Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 35C (Electrodynamics)

For compactly supported variations, antisymmetry of the electromagnetic field tensor gives $\delta (-F_{ab}{F}^{ab}/4)=-F^{ab}{\partial }_a\delta A_b$. Integrate by parts in the action to obtainArbitrariness of the variation gives the Proca equationtogether with the unchanged identity ${\partial }_{[a}{F}_{bc]}=0$ coming from $F=dA$. The signs follow from the source and mass terms as printed.

Take ${\partial }_b$ of the Proca equation. Commuting partial derivatives and antisymmetry make ${\partial }_b{\partial }_a{F}^{ab}=0$, while current conservation gives ${\partial }_b{J}^b=0$. Thus $m^2{\partial }_b{A}^b=0$. For $m\ne 0$, $\boxed{{\partial }_b{A}^b=0}$. This Lorenz constraint in Proca theory is a consequence of the massive equations, rather than a gauge choice.

The stated “arbitrary constant” includes a genuine exception at $m=0$. Then the divergence condition is not implied: with $J=0$ and $A_a={\partial }_a\chi$, $F=0$ solves the equations for any smooth $\chi$, whereas ${\partial }_a{A}^a=\square \chi$ need not vanish. In the massless theory the Lorenz condition can instead be imposed as a gauge fixing.

Use signature $(-,+,+,+)$, for which $\square =-{\partial }_t^2+\Delta$. With the Lorenz constraint in Proca theory, the Proca equation is $(\square -m^2)A^a=-{\mu }_0{J}^a$. A static solution therefore satisfies $(\Delta -m^2)A^a=-{\mu }_0{J}^a$.

For $R>0$ direct radial differentiation gives $(\Delta -m^2)(e^{-mR}/R)=0$. At the origin its $1/R$ singularity contributes $-4\pi {\delta }_0$: integrating the Laplacian over a small sphere gives limiting outward flux $-4\pi$, while the mass term's integral tends to zero. Thus $G_m(R)=e^{-mR}/(4\pi R)$ obeys $(\Delta -m^2)G_m=-{\delta }_0$. Convolution proves the printed Yukawa potential solutionThe integral requires suitable source decay, for example compact support. For conserved static sources, differentiating the convolution and integrating by parts gives ${\partial }_a{A}^a=0$ as well; its spatial divergence is the convolution with ${\partial }_i{J}^i=0$.

For the usual nonnegative mass parameter this is the decaying static solution, and $m=0$ gives the Coulomb kernel with Lorenz gauge imposed. If the arbitrary real parameter is negative, the printed kernel still gives a particular solution for compact sources, but grows at infinity; the decaying boundary condition selects $e^{-|m|R}/(4\pi R)$ instead. Static sources also allow homogeneous time-dependent solutions, so this construction establishes a static solution, not that every solution is static.