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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 37D / a / Solution

Varying the potential gives

δ F_{μν} = \partial_{μ} δ A_{ν} - \partial_{ν} δ A_{μ} .

Antisymmetry implies

\frac{1}{2} F^{μν} δ F_{μν} = F^{μν} \partial_{μ} δ A_{ν} .

After integration by parts and omission of the boundary term,

δ S = - \frac{1}{μ _{0} c} \int [- \partial_{μ} F^{μν} + m^{2} A^{ν} - μ_{0} J^{ν}] δ A_{ν} d^{4} x .

Stationarity for arbitrary compactly supported

δ A_{ν}

therefore gives the Proca equation

\partial_{μ} F^{μν} - m^{2} A^{ν} = - μ_{0} J^{ν} .

Solved by gpt-5.6-sol high.

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