Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 1 / c

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- Solution c

Solution

Varying

\overset{ˉ}{ψ}

and

ψ

independently gives

(i \neq D - iλ [γ^{μ}, γ^{ν}] F_{μν}) ψ = 0

and its adjoint

i (D_{μ} \overset{ˉ}{ψ}) γ^{μ} + iλ \overset{ˉ}{ψ} [γ^{μ}, γ^{ν}] F_{μν} = 0,

where

D_{μ} \overset{ˉ}{ψ} = \partial_{μ} \overset{ˉ}{ψ} - i e A_{μ} \overset{ˉ}{ψ}

. Varying

A_{ν}

and integrating the Pauli term by parts gives the modified Maxwell equations

\partial_{μ} F^{μν} = e \overset{ˉ}{ψ} γ^{ν} ψ - 2 iλ \partial_{μ} (\overset{ˉ}{ψ} [γ^{μ}, γ^{ν}] ψ) .

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