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A Reformulation of Schrödinger and Dirac Equations in Terms Of A reformulation of Schrödinger and Dirac equations in terms of observable local densities and electromagnetic fields : a step towards a new interpretation of quantum mechanics ? J. Des Cloizeaux To cite this version: J. Des Cloizeaux. A reformulation of Schrödinger and Dirac equations in terms of observable local densities and electromagnetic fields : a step towards a new interpretation of quantum mechanics ?. Journal de Physique, 1983, 44 (8), pp.885-908. 10.1051/jphys:01983004408088500. jpa-00209672 HAL Id: jpa-00209672 https://hal.archives-ouvertes.fr/jpa-00209672 Submitted on 1 Jan 1983 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LE JOURNAL DE PHYSIQUE J. Physique 44 (1983) 885-908 AOÛT 1983, 885 Classification Physics Abstracts 03.65 A reformulation of Schrödinger and Dirac equations in terms of observable local densities and electromagnetic fields : a step towards a new interpretation of quantum mechanics ? J. des Cloizeaux SPHT-CEN Saclay, 91191 Gif sur Yvette Cedex, France (*) and Institute for Theoretical Physics, University of California, Santa Barbara, CA. 93117, U.S.A. (Reçu le 27 septembre 1982, révisé le 22 dicembre, accepté le 14 avril 1983) Résumé. 2014 On montre que l’équation de Schrödinger, l’équation de Dirac à 3 dimensions et l’équation de Dirac à 4 dimensions peuvent être transformées de manière à éliminer tous les effets de phase et de jauge. Des densités locales réelles sont définies et l’on montre que les tenseurs obéissent à des « équations probabilistes » réelles qui sont équivalentes aux équations de la mécanique quantique. Ces nouvelles équations, qui sont construites expli- citement, ne font intervenir que les densités locales et le champ électromagnétique extérieur. De nouvelles condi- tions aux limites sont aussi établies. L’expérience d’Aharonov-Bohm est réinterprétée dans le cadre de ce nouveau formalisme. Abstract. 2014 It is shown that the Schrödinger equation, the 3-dimensional Dirac equation and the 4-dimensional Dirac equation can be transformed so as to eliminate all phase and gauge effects. Real local densities are defined and it is shown that these tensors obey real « probabilistic equations » which are equivalent to the equations of quantum mechanics. These new equations, which are explicitly constructed, involve only local densities and the external electromagnetic field. New boundary conditions are also derived. The Aharonov-Bohm experiment is reinterpreted in the framework of this new formalism. « I think one can make a safe will present his views and his motivations. If the guess that uncertainty reader does not agree with them and is just interested relations in their present in facts, he should pass directly to the following form will not survive in the sections. physics of the future. » The author thinks that some difficulties may be P.A.M. Dirac, 1962 connected with the fact that a wave function is not (Scientific American, 1963) an observable. Of course, physicists working now on particle do not use wave functions but more 1. Introduction. theory explicitly powerful tools like functional integrals or the S Quantum mechanics is extremely successful but many matrix. However, when they calculate a physical physicists [1], even young ones, do not consider its quantity, they generally use, in the course of the orthodox interpretation as completely satisfactory. calculation process, quantities which are gauge and The difficulties are purely philosophical but real. phase dependent. Quantum mechanics is an efficient recipe. Is that The situation would be clearer if quantum mecha- enough ? nics could be directly formulated in terms of proba- Actually, some physicists think that quantum bilities which are observables with a physical meaning. mechanics should also present a picture of reality. Let us dream. These probabilities should obey However, concerning this question, physicists have partial derivative equations just as the probability different opinions. In this introduction the author distributions of classical statistical mechanics obey a Fokker-Planck equation. These « probabilistic equa- (*) Permanent address. tions » cannot be trivial because the vacuum is not Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004408088500 886 a simple system; it is certainly as complex as a heat flux of an external field through the hole of the torus. bath. As the particle propagates in a domain where the Of course dissipation is not present since a particle magnetic field is always zero, we see that apparently may have a constant motion in the vacuum but the particle does not feel only the local electro- polarization effects should be important (think of magnetic field but also the local potential vector the complexity of the vacuum-vacuum diagrams in which is gauge dependent. Thus, the classical inter- field theory !). These probabilistic equations should pretation of the Aharonov-Bohm experiment [4] describe the evolution of real probabilities or obser- seems to contradict our answer to the preceding vables. They should also present a more realistic questions. view of nature than ordinary quantum mechanics. We shall justify our point of view by introducing Therefore, we may expect that it would be easier to real « local densities » and by showing that these interpret them. observable tensors obey « probabilistic equations ». In statistical mechanics, we describe the motion of These equations, which determine the space-time a particle interacting with an external potential and dependence of the local densities, will be explicitly a heat bath by writing a Langevin equation and, written down. Thus, the contradiction with the current from this Langevin equation, we deduce the cor- interpretation of the Aharonov-Bohm experiment responding Fokker-Planck equation; in this case, cannot be real and in fact the paradox will be dis- the Langevin equation provides a physical inter- cussed and solved at the end of the article. pretation of the Fokker-Planck equation. We may Actually, D. Bohm [5] himself found the « proba- expect a similar situation in quantum mechanics [2]. bilistic equations » which can be associated with an At the present time, this is only a dream. However, ordinary Schr6dinger equation describing the motion we may ask the question « Is it possible to replace a of a particle in an external potential. (D. Bohm, who Schr6dinger equation or a Dirac equation by an assumed the existence of « hidden variables », was equivalent probabilistic equation describing how trying to find out a new interpretation of quantum local densities (local mean values) vary with the mechanics.) However, the simple but important fact local electromagnetic field ? ». Another formulation that the equations found by Bohm are equivalent to of the same question is « Can we transform quantum the Schr6dinger equation but are more « physica.l », mechanics so as to eliminate all phase and gauge has not been sufficiently emphasized. effects ? ». In his famous lectures, R. P. Feynman [6] has written Our answer is « Yes » and the aim of the present down these « probabilistic equations » in the case article is to prove the validity of this affirmation. The where an external electromagnetic field produced by reader, however, might be tempted to say « No ». In a potential and a vector potential interacts with the fact, when we try to interpret the famous « Aharonov- particle. These results are reproduced and discussed Bohm experiment » [3] (see Fig. 1) we discover that in section 2. the interference pattern of a particle interfering with We believe however that the Schr6dinger equation itself can be modified without applying any electro- cannot lead to a deep understanding of reality. In magnetic field to the particle. In the experiment, the fact it is deduced from the Dirac equation by a com- particle beams define a toroidal space and to modify plicated unitary transformation (the Foldy-Wouthui- the interference pattern we have only to change the sen transformation [7]) which destroys simple phy- sical aspects of the Dirac equation. For instance, the operator vj, which represents the velocity of an electron should not depend on the external potential vector. Dirac equation satisfies this requirement since in this case, we have vop = cy i (c = velocity of light). However, when we deal with the Schrodinger equation, we have to use the unpleasant formula : We also note that the « Zitterbewegung » is a very important phenomenon [7] related to the exis- tence of spins, and that it requires an adequate inter- pretation. Of course, Dirac equation presents diffi- culties with respect to negative energies and some physicists think that it has a meaning only in field theory. We remark however that one-electron states - exist and that the motion of an electron in an external Fig. 1. Propagation of a wave in a toroidal domain. The circuit C is represented by the dashed line. 0 is the electromagnetic field is well described by Dirac magnetic flux through the hole. equation. Moreover, the effect known as the Klein 887 paradox also occurs in semiconductors and in that tions of x, y, z, and t. We shall also use the complex case, the situation is quite clear. The anomaly which conjugate of equation (1) : is observed in strongly repulsive potential is only a mark of the instability of the vacuum (or ground state) with respect to the production of pairs of particles of opposite signs. Thus, all these arguments brought us to believe that Dirac equation may be The P and the current J at a more realistic that Schrodinger equation.
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