Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 8 / b / Solution

Use ${\eta }_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ and define the physical electromagnetic source by ${\partial }_{\nu }{F}^{\nu \mu }=J^{\mu }$. For a constant infinitesimal phase $\delta \phi =ieϵ\phi$, the matter Noether current isA definition using phase parameter $\delta \phi =iϵ\phi$ instead divides this expression by $e$; the Maxwell source includes the electric coupling shown here. Current signs must be tied to the printed $D_{\mu }={\partial }_{\mu }-ie{A}_{\mu }$ convention.

To verify the equation of motion, vary the matter kinetic term with respect to $A_{\mu }$:Varying $-F^2/4$ and integrating by parts gives $({\partial }_{\nu }{F}^{\nu \mu })\delta A_{\mu }$. The stationary action therefore requiresThe potential has no explicit dependence on $A_{\mu }$ and contributes no extra source term. Antisymmetry of $F^{\nu \mu }$ implies ${\partial }_{\mu }{J}^{\mu }=0$ on the equations of motion; the same conservation law follows from the scalar equations. The source current is gauge-invariant although its expanded derivative and vector-potential pieces are not separately invariant.^[ref]