Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 301 / 2 / vii / Solution

Differentiate the Noether charge and use ${\dot{\varphi }}_a={\pi }_a$ and the Klein-Gordon equation for each species. The momentum products cancel because the two species commute:The first integral is the boundary flux of ${\varphi }_2\nabla {\varphi }_1-{\varphi }_1\nabla {\varphi }_2$, so it vanishes under the same boundary condition used for charge conservation. HenceConservation must hold for arbitrary configurations, rather than only a state in which this particular integral happens to vanish. Thus the internal rotation symmetry of two real scalar fields requiresEquivalently, the local divergence is ${\partial }_{\mu }{j}^{\mu }=(m_2^2-m_1^2){\varphi }_1{\varphi }_2$, exhibiting exactly how unequal masses break the symmetry.