SYMMETRIES IN SCIENCE V Algebraic Systems, Their Representations, Realizations, and Physical Applications
SYMMETRIES IN SCIENCE V Algebraic Systems, Their Representations, Realizations, and Physical Applications Edited by Bruno Gruber Southern Illinois University Carbondale, Illinois L.C. Biedenharn Duke University Durham, North Carolina and H.D. Doebner Technical University Clausthal, Germany SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging in Publication Data Symmetries in science V: algebraic systems, their representations, realizations, and physical applications / edited by Bruno Gruber, L.C. Biedenharn, and H.D. Doebner. p. cm. Proceedings of a symposium held July 30-Aug. 3, 1990, Landesbildungszentrum Schloss Hofen, Lochau, Vorarlberg, Austria. Includes bibliographical references and index. ISBN 978-1-4613-6643-0 ISBN 978-1-4615-3696-3 (ebook) DOI 10.1007/978-1-4615-3696-3 1. Algebra, Abstract-Congresses. I. Gruber, Bruno, 1936-. II. Biedenharn, L. C. III. Doebner, H. D. (Heinz Dieter), date. IV. Title: Symmetries in science 5. V. Title: Symmetries in science five. QA162.S94 1991 91-9900 512'.02-dc20 CIP Proceedings of a symposium entitled Symmetries in Science V: Algebraic Systems, Their Representations, Realizations, and Physical Applications, held July 30-August 3, 1990, at Landesbildungszentrum Schloss Hofen, Lochau, Vorarlberg, Austria ISBN 978-1-4613-6643-0 1991 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1991 Softcover reprint of the hardcover 1st edition 1991 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher
FOREWORD The Symposium "Symmetries in Science V: Algebraic Systems: their Representations, Realizations and Physical Applications" was held at the Landesbildungszentrum Schloss Hofen, Vorarlberg, Austria, during the period July 30 - August 3, 1990. Leading scientists presented summaries, as well as recent research developments, concerning their own field of specialization to the assembled international group of researchers. These review-oriented presentations are included in this volume in order to make them available to the general scientific community. As was the case for the preceding Symposia we need to thank those who made Symposium V possible through their continued and generous support. These include, on part of the State of Vorarlbergj Dr. Martin Purtscher, Landeshauptmann Dr. Guntram Lins, Landesrat Ms. Elisabeth Gehrer, Landesratin and on part of Southern Illinois University at Carbondale; Dr. John C. Guyon, President Dr. Benjamin A. Shepherd, Vice President for Academic Affairs and Research Dr. John H. Yopp, Dean of the Graduate School Dr. Charles B. Klasek, Executive Assistant to the President for International and Economic Development Dr. Russell R. Dutcher, Dean of the College of Science Dr. Victoria J. Molfese, Director, Office of Research Development and Administration Dr. James D. Quisenberry, Acting Director of International Programs and Services Dr. Frank C. Sanders, Chairman of the Department of Physics. Finally, we need to especially thank Dr. Hubert Regner, Vorstand der Abteilung Wissenschaft und Weiterbildung, whose support and cooperation - on a daily basis - has been the single most essential ingredient for the success of Symposium V. Bruno Gruber L.C. Biedenharn H.D. Doebner v
CONTENTS Dynamical Symmetries of Relativistic Two - and Many Body Systems... 1 A.O. Barut Algebraic Treatment of Multistep Processes in Electron-Molecule Scattering... 15 R. Bijker Symmetry, Constitutive Laws of Bounded Smo<;>thly Deformable Media and Neumann Problems... 31 E. Binz Imitation of Symmetries in Local Quantum Field Theory H.J. Borchers 67 Representations of Quantum Groups... 93 V.K. Dobrev Algebras and Symmetries - Quantum Mechanical Symmetry Breaking... 137 H.D. Doebner and J. Tolar q-analysis and Quantum Groups D.B. Fairlie Orthogonal Polynomials and Coherent States Ph. Feinsilver 147 159 Algebraic Model for Molecular Electronic Spectra... 173 A. Frank, R. Lemus and F. Iachello Highest Weight Unitary Modules for Non-Compact Groups and Applications to Physical Problems... 187 J. Garcia-Escudero and M. Lorente Scattering Theory and the Group Representation Matrix... 233 J.N. Ginocchio vii
Predicting "Anyons": The Origins of Fractional Statistics in Two-Dimensional Space... 259 G.A. Goldin The Role of Parabose-Statistics in Making Abstract Quantum Theory Concrete T. Gornitz 279 On Quantized Verma Modules... 293 B. Gruber and Yu.F. Smirnov The Role of Spectrum Generating Algebras and Dynamic Symmetries in Molecular Physics... 305 F. Iachello O( 4) Symmetry and Angular Momentum Theory in Four Dimensions... 323 P. Van Isacker Representations of the Quantum Algebras U q ( su(2)) and U q(su(l, 1))... 341 A.U. Klimyk, Yu.F. Smirnov and B. Gruber From "Quantum Groups" to "Quasi-Quantum Groups" Y. Kosmann-Schwarzbach 369 Symmetries of Icosahedral Quasicrystals... 395 P. Kramer, Z. Papadopolos and D. Zeidler Molecular Symmetry Adapted Bases in the Born-Oppenheimer Approximation... 429 R. Lemus and A. Frank On Certain Submodules of the Enveloping Fields R. Lenczewski 447 Invariants and States Generating Symmetry of Nonstationary Systems... 453 V. Man'ko Origins of Nuclear and Hadron Symmetries Y. Ne'eman, Dj. Sijaki 475 Projection Operator Method and Q-Analog of Angular Momentum Theory... 487 Yu.F. Smirnov, V.N. Tolstoy and Yu.I. Kharitonov Symmetry Breaking and Fractional Quantization of Quantum Systems... 519 R. Tao viii
New Phases of D ~ 2 Current and Diffeomorphism Algebras in Particle Physics... 527 C-H. Tze Algebra and Geometry in the Theory of Mixed States... 565 A. Uhlmann Group Structures and the Interacting Boson Approximation for Nuclei............................................. 585 B.G. Wyboume Paradigms of Quantum Algebras C. Zachos 593 Index... 611 ix