Symmetries In Nuclear Structure: An Occasion To Celebrate The 60th Birthday Of Francesco Iachello - Proceedings Of The Highly Specialized Seminar: An Occasion to Celebrate the 60th Birthday of Francesco Iachello - Proceedings of the Highly Specia... 9789812702760, 9789812388124

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THE SCIENCE AND CULTURE SERIES — PHYSICS Series Editor: A. Zichichi

Proceedings of the Highly Specialized Seminar on

SYMMETRIES IN NUCLEAR STRUCTURE

An Occasion to Celebrate the 60th Birthday of Francesco lachello Editors

Andrea Vitturi and Richard F. Casten

SYMMETRIES IN NUCLEAR STRUCTURE

THE SCIENCE AND CULTURE SERIES — PHYSICS Series Editor: A. Zichichi, European Physical Society, Geneva, Switzerland Series Editorial Board: P. G. Bergmann, J. Collinge, V. Hughes, N. Kurti, T. D. Lee, K. M. B. Siegbahn, G. 't Hooft, P. Toubert, E. Velikhov, G. Veneziano, G. Zhou

1. Perspectives for New Detectors in Future Supercolliders, 1991 2. Data Structures for Particle Physics Experiments: Evolution or Revolution?, 1991 3. Image Processing for Future High-Energy Physics Detectors, 1992 4. GaAs Detectors and Electronics for High-Energy Physics, 1992 5. Supercolliders and Superdetectors, 1993 6. Properties of SUSY Particles, 1993 7. From Superstrings to Supergravity, 1994 8. Probing the Nuclear Paradigm with Heavy Ion Reactions, 1994 9. Quantum-Like Models and Coherent Effects, 1995 10. Quantum Gravity, 1996 11. Crystalline Beams and Related Issues, 1996 12. The Spin Structure of the Nucleon, 1997 13. Hadron Colliders at the Highest Energy and Luminosity, 1998 14. Universality Features in Multihadron Production and the Leading Effect, 1998 15. Exotic Nuclei, 1998 16. Spin in Gravity: Is It Possible to Give an Experimental Basis to Torsion?, 1998 17. New Detectors, 1999 18. Classical and Quantum Nonlocality, 2000 19. Silicides: Fundamentals and Applications, 2000 20. Superconducting Materials for High Energy Colliders, 2001 21. Deep Inelastic Scattering, 2001 22. Electromagnetic Probes of Fundamental Physics, 2003 23. Epioptics-7, 2004 24. Symmetries in Nuclear Structure, 2004

THE SCIENCE AND CULTURE SERIES — PHYSICS

Proceedings of the Highly Specialized Seminar on

SYMMETRIES IN NUCLEAR STRUCTURE An Occasion to Celebrate the 60th Birthday of Francesco lachello

Erice, Italy

23 - 30 March 2003

Editors

Andrea Vitturi and Richard F. Casten Series Editor

A. Zichichi

\jjp World Scientific NEW JERSEY • LONDON

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Cover Illustrations Front: Roman mosaic ca 200 BC Back: Seventeenth century map of Sicily by Dutch cartographer Jansson

SYMMETRIES IN NUCLEAR STRUCTURE: AN OCCASION TO CELEBRATE THE 60TH BIRTHDAY OF FRANCESCO IACHELLO Copyright © 2004 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 or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-238-812-5

Printed in Singapore by World Scientific Printers (S) Pte Ltd

••••••HHHIIHI

(Pi ^••111^

•••I

^^^^^P

Professor Francesco Iachello

This page is intentionally left blank

CONTENTS

Preface A. Vitturi, R. F. Casten, editors Symmetry: the search for order in nature F Iachello

xiii

1

Additional quantum numbers, dynamical symmetries and degeneracies /. Talmi

10

Partial dynamical symmetry as an intermediate symmetry structure A. Leviatan

22

Vector coherent state theory: a powerful tool for solving algebraic problems in physics D. J. Rowe

32

Symmetries in strongly deformed nuclei J. P. Draayer. G. Popa, J. G. Hirsch, C. E. Vargas

42

Shape coexistence and its symmetries K. Heyde. R. Fossion

52

Shape-invariance and many-body physics A. B. Balantekin

62

Some new perspectives on pairing in nuclei S. Pittel. J. Dukelsky

72

Pairing and quartetting in the Interacting Boson Model P. Van Isacker

79

Challenges from symmetry on the drip lines D. D. Warner

87

VII

viii High accuracy atomic mass measurements. Application to neutron-rich zirconium isotopes J. Aysto. V. Kolhinen, S. Rinta-Antila, S. Kopecky, J. Hakala, J. Huikari, A. Jokinen, A. Nieminen, J. Szerypo

97

Nuclear supersymmetry: new tests and extensions A. Frank. J. Barea, R. Bijker

107

Everything you always wanted to know about SUSY, but were afraid to ask R. Bijker. J. Barea, A. Frank

117

Supersymmetry and identical bands P. von Brentano

125

Searching for boson-fermion symmetries in neutron-rich nuclei J. A. Cizewski. K. L. Jones, J. S. Thomas, D. W. Bardayan, J. C. Blackmon, C. J. Gross, F. Liang, D. Shapira, M. S. Smith, D. W. Stracener, R. L. Kozub, C. D. Nesaraja, U. Greife, R. J. Livesay, Z. Ma

134

Supersymmetry in nuclear clusterization G. Levai

140

Cluster effects in alternating parity and superdeformed bands of medium mass and heavy nuclei G. G. Adamian, N. V. Antonenko, R. V. Jolos. Yu. V. Palchikov, W. Scheid, T. M. Shneidman

146

Thermal signatures of phase transitions in finite nuclei Y. Alhassid

156

Integrability and quantum phase transitions in Interacting Boson Models 7. Dukelsky, J. M. Arias, J. E. Garcia-Ramos, S. Pittel

166

Phase transitional behavior in spherical-deformed transitions regions R. F. Casten

172

IX

Critical point nuclei in the Interacting Boson Model N. V. Zamfir, E. A. McCutchan, R. F Casten

182

Critical points in nuclei and Interacting Boson Model intrinsic states J. N. Ginocchio. A. Leviatan

191

The critical point symmetry E(5) and the IBM J. M. Arias, C. E. Alonso, A. Vitturi, J. E. Garcia-Ramos, J. Dukelsky, A. Frank

201

Finite well solutions for the E(5) and X(5) Hamiltonians M. A. Caprio

211

Analytical solutions of Bohr collective Hamiltonian with y-instability L. Fortunate. A. Vitturi

217

Phase transitions and critical points in the Interacting Boson Model J. E. Garcia-Ramos. J. M. Arias, J. Barea, A. Frank

223

Test of the empirical realization of the X(5) symmetry in 150Nd and 104Mo R. Kriicken

229

The rich structures of a very simple Hamiltonian J. Jolie

239

The excited 0+ states in 162Yb and the critical point phase/shape transition E. A. McCutchan. N. V. Zamfir, R. F. Casten

248

Test of the critical point symmetry X(5) in N - 90 nuclei and A ~ 180 Os isotopes A. Dewald. O. Moller, D. Tonev, A. Fitzler, B. Saha, K. Jessen, S. Heinze, A. Linnemann, J. Jolie, K. O. Zell, P. von Brentano, P. Petkov, R. F Casten, M. Caprio, N. V. Zamfir, R. Kriicken, D. Bazzacco, S. Lunardi, C. Rossi-Alvarez, F. Brandolini, C. Ur, G. de Angelis, D. R. Napoli, E. Farnea, N. Marginean, T. Martinez, M. Axiotis

254

Phase transitions in the octupole degree of freedom P. G. Bizzeti

262

Prompt particle decay in nuclei: present status and future perspectives C. Fahlander. D. Rudolph

272

Quadrupole moments and deformations of "shears" states in the Z = 82 Pb nuclei D. Balabanski, G. Neyens, K. Vyvey

280

Structure of bands in neutron-rich even palladium isotopes A. Giannatiempo. A. Nannini, P. Sona

287

Octupole two-phonon states in deformed nuclei G. Grow. Y. Eisermann, R. Hertenberger, H.-F. Wirth, S. Christen, O. Moller, D. Tonev, J. Jolie, C. Giinther, A. I. Levon, N. V Zamfir

293

IBFM2 study of odd-A Cs and Xe isotopes and beta decay in Xe-Cs N. Yoshida. L. Zuffi, S. Brant

303

Algebraic description of high angular momentum states in nuclei D. Vretenar. S. Brant, G. Bonsignori

309

Search for seniority isomers: lifetime measurements in 93Tc and 95Ru 319 K. P. Lieb. E. Galindo, M. Hausmann, A. Jungclaus, I. P. Johnstone, R. Schwengner, A. Dewald, A. Fitzler, O. Moller, G. De Angelis, A. Gadea, T. Martinez, D. R. Napoli, C. Ur Magnetic moments from the Mediterranean to Mt. Fuji N. Bencz.er-Koller. M. J. Taylor, G Kumbartzki, Y. Y. Sharon, L. Zamick, T. J. Mertzimekis, A. E. Stuchbery

325

Probing nuclear structure by real photons: systematics of low-lying dipole modes in heavy nuclei U. Kneissl

331

Electric dipole excitations close to the particle threshold A. Zilges

341

XI

Mixed-symmetry multiphonon structures and first evidence for F-vector El transitions N. Pietralla. C. Fransen, P. Von Brentano

351

Spin-isospin excitations, pairing and shape coexistence E. Moya de Guerra. P. Sarriguren, R. Alvarez-Rodriguez, A. Escuderos

361

Dipole symmetry near threshold M. Gai

367

Measurement of the spin entanglement of two-proton system H. Sakai. T. Saito, A. Tamii, T. Kawabata, Y. Satou

372

SU(6)-breaking symmetry and the ratio of proton momentum distributions M. M. Giannini, E. Santopinto. A. Vassallo, M. Vanderhaeghen

378

The hypercentral constituent quark model and its symmetry M. M. Giannini. E. Santopinto

384

Manifestation of symmetry properties of nucleon structure in strong and electromagnetic processes E. Tomasi-Gustafsson. M. P. Rekalo

390

Regularity and chaos in low-lying 2+ states of even-even nuclei A. Y Abul-Magd, H. L. Harney, M. H. Simbel, H. A. Weidenmiiller

398

Shape-phase and order-to-chaos transitions in nuclei G. Maino

406

Some remarks on the symmetry of the superconducting wavefunction in the cuprates K. A. Muller

414

Algebraic description of n-alkane molecules S. Oss

415

XII

Algebraic approach to vibrationally highly excited polyatomic molecules K. Yamanouchi, T. Sako

423

Molecular quasilinearity under the prism of dynamical symmetry breaking: a detailed study of the methinophosphide (HCP) A-X system F Perez-Bernal. F Iachello, P. H. Vaccaro, H. Ishikawa, H. Toyosaki, N. Mikami

431

Algebraic methods for the quantitative interpretation of vibronically-resolved molecular spectra: the structure and dynamics of disulfur monoxide (S20) P. H. Vaccaro. T. Miiller, F. Iachello, P. Perez-Bernal

441

Franco, the early days R. H. Siemssen

451

Poem R. F. Casten

455

List of participants

459

Photographs

465

PREFACE These Proceedings contain the papers presented at the Highly Specialized Seminar on "Symmetries in Nuclear Structure", which was held at the Ettore Majorana Centre in Erice, Italy, 23-30 March 2003. The meeting was intended to celebrate, on the occasion of his 60th birthday, the career and the remarkable achievements of Francesco Iachello. Since the development of the Interacting Boson Model in the early 1970s, the ideas of Francesco Iachello have provided a variety of frameworks for understanding collective behaviour in nuclear structure, founded in the concepts of dynamical symmetries and spectrum generating algebras. The original ideas, which were developed for the description of atomic nuclei, have now been successfully extended to cover spectroscopic behaviour in other fields, such as molecular or hadronic spectra. More recently, the suggestion by Iachello of Critical Point Symmetries to treat nuclei in shape/phase transitional regions has opened an exciting new front for both theoreticians and experimentalists. The talks presented at the meeting covered many of the most active forefront areas of nuclear structure as well as other fields where ideas of symmetries are being explored. Topics in nuclear structure included extensive discussions of dynamical symmetries, critical point symmetries, phase transitions, statistical properties of nuclei, supersymmetry, mixed symmetry states, shears bands, pairing and clustering in nuclei, shape coexistence, exotic nuclei, dipole modes, and astrophysics, among others. In addition, important sessions focused on talks by European Laboratory Directors (or their representatives) outlining future prospects for nuclear structure, and the application of symmetry ideas to molecular phenomena. Finally, a special lecture by Nobel Laureate Alex Mueller, on s and d wave symmetry in superconductors, presented a unique insight into an allied field. On behalf of the Organizing Committee and of all participants we thank the Ettore Majorana Centre and Professor Antonino Zichichi for providing such an ideal venue for the meeting. The location fitted perfectly into the spirit of the reunion, since precisely in this place so many of the early first successes of the Interacting Boson Model were announced in the late 1970s-early 1980s. Finally, we thank Raffaella Ruggiu and Pino Aceto from the Majorana Centre for the perfect organization in Erice and Annarosa Spalla from INFN, Padova, for the highly professional management of the conference and her fundamental help in the collection and the editing of these proceedings.

Andrea Vitturi and Richard F. Casten

XIII

SYMMETRY: THE SEARCH FOR ORDER IN N A T U R E FRANCESCO IACHELLO Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, CT 06520-8120

1

Symmetry

The subject that is being discussed at this Seminar is the role of symmetry in nuclear, molecular and hadronic physics. Symmetry is the unifying concept in many human endeavors. The word symmetry, from the Greek avfifierpoi;, describes an object that is well-ordered, well-organized. It began to appear systematically in many writings during the Greek and Roman periods. The Greek sculptor Polykleitos in his book on the art of construction (TLepb psAoiruKUjv) uses the word extensively. The Roman architect Vitruvius states that symmetry is "the result of proportions between different parts". Artifacts with symmetric properties have been present since the dawn of civilization. Figure 1 shows a portion of a decoration, dated circa 2000 B.C., which is symmetric with respect to translations along the horizontal axis. During the Greek and Roman periods symmetry was brought to perfection. Figure 2 shows a Roman mosaic from Corynth, Greece. This figure has a sixteen-fold rotation symmetry, £>i6/i, in the outer part and a broken symmetry in the inner part. (Because of its complex structure this figure has been taken as symbol for this Workshop.) Symmetry became an essential part of all works of art and reached its peak in the Greek temple, as it can be seen here in Sicily at Agrigento, Selinunte and Segesta. The symbol of Sicily, Fig. 3, has a threefold symmetry, T>3h, (similar to the symmetry structure of the Interacting Boson Model).

Figure 1. Decorative motif (Sumerian, circa 2000 B.C.). 1

2

Figure 2. A complex symmetry pattern in art (Roman mosaic, from Corynth, Greece, circa 200 B.C.).

U>1

• • ^ • s *

41Figure 3. The symbol of Sicily (Trinacria, from antiquity) is shown as an example of threefold symmetry.

Together with the development of art, there was in the Greek time, a development of Mathematics, especially geometry. The discovery of the five regular polyhedra, the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron, with their symmetric shapes, led the ancient Greeks to think that

3

Figure 4. Dodecahedron, after Leonardo da Vinci and Piero della Francesca (from Luca Pacioli, De Divina Proportione, Venice, 1509).

the entire Universe is built out of these polyhedra. The tetrahedron, octahedron, cube, icosahedron were associated with fire, air, earth and water, respectively (the building blocks of the Universe), while the (penta)dodecahedron was the image of the Universe itself. The study of symmetry took another step forward in the Italian Renaissance, when artists and mathematicians developed further the Greek concept. The five regular polyhedra were complemented with other types, such as the Archimedean polyhedra, Fig. 4. (Archimedes, 287-212 B.C. was also a native of Sicily.) The works of Piero della Francesca (Libellus de Quinque Corporibus Regularis, 1482), Luca Pacioli (De Divina Proportione, 1509) and others describe in full detail the symmetries of these bodies and begin to introduce what in modern mathematical language is called the theory of group transformations, or, simply, group theory. Once more, in these works, the basic idea is that the Universe is proportionate and thus follows strict mathematical (geometric) rules (simmetria). Symmetry is assumed to be the fundamental law of Nature, and is therefore above all (divina). In 1595, the German astronomer Kepler, in his book Mysterium Cosmographicum, noticed that the planetary systems known at the time, Saturn, Jupiter, Mars, Earth, Venus and Mercury, could be reduced to regular bodies which are alternatively inscribed and circumscribed in spheres, Fig. 5. He was so impressed that he concluded his book with the famous sentence " Credo spatioso numen in orbe", that is "I believe in a geometric order of the Universe". Again, order and symmetry are used in an interchangeable way.

4

Figure 5. Construction of the Universe according to Kepler. (From Mysterium Tubingen, 1595).

2

Cosmographicum,

S y m m e t r y in P h y s i c s

The idea of order (and symmetry) began then its entry into Physics. By the end of the 19th Century, as Physics changed more and more from the macroscopic to the microscopic level, it became clear that symmetries play a fundamental role in Physics. Many aspects of Nature are observed to be ordered. The best example are molecules and crystals. Figure 6 shows the molecule H3-C-C-CI3, whose symmetry is evident (C3, rotations of angles multiple of 120°). The symmetries encountered in molecules and crystals are called "geometric" symmetries. They are similar to those encountered in art (rotations, reflections and translations). These symmetries were the first ones to be recognized in the microscopic world. But, as time went on, it became clear that other types of symmetry occur in Nature, not related to the geometric arrangement of atoms in a molecule or in a crystal but rather to the dynamic laws of Nature. The role of these other symmetries emerged with the development of quantum mechanics, which occurred around 1920. Contrary to what happens in classical mechanics, in quantum mechanics the bound states of a physical system are discrete. The search for order in quantum mechanics therefore becomes the search for order in the states of a physical system. The manifestation of symmetry is the observed regularity of the quantum levels of the system. As in the previous cases of "geometric" symmetries, "dynamic" symmetries are characterized by groups of transformations. The groups of transformations appropriate to describe dynamic symmetries are called Lie groups, after the Norwegian mathematician Sophus Lie who introduced them towards the end of the 19th Century. Lie groups characterize how the wave functions describing the quantum states of the system are related by symmetry transformations. If there is a dynamic symmetry, states are simply related by the action of the elements of the Lie algebra associ-

5

Figure 6. The molecule H3-C-C-CI3 (C3 symmetry).

ated to the Lie group, called generators. In addition to geometric and dynamic symmetries, other types of symmetry, ("kinematic" symmetries, gauge symmetries, permutation symmetries,...) have been shown to play an important role in physics. Especially gauge symmetries have played in the last 30 years a major role in Physics after the discovery that the fundamental laws of Nature appear to be governed by non-Abelian gauge symmetries. These other symmetries will not be discussed here. In the course of the last 60 years, dynamic symmetries have been discovered at every level of quantum physics (in molecules, in atoms, in atomic nuclei and in hadrons). In 1926, Wolgang Pauli (Uber der Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik, Z. Physik 36, 2336 (1926)) noted that the regularity in the discrete spectrum of the hydrogen atom, Fig. 7, is due to the occurrence of a dynamic symmetry, described by the group SO (4). In 1961, Murray Gell'Mann (Symmetry of Baryons and Mesons, Phys. Rev. 125, 1067 (1962)) and Yuval Ne'eman (Derivation of Strong Interactions from a Gauge Invariance, Nucl. Phys. 26, 222 (1961)) discovered that the spectrum of hadrons, Fig. 8, is very regular, and could be described by the group SU(3). In 1974, Akito Arima and I (Collective Nuclear States as a Representation of a SU(6) Group, Phys. Rev. Lett. 35, 1069 (1975)), discovered that the discrete spectra of many atomic nuclei are regular and proposed that these spectra be described by the group of transformations U(6). An example is shown in Fig. 9. In 1981, I (Algebraic Methods for Molecular Rotation-vibration Spectra, Chem. Phys. Lett. 78, 581 (1981)) observed that the discrete spectra of many molecules are also regular and suggested that these spectra be described by the group of transformations U(4). An example is shown in Fig. 10. This suggestion led to the formulation, together with Raphael D. Levine and others, of a comprehensive

6

observed spectrum 4« 3s

4p 3p

2t

2p

4d 3d

predicted spectrum 4f

4» 3»

4p 3p

2>

2p

4d 3d

4f

-5

II

It

-15

Figure 7. A portion of the discrete spectrum of the hydrogen atom. Levels are characterized by the spectroscopic notation Is, 2s, 2p,.... The scale of energy is electron Volt (eV). On the left the experimental spectrum, on the right the calculated spectrum.

predicted spectrum

observed spectrum 1.8 >

1-6

V

2 "»i 1.4 0)

l5 1.2 1.0

-

a

-

a a

m

_z"

s' A

A

•

Figure 8. The energy spectrum of the baryon decuplet. States are characterized by the notation A, ...The energy scale is in GeV=10 9 eV. On the left the experimental spectrum, on the right the calculated spectrum.

theory of molecules. The concept of symmetry was enlarged even further in the 1970's by Julius Wess, Bruno Zumino and others through the introduction of a more complex type of symmetry, called supersymmetry, involving simultaneously bosons and fermions. There are here again several types of supersymmetries ("kinematic" supersymmetries, gauge supersymmetries,...). In the course of the last 20 years, dynamic supersymmetries have been discovered in atomic nuclei. In 1980, I (Dynamical Supersymmetries in Nuclei, Phys. Rev. Lett. 44, 772 (1980) observed that the spectra of certain even and odd nuclei could be classified in terms of the group of supersymmetry transformations U(6/4). An example is shown in Fig. 11. A mathematical theory was soon developed with Baha Balantekin and Ithzak Bars. The idea was further enlarged to include protons and neutrons. The latter type has also been found recently.

2.0 observed spectrum

predicted spectrum

— 6 1.5-

~

6

~ _

4

1 0

> 0)

2

- 5

— 6 10

- 4 - 3 — 2

— o

2. 5;l-0

— 6 5

4

4

2

3 2

— o

8

8

6

6

4

4

2 0

0

D) ©

c UJ

0.5

2

Figure 9. The energy spectrum of the nucleus 1 5 6 Gd is shown as an example of the dynamic symmetry [/(6) D SU(3). States are characterized by their angular momenta 0 , 2 , 4 , — The energy scale is in MeV=10 6 eV. On the left the experimental spectrum, on the right the theoretical spectrum.

One may note in all cases shown in the figures the arrangement of the energy levels into patterns characteristic of the dynamical symmetry which governs the quantum motion of constituent particles (electrons in the case of atoms, quarks in the case of hadrons, protons and neutrons in the case of nuclei and atoms in the case of molecules). The occurrence of these regular patterns is the manifestation of dynamic symmetry. One may wonder whether or not the study of dynamical symmetries and supersymmetries in physics has been concluded. The answer is no. The observation that these symmetries occur at all levels indicates that they may be present in other systems, including condensed matter systems. Indeed, in 2002 Alex Muller and I (^4 Model of Cuprate Superconductors Based on the Analogy with Atomic Nuclei, Phil. Mag. Lett. 82, 279, 289 (2002)) have suggested that dynamic symmetries may be responsible for high-temperature superconductivity in cuprate materials. This idea is presently being explored.

8 E x d p

(11)

The authors pointed out that it is possible to make the choice s=r. Since there is only one independent operator for any possible value of L, it is possible to take only one value of r. For L=0, r=2 is the only possible choice. The interaction (11) may be expressed by tensor products of U(6) generators. The pure three-boson interaction (11), with s=r, can be transformed by change of coupling transformations on the d+ and d operators. It can be expressed as a linear combination of various two-boson terms and terms which contain the complete three-boson interaction. The latter have the form I 22 k" \[[d+ x d p ' * x [d+ x dp")]W.[d+ x d]W

I rrk I

(12)

20 A three body interaction with non-vanishing eigenvalues for 3 d-boson states coupled to L—0 would have matrix elements which depend on values of « A and hence, could lift the degeneracy. Such a 3-body interaction is obtained by putting L=0 in (110, ({d+ x d+)V> • d+)([d x d\W • d)

(13)

To see if such an interaction can be constructed from 0(5) generators, consider its expression as a linear combination of terms (12). The 9j-symbol in the coefficient (13) has two equal columns and hence, vanishes if (—1)* +k"+k = —1. This implies that the 3-body part of the interaction considered, for any value of L, vanishes if all 3 tensors are generators of 0(5) with odd ranks. This is consistent with the results of Szpikowski and Gozdz that "the numbers « A and v cannot be, in general, simultaneously defined as good quantum numbers". 16 The reason for this vanishing is simple. If only odd tensors from (8) are used in (12), we find due to the symmetry of 3 tensors coupled to a scalar, that the only choices are k! — k" — k — 1 and k! — k" = 3,k — 1,3. The coupling of two equal rank tensors into an odd tensor is antisymmetric and hence, it contains only commutation relations between the components. Since the components of odd tensors are generators of 0(5), their commutators are equal to linear combinations of components of odd tensors. Hence, the result is equal to a two-boson interaction. To construct real 3-body interactions of this kind it is necessary to use also even rank tensors which are generators of U(5) but not of 0(5). If the three-boson interaction (13) is added to the Hamiltonian (9), the latter is no longer diagonal in the 0(5) scheme. Still, some 0(5) states with 71A=0 may be eigenstates of (13) with vanishing eigenvalues . Hence, such states are also eigenstates of (9) with eigenvalues given by (10). The Hamiltonian (9) is a linear combination of commuting Casimir operators and hence, such states with TIA=0 exhibit partial dynamical symmetry 12 . A simple argument shows that there are not many such states. None exist if rid — v >4 (nd >6 if v —0). The creation operator of such a state has a factor of the form (S + ) 2 d+ where S+ is the creation operator of two d-bosons coupled to L =0. This factor may be transformed by changes of the order of coupling into a linear combination of terms [[[d+ x d + p ) x d+](fc'> x [d+ x d+] ( f c ) ]^

(14)

The term with k =2, kl =0 yields a component of the state for which « A is at least equal to 1. Applying to this component the operator (13), the result does not vanish and the state does not have zero eigenvalue. This feature holds also for most states with rid — v+2. Apart from the state with L = 2v, seniority v, all of them are due to creation operators which may be put in a form where S+[d+ x d+]( 2 ' is a factor. This factor may be transformed into a linear combination of terms [[[d+ X d+]( 2 ) X d+]W X d+]< 2 '

(15)

21 The term with k =0 shows that the state under consideration cannot have a vanishing eigenvalue of (13) and partial dynamical symmetry. States with seniority v with vanishing eigenvalues of (13) may be found among those with n=C , 2 s|0> = C,32|0> = c2|0> = es|0) = 0.

(16)

The Hilbert space for the representation T of highest weight (A, 0) is then spanned by the states {\z) = ezif2+z^310)} for a suitable range of a pair of complex variables 22 and z3. For a general irrep T of highest weight (XfJ,), the states that are annihilated by the e2 and e 3 raising operators [ / = { | 0 ) G H | e 2 | 0 ) = e 3 | 0 > = O}

(17)

are not also annihilated by elements of the u(2) C su(3) subalgebra. However, they span a u(2)-invariant subspace U C H of highest grade states. Moreover, if {£„ = \sv)} is an orthornormal basis for U indexed by v, then the Hilbert space for the su(3) representation T is spanned by the states {ez^^+Z3^3\sv)} for a suitable range of the complex variables z2 and z 3 . Thus, any state \ip) in the Hilbert space is represented by the VCS wave function *(*) = X ^ ^ l e i V ' } ,

z = z2e2+z3e3.

(18)

An element X of the su(3) algebra is then represented as a linear operator on the VCS wave functions that is defined by \T(X)9](z)

= YJZAsv\eiX\i>) V

= Y1U8v\X{z)etW),

Y(X) (19)

V

where X(z) = ezXe~z

=X + [S,X] + §[J, [z, X]].

(20)

Explicit expressions for the T(X) operators are obtained by first observing that X(z) is an element of sit (3) and that Y,AA*"\fie?W> £ „ Usv\Hie*\i>)

= V,

= (A + s)9(z),

EvU™\C23ezm

= s+9(z),

T.Av^\ei^\rl>)=di9{z),

(21)

£ „ Usv\Hiez\iP)

= 2i0*(z),

J2^AMC32ezW

= !_*(*),

(22) (23)

where §0 and s± are intrinsic spin operators defined such that so6/ = v £„ ,

s± f „ = V / ( S T ^ ) ( S ± " + 1 ) &,±i,

(24)

with s = \ij1. It follows that r( J ffi) = A + a - | n >

T(H2)=2(s0+jo),

(25)

T(C23) = s++j+, T(C3a) = i-+j-, T(ei)=di, T(/ 2 ) = [A - s0]z2 - s+z3 - z2 Y,i Zidi, r

( / 3 ) = [A + s0]z3 - s-z2 - z3 J2i z^

,

(26) (27) (28)

36 where n = ^2 Zidi,

jo = \{z2d2 - z3d3),

j+ - z2d3 ,

j - = z3d2 .

(29)

i

It is seen that all the operators are simply expressed in terms of the elements l» and ji of two su(2) algebras, one of which of which is regarded as an intrinsic spin. The most complicated operators in the set are T(/i) and r ( / 2 ) . However, their matrix elements are easily determined by expressing them in the form

r(fl) = [k,zi],

(30)

A = (A + s)n - i n ( n - l ) - 2 s - j .

(31)

where

The expressions suggest defining orthonormal basis states for the su(3) irrep in the su(2)-coupled form ipjjM (z) = Kjj [f ipj {z)]jM ,

(32)

where fim(z)

= -7=. = , m = -3,...,+j, (33) V U +my.{j -m)\ and the norm factors {Kjj} remain to be determined. It is seen that T ( i / i ) , T(H2) and A are diagonal in this basis with eigenvalues given by r ( H i ) ipjjM = (A + s - 3j) 4>jJM ,

r{H2) i>jJM = 2M i/jjjM ,

A ipjjM = H(sjJ) ipjjM ,

(34) (35)

and n(sjj)

= 2(A + s)j + s(s + 1) - j(j - 2) - J(J + 1).

(36)

The operators T(C23) and r(C32) are simply the su(2) raising and lowering operators T(C23)=J+

= 8++j+,

T{C3a) = J-=a-+3-,

(37)

with the usual su(2) actions M±I

•

(38)

The matrix elements of di and z, can be evaluated explicity for the given basis wave functions. Since the z* are components of an su(2) spin-1/2 tensor, the result is conveniently expressed in terms of reduced matrix elements. With some Racah recoupling, we obtain (sjJ\\e\\s,j

+ i , J') = -y/(2j x

(s,j

+ lJ'\\f\\sjJ)

^

+ l){2j + 2){2J+l)(2J'

(KKJJ

:j +

,

\J) (39)

= (-l)J'-J+i x

+ l) W^jJ's

(8JJ\\e\\S,j )

&2X

+ l,J')

+ ^ + J ( J +x) " J ' ( J ' + ! ) " i

(40) + f]•

37 Thus, by setting 2

K,.

,

= i ( 2 A + /i) + J ( J + l ) - J ' ( J ' + l ) - i + | ,

(41)

we obtain the reduced matrix elements of a unitary representation with j and J running over all integer and half-odd integer values for which the Kjj coefficients are non-zero. 5

R e p r e s e n t a t i o n of sw(3) in a n so(3) basis

For applications in nuclear physics, one needs the su(3) representations in an angular momentum basis. They are easily constructed in coherent state theory as a result of the well-known observation ? : //10) is a highest weight state for an su(3) irrep, then the rotated states {R(n)\Xfi);Q

e SO{3)}

(42)

span the Hilbert space H of this irrep. Suppose the Hilbert space H has an orthonormal basis of angular-momentum coupled states {\aLM)}. Then, these states are represented by coherent state wave functions of the form VaLMW

= {Xn\R(Sl)\aLM)

= ^(XnlaLK)

VLKM{tt).

(43)

K

An element X of the su(3) Lie algebra then has coherent state representation as a linear operator T(X) on the coherent state wavefunctions, defined by [r(X)*](fi) = (Xfi\R{Q)X\il>) = (Xn\X((l)R{Sl)\rP),

(44)

where (with Ct denoting the transpose of fi) X(fl)

= R(n)XR(Cl).

(45)

In an angular-momentum basis, the su(3) algebra is spanned by the angular momentum and quadrupole operators with components given in terms of the root vectors shown in fig. 1 by Lo = - i ( C 2 3 - C32), Qo = 2tf!,

L± = i(e 3 - / 3 ) ± (e 2 - / 2 ) ,

(46)

Q±1 = T ^ / f [e2 + h ± i(es + fa)],

Q±2 = y/\ [H2 ± i(C23 + C 32 )] •

(47)

From the definition (44), the coherent state representation of a quadrupole operator is given by [r(Qm)*KiM](fi) =

(Xii\R(tt)Qm\KLM)

= 52(Xr\Q„\KLK) Vlm{V)VLKM{V).

(48)

38 The matrix elements (X/J,\Q^\KLK) and the identities

are inferred from the expansions (46) and (47)

(\n\Lo\aLK) (\/J.\L±\aLK)

(\n\Hi\aLK)

= K (Xn\L0\aLK),

= S/{LTK){L±K

= \(2X + fj,)(XiJ,\aLK), {Xti\C32\aLK)

One finds

10

(49)

+ \) (Xfjt\aL,K±l),

(Xn\H2\aLK)

= (\fi\fi\aLK)

= ^(Xfi\aLK),

= 0.

(50)

(51) (52)

that, if

VocLM = J2 aK ( a L ) VKM >

( 53 )

K

then [r(Q) ® *aL\uM

= Y. M{fKL)aK(aL)

V%M

(54)

with M{XL)

= SK.,K [(2A + ix + 3) + 6K1aL.L + 5 K < , K + 2 y § (/* - K){LK,22\L'K,

- \L\V

+ 1) + \L{L + 1)] (Lis,

20\L'K)

+ 2)

+O(:E), for a one-dimensional bound system. Since V'o(^) has no nodes it can be written as ipo(x) - exp I

— / W{x)dx J

(1)

where the function W(x) is related to the potential energy of the system. Introducing the operators A = W{x) + it

= w{i)

_

-4=P, * Pt V2m

one can write the Hamiltonian of the system as H-E0 = i f i ,

(2)

(3)

where EQ is the ground state energy. The ground state wavefunction satisfies the condition i|V>o> = 0. (4) It is straightforward to show that the supersymmetric partner potentials Hi = A* A

H2 = i i t

(5)

have the same energy spectra except the ground state of Hi, the energy of which is zero. Potentials corresponding to these Hamiltonians are Fi(ar) = [W{x)Y 2

/2m dx

V2(x) = [W(x)} + ^ = ^ . \/2m ax 62

(6)

63 The partner potentials in Eq. (6) are called shape-invariant 3 if they can be obtained from one another by changing their parameters: V2(x;a1)

= V1(x;a2) + R(a1),

(7)

where a2 is a function of o i , and the remainder R(ai) is independent of x. Eq. (7) is equivalent to the operator relation i ( 0 l ) i + ( o i ) = i t ( a 2 ) i ( o 2 ) + R{ai). 1.1

(8)

Algebraic Approach

Shape-invariance problem was formulated in algebraic terms in Ref. [4]. In this formulation one introduces an operator which transforms the parameters of the potential: f(oi)0(oi)f-1(ai) = 0(o2). (9) Defining the operators

B+=A^(ai)f(ai) B- = B | = ^ ( 0 1 ) ^ ( 0 1 )

(10)

one can show that the Hamiltonian can be written as H-Eo

= A^A = B+B-.

(11)

Using the definitions given in Eq. (10), the shape-invariance condition of Eq. (8) takes the form [B.,B+] = R(ao), (12) where R(ao) is defined via R(an) = f(a1)R(an-1)fHa1).

(13)

In terms of these new operators Eq. (4) takes the form B-\*Po)= 0,

(14)

i.e. the ground state is annihilated by the lowering operator B _ . One can easily establish the commutation relations 4 [H, Bl) = (R(ai) [H,B'L] = -B!(R{ai)

+ R(a2) + • • +R(an))Bl + R(a2) + • • +R(an)).

i.e., B+\ip0) is an eigenstate of the Hamiltonian with the eigenvalue R(a,i)+R(a2) • • +R(a„). The normalized eigenstate is

(15) (16) +

64 To identify the algebra we consider the commutation relations [B„,B+]

= R(a0)

(18)

[B+,R(a0)] = (iE(ai) - R(a0))B+, [B+, (R(ai)

- R(a0))B+]

= {(R(a2) - R(ai))

- (R(ai)

(19) - R(a0))}B+,

(20)

and so on. In general there are an infinite number of such commutation relations. If the quantities R(an) satisfy certain relations one of the commutators in this series may vanish. For such a situation the commutation relations obtained up to that point plus their complex conjugates form a Lie algebra with a finite number of elements. For example if the condition (R(a2) - fl(ai)) - (R(ai)

- R(a0)) = 0

(21)

is satisfied then the algebra is 4 either SU(2) or SU(1,1). Most of the exactly solvable one-dimensional problems in quantum mechanics can be described by this algebra 5 . It can be shown that this algebra also describes for example both the bound and scattering states of the Poschl-Teller potential 6 as well as associated transfer matrices. 1.2

Outlook on future

applications

Almost all exactly solvable one-dimensional potential problems encountered in quantum mechanics textbooks are shape invariant where the parameters are related by a translation 2 a2 = ai + T). (22) It should be emphasized that shape-invariance is not the most general integrability condition one can write for such potentials as there are exactly solvable problems which are not shape invariant 7 . There is a second class of shape invariant potentials where the parameters of the partner potentials are related by a scaling 8 ' 9 a 2 = qai.

(23)

In this latter class, corresponding one-dimensional potentials are defined implicitly, but explicit expressions are not given. In searching for integrable models in two-dimensional statistical mechanics a relationship was uncovered between those models, three-dimensional Chern-Simons gauge theory and quantum groups 1 0 . These models, being completely integrable, can be written in a shape-invariant way 11 , corresponding to a shift in the parameters a2 — qai + r\.

(24)

The associated algebras are called up-down algebras 12 . These developments suggest that there may be shape-invariant potentials where the parameters are related by linear-fractional transformations: o 2 = {qai +T))/(a1 + TJ')

(25)

65 This is a completely unexplored direction of research as nothing is known about such integrable systems. Recall that the notation oi, a?, etc. may represent not only single parameters, but also a set of them. In general one may suggest to simply relate these parameters by the transformation fia^Oia^f-1^)

= O(oa).

(26)

where T is an element of any group, not just of SL(2,R) as suggested by the linearfractional transformation and its limits that were so far employed. What kind of exactly solvable problems do we obtain? At the moment this is an open question. The basic philosophy of this approach is to consider the parameters of the Hamiltonians as auxiliary dynamical variables. This is reminiscent of the path leading to the Interacting Boson Model 1 3 . To describe the quadrupole collectivity in nuclei one needs t o consider a five-dimensional space. It is possible to formulate this problem in terms of boson variables 14 , however the problem is nonlinear written in terms of quadrupole bosons. By considering a parameter of the problem (boson number) as an additional degree of freedom, Interacting Boson Model introduced a scalar boson as a dynamical variable. This has led to the subsequent realization 15 of s and d bosons as pairs of nucleons coupled to the angular momentum L = 0 and L =2 So far we talked about considering parameters of the shape-invariant problem as auxiliary dynamical variables. One can imagine an alternative approach of classifying some of the dynamical variables as "parameters". An example of this is provided by the supersymmetric approach to the spherical Nilsson model of single particle states 1 6 . The Nilsson Hamiltonian is given by

H = Y^a\ai-2kLS + kvl?.

(27)

i

Introducing the variable

^+ = E ^ a i

(28)

i

one can show that the "Hamiltonians" Hi =F*F

= YJ a\cn -