Causal Dirac spinor current ($V^0={ϵ}^{†}ϵ,V^2\le 0$)

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.