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THE FUSION OF TWO PARTICLES IN FUNCTIONAL THEORY

Florence A. Aeschlimann

Published Dec 1, 1961• 0 citations
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Abstract

A study is made of a system of two particles with spin in relativistic functional theory. Equation of the barycentric wave, equations of the relative motion, and the reduced equation are obtained. In the states of the system, in some cases there exist states where the relative waves are constants. This case corresponds to vanishing relative motion. In the case of two particles of spin 1/ 2 of the same mass, a fused system is obtained which is equivalent to a particle with spin 1. From this result, by the approximation giving the usual wave mechanics, the equations given by the fusion method are obtained. The fusion method is thus justified and its physical significance is apparent. (auth.)

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A study is made of a system of two particles with spin in relativistic functional theory. Equation of the barycentric wave, equations of the relative motion, and the reduced equation are obtained. In the states of the system, in some cases there exist states where the relative waves are constants. This case corresponds to vanishing relative motion. In the case of two particles of spin 1/ 2 of the same mass, a fused system is obtained which is equivalent to a particle with spin 1. From this result, by the approximation giving the usual wave mechanics, the equations given by the fusion method are obtained. The fusion method is thus justified and its physical significance is apparent. (auth.)
A definition is given of the barycentric wave for a system of n particles in the functional theory, in the relativistic case, and in the non- relativistic case with spin and isospin or without spin. The equation for the barycentric wave in the relativistic case and in the non-relativistic case is derived. The case of particles of the same mass is considered. Motion around the barycenter is discussed, and equations for the relative motion are obtained. The particular case of a two-particle system is considered. (auth)
S>Several transformations of the fundamental equations for a fused part of a physical system are given in the functional theory of particles. Objectivity of the values of the mj coefficients of the particles in the fusion process is shown. Properties of the diatinguishable waves in a fused pant of a system are derived, and linear phase waves are obtained. The necessary and sufficient conditions for the fusion of two particles are determined, and the properties of the mean wave for a set of fused systems are obtained. The linear equations placed by de Broglie at the base of the fusion theory are deduced. Properties of the mean wave were determined by measurement, and the linear de Broglie equations were obtained (tr-auth)
A study was made of the general conditions required to set up a non- relativistic functional theory for systems of particles. Representation of each particle of a system by a space-time function uj(P,T). Formal classification for the particles of the system with respect to indiscernability and the exclusion principle for the fermions. Using the general theory of predictions, the arguments of the functional for predictions are fixed, functional configuration- space is introduced, and the general form of the equation for the predictional elements is determined. Conditions incurred by the limiting cases of the usual wave mechanics for systems and of the single particle in the functional theory of particles. Fundamental principle for the equations of motion concerning the particles of a system: each function uj(P,T) describing a particle of the system obeys a non-linear equation which depends upon the functions u/sub k/(P,T) of the other particles of the system. Brief investigation of the limiting case of classical mechanics. (auth)
New relativistic mechanics of two-particle system is set forth, where the two constituent particles are interacting by an arbitrary (central) action-at-a-distance. The fundamental equations are presented in a form covariant under general transformation of parameters parametrizing the world lines of constituent particles. The theory represents the proper relativistic generalization of the usual Newtonian mechanics in the sense that it tends in the non-relativistic (and weak interaction) limit to the usual mechanics of two particles moving under a corresponding non-relativistic potential. For the analysis of theory it is convenient to choose a certain particular gauge (i.e., parametrization) fixed by two gauge relations. This brings the theory to a canonical formalism accompanied by two weak equations, and in this gauge quantization can be performed. The result verifies that the relativistic quantum mechanics for two particles interacting by an action-at-a-distance is just represented by a bilocal wave equation and a subsidiary condition, with the clarification of its correspondence-theoretical fundation and internal dynamics. As an example the case of Hooke-type force is illustrated, where the internal motions are elliptic oscillations in the center-of-mass frame. Its quantum theory just reproduces the original form of bilocal theory giving bound states lying on a straightly rising trajectory and on its daughter trajectories.
Étude d'un système de deux corpuscules à spin en théorie fonctionnelle relativiste ; équation de l'onde barycentrique, équations relatives et équation réduite. Parmi les états de ce système, il y a, en certains cas, ceux pour lesquels les ondes relatives sont des corstantes ; ceci exprime que le mouvement relatif autour du barycentre est évanescent. Pour deux corpuscules de spin 1/2 et de même masse, on obtient ainsi un système fondu qui se comporte ccmme un corpuscule de spin 1. De là, par l'approximation conduisant à la mécanique ondulatoire usuelle, on retrouve les résultats de la méthode de fusion ; on obtient donc une justification physique du procédé formel de la fusion.
The correspondence of the previously given relativistic quantum mechanics of spinless or spin 1/2 particle with the quantum nonrelativistic and the classical relativistic mechanics is proved. In the Galilean quantum case the appearance of the familiar transformation phase factor is explained and the origin of the mass superselection rule indicated. In the relativistic classical case the equation for trajectories and the covariant equation of motion for spin are obtained. The conditions are found under which the usual theory of relativistic wave equations is of relevance (in some particular reference frame).