Dirac current ($j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$)

The global phase symmetry of a Dirac field gives the conserved Dirac current $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$.

For a commuting spinor in mostly-plus signature, orient its Dirac current to the future and write $V^i={ϵ}^{†}{\alpha }^iϵ$, with Hermitian ${\alpha }^i=-{\gamma }^0{\gamma }^i$. For every spatial unit vector $u$, $(u\cdot \alpha {)}^2=1$ implies $|u\cdot V|\le {ϵ}^{†}ϵ=V^0$. Choosing $u$ along the spatial current proves $V^2\le 0$. Thus spinor bilinears supply causal vectors suitable for energy-condition estimates. A Grassmann-valued spinor does not have this ordinary positivity interpretation.

In four dimensions a single commuting Majorana spinor is specified by a two-component complex spinor $\xi$. Its future Dirac current is proportional to $({\xi }^{†}\xi ,{\xi }^{†}\mathbf{\sigma }\xi )$, and the Pauli matrix multiplication law implies $|{\xi }^{†}\mathbf{\sigma }\xi {|}^2=({\xi }^{†}\xi {)}^2$. Thus it is null. A Dirac spinor or a sum of independent Majorana currents can be timelike. This distinction matters when applying a timelike-current classification to minimal real supersymmetry.

The Gordon identity rewrites a vector-current matrix element between equal-mass on-shell Dirac spinors using a momentum term and an antisymmetric spin term. In the mostly-minus convention $\{{\Gamma }^a,{\Gamma }^b\}=2{\eta }^{ab}$, $(/p-m)u=0$, and ${\Gamma }^{ab}=[{\Gamma }^a,{\Gamma }^b]/2$, it reads $2m{\bar{u}}^{\prime }{\Gamma }^{a}u={\bar{u}}^{\prime }[(p^{\prime }+p{)}^a-{\Gamma }^{ab}(p_b^{\prime }-p_b)]u$. Derive it by adding the two external Dirac equation identities. The equivalent formula in the mostly-plus Dirac convention is ${\bar{u}}^{\prime }[{\gamma }^{ab}(p_b^{\prime }-p_b)+2m{\gamma }^a+p^a+p^{\prime a}]u=0$. The identity before division by $m$ remains valid for massless equal-mass states.

The Dirac electromagnetic current is the electric charge times the Dirac current. Its coupling $-e{j}^{\mu }{A}_{\mu }$ to the electromagnetic four-potential is the matter interaction of quantum electrodynamics.

An interaction $-\lambda j^{\mu }{\partial }_{\mu }\varphi$ with an exactly conserved current is redundant: integration by parts gives $\lambda \varphi {\partial }_{\mu }{j}^{\mu }$ up to a boundary term. For the Dirac current, the local phase redefinition $\psi =e^{-i\lambda \varphi }\chi$ removes the coupling, so its on-shell scattering amplitudes vanish.