= Dirac plane waves in the standard representation
{c}
{title2=$u_s=\sqrt{E+m}\binom{\chi_s}{(\boldsymbol\sigma\cdot\mathbf p)\chi_s/(E+m)}$}
For $g=\operatorname{diag}(1,-1,-1,-1)$ and the block-diagonal standard representation of the <gamma matrices>, write a positive-energy <Dirac spinor> as $(\xi,\eta)^T$. The <Dirac equation> gives $\eta=(\boldsymbol\sigma\cdot\mathbf p)\xi/(E+m)$. Choosing two orthonormal two-spinors $\chi_s$ yields the displayed positive-frequency solutions. Negative-frequency solutions are $v_s=\sqrt{E+m}((\boldsymbol\sigma\cdot\mathbf p)\eta_s/(E+m),\eta_s)^T$ multiplying $e^{ip\cdot x}$. They obey $(\not p+m)v_s=0$. Both have norm $2E$; their <Dirac adjoints> give $\overline u_ru_s=2m\delta_{rs}$ and $\overline v_rv_s=-2m\delta_{rs}$. Spatial rotations act on each two-spinor with generators $\sigma_j/2$, explaining <spin one-half>.
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