 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-301/2/d/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

The commutator with the scalar metric term vanishes. Reorder the third gamma matrix using the Clifford algebra twice:

γ^{μ} γ^{ν} γ^{ρ} = 2 η^{ν ρ} γ^{μ} - 2 η^{μ ρ} γ^{ν} + γ^{ρ} γ^{μ} γ^{ν} .

Subtract the last term and multiply by

1/2

to obtain

[S^{μν}, γ^{ρ}] = η^{ν ρ} γ^{μ} - η^{μ ρ} γ^{ν}, C = 1, D = - 1.

η^{ν ρ}

as printed in the PDF; the TeX conversion's

η^{ρρ}

is erroneous and would fail, for example, when

μ = 0, ν = 1, ρ = 2

: the actual commutator is zero.

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