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arXiv:1605.09755 · Physical Review A

Exact form of the exponential Foldy-Wouthuysen transformation operator for an arbitrary-spin particle

Alexander J. Silenko

Published Sep 6, 2016• 34 citations
arXivHTMLPDF

Abstract

The exact exponential Foldy-Wouthuysen transformation operator applicable for a particle with an arbitrary spin is derived. It can be successfully utilized for verifying any Foldy-Wouthuysen transformation method based on the exponential operator. When a verified method is relativistic, the relativistic exponential operator should be expanded in the semirelativistic power series. The obtained exponential operator can be also used for a derivation of the Foldy-Wouthuysen Hamiltonian and its comparison with Hamiltonians found by other methods. This procedure makes it possible to check the validity of any other method of the Foldy-Wouthuysen transformation.

DOI
10.1103/physreva.94.032104

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The exact exponential Foldy-Wouthuysen transformation operator applicable for a particle with an arbitrary spin is derived. It can be successfully utilized for verifying any Foldy-Wouthuysen transformation method based on the exponential operator. When a verified method is relativistic, the relativistic exponential operator should be expanded in the semirelativistic power series. The obtained exponential operator can be also used for a derivation of the Foldy-Wouthuysen Hamiltonian and its comparison with Hamiltonians found by other methods. This procedure makes it possible to check the validity of any other method of the Foldy-Wouthuysen transformation.
The exact exponential Foldy-Wouthuysen transformation operator applicable for a particle with an arbitrary spin is derived. It can be successfully utilized for verifying any Foldy-Wouthuysen transformation method based on the exponential operator. When a verified method is relativistic, the relativistic exponential operator should be expanded in the semirelativistic power series. The use of the obtained exponential operator for a derivation of the Foldy-Wouthuysen Hamiltonian makes it possible to check the validity of any other method of the Foldy-Wouthuysen transformation, while this possibility may need a greater computational effort.
Since the important work of Newton and Wigner on the position operator and the localized states, several methods have been developed in the literature to deal with the problem of finding a position operator for relativistic systems. One of the most relevant has been the method based on the use of "canonical" transformations such as the Foldy-Wouthuysen transformation. In this paper, we strictly consider the Foldy-Wouthuysen transformation as a procedure which allows one to write the Bargmann-Wigner equations in a form which in the nonrelativistic limit leads to the Galilei-invariant Schr\"odinger equations for arbitrary spin. As a result of this interpretation, we derive for the position operator and the localized states of elementary systems the same expressions as Newton and Wigner; the spin operator is also obtained. Finally, the Chakrabarti transformation is considered in the same spirit.
The Foldy-Wouthuysen transformation for relativistic spin-1 particles interacting with nonuniform electric and uniform magnetic fields is performed. The Hamilton operator in the Foldy-Wouthuysen representation is determined. It agrees with the Lagrangian obtained by Pomeransky, Khriplovich, and Sen'kov. The validity of the Corben-Schwinger equations is confirmed. However, an attempt to generalize these equations in order to take into account the own quadrupole moment of particles was not successful. The known second-order wave equations are incorrect because they contain non-Hermitian terms. The correct second-order wave equation is derived.
A general method of the Foldy-Wouthyusen (FW) transformation for relativistic particles of arbitrary spin in strong external fields has been developed. The use of the found transformation operator is not restricted by any definite commutation relations between even and odd operators. The final FW Hamiltonian can be expanded into a power series in the Planck constant which characterizes the order of magnitude of quantum corrections. Exact expressions for low-order terms in the Planck constant can be derived. Finding these expressions allows one to perform a simple transition to the semiclassical approximation which defines a classical limit of the relativistic quantum mechanics. As an example, interactions of spin-1/2 and scalar particles with a strong electromagnetic field have been considered. Quantum and semiclassical equations of motion of particles and their spins have been deduced. Full agreement between quantum and classical theories has been established.