A. Zee, Quantum Field Theory in a Nutshell · Problem II.1.3

The Dirac equation in momentum space reveals the internal structure of the spinor $\psi(p) = \begin{pmatrix} \phi \\ \chi \end{pmatrix}$ by exploiting rotational symmetry—you can always choose coordinates so momentum points along the $z$-axis, which simplifies the $\gamma$-matrices and decouples the equations. The key insight is that $\phi$ and $\chi$ are related through the energy-momentum constraint, and their relative size depends on how fast the electron moves: at rest or low speeds, the "small component" $\chi$ is suppressed by factors of $v/c$ or $p/m$, but as the electron approaches light speed, both components become comparable in magnitude.

The Dirac equation in momentum space reveals the internal structure of the spinor $\psi(p) = \begin{pmatrix} \phi \\ \chi \end{pmatrix}$ by exploiting rotational symmetry—you can always choose coordinates so momentum points along the $z$-axis, which simplifies the $\gamma$-matrices and decouples the equations. The key insight is that $\phi$ and $\chi$ are related through the energy-momentum constraint, and their relative size depends on how fast the electron moves: at rest or low speeds, the "small component" $\chi$ is suppressed by factors of $v/c$ or $p/m$, but as the electron approaches light speed, both components become comparable in magnitude.