Paravector - Matrix Representation

Matrix Representation

The algebra of the space is isomorphic to the Pauli matrix algebra such that

	Matrix Representation 3D	Explicit matrix
		$\begin{pmatrix} 1 && 0 \\ 0 && 1 \end{pmatrix}$
		$\begin{pmatrix} 0 && 1 \\ 1 && 0 \end{pmatrix}$
		$\begin{pmatrix} 0 && -i \\ i && 0 \end{pmatrix}$
		$\begin{pmatrix} 1 && 0 \\ 0 && -1 \end{pmatrix}$

from which the null basis elements become ${ P_3} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \,; \bar{ P}_3 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \,; { P_3} \mathbf{e}_1 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \,;\mathbf{e}_1 { P}_3 = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}.$

A general Clifford number in 3D can be written as

$\Psi = \psi_{11} P_3 - \psi_{12} P_3 \mathbf{e}_1 + \psi_{21} \mathbf{e}_1 P_3 + \psi_{22} \bar{P}_3,$

where the coefficients are scalar elements (including pseudoscalars). The indexes were chosen such that the representation of this Clifford number in terms of the Pauli matrices is

$\Psi \rightarrow \begin{pmatrix} \psi_{11} & \psi_{12} \\ \psi_{21} & \psi_{22} \end{pmatrix}$

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