Past exam of the theoretical physics I course of the University of Cambridge / 2022 / 4 / b / Solution

The infinitesimal connected internal symmetry of a classical field theory is $\delta {\varphi }_1=ϵ{\varphi }_1$, $\delta {\varphi }_2=-ϵ{\varphi }_2$. For this exact invariance of the Lagrangian density, the Noether current is $J^{\mu }={\sum }_j{\mathcal{L}}_{{\partial }_{\mu }{\varphi }_j}\delta {\varphi }_j/ϵ$. In coordinates $(t,x)$ it isVarying the two fields gives the two wave equations ${\varphi }_{1,tt}-c^2{\varphi }_{1,xx}=0$ and ${\varphi }_{2,tt}-c^2{\varphi }_{2,xx}=0$. Direct differentiation cancels the first-derivative cross terms:The Noether charge $Q=\int J^{0}dx$ is constant when the spatial boundary flux vanishes. The disconnected transformations from part (a) do not have independent infinitesimal generators of this type.