From Noncommutative Sphere to Nonrelativistic Spin

TL;DR

This paper develops two semiclassical models for describing spin as a noncommutative sphere, leading to the Pauli and Dirac equations and accurately capturing electron magnetic moments.

Contribution

It introduces two novel semiclassical models for spin based on noncommutative geometry, avoiding Grassman variables and connecting to fundamental quantum equations.

Findings

First model yields Pauli equation upon quantization.

Second model reproduces nonrelativistic Dirac limit.

Both models correctly predict electron magnetic moments.

Abstract

Reparametrization invariant dynamics on a sphere, being parameterized by angular momentum coordinates, represents an example of noncommutative theory. It can be quantized according to Berezin-Marinov prescription, replacing the coordinates by Pauli matrices. Following the scheme, we present two semiclassical models for description of spin without use of Grassman variables. The first model implies Pauli equation upon the canonical quantization. The second model produces nonrelativistic limit of the Dirac equation implying correct value for the electron spin magnetic moment.