User:Jheald/sandbox/GA/Spinors in 4-dimensional Euclidean space

Representation of the Clifford Algebra
The signature of the algebra requires
 * $$e_1^2 = e_2^2 = e_3^2 = e_4^2 = 1\;$$

and the Clifford condition requires that
 * $$e_i e_j + e_j e_i = 0, \; \mbox{for } i \neq j$$

making
 * $$e_{1234}^2 = +1\;$$

A suitable set of matrices (Lounesto, p. 86) is Mat(2,ℍ) with quaternions as entries:

e_1 \simeq \begin{pmatrix} 0&i\\-i&0 \end{pmatrix};\; e_2 \simeq \begin{pmatrix} 0&j\\-j&0 \end{pmatrix};\; e_3 \simeq \begin{pmatrix} 0&k\\-k&0 \end{pmatrix};\; e_4 \simeq \begin{pmatrix} 0&1\\1&0 \end{pmatrix};\; $$ giving

e_{12} \simeq \begin{pmatrix} -k&0\\0&-k \end{pmatrix};\; e_{23} \simeq \begin{pmatrix} -i&0\\0&-i \end{pmatrix};\; e_{34} \simeq \begin{pmatrix} k&0\\0&-k \end{pmatrix};\; e_{41} \simeq \begin{pmatrix} -i&0\\0&i \end{pmatrix};\; $$

e_{13} \simeq \begin{pmatrix} j&0\\0&j \end{pmatrix};\; e_{24} \simeq \begin{pmatrix} j&0\\0&-j \end{pmatrix};\; $$

e_{123} \simeq \begin{pmatrix} 0&1\\-1&0 \end{pmatrix};\; e_{234} \simeq \begin{pmatrix} 0&-i\\-i&0 \end{pmatrix};\; e_{341} \simeq \begin{pmatrix} 0&j\\j&0 \end{pmatrix};\; e_{412} \simeq \begin{pmatrix} 0&-k\\-k&0 \end{pmatrix};\; $$ and

e_{1234} \simeq \begin{pmatrix} 1&0\\0&-1 \end{pmatrix}$$

Spinor
A spinor (or, at least, a vector of quaternions) can be projected by right-multiplying by ½(1+e1234); the other column can be projected by ½(1-e1234).

Dimensionality of the subspace?