A Universe of Sorts

§ Concrete description of spinors

Consider how in physical space, $|\uparrow\rangle$ and $\downarrow\rangle$ in physical space are 180 degrees away (pointing up or down), but in the qubit state space, they are orthogonal. So there is an "angle doubling", where 180 degrees in physical space is 90 degrees in state space. How to encode this?
Idea: some matrices can be written as a product of two vectors, as [x y] [a b]^T. Do the same to a vector!
Take a vector [x, y, z]. Convert to a pauli matrix. [z x-yi; x+yi z]. This can be written as [s1 s2] [-s2 s1]^T.
Just as we can rotate a 3D vector with a rotation matrix, we can rotate a pauli vector with 2x2 matrices:

\begin{bmatrix} \cos \theta/2 & i \sin \theta/2 \\ i \sin \theta/2 & \cos \theta/2 \end{bmatrix} \begin{bmatrix} z & x - yi \\ x+yi & -z \end{bmatrix} \begin{bmatrix} \cos \theta/2 & i \sin \theta/2 \\ i \sin \theta/2 & \cos \theta/2 \end{bmatrix}^\dagger

\begin{bmatrix} \cos \theta/2 & i \sin \theta/2 \\ i \sin \theta/2 & \cos \theta/2 \end{bmatrix} \begin{bmatrix} s_1 \\ s_2 \end{bmatrix} \begin{bmatrix} -s_2 & s_1 \end{bmatrix} \begin{bmatrix} \cos \theta/2 & i \sin \theta/2 \\ i \sin \theta/2 & \cos \theta/2 \end{bmatrix}^\dagger

So a spinor is like a "rank 1/2 object", between scalars and vectors. We can combine two spinors to get a vector.
Pauli spinors are associated with 3D space (what was explained now).
Weyl spinors are associated with 4D spacetime.

Geometric algebra, nuff said.
Spinors arise for free in a geometric algebra.
For example, the spin up state is $1/2(1 + e_z)$ where $e_z$ is the basis $z$ vector. The spin down state is $1/2(e_x + e_x e_y)$ .
Apparently, we can say that "spinors are elements of minimal left ideals in clifford algebras".

For tensors, we need only co and contravariant indeces.
For spinors, they have chirality as well, so we get {left, right} x {co, contra}.
This notation for denoting chirality based on dots is the Van der waerden notation .