Covariant version of Pauli Hamiltonian, spin-induced non commutativity, Thomas precession and precession of spin

TL;DR

This paper develops a covariant form of the Pauli Hamiltonian that incorporates spin-induced noncommutativity and clarifies its relation to the classical and relativistic descriptions of spin precession, including Thomas precession.

Contribution

It introduces a manifestly covariant Hamiltonian for spin-1/2 particles that naturally accounts for noncommutative positions and explains the connection to the traditional Pauli Hamiltonian without Thomas precession assumptions.

Findings

Covariant Hamiltonian quadratic in spin and field strength.

Spin-induced noncommutativity explains the transition to Pauli Hamiltonian.

Clarifies the role of Thomas precession and spin transformations in different frames.

Abstract

We show that there is a manifestly covariant version of the Pauli Hamiltonian with equations of motion quadratic on spin and field strength. Relativistic covariance inevitably leads to noncommutative positions: classical brackets of the position variables are proportional to the spin. It is the spin-induced noncommutativity that is responsible for transforming the covariant Hamiltonian into the Pauli Hamiltonian, without any appeal to the Thomas precession formula. The Pauli theory can be thought to be $1/c^2$ approximation of the covariant theory written in special variables. These observations clarify the long standing question on the discrepancy between the covariant and Pauli Hamiltonians. We also discuss the transformational properties of spin axis in the passage from laboratory to comoving and instantaneous frames, and reveal the role of Thomas spin-vector in the covariant scheme.