Weyl equation ($i({\partial }_t\mp \mathbf{\sigma }\cdot \nabla ){\chi }_{L/R}=0$)

The free massless relativistic field equation for one chiral two-component spinor. With mostly-minus metric, the left-chiral equation has the minus sign and the right-chiral equation the plus sign. The positive-energy particle helicities are respectively $-1/2$ and $+1/2$; antiparticle helicities are opposite.

For a coherent two-component field $\psi =e^{-i{V}_{0}t}(\eta (t-z),\rho (t+z){)}^T$, the physical current is $\mathbf{J}=(2\mathrm{Re}{\eta }^{\ast }\rho ,2\mathrm{Im}{\eta }^{\ast }\rho ,|\eta {|}^2-|\rho {|}^2)$. The density is $|\eta {|}^2+|\rho {|}^2$. Transverse components generally survive interference even though the field has no transverse coordinate dependence. They have zero transverse divergence; together with the longitudinal current they obey local conservation and $|\mathbf{J}{|}^2=({\psi }^{†}\psi {)}^2$.