= Vacuum subtraction of a complex scalar charge
{title2=$Q=\int d\Pi_p(a^\dagger a-b^\dagger b)$}
Substituting the free <complex scalar field> modes into $i\int(\phi^\dagger\dot\phi-\dot\phi^\dagger\phi)d^3x$ gives the formally ordered expression $Q_{\rm bare}=\int d\Pi_p(a^\dagger a-bb^\dagger)$. Its divergent vacuum constant is removed by <normal ordering>, equivalently by choosing $Q|0\rangle=0$. The resulting <complex scalar charge operator> has $[Q,a^\dagger]=a^\dagger$ and $[Q,b^\dagger]=-b^\dagger$. Subtracting the constant changes neither its action on fields nor current conservation, but it is essential when identifying the vacuum charge and the charges of one-particle states.
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