Classical Origin of the Spin of Relativistic Pointlike Particles and Geometric interpretation of Dirac Solutions

TL;DR

This paper explores the classical origins of particle spin by analyzing Lorentz group representations acting on velocities, providing a geometric interpretation of Dirac solutions that bridges classical and quantum descriptions.

Contribution

It introduces a classical framework for spin using velocity transformations, offering a geometric interpretation of Dirac spinors and connecting classical velocities with quantum spin.

Findings

Spin can be described classically via velocity transformations.

Dirac spinors correspond to rotations of velocity vectors.

The approach yields a compact group with finite-dimensional unitary representations.

Abstract

Spin of elementary particles is the only kinematic degree of freedom not having classical corre- spondence. It arises when seeking for the finite-dimensional representations of the Lorentz group, which is the only symmetry group of relativistic quantum field theory acting on multiple-component quantum fields non-unitarily. We study linear transformations, acting on the space of spatial and proper-time velocities rather than on coordinates. While ensuring the relativistic in- variance, they avoid these two exceptions: they describe the spin degree of freedom of a pointlike particle yet at a classical level and form a compact group hence with unitary finite-dimensional rep- resentations. Within this approach changes of the velocity modulus and direction can be accounted for by rotations of two independent unit vectors. Dirac spinors just provide the quantum description of these rotations.