The spinors – Geometric Algebra

We have seen that if we take a vector $\pmb{v}$ and reflect it with respect to the vector $\pmb{a}$ and then to the vector $\pmb{b}$ (in this order) we obtain a rotation of the vector of an angle double the one between $\pmb {a}$ and $\pmb{b}$ (in that order).

Algebraically it is a question of applying reflection twice with respect to a vector, with the known sandwich product: $v '= b (ava ^ {-1}) b^{-1} = (ba) v (ba) ^ {-1} = e^{- i \alpha} ve^{i \alpha}$

So if we want a rotation equal to $\alpha$ we will have to use a rotor $e^{i \frac {\alpha} {2}}$ which was named spinor by Wolfgang Pauli for the reason that we will see shortly.

So we have this strange equality:

$\begin{equation}e^{-B\frac{\alpha}{2}} v e^{B\frac{\alpha}{2}} = (-e^{-B\frac{\alpha}{2}}) v (-e^{B\frac{\alpha}{2}})$

The two minus signs cancel out in the sandwich, so the rotor $R = e^{B\alpha/2}$ and its opposite $-R$ produce exactly the same rotation: same angle, same direction. And yet $R$ and $-R$ are two distinct algebraic objects. Here is the surprising fact: to each physical rotation there correspond two rotors, one the opposite of the other. (Mathematicians call this a double cover: the group of rotors “covers the group of rotations twice”.)

To see how this is possible, let us follow the rotor as the angle $\alpha$ grows:

for $\alpha \in [0, 2\pi]$ the body completes a full turn and returns to its starting position. But the rotor $e^{B\alpha/2}$ has travelled only halfway (its exponent has reached $B\pi$ ), and now equals $-1$ : the rotor has become its own opposite!
for $\alpha \in [2\pi, 4\pi]$ the body completes a second turn, physically identical to the first. Only at the end of this second turn ( $\alpha = 4\pi$ ) does the rotor finally return to its initial value $+1$ .

This follows from the computation: adding a physical turn $2\pi$ to the angle amounts to multiplying the rotor by $-1$ ,

$\begin{equation}e^{B\frac{\alpha+2\pi}{2}} = e^{B\pi}e^{B\frac{\alpha}{2}}= - e^{B\frac{\alpha}{2}}$

and only after two turns do we return to the identity:

$\begin{equation}e^{B\frac{\alpha+4\pi}{2}} = e^{B\frac{\alpha}{2}}$

and so rotors have a periodicity of $4\pi$ , twice that of the rotations they represent! If spinors seem strange to you… it is because they are!!

It is carved into everyone’s beliefs that an object returns to its initial state after a rotation of $2 \pi$ , but this time we are dealing with an object that returns to itself after two rotations.
This weirdness can be demonstrated with Dirac’s trick of the saucer or the belt, which will leave the spectators astounded as if it were a magic show.

This makes the half-integer spin of electrons a little less strange, after all.