Past exam of the theoretical physics I course of the University of Cambridge / 2015 / Paper 1 / 5

Use $\hslash =c=1$, signature $(+---)$, and define $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$. Under a local phase transformation,the derivative of the phase cancels the change of the potential, giving $D_{\mu }^{\prime }{\varphi }^{\prime }=e^{-ie\alpha }{D}_{\mu }\varphi$. Therefore a gauge-invariant scalar-electromagnetic Lagrangian density isThe mass term is invariant, the covariant-derivative product has canceling phases, and $F_{\mu \nu }^{\prime }=F_{\mu \nu }$ because mixed partial derivatives commute. A potential depending only on ${\varphi }^{\ast }\varphi$ may be added without changing the symmetry. The Maxwell term makes the electromagnetic field dynamical; it can be omitted when $A_{\mu }$ is a prescribed external background.

Expanding the covariant-derivative term gives the interaction linear in $e$ plus $e^2{A}_{\mu }{A}^{\mu }|\varphi {|}^2$. This quadratic term is required for exact local gauge invariance. The later part (c) deliberately gives only the linear interaction, and must be solved as that different stated Lagrangian rather than silently adding the missing term.

For fields ${\phi }_A$ and a continuous infinitesimal internal symmetry $\delta {\phi }_A=ϵ\Delta {\phi }_A$, suppose the density changes by a divergence, $\delta \mathcal{L}=ϵ{\partial }_{\mu }{K}^{\mu }$. Varying the action and using the Euler-Lagrange equations gives the Noether currentThe conserved charge is $Q=\int j^0{d}^{3}x$ when the flux through the boundary vanishes; more generally $dQ/dt=-\oint \mathbf{j}\cdot d\mathbf{S}$. If the Lagrangian is exactly invariant, $K^{\mu }=0$. Treat a complex field and its conjugate as independent variables during variation, imposing conjugacy on physical solutions. Time and space translations give energy and momentum through the corresponding spacetime version of the theorem. Continuous global symmetry gives an ordinary charge; local gauge redundancy requires additional care and is not an independent conserved charge for every arbitrary function.

Treat $\varphi$ and ${\varphi }^{\ast }$ as independent fields, keep $A_{\mu }(x)$ as the prescribed background, and use the source's interaction with no added $e^2{A}^2|\varphi {|}^2$ term. The momentum derivatives areVarying ${\varphi }^{\ast }$ and $\varphi$ gives respectivelyNo Lorenz gauge condition has been assumed. The background divergence term must be retained unless such a condition is separately imposed.

The canonical momenta conjugate to the two scalar fields areso $\dot{\varphi }={\pi }^{\ast }-ie{A}_0\varphi$ and ${\dot{\varphi }}^{\ast }=\pi +ie{A}_0{\varphi }^{\ast }$. Define the physical three-vector potential by $A_{\mu }=(A_0,-\mathbf{A})$. The Legendre transform in the proper canonical variables isEquivalently the first product is $\pi {\pi }^{\ast }+ie{A}_0({\varphi }^{\ast }{\pi }^{\ast }-\varphi \pi )+e^2{A}_0^2|\varphi {|}^2$. The last quadratic term arises from replacing velocities by canonical momenta even though the original Lagrangian has no quadratic interaction. It is not permission to modify the specified Lagrangian. An explicitly time-dependent background generally makes the scalar-field energy time dependent.

For $\Delta \varphi =-i\varphi$ and $\Delta {\varphi }^{\ast }=i{\varphi }^{\ast }$, the interaction is invariant under the constant phase because every product has net zero phase. Applying the Noether formula to the two independent fields givesThe sign corresponds to the printed transformation $\varphi \to e^{-iϵ}\varphi$. Reversing the phase parameter would reverse the entire current, with no change in conservation.

The derivative products in the divergence of the first term cancel pairwise:The two field equations from part (c) implyMultiplication by $i$ exactly cancels the remaining background terms, proving ${\partial }_{\mu }{J}^{\mu }=0$ without requiring $A_{\mu }$ to be constant or divergence free. The global phase symmetry survives even though this truncated interaction is not exactly invariant under arbitrary local gauge transformations.