Correspondence between classical and Dirac-Pauli spinors in view of the Foldy-Wouthuysen transformation

Abstract

The classical dynamics for a charged spin particle is governed by the Lorentz force equation for orbital motion and by the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation for spin precession. In static and homogeneous electromagnetic fields, it has been shown that the Foldy-Wouthuysen (FW) transform of the Dirac-Pauli Hamiltonian, which describes the relativistic quantum theory for a spin-1/2 particle, is consistent with the classical Hamiltonian (with both the orbital and spin parts) up to the order of $1/m^{14}$ ($m$ is the particle's mass) in the low-energy/weak-field limit. In this paper, we extend this correspondence to the case of inhomogeneous fields. Regardless of the field gradient (e.g., Stern-Gerlach) force, the T-BMT equation is unaltered and thus the classical Hamiltonian remains the same, but subtleties arise and need to be clarified. For the relativistic quantum theory, we apply Eriksen's method to obtain the exact FW transformations for the two special cases, which in conjunction strongly suggest that, in the weak-field limit, the FW transformed Dirac-Pauli Hamiltonian (except for the Darwin term) is in agreement with the classical Hamiltonian in a manner that classical variables correspond to quantum operators via a specific Weyl ordering. Meanwhile, the Darwin term is shown to have no classical correspondence.

14 pages, 1 table; one subsection added; version to appear in PRA

References in corpus (4)

• Topological spin transport of relativistic electron
• Semiclassical Diagonalization of Quantum Hamiltonian and Equations of Motion with Berry Phase Corrections
• Semiclassical Dynamics of Dirac particles interacting with a Static Gravitational Field
• Quantum-mechanical description of spin-1 particles with electric dipole moments

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• General method of the relativistic Foldy-Wouthuysen transformation and proof of validity of the Foldy-Wouthuysen Hamiltonian
• Spin-one-half particles in strong electromagnetic fields: spin effects and radiation reaction
• Identifying the Stern-Gerlach force of classical electron dynamics
• Energy expectation values of a particle in nonstationary fields
• Fully relativistic kinetic equation for spin-1/2 particles in the long scale-length approximation
• Lagrangian geometrical optics of nonadiabatic vector waves and spin particles
• High-order Foldy-Wouthuysen transformations of the Dirac and Dirac-Pauli Hamiltonians in the weak-field limit
• Exact Foldy-Wouthuysen transformation of the Dirac-Pauli Hamiltonian in the weak-field limit by the method of direct perturbation theory
• Exact Foldy-Wouthuysen transformation for a Dirac equation describing the interaction of spin-1/2 relativistic particles with an external electromagnetic field
• Stationary Schrödinger Equation and Darwin Term from Maximal Entropy Random Walk