## § 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:
$\begin{bmatrix} \cos \theta/2 & i \sin \theta/2 \\ i \sin \theta/2 & \cos \theta/2 \end{bmatrix} \begin{bmatrix} z & x - yi \\ x+yi & -z \end{bmatrix} \begin{bmatrix} \cos \theta/2 & i \sin \theta/2 \\ i \sin \theta/2 & \cos \theta/2 \end{bmatrix}^\dagger$
• But this can be written as:
$\begin{bmatrix} \cos \theta/2 & i \sin \theta/2 \\ i \sin \theta/2 & \cos \theta/2 \end{bmatrix} \begin{bmatrix} s_1 \\ s_2 \end{bmatrix} \begin{bmatrix} -s_2 & s_1 \end{bmatrix} \begin{bmatrix} \cos \theta/2 & i \sin \theta/2 \\ i \sin \theta/2 & \cos \theta/2 \end{bmatrix}^\dagger$
• 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.

#### § Clifford algebras

• Geometric algebra, nuff said.
• 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 $1/2(e_x + e_x e_y)$.
• For spinors, they have chirality as well, so we get {left, right} x {co, contra}.