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International Centre for Theoretical Physics INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS OUTLINE OF A NONLINEAR, RELATIVISTIC QUANTUM MECHANICS OF EXTENDED PARTICLES Eckehard W. Mielke INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1981 MIRAMARE-TRIESTE IC/81/9 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization 3TERKATIOHAL CENTRE FOR THEORETICAL PHTSICS OUTLINE OF A NONLINEAR, KELATIVXSTTC QUANTUM HECHAHICS OF EXTENDED PARTICLES* Eckehard W. Mielke International Centre for Theoretical Physics, Trieste, Italy. MIRAHABE - TRIESTE January 198l • Submitted for publication. ABSTRACT I. INTRODUCTION AMD MOTIVATION A quantum theory of intrinsically extended particles similar to de Broglie's In recent years the conviction has apparently increased among most theory of the Double Solution is proposed, A rational notion of the particle's physicists that all fundamental interactions between particles - including extension is enthroned by realizing its internal structure via soliton-type gravity - can be comprehended by appropriate extensions of local gauge theories. solutions of nonlinear, relativistic wave equations. These droplet-type waves The principle of gauge invarisince was first formulated in have a quasi-objective character except for certain boundary conditions vhich 1923 by Hermann Weyl[l57] with the aim to give a proper connection between may be subject to stochastic fluctuations. More precisely, this assumption Dirac's relativistic quantum theory [37] of the electron and the Maxwell- amounts to a probabilistic description of the center of a soliton such that it Lorentz's theory of electromagnetism. Later Yang and Mills [167] as well as would follow the conventional quantum-mechanical formalism in the limit of zero Sharp were able to extend the latter theory to a nonabelian gauge particle radius. At short interact?cm listances, however, a promia.ag nonlinear theory with SU{f) as local symmetry group. This became necessary in order and nonlocal theory emerges. This model is not only capable of achieving a to account for the postulated "internal" degrees of freedoa - auch aa conceptually satisfying synthesis of the particle-wave dualism, but may also isospin and strangeness - of the then newly observed baryons. After the recent lead to a rational resolution of epistemological problems in the quantum- acclaim of the Weinberg-Salam model of weak interactions and some achievements theoretical measurement process. Within experimental errors the results for, of the so-called quantum chromodynamlcs (QCD) {see Ref. 91 for a review) a e.g., the hydrogen atom can be reproduced by appropriately specifying the nature "grand unification" of all particle forces seems to be in reach [130], of the nonlinear self-interaction. It is speculated that field theoretical issues raised by such notions as identical particles, field quantization and However, it should be noted that in order to apply the methods of renormalization are already incorporated or resolved by this nonlocal theory, quantum field theory (QFT), in particular those of renormalization [12, 109], at least in principle. all these gauge models necessarily have to make reference to structureless elementary fields. CONTENTS Besides several electrons and neutrinos, quarks* [56]-which nowadays I. Introduction and Motivation 2 come in at least three colours - intermediate vector bosons, gluons and Higgs II. Basic Concepts of the Theory of the Double Solution 7 fields have to be postulated as elementary building blocks of matter. III. Droplet Properties of the u-solitons 11 Furthermore, the elusive quarks as well aa the gluona (which appear to be late IV. Synthesis of the Particle-Wave Duality 18 revenge of Pauli, who despite of his invention of the neutrino, was known for V. Correspondence Principles 23 bi9 aversion againat introducing hypothetical particles into physics) supposedly VI. Relativiatic Hydrogen Atom in Nonlinear Models 26 constituting hadrons need to be confined in stable hadrons by a yet unknown VII. Identical Particles 28 mechanism. Otherwise, they would have been observed at a detectable rate. VIII. In Quest of a Universal Nonlinear Field Equation 31 With "grand unified theories" becoming more Into fashion the number of IX. Remarks on Epistemological Problems of the Measurement Process... hO constituent fields has become "positively baroque" [130]. This is admitted even by X. Prospects: Quantum Field Theory Without Quantum Field Theory? ... 1*3 the standards of the "orthodox view" in particle physics. Acknowledgments , I18 In order to dissolve the dilemma of quark-lepton proliferation, the References 49 further speculation [113] has been put on top of the former, that quarks and Figures 56 gauge fields are themselves composite and in "reality" built out of preons, -tfiese themselves of prepreons etc.If With regard to this retreat to preons and •James Joyce's "Three quarkB make a muster mark" [60] should be compared vith -1- wilhelm Busch's earlier statement "Niehts als Quark!" [25, p.6d]. -2- prepreons the questions may be allowed if high-energy physics intends to mimic the scattering matrix converges for high energies [12, p.37U], Therefore, even. 1 Lakatos ironical example of a research program which is typical for a the proponents of canonical relativistic quantum field theory in which leptona nonfalsifiable theory? [85, p.100]. (and quarks) are regarded aa intrinsically geometrical points* admit [11, p.5] For one of the founders of quantum mechanics (QM), Heisenberg, the quest the possibility that local field theory ia a correct description of for elementarity "has a tolerably clear aeaning only if the particle can be microphy3ica only if limited to distances bigger than 10" cm- divided into pieces vith a small amount of energy, much smaller than the rest of the mass particle itself". Furthermore, the heuristic value of higher Conaequently, a foundation of the notion of -particle extension being symmetries "could probably be compared vith the heuristic value of cycles and compatible with (general) relativity as well as with the basic features o* QM epicycles in Ptolemy's astronomy" [69]. strikes us as the j»ost crucial point of departure from existing theories. Therefore, a refreshing attempt of redirecting the course of the To be sure, the concept of a finite size of a particle ia an old one, history of particle physics has been undertaken by Barut [It]. In this theory being first proposed by Abraham and Lorenta in the case of an electron. Later all matter is made up of the absolutely stable particles: protons, electrons, on, an extensible model of electron and the muon - regarded as the next excited neutrinos and photons. Strong interactions are entirely of a magnetic type. state - has been devised by Dirac [38] (see also [6]), anticipating to some When treated non-perturbatively, magnetic forces are supposed to become very extent the "bag" models [63] in QCD. strong at 3hort distances. This,approach, to a certain extent, seems to work well. However, questions about the nature of the stable particles themselves Intuitively as well as physically Justifiable by nuclear models a first remain unanswered. Also it mainly rests upon the notions of quantum step toward the inclusion of spatial extension is to regard a particle as a electrodynamics (QED) [11] which also has been deeply questioned! relativistic fluid droplet. This proposal has been considered before (see In the introduction to a perspective paper, Born, the founder of the Kef. 17 and references therein) and In connection vith the quark hypothesis statistical interpretation of quantum mechanics critically comments that taken up recently [155]. The conceptual difficult problem is that of a "existing attempts to apply the quantum theory to the electromagnetic field are reconciliation of such mechanical models with quantum theory, open to serious objections", ",., relativistic invariance cannot be derived *The absurdity of the point 1-particleJ concept from a intuitive point of view simply from the symmetry of the formulae in the four world co-ordinates, but must be artificially imposed and demonstrated by a complicated proof. Further, has probably never been more vividly expressed than 1B90 by Wilhelm Buschi it is not a self-contained theory of electromagnetic field but a superposition "Hoch gewandter und vindiger als vir, und das vill doch was sagen, trieben die of Maxwell's electromagnetic field on the material field of Schrodinger or nur gedachten, die rein mathematischen Punkte ihre terpsichorischen Kunste. Dirac, in which the elementary particles occur as point-charges. Thus there is Sie waren aber derma&en schuchtem, dass sie imnser kleiner und kleiner vurden, no idea of the radius of the particle, and consequently no rational motion of Je mehr man sie anaah; ja, einer verschwand g'anzlich, als ich ihn sch'arfer mass, not to mention a theory of the ratio of the mass of a proton to that of in's Auge faBte- Celungene Burschen, diese Art Punkte! Der alte Brenneke, mein Mathematiklehrer, an electron. In addition to these fundamental difficulties there are others, pflegte freilich zu sagen: "Wer sich keinen Punkt denken kann, der i3t einfach such as that of infinitely great "Hullpunktsenergie", which i3 avoided by an zu faul daaul" Ich hab's oft versucht seitdem. Aber just dann, vena ich denke, artificial modification of the formalism". (Citation from Eef. 21 without the underlining) ich hatt'ihn, Just dann hab ich gar nichts. Und uberhaupt,
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