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

Solution

The Dirac spinor representation is the four-dimensional representation

(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})

of the connected Lorentz group. It is characterized by

S [Λ]^{- 1} γ^{μ} S [Λ] = Λ^{μ}_{ν} γ^{ν} .

For

Λ = exp (\frac{1}{2} Ω_{ρ σ} M^{ρ σ}),

one may take

S [Λ] = exp (\frac{1}{8} Ω_{ρ σ} [γ_{ρ}, γ_{σ}]),

with the signs of

Ω_{ρ σ}

fixed by the displayed vector-representation convention. This formula follows by differentiating the covariance relation and using the Clifford algebra

{γ^{μ}, γ^{ν}} = 2 η^{μν}

; moving both gamma indices upstairs changes the apparent parameter signs according to the metric.

γ^{0} = (0 I_{2} I_{2} 0), γ^{i} = (0 - σ^{i} σ^{i} 0) .

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.