Vacuum subtraction of a complex scalar charge ($Q=\int d{\Pi }_p(a^{†}a-b^{†}b)$)

Substituting the free complex scalar field modes into $i\int ({\varphi }^{†}\dot{\varphi }-{\dot{\varphi }}^{†}\varphi )d^{3}x$ gives the formally ordered expression $Q_{\mathrm{bare}}=\int d{\Pi }_p(a^{†}a-b{b}^{†})$. Its divergent vacuum constant is removed by normal ordering, equivalently by choosing $Q|0⟩=0$. The resulting complex scalar charge operator has $[Q,a^{†}]=a^{†}$ and $[Q,b^{†}]=-b^{†}$. 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.