 Left-handed massless plane-wave spinor

In the standard Weyl representation of the gamma matrices and mostly-minus metric signature, the left chirality component obeys $i({\partial }_t-\mathbf{\sigma }\cdot \nabla ){\chi }_L=0$. A plane wave with $p^0>0$ exists precisely on the massless mass shell $p^0=|\mathbf{p}|$. Its coefficient is a negative helicity eigenspinor. For direction angles $\vartheta ,\phi$, one normalized choice isThe nonzero solution is unique up to overall normalization and phase. Different patches are required for a continuous phase choice over all directions; the direction of zero momentum is undefined.