= Solution
For nonzero <momentum>, physical <helicity> is the <Spin>[spin projection] along the direction of motion,
$$
\boxed{\hat h=\frac{\mathbf S\cdot\mathbf p}{|\mathbf p|}=\frac12\begin{pmatrix}\boldsymbol\sigma\cdot\hat{\mathbf p}&0\\0&\boldsymbol\sigma\cdot\hat{\mathbf p}\end{pmatrix},\qquad h=\pm\frac12.}
$$
The convention without $1/2$ labels the same two <eigenspaces> by $\pm1$. Write $\hat{\mathbf p}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$. <Normalization>[Normalized] <Spinor>[two-component eigenvectors] are
$$
\chi_+=\begin{pmatrix}\cos(\theta/2)\\e^{i\varphi}\sin(\theta/2)\end{pmatrix},\qquad
\chi_-=\begin{pmatrix}-e^{-i\varphi}\sin(\theta/2)\\\cos(\theta/2)\end{pmatrix},\qquad
(\boldsymbol\sigma\cdot\hat{\mathbf p})\chi_\pm=\pm\chi_\pm.
$$
Each can carry an arbitrary <phase>. Before imposing an <energy> equation, the general <Dirac spinor>[four-component] <helicity> <eigenvector> is $(a\chi_\pm,b\chi_\pm)^{\mathsf T}$, with arbitrary complex $a,b$; unit <Norm> imposes $|a|^2+|b|^2=1$. For a <Energy eigenvalue>[positive-energy] free <electron>, the <Dirac equation> fixes their ratio and gives
$$
\boxed{u_\pm(\mathbf p)=\sqrt{\frac{E+m}{2E}}\begin{pmatrix}\chi_\pm\\\displaystyle\pm\frac{|\mathbf p|}{E+m}\chi_\pm\end{pmatrix},\qquad u_\pm^\dagger u_\pm=1.}
$$
Multiplying by an arbitrary overall <Probability amplitude>[amplitude] and the common <plane wave> supplies the corresponding free <helicity> eigenstate. <Helicity> has no preferred direction at $\mathbf p=0$. For a massive particle a <Lorentz boost>[boost] can overtake its <momentum> and change its <helicity>; it should not be identified with <Lorentz invariance>[Lorentz-invariant] <chirality>.
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