= Complex scalar charge operator
{title2=$Q=N_a-N_b$}
For the global phase transformation $\delta\phi=-i\alpha\phi$, the <Noether current> is $j^\mu=i(\phi^\dagger\partial^\mu\phi-\partial^\mu\phi^\dagger\,\phi)$. The <Klein-Gordon equation> gives $\partial_\mu j^\mu=0$. With zero vacuum charge, its <normal-ordered> charge is $Q=\int d\Pi_p(a^\dagger a-b^\dagger b)$. Thus $[Q,a^\dagger]=a^\dagger$, $[Q,b^\dagger]=-b^\dagger$ and $[Q,\phi]=-\phi$. Particle and <antiparticle> creation carry opposite charges while both increase energy.
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