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Compt. rend.

A GENERALIZATION OF THE FOLDYWOUTHUYSEN TRANSFORMATION TO PARTICLES WITH ANY SPIN WITHOUT EXTERIOR FIELD AND SOME APPLICATIONS

M. Baktavatsalou

Published May 1, 1961• 0 citations
OpenAlex

Abstract

It is shown that the general Foldy-Wouthuysen transformation for a particle of any spin is obtained from different elementary transformations corresponding each to the particle of spin h/2. It was applied to hamiltonian and position operators. The last result was compared with that of Newton-Wigner. (tr-auth)

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•May 31, 2016•0 citations
A canonical transformation similar to that given by Foldy and Wouthuysen is obtained for the spin zero and one theories. For free particles positive and negative charge states are separately described by one or three component wave functions. It is found that the resulting forms for operators and the non-relativistic limits are quite similar to that for spin \textonehalf{}. The resulting similarity is found to carry over to still higher spins. Particles moving in magnetic fields are described most simply using the transformed representation.
An examination is made of the consequences for the quantum mechanics of spinning particles of equations characteristic of Lorentz-covariant position variables. These equations are commutator analogs of the Poisson bracket equations that express the familiar transformation properties of space-time events in classical mechanics. For a particle of zero spin it is found that the usual canonical coordinate is the unique solution of these equations. For a particle with positive spin there is no position operator which satisfies these equations and has commuting components. For a particle and antiparticle there is a unique solution with commuting components which is valid for all values of the spin and reduces for zero spin to the canonical coordinate. For spin \textonehalf{} this is the Foldy-Wouthuysen transform of the position operator of the Dirac equation. A generalization of the inverse Foldy-Wouthuysen transformation, valid for any value of the spin, appears as a unique unitary transformation which takes this generalized Dirac position to the canonical coordinate. The application of this transformation to the canonical form of the Hamiltonian gives a generalization of the Dirac equation Hamiltonian. This is developed and compared with the literature for spin 1. It gives a nonlocal equation as the spin 1 analog of the Dirac equation.
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.
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 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.