= Local fermion density commutator with the Dirac Hamiltonian
{title2=$[\psi^\dagger\psi,H]=-i\boldsymbol\nabla\cdot(\bar\psi\boldsymbol\gamma\psi)$}
The equal-time <canonical anticommutation relations> imply
$$
[n(\mathbf x),\psi_b^\dagger(\mathbf y)\psi_c(\mathbf z)]
=[\delta^{(3)}(\mathbf x-\mathbf y)-\delta^{(3)}(\mathbf x-\mathbf z)]\psi_b^\dagger(\mathbf y)\psi_c(\mathbf z),
\qquad n=\psi^\dagger\psi.
$$
Apply $-i\alpha^k_{bc}\partial_{z^k}+m\beta_{bc}$, set $\mathbf z=\mathbf y$, and integrate over $\mathbf y$. The <mass term> cancels, leaving $i\int d^3y\,\psi^\dagger\alpha^k\psi\,\partial_{y^k}\delta^{(3)}(\mathbf x-\mathbf y)$. <Integration by parts> proves the title formula as a <distribution> identity. The <Heisenberg equation of motion> then yields $\partial_t n=-\boldsymbol\nabla\cdot\mathbf j$ for the physical <Dirac current> $j^k=\bar\psi\gamma^k\psi$. The factor $-i$ is required because the <commutator> of <Hermitian operators> is anti-Hermitian.
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