11 Appendix: Mathematical theory

 

 

Table 3: Algebraic entities and their notation as used in this paper

Entity	Vector	$2 \times 2\$Matrix	Stokes vector	Quaternion
Coherency	$\vec{e}_{jk \vphantom{'}}= \vec{e}_{j \vphantom{'}}\otimes \vec{e}_{k \vphantom{'}}^*$	$\! \vec{E}_{jk \vphantom{'}}= \vec{e}_{j \vphantom{'}}\vec{e}_{k \vphantom{'}}^{\dagger}$	$\vec{s}_{jk \vphantom{'}}\!=\! \left( \begin{array}{cccc} 1 &0 &0 &1 \\ 1 ... ...\ 0 &1 &1 &0 \\ 0 &-i &i &0 \end{array} \right) \!\vec{e}_{jk \vphantom{'}}$
General element	$\left( \begin{array}{l} a_{11}\\ a_{12}\\ a_{21}\\ a_{22} \end{array} \right)$	$\! \begin{array}{l} \vec{A}= \left( \begin{array}{cc} a_{11} &a_{12} \\ a_{21... ... +a_1 & a_2-ia_3 \\ a_2+ia_3 & a - a_1\end{array} \right) \end{array} \!\!\!$	$\left( \begin{array}{c} a \\ \!\mbox{- -}\! \\ a_1\\ a_2\\ a_3 \end{array} \... ...left( \begin{array}{c} a \\ \!\!\mbox{- -}\!\! \\ \vec{a}\end{array} \right)$	$\begin{array}{l} [a+\vec{a}{]}, \\ % \qquad \vec{a}\equiv \left( \begin{array}{ccc} a_1 &a_2 &a_3 \end{array} \right) \end{array}$
Scalar part			intensity a	$[a+\vec{0}] \equiv [a{]} \equiv a$
Vector part			polvector $\vec{a}$	$[0+\vec{a}] \equiv [\vec{a}{]} \not\equiv \vec{a}$
Base for vector part		$\mathbf{Q}, \mathbf{U}, \vec{V}, \mbox{\ \ Eq. (\ref{.eq.Pauli}) }$	$\vec{1}_{\mathbf{q}}, \vec{1}_{\mathbf{u}}, \vec{1}_{\mathbf{v}}, \mbox{\ \ Eq. (\ref{.eq.quv}) }$	$[\vec{1}_{\mathbf{q}}{]}, [\vec{1}_{\mathbf{u}}{]}, [\vec{1}_{\mathbf{v}}{]}$
Alternate base		$\vec{V}, \mathbf{Q}, \mathbf{U}, \mbox{\ \ Eq. (\ref{.eq.A=a0I+a1V...}) }$	$\vec{1}_{\mathbf{v}}, \vec{1}_{\mathbf{q}}, \vec{1}_{\mathbf{u}}$	$[\vec{1}_{\mathbf{v}}{]}, [\vec{1}_{\mathbf{q}}{]}, [\vec{1}_{\mathbf{u}}{]}$
Unit element		I		[1] = 1
Conjugation		$\vec{A}^{\dagger}$	$\left( \begin{array}{c} a^* \\ \!\!\mbox{- -}\!\! \\ \vec{a}^* \end{array} \right)$	$[a^* + \vec{a}^*{]}$
Multiplication		$\vec{A}\vec{B}$		Eq. (25)
Unimodular unitary matrix		$\vec{Y}$		$\cos\eta + i\vec{1}_{\vec{y}}\sin\eta$
Unimodular positive hermitian matrix		$\vec{H}$		$\cosh\gamma + \vec{1}_{\vec{h}}\sinh\gamma$