A Transport Theory Route to the Dirac Equation

Abstract

Starting from the similarity between the spherical harmonics approximation of order one to the linear transport equation (usually referred as P₁ approximation) and the Klein-Gordon equation of the quantum physics, an extended set of equations is introduced, which is proved to be equivalent to the Dirac equation with imaginary mass. Conversely, when a real mass is restored into the extended P₁ system, a new equation is obtained, whose solutions are superposition of the spinors for a $\frac{1}{2}-$ spin particle and the corresponding antiparticle.

KEYWORDS:

• Linear transport theory
• P₁ approximation
• Dirac equation

Notes

1 The Dirac matrices are defined as: ${\alpha }_i=(\begin{array}{cc}0& {\sigma }_i\\ {\sigma }_i& 0\end{array}),$ i = 1, 2, 3, where ${\sigma }_i$ are the Pauli matrices, and ${\alpha }_m=(\begin{array}{cc}{I}_{\mathbf{2}\times \mathbf{2}}& 0\\ 0& -I_{\mathbf{2}\times \mathbf{2}}\end{array}).$