Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 37D (Electrodynamics)

Varying the potential givesAntisymmetry impliesAfter integration by parts and omission of the boundary term,Stationarity for arbitrary compactly supported $\delta A_{\nu }$ therefore gives the Proca equation

Lorentz invariance requires $A^{\mu }$ and $J^{\mu }$ to transform as four-vectors, $\chi$ as a scalar, $m$ and ${\mu }_0$ as scalars, and the integration domain and boundary conditions to respect the transformation. Then every contracted term and $d^{4}x$ are Lorentz invariant.

Under $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\chi$, the field strength is unchanged. The source term changes by a boundary term pluswhich vanishes for a conserved current and suitable boundary behaviour of $\chi$. The mass term is not gauge invariant. Thus the action has the usual local gauge invariance whenwith the gauge function or its boundary contribution suitably controlled. For $m>0$, the mass term breaks this gauge symmetry.

Take the divergence of the field equation. Commuting derivatives and antisymmetry of $F^{\mu \nu }$ giveCurrent conservation then yieldsFor $m>0$,This is the Lorenz constraint in Proca theory; it follows dynamically rather than being a gauge choice.

Usingand writingthe temporal and spatial components give, with the stated metric conventions,For $m>0$, part (c) sets $\alpha =0$. For $m=0$, $\alpha =0$ is the Lorenz gauge condition.

For the static point source, take $A=0$ and $\alpha =0$. The scalar equation becomesIts spatial Fourier transform satisfiesThe stated inverse transform therefore giveswhere $\lambda$ absorbs the transform normalization and source strength. This is the Yukawa potential: a nonzero vector-field mass screens the interaction beyond the range $m^{-1}$. As $m\to 0$, it tends to the long-range Coulomb potential $\lambda /r$. The same limit removes the mass term that broke gauge invariance in part (b).