= Solution
Let $\int_{\mathbf p}=\int d^3p/(2\pi)^3$. The <conjugate momentum> <fields> are $\pi_\phi=\dot\phi^\dagger$ and $\pi_{\phi^\dagger}=\dot\phi$. Inserting the given <mode expansion of a free field> into $i\int d^3x(\dot\phi^\dagger\phi-\phi^\dagger\dot\phi)$ produces spatial <Dirac delta> factors. The $b^\dagger c^\dagger$ and $cb$ terms vanish after integration: they contain $\omega_{\mathbf p}-\omega_{\mathbf q}$ on $\delta^3(\mathbf p+\mathbf q)$, and $\omega_{-\mathbf p}=\omega_{\mathbf p}$. The number-conserving terms instead add their two frequency factors, cancelling the mode normalization. The un-subtracted ordered expression is
$$
Q_{\mathrm{raw}}=\int_{\mathbf p}\left(c_{\mathbf p}c_{\mathbf p}^\dagger-b_{\mathbf p}^\dagger b_{\mathbf p}\right)
=\int_{\mathbf p}\left(c_{\mathbf p}^\dagger c_{\mathbf p}-b_{\mathbf p}^\dagger b_{\mathbf p}\right)
+\int_{\mathbf p}(2\pi)^3\delta^3(0).
$$
The last term is an infinite <vacuum> constant, not a finite physical <vacuum> charge. Choosing the <Fock vacuum> to have zero <Noether charge>, equivalently using <normal ordering>, gives
$$
\boxed{Q=\int_{\mathbf p}\left(c_{\mathbf p}^\dagger c_{\mathbf p}-b_{\mathbf p}^\dagger b_{\mathbf p}\right)=N_c-N_b.}
$$
A state containing $n_c$ quanta created by $c^\dagger$ and $n_b$ by $b^\dagger$ has <Noether charge> $n_c-n_b$. Thus the two families describe opposite unit <Noether charges>; with this source's current the $c^\dagger$ quantum has positive <Noether charge> and the $b^\dagger$ quantum negative <Noether charge>. This is an internal <Noether charge>; identifying it with <electric charge> requires a specified electromagnetic <coupling>. The <vacuum subtraction of a complex scalar charge> leaves all <commutators> unchanged.<Quantum Field Theory by David Tong>[Section 2.5, complex scalar fields and their opposite-charge oscillator families; the phase convention fixes the overall sign]{ref}
\b[Scope of the expansion:] a fixed-frequency expansion with time-independent oscillator coefficients is the free-field or asymptotic-field form. For arbitrary interacting $f$, it can be used as canonical oscillator coordinates at a chosen time, rather than as the exact interacting time dependence. For example, $f(s)=m^2s+\lambda s^2$ gives $(\Box+m^2)\phi+2\lambda\phi^*\phi\phi=0$, which a generic free plane-wave superposition does not satisfy. The <Noether charge> and its generator <action> remain valid in the interacting theory; the <symmetry> does not depend on using a free <Hamiltonian>.
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