= Left-handed massless plane-wave spinor
{title2=$\lambda_L(p)=N\binom{\chi_-(\widehat{\mathbf p})}{0}$}
In the standard <Weyl representation of the gamma matrices> and mostly-minus <metric signature>, the left <chirality> component obeys $i(\partial_t-\boldsymbol\sigma\cdot\boldsymbol\nabla)\chi_L=0$. 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 $\vartheta,\varphi$, one normalized choice is
$$
\chi_-=\begin{pmatrix}-e^{-i\varphi}\sin(\vartheta/2)\\\cos(\vartheta/2)\end{pmatrix},\qquad
(\boldsymbol\sigma\cdot\widehat{\mathbf p})\chi_-=-\chi_-.
$$
The 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.
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