Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 35C / i / Solution

For compactly supported variations, antisymmetry of the electromagnetic field tensor gives

δ (- F_{ab} F^{ab} /4) = - F^{ab} \partial_{a} δ A_{b}

. Integrate by parts in the action to obtain

δ I = \int d^{4} x [\partial_{a} F^{ab} - m^{2} A^{b} + μ_{0} J^{b}], δ A_{b} .

Arbitrariness of the variation gives the Proca equation

\partial_{a} F^{ab} - m^{2} A^{b} = - μ_{0} J^{b},

together with the unchanged identity

\partial_{[a} F_{b c]} = 0

coming from

F = d A

. The signs follow from the source and mass terms as printed.

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