗ 3] 


E 3 E ∗ 2 − E 2 E ∗ 3 + cB 2 cB ∗ 3 − cB 3 cB ∗ 2, 0] (5.19)

5.3. THE K AND K TENSORS 43 

and 

K µν = 1 2 

[0, −cB 2 cB ∗ 3 + cB 3 cB ∗ 2 − E 3 E ∗ 2 + E 2 E ∗ 3, cB 1 cB ∗ 3 − cB 3 cB ∗ 1 + E 3 E ∗ 1 − E 1 E ∗ 3, 

− cB 1 cB ∗ 2 + cB 2 cB ∗ 1 − E 2 E ∗ 1 + E 1 E ∗ 2] 

[E 3 E ∗ 2 − E 2 E ∗ 3 + cB 2 cB ∗ 3 − cB 3 cB ∗ 2, 0, cB 1 E ∗ 2 − E 2 cB ∗ 1 − cB 2 E ∗ 1 + E 1 cB ∗ 2, 

cB 1 E ∗ 3 − E 3 cB ∗ 1 − cB 3 E ∗ 1 + E 1 cB ∗ 3] 

[−E 3 E ∗ 1 + E 1 E ∗ 3 − cB 1 cB ∗ 3 + cB 3 cB ∗ 1, cB 2 E ∗ 1 − E 1 cB ∗ 2 − cB 1 E ∗ 2 + E 2 cB ∗ 1, 0, 

cB 2 E ∗ 3 − E 3 cB ∗ 2 − cB 3 E ∗ 2 + E 2 cB ∗ 3] 


respectively. 

cB 3 E ∗ 2 − E 2 cB ∗ 3 − cB 2 E ∗ 3 + E 3 cB ∗ 2, 0], (5.20)

44 CHAPTER 5. COVARIANT POLARIZATION TENSORS

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) 

45

46 CHAPTER 6. REDUCIBLE REPRESENTATION OF τ αβγδ 

6.2 The irreducible tensors 

The decomposition we make is in terms of proper Lorentz tensors, i.e. the 

tensors are manifestly covariant under Lorentz transformations. The decomposition 

is similar to Equation (2.123) but here we construct the tensors in the 

decomposition from the original tensor. By considering which tensors we can 

construct using τ αβγδ and the invariant tensors of the group SO(3, 1) (see Section 

2.4 Equation (2.44)) we obtain a decomposition of the tensor. From the 

covariant spectral density we form the following tensors 

(p) S = τ αβγδ ɛ αβγδ , 

S = τ αβγδ g αγ g βδ , 

T αγ = τ αβγδ g βδ − 1 4 Sgαγ , 

(6.4) 

(p) T µν = τ αβγδ g αγ ɛ βδµν , 

where S is an ordinary scalar, (p) S is a pseudo-scalar, T αγ and (p) T µν are tensor 

and pseudo tensor, respectively. We can construct several similar tensors, but 

due to the symmetries of τ αβγδ several of them will be zero and for simplicity, 

we decompose τ αβγδ into a reducible representation. 

Calculating the scalars, S and (p) S, in the representation, they are found to 

be 

S = −2(E 2 − c 2 B 2 ), 

(p) S = 4( ⃗ E · ⃗B † + ⃗ B · ⃗E † ). 

(6.5) 

These are recognised as the Lagrangian for a free electromagnetic field and the 

real part of the scalar product ⃗ E · ⃗B † , respectively, both known invariants. The 

first tensor, T αγ , is the complex electromagnetic energy-momentum tensor (see 

Section 2.1.4 Equation (2.44)). The second tensor, (p) T µν , is found to be −2K µν , 

a tensor introduced in the previous Chapter (Chapter 5, Equation (??)). 

In Chapter 5 we introduced ten covariant polarization tensors, the X and Y 

tensors together with the two tenors K and K. These tensors can be constructed 

from the covariant spectral density tensor directly, using the metric tensor, g µν 

and the totally anti-symmetric tensor, ɛ µναβ . In terms of the covariant spectral 

density tensor, 3 X νδ and 4 X νδ (Eq. (??)) are given by 

and 

3 X νδ = F αν (F αµ ) ∗ g µδ = 1 2 g µγɛ µν 

αβ τ αβγδ (6.6) 

4 X νδ = F αν (F αµ ) ∗ g µδ = 1 2 g αµɛ µν 

γδ τ αβγδ . (6.7) 

From these expressions we get the expressions for χ Y νδ (plus sign) and ψ Y νδ 

(minus sign), and we have 

χ νδ 1( ψ Y = 3 

X νδ ± 4 X νδ ) 1 = 

2 

4 τ αβγδ( g µγ ɛ µν 

αβ ± g αµɛ µν ) 

γδ . (6.8) 

For 1 X and 2 X, the expressions are 

1 X νδ = F αν F ∗ αµg µδ = τ ανβδ g αβ (6.9)

6.3. REDUCIBLE REPRESENTATION 47 

and 

2 X νδ = F αν Fαµg ∗ µδ = 1 4 ɛαν α ′ β ′ɛ α δ 

γ ′ σ ′τ α′ β ′ γ ′ σ ′ , (6.10) 

where the prime, ” ′ ” denotes dummy indices. The expression for the corresponding 

Y tensors is 

Σ νδ 1( ∆ Y = 3 

X νδ ± 4 X νδ ) 1 = 

2 

2 τ α′ β ′ γ ′ δ ′( gβ ν ′gδ δ ′g α ′ γ ′ ± 1 4 ɛαν α ′ β ′ɛ α δ ) 

δ ′ γ . (6.11) 

′ 

These equations are examples of the variety of tensors one is able to construct, 

just using the metric tensor and the totally anti-symmetric tensor and combine 

them into new tensors. 

6.3 Reducible representation 

In this Chapter we use the covariant spectral density tensor with purely contravariant 

components. The tensors and decompositions can clearly be altered 

into any mixed or covariant form using the metric, g µν . Having all indices in 

the same position (upper or lower) makes it simple to combine the elements in a 

correct manner, in contrast to a mixed representation where one must be careful 

in which order one has the indices. Furthermore it is only in the purely covariant 

or contravariant notation one can form symmetric and anti-symmetric parts 

directly by adding and subtracting the transposed tensor (see Section 2.4). One 

reason to use a mixed representation is that the metric reduces to the Kronecker 

delta, g µ ν = δ µ ν , which can be more convenient in some cases. 1 

The tensors in Equation (6.4) are, due to their construction, manifestly 

covariant and irreducible under SO(3, 1). With these tensor we can decompose 

τ αβγδ in terms of irreducible tensors and a rest tensor, which will be reducible. 

We have 

τ αβγδ = F αβ (F γδ ) ∗ = 

1 

16 Sgαγ g βδ + 1 4! S(p) ɛ αβγδ + 1 (1) T αγ g βδ + 1 (2) T µν g αγ ɛ βδµν + Ψ αβγδ . (6.12) 

4 8 

The rest tensor Ψ will be a proper rank four tensor since all other tensors are 

proper tensors, and it is formed by a linear combination of proper tensors. 

Since τ αβγδ is constructed from the field strength tensor it is antisymmetric 

in the first two indices and also in the last two, but is symmetric between the 

first two indices and the last two indices. By looking at the symmetries we can 

see how we can decompose Ψ further, using similar tensors but permuting the 

indices. 2 This process of performing a simple decomposition to get a reduced 

rest tensor, can be applied to a general rank four tensor. 

The covariant spectral density tensor is a basic and fundamental object 

considering polarization. Because of its construction the components of this 

tensor only contains one correlation component, and from it one can construct 

a variety of tensors consisting of different components. 

1 See Sections 2.1 and 2.4 for more details on mixed form. 

2 For further info and examples of this ”index gymnastics” see Misner, Thorne & Wheeler [7] 

(”the big black book”) or Schutz [21].

48 CHAPTER 6. REDUCIBLE REPRESENTATION OF τ αβγδ

Chapter 7 

Conclusions 

The purpose of this work was to derive covariant a representation for the polarization 

properties of an electromagnetic field. I.e. the parameters in the 

representation should have known transformation properties. The basic idea 

has been to make a decomposition of each of the correlation matrices into a 

scalar, a vector and a tensor, an irreducible tensor representation under SO(3). 

Performing this decomposition, one gets parameters with components very similar 

to the generalized polarization parameters, defined by Carozzi, Karlsson 

and Bergman [6] who used the generators of the SU(3) symmetry group (see 

Gell-Mann et. al. [17]). 

The representation of the electromagnetic field given by Storey and Lefeuvre 

[23] uses a six dimensional ”quasi-vector” which gives a 6×6 spectral density 

matrix, with 36 correlation parameters. The problem with this representation 

is that the six-vector is not a genuine vector (see Section 2.2) and that it does 

not give a geometrical interpretation for the field. 

The decomposition in Chapter 3 has some similar problems. We obtain 36 

parameters but they are not combined in such a way that one can use them to 

get a geometrical interpretation of how the field behaves. 

The new set of polarization parameters introduced and defined in Chapter 

4 does not exhibit these problems. These parameters represent well known geometrical 

quantities, covariant under the group SO(3), and they have separately 

been found in the literature, i.e. they have a known physical meaning. E.g., the 

Poynting vector and the total energy density of the field. Two of the parameters 

have not been found in the literature. These are the tensors T χ and T ψ . These 

parameters has so-far no physical interpretation but can be thought of like two 

cross-correlated Maxwell stress tensors. 

A set of Lorentz covariant tensors, i.e. covariant under the group SO(3, 1), 

are constructed in Chapter 5, the Y tensors. These tensors have the S, V, T 

parameters as components and the tensors are valid in any curved space-time. 

In Chapter 3 we introduce and define the covariant spectral density tensor, a 

manifestly Lorentz covariant rank four tensor, consisting of all quadratic correlation 

components. The Y tensors can be constructed from the covariant spectral 

density tensor, τ αβγδ (Chapters 3 and 6), using the algebra of general relativity. 

The covariant spectral density tensor is a more fundamental object than the 

Y tensors and for a further investigation of polarization τ αβγδ is a favourable 

object to consider. 

49

50 CHAPTER 7. CONCLUSIONS 

The new parameters defined in this work give a representation that consists 

of SO(3)-covariant, and in some cases relativistically covariant, parameters that 

give a full description of the polarization state of an electromagnetic wave. 

From the relativistic tensors in Chapter 5 we deduce how the S, V, T parameters 

transform under proper Lorentz transformations. 

In the calculations we have considered deterministic fields, which means that 

we have a field which is completely known. In measurements one considers bandlimited 

stochastical fields, i.e., one considers the correlation in a bandwidth, in 

contrast to the deterministic fields considered in this work. In this process of averaging 

over a bandwidth, The formulas for S, V, T parameters are not affected 

which means that the parameters are directly applicable to measurements. 

A direct continuation of this work would be to find a physical interpretation 

of T χ and T ψ , the previously unknown tensors, and to calculate the parameters 

for different waves with known dispersion relations. This would make the parameters 

useful in identification of wave modes in space plasma or other media 

supporting electromagnetic wave propagation. Also lacking is the irreducble 

representation of the covariant spectral density tensor and calculation of all 

non-zero components in this representation, which would be convenient for a 

deeper investigation of the gravitational effect on wave polarization. An interesting 

question for the future is if it is possible to detect gravitational waves by 

polarization effects in space plasma.

Appendix A 

Generalized polarization 

parameters and generators 

of SU(3) symmetry group 

Given the spectral density matrix for a vector field ⃗ F (using the notation x, y, z 

instead of 1,2,3) 

⎛ 

⎞ 

S d = F ⃗ ⊗ F ⃗ F x F † x ∗ F x Fy ∗ F x Fz 

∗ 

= ⎝F y Fx ∗ F y Fy ∗ F y Fz 

∗ ⎠ . 

F z Fx ∗ F z Fy ∗ F z Fz 

∗ 

(A.1) 

The spectral density matrix can be written as a linear combination, using the 

generators of the SU(3) symmetry group, λ i , and the three dimensional unit 

matrix, I 3 , as a basis. In the form given by Gell-Mann [17], these matrices are 

given by 

⎛ 

I 3 = ⎝ 1 0 0 ⎞ ⎛ 

0 1 0⎠ , λ 1 = ⎝ 0 1 0 ⎞ ⎛ 

1 0 0⎠ , λ 2 = ⎝ 0 −i 0 ⎞ 

i 0 0⎠ , 

0 0 1 0 0 0 0 0 0 

⎛ 

λ 3 = ⎝ 1 0 0 ⎞ ⎛ 

0 −1 0⎠ , λ 4 = ⎝ 0 0 1 ⎞ ⎛ 

0 0 0⎠ , λ 5 = ⎝ 0 0 −i ⎞ 

0 0 0 ⎠ , 

0 0 0 1 0 0 i 0 0 

(A.2) 

⎛ 

λ 6 = ⎝ 0 0 0 ⎞ ⎛ 

0 0 1⎠ , λ 7 = ⎝ 0 0 0 ⎞ ⎛ 

0 0 −i⎠ , λ 8 = √ 1 ⎝ 1 0 0 ⎞ 

0 1 0 ⎠ . 

0 1 0 0 i 0 

3 0 0 −2 

51

52 APPENDIX A. GENERALIZED POLARIZATION PARAMETERS 

Denoting the coefficients in this expansion by Λ i , we obtain 

S d = 1 3 I 3 + 1 8∑ 

Λ i λ i = 

2 

i=1 

⎛ 

1 

3 

⎜ 

Λ 0 + 1 2 Λ 3 + 1 

1 

⎝ 2 Λ 1 + i 2 Λ 2 

1 

2 Λ 4 + i 2 Λ 5 

2 √ 3 Λ 8 

1 

2 Λ 1 − i 2 Λ 2 

1 

3 Λ 0 − 1 2 Λ 3 + 1 

1 

2 Λ 6 − i 2 Λ 7 

2 √ 3 Λ 8 

1 

2 Λ 4 − i 2 Λ ⎞ 

5 

1 

2 Λ 6 − i 2 Λ ⎟ 

7 ⎠ , 

1 

3 Λ 0 − √ 1 

3 

Λ 8 

(A.3) 

where the Λ i ’s are 

Λ 0 = F 2 x + F 2 y + F 2 z , 

Λ 1 = F x F ∗ y + F y F ∗ x = 2Re[F x F ∗ y ], 

(A.4) 

(A.5) 

Λ 2 = i [ F x F ∗ y − F y F ∗ x 

] 

= −2Im[Fx F ∗ y ], (A.6) 

Λ 3 = F 2 x − F 2 y , 

Λ 4 = F x F ∗ z + F z F ∗ x = 2Re[F x F ∗ z ], 

(A.7) 

(A.8) 

Λ 5 = i [ F x F ∗ z − F z F ∗ x 

] 

= −2Im[Fx F ∗ z ], (A.9) 

Λ 6 = F y F ∗ z + F z F ∗ y = 2Re[F y F ∗ z ], 

(A.10) 

Λ 7 = i [ F y Fz ∗ − F z Fy ∗ ] 

= −2Im[Fy Fz ∗ ], (A.11) 

Λ 8 = √ 1 [ 

F 

2 

x + F 2 ] 2 

y − √3 Fz 2 . 

3 

(A.12) 

These Λ i ’s are the generalized polarization parameters.

Appendix B 

Parameters from Dirac 

gamma matrices 

In Chapter 3, the Dirac gamma matrices are used to obtain a decomposition of 

the electromagnetic field, which is represented by the field strength tensor 

⎛ 

⎞ 

0 −E 1 −E 2 −E 3 

F µν = ⎜E 1 0 −cB 3 cB 2 

⎟ 

⎝E 2 cB 3 0 −cB 1 

⎠ . 

E 3 −cB 2 cB 1 0 

(B.1) 

In Lorentz space ((ct, x, y, z) notation) the original four gamma matrices are 

given by 

γ 0 = 

( ) 

I2 0 

0 −I 2 

( ) 

γ i 0 σ 

i 

= 

−σ i , i = 1, 2, 3 , (B.2) 

0 

where σ i are the Pauli spin matrices and I 2 is the two dimensional unit matrix, 

Equation (3.3). These are the four matrices which appear in the relativistic 

treatment of a spin - 1 2 particle. 

The four gamma matrices are then expanded into a 16 dimensional basis 

defined as 

I 4 , γ 5 = iγ 0 γ 1 γ 2 γ 3 , γ µ , γ 5 γ µ , σ µν = i(γ µ γ ν − γ ν γ µ )/2, (B.3) 

where µ, ν = 0, 1, 2, 3. Explicitly, the original four gamma matrices are 

⎛ 

⎞ ⎛ 

⎞ 

1 0 0 0 

0 0 0 1 

γ 0 = ⎜0 1 0 0 

⎟ 

⎝0 0 −1 0 ⎠ , γ1 = ⎜ 0 0 1 0 

⎟ 

⎝ 0 −1 0 0⎠ , 

0 0 0 −1 

−1 0 0 0 

⎛ 

⎞ ⎛ 

⎞ 

0 0 0 −i 

0 0 1 0 

γ 2 = ⎜ 0 0 i 0 

⎟ 

⎝ 0 i 0 0 ⎠ , γ3 = ⎜ 0 0 0 −1 

⎟ 

⎝−1 0 0 0 ⎠ . 

−i 0 0 0 

0 1 0 0 

(B.4) 

53

54 APPENDIX B. PARAMETERS FROM DIRAC GAMMA MATRICES 

The four dimensional unit matrix and the γ 5 are given by 

⎛ ⎞ ⎛ ⎞ 

1 0 0 0 

0 0 1 0 

I 4 = ⎜0 1 0 0 

⎟ 

⎝0 0 1 0⎠ , γ5 = ⎜0 0 0 1 

⎟ 

⎝1 0 0 0⎠ . 

0 0 0 1 

0 1 0 0 

(B.5) 

And the ten constructed matrices, γ 5 γ µ and σ µν , are found to be 

⎛ 

⎞ 

⎛ 

⎞ 

0 0 −1 0 

0 −1 0 0 

γ 6 = γ 5 γ 0 = ⎜0 0 0 −1 

⎟ 

⎝1 0 0 0 ⎠ , γ7 = γ 5 γ 1 = ⎜−1 0 0 0 

⎟ 

⎝ 0 0 0 1⎠ , 

0 1 0 0 

0 0 1 0 

⎛ 

⎞ 

⎛ 

⎞ 

0 i 0 0 

−1 0 0 0 

γ 8 = γ 5 γ 2 = ⎜−i 0 0 0 

⎟ 

⎝ 0 0 0 −i⎠ , γ9 = γ 5 γ 3 = ⎜ 0 1 0 0 

⎟ 

⎝ 0 0 1 0 ⎠ , 

0 0 i 0 

0 0 0 −1 

⎛ ⎞ 

⎛ 

⎞ 

0 0 0 i 

0 0 0 1 

γ 10 = σ 01 = ⎜0 0 i 0 

⎟ 

⎝0 i 0 0⎠ , γ11 = σ 02 = ⎜ 0 0 −1 0 

⎟ 

⎝ 0 1 0 0⎠ , (B.6) 

i 0 0 0 

−1 0 0 0 

⎛ 

⎞ 

⎛ 

⎞ 

0 0 i 0 

1 0 0 0 

γ 12 = σ 03 = ⎜0 0 0 −i 

⎟ 

⎝i 0 0 0 ⎠ , γ13 = σ 12 = ⎜0 −1 0 0 

⎟ 

⎝0 0 1 0 ⎠ , 

0 −i 0 0 

0 0 0 −1 

⎛ 

⎞ 

⎛ ⎞ 

0 i 0 0 

0 1 0 0 

γ 14 = σ 13 = ⎜−i 0 0 0 

⎟ 

⎝ 0 0 0 i⎠ , γ15 = σ 23 = ⎜1 0 0 0 

⎟ 

⎝0 0 0 1⎠ . 

0 0 −i 0 

0 0 1 0 

The 36 polarization parameters are obtained by the contractions 

τ αβµν γ n αβγ m µν = F αβ (F µν ) ∗ γ n αβγ m µν, 

(B.7) 

and they are found to be: 

1 

4 F αβ (F µν ) ∗ γ 1 γ 1 = E 3 E3 ∗ + E 3 cB3 ∗ + cB 3 E3 ∗ + cB 3 cB3, ∗ (B.8) 

1 


1 

4 F αβ (F µν ) ∗ γ 6 γ 6 = E 2 E2 ∗ − E 2 cB2 ∗ − cB 2 E2 ∗ + cB 2 cB2, ∗ (B.10) 

1 

4 F αβ (F µν ) ∗ γ 8 γ 8 = −E 1 E1 ∗ + E 1 cB1 ∗ + 4cB 1 E1 ∗ − cB 1 cB1, ∗ (B.11) 

1 

4 F αβ (F µν ) ∗ γ 11 γ 11 = E 3 E3 ∗ − E 3 cB3 ∗ − cB 3 E3 ∗ + cB 3 cB3, ∗ (B.12) 

1 

4 F αβ (F µν ) ∗ γ 14 γ 14 = −E 1 E1 ∗ − E 1 cB1 ∗ − cB 1 E1 ∗ − cB 1 cB1, ∗ (B.13)

1 


1 

4 F αβ (F µν ) ∗ γ 1 γ 6 = −E 3 E2 ∗ − E 3 cB2 ∗ + cB 3 E2 ∗ + cB 3 cB2, ∗ (B.15) 

1 

4 F αβ (F µν ) ∗ γ 1 γ 8 = i(E 3 E1 ∗ − E 3 cB1 ∗ + cB 3 E3 ∗ − cB 3 cB1), ∗ (B.16) 

1 

4 F αβ (F µν ) ∗ γ 1 γ 11 = E 3 E3 ∗ − E 3 cB3 ∗ + cB 3 E3 ∗ − cB 3 cB3, ∗ (B.17) 

1 

4 F αβ (F µν ) ∗ γ 1 γ 14 = i(E 3 E3 ∗ + E 3 cB3 ∗ + cB 3 E3 ∗ + cB 3 cB3), ∗ (B.18) 

1 

4 F αβ (F µν ) ∗ γ 3 γ 6 = −E 2 E2 ∗ + E 2 cB2 ∗ − cB 2 E2 ∗ + cB 2 cB2, ∗ (B.19) 

1 


1 

4 F αβ (F µν ) ∗ γ 3 γ 11 = E 2 E3 ∗ − E 2 cB3 ∗ + cB 2 E3 ∗ − cB 2 cB3, ∗ (B.21) 

1 


1 

4 F αβ (F µν ) ∗ γ 6 γ 8 = i(−E 2 E1 ∗ + E 2 cB1 ∗ + cB 2 E1 ∗ − cB 2 cB1), ∗ (B.23) 

1 

4 F αβ (F µν ) ∗ γ 6 γ 11 = −E 2 E3 ∗ + E 2 cB3 ∗ + cB 2 E3 ∗ − cB 2 cB3, ∗ (B.24) 

1 

4 F αβ (F µν ) ∗ γ 6 γ 14 = i(−E 2 E1 ∗ − E 2 cB1 ∗ + cB 2 E1 ∗ + cB 2 cB1), ∗ (B.25) 

1 

4 F αβ (F µν ) ∗ γ 8 γ 11 = i(E 1 E3 ∗ − E 1 cB3 ∗ − cB 1 E3 ∗ + cB 1 cB3), ∗ (B.26) 

1 

4 F αβ (F µν ) ∗ γ 8 γ 14 = −E 3 E3 ∗ + E 3 cB3 ∗ + cB 3 E3 ∗ + cB 3 cB3, ∗ (B.27) 

1 

4 F αβ (F µν ) ∗ γ 11 γ 14 = iE 3 E1 ∗ + E 3 cB1 ∗ − cB 3 E1 ∗ − cB 3 cB1, ∗ (B.28) 

1 


1 

4 F αβ (F µν ) ∗ γ 14 γ 8 = −E 1 E1 ∗ + E 1 cB1 ∗ − cB 1 E1 ∗ + cB 1 cB1, ∗ (B.30) 

1 

4 F αβ (F µν ) ∗ γ 14 γ 6 = i(−E 1 E2 ∗ + E 1 cB2 ∗ − cB 1 E2 ∗ + cB 1 cB2), ∗ (B.31) 

1 


1 


1 

4 F αβ (F µν ) ∗ γ 11 γ 8 = i(E 3 E1 ∗ − E 3 cB1 ∗ − cB 3 E1 ∗ + cB 3 cB1), ∗ (B.34) 

1 

4 F αβ (F µν ) ∗ γ 11 γ 6 = −E 3 E2 ∗ + E 3 cB2 ∗ + cB 3 E2 ∗ − cB 3 cB2, ∗ (B.35) 

55

56 APPENDIX B. PARAMETERS FROM DIRAC GAMMA MATRICES 

1 

4 F αβ (F µν ) ∗ γ 11 γ 3 = E 3 E2 ∗ + E 3 cB2 ∗ − cB 3 E2 ∗ − cB 3 cB2, ∗ (B.36) 

1 

4 F αβ (F µν ) ∗ γ 11 γ 1 = E 3 E3 ∗ + E 3 cB3 ∗ − cB 3 E3 ∗ − cB 3 cB3, ∗ (B.37) 

1 

4 F αβ (F µν ) ∗ γ 8 γ 6 = i(−E 1 E2 ∗ + E 1 cB2 ∗ + cB 1 E2 ∗ − cB 1 cB2), ∗ (B.38) 

1 

4 F αβ (F µν ) ∗ γ 8 γ 3 = i(E 1 E2 ∗ + E 1 cB2 ∗ − cB 1 E2 ∗ − cB 1 cB2), ∗ (B.39) 

1 

4 F αβ (F µν ) ∗ γ 8 γ 1 = i(E 1 E3 ∗ + E 1 cB3 ∗ − cB 1 E3 ∗ − cB 1 cB3), ∗ (B.40) 

1 


1 


1 

4 F αβ (F µν ) ∗ γ 3 γ 1 = E 2 E3 ∗ + E 2 cB3 ∗ + cB 2 E3 ∗ + cB 2 cB3. ∗ (B.43)

Appendix C 

⃗E, ⃗ B component form of the 

S, V, T parameters 

In Chapter 4 we defined and labelled the S, V, T parameters in the following 

way 

Σ : ⃗ E ⊗ ⃗ E † + ⃗ B ⊗ ⃗ B † = 1 3 δ ijS Σ + iɛ ijk V Σ 

k + T Σ ij , (C.1) 

∆ : ⃗ E ⊗ ⃗ E † − ⃗ B ⊗ ⃗ B † = 1 3 δ ijS ∆ + iɛ ijk V ∆ 

k + T ∆ ij , (C.2) 

χ : ⃗ E ⊗ ⃗ B † + ⃗ B ⊗ ⃗ E † = 1 3 δ ijS χ + iɛ ijk V χ 

k + T χ ij , (C.3) 

ψ : ⃗ E ⊗ ⃗ B † − ⃗ B ⊗ ⃗ E † = 1 3 δ ijS ψ + iɛ ijk V ψ 

k + T ψ ij . (C.4) 

Here we list the parameters in component form. 

C.1 Σ 

S Σ = E 2 + c 2 B 2 = E 1 E ∗ 1 + E 2 E ∗ 2 + E 3 E ∗ 3 + c 2 B 1 B ∗ 1 + c 2 B 2 B ∗ 2 + c 2 B 3 B ∗ 3, (C.5) 

( 

⃗V Σ = − i 2 

E 2 E3 ∗ −E 3 E2 ∗ +c 2 B 2 B3 ∗ −c 2 B 3 B2, ∗ E 3 E1 ∗ −E 1 E3 ∗ +c 2 B 3 B1 ∗ −c 2 B 1 B3 

∗ 

) 

, E 1 E2 ∗ − E 2 E1 ∗ + c 2 B 1 B2 ∗ − c 2 B 2 B1 

∗ , (C.6) 

Tij Σ = 

[E1+c 2 2 B1− 2 1 3 SΣ , 1 2 (E 1E2+E ∗ 2 E1+c ∗ 2 B 1 B2+c ∗ 2 B 2 B1), ∗ 1 2 (E 1E3+E ∗ 3 E1+c ∗ 2 B 1 B3+c ∗ 2 B 3 B1)] 

∗ 

[ 1 2 (E 1E2+E ∗ 2 E1+c ∗ 2 B 1 B2+c ∗ 2 B 2 B1), ∗ E2+c 2 2 B2− 2 1 3 SΣ , 1 2 (E 2E † 3 +E 3E † 2 +c2 B 2 B3+c ∗ 2 B 3 B2)] 

∗ 

[ 1 2 (E 1E3+E ∗ 3 E1+c ∗ 2 B 1 B3+c ∗ 2 B 3 B1), ∗ 1 2 (E 2E3+E ∗ 3 E2+c ∗ 2 B 2 B3+c ∗ 2 B 3 B2), ∗ E3+B 2 3− 2 1 3 SΣ ]. 

(C.7) 

57

58 APPENDIX C. ⃗ E, ⃗ B COMPONENT FORM OF S, V, T PARAMETERS 

C.2 ∆ 

S ∆ = E 2 −c 2 B 2 = E 1 E ∗ 1 +E 2 E ∗ 2 +E 3 E ∗ 3 −c 2 B 1 B ∗ 1 −c 2 B 2 B ∗ 2 −c 2 B 3 B ∗ 3, (C.8) 

( 

⃗V ∆ = − i 2 

E 2 E3 ∗ −E 3 E2 ∗ −c 2 B 2 B3+c ∗ 2 B 3 B2, ∗ E 3 E1 ∗ −E 1 E3 ∗ −c 2 B 3 B1+c ∗ 2 B 1 B3 

∗ 

) 

, E 1 E2 ∗ − E 2 E1 ∗ − c 2 B 1 B2 ∗ + c 2 B 2 B1 

∗ , (C.9) 

Tij ∆ = 

[E1−c 2 2 B1− 2 1 3 S∆ , 1 2 (E 1E2+E ∗ 2 E1−c ∗ 2 B 1 B2−c ∗ 2 B 2 B1), ∗ 1 2 (E 1E3+E ∗ 3 E1−c ∗ 2 B 1 B3−c ∗ 2 B 3 B1)] 

∗ 

[ 1 2 (E 1E2+E ∗ 2 E1−c ∗ 2 B 1 B2−c ∗ 2 B 2 B1), ∗ E2−c 2 2 B2− 2 1 3 S∆ , 1 2 (E 2E3+E ∗ 3 E2−c ∗ 2 B 2 B3−c ∗ 2 B 3 B2)] 

∗ 

[ 1 2 (E 1E3+E ∗ 3 E1−c ∗ 2 B 1 B3−c ∗ 2 B 3 B1), ∗ 1 2 (E 2E3+E ∗ 3 E2−c ∗ 2 B 2 B3−c ∗ 2 B 3 B2), ∗ E3−c 2 2 B3− 2 1 3 S∆ ]. 

(C.10) 

C.3 χ 

S χ = c ⃗ E· ⃗B † +c ⃗ B· ⃗E † = c ( E 1 B ∗ 1 +E 2 B ∗ 2 +E 3 B ∗ 3 +B 1 E ∗ 1 +B 2 E ∗ 2 +B 3 E ∗ 3) 

, (C.11) 

( 

⃗V χ = −c i 2 

E 2 B3 ∗ − E 3 B2 ∗ + B 2 E3 ∗ − B 3 E2, ∗ E 3 B1 ∗ − E 1 B3 ∗ + B 3 E1 ∗ − B 1 E3 

∗ 

) 

, E 1 B2 ∗ − E 2 B1 ∗ + B 1 E2 ∗ − B 2 E1 

∗ , (C.12) 

T χ ij = 

[cE 1 B1+cB ∗ 1 E1− ∗ 1 3 Sχ , c 2 (E 1B2+E ∗ 2 B1+B ∗ 1 E2+B ∗ 2 E1), ∗ c 2 (E 1B3+E ∗ 3 B1+B ∗ 1 E3+B ∗ 3 E1)] 

∗ 

[ c 2 (E 1B2+E ∗ 2 B1+B ∗ 1 E2+B ∗ 2 E1), ∗ cE 2 B2+cB ∗ 2 E2− ∗ 1 3 Sχ , c 2 (E 2B3+E ∗ 3 B2+B ∗ 2 E3+B ∗ 3 E2)] 

∗ 

[ c 2 (E 1B3+E ∗ 3 B1+B ∗ 1 E3+B ∗ 3 E1), ∗ c 2 (E 2B3+E ∗ 3 B2+B ∗ 2 E3+B ∗ 3 E2), ∗ cE 3 B3+cB ∗ 3 E3− ∗ 1 3 Sχ ]. 

(C.13) 

C.4 ψ 

S ψ = c ⃗ E· ⃗B † −c ⃗ B· ⃗E † = c ( E 1 B ∗ 1 +E 2 B ∗ 2 +E 3 B ∗ 3 −B 1 E ∗ 1 −B 2 E ∗ 2 −B 3 E ∗ 3) 

, (C.14) 

( 

⃗V ψ = −c i 2 

E 2 B3 ∗ − E 3 B2 ∗ − B 2 E3 ∗ + B 3 E2, ∗ E 3 B1 ∗ − E 1 B3 ∗ − B 3 E1 ∗ + B 1 E3 

∗ 

) 

, E 1 B2 ∗ − E 2 B1 ∗ − B 1 E2 ∗ + B 2 E1 

∗ , (C.15)

C.4. ψ 59 

T ψ ij = 

[cE 1 B1−cB ∗ 1 E ∗ − 1 3 Sψ , c 2 (E 1B2+E ∗ 2 B1−B ∗ 1 E2−B ∗ 2 E1), ∗ c 2 (E 1B3+E ∗ 3 B1−B ∗ 1 E3−B ∗ 3 E1)] 

∗ 

[ c 2 (E 1B2+E ∗ 2 B1−B ∗ 1 E2−B ∗ 2 E1), ∗ cE 2 B2−cB ∗ 2 E2− ∗ 1 3 Sψ , c 2 (E 2B3+E ∗ 3 B2−B ∗ 2 E3−B ∗ 3 E2)] 

∗ 

[ c 2 (E 1B3+E ∗ 3 B1−B ∗ 1 E3−B ∗ 3 E1), ∗ c 2 (E 2B3+E ∗ 3 B2−B ∗ 2 E3−B ∗ 3 E2), ∗ cE 3 B3−cB ∗ 3 E3− ∗ 1 3 Sψ ]. 

(C.16)

60 APPENDIX C. ⃗ E, ⃗ B COMPONENT FORM OF S, V, T PARAMETERS

Appendix D 

Covariant polarization 

tensors in E,B-component 

form 

In Chapter 5 we saw that the S, V, T parameters can be found in Lorentz covariant 

tensors. These tensors were constructed according to 

where 

is the metric, 

and 

1 X µν = F αµ (F αδ ) ∗ g δν , (D.1) 




⎛ 

⎞ 

1 0 0 0 

g µν = g µν = ⎜0 −1 0 0 

⎟ 

⎝0 0 −1 0 ⎠ , 

0 0 0 −1 

⎛ 

⎞ 

0 −E 1 −E 2 −E3 

F µν = ⎜E 1 0 −cB 3 cB 2 

⎟ 

⎝E 2 cB 3 0 −cB 1 

⎠ 

E 3 −cB 2 cB 1 0 

⎛ 

⎞ 

0 E 1 E 2 E 3 

F µν = ⎜−E 1 0 −cB 3 cB 2 

⎟ 

⎝−E 2 cB 3 0 −cB 1 

⎠ 

−E 3 −cB 2 cB 1 0 

(D.5) 

(D.6) 

(D.7) 

are the contravariant and covariant field strength tensors, respectively, and 

⎛ 

⎞ 

0 −cB 1 −cB 2 −cB 3 

F µν = ⎜cB 1 0 E 3 −E 2 

⎟ 

⎝cB 2 −E 3 0 E 1 

⎠ 

(D.8) 

cB 3 E 2 −E 1 0 

61

62 APPENDIX D. COVARIANT POLARIZATION TENSORS 

and 

⎛ 

⎞ 

0 cB 1 cB 2 cB 3 

F µν = ⎜−cB 1 0 E 3 −E 2 

⎟ 

⎝−cB 2 −E 3 0 E 1 

⎠ 

−cB 3 E 2 −E 1 0 

are the corresponding dual tensors. 

(D.9) 

D.1 The X tensors 

In E, B-component form the n X µν tensors are given by 

1 X µν = 

[−E 1 E ∗ 1 − E 2 E ∗ 2 − E 3 E ∗ 3, −E 2 cB ∗ 3 + E 3 cB ∗ 2, E 1 cB ∗ 3 − E 3 cB ∗ 1, −E 1 cB ∗ 2 + E 2 cB ∗ 1] 

[−cB 3 E ∗ 2 + cB 2 E ∗ 3, E 1 E ∗ 1 − c 2 B 3 B ∗ 3 − c 2 B 2 B ∗ 2, E 1 E ∗ 2 + c 2 B 2 B ∗ 1, E 1 E ∗ 3 + c 2 B 3 B ∗ 1] 

[cB 3 E ∗ 1 − cB 1 E ∗ 3, E 2 E ∗ 1 + c 2 B 1 B ∗ 2, E 2 E ∗ 2 − c 2 B 3 B ∗ 3 − c 2 B 1 B ∗ 1, E 2 E ∗ 3 + c 2 B 3 B ∗ 2] 

[−cB 2 E ∗ 1 +cB 1 E ∗ 2, E 3 E ∗ 1 +c 2 B 1 B ∗ 3, E 3 E ∗ 2 +c 2 B 2 B ∗ 3, E 3 E ∗ 3 −c 2 B 2 B ∗ 2 −c 2 B 1 B ∗ 1] 

(D.10) 

2 X µν = 

[−c 2 B 1 B ∗ 1−c 2 B 2 B ∗ 2−c 2 B 3 B ∗ 3, −cB 3 E ∗ 2+cB 2 E ∗ 3, cB 3 E ∗ 1−cB 1 E ∗ 3, −cB 2 E ∗ 1+cB 1 E ∗ 2] 

[−E 2 cB ∗ 3 + E 3 cB ∗ 2, c 2 B 1 B ∗ 1 − E 3 E ∗ 3 − E 2 E ∗ 2, E 2 E ∗ 1 + c 2 B 1 B ∗ 2, E 3 E ∗ 1 + c 2 B 1 B ∗ 3] 

[E 1 cB ∗ 3 − E 3 cB ∗ 1, E 1 E ∗ 2 + c 2 B 2 B ∗ 1, c 2 B 2 B ∗ 2 − E 3 E ∗ 3 − E 1 E ∗ 1, E 3 E ∗ 2 + c 2 B 2 B ∗ 3] 

[−E 1 cB ∗ 2 + E 2 cB ∗ 1, E 1 E ∗ 3 + c 2 B 3 B ∗ 1, E 2 E ∗ 3 + c 2 B 3 B ∗ 2, c 2 B 3 B ∗ 3 − E 2 E ∗ 2 − E 1 E ∗ 1] 

(D.11) 

3 X µν = 

[−cB 1 E ∗ 1−cB 2 E ∗ 2−cB 3 E ∗ 3, −c 2 B 2 B ∗ 3+c 2 B 3 B ∗ 2, c 2 B 1 B ∗ 3−c 2 B 3 B ∗ 1, −c 2 B 1 B ∗ 2+c 2 B 2 B ∗ 1] 

[E 3 E ∗ 2 − E 2 E ∗ 3, cB 1 E ∗ 1 + E 3 cB ∗ 3 + E 2 cB ∗ 2, cB 1 E ∗ 2 − E 2 cB ∗ 1, cB 1 E ∗ 3 − E 3 cB ∗ 1] 

[−E 3 E ∗ 1 + E 1 E ∗ 3, cB 2 E ∗ 1 − E 1 cB ∗ 2, cB 2 E ∗ 2 + E 3 cB ∗ 3 + E 1 cB ∗ 1, cB 2 E ∗ 3 − E 3 cB ∗ 2] 

[E 2 E ∗ 1 − E 1 E ∗ 2, cB 3 E ∗ 1 − E 1 cB ∗ 3, cB 3 E ∗ 2 − E 2 cB ∗ 3, cB 3 E ∗ 3 + E 2 cB ∗ 2 + E 1 cB ∗ 1] 

(D.12) 

4 X µν = 

[−E 1 cB ∗ 1 − E 2 cB ∗ 2 − E 3 cB ∗ 3, −E 3 E ∗ 2 + E 2 E ∗ 3, E 3 E ∗ 1 − E 1 E ∗ 3, −E 2 E ∗ 1 + E 1 E ∗ 2] 

[c 2 B 2 B ∗ 3 −c 2 B 3 B ∗ 2, E 1 cB ∗ 1 +cB 3 E ∗ 3 +cB 2 E ∗ 2, −cB 2 E ∗ 1 +E 1 cB ∗ 2, −cB 3 E ∗ 1 +E 1 cB ∗ 3] 

[−c 2 B 1 B ∗ 3+c 2 B 3 B ∗ 1, −cB 1 E ∗ 2+E 2 cB ∗ 1, cB 3 E ∗ 3+E 2 cB ∗ 2+cB 1 E ∗ 1, −cB 3 E ∗ 2+E 2 cB ∗ 3] 

[c 2 B 1 B ∗ 2−c 2 B 2 B ∗ 1, −cB 1 E ∗ 3+E 3 cB ∗ 1, −cB 2 E ∗ 3+E 3 cB ∗ 2, cB 2 E ∗ 2+E 3 cB ∗ 3+cB 1 E ∗ 1] 

(D.13)

D.2. THE Y TENSORS 63 

D.2 The Y tensors 

The Y tensors, constructed by the combinations 

Σ Y µν = 1 2 

( 1 

X µν + 2 X µν ) 

, 

χ Y µν = 1 ( 3 

X µν + 4 X µν ) 

, 

2 

∆ Y µν = 1 ( 1 

X µν − 2 X µν ) 

(D.14) 

, 

2 

ψ Y µν = 1 ( 3 

X µν − 4 X µν ) 

, 

2 

and are then found to be 

Σ Y µν = 1 2 

[−E 1 E ∗ 1−E 2 E ∗ 2−E 3 E ∗ 3−cB 1 cB ∗ 1−cB 2 cB ∗ 2−cB 3 cB ∗ 3, −cB 3 E ∗ 2+cB 2 E ∗ 3−E 2 cB ∗ 3+E 3 cB ∗ 2, 

E 1 cB ∗ 3 − E 3 cB ∗ 1 + cB 3 E ∗ 1 − cB 1 E ∗ 3, −E 1 cB ∗ 2 + E 2 cB ∗ 1 − cB 2 E ∗ 1 + cB 1 E ∗ 2] 

[−cB 3 E ∗ 2+cB 2 E ∗ 3−E 2 cB ∗ 3+E 3 cB ∗ 2, E 1 E ∗ 1−cB 3 cB ∗ 3−cB 2 cB ∗ 2+cB 1 cB ∗ 1−E 3 E ∗ 3−E 2 E ∗ 2, 

E 2 E ∗ 1 + cB 1 cB ∗ 2 + E 1 E ∗ 2 + cB 2 cB ∗ 1, E 3 E ∗ 1 + cB 1 cB ∗ 3 + E 1 E ∗ 3 + cB 3 cB ∗ 1] 

[E 1 cB ∗ 3 − E 3 cB ∗ 1 + cB 3 E ∗ 1 − cB 1 E ∗ 3, E 2 E ∗ 1 + cB 1 cB ∗ 2 + E 1 E ∗ 2 + cB 2 cB ∗ 1, 

E 2 E ∗ 2−cB 3 cB ∗ 3−cB 1 cB ∗ 1+cB 2 cB ∗ 2−E 3 E ∗ 3−E 1 EC 1 , E 3 E ∗ 2+cB 2 cB ∗ 3+E 2 E ∗ 3+cB 3 cB ∗ 2] 

[−E 1 cB ∗ 2 + E 2 cB ∗ 1 − cB 2 E ∗ 1 + cB 1 E ∗ 2, E 3 E ∗ 1 + cB 1 cB ∗ 3 + E 1 E ∗ 3 + cB 3 cB ∗ 1, 

E 3 E ∗ 2+cB 2 cB ∗ 3+E 2 E ∗ 3+cB 3 cB ∗ 2, E 3 E ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+cB 3 cB ∗ 3−E 2 E ∗ 2−E 1 E ∗ 1] 

(D.15) 

χ Y µν = 1 2 

[−cB 1 E ∗ 1−cB 2 E ∗ 2−cB 3 E ∗ 3−E 1 cB ∗ 1−E 2 cB ∗ 2−E 3 cB ∗ 3, −cB 2 cB ∗ 3+cB 3 cB ∗ 2−E 3 E ∗ 2+E 2 E ∗ 3, 

cB 1 cB ∗ 3 − cB 3 cB ∗ 1 + E 3 E ∗ 1 − E 1 E ∗ 3, −cB 1 cB ∗ 2 + cB 2 cB ∗ 1 − E 2 E ∗ 1 + E 1 E ∗ 2] 

[E 3 E ∗ 2−E 2 E ∗ 3+cB 2 cB ∗ 3−cB 3 cB ∗ 2, cB 1 E ∗ 1+E 3 cB ∗ 3+E 2 cB ∗ 2+E 1 cB ∗ 1+cB 3 E ∗ 3+cB 2 E ∗ 2, 


[−E 3 E ∗ 1 + E 1 E ∗ 3 − cB 1 cB ∗ 3 + cB 3 cB ∗ 1, cB 2 E ∗ 1 − E 1 cB ∗ 2 − cB 1 E ∗ 2 + E 2 cB ∗ 1, 

cB 1 E ∗ 1+E 3 cB ∗ 3+E 2 cB ∗ 2+E 1 cB ∗ 1+cB 3 E ∗ 3+cB 2 E ∗ 2, cB 2 E ∗ 3−E 3 cB ∗ 2−cB 3 E ∗ 2+E 2 cB ∗ 3] 


cB 3 E ∗ 2−E 2 cB ∗ 3−cB 2 E ∗ 3+E 3 cB ∗ 2, cB 1 E ∗ 1+E 3 cB ∗ 3+E 2 cB ∗ 2+E 1 cB ∗ 1+cB 3 E ∗ 3+cB 2 E ∗ 2] 

(D.16)

64 APPENDIX D. COVARIANT POLARIZATION TENSORS 

∆ Y µν = 1 2 

[−E 1 E ∗ 1−E 2 E ∗ 2−E 3 E ∗ 3+cB 1 cB ∗ 1+cB 2 cB ∗ 2+cB 3 cB ∗ 3, cB 3 E ∗ 2−E 2 cB ∗ 3−cB 2 E ∗ 3+E 3 cB ∗ 2, 


[cB 2 E ∗ 3−E 3 cB ∗ 2−cB 3 E ∗ 2+E 2 cB ∗ 3, E 1 E ∗ 1−cB 3 cB ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+E 3 E ∗ 3+E 2 E ∗ 2, 

− cB 1 cB ∗ 2 + cB 2 cB ∗ 1 − E 2 E ∗ 1 + E 1 E ∗ 2, −E 3 E ∗ 1 + E 1 E ∗ 3 − cB 1 cB ∗ 3 + cB 3 cB ∗ 1] 

[cB 3 E ∗ 1 − E 1 cB ∗ 3 − cB 1 E ∗ 3 + E 3 cB ∗ 1, E 2 E ∗ 1 − E 1 E ∗ 2 + cB 1 cB ∗ 2 − cB 2 cB ∗ 1, 

E 1 E ∗ 1−cB 3 cB ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+E 3 E ∗ 3+E 2 E ∗ 2, −cB 2 cB ∗ 3+cB 3 cB ∗ 2−E 3 E ∗ 2+E 2 E ∗ 3] 


E 3 E ∗ 2−E 2 E ∗ 3+cB 2 cB ∗ 3−cB 3 cB ∗ 2, E 1 E ∗ 1−cB 3 cB ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+E 3 E ∗ 3+E 2 E ∗ 2] 

(D.17) 

ψ Y µν = 1 2 

[−cB 1 E ∗ 1−cB 2 E ∗ 2−cB 3 E ∗ 3+E 1 cB ∗ 1+E 2 cB ∗ 2+E 3 cB ∗ 3, E 3 E ∗ 2−E 2 E ∗ 3−cB 2 cB ∗ 3+cB 3 cB ∗ 2, 

cB 1 cB ∗ 3 − cB 3 cB ∗ 1 − E 3 E ∗ 1 + E 1 E ∗ 3, −cB 1 cB ∗ 2 + cB 2 cB ∗ 1 + E 2 E ∗ 1 − E 1 E ∗ 2] 

[E 3 E ∗ 2−E 2 E ∗ 3−cB 2 cB ∗ 3+cB 3 cB ∗ 2, cB 1 E ∗ 1+E 3 cB ∗ 3+E 2 cB ∗ 2−E 1 cB ∗ 1−cB 3 E ∗ 3−cB 2 E ∗ 2, 

cB 2 E ∗ 1 − E 1 cB ∗ 2 + cB 1 E ∗ 2 − E 2 cB ∗ 1, cB 3 E ∗ 1 − E 1 cB ∗ 3 + cB 1 E ∗ 3 − E 3 cB ∗ 1] 

[cB 1 cB ∗ 3 − cB 3 cB ∗ 1 − E 3 E ∗ 1 + E 1 E ∗ 3, cB 2 E ∗ 1 − E 1 cB ∗ 2 + cB 1 E ∗ 2 − E 2 cB ∗ 1, 

cB 2 E ∗ 2+E 3 cB ∗ 3+E 1 cB ∗ 1−cB 3 E ∗ 3−E 2 cB ∗ 2−cB 1 E ∗ 1, cB 3 E ∗ 2−E 2 cB ∗ 3+cB 2 E ∗ 3−E 3 cB ∗ 2] 

[−cB 1 cB ∗ 2 + cB 2 cB ∗ 1 + E 2 E ∗ 1 − E 1 E ∗ 2, cB 3 E ∗ 1 − E 1 cB ∗ 3 + cB 1 E ∗ 3 − E 3 cB ∗ 1, 

cB 3 E ∗ 2−E 2 cB ∗ 3+cB 2 E ∗ 3−E 3 cB ∗ 2, cB 3 E ∗ 3+E 2 cB ∗ 2+E 1 cB ∗ 1−cB 2 E ∗ 2−E 3 cB ∗ 3−cB 1 E ∗ 1] 

(D.18) 

D.3 The K and K tensors 

The K and K tensors are formed by removing the trace from ∆ Y µν and χ Y µν , 

respectively, and are given by 

K µν = 1 2 

[0, cB 3 E ∗ 2 − E 2 cB ∗ 3 − cB 2 E ∗ 3 + E 3 cB ∗ 2, cB 1 E ∗ 3 − E 3 cB ∗ 1 − cB 3 E ∗ 1 + E 1 cB ∗ 3, 

cB 2 E ∗ 1 − E 1 cB ∗ 2 − cB 1 E ∗ 2 + E 2 cB ∗ 1] 

[cB 2 E ∗ 3 − E 3 cB ∗ 2 − cB 3 E ∗ 2 + E 2 cB ∗ 3, 0, −cB 1 cB ∗ 2 + cB 2 cB ∗ 1 − E 2 E ∗ 1 + E 1 E ∗ 2, 

− E 3 E ∗ 1 + E 1 E ∗ 3 − cB 1 cB ∗ 3 + cB 3 cB ∗ 1] 

[cB 3 E ∗ 1 − E 1 cB ∗ 3 − cB 1 E ∗ 3 + E 3 cB ∗ 1, E 2 E ∗ 1 − E 1 E ∗ 2 + cB 1 cB ∗ 2 − cB 2 cB ∗ 1, 0, 

− cB 2 cB ∗ 3 + cB 3 cB ∗ 2 − E 3 E ∗ 2 + E 2 E ∗ 3] 


E 3 E ∗ 2 − E 2 E ∗ 3 + cB 2 cB ∗ 3 − cB 3 cB ∗ 2, 0] 

(D.19)

D.3. THE K AND K TENSORS 65 

and 

K µν = 1 2 

[0, −cB 2 cB ∗ 3 + cB 3 cB ∗ 2 − E 3 E ∗ 2 + E 2 E ∗ 3, cB 1 cB ∗ 3 − cB 3 cB ∗ 1 + E 3 E ∗ 1 − E 1 E ∗ 3, 

− cB 1 cB ∗ 2 + cB 2 cB ∗ 1 − E 2 E ∗ 1 + E 1 E ∗ 2] 

[E 3 E ∗ 2 − E 2 E ∗ 3 + cB 2 cB ∗ 3 − cB 3 cB ∗ 2, 0, cB 1 E ∗ 2 − E 2 cB ∗ 1 − cB 2 E ∗ 1 + E 1 cB ∗ 2, 

cB 1 E ∗ 3 − E 3 cB ∗ 1 − cB 3 E ∗ 1 + E 1 cB ∗ 3] 

[−E 3 E ∗ 1 + E 1 E ∗ 3 − cB 1 cB ∗ 3 + cB 3 cB ∗ 1, cB 2 E ∗ 1 − E 1 cB ∗ 2 − cB 1 E ∗ 2 + E 2 cB ∗ 1, 0, 

cB 2 E ∗ 3 − E 3 cB ∗ 2 − cB 3 E ∗ 2 + E 2 cB ∗ 3] 


respectively. 

cB 3 E ∗ 2 − E 2 cB ∗ 3 − cB 2 E ∗ 3 + E 3 cB ∗ 2, 0], 

(D.20)

66 APPENDIX D. COVARIANT POLARIZATION TENSORS

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