= Particle and antiparticle occupation operators of a Dirac field
{title2=$N_{\rm total}=N_e+N_{\bar e},\quad Q=N_e-N_{\bar e}$}
The positive occupation operators are $N_e=\sum_s\int d^3p\,a_{\mathbf p}^{s\dagger}a_{\mathbf p}^s/(2\pi)^3$ and $N_{\bar e}=\sum_s\int d^3p\,b_{\mathbf p}^{s\dagger}b_{\mathbf p}^s/(2\pi)^3$. Both commute with the free Hamiltonian. The <normal-ordered> local density integral $\int:\psi^\dagger\psi:\,d^3x$ is their difference, not their sum. It generates the global phase symmetry. For electron charge $-e$, electric charge is $-eQ$. Pair creation can change total occupation while preserving this net charge.
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