Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 301 / 3 / i

Table of contents
- Solution i

Solution

Define the spinor generators

S^{μν} = \frac{1}{4} [γ^{μ}, γ^{ν}] .

Repeated use of the Clifford algebra gives

[S^{μν}, γ^{ρ}] = \frac{1}{4} (γ^{μ} γ^{ν} γ^{ρ} - γ^{ν} γ^{μ} γ^{ρ} - γ^{ρ} γ^{μ} γ^{ν} + γ^{ρ} γ^{ν} γ^{μ}) = γ^{μ} η^{ν ρ} - γ^{ν} η^{ρ μ} .

The Spinor representation of the Lorentz group is

S [Λ] = exp (\frac{1}{2} Ω_{ρ σ} S^{ρ σ}) .

The adjoint relation for the Dirac matrices implies

(S^{ρ σ})^{†} = - γ^{0} S^{ρ σ} γ^{0} .

Exponentiating yields

S [Λ]^{†} = γ^{0} S [Λ]^{- 1} γ^{0} .

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