= Fixed-axis spin obstruction for positive-energy Dirac spinors
{c}
{title2=$p_y^2+p_z^2=0$}
A positive-energy standard-representation <Dirac spinor> has upper component $\xi$ and lower component $(\boldsymbol\sigma\cdot\mathbf p)\xi/(E+m)$. To be an eigenstate of $\operatorname{diag}(\sigma_x,\sigma_x)/2$, both components must have the same $\sigma_x$ eigenvalue. With $\xi=\chi_\pm$, the commutator obstruction $2i(p_y\sigma_z-p_z\sigma_y)\chi_\pm$ has squared norm $4(p_y^2+p_z^2)$. It vanishes precisely for <momentum> parallel to x, including zero. Exact transverse-spin eigenstates therefore fail at nonzero transverse <momentum>; the obstruction is not a universal failure at every relativistic <energy>.
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