I've come across two definitions of "spinors" that I'm having a hard time reconciling:
- Spinors are the "square root" of a null vector (see here, and also Cartan's book "The Theory of Spinors")
- Spinors are minimal ideals in a Clifford algebra (see here, and several other texts like "Clifford Algebras and Spinors" by Lounesto)
I'll give a run-down of these definitions, and then I have some questions at the end.
For definition #1, we take a vector $\vec{v} = x \vec{e_y} + x \vec{e_y} + z \vec{e_z}$ and write it as a linear combination of the sigma matrices to get a $2 \times 2$ matrix. This gives us:
$$\vec{v} \rightarrow V = x\sigma_x + y\sigma_y + z\sigma_z$$ $$ = x \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix} + y \begin{bmatrix} 0 & -i \\ i & 0\end{bmatrix} + z \begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix} $$ $$ = \begin{bmatrix} z & x-iy \\ x+iy & -z\end{bmatrix} $$
We can check that $det(V) = - (length(\vec{v}))^2 = - (x^2 + y^2 + z^2) $. If the vector is "null", this quantity is zero. If the determinant of the matrix is zero, we can "factor" it as the multiplication of a column and row of complex numbers, which are the left and right spinors:
$$ V = \begin{bmatrix} z & x-iy \\ x+iy & -z\end{bmatrix} = 2 \begin{bmatrix} \xi_1 \\ \xi_2 \end{bmatrix} \begin{bmatrix} -\xi_2 & \xi_1 \end{bmatrix} $$
In this representation, when we want to rotate the components of a vector, we use a double-sided transformation $V \rightarrow UVU^\dagger$ with $U \in SU(2)$ being a $2 \times 2$ unitary matrix. So if a null vector is "split" into a (left) column spinor and a (right) row spinor, each spinor gets acted on by a single-sided transformation involving $U$ (for left) or $U^\dagger$ (for right).
Now for Definition #2, we look for minimal ideals of a Clifford Algebra. We can start with the Clifford Algebra $Cl(3,0)$ with the 3 basis vectors $\{ \sigma_x, \sigma_y, \sigma_z \}$ with $(\sigma_i)^2=+1$, and all anti-commuting with each other, $\{\sigma_i, \sigma_j\} = \sigma_i \sigma_j + \sigma_j \sigma_i = 0$ for $i\neq j$.
My understanding is that every (minimal) ideal has a (minimal) projector element $(p^2 = p)$ associated with it. So we can use the projector $p_+ = \frac{1}{2}(1 + \sigma_z) = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} $ to get a left ideal, by left-multiplying the algebra on it, $Cl(3,0)p_+$. The basically gives us the set of $2 \times 2$ matrices with only the left column being non-zero. There is also the projector $p_- = \frac{1}{2}(1 - \sigma_z) = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} $, which gives us another left ideal: the set of $2 \times 2$ matrices with only the right column being non-zero.
So in this interpreation, some examples of spinors as members of a minimal ideal would be:
$$\sigma_x p_+ =\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} $$
$$\sigma_x p_- =\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} $$
In this case, spinors actually add together to give us a vector... for example $(\sigma_x p_+) + (\sigma_x p_-) = \sigma_x$.
We could also act on projectors from the right to get $p_\pm Cl(3,0)$, which would give right ideals (the sets of $2 \times 2$ matrices with either the top row non-zero or bottom row non-zero).
I don't understand the connection between these two definitions. I realize we can write
$$\begin{bmatrix} \xi_1 \\ \xi_2 \end{bmatrix} \begin{bmatrix} -\xi_2 & \xi_1 \end{bmatrix} = \begin{bmatrix} \xi_1 & 0 \\ \xi_2 & 0 \end{bmatrix} \begin{bmatrix} -\xi_2 & \xi_1 \\ 0 & 0 \end{bmatrix} $$
which makes column/row spinors look a bit more like members of left and right ideals, using the projectors I used in definition #2.
- Do the two definitions describe the exact set same of objects? Or is one more general?
- Is there a procedure for factoring null elements of a clifford algebra, like $\sigma_x + i \sigma_y$ into a product of spinors?
- What motivates the notion of a "single-sided multiplication" on spinors in definition #2? There's no mention of "factoring a vector into spinors" anywhere, so I don't understand why we'd be motivated to do single-sided transformations on spinors.
