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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-301/1/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 301 / 1 / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

After using the Clifford algebra, integrations by parts and the Dirac equation, every allowed Lorentz-covariant symmetric rank-two expression with one derivative reduces, modulo terms vanishing on shell and identically conserved improvements, to an overall multiple of

\overset{ˉ}{ψ} γ^{(μ} \partial^{ν)} ψ .

(1)

Thus the general nontrivial tensor in the stated class is

T^{μν} = C [\overset{ˉ}{ψ} γ^{μ} \partial^{ν} ψ + \overset{ˉ}{ψ} γ^{ν} \partial^{μ} ψ - (\partial^{ν} \overset{ˉ}{ψ}) γ^{μ} ψ - (\partial^{μ} \overset{ˉ}{ψ}) γ^{ν} ψ] .

(2)

It is manifestly symmetric. Differentiating it, commuting partial derivatives and using

i γ^{ρ} \partial_{ρ} ψ = m ψ, i (\partial_{ρ} \overset{ˉ}{ψ}) γ^{ρ} = - m \overset{ˉ}{ψ},

(3)

makes the terms cancel pairwise, proving

\partial_{μ} T^{μν} = 0

on shell. The normalization

C

remains free at this stage.

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