Dirac Spinor - Four-spinor For Particles

Four-spinor For Particles

Particles are defined as having positive energy. The normalization for the four-spinor ω is chosen so that . These spinors are denoted as u:

$u(\vec{p}, s) = \sqrt{E+m} \begin{bmatrix} \phi^{(s)}\\ \frac{\vec{\sigma} \cdot \vec{p} }{E+m} \phi^{(s)} \end{bmatrix} \,$

where s = 1 or 2 (spin "up" or "down")

Explicitly,

$u(\vec{p}, 1) = \sqrt{E+m} \begin{bmatrix} 1\\ 0\\ \frac{p_3}{E+m} \\ \frac{p_1 + i p_2}{E+m} \end{bmatrix} \quad \mathrm{and} \quad u(\vec{p}, 2) = \sqrt{E+m} \begin{bmatrix} 0\\ 1\\ \frac{p_1 - i p_2}{E+m} \\ \frac{-p_3}{E+m} \end{bmatrix}$

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