Past exam of the theoretical physics I course of the University of Cambridge / 2014 / Paper 1 / 4

Use metric signature $(+,-,-,-)$ and units $\hslash =c=1$. Varying ${\varphi }^{\ast }$ and $\varphi$ independently givesIn the divergence of the stated current, the products of first derivatives cancel, leavingThe two identical mass terms cancel on the field equations.

An infinitesimal constant phase change has $\delta \varphi =-iq\alpha \varphi$ and $\delta {\varphi }^{\ast }=iq\alpha {\varphi }^{\ast }$. Both the derivative product and the mass term are invariant. Noether's theorem gives, per unit $\alpha$,Thus the global phase symmetry supplies the conserved current in part a.

The derivatives needed for varying $A_{\nu }$ areTherefore the electromagnetic Euler-Lagrange equation isAntisymmetry of $F$ and commutation of derivatives imply ${\partial }_{\nu }{\partial }_{\mu }{F}^{\mu \nu }=0$, hence ${\partial }_{\nu }{J}^{\nu }=0$. In ordinary derivatives, $J^{\mu }=j^{\mu }-2q^2{A}^{\mu }|\varphi {|}^2$; the interaction contribution must not be omitted.

Under the combined local transformation,The derivative of the phase cancels the added gauge potential. The conjugate derivative gains the opposite phase, so their product and $|\varphi {|}^2$ are invariant. AlsoEvery term of the interacting Lagrangian is therefore invariant.

The local gauge symmetry contains the constant-phase subgroup. For constant $\alpha$, $\delta A_{\mu }=0$, and Noether's theorem applied to the matter fields givesThus ${\partial }_{\mu }{J}^{\mu }=0$ on the matter field equations. More explicitly, allowing $\alpha (x)$ in the matter variation while holding $A$ fixed gives $\delta \mathcal{L}=J^{\mu }{\partial }_{\mu }\alpha$. On shell it also equals ${\partial }_{\mu }(\alpha J^{\mu })$, so equality for arbitrary $\alpha$ implies conservation. In the full gauge transformation, varying $A$ cancels the $J^{\mu }{\partial }_{\mu }\alpha$ term; this is the related Noether second-theorem identity, not a new independent charge for every gauge function.