Foldy-Wouthuysen Transformation and Semiclassical Limit for Relativistic Particles in Strong External Fields

Abstract:
The Foldy–Wouthuysen (FW) representation occupies a special place in the quantum theory. For relativistic particles in external fields, operators in this representation have the same form as in the nonrelativistic quantum theory. The relations between the operators in the FW representation are similar to those between the respective classical quantities. Only the FW representation possesses these properties considerably simplifying the transition to the semiclassical description. As a result, this representation provides the best possibility of obtaining a meaningful classical limit of the relativistic quantum mechanics.
There is a limited class of Hamiltonians those transformation to the FW representation is exact [1,2]. In the general case, the exact FW transformation has been performed in Ref. [3]. However, the derived FW Hamiltonian cannot be used for obtaining a semiclassical limit of the Dirac equation for relativistic particles because it is very cumbersome and contains roots of Dirac matrix operators. The methods of performing the FW transformation and finding the semiclassical limit of wave equations for relativistic spin-1/2 and spin-1 particles have been developed in Refs. [2] and [4], respectively. These methods can be used only in the weak-field approximation.
In the present work, a new method of the FW transformation for relativistic particles of arbitrary spin in strong external fields is proposed. This method is based on the previous developments [2,4]. However, it does not need any definite commutation relations between even and odd operators. The general form of a transition operator has been found. The final FW Hamiltonian can be expanded into a power series in the Planck constant. Since just this constant defines the order of magnitude of quantum corrections, the transition to the semiclassical approximation becomes trivial. As an example, interactions of scalar particles and Dirac ones with a strong electromagnetic field have been analyzed. The equation of spin motion of Dirac particles in the strong electromagnetic field has been derived.

1. A. G. Nikitin, J. Phys. A 31, 3297 (1998).
2. A. J. Silenko, J. Math. Phys. 44, 2952 (2003).
3. E. Eriksen, Phys. Rev. 111, 1011 (1958).
4. A. J. Silenko, Zs. Eksp. Teor. Fiz. 123, 883 (2003) [JETP 96, 775 (2003)].