Quantum Non-integrability 9810206224, 9789810206222

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QUANTUM NON-INTEGRABILITY

DIRECTIONS IN CHAOS

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Editor-in-Chief: HaoBai-lin

Published Directions in Chaos Vol. 1 (published as Vol. 3 of Directions in Condensed Matter Physics series) edited by Hao BaHin Directions in Chaos Vol. 2 (published as Vol. 4 of Directions in Condensed Matter Physics series) edited by Hao Bai-lin Vol. 3: Experimental Study and Characterization of Chaos edited by Hao Ba'hiin Vol. 5: Bibliography on Chaos compiled by Zhang Shu-yu

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DIRECTIONS IN CHAOS - VOL. 4

QUANTUM NON-INTEGRABILITY Edited by

Da Hsuan Feng M. Russell Wehr Professor of Physics

Jian-Min Yuan Professor of Physics Department of Physics and Atmospheric Science Drexel University, Philadelphia

fe World Scientific !■

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

UK office: 73 Lynton Mead, Totteridge, London N20 8DH

Library of Congress Cataloging-in-Publication Data Quantum nonintegrability / edited by Da Hsuan Feng and Jian-Min Yuan. p. cm. — (Directions in chaos ; vol. 4) Includes bibliographical references. ISBN 9810206224 1. Integral equations. 2. Quantum theory. I. Feng, Da Hsuan, 1945. II. Yuan, Jian-Min, 1944III. Series. QC174.17.I58Q36 1992 530.1'2--dc20 92-24407 CIP

Copyright © 1992 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording orany information storage and retrieval system now known or to be invented, without written permission from the Publisher.

Printed in Singapore.

V

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PREFACE

It was the failure of classical mechanics to reproduce certain experimental re­ sults which resulted in the birth of quantum mechanics about sixty years ago. But the interest in classical mechanics, even among researchers working on microscopic systems, has all but diminished now. It is more than the formal similarity between classical and quantum mechanics and the asymptotic con­ nection through the correspondence which has kept up the interest in classical mechanics. The powerful visualization and physical intuition supplied by the classical theories and the inescapability of human beings from their earthly shells form the cornerstone of the close tie between classical and quantum mechanics. The recent surge of interests in classical dynamics, however, arises from the fasinating and colorful behavior of nonlinear systems, whose exploration has been stimulated by the developments of high-speed computers and their graphical capability. Complicated phenomena, related to periodic orbits, bi­ furcations, chaos, homoclinic tangles, diffusion, etc., were foreseen by Poincare at the turn of the century. A natural question that follows for a microscopic system is what becomes of such rich nonlinear behavior when one seeks the quantum solutions of such a system. In the chaotic regime fractal structures often appear, either in phase space or in parameter space, which reveal ever finer structure at ever smaller scale. We may similarly ask how such fractal structures manifest themselves in corresponding quantum systems, where there exists a natural 'length* limit, the square root of Planck's constant. Another development, dating back also to the early days of quantum me­ chanics, was prompted by the desire to preserve the intuitive power of classical mechanics in solving quantum mechanical problems and by an attempt to ex­ tend the old quantum theory of Bohr and Sommerfeld to more complicated systems than atomic hydrogen. This has resulted in the development of a semiclassical mechanics, known as the Einstein-Brillouin-Keller (EBK) quan­ tization scheme. This quantization scheme provides a firm foundation for the quantum-classical correspondence in the regular regime. With the hope that one can solve the vibrational eigenmode problems of polyatomic molecules in a more efficient and accurate way, its applications to nonseparable multi­ dimensional systems was vigorously pursued in the 1970's and 80's. The con­ nection between the Kolmogorov-Arnold-Moser (KAM) tori and eigenstates established by the EBK scheme has had, however, limited success when ex-

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vi tended to the chaotic regime. Fortunately, the periodic orbit theory developed by Gutzwiller, Balian and Bloch in the 1970's comes to the rescue and an in­ creasing number of examples show that the semiclassical periodic-orbit theory works nicely in this regime. From the many points of view the field of quantum-classical correspon­ dence is still very young. The field still lacks of a commonly accepted definition for "quantum chaos". This turns out not to be an obstacle for this budding new field. It only makes the field rich in the spirit of exploration which has spurred the discoveries of many interesting phenomena, such as the "scars" of periodic orbits which form the skeletons for the probability densities of the eigenfunctions, the relation of level spacing distributions to the time-honored random matrix theory, etc. This spirit of exploration continues in the articles collected in this monograph, which reviews many of the interesting new results and phenomena recently unearthed. The articles collected in this volume can be divided into two categories. The articles in the first category deal with the more formal aspects of the quantum-classical correspondence while those in the second investigate mainly specific systems. Each category contains three and five articles, respectively. Within the first category, Lay-Nam Chang of VPI and State University and Yigao Liang of the University of Rochester discuss the topological structures in parameter space around the avoided crossings and crossings of eigenval­ ues, the knot patterns and their relation to eigenfunctions. Wei-Min Zhang of Ohio State University, Da Hsuan Feng and Jian-Min Yuan of Drexel University have utilized the concepts of dynamical symmetry and the generalized coher­ ent states in quantum mechanics to develop formal quantum structures which strongly resemble the classical ones. Following his long dedication to the field, George Zaslavsky of the Courant Institute in New York University discusses the Planck-constant-dependence of the break time (of the quantum behavior from the classical one) and some observations in level spacing distributions. In the second category, Leo Moorman of the University of Wyoming and Peter Koch from SUNY at Stony Brook have made an extensive review on the current understanding of a milestone problem — microwave ionization of Rydberg hydrogen atoms and on some recent experimental results of helium, alkali and alkaline-earth atoms. Atomic hydrogen has long been the only realistic test ground for quantum non-integrability and recent extension to more com­ plicated systems, such as helium is a development which is long overdue. Along this same line, application of nonlinear dynamics to the three-body problem of a field-free helium atom is another exciting research direction. Reinhold Blumel and William P. Reinhardt of the University of Pennsylvania have con-

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VII

tributed an original article in this research area with a review of the works by others in the field. Work done on a prototype molecular system has been reviewed by James F. Heagy of Naval Surface Weapons Center, Zi-Min Lu of the University of Oregon, Jian-Min Yuan and Michel Vallieres of Drexel University who discuss the phase space structures, their organization in terms of symmetry lines, bifurcations, chaotic and fractal behavior associated with photodissociation, and finally, chaotic scattering of.the molecular system. This last item, chaotic scattering, is another relatively recent development in the application of nonlinear dynamics to physical systems. Some relevant quantum theories are here reviewed by Reinhold Blumel. Last but not the least, the rel­ evance of chaos to nuclear systems has received wide attention in the nuclear physics community. Here the relation of chaos to collective nuclear motion is extensively reviewed by Fumihiko Sakata and Toshio Marumori from the In­ stitute of Nuclear Studies of the University of Tokyo and Tsukuda University, respectively. We would like to acknowledge the editorial support we received from our colleague Michel Vallieres. Finally, we must thank the Editor-in-Chief of the Directions in Chaos series, Professor Hao Bai-lin for his infinite patience and for giving us the opportunity to edit this volume.

Jian-Min Yuan and Da Hsuan Feng Philadelphia, June 1992

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CONTENTS

Preface

v

Topology in Quantum Mechanics Lay Nam Chang and Yigao Liang

3

Quantum Classical Correspondence and Quantum Chaos Wei-Min Zhang, Da Hsuan Feng and Jian-Min Yuan

66

From Classical to Quantum Chaos George M. Zaslavsky

109

Microwave Ionization of Rydberg Atoms Leo Moorman and Peter M. Koch

143

Where is the Chaos in Two-Electron Atoms? Reinhold Blumel and William P. Reinhardt

245

Dynamics of Driven Molecular Systems James F. Heagy, Zi-Min Lu, Jian-Min Yuan, and Michel Vallieres

322

Quantum Chaotic Scattering Reinhold Blumel

397

Nuclear Collective Dynamics and Chaos Fumihiko Sakata and Toshio Marumori

459

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3

Topology in Quantum Mechanics Lay Nam Chang Department of Physics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061, USA Yigao Liang Department of Physics and Astronomy University of Rochester Rochester, New York 14627, USA

4 Abstract The behaviors of energy levels and eigenfunctions for parametrized Hamiltonians

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are studied. We review the codimensions for degeneracies for general parameter spaces, thereby establishing the degree to which specific degeneracies are generic. We then present a scheme for systematically investigating the dependences of energy eigenvalues and eigenfunctions on the topological structure of the parameter space, and examine in detail how this structure affects degeneracies in the system. The relationship of the degeneracies to the twistings of eigenfunctions and their topological characterization is demonstrated; in particular, we show how topological indices can knot energy bands together. We also discuss the metamorphoses of the knotting patterns and the corresponding twists in the eigenfunctions, and obtain general rules which govern these changes. Some necessary mathematical concepts and results in algebraic topology are reviewed.

5

Contents

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1 Introduction 2 Generic Degeneracies and Co dimensions 3 Topological Behaviors of Energy Levels 3.1 Level rearrangement 3.2 Van Hove singularities and other critical points 3.3 Non-trivial phases in eigenfunctions 3.4 Non-abelian phases and degeneracies 4 A Synopsis of Algebraic Topology 4.1 Homotopy and homotopy type 4.2 Homotopy groups 4.3 Homology and cohomology 4.4 Fiber bundles 4.5 Vector bundles 5 Characterization of Topological Twists of Eigenfunctions of Energy Bands 5.1 Topological twists 5.2 Topological knotting of energy bands 6 Metamorphoses of Energy Bands 6.1 An Explicit Example 7 Conclusion

6

1

Introduction

Characteristics of physical systems can undergo drastic changes as parameters in

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their Hamiltonians are varied. For example, the classical trajectories of a system can either be regular or chaotic, depending upon the choice of values for certain of these parameters. In other cases, where there are phase transitions, the states of a system may actually manifest different symmetry patterns for different ranges of the parameters in the Hamiltonian. A general understanding of what can take place for a system therefore requires an analysis based upon the entire range of variation of these parameters. The behaviors of quantum systems are controlled by the eigenfunctions and eigen­ values of the Hamiltonian. In this article, we will review what is known about these behaviors under variation of the parameters. We will concentrate particularly on dependences on the topological properties of the parameter space. Parameter spaces with non-trivial topologies are quite common in quantum theo­ ries. In condensed matter physics, the Brillouin zone is a well-studied example, while in particle physics, the space of background gauge fields with unusual topologies have become rather important in studies of non-perturbative effects in the standard model. We will show that different topologies can give rise to various topological charges which cause twists to appear in energy eigenfunctions in regions of parameter space, thereby knotting together energy bands to produce degeneracies unrelated to any symmetries of the original Hamiltonian. We may think of these degeneracies as signals of dynamically generated symmetries. We will also derive rules which govern the metamorphoses of the knotting patterns of these energy bands under deformations of the Hamiltonian. Degeneracies are important because they are frequently responsible for phenomena

7 peculiar to quantum systems. Usually, they occur only in subspaces of the total parameter space. The difference between the dimension of the total space and that of this subspace is called the codimension of the degeneracies. This number measures Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

how typical specific degeneracies are for the system. The lower the codimension, the more frequently one encounters the degeneracy in question as one varies the parameters of the Hamiltonian. We will compute this codimension for different kinds of degeneracies, and point out how these results are expected to hold for systems with arbitrarily high degrees of freedom. The article is organized as follows. We will first review, in Sect.2, the general codimension counting rules for degeneracies for families of general Hamiltonians, and then examine the special cases of real symmetric Hamiltonians, and the complemen­ tary cases of imaginary Hamiltonians. These results are simple, but important, and have already been discussed in the literature]!, 5]. They provide the background for our subsequent discussions. In Sect.3, we describe, without going into technical de­ tails, the role played by the topology of the parameter space in fixing the behavior of the energy levels and eigenfunctions. We will show how degeneracies, level rearrange­ ments and critical points in the spectra can arise within this context. We will pay particular attention to degeneracies that are related to the twisting of eigenfunctions induced by non-trivial topological phases. The codimensions for these degeneracies are generally different from those of ordinary degeneracies. Sect.4 is devoted to a review of the topological machinery and results which are needed for systematic de­ scriptions of these parametric dependences. In Sect.5, we show how these techniques can be used to characterize twisting of eigenfunctions and the knotting together of bands of energy levels by topological indices. A different type of codimension count­ ing for degeneracies based on topological constraints is then established. In Sect.6, we discuss rules governing the metamorphoses of the degeneracies, i. e. possible changes

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8

in the knotting patterns of the spectrum under deformations of the Hamiltonian. We describe an explicit example illustrating how the rules work for a specific Hamiltonian.

9

2

Generic Degeneracies and Co dimensions

What are the conditions under which specific numbers of energy levels of a HamilQuantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

tonian become degenerate? This question was first considered by von Neumann and Wigner[l], but their answer was given in terms of the dimensionality of the subspace on which the degeneracies occurred. For a sufficiently large number of parameters in the Hamiltonian, one can always find a degeneracy of the desired type by fine-tuning a certain number of them. This number gives a measure of the difficulty in locating the degeneracy. Suppose that we are given an arbitrary self-adjoint operator which can be thought of as the Hamiltonian of a certain system. For the discrete part of the spectrum, it is unlikely that this arbitrary Hamiltonian has a degenerate eigenvalue. We may how­ ever deform the system by tuning a number of parameters in the operator, and for a sufficiently large number, we will encounter a point in parameter space at which de­ generacy occurs. For example, in 3-space, the degeneracy may occur on a line. Then starting from an arbitrary point in this space, we will have to search on surfaces in order to locate points on this curve. The relevant quantity we will be interested in is the dimension of the bulk space minus the dimension of the subspace on which the degeneracy occurs. This number is called the codimension. In the example just quoted, this codimension is two. It is the minimum number of parameters that needs to be tuned to find a degeneracy of a certain degree. For definiteness, we consider in what follows Hamiltonians of a finite dimensional spectrum n. The resulting codi­ mensions however will turn out not to depend on n and are therefore valid even for arbitrarily high dimensions. We consider the following three general cases with their corresponding codimensions.

10 1. In the space oinxn

complex hermitian matrices, those matrices with degenerate

eigenvalues of degrees mi, m 2 , m 3 , . . . have codimension £,-(m? — 1);

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2. in the space of n x n real symmetric matrices, those matrices with degenerate eigenvalues of degrees mi, m 2 , m 3 , . . . have codimension £ » ( m t — l)( m « + 2)/2; 3. in the space o f n x n imaginary skew matrices, those matrices with degenerate eigenvalues of degrees mi,7712,7713,... have codimension X2,-(mJ — 1) ( since the spectrum is symmetric with respect to 0, we look at the positive eigenvalues only ). These results are obtained by the following arguments. Let us first look at an n x n hermitian matrix with one degenerate eigenvalue of degree m. The diagonalized form is ^llrr

(i)

\

^n-m+l

/

A general unitary transformation will leave the spectrum unchanged. This transfor­ mation is an element of the U(n) group. For the above matrix a U(m) x Un~m(l) subgroup of U(n) leaves this whole matrix unaltered. Therefore the set of matrices with the same spectrum is parametrized by the coset U(n)/[U(m) x Un~m(l)]. (Such a construction is used in defining the field space of the Nambu-Goldstone bosons.) There are (h — 771 + 1) eigenvalues that can be changed without lifting the degener­ acy. Therefore, the matrices with an m-fold degeneracy form a subspace of dimension

11 dimU(n)/[U(m) x Un~m(l)] + n - m + l = r i 2 - r a 2 + i. The codimension is therefore n2 — (n 2 — m 2 + 1) = ra2 — 1, independent of n. The matrices with degeneracies of

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degrees

roi,ra2,...

have codimension J2i(m 2. The fundamental group, on the other hand, may be non-abelian. One may introduce TTQ(Y) as the set of path components of Y. The 0-sphere is the set of two points. Since one of the points is always mapped to the designated point in y , the homotopy classes of the maps are just the path components of Y. The set 7r0(y) in general cannot be made into a group, however. If Y is a topological group, then 7T0(y) is a group, and ^i(Y) is abelian. If the space Y has more than one path component, the homotopy groups may depend on which component the base point is in.

23 If the space is connected and simply connected, 7r0, 7rx each has only one element, and we can make the identification 7r t (y) = [5',V].

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If the space is not simply connected, then ft\{Y) can be made to act on irn(Y).

An

element of 7rn(Y) can be represented by the image of an n-sphere with its north pole attached to the base point while that of 7r 1 (y) a loop that starts from and ends at the base point(Fig.2). All the spheres obtained by continuous deformations that have the same base point represent the same element in the homotopy group. The result of the action of [o^] 6 Wi(Y) on [a2] 6 Kn(X) is described as follows. Take the (image of the) sphere a2. Continuously move the base point along ax(which is oriented). The instant we move the base point, the sphere does not represent an element of 7rn(Y) anymore. However, at the end of the trip around the loop, when the base point has returned to the original point, it will represent an element in 7rn(Y) once again . This element can be different from the original one([a 2 ]). This difference will be identified as the the result of the action of [ai] on [a 2 ], which

we

denote by [ai][a 2 ]. The

1

homotopy classes [S' ,!'] without base point identify all deformed spheres along the way and do not necessarily form a group. In fact [5*,!^] can be identified with the quotient ic%(Y)liri{Y) with respect to the above action. The action of ft\(Y) on itself is simply the conjugation operation, so that [5 ,1 ,F] is the set of conjugate classes of

*i{Y). We list some useful facts about homotopy groups. Let Z be the group of integers(free group in one generator, the infinite cyclic group) and Zm the cyclic group of order m. 1. For spaces X and Y, the homotopy groups for the product space X x Y are 7Ti(X xY)

= 7Ti(X) © 7r t (y), where we have used the notation for abelian groups

even though TTI can be non-abelian.

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24

Figure 2: Action of 7Ti on irn: here, [ai] is a generator of 7Ti(y); [mCim\a>m € A} . The most important coefficient group in many cases

will be the group of integers Z. The boundary of a chain is defined linearly so that the boundary of £ m o m c tm is J2m am9cim, where 9c tm is the boundary of the individual i-cell. This boundary is defined to be an (i — 1)-chain with the coefficients of the (i — l)-cells being the multiplicity of the cell in the boundary of the i-cell. Finally, for the multiplicity nm € Z and am G A, nmam = a m 4 - - - - + am € A. The boundary of the boundary of a chain vanishes.

But chains with vanishing

boundaries (kernel of the boundary operator) may not be the boundaries of of some chains (image of the boundary operator). Homology groups give a measure of this distinction.

g

^ A )'wa-c^r

(12)

im(a : Gt+i -> Gt) As a simple example of homology groups, consider the Klein bottle. It can be decomposed into a 0-cell v( a point), two l-cells bx and 62 , and a 2-cell c(Fig.3). dc = 26i, dbi = db2 = 0. Therefore, for integer coefficients A= Z, ker(9 : C2 —> C\) = 0, im(d : C 2 -> d) = 2Z[b1], ker(9 : Cx -> C0) = Z[bx] © Z[62], im(0 : CX -> C0) = 0, ker(d : Co —> C_i) = Co = Z[v\. Hence, the integral homology groups are given by H2(Z) = 0,

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27

b( V

Figure 3: The Klein bottle. The top and bottom edges, the left and right edges are

glued together with the orientations indicated.

28

HX{Z) =

(Z[b1}eZ{b2})/2Z[bl}^Z2[b1]®Z[b2]i

Ho(Z) = Z[v}.

(13)

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For Z2 coefficients, we have ker(9 : C2 —» C\) = Zjfc], im(3 : C2 —> Ci) = 0, ker(d : d -> C0) = Ci = Z2[6X] 0 Z 2 N , im(5 : Ci -* [F, BG]. The Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

simplest of such universal bundles is the helix structure for the group of integers Z: Z —* R —> S1.

Z acts on the real line by addition. Therefore, [Yi*?1] classifies Z

bundles. There are some special cases where the classification [Y, BG] can be reduced to cohomology groups and homotopy groups. For example, it can be shown that, if G is an Eilenberg-MacLane space of type G = K(A,i From obstruction theory, one finds[9] [Y, K(A,i)] G = Z,

1

frS ]

= [Y,K(l,Z)]

[ S \ BG] = 7ri(BG)/ir1(BG)

X

— 1), then BG =

K(A,i).

= //*(Y, A). In the example above,

= H {Y,Z).

On the other hand, if Y S £•', then

= ni.1(G)/7r0(G).

The last equality can be shown by the

fact that EG is contractible and the associated exact sequence. For each bundle structure, the homotopy groups of the spaces F, B and Y are related by an exact sequence, • ■ • -» *»(F) - *■»(*) -» *.(Y) -»7r n _x(F) - » . . - .

(17)

An exact sequence of groups means that at every group in the sequence, the image of the previous homomorphism is equal to the kernel of the next homomorphism. If 0 —► G\ —> &2 —► ^ 3 —> 0 is exact, one can conclude that G3 = G2/G1, from which the result above follows. Exact sequences are useful in the calculation of homotopy groups.

4.5

Vector bundles

In this section, we apply the results discussed so far to vector bundles. These appli­ cations will produce the main mathematical tool we will be using in the remaining

34 sections. Vector bundles are fiber bundles whose fiber is a vector space V, with the struc­ ture group being the linear transformation group GL(V) on this vector space. From Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

the previous section, the classification of vector bundles amounts to the classification of the associated principal GL(V) bundle. For real vector spaces GL(n, R) is homotopy equivalent to the orthogonal group 0(n), while for complex vector spaces GL(n,C)

is homotopy equivalent to the unitary group U(n).

Universal bundles

for 0(n) can be roughly described as follows. Take EMO(TI) to be the coset space 0(n -f M)/0(M)

= Vn+Mtn, this space is called a Stiefel manifold. Since there is an­

other subgroup 0(n) which commutes with O(M), 0(n) acts freely on V(n + M,n) by the multiplication in 0(n -f M). Therefore, we have a principal 0(n) bundle: 0(n) -+ EM0(n) i < M.

-» GnM{R)

= 0(n + M)/[0{n)

For M —> oo, EMO(TI)

x 0{M)].

7n{EM0{n))

= 0 for

= EO(n) becomes contractible and the bundle

0(n) —> EO(n) —► Gny00 becomes the universal bundle for 0(n). Here, Gn,oo is called an infinite real Grassmannian. The 0(n) bundles, and therefore n-dimensional real vector bundles, are classified by \Y, Gn}QO(R)]. Similarly, we have the complex Grass­ mannian Gn}\f{C) = U(n + M)/[U(n)

x U(M)} and complex n-dimensional vector

bundles are classified according to [Y, G n|00 (C)]. Let Vect(Y,F) phism classes of vector bundles on Y. Let Vectn(Y,F)

be the set of isomor­

be the subset of isomorphism

classes of vector bundle with fiber dimension n. Then Vectn{Y,R)

~ [Y", Gny00(R)\

(1 — 26)/(2 — 26), while the other two degeneracies remain. In Frame 8, the degeneracy that appeared in the beginning has disappeared all together, but the other two degeneracies are still there, and are located near the edge of the picture. At this point, we have two 2-bands. From Frame 8 to Frame 9, there are no further qualitative changes. The initial four 1-bands have undergone a metamorphosis into two 2-bands. We now examine how these behaviors are governed by the evolution of the topological indices, consistent with the conservation law. The parameter space is the 4-sphere Y = S4. From sect. 5.2, Vect(S4,C)

= {(d,c)\d,c

G Z,d > 1} U {(1,0)}.

50

Initially, we have four 1-bands and each is associated with a trivial bundle, v = (1,0). When the inner two bands merge into a 2-band, the associated 2-dimensional bun­ dle V293 = (1^0) ® (1,0) = (2,0) is actually trivial, consistent with the conservation Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

law enunciated previously. Nothing really prevents this 2-band from dissociating because c = 0. As we will see, this degeneracy is present to prepare for the ex­ change of topological indices a little later between the upper and the lower pairs of bands. When the upper and lower pairs become connected, the 4-band is still a trivial bundle V(i©2)®(3©4) = (2,0) © (2,0) = (4,0). However, after the middle de­ generacy is lifted, the upper and lower pairs become two 2-bands characterized by (2,1) and (2,-1). Each 2-band is now represented by a non-trivial 2-dimensional vector bundle that cannot be split by itself(see sect. 5.2). It is easy to check how the conservation law on topological indices is satisfied. On the level of Vect(S4, C), we have V\ = V2 = i>3 = v4 = (1,0) and wi = (2,1), u>2 = (2,-1). Therefore, vi © t>2 © vs © v4 = tui © u>2- The second Chern character which is operative in this case is simply the second index in (d, c) and is obviously conserved. The upper two bands are knotted together by the second Chern character, and so are the lower ones. They will not dissociate unless this topological charge is given to another band, or un­ less they undergo the inverse process to annihilate the second Chern character. Fig.5 gives a schematic diagram of the metamorphosis of four 1-bands into two 2-bands.

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51

©

©

E♦

CM

T

Figure 5: Metamorphosis of energy bands: four 1-bands become two 2-bands.

52

7

Conclusion

In this paper, we have described a way to characterize the topological properties of Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

the dependences of eigenvalues and eigenfunctions on parameters. In the simplest cases when there is no degeneracy for all values of parameters, non-trivial phases may nonetheless appear. In more complicated cases, degeneracies that occur in pa­ rameter space can be related to the twisting of eigenfunctions around the degenerate subspaces. When such situations arise, non-trivial topological charges characteriz­ ing the twists will knot the eigenvalue bands together. A general characterization of topological charges for both the non-degenerate cases and cases with degeneracies is given. Metamorphoses of the degenerate patterns take place when the topological charges are exchanged among the bands. Conservation laws involving topological charges must hold during these exchanges. Acknowledgements. We thank Professor Da Hsuan Feng for inviting us to contribute to this volume and to participate in the Drexel Workshops on Quantum Non-integrability. Parts of the work reported here have been completed while YL was at the University of Kentucky. Partial support for this work has come from DOE, DE-FG05-84ER40154, DE-AC02-76ER13065 and DE-AS05-80ER10713. Useful dis­ cussions with Professors H.B. Nielsen and K.F. Liu are also acknowledged.

53

References [1] John von Neumann and Eugene Wigner, English transl. in Symmetry in the Solid

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State. R.S., Knox, and A. Gold. New York: W.A. Benjamin. Originally in Phys. Z. 30 , 467-470 (1929) [2] H.B. Nielsen, in Proc. of the Seventeenth Scottish Universities Summer School in Physics , St. Andrews, August 1976. Dual Strings, section 6, p 528(eds. I.M. Barbour and A.T. Davies), Univ. of Glasgow, publ. by Scottish Univ. Summer School in Physics. [3] G. A. Hagedorn, "Classification and Normal Forms for Quantum Mechanical Eigenvalue Crossings", Mathematics preprint, Virginia Polytechnic Institute and State University, 1991. [4] J. M. Ziman, Principles of the Theory of Solids, (Cambridge University Press, 1972). [5] C.A. Mead, Chem. Phys. 49, 23-32; 33-38(1980). [6] M.V. Berry, Proc. Royal Society, A392, 45-57(1984) [7] M. Stone, Phys. Rev. D33, 1191-1194(1986). [8] F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111-2114(1984). [9] See for example, G.W. Whitehead, Elements of Homotopy Theory ,(SpringerVerlag, 1978), pp.243-244. [10] J.W. Milnor and J.D. StashefF, Characteristic Classes, ( Princeton University Press,1974).

54 [11] M. Atiyah, K-Theory, (W. A. Benjamin, 1967). [12] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, (Springer-

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Verlag, 1982). [13] N. Steenrod, The Topology of Fibre Bundles, (Princeton University Press, 1951). [14] T. Eguchi, P. B. Gilkey, and A. J. Hanson, Gravitation, gauge theories and differential geometry, Phys. Rep. 66, No.6, 213-393(1980). [15] D. Thouless, M. Kohmoto, M.Nightingle and M. den Nijs, Phys. Rev. Lett. 49, 405-408(1982) [16] J.E. Avron, R. Seiler and B. Simon, Phys. Rev. Lett. 5 1 , 51-53(1983) [17] L.N. Chang and Y. Liang, Mod. Phys. Lett. 3, 1839-1845(1988). [18] L. N. Chang and Y. Liang, Comm. Math. Phys., 108, 139-152(1987). [19] Chang, L.N., Liang, Y., A topological theory of parametrized quantum mechan­ ics. preprint.

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Wei-Min Zhang Wei-Min Zhang was raised in the rural area near Suzhou, China and received his Ph.D. in theoretical physics from Drexel University in 1990. In 1990/92, he was a research associate with the nuclear theory group of the University of Washington. Currently he is a research associate with the nuclear theory group of Ohio State University. Dr. Zhang's main research interests are in light-front quan­ tum chromodynamics, nuclear structure physics, relativistic heavy- ion collisions, quantum nonintegrability and co­ herent states theory. (e-mail: [email protected])

Da Hsuan Feng Da Hsuan Feng received his Ph.D. in theoretical nuclear physics from the University of Minnesota in 1972. From 1972 until 1976, he was a United Kingdom Science Re­ search Council postdoctoral fellow at the Department of Theoretical Physics of the University of Manchester in England and a Research Scientist at the Center of Nuclear Studies of the University of Texas at Austin. In 1976, he came to Drexel University as a faculty member and is cur­ rently M. Russell Wehr Professor of Physics. A frequent visitor to China, he holds also the title Guest Professor of the Department of Modern Physics of Lanzhou Univer­ sity. In 1979/80, he was a visiting scientist at the Niels Bohr Institute and from 1983/85, a Program Director in Theoretical Physics in the Physics Division of the National Science Foundation. Dr. Feng's research interests are in nuclear spectroscopy, mathematical physics and coherent states theory, quantum nonintegrability and nuclear astro­ physics. He is also currently Chairman-elect of the Com­ mittee of the Arts and the Sciences of the Franklin Institute in Philadelphia. (e-mail: [email protected])

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J i a n - M i n Yuan Jian-Min Yuan received his Ph.D. in chemical physics from the University of Chicago in 1973. He was a postdoc­ toral research fellow at the Quantum Theory Project of the University of Florida and an instructor and research asso­ ciate at the Department of Chemistry at the University of Rochester. Since 1978, he has been a faculty member at Drexel University and is now Professor of Physics. Over the years he has been a visiting scientist at the Institute of Atomic and Molecular Sciences of Academia Sinica, Uni­ versity of Utrech, University of Kaiserslautern, University of Pennsylvania, University of Kansas, University of New Mexico, Lanzhou University and the Univeristy of Science and Technology of China. His main research interests have been in few-body problems, molecular reaction dynamics, collisions in presence of external fields, interaction between radiation and matter, non-linear dynamics and chaos, and applications of group theory. (e-mail: [email protected])

66

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Quantum-Classical Correspondence and Quantum Chaos W e i - M i n Zhang Department of Physics, FM-15, University of Washington Seattle, Washington 98195, USA k D a H s u a n Feng 1 & J i a n - M i n Y u a n Department of Physics and Atmospheric Sciences, Drexel University Philadelphia, Pennsylvania 19104, USA

Abstract In this article, a theory which we have recently developed is reviewed. This theory addresses the long standing problem of quantum-classical correspondence. It begins with the axiomatic structure of quantum mechanics, from which the basic concept of dynamical degrees of freedom emerges and the associated physical geometry of an arbi­ trary quantum system is constructed. Such a geometrical space possesses both complex and symplectic structures, the two basic ingredients to guarantee the rigorous corre­ spondence between quantum and classical kinematics. Also, it is remarkable that the quantum-classical correspondence of dynamics can be mediated by a quenched index obtained from the geometry. The intermediate stage from quantum to classical dynam­ ics is described by the semiquantal dynamics. The main property of the semiquantal dynamics is that it manifests the effect of quantum fluctuation on classical trajecto­ ries, and quantum fluctuation depends explicitly on the quenched index. This offers a way to study "chaos in quantum systems", We also prove a theorem which links dynamical symmetry and quantum integrability, and conclude that "quantum chaos" is related to dynamical symmetry breaking. The generic rule of quantum chaos is ex­ plored. Throughout the article, many basic quantum systems are used to illustrate these important points.

1

E-mail address: FENGQDUVM

67

1

Introduction

Half a century ago, Einstein argued that "The t r u e laws c a n n o t b e linear nor can Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

t h e y derived from s u c h " [ l ] . This is obviously a reasonable point of view since nature is far too complicated to be linear. Yet, a common belief among the scientific community is that quantum mechanics, which is at least formally linear, is the underlying methodology to understand nature. These seemingly conflicting views naturally bring one to ask whether quantum mechanics can encompass nonlinear phenomena [2]. As the understanding of classical nonlinear phenomena deepens, a part of the above raised question is what are the generic behaviors (if any) of a quantum system when its classical counterpart (if exists) is chaotic. In the past decade, seeking the answer to this part of the question has become an exciting research field of theoretical and experimental physics. In this review, we will discuss our attempt to answer the question. It is well-known that Hilbert space is the framework of quantum mechanics from which physical states are constructed and represented by wave functions.

A typical quantum

observation is the spectrum. Therefore, in order to explore the quantum manifestation of classical chaos, it seems natural to choose spectrum and wave functions as starting points and then explore the classical pattern embedded in these basic quantum building blocks [3]. However, until now, there is generally no commonly accepted definition of quantum chaos! The lack of such a definition is because many of the concepts which are essential in analyzing classical chaos are meaningless in quantum mechanics. Still, intuitively, in the nearly classical limit, the quantum phenomena cannot differ much from those of classical mechanics. Therefore, addressing the problem of "chaos" in quantum mechanics demands a more rigorous formulation from the fundamental structure of quantum theory [4]. Undoubtedly, to meet such a demand, one needs to formulate quantum-classical cor­ respondence. Unfortunately, there is no clear formulation at this point. A common (a la textbooks) understanding of the classical limit of quantum mechanics is Tt -* 0. This limit

68 is in fact one of the most ambiguous limits in physics. To illustrate this point, let us con­ sider the well-known non-relativistic limit of Einstein's theory of special relativity, which is achieved by a small speed ( small compare to the universal constant c, namely, the speed of Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

light). However, the classical-limit of quantum mechanics cannot be obtained analogously because no such physical quantity, like speed, exists to compare with the universal constant, ft. It should be mentioned that sometimes the classical limit can be obtained when the "ac­ tion" is large compare to h. However, such a comparison is not ambiguous free (except for one-dimensional systems). On the other hand, most attempts to study "quantum chaos" have concentrated on quantizing classical nonintegrable systems [3,5] while not enough attention is focussed on studying the classical limit of quantum integrable and nonintegrable systems (the concepts quantum integrability and non-integrability will be discussed later). Yet, since in principle the former is only a limiting case of the latter, and most realistic quantum systems do not have classical counterparts, the latter approach is more general and natural. The central theme of this review is to exploit such a "natural" approach. To appreciate what will be discussed in this article, it is useful to remind ourselves the historical developments in studying this problem. In most textbooks on quantum mechanics, the so-called classical limit has been formally discussed by using the Ehrenfest's theorem 2 . The rigorous investigations given in the literature for seeking the classical limit of quantum mechanics can be classified by the following three approaches. The first, which is due to Schroedinger [6], is to develop a wave packet whose time evolution follows the classical trajectories (namely, the time evaluation of the coordinate and momentum expectation values are the solution of the Hamilton equation) and at the same time satisfies the Schroedinger equation. Although this approach was extremely useful in developing quantum optics [7], it was only successful in constructing the wave packets (a set of coherent states) of harmonic oscillators. 2

The derivation of Ehrenfest's theorem is, however, not rigorous

69 The second approach, proposed by Dirac [8], is to construct a quantum Poisson bracket such that the basic structures of quantum and classical mechanics can be in one-to-one correspondence.

It is interesting to note that such a procedure was in fact "inversely"

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realized by the so-called geometric quantization procedure of Konstant, Sourian and Krillov in 1970's [9]. The essence of this work can roughly be summarized as follows. One begins with a symplectic geometry 3 and then uses the coadjoint representation of an algebra g to construct a quantum system.

In our opinion, this process is directionally unnatural

(from classical to quantum and not vice versa). Quite recently, using coherent states [12], YafFe retraced the geometric quantization path and obtained "large iV-limit as classical mechanics" [10]. Unfortunately, this procedure suffers from the fact that for a quantum system, there exists a number of corresponding classical systems. Thus, the one-to-one correspondence between quantum and classical limits is lost. The third approach, namely the well-known Feynman path integral formalism, is to express quantum mechanics in terms of classical concept by integrating over all possible paths for given initial and final states. This approach is based on the underlying classical mechanics and therefore not applicable to quantum systems without classical counterparts. To overcome this difficulty, Klauder extended the path integral formalism to phase space representation via coherent state basis [11]. This formalism provides, at least formally, a connection between quantum and classical mechanics. However, before it can become practically useful, the phase space structure must be constructed first. These three approaches are in fact based on the three basic quantum mechanics pictures, namely Schrodinger picture, Heisenberg picture and Feynman path integral representation. None of them, however, can completely answer the question of quantum-classical correspon­ dence. Yet all of them points to the fact that coherent states [12] play a cardinal role in studying the classical limit of an arbitrary quantum system. This is quite reasonable, espe­ cially when one recalls that coherent states are constructed explicitly to exhibit much of the 3

In layman terms, this means that the system possesses the hamilton dynamics.

70 classical features within the quantum context and embed them into a phase space structure. Still, one must be warned that a coherent state basis is only one of many representations of quantum mechanics, whereas the phase space of a quantum system must be the basic Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

structure in determining the classical limit and should logically exist, independent of the coherent states. Therefore, it is interesting to inquire how explicit phase space structures of quantum systems can be constructed without relying on any classical basis and independent of the process of parameterizing quantum states. Only after this step can one then study the dynamics on the quantum phase space and its classical limit, from which to explore the manifestation of chaos in general. Our approach to this problem is based on the axiomatic structure of quantum mechanics, from which we define the quantum dynamical degrees of freedom (QDDF) and construct the quantum phase space (QPS). A concept of quantum integrability is then proposed. Also, a theorem and its proof about the relationship between quantum integrability and dynamical symmetry [13,14,15] are presented. The quantum-classical correspondence of kinematics and dynamics are explicitly obtained via phase space structure and a quenched index. Furthermore, we will show also how to construct systematically the canonical form of the coordinates of the quantum phase space for all semisimple Lie groups with Cartan decomposition. From such a canonical form, one can realize and compute explicitly the "classical limit". This offers a natural route to investigate the quantum manifestation of chaos. After that, we shall concentrate on the subject of seeking the generic rule of quantum fluctuation on classical chaos, in order to obtain the signature of "quantum chaos" [16,17,18].

2

Quantum-Classical Correspondence of Kinematics

As we have emphasized, at the heart of "quantum chaos" lies the question of quantumclassical correspondence. This correspondence is operative on two levels: kinematical and dynamical. The study of "quantum chaos" should probe the latter. To this end, it is

71 necessary to first clarify what is the quantum-classical kinematical correspondence. In a nutshell, a kinematical quantum-classical correspondence is to reconcile the quantum and classical degrees of freedom and the associated geometrical structures. The following is a

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succinct discussion of this reconciliation. Details of our construction can be found in [14,15]. It is well known that time evolution of a classical system is represented by a trajectory in phase space (a symplectic structure), denoted here by p , and dynamical observables are also functions defined in this space. The dimension of p is of course twice the degrees of freedom and a point in the space represents a physical state. Likewise, for a quantum system, dynamical properties are described within a Hilbert space, 9£, and dynamical observables are self-adjoint operators acting on this space and a physical state of the system is represented by a wave function, or more precisely a ray, (i.e., a set of wave functions: {e* a ^, i/> 6 3£}). Thus, $ in quantum mechanics serves the analogous role as p in classical mechanics. However, the problem is that a Hilbert space cannot be directly defined as a phase space because its dimension does not have any physical interpretation, nor can it be directly reduced to the phase space in classical limit. In fact, one knows that a quantum system does not a priori require a phase space structure. What is needed is only a complex structure 4 . Yet, in order for a quantum system to have a classical limit, a phase space is required. Historically, an attempt was made [22] to formally transfer quantum mechanics to clas­ sical mechanics in terms of the language of phase space. Roughly it is as follows. Let | ip) be an arbitrary state N

W = 2>#;>,

(2.1)

t=i

where ut (i — 1,..., N) are complex coefficients which, once given, will completely determine | V7). If one denotes ttt = (qi+ipi)/y/2

and u* = (qi — ipi)/y/2,

then the Schroedinger equation

is immediately reduced to Hamilton-like equations dqi, _ dt " 4

dH{p,q) dPi '

(

]

A Hilbert space can be defined on a phase space which must have a complex structure. Definitions of

complex and symplectic structures are given, for example, in Arnold's book, p.224 (see Ref.[21])

72 dpi

_

dH{p,g)

It

~

W~

(

]

with the Hamiltonian function being the matrix element of the Hamiltonian operator in the Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

state | tp) H(q,p) = (^ \H\ *> = \ £ { * | f f |*fc)(ttPk + mk + tpfcft - • « » )

(2.4)

Also, in the above formulation, the commutation relationship [A, B] of any two physical observables A and J9 can be expressed as a Poisson bracket. Thus, it appears that quantum mechanics can be embedded formally into a 2N-dimensional classical mechanics. So, it does appear that for a given quantum system, there is always a phase space whose dimension is twice the dimension of the quantum state space and the quantum dynamics is equivalent to a classical system of N coupled harmonic oscillators. Furthermore, since the Hamiltonian function is in quadrature, Eq.(2.2) is, by definition, integrable. From this result, one may conclude that a quantum system is always integrable (and hence no chaos!) [3]. The above conclusion is obviously wrong. In fact such a procedure cannot be carried out in quantum mechanics. The process of a quantum mechanical measurement is based on its probability interpretation. This interpretation demands that the Hilbert space is integrable [23]:

JU)\2dp { i 2 = i,

(2-7)

t=i

where u\ — cu{. The integrability constraint of the Hilbert space reduces at least the 2iV-dimensional phase space to a complex projected space which mathematically is the CP(N - 1) space (see the latter derivation). In this space, the Hamiltonian function is no

73 longer a TV-coupled harmonic oscillator system in quadrature. Also, since the CP(N

— 1)

is compact, its topology is highly non-trivial and the corresponding phase space structure must be reconstructed. Eq.(2.2) defined on the original flat 2iV-dimensional phase space do

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not represent a formulation of either quantum mechanics or classical mechanics. In general, for a quantum system, one cannot even a priori specify its Hilbert space. Hence, there is no direct way to consider such a complex projected space as the geometry of a quantum system. Fortunately, consistency of quantum theory implies that there must exist a fundamental structure which can be used to determine the system's Hilbert space structure a priori before solving the quantum dynamical equation (Schrodinger equation, Heisenberg equation, and etc.). This can be seen as follows. According to the axiomatic structure of quantum mechanics, such a fundamental structure is simply the algebraic structure of a quantum system [23]. In fact, the quantum mechanical Hilbert space is realized as a unitary irreducible representation (irrep) of an algebra g. Some examples are listed here. A harmonic oscillator is given by Heisenberg algebra h*: {a,a\a^ayI},

where a,a^ are

the creation and annihilation operators of a harmonic oscillator; a spin system is given by su(2) : { 5 _ , 5 + , S0}j the spin operators; a hydrogen atom and a Dirac particle are given by 5o(4, 2) with operators {Lrs}

=:

L = rxp,

A = -rV-l)-p(r-p),

M = ^r(p2+l)-p(r.p),

T = rp,

S = ~2r(p2 - 1), T = r . p - i ,

(2.8)

r0 = - r ( p 2 - l )

)m and for the hydrogen atom

Lk = Lt] = \ t i n i Ai

=L ^ \

t

^

(spin), (analog of Lenz vector),

Mi = Li5 = -rya0

(pure Lorentz transformations)

TM = L^6 = - 7 ^

(algebraic current operators),

S = L4i = - ^ 7 5 7 o

('tilt'),

(2.9)

74 T = L56 = -~75

(Lorentz scalar)

for Dirac particles. (A general procedure to construct a group structure of a quantum

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system is discussed in Chapters 12-13 of the book by Barut and Raczka [20]). Furthermore, the associated covering group G of g carries a natural geometrical manifold [20] and all representations of quantum mechanics can be represented on such a geometrical manifold. Note that the covering group G is defined by a exponential mapping of g. Thus, without any loss of generality, we can construct kinematical correspondence from a dynamical group structure of the system. The general solution [14,15] is:

• A quantum system possesses a well-defined dynamical group structure G with a Hilbert space & (an irrep space). The number of quantum dynamical degrees of freedom (QDDF) of the system is just the same number as the M independent non-fully degen­ erate quantum numbers necessary to specify ft. The quantum phase space} denoted by p, can be uniquely realized on a 2M-dimensional coset space G/H, where H (C G) is the maximal stability subgroup of a fixed state \ip0) € ft of the system. The coset space G/H and its global property gives a precise realization of the kinematical quantumclassical correspondence.

Note that the fixed state, \rp0) € &, can be found as follows. If G is compact, |V>0) 6 ft is the lowest (highest) weight state of & When G is non-compact, \^0) G ft is the lowest bound state of ft. To see what we can deduce from the above statement, we will now show some specific but nontrivial examples. A detail and rigorous proof can be found in Refs.[14,15]. First of all, we will illustrate the concept of the QDDF. The non-fully degenerate quantum numbers here are defined by the non-constant eigenvalues of a complete set of commuting operators in the associated basis. The simplest and trivial example is of course harmonic oscillator whose dynamical group is Heisenberg-Wyle group /f4 and the Hilbert space is the Fock space in which a basis {|n)}, as it is well-known, is specified by the non-fully degenerate quantum number n. The next but nontrivial example is a spin system. It is also well-known

75 that the dynamical group for this system is SU(2). Its QDDF is one! This is because in its Hilbert space {\jm)}

(a given irrep space of 5*Z7(2)), the total spin quantum number j is a

constant (i.e., fully-degenerate quantum number). The only non-fully degenerate quantum Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

number to specify the basis {\jm)}

is the z-component quantum number m.

Another

important example is the central potential problem which has the well-known dynamical group 5(7(1,1). Its QDDF is also one. Here, the irrep space (Hilbert space) of SU(1,1)

is

specified by two quantum numbers, k (related to angular momentum) and n, the principal quantum number. Since in the central potential the angular momentum is conserved, k is a fully degenerate quantum number. Note that although the QDDF of the spin and central potential are the same, their phase space structures are vastly different (see later discussion). The simplest and realistic quantum systems are hydrogen atom (non-relativistic) and/or a free Dirac particle (relativistic). Both systems have the dynamical group 5 0 ( 4 , 2 ) , but the QDDF for each are three and two, respectively. For the hydrogen atom, the three QDDF correspond to three non-fully degenerate quantum numbers: the principal quantum number n, angular momentum j and its z-component m which completely label its Hilbert space basis {\njm)}.

For the free Dirac electron, the angular momentum (spin) is fixed so that

we only have two non-fully degenerate quantum numbers. The results quoted here are all derived from pure quantum mechanics [14]. Some remarks on the above construction of the QDDF should be added:

1. Consistency: Such a construction is fully consistent with the corresponding classical concept.

For example, a canonically quantized system of a classical Hamiltonian

system with M degrees of freedom requires M quantum numbers to fully specify a basis of its Hilbert space. Let us consider a system with n structureless particles. The corresponding degrees of freedom are known to be 3n. Its Hilbert space is spanned by the simultaneous eigenstate set {0(ri,r2, ...,r n )} of Sn position observables: (Xiil>)(rur2l (Y^)(ru

- . , r n ) = x^(rur2,...,rn),

(2.10)

r 2 ,..., r n ) = ^ ( n , r 2 ,..., r„),

(2.11)

76 (Zii))(ru r2,..., r„) = *,-^(ri, r 2 ,..., r„), i = 1,..., n.

(2.12)

This clearly shows that there are 3n non-fully degenerate quantum numbers (#,, j/ t , Z{

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with t = 1, ...,n) to specify a basis vectors of its Hilbert space for such a system. 2. Universality: Since the construction is carried out without any explicit or implicit assumption about the system possessing a classical counterpart, therefore it is appli­ cable to any quantum system containing both classical-like and additional internal degrees of freedom. 3. Uniqueness: The procedure is based on the algebraic structure of the basic physical observables without specifying the details of a Hamiltonian. It therefore depends only on the global structure of the system and so is unique within a specific Hilbert space.

The above discussions show that once an explicit definition of the QDDF is given, it is rather trivial to determine the QDDF number for any arbitrary quantum system. What is of course not obvious is whether one can indeed construct a quantum phase space whose dimension is that number. One major difficult in finding a unique and useful phase space structure with this global property. Here we would like to point out that by using the well known coherent state theory [12], one can achieve this aim. Indeed, we find that in the coherent state theory, the most crucial step is to determine the so-called fixed state \0) [15] because as we shall show, such a state will completely determine the global properties of space. We will illustrate this point by several explicit examples. Example 1. Harmonic oscillator. As we have pointed out before, the dynamical group for this system is the Heisenberg-Weyl group H4. To construct its phase space, we must first find its fixed state, which turns out to be the vacuum state. The phase space is then H±IU(l) ® 17(1), where (7(1) ® U{\) is invariant with respect to the vacuum. The phase space structure is determined by the coherent state, | z), of # 4 /£/(l) ® U(l): D(a)\ z) = \z + a),

D(a) e H4/U(l) ® U(\)

(2.13)

77 This shows that for this system the phase space is rather trivial as in classical mechanics, namely, a one-dimensional complex space or two-dimensional real space.

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Example 2. Spin systems. Since the dynamical group here is SU(2), in an irrep space {\jm)}

the fixed state is \j - j). The phase space is then SU(2)/U(\),

which is isomorphic

to a 2-sphere. To learn more about the structure, we can begin with the coherent state I «) = D(z)\j-i)

= exp (zS+ - z'S-)\j - j).

(2.14)

where S+ ( £ - ) is the spin-up (spin-down) operator. In the geometrical representation, it is

D{z)

0

z

-z*

0

(2.15)

exp

coslzl

13!

L —TTT " M sin \z\

X0

X

(2.16)

cos z

where xQ is real and x = x\ + zx 2 . Unitarity of D(z) 6 SU(2)/U(1)

requires that

a£ + s? + a£ = 1

( 2 - 17 )

which is a 2-sphere, and z = | e ~ , v . It must be pointed out that this phase space structure is non-trivial. Its canonical coordinates can be found from the coherent state basis [13,15] as

I

q = yJAjh sin | cos

ft€G/H) [15]. It turns out that G / H possesses both symplectic and complex structures. We write its closed nondegenerate 2-form u as w = t n

^ § i l £ l ^ A ^

(2.26)

where z is the local coordinate of G/H. The quantity K is the Bergmann kernel of G/H. In summary, we show in this section that the complex and symplectic structures of G/H insure the kinematic correspondence of quantum-classical mechanics. Namely, one can defined both quantum and classical mechanics on G/H.

3

Quenched Quantum Dynamics

The dynamical correspondence of quantum-classical mechanics is a fundamental issue which has not so far been fully understood. The recent interests in searching for the possible quantum chaotic behaviors have revived this important question. Quite recently, in order to address this problem, we (in collaboration with Lay Nam Chang) have introduced a theoretical formulation called quenching of quantum mechanics (QQM) [17]. The usual statement in textbooks is that the classical limit is achieved by taking the h -» 0 limit. However, as was pointed out in the introduction, this is in fact a misleading process. In order to better study the resultant singularity structures which are likely to accompany a transition to chaos, one must be a little more precise as to what this limit entails. It was suggested by YafFe [10] that the large N limit can be taken only for wave functions expressed in the coherent state basis, and for matrix elements expressed in this

81 same basis. However, the origin of the parameter N which mediates the approach to the classical limit is not clear.

Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

As we shall see later, the QQM will reveal a possible origin of this parameter, which is based upon our work discussed in the previous section (also see [15]), where a specific geometric phase space framework is provided for this basis. The passage to the classical limit then makes use of an alternative approach to the usual h —► 0. It involves letting a parameter, called the quenched index 5, tend to infinity. The quenched index is in this case dimensionless, which in many cases can be proven advantageous. Also, in cases that E turns out to be a fixed number the system does not have a classical limit. To understand the inherent property of E, we shall consider the case that the associated Lie algebra can be split up as follows: g = h + k with [h, h] € h, [h, k] 6 k, [k, k] € h, where h is the Lie algebra of H. Then the explicit form for JC is JC(z,zm) = det(/± zh)±E

(3.1)

where the sign + ( - ) refers to the case where G is compact (non-compact), Z a matrix with elements z, and the parameter E is related to (0|/it|0) with h{ £ h. This offers us a geometrical interpretation of E. The QQM can be carried out by expressing first the quantum equations (propagator) on G / H U = Jvexp(iS/H)

(3.2)

where U is the propagator, V the measure of path integral and S the effective action

= f(d - Hdt)

(3.3)

In Eq.(3.3), tf is the one-form of G / H and H the expectation value of the Hamiltonian operator H = H(Ti),Ti € G, as given by =

H =

MfdMC^-^dA 2 V dz' dz" ) (il\H(Ti)\U)

(3.4) ' (3.5)

v

82 Obviously, the quantum equations presented here are dependent on H. When one expands around this parameter (not h), one obtains the semiquantal dynamical equations which describe a classical-like system. Note that this comes purely from quantum structures and Quantum Non-integrability Downloaded from worldscientific.com by Daniel Yang on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

therefore provides a explicit way to reach to classical dynamics when 5 —► oo (if exist) lim 7i = WesJr((ft|Ti|ft)).

(3.6)

E-*oo

It should be pointed out that the above limit is actually divergent. This is because the phase space derived from the quantum geometry is not yet scaled. To obtain a convergent limit, one must introduce a scaled canonical coordinates,

v/

=(