 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-305/1/a/solution

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

g^{μν} = diag (1, - 1, - 1, - 1)

throughout. Taking the Hermitian conjugation of the massive Dirac equation and using the Dirac adjoint gives

i (\partial_{μ} \overset{ˉ}{ψ}) γ^{μ} + m \overset{ˉ}{ψ} = 0.

C

yields

i C γ^{μ T} C^{- 1} \partial_{μ} (C \overset{ˉ}{ψ}^{T}) + m C \overset{ˉ}{ψ}^{T} = 0, - i γ^{μ} \partial_{μ} ψ^{c} + m ψ^{c} = 0.

(i γ^{μ} \partial_{μ} - m) ψ^{c} = 0.

C

and

γ^{0} C^{- 1} γ^{0} = - C^{- 1} γ^{0 T}

also give

\overline{ψ^{c}} = (C \overset{ˉ}{ψ}^{T})^{†} γ^{0} = ψ^{T} γ^{0 T} C^{- 1} γ^{0} = - ψ^{T} C^{- 1} .

Consequently

\hat{C} \overset{ˉ}{ψ} (x) \hat{C}^{- 1} = - ψ^{T} (x) C^{- 1} .

Here

\hat{C}

C

acts on Dirac spinor indices.

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