Local fermion density commutator with the Dirac Hamiltonian ($[{\psi }^{†}\psi ,H]=-i\nabla \cdot (\bar{\psi }\mathbf{\gamma }\psi )$)

The equal-time canonical anticommutation relations implyApply $-i{\alpha }_{bc}^k{\partial }_{z^k}+m{\beta }_{bc}$, set $\mathbf{z}=\mathbf{y}$, and integrate over $\mathbf{y}$. The mass term cancels, leaving $i\int d^{3}y{\psi }^{†}{\alpha }^k\psi {\partial }_{y^k}{\delta }^{(3)}(\mathbf{x}-\mathbf{y})$. Integration by parts proves the title formula as a distribution identity. The Heisenberg equation of motion then yields ${\partial }_{t}n=-\nabla \cdot \mathbf{j}$ 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.