Table Of ContentSYMMETRIES
IN SCIENCE IX
SYMMETRIES
IN SCIENCE IX
Edited by
Bruno Gruber
Southern Illinois University at Carhondale
Carhondale, Illinois
and
Michael Ramek
Technische Universitiit Graz
Graz, Austria
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging-in-Publication Data
On file
Proceedings of a Symposium on Symmetries in Science IX, held August 6-10, 1996,
in Bregenz, Austria
ISBN 978-1-4613-7715-3 ISBN 978-1-4615-5921-4 (eBook)
DOI 10.1007/978-1-4615-5921-4
© 1997 by Springer Science+Business Media New York
Origina11y published by Plenum Press New York in 1997
Softcover reprint of the hardcover 1s t edition 1997
http://www.plenum.com
10987654321
Ali 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
PREFACE
The international Symposium "Symmetries in Science IX" was held at the Col
legium Mehrerau in Bregenz, Austria, during the period August 6-10, 1996. Some 40
invited scientists from 13 countries participated in the meeting.
The Symposium was sponsored by
Southern Illinois University at Carbondale,
the Land Vorarlberg, Austria, and
the Landeshauptstadt Bregenz, Vorarlberg, Austria.
I wish to thank Chancellor Emeritus John C. Guyon of Southern Illinois Univer
sity at Carbondale, Dr. Hubert Regner of the Amt der Vorarlberger Landesregierung,
and Biirgermeister and Landtagspriisident Dipl.-Vw. Siegfried Gasser for financial as
sistence for the meeting and Frater Albin Printschler of the Cloister Mehrerau for his
continued support of the Symposium series.
Bruno Gruber
v
CONTENTS
Classical vs. Quantum Groups as Symmetries of Quantized Systems 1
M. Arik and C. Vnel
Algebraic Model of an Oblate Top 9
R. Bijker and A. Leviatan
The Mass-Squared Operator and the Einstein-Hilbert Action for Rescaled
Lorentz Metrics ..................... 25
E. Binz and P. Oellers
Multichannel Dynamic Symmetry 37
J. Cseh
Kazhdan-Lusztig Polynomials, Subsingular Vectors and Conditionally
Invariant (q-Deformed) Equations ........... . 47
V. K. Dobrev
On a Path to Nonlinear QuantuIll Mechanics 81
H.-D. Doebner and J.-D. Hennig
Quantum Mechanical Problems with q-Deformations and over the p-Adic
Number Fields . . . . . . . . . . . . . . . . . . . 93
1. H. Duru
A Symmetry Adapted Algebraic Approach to Molecular Spectroscopy 99
A. Frank, R. Lemus, R. Bijker, F. Perez-Bernal, and J. M. Arias
Dyson Boson-Fermion Realization of Lie (Super)Algebras 117
D. V. Fursa, Yu. F. Smirnov, and V. N. Tolstoy
Formal Languages for Quasicrystals 139
J. Garda Escudero
vii
On Quadratic arid Nonquadratic Forms: Application to R2m ----t R2m-n
Nonbijective Transformations .............. 153
M. Kibler
Quantization of Systems with Constraints . . . . . . . . . . . . . .. 167
J. R. Klauder
Automorphisms and Discrete Fiber Bundles 179
P. Kramer, U. Beck, and S. I. Ben-Abraham
Algebraic Approach to Baryon Structure 193
A. Leviatan and R. Bijker
Discrete Reflection Groups and Induced Representations of Poincare Group
on the Lattice . . . . . . . . . . . . . . . . . . . . . . .. 211
M. Lorente
Quantum Mechanics and Classical Probability Theory 225
V. I. Man'ko
Supermultiplet Formulation of the Dirac Oscillator of Arbitrary Spiin 243
M. Moshinsky and Yu. F. Smirnov
Construction of Schrodinger- and q-Schrodinger Invariant Differential
Operators ...................... . 251
C. Mrugalla
Automorphisms of Free Groups and Universal Coxeter Groups 263
B. M iihlherr
Generalized Quantum Mechanics and Nonlinear Gauge Transformations 269
P. N attermann
A Solution of an Extended Hartree-Bogoliubov Equation on the Coset Space
SO(2N+2)/U(N+1) for Unified Description of Bose and Fermi Type
Collective Excitations . . . . . . . . . . . . . . . . . . . .. 281
S. Nishiyama
Quantum Space-Time Groups and Beyond 297
M. A. del Olmo
Quantum Mechanics on the Torus, Klein Bottle and Projective Sphere 313
C. Schulte
Application of the Supersymmetric Model to Exotic Oxygen Nuclei 325
S. Szpikowski, L. Pr6chniak, and W. Berej
viii
q-Quantum Mechanics on Sl 331
R. Twarock
List of Participants 345
Contributors 349
Index ... 351
ix
CLASSICAL VS. QUANTUM GROUPS AS
SYMMETRIES OF QUANTIZED SYSTEMS
Metin Arik and Gokhan Un el
Bogazi<;i Gniversity
Department of Physics
and
Centre for Turkish-Balkan Physics
Research and Applications
istanbul, Turkey
INTRODUCTION
Quantization of nonlinear systems has been one of the fundamental problems of
quantum mechanics. The idea that classical integrable nonlinear systems should be
solvable after quantization, has lead to the introduction of quantum groups as "gen
eralized symmetries" of a physical system. Quantum groups [1] can he obtained by
applying the idea of q-deformation either to the Lie algebra or to a matrix represen
tation of the corresponding classical group. However, as we will show a q-deformation
of the algebra of quantum observables does not necessarily imply a q-deformation of
the symmetry group. To clarify this point, we discuss the quantum group invariant
and the classical group invariant q-oscillators separately in the second section. In the
third section, we consider a specific quadratically constrained model [2J and show that
the classical group invariant q-oscillator can be derived using Dirac bracket (diracket)
quantization [3J with some modifications.
QUANTUM GROUP AND CLASSICAL GROUP INVARIANT q-OSCIL
LATORS
A q-deformation which seems to be the most relevant from the physical point of
view, concerns the operator algebra in Fock space. In this context, the standard al
gebraic relations among the harmonic oscillator's creation and annihilation operators
are modified to include a dimensionless parameter q. One possible multidimensional
realization of such a generalization is the quantum group invariant q-oscillator. The
easiest way to understand these oscillators is to start with d commuting copies of the
Symmetries in Science IX, edited by Gruber and Ramek
Plenum Press, New York, 1997
one dimensional simple q-oscillator [4] defined by
q E R, (1)
to obtain
a) aia; - qa;ai = 1
b) aia} = a}ai' i =I j (2)
c) aiaj = ajai, i =I j
Here the algebra is deformed in such a way that the different excitation directions
commute. In particular, this implies that the U (d) invariance of the standard oscillator
is broken. The number operators Ni of this q-deformed system are then given by
1-qNi
a;ai = (3)
1-q
Obtaining the unitary quantum group invariant q-oscillator from this simple system
requires the transformation [5]
=
(N1 +N2+···+Ni 1)
Ci q 2 ai (4)
It then follows that under the unitary quantum group U,,;q( d), the annihilation operators
Ci [6] transform as:
e --> Me (5)
where
(6)
Here the elements of the unitary quantum group matrix U are themselves operators
which satisfy the well known commutation relations of a quantum group matrix. They
are taken to commute with Ci. Or, equivalently, the elements of the matrix M act on
Ci by the standard tensor product. It turns out that these non-commuting elements of
the matrix M can also be parametrized in terms of d(d-1)/2 q-oscillators in (1) where
different oscillators commute [7].
The number of n-particle states are degenerate since different directions essentially
commute. Indeed there is a one to one correspondence between the states of the two
oscillators:
(7)
It follows that the number of n-particle states is given by
n + d -1 ) = (n + l)(n + 2)··· (n + d - 1)
( (8)
d-l 1·2···(d-l)
which counts the number ofi ndependent components of an nth-rank symmetricaltensor
in d dimensions.
2
Having discussed quantum group invariant q-oscillators, we now turn to discuss
classical group invariant q-oscillators [8, 9J. A straightforward multidimensional gener
alization of (1) is obtained by keeping (2.a) and modifying (2.b) such that these relations
can be combined into a single relation which is invariant under the group U( d):
a, b) aia; - q a}ai = Dij (9)
It then follows that for q #- 1, the n-particle states created from the vacuum are
linearly independent. Hence the relation (2.c) is replaced by
c) if q #- 1
if q = 1
The key point is that, for a Fock space with a unique ground state the commutation
relation (2.c) or (9.c) are redundant. The commutation relations (9.a) and (9.b) are
invariant under the classical group U( d) acting on the annihilation operators by
a-+ Ma ME U(d) (10)
where
(ll)
The number of linearly independent n-particle states for q #- 1 is given by dn. The Fock
space defined by (9) is a Hilbert space for -1 < q < 1, the scaJar product is positive
definite and the operators ai and a; are bounded [9J. For this oscillator the structure of
the Hilbert space is dramatically different for q = 1 and for q #- 1. For the former case,
the n-particle states consist of the totally symmetric representation occuring in the n
fold product of the d-dimensional representations of U( d). Whereas in the latter case,
they consist of all representations occuring in the n-fold product of the d-dimensional
representation of U( d). In the q # 1 case some additional states iu the Hilbert space
appear, i.e. the states are split when q is introduced. These extra states created from
the vacuum become linearly dependent on the remaining ones in the q = 1 case.
A SPECIFIC QUADRATIC CONSTRAINED MODEL
In [2J a quadratic constrained model with a space-time metric having the signature
(+, -, -, -, -) or (+, -, -, -, +) was considered. We find that, in this model after
elimination of the first and fifth coordinates using the constraints and the gauge condi
tion, we can obtain a proper Hilbert space with a positive definite inner product. The
lagrangian of this quadratic constrained system is given by
(12)
where f1 = 0,1,2,3,5, m and r are constants and N and .A are Lagrange multipliers
which will be used to generate primary constraints. We will mainly be interested in the
3
