High-order Foldy-Wouthuysen transformations of the Dirac and Dirac-Pauli Hamiltonians in the weak-field limit

Abstract

The low-energy and weak-field limit of Dirac equation can be obtained by an order-by-order block diagonalization approach to any desired order in the parameter $\pi /mc$ ($\pi$ is the kinetic momentum and $m$ is the mass of the particle). In the previous work, it has been shown that, up to the order of $(\pi /mc{)}^8$, the Dirac-Pauli Hamiltonian in the Foldy-Wouthuysen (FW) representation may be expressed as a closed form and consistent with the classical Hamiltonian, which is the sum of the classical relativistic Hamiltonian for orbital motion and the Thomas-Bargmann-Michel-Telegdi (T-BMT) Hamiltonian for spin precession. In order to investigate the exact validity of the correspondence between classical and Dirac-Pauli spinors, it is necessary to proceed to higher orders. In this paper, we investigate the FW representation of the Dirac and Dirac-Pauli Hamiltonians by using Kutzelnigg's diagonalization method. We show that the Kutzelnigg diagonalization method can be further simplified if nonlinear effects of static and homogeneous electromagnetic fields are neglected (in the weak-field limit). Up to the order of $(\pi /mc{)}^{14}$, we find that the FW transformation for both Dirac and Dirac-Pauli Hamiltonians is in agreement with the classical Hamiltonian with the gyromagnetic ratio given by $g=2$ and $g\ne 2$ respectively. Furthermore, with higher-order terms at hand, it is demonstrated that the unitary FW transformation admits a closed form in the low-energy and weak-field limit.

18 pages; follow-up to arXiv:1005.4128, arXiv:1310.8513; v2: more materials added

References in corpus (5)

• 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
• Comparative analysis of direct and "step-by-step" Foldy-Wouthuysen transformation methods
• Correspondence between classical and Dirac-Pauli spinors in view of the Foldy-Wouthuysen transformation

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• Energy expectation values of a particle in nonstationary fields
• Fully relativistic kinetic equation for spin-1/2 particles in the long scale-length approximation
• Emergence of chirality by multipole interconversion
• Non-relativistic expansion of single-nucleon Dirac equation: Comparison between Foldy-Wouthuysen transformation and similarity renormalization group
• Non-relativistic expansion of Dirac equation with spherical scalar and vector potentials by reconstituted Foldy-Wouthuysen transformation
• Exact Foldy-Wouthuysen transformation of the Dirac-Pauli Hamiltonian in the weak-field limit by the method of direct perturbation theory