Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 301 / 3 / i

Table of contents
- Solution i

Solution

The displayed

R (φ)

rotates the coordinate components in the

x y

- φ

. Since

γ^{1} γ^{2} = - i (σ^{3} 0 0 σ^{3}),

its spin representative is

S [R (φ)] = exp (- \frac{φ}{2} γ^{1} γ^{2}) = (e^{i φ σ^{3} /2} 0 0 e^{i φ σ^{3} /2}) .

At a full turn,

S [R (2 π)] = - I_{4} .

Thus a

2 π

spatial rotation changes the sign of a spin-one-half state, while a

4 π

rotation returns it to itself. This realizes the fact that the Spin group is a double cover of the proper orthochronous Lorentz group; observable spinor bilinears are unchanged by the sign.

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