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This book is a collection of reviews and essays about the recent developments in the area of Symmetries and applications of Group Theory. Contributions have been written mostly at the graduate level but some are accessible to advanced undergraduates. The book is of interest to a wide audience and covers a broad range of topics with a strong degree of thematical unity.
The book is part of a Series of books on Symmetries in Science and may be compared to the published Proceedings of the Colloquia on Group Theoretical Methods in Physics. Here, however, prevails a distinguished character for presenting extended reviews on present applications to Science, not restricted to Theoretical Physics.
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SYMMETRIES IN SCIENCE XI
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Symmetries in Science XI
Edited by
Bruno J. Gruber College of Science, Southern Illinois University at Carbondale, Carbondale, Illinois, U.S.A.
Giuseppe Marmo Università di Napoli “Federico II”, Instituto Nazionale di Fisica Nucleare, Napoli, Italy
and
Naotaka Yoshinaga Saitama University, Saitama, Japan
KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
eBook ISBN: Print ISBN:
1-4020-2634-X 1-4020-2633-1
©2005 Springer Science + Business Media, Inc. Print ©2004 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: and the Springer Global Website Online at:
http://ebooks.springerlink.com http://www.springeronline.com
This volume of the proceedings "Symmetries in Science XI" is dedicated to my colleagues and friends Akito Arima, Francesco Iachello, Marcos Moshinsky, and to my wife Burghilde, all of whom have significantly contributed, in various ways, to the series of symposia "Symmetries in Science". Bruno J. Gruber, Chairman and Organizer
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Contents
List of Figures
xvii
List of Tables
xxi
Preface
xxiii
Why symmetry? P. Roman
1
J -pairing Interactions of Fermions in a Single-j Shell
13
A. Arima 1 2 3 4 5 6
13 13 16 18 19 20
Introduction 0+ ground state dominance Pair Approximation for Fermions in a single-j shell Regularities of states in the presence of HJmax Solutions for the case of n = 3 Summary
Supersymmetry in nuclei F. Iachello 1 Introduction 2 Symmetries 2.1 Geometric symmetries 2.2 Space-time symmetries 2.3 Gauge symmetries 2.4 Dynamic symmetries 3 Dynamic symmetries of the Interacting Boson Model 4 Supersymmetry 4.1 Geometric supersymmetries 4.2 Space-time supersymmetries 4.3 Gauge supersymmetries 4.4 Dynamic supersymmetries 5 Dynamic Supersymmetries of the Interacting Boson-Fermion Model 5.1 Supersymmetry in nuclei found
vii
23 23 23 23 24 24 24 26 28 28 29 30 30 30 32
viii
XIII SYMPOSIUM ON “SYMMETRIES IN SCIENCE”
6 7
5.2 Supersymmetry in nuclei confirmed Implications of supersymmetry in nuclei Conclusions
The relativistic many body problem in QM M. Moshinsky 1 Introduction 2 A formulation of the relativistic many body problem 3 The Hamiltonian of the n-body relativistic problem and its FoldyWouthuysen transformation 4 The particular case when n = 2 5 Conclusion Gauge invariance and the E1 sum rule in nuclei W. Bentz, A. Arima 1 Introduction 2 The orbital g-factor 3 The E1 sum rule and the κ − g relation 4 Summary and conclusions Applications of the Heisenberg Group E. Binz and S. Pods 1 Introduction 2 The Geometric Setting 2.1 The