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Physical Review

Reduction of Relativistic Two-Particle Wave Equations to Approximate Forms. II

Zeno V. Chraplyvy

Published Dec 1, 1953• 70 citations
Publisher

Abstract

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 "$\mathrm{uU}$ separating" or an "$\mathrm{lL}$ separating" ${\mathcal{H}}_{\mathrm{tr}}$ 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 ${\mathcal{H}}_{\mathrm{tr}}$, 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.

DOI
10.1103/physrev.92.1310

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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 mI\ensuremath≠mII{m}_{\mathrm{I}}\ensuremath{\ne}{m}_{\mathrm{II}}mI​\ensuremath=mII​. 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 \ensuremathβI{\ensuremath{\beta}}^{\mathrm{I}}\ensuremathβI and \ensuremathβII{\ensuremath{\beta}}^{\mathrm{II}}\ensuremathβII are replaced by 1 (or ---1). Consequently the reduced wave equation is the same in all cases.
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.
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General properties of the Foldy-Wouthuysen transformation which is widely used in quantum mechanics and quantum chemistry are considered. Merits and demerits of the original Foldy-Wouthuysen transformation method are analyzed. While this method does not satisfy the Eriksen condition of the Foldy-Wouthuysen transformation, it can be corrected with the use of the Baker-Campbell-Hausdorff formula. We show a possibility of such a correction and propose an appropriate algorithm of calculations. An applicability of the corrected Foldy-Wouthuysen method is restricted by the condition of convergence of a series of relativistic corrections.
Uses the non-relativistic limit of the integro-differential formulation of the Dirac equation in order to find an effective Schrodinger equation which, solved in the well known non-relativistic free-particle basis, gives approximately the same results (the probability amplitudes) as the Dirac equation for non-relativistic states written in terms of the free-particle spinors. Effective Hamiltonians for the electromagnetic and pseudoscalar interactions are derived. Comparison with the results of the Foldy-Wouthuysen method (1950) for the electromagnetic case is also achieved.
We construct the Foldy-Wouthuysen transformation operator for wave equations describing two interacting particle systems, and possessing Poincare invariance and manifest covariance. We consider systems composed of one spin-1/2 fermion and one spin-0 boson, and also of one spin-1/2 fermion and one spin-1/2 antifermion. We first construct the above operator in the free case. Then, we show that for several classes of interaction potential it can also be constructed in a compact form. As compared to the one-particle case, manifest covariance facilitates here the construction of the Foldy-Wouthuysen transformation operator, although the number of particles is larger On construit l'operateur de la transformation de Foldy-Wouthuysen pour des equations d'onde decrivant des systemes de deux particules en interaction, et possedant l'invariance de Poincare et la covariance manifeste. On considere des systemes composes d'un fermion de spin 1/2 et d'un boson de spin O, et aussi d'un fermion de spin 1/2 et d'un antifermion de spin 1/2. On construit d'abord cet operateur dans le cas libre. Puis on montre que plusieurs classes de potentiels d'interaction, il peut aussi etre construit sous une forme compacte. En comparaison avec le cas a une particule, la covariance manifeste facilite ici la construction de l'operateur de la transformation de Foldy-Wouthuysen, bien que le nombre de particules soit plus grand