Here, J and ρ are the current and charge density of the matter field. If the matter field is taken so as to describe the interaction of electromagnetic fields with the Dirac electron given by the four-component Dirac spinor field ψ, the current and charge densities have form: J = − − e ψ ψ † † α α ψ ψ ρ ρ = − − e ψ ψ † † ψ ψ , where α are the first three Dirac matrices. Using this, we can re-write Maxwell's equations as: