Arbitrary Spin Galilean Oscillator

TL;DR

This paper extends the Dirac oscillator model to arbitrary spin within Galilean relativity, demonstrating how spin can be incorporated into non-relativistic quantum frameworks and analyzing the spin-orbit coupling behavior.

Contribution

It introduces a method to include arbitrary spin in the Galilean oscillator framework using symmetric multispinors, expanding the applicability of the Dirac oscillator model.

Findings

Explicit formulations for spin zero and one cases.

The spin-orbit coupling coefficient scales as 1/S for arbitrary spin.

The approach parallels historical techniques used for g-factor derivations.

Abstract

The so-called Dirac oscillator was proposed as a modification of the free Dirac equation which reproduces many of the properties of the simple harmonic oscillator but accompanied by a strong spin-orbit coupling term. It has yet to be extended successfully to the arbitrary spin S case primarily because of the unwieldiness of general spin Lorentz invariant wave equations. It is shown here using the formalism of totally symmetric multispinors that the Dirac oscillator can, however, be made to accommodate spin by incorporating it into the framework of Galilean relativity. This is done explicitly for spin zero and spin one as special cases of the arbitrary spin result. For the general case it is shown that the coefficient of the spin-orbit term has a 1/S behavior by techniques which are virtually identical to those employed in the derivation of the g-factor carried out over four decades ago.