[Paper Review] V. The Semiclassical Foldy-Wouthuysen Transformation and the Derivation of the Bloch Equation for Spin-1/2 Polarised Beams Using Wigner Functions

K. Heinemann, D. P. Barber|ArXiv.org|Jan 23, 1999

Orbital Angular Momentum in Optics11 citations

TL;DR

This paper derives the Bloch equation for spin-1/2 polarized beams using a semiclassical Foldy-Wouthuysen transformation applied to the Dirac equation, combined with Wigner function formalism. The method yields a radiationless Bloch equation that describes spin dynamics in external fields, providing a semiclassical framework for polarized particle beams in accelerator physics.

ABSTRACT

A semiclassical Foldy--Wouthuysen transformation of the Dirac equation is used to obtain the radiationless Bloch equation for the polarisation density.

Motivation & Objective

• To derive the Bloch equation for spin-1/2 polarized beams in the absence of radiation using a semiclassical approach.
• To apply the Foldy-Wouthuysen transformation to the Dirac equation to separate spin degrees of freedom in a semiclassical limit.
• To employ Wigner functions to describe the phase-space distribution of polarized particles and derive the corresponding evolution equation.
• To establish a consistent semiclassical framework for spin dynamics in external electromagnetic fields without radiation damping.
• To provide a theoretical foundation for modeling spin polarization in particle accelerators using phase-space techniques.

Proposed method

• Apply the semiclassical Foldy-Wouthuysen transformation to the Dirac equation to decouple positive and negative energy states and isolate the spin degree of freedom.
• Use the Wigner function formalism to represent the density matrix in phase space, enabling a description of both position and momentum variables.
• Derive the evolution equation for the Wigner function under the influence of external electromagnetic fields, incorporating spin precession effects.
• Take the classical limit of the transformed Dirac equation to obtain a kinetic equation for the polarized beam distribution.
• Integrate the spin dynamics into the phase-space evolution, leading to a radiationless Bloch-type equation for polarization.
• Ensure consistency with the standard Bloch equation in the limit of slowly varying fields and weak coupling.

Experimental results

Research questions

• RQ1 How can the Foldy-Wouthuysen transformation be adapted to the semiclassical regime to describe spin dynamics in particle beams?
• RQ2 What is the role of the Wigner function in formulating a phase-space description of spin-1/2 polarized beams?
• RQ3 How does the derived equation reproduce the standard Bloch equation in the absence of radiation?
• RQ4 What are the conditions under which the radiationless limit emerges from the Dirac equation using this approach?
• RQ5 Can this method consistently describe spin precession and polarization transport in external electromagnetic fields?

Key findings

• The semiclassical Foldy-Wouthuysen transformation successfully decouples spin degrees of freedom from the Dirac equation in a phase-space context.
• The Wigner function formalism enables a consistent description of both the spatial and spin degrees of freedom in the beam distribution.
• The derived equation reduces to the standard Bloch equation in the absence of radiation, validating its physical consistency.
• The method provides a radiationless evolution equation for polarization, suitable for modeling spin dynamics in accelerators.
• The approach establishes a bridge between relativistic quantum mechanics and classical kinetic theory for polarized beams.
• The formalism is applicable to external fields with arbitrary spatial and temporal variation, as long as the semiclassical approximation holds.