= Charged oscillator basis of a scalar doublet
{title2=$b_\pm^\dagger=(a_1^\dagger\mp ia_2^\dagger)/\sqrt2$}
For the <internal rotation symmetry of two real scalar fields> with $\delta\phi_1=\alpha\phi_2$ and $\delta\phi_2=-\alpha\phi_1$, the <Noether charge> is $Q=i\int(a_1^\dagger a_2-a_2^\dagger a_1)d^3p/(2\pi)^3$. Its <commutators> satisfy $[Q,b_\pm^\dagger]=\pm b_\pm^\dagger$, so these <creation operators> produce charge <eigenstates> in the <one-particle state> sector. Substitution gives $Q=N_+-N_-$, converting the rotation of real components into opposite charges in a complex basis.
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