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

The continuous internal symmetry is local $U(1)$ gauge invariance. With the source's gauge covariant derivative convention it acts asso $D_{\mu }\phi$ transforms like $\phi$ and $F_{\mu \nu }$ is unchanged. Constant phases are its global subgroup, not an additional independent internal rotation group.

In flat four-dimensional space-time the continuous external symmetry is the Poincaré group: four translations and six Lorentz transformations. Noether's theorem associates translations with conserved energy and momentum, spatial rotations with angular momentum, and boosts with the relativistic center-of-energy/boost charges. In tensor notation the latter two are the six components of the conserved total angular-momentum tensor $M^{\mu \nu }$, including field spin contributions where appropriate. The nonzero dimensionful parameter ${\mu }^2$ prevents exact scale invariance. Discrete transformations, such as parity, are outside the requested continuous-symmetry list.