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?