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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Foldy-Wouthuysen transformations in an indefinite-metric space. IV. Exact, closed-form expressions for first-order wave equations

Richard A. Krajcik, Michael Martin Nieto

Published Jan 15, 1977• 8 citations
Publisher

Abstract

We use a powerful matrix theorem to derive the closed-form, finite-polynomial, matrix expression of a Lorentz transformation for the class of relativistic, first-order wave equations described in the preceding paper. Combined with the method developed there, this theorem allows the exact, closed-form expression to be given for a Foldy-Wouthuysen (FW) transformation. An algorithm is given which is a simple procedural prescription for writing down the FW transformation results we have obtained. We discuss two specific examples which illustrate our results, and in a third example show that our method must be modified if the wave equation is not first-order in space.

DOI
10.1103/physrevd.15.426

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We use a powerful matrix theorem to derive the closed-form, finite-polynomial, matrix expression of a Lorentz transformation for the class of relativistic, first-order wave equations described in the preceding paper. Combined with the method developed there, this theorem allows the exact, closed-form expression to be given for a Foldy-Wouthuysen (FW) transformation. An algorithm is given which is a simple procedural prescription for writing down the FW transformation results we have obtained. We discuss two specific examples which illustrate our results, and in a third example show that our method must be modified if the wave equation is not first-order in space.
The method of reduction of two-particle relativistic wave equations (an extension of the Foldy-Wouthuysen method), as given in Part I, was applicable only if ${m}_{\mathrm{I}}\ensuremath{\ne}{m}_{\mathrm{II}}$. Other variants of the procedure, free from this restriction, are developed now. On the basis of a discussion of properties of the matrices involved, it is found that the postulate of an "even-even" transformed Hamiltonian was too far-reaching. The less stringent requirement of a "uU\mathrm{uU}uU separating" or an "lL\mathrm{lL}lL separating" Htr{\mathcal{H}}_{\mathrm{tr}}Htr​ leads to a whole class of usable transformations, which includes the transformation of Part I as a special case. Another important special case, (that of the "least change" transformation) has been calculated through in detail. Different transformations give different expressions for Htr{\mathcal{H}}_{\mathrm{tr}}Htr​, but they coincide after (as a part of the next step of the procedure) the matrices ${\ensuremath{\beta}}^{\mathrm{I}}$ and ${\ensuremath{\beta}}^{\mathrm{II}}$ are replaced by 1 (or ---1). Consequently the reduced wave equation is the same in all cases.
We apply Kutzelnigg's method for the Foldy-Wouthuysen (FW) transformation of the Dirac-Pauli Hamiltonian. The exact FW transformations exist and agree with those obtained by Eriksen's method for two special cases. In the weak-field limit of static and homogeneous electromagnetic fields, the long-held speculation is rigorously proven, by mathematical induction on the orders of 1/c1/c1/c in the power series, that the FW transformed 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.
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 Dirac Hamiltonian for a particle in a nonexplicitly time-dependent field is converted to an even Dirac matrix by means of a single canonical transformation. When the interaction term is an odd Dirac matrix, the transformed Hamiltonian is expressed in a very simple form. An exact transformation is also found for two-particle wave equations of Breit's type. The transformed Hamiltonian is then a uU\mathrm{uU}uU-separating matrix, in Chraplyvy's sense.In the nonrelativistic limit expansions in powers of 1m\frac{1}{m}m1​ or 1c\frac{1}{c}c1​ are made. The approximate wave equations are in agreement with previous transformation results.
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.