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arXiv:math-ph/0612045 · Physics of Particles and Nuclei Letters

Connection between wave functions in the Dirac and Foldy-Wouthuysen representations

A. J. Silenko

Published Nov 1, 2008• 21 citations
arXivHTMLPDF

Abstract

When the Foldy-Wouthuysen (FW) transformation is exact and the particle energy is positive, upper spinors in the Dirac and FW representations differ only by a constant factor, and lower spinors in the FW representation are zero. Deducing FW wave eigenfunctions directly from Dirac wave eigenfunctions allows one to use the FW representation to calculate expectation values of needed operators and to derive quantum and semiclassical equations of motion.

DOI
10.1134/s1547477108060034

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When the Foldy-Wouthuysen (FW) transformation is exact and the particle energy is positive, upper spinors in the Dirac and FW representations differ only by a constant factor, and lower spinors in the FW representation are zero. Deducing FW wave eigenfunctions directly from Dirac wave eigenfunctions allows one to use the FW representation to calculate expectation values of needed operators and to derive quantum and semiclassical equations of motion.
The paper describes conditions for transformation from the Dirac representation to the Foldy-Wouthuysen representation. The necessary condition is the block-diagonal transformation of Hamiltonian relative to the upper and lower components of the wave function. The sufficient condition is the wave function transformation law described by relation (6). It has been demonstrated that the unitary transformations offered by the authors of the papers [14], [15], [16], [17] do not satisfy the sufficiency condition (6) and, hence, they are not the Foldy-Wouthuysen transformations. In applications, the matrix elements of any operator in the FW representation can be calculated, according to (6), using the normalized two-component wave functions in the Dirac representation known for the given problem.
It has long been speculated that the Dirac or, more generally, the Dirac-Pauli spinor in the Foldy-Wouthuysen (FW) representation should behave like a classical relativistic spinor in the low-energy limit when the probability of particle-antiparticle pair creation and annihilation is negligible. In the weak-field limit of static and homogeneous electromagnetic fields, by applying the method of direct perturbation theory inductively on the orders of 1/c1/c1/c in the power series, we rigorously prove that it is indeed the case: the FW transformation of the Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.
Nonlocalities of Foldy-Wouthuysen and related transformations, which are used to separate positive- and negative-energy states in the Dirac equation, are investigated. Second moments of functional kernels generated by the transformations are calculated, and the transformed functions and their variances are computed. It is shown that all the transformed quantities are smeared in the coordinate space by an amount comparable to the Compton wavelength ${\ensuremath{\lambda}}_{c}=\ensuremath{\hbar}/mc$.