Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 301 / 2 / d / ii

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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