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arXiv:math-ph/0404067 · Journal of Mathematical Physics

Foldy–Wouthuysen transformation for relativistic particles in external fields

Alexander J. Silenko

Published Jun 19, 2003• 111 citations
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Abstract

A method of Foldy–Wouthuysen transformation for relativistic spin-1/2 particles in external fields is proposed. It permits the determination of the Hamilton operator in the Foldy–Wouthuysen representation with any accuracy. Interactions between a particle having an anomalous magnetic moment and nonstationary electromagnetic and electroweak fields are investigated.

DOI
10.1063/1.1579991

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A method of Foldy–Wouthuysen transformation for relativistic spin-1/2 particles in external fields is proposed. It permits the determination of the Hamilton operator in the Foldy–Wouthuysen representation with any accuracy. Interactions between a particle having an anomalous magnetic moment and nonstationary electromagnetic and electroweak fields are investigated.
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 classical and quantum formulae for the Hamiltonian agree. The validity of the Corben-Schwinger equations is confirmed. However, it is difficult to generalize these equations in order to take into account the quadrupole moment defined by a particle charge distribution. The known second-order wave equations are not quite satisfactory because they contain non-Hermitian terms. The Hermitian second-order wave equation is derived.
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.
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.