A. Quaternion algebra

Quaternions are not part of the standard mathematical toolkit of physicists and engineers. They do find applications in the practical calculation of rotations, e.g. in computer animation, but the relevant texts concentrate on that particular application (e.g. [Kuipers 1998]; see also many entries on the World-Wide Web). In scientific textbooks (e.g. [Cornbleet 1976]; [Korn & Korn 1961]) one may find brief references to them but hardly anything more. Hestenes (1986) mentions them as a variant of the concepts of bivectors and spinors that have a more central place in his text, and most of the concepts that we need are to be found there in one form or another; however, his work is not very accessible as a quick reference. For lack of better, I give here a brief summary of quaternion theory in the form in which I use it.

In the algebraic view, the vector part of a quaternion is a generalisation of the imaginary part of a complex number; correspondingly, the scalar and vector components are real and the square of a unit vector equals -1. Hestenes' version of quaternions emphasises the geometrical viewpoint that also underlies my work; it is then more appropriate for a unit vector squared to equal +1, which is achieved by inserting factors $i=\sqrt{-1}$ in the Pauli-matrix definitions. Another departure from the algebraic approach is that we explicitly allow the coefficients to be complex. The reader consulting literature on quaternions should be aware of these possible differences.

Another connection that looks interesting is that with the theory of special relativity, in which four-vectors appear with three spatial and one temporal component. The discussion of the Lorentz transformation by Feynman et al. (1975) shows a close formal analogy with my poldistortions (which was recently also noted by [Britton 2000]), but I have found no useful inspiration in it. I derive here the algebraic rules that quaternions must follow to make them behave in the same way as the equivalent matrices.

Once we have these rules in place we may proclaim, in mathematical language, that the multiplicative quaternion group and the multiplicative group of $2 \times 2\$ matrices are isomorphous ([Korn & Korn 1961]). In simpler terms, we may consider " $2 \times 2\$ matrix'' and "quaternion'' as names for the same object in two different languages, or even as synonyms. Such phrases as "the scalar and vector parts of a matrix'' will then make sense.

In Table 3 I present a dictionary for translating from matrix to quaternion language.

  
A.1. Addition

I start from the matrix/quaternion equivalence of Sect. 2.5:

$$\begin{eqnarray*}\vec{A}= [\, a + \vec{a}\,] \equiv [\, a_0 + (a_1, a_2, a_3) \,]. \end{eqnarray*}$$

The use of the "+'' sign is justified by the expansion that the right-hand side actually represents:

 

$$\vec{A}= a_0\,\mathbf{I}+ a_1\,\mathbf{Q}+ a_2\,\mathbf{U}+ a_3\,\vec{V}.$$ (22)

The addition rules for quaternions are the obvious ones as can be shown in the same way.

  
A.2. Transposition and conjugation

The definition of the vector part of a quaternion does not specify whether it is a row or column vector and the same is true for the dot and cross products that we will need later.

Since the Pauli matrices are hermitian, it follows that

 

$$\vec{A}^{\dagger}= [\, a^* + \vec{a}^* \,] .$$ (23)

  
A.3. Multiplication

The following identities follow directly from the definitions Eq. (7) of the Pauli matrices:

 

$$\begin{array}{lcl} \mathbf{Q}^{\dagger}&=& \mathbf{Q}\,,\quad \ma... ...f{Q} \\ \vec{V}\,\mathbf{Q}&=& - \mathbf{Q}\vec{V}= i\,\mathbf{U}. \end{array}$$ (24)

Now consider the product

$$\begin{eqnarray*}\vec{A}\vec{B}= [\, a+\vec{a}\,]\,[\, b+\vec{b}\,] . \end{eqnarray*}$$

Writing out the expansions Eq. (22) for $\vec{A}$ and $\vec{B}$ and multiplying term by term using Eq. (24), one finds the multiplication rule

 

$$[\, a+\vec{a}\,]\,[\, b+\vec{b}\,] = [\, (ab + \vec{a}\!\cdot\!\vec{b}) + (a\vec{b}+ b\vec{a}+ i\vec{a}\times\vec{b}) \,] .$$ (25)

An important corrolary is that $[\, \vec{1}_{{\vec{x}}} \,]^2 = [\, 1 \,]$ for any unit vector $\vec{1}_{{\vec{x}}}$. Like the multiplication of the equivalent matrices, quaternion multiplication is generally non-commutative. An exception occurs when $\vec{a}$ and $\vec{b}$ are collinear so that $\vec{a}\times \vec{b}= \vec{0}$. In that case I also call the matrices/quaternions $[\, a+\vec{a}\,]$ and $[\, b+\vec{b}\,]$ collinear.

Also note that the product of two real quaternions is not real unless they commute. The corresponding matrix property is that the product of two hermitian matrices is generally non-hermitian.

  
A.4. Scalars as quaternions

The scalar quaternion $[\,a\,]$ represents the $2 \times 2\$matrix aI, and since $\mathbf{I}\,\mathbf{I}= \mathbf{I}$, both the addition and the multiplication rule for scalar quaternions are the same as those for scalars. We may consider scalars as a subset of the quaternions:

$$\begin{eqnarray*}a\mathbf{I}\equiv [\,a\,] \equiv a . \end{eqnarray*}$$

  
A.5. Determinant, trace and variance

Every familiar property of a $2 \times 2\$matrix has a quaternion counterpart. Thus a quaternion has a determinant

 

$$\det [\, a+\vec{a}\,] = a^2 - \vec{a}^2$$ (26)

and $\det \vec{A}\vec{B}= \det\vec{A}\det\vec{B}$. A unimodular matrix/quaternion is one whose determinant equals 1.

The trace is

$$\begin{eqnarray*}\mbox{Tr}\ [\, a+\vec{a}\,] = 2a \end{eqnarray*}$$

$\mbox{Tr}\ \vec{A}= \mbox{Tr}\ \vec{A}^{\rm T}$, $\mbox{Tr}\ (\vec{A}+ \vec{A}^{\dagger}) = 2 \mbox{Tr\,Re\ }\vec{A}$, and $\mbox{Tr}\ (\vec{A}\vec{B}) = \mbox{Tr}\ (\vec{B} \vec{A})$. The coefficients of the Pauli matrices in Eq. (7) are given by expressions such as

$$\begin{eqnarray*}a_0 = { \textstyle \frac{1}{2} }\mbox{Tr}\ \vec{A}\mathbf{I}; ... ...le \frac{1}{2} }\mbox{Tr}\ \vec{A}\mathbf{Q}; \qquad\mbox{etc.} \end{eqnarray*}$$

The trace is invariant under a unitary transformation:

$$\begin{eqnarray*}\mbox{Tr}\ (\vec{Y}\vec{A}\vec{Y}^{\dagger}) = \mbox{Tr}\ \vec{A}. \end{eqnarray*}$$

I define the variance of a matrix as the sum of the moduli squared of its elements. It is the square of the "Frobenius norm'' ([Lancaster & Tismenetsky 1985]) and given by

 

$$\mbox{Var}\ \vec{A}= \mbox{Tr}\ (\vec{A}\vec{A}^*) = aa^* + \vec{a}\cdot\vec{a}^*$$ (27)

$\mbox{Var}\ (\vec{A}-\vec{B})$ may be used as a measure of "how different'' $\vec{A}$ and $\vec{B}$ are. It is readily shown that, like the trace, the variance is also invariant under unitary transformations.

  
A.6. Coordinate systems

The vector parts of quaternions form a three-dimensional quaternion-vector space. It is convenient to choose the coordinates in this space in accordance with our definition of the Stokes quaternion. When we express the electric field vectors in geometric xy coordinates, and use the conventional definition of the Stokes vector (cf. Paper I), the definition Eq. (6) follows. The corresponding base vectors are the quaternion vectors corresponding to the Pauli matrices Q, U and $\vec{V}$:

 

$$\vec{1}_{\mathbf{q}}=\left( \begin{array}{c} 1,0,0\end{array} \ri... ..., \quad \vec{1}_{\mathbf{v}}=\left( \begin{array}{c} 0,0,1\end{array} \right) .$$ (28)

Analogously to Eq. (24) we have

 

$$[\vec{1}_{\mathbf{q}}] \,[\vec{1}_{\mathbf{u}}] = -[\vec{1}_{\mathbf{u}}] \,[\vec{1}_{\mathbf{q}}] = i \,[\vec{1}_{\mathbf{v}}] \qquad\mbox{etc.}$$ (29)

If, instead, we use circular rl coordinates to describe the electric field, this results in a cyclic permutation of the coordinate axes (Paper I) and instead of Eq. (22) we have

 

$$\vec{B}= a\,\mathbf{I}+a_1\,\vec{V}+a_2\,\mathbf{Q}+a_3\,\mathbf{U}.$$ (30)

This form is convenient for analysing systems with nominally circular feeds.