Quantum-classical correspondence of the Dirac matrices: The Dirac Lagrangian as a Total Derivative

TL;DR

This paper offers a physical interpretation of the Dirac equation by linking Dirac spinors to classical vectors representing particle speed and proper time change, revealing the Dirac Lagrangian as a total derivative.

Contribution

It introduces a novel classical-quantum correspondence for Dirac matrices, providing physical insight into spin, antiparticles, and the Dirac Lagrangian's meaning.

Findings

Dirac spinors correspond to two classical vectors.

Spin and antiparticles relate to rotation symmetry of vectors.

Dirac Lagrangian is a total derivative with physical meaning.

Abstract

The Dirac equation provides a description of spin 1/2 particles, consistent with both the principles of quantum mechanics and of special relativity. Often its presentation to students is based on mathematical propositions that may hide the physical meaning of its contents. Here we show that Dirac spinors provide the quantum description of two unit classical vectors: one whose components are the speed of an elementary particle and the rate of change of its proper time and a second vector which fixes the velocity direction. In this context both the spin degree of freedom and antiparticles can be understood from the rotation symmetry of these unit vectors. Within this approach the Dirac Lagrangian acquires a direct physical meaning as the quantum operator describing the total time-derivative.