= Solution
The <Hamiltonian operator> is invariant under the constant <global phase symmetry> $\psi\mapsto e^{i\theta}\psi$, $\bar\psi\mapsto\bar\psi e^{-i\theta}$. The <Noether current> of this <unitary group> $U(1)$ is the <Dirac current> obtained in <3/b>. For fields with vanishing flux at infinity, or periodic boundary conditions, the <divergence theorem> and that <continuity equation> imply
$$
\boxed{Q=\int d^3x\,:\!\psi^\dagger\psi\!:\ ,\qquad [Q,H]=0,\qquad \frac{dQ}{dt}=0.}
$$
The <canonical anticommutation relations> also give $[Q,\psi]=-\psi$ and $[Q,\psi^\dagger]=\psi^\dagger$, hence $e^{i\theta Q}\psi e^{-i\theta Q}=e^{-i\theta}\psi$. Reversing the parameter sign recovers the stated phase convention. Thus $Q$ generates the <global phase symmetry>. With <normal ordering>, $Q$ counts <fermions> minus <antiparticles>, rather than the sum of both populations. Its <conservation law> is <Dirac fermion number conservation>. A position-dependent phase produces derivative terms in this free <Hamiltonian operator>; the symmetry under discussion is global.
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