Group Theoretical Methods in Physics: Proceedings of the XVIII International Colloquium Held at Moscow, Ussr, 4-9 June 1990 (International Colloquium on ... Theoretical Methods in Physics: Proceedings)

Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, M[~nchen, K. Hepp, ZL~rich R. L. Jaffe, Cambridge, MA, R...

Author: V. V. Dodonov and V. I. Manko (eds.)

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Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, M[~nchen, K. Hepp, ZL~rich R. L. Jaffe, Cambridge, MA, R. Kippenhahn, MSnchen, D. Ruelle, Bures-sur-Yvette H.A. WeidenmQller, Heidelberg, J. Wess, Karlsruhe and 1. Zittartz, K61n Managing Editor: W. Beiglb6ck

382 iii

V.V. Dodonov

V.I. Man'ko (Eds.)

Group Theoretical Methods in Physics Proceedings of the XVIII International Colloquium Held at Moscow, USSR, 4-9 June 1990 ml

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Editors

Victor V. Dodonov M o s c o w Physics Technical Institute 141 ?00 Dolgoprudny, M o s c o w Region, USSR Vladimir I. Man'ko Lebedev Physics Institute Leninsky Prospect 53, 117 924 Moscow, USSR

ISBN 3-540-54040-? Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-54040-? Springer-Verlag N e w Y o r k Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Printing: Druckhaus Beltz, Hemsbach/Bergstr. Bookbinding: .1. Sch&ffer GmbH & Co. KG., GrL~nstadt 2153/3140-543210 - Printed on acid-free paper

Dedicated to the late A. D. Sakharov

Colloquia on Group Theoretical Methods in Physics year editor/publisher 1972 3oint report of the University of Provence, The University at AixMarseille, and the CNRS. II Nijmegen, 1973 Printed by the Faculty of Science, University of Nijmegen. The Netherlands III Marseille, 1974 Printed by the Faculty of Science, University of Nijmegen. France IV Nijmegen, 1975 Eds. A. Janner, T. Janssen and M. Boon, Lecture Notes in Physics 50, The Netherlands Springer, Heidelberg, 1976. V Montreal, 1976 Eds. R.T. Sharp and B. Kolman, Academic Press, 1977. Canada VI Tfibingen 1977 Eds. P. Kramer and A. Rieckers, Lecture Notes in Physics 79, Germany F.R. Springer, Heidelberg, 1978 VII Austin, 1978 Eds. W. BeiglbSck, A. Bohm and E. Takasugi, Lecture Notes in Texas, USA Physics 94, Springer, Heidelberg, 1979. VIII Kibbutz Kiriat 1979 Eds. L.P. Horwitz and Y. Ne'eman, A. Hilger and the Israel Physical Society, Anavim, Israel The American Institute of Physics, 1980. IX Cocoyoc, 1980 Ed. K.B. Wolf, Lecture Notes in Physics 135, Springer, Heidelberg, 1980. Mexico X Canterbury, 1981 Eds. L.L. Boyle and A.P. Cracknell, Physica A, Vol. l14A, No. 1-3, 1982 England XI Istanbul, 1982 Eds. M. Serdaroglu and E. InSnfi, Lecture Notes in Physics 180, Turkey Springer, Heidelberg, 1983. XlI Trieste, 1983 Eds. G. Denardo, G. Ghirardi and T. Weber, Lecture Notes in Physics 201, Italy Springer, Heidelberg, 1984. XIII College Park, 1984 Ed. W.W. Zachary, World Scientific, Hongkong, 1984. Maryland, USA XIV Seoul, 1985 Ed. Y.M. Cho, World Scientific, Hongkong, 1986. Korea XV Philadelphia, 1986 Ed. R. Gilmore, World Scientific, Hongkong, 1987. Pennsylvania, USA XVI Varna, 1987 Eds. H.D. Doebner, J.D. Hennig and T.D. Paler, Lecture Notes in Physics 313, Bulgaria Springer, Heidelberg, 1988.

No. place I Marseille, France

V

Colloquia on Group Theoretical Methods in Physics NO.

place

year editor/publisher 1988 World Scientific, Hongkong, 1989 1990 Eds. V.V. Dodonov and V.I. Man'ko Lecture Notes in Physics Springer, Heidelberg, 1991. Salamanca, 1992 Spain

XVII Montreal, Canada XVIII Moscow, USSR XIX

¥1

STANDING COMMITTEE

A° H.

L.C. J.L. K. A. L.L. B.H.

H.D. G.C. R. B.

Zhou M. A. Y.S. B.

P. H.J.

V.I. D. L. M. T.D. A.I. P. J.A.

Arima Bacry Biedenharn Birman Bleuler Bohm Boyle Cho Doebner Ghirardi Gilmore Gruber Guangzhao Hamermesh Janner Kim Kostant Kramer Lipkin Man'ko Mermin Michel Moshinsky Palev Solomon Winternitz Wolf

Tokyo, Japan Marseille,France Durham, USA New York, USA Bonn, Germany Austin, USA Canterbury, England Seoul, South Korea Clausthal, Germany Trieste, Italy Phitadelhia, USA Carbondale, USA Beijing, China Minneapolis,USA Nijmegen, Netherlands College Park, U S A Cambridge, USA Tfibingen,Germany Rehovot, Israel Moscow, USSR Ithaca, USA Bures-sur-Yvette, France Mexico, Mexico Sofia, Bulgaria Milton Keynes, England Montreal, Canada Berkeley, USA

VII

ADVISORY COMMITTEE L. C. Biedenharn

A.

Bohm I . M . Gelfand R. Glauber A.A. Kiriltov Michel L. Moshinsky M. A.D. Sakharov

Duke University, Durham, USA University of Texas, Austin, USA Moscow State University, Moscow, USSR Harvard University, Cambridge, USA Moscow State University, Moscow, USSR IHES, Bures-sur-Yvctte, France UNAM, Mexico, Mexico Lebedev Physics Institute, Moscow, USSR

ORGANIZING V. V. E.S. V. A. Yu. I. V.I. M. A. V.I.

Moscow Physics-Technical Institute Lebedev Physics Institute Moscow State University Steklov Mathematical Institute Lebedev Physics Institute Lebedev Physics Institute Joint Institute of Nuclear Research

Dodonov (scientific secretary) Fradkin Koptsik Manin Man'ko (vice chairman) Markov (chairman) Ogievetsky

WORKING E.A. Akhundova V. A.. Andreev S. M. Chumakov V. V. Dodonov V. P. Karassiov A. B. Klimov E.V. Kurmyshev A. A. Mamedov M. A. Man'ko O. V. Man'ko A. V. Naumov D.L. Ossipov E.S. Tartynskaja V. N. Zaikin

COMMITTEE

GROUP

Institute of Physics, Baku Lebedev Physics Institute, Moscow Central Bureau of Unique Device Designing, Moscow Moscow Physics-Technical Institute Lebedev Physics Institute, Moscow Moscow Physics Technical Institute Central Bureau of Unique Device Designing, Moscow Institute of Physics, Baku Lebedcv Physics Institute, Moscow Institute of Nuclear Research, Moscow Institute of Applied Mathematics, Vladivostok Moscow Physics Technical Institute Lebedev Physics Institute, Moscow Lebedev Physics Institute, Moscow

VIII

Preface

The XVIII International Colloquium on Group Theoretical Methods in Physics took place in Moscow on June 4-9, 1990 under the sponsorship of IUPAP, the Nucleax Physics Depaxtment of the USSR Academy of Sciences, and the Lebedev Physics Institute of the USSR Academy of Sciences. It gathered more than 300 participants from 30 countries, including developing countries. The general rules, terms, and places of all colloquia axe established by an International Standing Committee. The program of the colloquium was compiled by the Local Organising Committee under the supervision of an Advisory Committee and some members of the Standing Committee. All the practical aspects of organizing the colloquium were seen to by the members of a Working Group. The participants of the colloquium gave more than 300 talks and repoits: 28 plenary (40min), 27 review (30rain), and 130 original (15rain) talks made in 18 parallel section sessions, and about 130 poster reports. In addition, a special session was devoted to the ceremony at which the Wigner medal was awarded. Academician A. D. Sakhaxov was the first to agree to be a member of the Advisory Committee of the colloquium. Unfortunately, his premature death prevented his paxticipating in the colloquium, as well as carrying out his other plans. We would like to dedicate this volume (by the consent of the Standing Committee) to the memory of the late A. D. Sakhaxov on the eve of his 70 th birthday (21 May 1991). Sakhaxov's memorial lecture was part of the closing ceremony of the colloquium. This volume contains about 70 papers devoted mainly to such subjects as spectrum generating groups, quantum groups, symmetries of equations, quantum mechanics, coherent states, and group representations. Other papers received by the editors (also about 70 papers) would constitute another volume with the emphasis on problems of quantum field theory, solid state physics, quantum physies~ integrable systems, algebras, and superalgebras. The realization of the colloquium would have been impossible but for the enormous and selfless work of the members of the Working Group, which had united on an informal basis physicists from different institutes and cities of the Soviet Union. Extremely important for the organization of the colloquium was the paxtnership with Prof. H. D. Doebner of the Technical University of Clausthai (Germany). Moreover, this volume was prepared for publication with the help of D. Kruse and M. Wehrhahn from the Technical University of Clausthal. We express our deep gratitude to all these people. V. V. Dodonov V. I. Man'ko

Moscow, November 1990

I×

Contents Opening a n d Closing Talks M.A. Maxkov :

Introductory Talk B.L. Altshuler :

Creating "Clouds" Wigner Medal Ceremony A. Bohm :

Opening Remarks

19

F. Gfirsey :

Lauda~io for Professor Francesco IacheUo

20

F. Iache~o : 22

Symmetry I. S p e c t r u m Generating Groups, D y n s m i c a l Symmetries a n d Their Applications Y. N e ' e m a n :

Dynamical and Fundamental Symmetries in Particles and Nuclei

35

A. Bohm :

Spectrum Generating Groups -Idea and Application

63

F. Iachello :

The Role of Dynamic Symmetries and Supersymme~ries in Nuclear Physics

85

M.E. Loewe :

50(3,2) for Oscillator and HydrogenIike Systems

98

J.F. Caxifiena, C. L6pe=, M.A. del Olmo, M. Santande~ :

Co,formal Groups in the Kepler Problem

106

A. Frank, F. Leyvra~, K.B. Wolf:

Potential Group in Optics : The Mazwell Fish-Eye System

111

P. Kielanowski :

Transition to Chaos in Hadronic Systems

120

G. Rose,steel :

The Dynamical Group o f Riemann Ellipsoids

124

G. L~vai, J. Cseh :

An Algebraic Model of Cluster States in Odd-Mass Nuclei

128

M. Mukerjee :

Compact to No,compact Transition and Nuclear Collective Levels P. T~ulni : SL(3, R) x T 6 as a Nuclear Collective Mo~ion Group

Xl

132 136

B.G. Wybourne :

Ezceptional Groups and the Interacting Boson Approzimation

140

IL Quantllm Groups a n d Algebras, N o n - C o m m u t a t i v e G e o m e t r y L.C. Biedenham :

An Overview o~ Quantum Groups

147

H.J. de Vega :

Integrable Theories, Yang-Bazter Algebras and Quantum Groups : An Overview

164

T.D. Paler, V.N. Tolstoy :

Finite-Dimensional Irreducible Representations of the Quantum Superalgebra Uq[gl(n/1)]

177

Yu.F. Smirnov, V.N. Tolstoy, Yu.I. Kharitonov :

Projection Operator Method and Q-Analog of Angular Momentum Theory

183

P.P. Kulish :

Quantum Algebras and Symmetries of Dynamical Systems

195

t

C. Ramlrez, H. Ruegg, M. Ruiz-Altaba :

Coulomb Gas Realization of Simple Quantum Groups

199

N. S~inchez :

S~ring Theory, Quantum Gravity and Quantum Groups

204

R. Floreanini, V.P. Spiridonov, L. Vinet :

q-Oscillator Realizations of the Quantum Superatgebras

208

R.M. Mi~-Kasimov :

Relativistic Oscillator = q-Oscillator

215

K. Srinivasa Rao, V. Rajeswari :

Some Aspects of the Angular Momentum Coefficients in SUq(2)

221

N.A. Gromov~ V.I. Man'ko :

Contractions and Analytic Continuations of ~he Irreducible Representations of the Quantum Algebra SUg(2)

225

S.M. Khoroshkin, V.N.Tolstoy :

Universal R-Ma~riz for Quantum Supergroups

229

It. Keener:

Recen~ Progress in Non-Commutative Geometry

233

G. Landi, G. Marmo :

Algebraic Gauge Theory and Noneommutative Geometry

241

HI. Symmetries of Equations a n d Fields J. Krause, L. Michel :

Classification of the Symmetries of Ordinary Differential Equations

Xll

251

P. Winternitz :

Conditional Symmetries and Conditional Integrability for Nonlinear

263

Systems J.F. Caxifiena, M.A. del Olmo, P. Winternit= :

Cohomology and Symmetry of Differential Equations

272

W. Snrlet, E. Ma~t{nez :

Derivations of Semi-Basic Forms and Symmetries of Second-Order Equations

277

¢

M.A. del Olmo, M.A. Rodngne=, P. Winternitz :

Hamilton-Jacobi Equations in SU(~,~} Homogenenous Spaces

281

G. Bluman :

Linearization of PDEs

285

M. Irne-Astand :

Perturbed Nonlinear Equations: Application to the Korteweg-de Vries Equation Considered as a Perturbed Euler Equation

289

E.V. Doktorov, I.N. Prokopenya :

Algebraic Version of the Soliton PerturbaZion Theory

294

I.T. Todorov :

Eztended Chiral Conformal Models with a Quantum Group Symmetry

299

L. O'Raifea~taigh :

Conformal Reduction of WZNW Theories and W-Algebras

314

IV. Q u a n t u m P h y s i c s H. Baery :

The Resurrection of ¢ Forgotten Symmetry : de Broglie'8 Symmetry

331

M. Moshinsky, G. Loyola, C. Villegas :

Relativistic Invariance of a Many Body System wi~k a Dirac Oscillator Interaction

339

A. Bohm, L.J. Boya, B. Kenckick :

Derivation of the Geometrical Berry Phase

346

R.R. Aldlnger :

Gauge Translations and the Berry Phase

351

G.G. Emch :

Geometric Quantization : Regular Representations and Modular Algebras

356

E. Prugovefiki :

Geometro-Stochastic Quantization and Quantum Geometry

365

W. Drechsler :

SO(~,])-Coherent States and the Geometro-Stochastic Ouantization of a Gauge Theory for Eztended Objects

Xlll

373

M.C. Lsnd, L.P. Horwitz :

Covariant Quantum Mechanics and the Symmetries of Its Radiation Fields

379

J.L. Birman, A.V. Gorokhov :

Double Stratonovich-Hubbard Trick and Novel Path Integral for a System of Interacting Fermions

383

J.A. de Azc~rraga, D. Ginestar :

Nonrelativis~ic Limi~ of Superfield Theories

394

A. I n o m ~ t a , A. Suparmi, S. K u r t h :

Remarks on the Supersymmetric WKB Quantization Formula

399

P.A. Horvathy :

Particle in a Self-Dual Monopole Field Quantum Mechanics

:

Ezample of Supcrsymme~ric

404

V.V. S e i n e , o r :

Supersymme~r~ and Electron Angular Momentum

410

J. Beckers, N. Debergh :

Parasupersymmetries and Lie Superalgebras

414

F. Leyvraz, J. Quezads, T . H . Seligmsn :

Quantum Chaos : Spectra and States

418

M. Pauri, A. Seotti :

Quantum Ergodicit9 and Eigenvalue Problems for Plane Polygons

434

K . - E . Hellwig, M. Singer :

"Classical" in Terms of General Statistical Models

438

Y.S. Kim, M.E. Noz :

Temporal Decoherence in Lorentz-squeezed Hadrons

442

V.V. Dodonov, A.B. Klimov, V.I. Man'ko :

Phgsical Significance of Correlated and Squeezed States

450

D.A. Tfifonov :

On the Symmetry and D~namics of Squeezed and Correlated States

457

O.V. Man'ko :

Correlated States of Quantum Chain

461

M. Kozierowski, A.A. Mamedov, S.M. Chumakov :

Interaction of Weak Coherent Light witl~ a System of Two-Level Atoms in a Lossless Cavity

469

G.J. PekI~dOpoulos :

Time dependent Quantum Tunnelling

475

A. Novoselsky, J. Katriel :

Non-Spurious Harmonic Oscillator Symmetrg

S~ates

with Arbitrary Permutational

483

B.I. Zhilinskii :

Symmetry Analysis of the Qualitative Intramolecular Phenomena

XIV

487

V. Group Representation Theory and Geometrical Aspects V.P. Kaxassiov : Algebras of the 5U(n) Vector In~a~iants and Some of Their Applications P.Kasperkovits : Product Formulas for Q-Representatives R.C. Kins, T.A. Welsh : Tensor P~'c~uc~s for A.~ne Kac-Moody Algebras J. Van der Jeugt, J.W.B. Hughes, R.C. King, J. Thierry-Mieg :

Atypical Modules of ~he Lie Superalgebra gl(m/n)

493 505 508

512

C. Quesne :

Vector Coherent S~ate Theory of ~he Non-Compact Or~hosymplec~ic S~peralgebras

516

R.C. King, J.W.B. Hughes, J. Van der Jeugt :

The Composition Factors of Kac Modules of sl(M/N)

522

R. Shsw:

Clifford Algebras, Spinors and Finite Geometries

527

V. Kopsk~ :

Reducibility of Euclidean Motion Groups

531

B.L. Davies, R. Difl :

Sofl~oare Packages : Space Groups and Their Representations

535

R. Did, B.L. Davies :

Software Packages : 23~znsforma~ion Coe~icien~s for Space Groups

539

M. Baake, P. Kramer, Z. Papadopolos, D. Zeidler :

Icosahedral Dissectable Tilings ~om the Root Lattice De

543

Y. Ohnuki, S. Ksmefuchi :

Groups ~flh G-Number Parameters

548

C. Holm, J.D. Hennig :

Regge Calculus u~i~h Torsion

556

L. Danzer :

Quasiperiodicity : Local and Global Aspects

561

E. P a ~ :

Derivative Moufang Transformations

573

J.A. de Aze/Lrraga, J.M. Izquierdo :

On the Ezplicit Form of Consistent Anomalies

575

J. Zak :

Berry Phases and Wyckoff Positions for Energy Bands in Solids

581

V.A. Koptsik :

Crystallography of Quasicrystals : The Problem of Restoration of Broken Symmetry

XV

588

Author index

Katriel, 3. 483 Kendrick, B. 346 Kezner, R. 233 Khazitonov Yu.I. 183 Khoroshkin, S.M. 229 Kidanowskl, P. 120 Kim, Y.S. 442 King, R.C. 508, 512, 522 Klimov, A.B. 450 Kopsk~, V. 531 Koptsik, V.A. 588 Kozierowski, M. 469 Kramer, P. 543 Krause, J. 251 Kulish, P.P 195 Kurth, S. 399 Land, M.C. 379 Landi, G. 241 L~vai, G. 128 Leyvraz, F. 111,418 Loewe, M.E. 98 L6pez, C. 106 Loyola, G. 339 Mamedov, A.A. 469 Man'ko, O.V. 461 Man'ko, V.I. 225, 450 Markov, M.A. 3 Marmo, G. 241 Mart~nez, E. 277 Michel, L. 251 Mir-Kasimov R.M. 215 Moshinsky, M. 339 Mukerjee, M. 132 Ne'eman, Y. 35 Novoselsky, A. 483 Noz, M.E. 442 Ohnuki, Y. 548 del Olmo, M.A. 106, 272, 281 Paal, E. 573 Palev, T.D. 177 Papadopolos, Z. 543 Papadopoulos, G.J. 475

Aldinger, R.R 351 Altshuler, B.L. 5 de Azc~rraga, J.A. 394, 575 Baake, M. 543 Bacry, H. 331 Beckezs, J. 414 Biedenharn, L.C. 147 Birman, J.L. 383 Bluman, G. 285 Bohm, A. 19, 63, 346 Boya, L.J. 346 C a ~ e n a , J.F. 106, 272 Chumakov, S.M. 469 Cseh, J. 128 Danzer, L. 561 Davies, B.L. 535, 539 Debergh, N. 414 Did, R. 535, 539 Dodonov, V.V. 450 Doktorov, E.V. 294 Drechsler, W. 373 Emch, G.G 356 Floreanini, R. 208 Frank, A. 111 Ginestar, D. 394 Gorokhov, A.V. 383 Gromov, N.A. 225 Gfirsey, F. 20 Hellwig, K.-E. 438 Hennig, J.D. 556 Holm, C. 556 Horvathy, P. 404 Ho~witz, L.P 379 Hughes, J.W.B. 512, 522 lachello,F. 22, 85 Inomata, A. 399 Izquierdo, J.M. 575 Imc-Astaud, M. 289 Van der Jeugt, J. 512, 522 Karnefuchi, S. 548 Karassiov, V.P. 493 Kasperkovitz, P. 505

XVl

Pauzi, M. Prokopenya, I.N. Prugove~ki, E. Quesne, C. Quezada, 3. O'Raifeartaigh L. Rajeswaxi, V. Ramilez, C. Rao, K.S. Rodr~gues, M.A. Rosensteel, G. Ruegg, H. Ruiz-Altaba, M. S~uehez, N. Santander, M. Saflet, W. Scotti, A. Seligman, T.H. Semenov, V.V. Shaw, R.

Singer, M. 438 Smirnov, Yu.F. 183 Spiridonov, V.P. 208 Suparmy, A. 399 Thierry-Mieg, ,1. 512 Todorov, I.T. 299 Tolstoy, V.N. 177, 183, 229 Tfifonov, D.A. 457 Truini, P. 136 de Vega, H.3. 164 Villegas, C. 339 Vinet, L. 208 Welsh, T. 508 Winternitz, P. 263, 272, 281 Wolf, K.B. 111 Wybourne, B.G. 140 Zak, J. 581 Zeidlez, D. 543 Zhilinskii, B.I. 487

434 294 365 516 418 314 221 199 221 281 124 199 199 204 106 277 434 418 410 527

XVII

Introductory Talk M. A. Markov

On behalf of the presidium of the USSR Academy of Sciences I would like to express a deep gratitude to the Standing Committee of International Colloquium on Group Theoretical Methods in Physics for the decision to hold the XVII colloquium due in this country. For the first time such a prestige conference is held in the Soviet Union. Great scientific possibilities of the Group Methods theory in different branehes of science were understood quite long ago. More than fifty years ago (1933), for example, I used the groups of rotation and mirror reflection in solving the problem of quantum mechanical stability of the benzene molecule. The pointed out symmetries (of rotation and mirror reflection) in the benzene molecule structure helped to reduce the 34-order equation written for the energy state of the benzene molecule in the frames of the homopolar bond theory (Heitler - London Ruiner) to the 8-order equation, which could already be numerically solved. In 1935-1936 for the first time, using G10, I obtained (in the frame of the E. Noether theorem) all ten all the ten conservation laws for the Dirac equation. By that time it was already known that the law of the Dirac current conservation disintegrates on two relativistically invariant laws:

-

a) the law of the conservation of the conductivity current S~: aS~ 0~ = 0

4

1

i

p

S~ = ~-~m~b( ~ - A;~)~b+ - ~b+(P~ -+ A~)~b

b) the law of. the conservation of the polarization current

a,S = o h

~4

0

+ a. h

~

OMok

It turns out that not only the law of the conservation of the Dirac current, but also all ten other laws disintegrate in the same relativistically invariant way. Some part, depending exclusively on various expressions, constructed from 7 matrices, "splits out". Thus, the corresponding divergence equation for the energy-momentum tensor, analogical to the expression for the polarization current, has the form 4 a0~ -0 i

~

A

p#v where A is the "splitted out" relativistically invariant spinor part of the complete Langrangian i A = --mc

P~+TVT~P~ 1

To tell the truth, till now I don't know~ if the splitted out components of all other observables in the Dirac theory as distinct from the holes current have any physical sense. True, I preserved the Pauli's letter, in which he had drawn my attention to the fact, that this disintegration of the conservation laws was reMly relativisticMly invariant. But in the presence of the electromagnetic field the relativistically invariant disintegration of the energy-momentum tensor and corresponding divergence equations turns out to be gradiently invariant. At present I turned back to the Gx0-group. The deal is that in our paper titled "Black Holes Collapse as a Possible Source of Closed and Semiclosed Universes" new (daughter) universes come into being in the original space, but in the new R2-space and in the absolute future in relation to the collapsing black hole. In such cases the "transitional area" exists, in which we can't exclude the violation of all the conservation laws.

This article was processed using the IKI~X macro package with ICM style

Creating "Clouds Boris L. Altshuler Lebedev Physical Institute Academy of Sciences of the U S S R Theoretical Physics Department I17 333 Moscow, Lcninsky pr.,53, U S S R Telex: 411 479 NEOD SU

i. There are telephone calls which you can never forget. It is evening, December 18, 1986. The telephone rings: Sakharov's friend Galina l~vtusehenko says in a happy voice: "Borya, do you want to have a talk with Andrey Dmitrievich?". I dile the number in Gorky which she gave me an~ hear Sakharov's voice after 7 years of absolute impossibility of any direct conta'~t. December 14, 1989, 11PM. The telephone rings and Efrem Yankelevich just said: "Borya, Andrey Dmitrierich died." This was two hours after his death and half an hour after Elena Bonner found him lying dead on the floor. An hour later my son and I came to the place. There were rather many people in the apartment: ambulance doctors, prosecutor, militia. Andrey Dmitrievich was lying on the sofa in the middle of the room, Elena Georgievna sat on the chair next to him. It is impossible to remember all it in detail. The death of Sakha~ov is the loss for all of us, but especially if you knew him personally. Perhaps the most surprising feature which you learn about him during personal contacts is that he never behaved in prosaic, everyday-type manner. Any moment among many guests and noisy talk he could begin to speak about physics - strings, the arrow of time, news of astrophysics. He could propose to solve the problem or recite his own new funny rhyme. Two years ago, on the 28 of June 1988, I called on the Sakharov's apartment in Chkalov street. Soon Andrey Dmitrievich came into the kitchen after a short evening nap. (He had to work at nights and tried to have some compensation in the evening.) Without saying "hello" he announced that the day before he wrote the verse and immediately recited it: " Nikita ~ was a merry guy, He launched the satellite to sky, The satellite circles in the height And makes Nikita giddy slight. And soon all over the land He ordered Indian corn to plant. Result of all this situation: Nikita died, but lives Stagnation." * Sakharov Memorial Talk at the closing of the colloquium, June 9, 1990. Khrushchev

"Did you write it down ?" - I asked. "Of course not. What for ?" - he smiled. But I remembered the verse and later in the evening put it down. And I remember well his short narration in the Lebedev Physical Institute in the very depth of the so called period of Stagnation, in 1973. He spoke about the "Black September" visit to his apartment the day before, on October 18. They threatened to kill him and his wife demanding of Sakharov to denounce his statement on Iom Kipur war. They hinted with m e n ~ e on the children and grandchild. Sakharov spoke about the episode calmly and seriously, but with some curiosity of explorer who faces rather unordinary phenomenon. One of the theoreticists asked: "Andrey Dmitrievich , do you think that if these so called "arab terrorists" are carefully washed they will become white ?" "Yes, I think so" answered Sakharov. (The same idea was expressed in Alexander Solzhenitsin's letter in support of Sakharov, October 28, 1973, published in the USSR seventeen years later in [1]. Of course, this remark on the "washed Palestinian terrorists" was just a joke 3, but this joke, alas, has not lost its meaning in our days. -

$. I got acquainted to Sakharov in 1968 when he agreed to be the opponent at the defense of my Thesis on general relativity. But I heard about him earlier because I spent all my school-years, until 1956, in the so far secret town (Installation) where Sakharov had worked on nuclear weapons and where my father worked in the same special group of physicists. My father Lev ~Atshuler is a specialist on the shock wave high pressure physics. In November 1988 Sakharov for the first time crossed the border of the USSR. He visited USA and France and met many people. In Washington there was the historical meeting of two "fathers" of H-bombs: Andrey Sakharov and Edward Teller. This was 80 years jubilee of Teller. In his talk Sakharov among other things said that he is sure that American and Soviet special groups of physicists worked with the same enthusiasm, with the same awareness of the absolute necessity of their work. Many times at the interview of the last years Sakharov kept saying that creation of nuclear weapons prevented the Third World war for a long period. (Although as he repeated again and again this "balance of fear" now became crucially dangerous.) But the following citation of Sakharov perhaps clarifies something in this pragmatic-moral dilemma: "We worked with enthusiasm and with the feeling that it is necessary. Greatness of the task and its difficulty strengthened the impression that we fulfill the heroic work. But every minute of my life I realize that if it happens - this terrible disaster, thermonuclear war - and if I have time to think about anything, then my estimation of my personal role may tragically change." (From the interview in January 1987 to "Literaturnaya Gazeta"; the publishing of this interview was forbidden, some pieces of it were published in January 1990 in [1]) Seriously speaking, Sakharov's opinion on the case is expressed in the following citation from his "Memoirs": " Without doubt all their actions were under strong control of KGB and may be on the initiative of KGB, but, perhaps, they did not know about it(all the time of the visit they looked as if they were afraid of something)".

3. We were the contemporaries of the person who, during m a n y years, permanently produced the "revolutions of conscience", created new systems of notions and perhaps - what is most miraculous - he transformed not only our picture of reality, but reality itself. I think t h a t his actions or concrete scientific results m a y be deduced, as a rule, from some most general hypothesis or postulates which were practically self-evident to Sakharov. But the conclusions which he drew out of these basic postulates, the ways and means to the full solution of the problem were practically always unexpected. And so m a n y times the standard scenario was repeated: disagreement, irritation, accusations and later came consent. Even on the day of his death some People Deputies, which were his colleagues-academicians and the members of the same opposition Mezhregionalnaya group accused him creating provocation (Sakharov initiated an appeal for two hours general political strike.) In science, of course, everything was "bloodless". When his article on the instability of proton was published in 1967 colleagues had j u s t shrugged their shoulders. But for Sakharov this idea was not occasional. It originated from the basic hypothesis which he liked very much: the hypothesis of the C P T s y m m e t r y of the Universe - not only of the dynamic equations but of the state of the Universe. But the observable Universe consists of m a t t e r (and not of antimatter) and it possesses the arrow of time. How can we match these two very strong C and T - a s y m m e t r i e s of the Universe with the general hypothesis of the C P T s y m m e t r y of the state of the Universe ? It is possible only if two "crazy" things are presumed : 1. the baryonic charge nonconservation [3] and 2. the existence in the history of the Universe of the turning point of the arrow of time (although this hypothesis is somehow wider t h a n the C P T - s y m m e t r y of the Universe). This turning point should be the m o m e n t of the minimal entropy ("the m o m e n t where T - n o n i n v a r i a n t statistical correlations are absent".) Sakharov always insisted, that the arrow of time is nothing but the growth of entropy. Another specific feature of his way of thinking: he was constructive. Construction must work - let it be the bomb, the package of proposals on nuclear disarmament and h u m a n rights or just an effort to save this one person. And he always saw not only the final goal but the whole chain of steps to reach it. To my mind his public activities may be "deduced" from two basic postulates: 1. Every action must be morally flawless, doubtless from the most simple human point of view not distorted by any sort of ideology. 2. The necessity of victory even in small. (To reach it he narrowed the problem, concentrated m a x i m u m of the effort to the minimal surface - at the point in the limit.) Liza Alekseeva's case in 1081 is one of the most illustrative examples. From the moral point of view everything was absolutely clear: Sakharov took personal responsibility for this girl, he promised that she will come to USA to her fiancee

Alexey Semenov and he respected his own word. He knew that he should do it in spite of KGB, of exile, of the condemnation by authorities, by the colleagues, by the strongly ideologized friends-dissidents. And the second postulate had worked here. The demand was rather narrow, but the final "decision making" was made at the very top of "Kremlin". And to reach this little victory the maximum concentration of efforts was necessary: Sakharov and Elena Bonnet staked their life in a 17-days hunger strike, the cumulative actions in their support from the many different directions - to the one point penetrated to the highest level of political power in the USSR. (At that time the visit to the USSR of French scientists Michel and Pekar who attacked Soviet tops not from the far off Paris but here in Moscow was very important; Sakharov writes about it in his "Memoirs".) Result: the girl has got the visa. But wasn't it stupid - such a global fuss with such a small profit ? No, I think it was not. Sakharov created "cloud" just in a sense as MichaelsonMorley experiment and the black body radiation problem were the "clouds" on the serene sky of classical physics. But they were signals of absolutely novel (for that time) relativistic, quantum-mechanical world. The question about "small profit" in Liza Alekseeva's case may be generalized to the whole body of the human rights struggle which as a rule aimed to save one single person. Why this "charity" should have global political, ideological, geostrategical consequences ? As I understand it, the reason is that the giant totalitarian dinosaurus has no brains, its dreadful mechanisms of suppression work automatically, with great inertia. To save somebody, to achieve the minimal loose means to force the behavior which contradicts the inner laws of the system. This may be possible only if you manage to influence the very top of the power. And there, in this singular point the final decision depends . . . . Who knows what it depends on ? Perhaps Sakharov guessed something. But if the absolutely untrivial result has followed - the girl has got the visa, Vladimir Bukowski was exchanged toCorvalan etc. - then this is a "cloud". In some, almost irrational way it effects the structure of the whole strongly centralized system. (Imagine the special sort of crystals - i n Russian they are called "Batavian tears" - w h i c h change their structure in the whole volume at the very moment when you brake off the very thin tip.) Sakharov understood this mechanics. But there were so many people who did not understand him, especially in his termless hunger strikes. ,f. Now let me make a few remarks on the Destiny, Freedom of will and the End of ~he world. In one of the letters, which I have got from Gorky in 1982, Sakharov discusses the inflationary cosmological models versus his manysheet (pulsating) cosmology. Then he writes the following words: "Ok, let us wait and see. Future will show us who is right. Future will show us many other things as well. Fortunately future is unpredictable and also (due to quantum effects) - it is not determind." This phrase reflects not only the probabilistic nature of quantum laws and it is not only about physics. In the history, as well as in the personal destiny, future is not only unpredictable - it does not exist at all. -

When we pronounce the word "Future" we speak about nothing. This word is not defined now. Different scenarios are possible, some of them with the directly opposite outcomes. Will the mankind survive or tomorrow is the Doomsday ? Will this one prisoner of justice die or not ? - Result may depend on the personal action (or inaction) now. This is the "Schr6dinger cat" situation of quantum mechanics, but the outcome depends not on the behavior of the, say, "stupid" electron, but on the "free will" of the observer, on his "decision making". With this sense of responsibility academician Sakharov lived. And in this connection I would like to say few words about History. Does it exist, the History ? Or, perhaps we deceive ourselves formulating its laws which in reality are nothing more than some net of coordinates which we put on the past on the events which happened after the "wave function reduction". "Only those things do matter which have already happened" - that was Sakharov's answer to my question "What lies ahead?". This conversation occurred in 1977 after the arrest of Orlov, Scharansky and other Helsinki group members. "Only things which have already taken place, do matter" and "Future is not determined" - this is Sakharov's position which I think is not the conventional one. The alternative objectively-materialistic attitude to the history says that inevitability of Perestroyka is dictated by the new technical revolution when the soviet system shows its noneffectiveness, and also by mortal threats to the whole mankind. Thus History knows where it goes. During the last 20 years Sakharov tried to persuade the world that reality is much more frightening: the fall into the abyss may happen any moment. Where does the History go? W h a t do they know there at the top of the power (which in fact is the volcano of the struggle for the power) about the menaces to the mankind. Perhaps some wisedom about the ways of the history is the well known joke about two cosmonauts. " W e rush with the speed approaching the speed of light",- one says with admiration. "Yes, but in the opposite direction", - the other answers. Let me remind you of the three examples from the past before I present here two citations of Sakharov which I think are very important. So the question is: "Does the History go the sensible ways?" 1) 1975, United States get out of the Indo-China peninsula, but the triumph of Sense is somehow clouded by the three-million hecatomb in Cambodia. 2) 1979,- the beginning of Afghanistan slaughter. 3) 1982 - Israeli troops found in the South Lebanon the underground PLO stores of soviet arms for the army of one million soldiers. Who at the Soviet tops gave sunction to have this storage ? -

In October 1980 Sakharov in his letter from Gorky to the President of Soviet Academy academician Alexandrov among other things wrote the following: "In 1955 I learned that our Near-East politics performed a sharp turn with the purpose to create the 'oil-dependence of the West'. During the following years this turn brought the enormous sufferings to the people of the Region arabs, Israel, Lebanon" [5,6]. The burden of this long-term politics still weighs

down Perestroika and brings suffering to many people - not only at the Near East. In the same letter Sakharov writes about the danger of expansionist geopolitical mentality. Here is the example of what Sakharov understood and what practically nobody else had realized. Sakharov knew who holds their finger on the Soviet nuclear button. He knew how conservative, how dangerously isolated of reality are these circles, which behavior is dictated by the blind gang-like interests. In 1975 in his book "About the Country and the World" he wrote in the following words about the danger of Nixon-Brezhnev agreement on the restricted Antimissile Defense: "The terrible suspicion creeps into your soul against your will. The picture is imagined that this defense system permits to sacrifice the most part of the territory and population of the country for the sake of advantage of the first nuclearrocket blow when the relative security of Moscow bureaucrats is guaranteed" ([7], see also [6]). Imagine the Colombian narcomafia multiplied many times - then perhaps we can realize who Sakharov opposed here in the USSR. "Spiritual renegade, provocator Sakharov with all his subversive actions has long ago put himself in the position of the traitor of his people and his State", [8]. How many times was it repeated! Even now in December 1989, a week before his death Sakharov has read about himself in the Journal of the Ministry of Defense: "Slanderer of Russia, p r o v o c a t o r . . . " , and many of the epithets of previous years were repeated there [9]. Why did this "Defense" attack Sakharov ? The direct reason was his remark in Canada in February 1989 on some Western mass-media reports of the first years of Afghanistan war that Soviet Army command ordered to kill the encirceled soviet soldiers to prevent their captivity. The fierce reaction to this remark followed on the First people Deputies Congress on June 2, 1989. And even in December they did not calm down. Andrey Dmitrievich was able to pronounce severM words in a low voice but somebody's nerves gave out. Like highly skilled surgeon Sakharov penetrated with his scalpel into the proper place and let out the puss. But now the new times had come. The wave function of the History reduced in 1987 in such a way that some soviet and american rockets are destroyed and perhaps we shall live. "To Live on the Earth and Live Long", - that is the title of Sakharov's interview published in 1989 ([10], see also [6]). But who knows what could be the outcome of the "reduction" without the efforts of rather few people in the USSR who began to struggle for human rights in this country or without Sakharov's hystorical "Danger of the Thermonuclear War" - the open letter to Sydney Drell, which cost too much for Sakharov and his wife. It was she who smuggled this letter from Gorky in May 1983 in spite of infarction, in spite of many other things (only the third attempt was successful). After the letter was published Sakharov was declared to be mentMly disturbed (President of Academy Alexandrov, Soviet leader Andropov). But "It's not Sakharov who's 'chukoo' ", - this is the title of the article by Harry J. Lipkin from the Weitzmann Institute published in 1983 [11]. This sort of support was then of vital importance.

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Now new times came and perhaps we have a Hope. But the same circIes who sent "arab terrorists" to Sakharov in 1973, who invaded Afghanistan and used PLO to pile up arms in Lebanon, who tortured Sakharov in Gorky in the eighties and attacked him half a year ago - they are strong enough today. The hell-like pictures of Sumgait, Fergana, Tbilisy and Osh (pogroms and massacres in 1988-90) - this is the modal. The model of the Doomsday, which academician Sakharov tried to prevent. 5. Andrey Dmitrievich never was a victim in the direct sense of the word. Even when hundreds of pages of his manuscript were stolen and he had no copies. He ~vas not ,a victim because after all he made everything what he had planned. He wrote the book anew again and again Elena Bonnet saved it. Now the huge book of Sakharov's memories is published. But it is another book. And in spite of many public appeals KGB refuses to give back the stolen manuscripts. The face of ~he enormous Sakharov's archives in KGB should be ~he gheme of ~he irtte'rna~ionaI cortcern. Sakharov was not a victim in Hospital although they tortured him. It was he who put KGB into the very difficult position. He was ~he unsolvable problem .for them. In May 1984 they locked Elena Bonner in Gorky and decided that the problem of Sakharov is finally solved - at last he is absolutely isolated and will stop sounding off all over the world. The fact is that during 4 years of exile his wife performed about 70 shuttle trips from Gorky to Moscow. She smuggled many Sakharov's documents, brought letters to him and owing to it he existed for the world. Thus when in May 1984 they brought up glena Bonner to trial they cut the last thread and buried Sakharov alive (Two visits of theoreticians from the Lebedev Institute - in November 1984 and February 1085 - did not changed, alas, this "black hole" situation). But it was impossible to "beat" Andrey Dmitrievich. I repeat that-he was the unsolvable problem for the KGB. He went on hunger strike, but doing it he did not only saved his wife and himself - he at the same time shaped the new world for all of us. In fact, there is nothing new in saying that Sakharov's situation was one of the most i m p o r t a n t problems of the pre-Perestroyka period. That is why KGB did their best to prevent any leakage of information from Gorky. In this way they tried to save themselves. I had a chance to witness some of these their efforts but this lecture is not the place for memories. The scenes of torture in Semashko hospital go in parallel with the Orwell's fantastic "1984" as Sakharov himself noted in his tragic letter to President Alexandrov (November, 1984, see e.g. in [12]). This was the struggle with the absolutely real, not fantastic " T h e Force of Devil" which is the menace for the life and health of all the people on the Earth. And when he pushed the food out of his m o u t h - month by month - this was not crazy stubbornness. This may be defined with a single word: he worked. Sakharov worked all his life and was highly professional in everything he did. When Sakharov refused to take food he performed something which he considered to be absolutely necessary. But of course he would be happy if he were given help in this very hard and harmful job. If Academy of Sciences responded to Sakharov's requests. Along with the efforts from abroad it could create the 11

necessary penetrating cumulative effect. But, it did not happen. Instead, all the energy was canalized to the irritation and accusations of Elena Bonner. This was in full accordance with the politics of authorities. Let us listen to what he himself tells about this tactics of KGB (from the interview in September 1988, " T h e main blow, the centre of pressure was shifted, directed mostly not against me, but against my wife, indirectly - against her children. This was rather crafty tactics, which put me in the very difficult psychological position and extremely hard for Elena Georgievna. The situation demanded of her the exceptional strength of will. And it seems to me that she showed it and at the same time managed to save me, to preserve me in the form in which I was to be. Thus our life in that period was not simple". Here is only one example of what was published in the USSR about Sakharoy's wife in 1983 in circulation about 10 million copies [14]: "In its attempts to undermine the soviet system from inside CIA have resorted to the service of the international Zionism. They used not only spy network of american, israeli and Zionist secret services and the jewish masonic lodge "Bnei Brit", but also the elements exposed to influence of the Zionist propaganda. Academician A. D. Sakharov was one of the victims of the Zionist agents of CIA . . . . The whole job was done under the firm E. Bonner and children." Mass-media was full of such slandering. The element of new reality now is that youth journal "Smena" published the strong apologizes in its No. 1 of 1990. A month ago, on May 2, I participated in Sakharov's memorial round table in Jerusalem organized by the same Bnei Brit lodge. The atmosphere of the meeting was very warm. Other speakers were Anatoly Scharansky, Nauru Meiman and Alexander Voronel. And now I am here in the USSR and telling you all this. It is Perestroyka. And at the same time the same terrible circles which I mentioned above spread the terrible wave of antisemitism all over the country through the so called Pamyat and other organizations. Situation here is really confused and even crazy. 6. Sakharov always thought about science. The only exception were the dreadful moments in Semashko hospital. Once, after his return from Gorky I asked him about it and he said that the most unbearable thing was that he had never been left alone there, not for a minute. All the time he was surrounded by the agents who pretended to be patients, sometimes rather aggresive. But at any other time he pondered over physics in spite of all mess and fuss surrounding him. At the beginning of this lecture I remembered him coming up with a poem after the evening nap. I witnessed a similar scene soon after he returned from Gorky. He came into the kitchen and said that he was not able to fall asleep but had a good rest. And with great pleasure he began to speak about spheres, handles and polygons. Resting on a sofa he pondered on two-dimensional surfaces of arbitrary topology, which is related to the unified string theory. And this was the time when he was extremely busy with other things. Sakharov liked astronomy and everything which is connected with the observations in space. Since 1988 he was Chairman of the Council on Cosmomicro-

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physics of the Soviet Academy, Being one of the members of this Council I was present at its General meeting at the G A I S H (State Astronomical Shternberg Institute) on November 29, 1989. It was two weaks before Sakharov's death. He was very active in questions and discussions. Special interest was shown on the Radioastron program (radio telescope with the cosmic base and unpreccdental resolution which perhaps will permit to see as far as the Universe horizon). In cosmology he liked the hypothesis of CPT-symmetry of the Universe and the Hypothesis of the reversal of the time arrow which I spoke above. He was not very enthusiastic about inflation which to his mind contradicted the idea of "manysheet" Universe. In 1984 Sakharov published in J E T P the fundamental paper: "Cosmological transitions with alteration of the metric signature" [15] where he came out with the idea of existence of m a n y "times" (extra "times" are compacti£ed) and also the idea of the quantum tunneling between the space with the different number of time axes. This is quantum cosmology and this is at the first line of modern science. The paper was submitted to J E T P in March 1984 just before the dreadful events had begun and it was published in August when Sakharov was in hospital for already three months. Another novel idea of this paper: the possibility of quantum-induced conforreal gravity in higher dimensions. (Einstein's theory in 4 dimensions is induced by the subsequent compactification). I shall not come here in details but just note that on this way the difficulty of cosmological "floating" fundamental constants which exists in the String theory m a y be overcomed. The quantum-induced gravity idea goes back to Sakharov's famous work in 1967 [17]. In his second book of rememberings [18] he writes: "String theory is, at the level, the realization of m y old idea of induced gravity! I can't help being proud of it." In his paper of 1984 Sakharov advocates the anthropic principle which assumes the plurality of universes with different fundamental characteristics. This lies in the line of what he said in his Nobel Prize Lecture (1975). Thus he backed the idea of pluralism not only for the human society but also for the Universe as a whole. The following arc some quotations from Sakharov's letters which I have got from Gorky. In 1983: " . . . As to compactification, this hope became quite fashionable now . . . I have an idea that compactification radius is possibly fixed at some constant value if we take into account quantum effects, in analogy with the hydrogen atom". In the modern higher-dimensional Kaluza-Klein type theories there is no first principle to fix the radius of eompactification of extra dimensions. This fixation is always done "by hands", i. e. by some fine-tuning; without fine-tuning we have "fiat potential" - hence the problem of cosmologically changing fundamental constants. Sakharov's idea lies in the intersection of higher-dimensional physics and quantum cosmology and I think it has not lost its meaning. The analogy with the hydrogen atom

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seems stimulating. However: " N o n r e a l i z e d idea - i~ is hog an idea yeF', as Sakharov used to say. From the letters of 1986: " . . . Great events of our time-superstrings, Kaluza-Klein . . . " " . . • I think that the works of Polyakov and subsequent works of Fradkin and Tseytlin are very important . . . " " W h a t do you think of Polyakov's idea that cosmological constant may become zero in analogy with the zero-charge in quantum electrodynamics, but due to the longwave fluctuations of scale scalar field ?" "You are interested in the mass hierarchy problem. I think that the almost conventional idea is correct, that "small" masses appear because of nonperturbative effects of type e x p ( - c o u s t / g 2 ) . Who knows however

7. Thus the Marvel of Sakharov. He became a legend still alive. Here are excerpts from his interview of 1988 [13] at the 38 Paguosh conference (Dagomys, USSR): Quesgion : Do you believe in destiny ? Sakharov : Practically I believe in nothing except some general feeling of an

inner meaning of the course of events. The course of events not only in the life of mankind,but more generally - in the whole Universe. I do not believe in destiny as something fatal. To my opinion future is unpredictable and it is not determined~ it is created by all of us - step by step - in our infinitely complicated interaction. Q u e s l i o n : If I understand you well, to your opinion everything is not "In the hand of God" but "In the hand of Man"? Sakharov : There is interaction of the both forces, but the freedom of choice is left to Man. That is why of such an importance may be the role of personality who was put by the destiny at some key points of history . . . .

References 1. "Dos'ye of Literaturnaya Gazeta", Sakharov's memorial issue, Moscow, January 1990. 2. Andrey Sakharov "Memoirs" Chekhov Publ. House, New York 1990. 3. A. D. Sakharov, JETP Pis'ma, 5, 32 (1967) (see [16], paper 7). 4. A. D. Sakharov, Preprint IAM, Moscow 1970; 3ETP, 79, 689 (1980) (see [16], papers 10, 12). 5. "On Sakharov', Ed. A. Babenyshev, New York 1981. 6. "Trevoga i Nadczhda ("Trouble and Hope", collection of some Sakharov's works), Inter-Verso, Moscow, 1990. 7. A. D. Sakharov 'O Strane i Mire" ("About the Country and the World"), Khronlka, New York, 1975. 8. "Komsomolskaya Pravda", Moscow, February 15, 1980. 9. "Military-historical journal", Moscow, November, 1989. 10. "The art of cinema" Journal, Moscow, August 1989.

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11. H. J. Lipkin in "The Guardian" December 25, 1983 (Reprinted from "Washington Post"). 12. Elena Bonner, "Postscrlptum", de la Presse Libre, Paris 1988. (English translation is titled "Alone together"). 13. "Molodezh Estonii" ("Youth of Estonia") newspaper, October 11, 1988. 14. "Chelovek i Zakon" ("Man and Law") Journal, October 1983. 15. A. D. Sakharov, J E T P 1984, 87, 375. 16. A. D. Sakharov "Collected Scientific Works", Ed. D. ter Haax, D. V. Chudnovsky, G. V. Chudnovsky, Marcel Dekker, Inc. New York and Basel~ 1982. 17. Doklady AN USSt~ 177, 70 (1967) (see [18] paper 14 and also papers 15, 16). 18. "Gorky, Moscow~ dalee vezde" ("Gorky, Moscva and further on everywhere"), Chekhov Publ. House, New York 1990.

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T h e 1990 W i g n e r M e d a l C e r e m o n y

On June 7, 1990 the Wigner Medal was awarded to Francesco Iachello in recognition of his contributions in the algebraic theory of nuclear and molecular physics.

Opening Remarks

Arno Bohm Physics Department University of Texas Austin, TX 78712 As chairman of the Group Theory Foundation I open this Ceremonail Session. The Wigner Medal is awarded for outstanding contributions in the mathematical foundation or in applications of group theoretical methods to physics. Previous recepients of the Wigner Medal are Eugene Wigner and Valentine Bargmann in 1978, Israil Moiseevich Gel'fand in 1980, Yuval Ne'eman in 1982, Luis Michel in 1984, Feza Gfirsey in 1986 and I.M. Singer in 1988. The international selection committee for the 1990 Wigner Medal consisted of Peter Kramer (Germany), Feza Gursey (USA), Luigi Radicati (Italy), Aloysio Janner (Netherlands) and E.P. Wigner. As Professor Kramer, the chairman of the committee, was unable to come to Moscow, I want to ask Professor G/irsey to deliver the encomium for Franco Iachello.

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Laudatio for Professor Francesco Iaehello

Feza Giirsey Center for Theoretical Physics Yale University New Haven, CT. 06511

It is a great honor and pleasure for me to be chosen to deliver the laudatio for a friend and colleague I admire and love, Professor Franco Iachello, who becomes this year's recipient of the biannual Wigner Medal of the Group Theory and Fundamental Physics Foundation. Let me sketch a few landmarks in the education and professional development of our laureate. Franco was born on January 11, 1942 in Francofonte, Italy. After a doctorate in nuclear engineering from the Turin Polytechnic he received a second doctorate in physics at MIT in 1969. He has held research and faculty positions at the Niels Bohr Institute, the Politecnico of Torino and Groningen University where he became professor in 1976. He has been a professor of physics at Yale University since 1978. In 1981 he was awarded the AKZO prize of the Dutch Academy of Sciences for his contributions to Nuclear Physics. Professor Iachello has been recognized for the Wigner Medal for his outstanding work in Theoretical Nuclear Physics. He has given new impetus and direction to this subject through the imaginative use of group theoretical and more generally algebraic methods to study nuclear spectra and electromagnetic transitions for certain families of nuclei. The models he has developed with coworkers have led to specific predictions that have been strikingly confirmed by experiment. The methods he has introduced are quite general and not confined to Nuclear Physics. In fact Professor Iachello has adapted his approach to develop new models for molecular spectra, hadronic physics and scattering theory. Here are a few words intended to put Iachello's work into historical perspective. The use of group theory in nuclear physics is as old as nuclear physics itself. Heisenberg defined the isotopic spin connected with an internal SU(2) group as soon as the neutron was discovered in 1932. Five years later Wigner introduced the SU(4) group which admits the isospin and spin groups as subgroups in connection with his theory of supermultiplets. In the fifties we find the pioneering contributions of Racah, Flowers, Elliott [with his SU(3) model that has the rotation group O(3) as a subgroup], Goshen and Lipkin with their group theoretical treatment of collective states through the symplectic groups. These papers ushered a decade of group theoretical models in nuclear physics by Radicati, Moshinsky, Biedenharn, Hamermesh and many others. While group theory was entering nuclear physics two other major developments took place in the fifties, namely the discoveries of the one particle shell model by Goeppert-Mayer and Iensen and the geometrical collective model by Bohr and Mottelson. In fact these breakthroughs led Blatt and Weisskopf to their famous diatribe against group theoretical methods in their book on nuclear theory. This groupophobia still persists in some reputable research centers to this day. However, since the towering work of Arima and Iachello

20

and the largebody of successful research that followed it, mostly under Iachello's leadership, the voices of opposition have largely been silenced. The IBM model and its aftermath constitute one of the most important developments in nuclear physics in the last decade. In a series of papers that appeared in Annals of Physics between 1976 and 1979, F. Iaehello together with A. Arima developed the interacting boson model (IBM) that was destined to have a profound influence on nuclear theory. The model provides a simple and fairly detailed description of nuclear properties including nuclear spectra and electromagnetic transitions. It also unifies different nuclear models mentioned above, like the independent particle model (the shell model) for which Maria Goeppert-Mayer and Hans Jensen received the Nobel prize in 1963 and the collective model of Aage Bohr and Ben Mottelson who also received a Nobel prize for their work in 1976. These algebraic models are based on solvable Hamiltonians that are related to a dynamical group that cascades through a chain of subgroups down to the rotation group. Each subgroup is represented by its Casimir operator multiplied by an undetermined constant. The Hamiltonian is the sum of this chain of Casimirs. In the IBM model the group is U(6) cascading down to 0(3) in three different ways through the subgroups U(5), O(6) or SU(3), leading to three types of dynamical symmetries that were subsequently discovered. The six bosons of the model (one with zero spin and five with spin 2) are interpreted as correlated pairs of nucleons, much like the Cooper pairs of superconductivity. In another series of papers, some of them in collaboration with I. Bars and A. B. Balantekin, published between 1980 and 1981, Iachello was responsible for the discovery of supersymmetrie generalizations of the U(6) model. The predictions of this boson-fermion model have been found to be realized in several nuclear species, showing a physical manifestation of supersymmela'y in nature. For more details of these algebraic models the reader is referred to a masterful summary in Professor Iachello's invited lecture in this volume. Algebraic models similar to those for nuclei were introduced in molecular physics by F. IacheUo and his collaborator R. D. Levine during 1981 and 1982. The most successful model based on the group U(4) is known as the vibron model which allows a relatively simple description of some molecular properties. Professor Iachello has also extended algebraic methods to include scattering problems in collaboration with Y. Alhassid and F. Giirsey. It has been a pleasure for me to be involved in this project. Professor IacheUo's work in nuclear theory is well explained and summarized in books coauthored with his collaborators. The include "The Interacting Boson Model" by F. Iachello and A. Arima (Cambridge U. Press, 1987) and "The Interacting Boson-Fermion Model" by F. Iachello and P. van Isacker (Cambridge U. Press, 1990). Professor Iachello's wide ranging interests also include musicology to which he has contributed wich a scholarly article on the musical notation of Monteverdi. I have promised not to mention his career as a racing car driver in Italy. Such activities illustrate Professor Iachetlo's multichanneled intellectual energy independently of his work on multichannel scattering. I am delighted to have participated in a double scattering experiment starting in New Haven and ending in Moscow for this special ceremony. It is also a very happy gathering due to the presence of Franco's wife Irene and his son Giovanni, both exceptional people who have given him loving support in his Odyssean shuttling between two continents. I offer my heartiest congratulations to Franco and his family on the occasion of the award of the Wigner Medal.

21

SYMMETEY

F. lachello Center for Theoretical Physics, Yale University New Haven, Connecticut 06511

It

is

a

great

honor

for

particularly

in view

of the

Wigner,

of

"founding

one

the

me

fact

to

that

receive this

fathers"

of

inspiration to generations of physicists.

the

Medal

1990

Wigner

is named

symmetry

in

Medal,

after

physics

Eugene and

an

I am also happy that the medal is

handed to me by my friend Arno BShm and that some very kind words have been said on my work by Feza G[irsey.

Feza's work

on symmetries

in elementary

particle physics has greatly influenced mine. It is a pleasure for me to briefly review the concept of symmetry and its applications

to physics.

This

is the unifying

theme of this Symposium

and the subject of my work for the last 16 years.

I.

THE NOTION OF SYMM TEY f

The word symmetry, ordered,

was originally

paintings, have

(about

especially

this

concept

3,000

in

B.C.)

translation

well-proportioned,

used to describe properties

sculptures and architectural designs.

employed

Sumeria

from the Greek a u ~ p o v ,

and

their

show

early

already

reflection

symmetries as spatial (or geometric) Greek world that symmetry became

of artifacts,

Most civilizations

developments. clear

symmetry.

symmetries.

the central

wellsuch as

Artifacts

examples I

will

However,

concept

seem to

of

symmetry,

refer

it was

in art.

from

to

these

only

in the

One of the

oldest examples of geometric symmetry in an artifact from Greece is shown in Fig. i.

Here, a floor pattern found at the Megaron

1,200 B.C.)

provides

an example

of reflection

in Tyrlns

symmetry

[i].

ideas were codified during the Golden Age of Greek civilization artists

and philosopher

wrote

treatises

on symmetry

(dated about These

early

and several

and proportions.

The

notion of symmetry has remained a guiding principle in art ever since and it is used even today in architectural design and painting.

22

Fig.l. Design found on a floor at the Megarons in Tyrins, Greece, ca. 1,200 B.C.

2.

2.1.

SYMMETRY IN PHYSICS

Geometric Symmetries The first applications of symmetry concepts in physics were along lines

similar

to those

of geometric

symmetries

in art.

Symmetry

was

used

to

describe certain geometric arrangements of atoms in molecules and crystals.

Fig.2. The twisted H3C-CCI 3 molecule.

Here the three key symmetry operations were: reflections, rotations.

translations and

Fig. 2 shows an example of a geometric symmetric,

H3G-CCI 3 molecule.

the twisted

The geometric arrangement of the atoms in the molecule

23

is invariant under rotations of angles multiple of 120 ° around the C-C axis. This set of transformations

forms a group,

then the appropriate mathematical

the discrete

group G3, which is

framework to describe the symmetry of this

molecule.

2.2

Permutational Symmetries A

somewhat

permutational

direct

symmetry.

set of identical some

of

the

generalization This

objects.

objects

in

of

symmetry

It describes the

discrete

is relevant

set.

geometric

what happens

Permutational

mathematically by the discrete permutatIQn group,

symmetry

to the description under

interchange

symmetry

is

is

of a of

described

Sn, where n is the number

of objects in the set.

2.3

Space-time

(or Fundamental Symmetries)

For a long time, discrete geometric symmetries,

such as those of Fig.2,

were the only ones to be extensively used in physics.

However,

as time went

on, the original concept of symmetry was enlarged. The main new idea was the introduction continuous on

of

continuous

symmetries.

Although

symmetries and indeed Aristotle

spheres

(the

most

symmetric

body

the

Greeks

had

employed

thought that the world was built

in

a

three-dlmensional

space),

the

extensive use of continuous symmetries in physics came only at the beginning of the 20th Century. Among the continuous symmetries, are

space-time

(sometimes

or

called also

non-relatlvistic

transformations, their

symmetries.

kinematic

symmetries)

quantum mechanics

quantum mechanics.

of

fundamental

Examples

are

often

these

invarlance

called

symmetries

invarlance

in

in relativistic

are described by continuous

the Lie groups S0(3) and S0(3,1)

importance,

of

are rotational

and Lorentz

These symmetries

particularly important

groups of

respectively and, because

fundamental.

They

have

played

a

crucial role in the development of physics in this Century.

2.4

Gau~e Symmetries Another

type

of symmetry

recent years is gauge symmetry.

that

has

become

particularly

important

in

This symmetry is also a continuous symmetry

and has to do with the transformation properties

of the basic interactions

of Nature. It has assumed a particularly important role in physics after the discovery

that also the weak and strong

24

interactions

(in addition

to the

electromagnetic one) are governed by gauge symmetries.

The gauge groups of

these interactions are continuous Lie groups, such as SU(2) and SU(3).

3.

DYNAMIC SYMMETRY

With

increasing

techniques,

sophistication

of both

theoretical

and

experimental

other types of symmetry have been introduced in physics.

important new type is that called dynamic symmetry. describing this symmetry,

An

I will spend more time

since it is this type of symmetry that is being

recognized here today with the award of the 1990 Wigner medal. of dynamic symmetry is schematically illustrated in Fig. 3.

The notion

Consider the

0(2)

C4

Fig.3. Illustration of dynamic symmetry by regular breaking of rotational invariance.

C2

perfect

figure

in the plane,

under the set of rotations

i.e.

the circle.

This

figure

around an axis perpendicular

going through the center of the circle, the group S0(2). that, because of some reason, broken,

but

in a very

smaller set of transformations.

to the page

leaving

For example,

the figure

and

Now, it may happen

usually of dynamic origin,

ordered way

is invariant

the symmetry

invarlant

is

under a

in Fig. 3 the figure in the

middle, a wheel with parts of one metal (white) and parts "of another metal (black) is invariant only under the set of rotations by angles multiple of

25

90 ° , the group C 4.

The C 4 symmetry may or may not be further broken down by

additional (dynamic) constraints, as indicated by the lower part of Fig. 5. The ordered breaking of a symmetry for dynamic reasons is usually called a dynamic

symmetry.

Dynamic

symmetries,

illustrated

in Fig.3

in geometric

terms, are actually used to describe physical systems composed of a certain number of particles interacting with each other. One can then give a precise mathematical definition to this concep~ by introducing a set of operators, called

invariant

Hamiltonian

(or

operator

Casimir) governing

operators the

[2]

dynamic

in

of

terms

the

system

of

which

is

the

expanded.

Although the first example of a dynamic symmetry in physics is rather old [3,4]

it was

only in the early

60's hhat

its

general

role was

clearly

recognized [5].

4s

4p

3s

~

3p

2s

~

2p

~

4d .....

~

4f

8d

-5 A

>, 0 fLU

-10

-15

Fig.4. Level diagram of the hydrogen atom.

Dynamic symmetries are particularly useful to describe the structure of physics.

Finite quantum mechanical systems are characterized by a set of

discrete energy levels (the role of dynamic symmetries in the unbound part of the spectra will not be discussed here).

Dynamic

patterns of energy levels and can be used at all layers of particular interest are:

26

symmetries of physics.

describe Layers

Molecules, Atoms, Nuclei, Hadrons, Quarks, .o*,

in order of decreasing dimensions.

3.I.

Dynamic symmetries in atoms (1926-35) The oldest example of a dynamic symmetry is provided by the Bohr model

of the hydrogen atom.

Pauli [3] and Fock [4] recognized that the observed

regularity in the energy levels of the hydrogen atom, Fig. 4, was due to the occurrence of a dynamic symmetry, described by the group S0(4). years several authors

In recent

[6] have generalized this dynamic symmetry to atoms

with more than one electron. Because of its clearness,

the spectrum of the

hydrogen

example

atom

remains

to

date

the

most

pedagogical

of

dynamic

symmetries in physics.

3.2. Dynamic symmetries in hadrons (1962-64) Dynamic symmetries were also used implicitly in elementary particle

E(GeV) n 1.6 1.4-

A

12-

Fig.5. Level diagram of the baryon decuplet.

27

physics

prior

to

their

explicit

formulation.

In

1961

Gell-Mann

[7]

and

Ne'eman [8] and later G~rsey and Radicatl [9] recognized that the pattern of energy

levels

symmetries,

of hadrons,

SU(3)

flavor degrees

and

Fig.

SU(6)

of freedom.

5, was

due

respectively, This

to the occurrence associated

led subsequently

to

with the

the

of dynamic so-called

quark model

of

hadrons and to the development of the present theory of strong interactions.

3.3.

Dynamic symmetries in nuclei (1974-78) Perhaps the best examples to date of dynamic symmetries in physics are

provided by the spectra of atomic nuclei.

It was recognized by Arima and

myself [i0] that the patterns of energy levels observed in several nuclei could be described in terms of dynamic symmetries arising from the regular breaking of the U(6) group (interacting boson model of nuclei), is shown in Fig. 6.

An example

Because of the complexity of nuclear spectra, dynamic

symmetries here provided a unique tool to decypher the intricate patterns observed

experimentally

and

led

to

a

deeper

understanding

of

nuclear

structure. 2,0 predicted spectrum

observed spectrum

--6 1.5

~1.0

--6

--5

--4

--4 --3

--4

--4

--2

--2

--2

--3 --2

--10 :> o)

--6

--6

--10

--0

--5

--0

--8

--8

--6

--6

--4

--4

--2 --0

--2 --0

tU

0.5

Fig.6. Level diagram of a medium mass nucleus. On the left the observed spectrum. On the right that predicted by dynamic symmetry.

28

3.4.

Dynamic symmetries in molecules Geometric

developments role

(1981-84)

symmetries had been used in molecular physics

of this

of dynamic

field.

symmetries

However, was

it

was

recognized.

not until Dynamic

from the early

the

80's

symmetries

that

the

arise here

from the properties of the interatomic interaction that binds atoms together to form molecules. that the patterns

In 1981,

It was

recognized by

myself

and Levine

[11]

of the observed energy levels could be described in terms

of a dynamic S0(41 symmetry originating from the regular breaking of a U(41 group

(vibron

model

of molecules).

An

example

of

dynamic

symmetries

in

molecules is shown in Fig. 7.

E(cm'l) H2

Exp.

IO,OOC 345

_ - - = = _ 0

-IO,OOC

-

-

-

-~

_-- - -

--

-= ---- -= I I

--

Z

--

--

--

--

=

=

=--

_

--

--

--

_----

~

--

i

(9)

--

_-- ---- --~ ca}

--

=

--- _ -- -

"20.00C

-"Z _

--

~

Ill

It

z "

(,=(1 -

171

- - --: '---- ---~ i (6) ---- - - --- ~" (~1 -

_

.

(,1

-30,00( =

Fig.7. Level diagram of a

-40,00C E ( c m "j)

Th

IO,OO0 --29__27

o

. --

--

- - ~ - - 2 3

--

--

--

___=-IO,O00

_

--

_

--

__19

i~i!~

--

=-

--

--

~

--

--

13ZII

~9

i

(to)

~,,,,

=:7

=It

Ira3 --

(?)

- - - = - -__--~61 -20,000

--

--

--

-~

141

(31

- 30,00c

-= =

(o~

-=

On top the observed

spectrum.

On bottom that

predicted by

I_

2

molecule.

(tl

iii

(I)

0(4)

-40,OOC

29

dynamic symmetry.

4.

ROLE OF DYNAMIC SYMMETRY As

one

can

see

from

the brief

description

given

in

above, symmetries are used in a variety of ways in physics. has proven to be very useful

the

paragraphs

One type which

(and which I have described somewhat

in more

detail) is dynamic symmetry. It provides patterns for measurable quantities. These patterns may be very intricate and difficult to recognize. intricate the pattern,

the more useful are symmetry concepts.

The more

The examples

shown in Figs. 4-7 appear to indicate that dynamic symmetries are present at all

scales

hadrons

with

universal of

of physics, energy

concept

physics.

One

from molecules

scales

of

10 9

with

eV.

energy

Dynamic

scales symmetry

which can be used to describe phenomena can

paraphrase

J.C.

Maxwell's

of

10 -2 eV is

in

sentence

loves symmetry above all" to "Nature loves symmetry above all".

to

thus the

a

whole

"Mathematics Perhaps the

reason why nature displays symmetry in many of its manifestation is similar to the motivation,

particularly present in the Greek world,

that stimulated

Fig.8. The Parthenon in Athens exemplifies beauty in Greek culture.

30

anclent civilizations

to produce artifacts with symmetry properties:

beauty

is bound up with symmetry.

This association,

exemplified by the beauty of

the Greek

can be made

stronger

sentence beautiful,

temple,

Fig.

attributed

8,

to

P.A.M.

it must be true".

even

Dirac

that

"if

To find a symmetry

in accordance

a

theory

of

with

nature

a is

is thus to find the key to

Nature.

References

[i]

H. Weyl,

"Symmetry",

[2]

See, for example, symmetries

Princeton University

Press,

Princeton

(1952).

the accompanying '~aper "The role of dynamic

and supersymmetries

in nuclear physics"

in these

Proceedings. [3]

W. Pauli,

"0her des Wasserstoffspekrum

Quantenmechanik,

yon Standpunkt

der neuen

Z. Physik 36, 336 (1926).

[4]

V. Fock,

"Zu Theorie des Wasserstoffatoms",

[5]

A.O. Barut and A. BShm,

Z. Physik 98, 145 (1935).

"Dynamical groups and mass formula",

Phys. Rev. 139B, 1107 (1965). [6]

See, for example,

D.H. Herrick and M.E. Kellman,

"Novel supermultiplet

energy levels for doubly excited He", Phys. Rev. A21, 418 (1980). [7]

M. Gell-Mann, physics",

[8]

M. Ne'eman, invariance",

[9]

"The eightfold way: a theory of strong interaction

California

"Derivation of strong interactions Nucl.

F. G~rsey and L. Radicati,

[Ii]

from a gauge

"Spin and unitary spin independence

"Collective

of a SU(6) group",

F. laehello and RoD. Levine, spectra.

(1961) q

of

Phys. Rev. Lett. 13, 173 (1964).

A. Arima and F. lachello, representations

Report CTSL-20

Phys. 26, 222 (1961).

strong interactions", [i0]

Institute of Technology,

nuclear states as

Phys. Rev. Lett.

"Algebraic

I. Diatomic molecules",

35, 1069 (1975).

approach to rotation-vibration

J. Chem. Phys. 77, 3046 (1982).

31

DYNAMICAL AND FUNDAMENTAL SYMMETRIES IN PARTICLES AND NUCLEI Yuval Ne'eman Wolfson Chair Extraordinary in Theoretical Physics Raymond and Beverly Sackler Faculty of Exact Sciences Te1-Aviv U n i v e r s i t y , Tel-Aviv, Israel 69978*¶ and Center for P a r t i c l e Theory, University of Texas, Austin, Texas 78712#

I_L. R e l a t i v i t y of the Dynamical/Fundamenta! Sometime in the S i x t i e s ,

Definition

our notions of elementarity and compositeness

in the Physics of Particles and Fields underwent a phase t r a n s i t i o n .

As a

matter of fact, even the name of t h i s sub-discipline of Physics was changed, from "Elementary Particle Physics" to the "Physics of Particles and Fields". I

recall

that

perhaps

the main psychological

difficulty

which

experienced in 1960-61, when I postulated SU(3) symmetry [ i ] , nucleon

had to be assigned to a representation

"fundamental"

representation

of

the group.

I

myself

was that the

(the 8) which was not the

The immediate

implication

was

that the nucleon is not an elementary e n t i t y - a view which contradicted the "common sense" of the day. In 1969, the SLAC experiments of deep-inelastic photon-nucleon scattering (fractional,

indicated the existence of point e l e c t r i c

according to l a t e r experimental

results)

charges

"floating"

inside the

nucleon, and with which the energetic photons were i n t e r a c t i n g .

Quarks [2]

became a r e a l i t y , as constituents of the nucleons. As a r e s u l t , the symmetries of the baryons and mesons, such as SU(3) and its

static-Lorentz

extensions

SU(6),

U(6)xU(6)

etc.,

ceased being

"fun-

damental, as was thought in 1961-73, and became "dynamical". I had, in fact, made the f i r s t an a r t i c l e

(and correct)

suggestion for these dynamics in 1964, when in

[3] named "The F i f t h

Interaction"

I resolved the paradox repre-

sented by a p e r t u r b a t i v e l y broken "strong" i n t e r a c t i o n symmetry [ 4 ] . strong

coupling

displayed, first

-

how could

the

breaking

obey perturbation

for instance, by the mass formulae, which were a l l

With a

theory,

as

based on the

order in the breaking ? My suggestion was that two e n t i r e l y d i f f e r e n t

¥Supported in part by the USA-Israel BNSF, contract 87-00009/1. ¶Supported in part by the FDR-Israel GIF, contract 1-52.212.7/87. #Supported in part by the USA DOE Grant DE-FG05-85 ER 40200.

35

interactions were involved. The true "strong interaction", the nonperturbative force responsible for Regge trajectories, is SU(3) invariant. The symmetry-breaking is a different "Fifth Interaction" (the name was later plagiarized for an extremely hypothetical correction to gravity), f i t t i n g a perturbative description. In the present view, the "Strong Interaction" is QCD, a Yang-Mills type local gauge field theory whose SU(3)c gauge group commutes with our original hadronic SU(3)f syn~netry. The Fifth Is the interaction responsible for the "generations" puzzle; I had, in fact suggested that the force which gives i t s mass to the s quark is also responsible for the mass of the muon. This Fifth interaction is s t i l l far from being understood. There is even a possibillty that i t relates to the existence of quark constituents, since the generations may well represent their (vibrational) excitations. In that case, QCD and i t s SU(3)c might wel] turn out some day to be dynamical, rather than fundamental.. Neither is the "gauge" nature of the interaction symetry a guarantee of "fundamentality".

True, such local-gauge Yang-Mills interaction groups -

mathematically the structure groups of a Principal Fibre Bundle whose base manifold is spacetime - are "dynamical" in a different semantic sense: Whereas the usual definition of a dynamical group relates to a symmetry group resulting "accidentally" from the dynamics (e.g. the SO(4) degeneracy of the hydrogen atom Hamiltonian), the Yang-Mills gauge group "generates" the dynamics, in the sense that the existence of the Yang-Mills (connection) field is imposed by the local group symmetry requirement. This would appear at f i r s t sight to represent a very fundamental feature. Yang-Mills interactions were inspired by General Relativity and the "fundamental" role of the Covariance group, the Diffeomorphism group Diff(4,R). And yet there now exist models in which General Relativity is not a fundamental interaction and i t s symmetries appear as effective, dynamically originated local gauge groups [5]. As to the Yang-Mills forces, the discovery of the (p+ p- pO mo K+ K- K° K° ¢0) 8+~ J=1- vector mesons, with their universal couplings to unitary spin, f i r s t suggested that they represent Yang-Mills connection fields and that SU(3)f is a local gauge group; but i t was soon demonstrated [6] that this feature could be "induced" by the conservation of electric charge, for instance, through a "vector dominance" mechanism. Thus, there is also no guarantee that the present "fundamental" gauge symmetries of the "Standard Model" SU(3)xSU(2)xU(1) could not become "dynamical" in the future,

36

when we reach energies at which the compositeness of quarks and leptons might be observed. ' t Hooft [7] has suggested that for a system of constituents q and their composites Q, assuming q-conflnement by a "fundamental '~ gauge group G (representation g) and an interaction of the Q-particles constrained by a non-gauge "flavor" group F (representations f ) , the systematics of the anomalies impose the following conditions: (a) Vanishing of the (GGG) and (GGF) anomalies for the q, AqGGG : O, AqFGG = 0

(1,1)

(b) Equalities of the anomalies due to the F-Noether currents, whether estimated through the q or through the Q, AQFFF = AqFFF

(1,2)

These requirements are solved by the algebraic condition dabc(g,f) = 0 which can be decomposed into g = ~ (the Q) and g $ ~ (the q) dabc(g,f ) + ~ dabc(1,f) = 0

gt~

(1,3)

f

2~ Dynamical ("Accidental Degeneracy") Symmetries Let u s now review the mechanism generating "accidental degeneracies". The best known example in Quantum Physics is Pauli's solution of the Hydrogen Atom [8] spectrum. I t was perfected a few months ahead of Schroedinger's analytic solution.

Pauli used an algebraic approach,

following MeisenbergJs Matrix Mechanics postu]ate.

I t is interesting that

the Physics community, trained in the classical tradition, adopted Schroedinger's treatment, hoping that the "group-pest" would go away,

37

Textbooks in Quantum Mechanics, written between 1926-1965, generally kept to the analytical treatment.

Matrix Mechanics were only mentionned in the

historical introductions.

Once Schroedinger had proved the equivalence bet-

ween the two approaches, physicists breathed in r e l i e f and forgot about the algebraic version. I t is instructive to follow Pauli's elegant algebraic solution. I t will serve us as a f i r s t example of the role played by a Dynamical Degeneracy Group (DDG). In the Kepler problem, Laplace already noticed that aside from the conservation of Angular Momentum ( F is the position vector, ~ the linear momentum vector) [ []

F × ~

(2.1)

there is another conserved quantity, the Laplace-Lenz vector []

(2

Z e' m)-~

{( [

x

~ ) - ( ~ x

[ )}

F/r

(2.2)

where we have introduced the Coulomb potential in place of Newtonian gravity. Z is the atomic charge number, e the electric charge of the electron, m its mass. defines the direction of the ellipse's long axis, its magnitude determines the eccentricity. product A.F. trajectory,

The trajectory can be recovered by taking the scalar

The angular momentum vector is orthogonal to the plane of the

R.[ = o

(2.3)

Pauli applied the Principle of Correspondence and quantized ~ ~ -iB/BF and defined a normalized Lenz vector M = (-2 H)-~/2 A. The [ and M close on the algebra of $0(4), the M behaving like rotations into the (fictive here) 4th dimension, with an energy dependence represented by the presence of the Hamiltonian H in the normalization of M. The unitary representations of SO(4) are given by the values ( j , k ) , where these represent the values of the two commuting abstract spins, -- i / 2

(Z+~),

R--

i/2

(Z-~) 38

(2.4)

From (2.3) Pauli got lJl = JKJ

(2.5)

thus constraining the spectrum to the representations ( j , j ) .

Pauli found

that each such representation should appear once, thus deriving the spectrum

(J,J) II "n"

"n,L"

. . . . .

. i . e .

(1,1)

Jj JJ

3 :

3s

3p

(1/2,1/2)JJ JJ

2 :

2s

2d

(0,0) Ji

1 :

ls

3d

(2.6)

Note that [L,H] so

=

O,

[M,H] : 0

(2.7)

that SO(4) is the "accidental" Dynamic Degeneracy Group. The derivation

is deductive, going from known dynamics to the algebraic structure, from which the Hilbert space spectrum is then deduced. The dynamics provide the constraints that select the particular representations of the DDG that correspond to the problem's spectrum of solutions. The physical Quantum Numbers are functions of the algebraic quantum numbers. In (2.6), we have [ = J + K from (2.4) and also n2 =

2 ( I~I'+

I~I = ) + I

=

( Z H ) -~

(2.8)

3. Spectrum Generating Groups(SGG) The success of SU(3) as a classification scheme for the hadrons [9], followed by the unexpected successes of static extensions such as SU(6), U(6) x U(6) [10] etc.. generated a very favourable climate for algebraic

39

treatments, at least in the world of particles.

The long sequences of

"Regge recurrences", like similar or even longer sequences of rotational and vibrational levels in nuclei, revived the interest in spectra such as that of Hydrogen, now revisited with an algebraic bias.

I t was then noticed

[11,12] that the spectrum in (2.6) forms one infinite-dimensional representation of the non-compact groups S0(1,4) or 50(2,4), and the precise expressions for the generators of these groups were indeed found [13]. non-compact generators connect the S0(4) levels.

The

In S0(2,4), the quantum

number n can also be identified as a quantum number of the group: i t is the eigenvalue of the compact subgroup S0(2), since the maximal compact subgroup is here S0(4) x S0(2). S0(2,4) is thus the Spectrum Generating Group (SGG), with the DDG S0(2) x S0(4) as the maximal compact subgroup. Another simple example is provided by the quantum Harmonic Oscillator. I t has long been known that the DDG here is

U(d), for a d-dimensional

oscillator. The spectrum is given by the representations of the type in,O):

(n,o) 11 .....

tl

(3,0) II

n

L =

0

1

2

3

(3.1)

. . . . . . . . . . . . . . . . .

3

3p

3f

II

(2,o) II It (1,o) II

z

23

1

2d

lp

II

(o,o) II

o

os

The DDG Is generated by the Mij = ai aj , where the ai* is the creation operator and aj the annihilation operator, ai : (I/2)-~/'

[ a i * , aj ] = Sij

{x i - Ip i }

(3.2)

The SGG was identified [14] as SU(I,3), with the non-compact generators Ai : ( I 1 2 ) - I / '

ai

(H - I12)~/'

, and Ai* 40

(3.3)

I t was in Nuclear Structure that the algebraic approach f i r s t scored seriously. tal results.

The shell-model i t s e l f was imposed inductively by the experimenIn the Forties, several physicists, impressed by the "magic

numbers" thought of a classification of nuclear states modeled upon the atom, but the concensus objected to this approach. The claim was that there was no central force here, and that the actual forces were strong and thus did not f i t such a picture. In time, the phenomenological resemblances had to be taken seriously and i t was understood that the short range two-body forces manage to produce an "effective" central force. The shell-model was not treated algebraically at the operator level. In 1958, E l l i o t t [15] introduced a DDG at the semi-deductive level.

For non-

spherical nuclei, he conjectured that the dynamics should be determined by the quadrupoles resulting from the non spherical distribution of the mass. In E l l i o t t ' s SU(3), the (compact) raising and lowering operators within the DDG are given by Q(iJ) = i / 2

~ [x(i

xj)

+ P(i

Pj)] -

Tr ~ [ ]

(3.4)

where the summation is over the constituent nucleons. Together with the Angular Momenta

[

=

(3.s)

we get the SU(3). The f i r s t to introduce the concept of a SGG were Lipkin and Goshen [16], following the success of E l l i o t t ' s SU(3) in nuclei [15].

They pointed to

the Symplectic Group Sp(3,R), with the inclusion U(3) c Sp(3,R), as the SGG relevant to that problem. Thls is the group generated by the non-compact operators Bij

=

ai

aj

, and

(3.6)

B* i j

The correspondence between the groups and Quantum Mechanics is as profound as was claimed by Weyl in his introduction to the f i r s t (1928) edition of his book "The Theory of Groups and Quantum Mechanics".

41

"The algebraist of the present day considers the continuum of complex numbers as merely one "field" including

among many...This newer mathematics,

the modern theory of groups and "abstract

motivated by a s p i r i t different found i t s

real or

algebra",

is

clearly

from that of "classical mathematics", which

highest expression in

the

theory of

functions

of

a

complex

variable. The continuum of real numbers has retained i t s ancient prerogative in Physics for the expression of physica] measurements, but i t can justly be maintained that the essence of the new Heisenberg Schroedinger-Dirac Quantum Mechanics is to be found in the fact that there is associated with each physical

system a set of quantities,

constituting a non-commutative algebra..

..the elements of which are the physical quantities themselves." [17]* Indeed,

it

is

straightforward

to see, for a one-dlmensiona] oscillator

that the zero-energy is a property of the representations of SU(1,1), the appropriate SGG [18].

One extension of

these results which has made high

energy physicists hold their breath in 1968-71 and again and even more so since 1984 is the Quantum String, an oscillator with d = 26 and an additional degree of freedom ranging over a countable i n f i n i t y .

4.Theory of the SGG and the, O'Raifeartaigh Theorem The emergence of tion

of

period. of

the

Physics

of

Particles

and Fields

that

to a c r l s i s ocurred

in

in

the evolu-

the

1955-1971

The merger of Quantum Mechanics with Special Relativity in the form

Relativistic

dication after

the SGG concept was related

in

Quantum Field

the

Theory (RQFT) had just

successful description

running into

of

achieved full

Quantum Electrodynamics (QED),

apparently insurmountable difficulties

And yet i t seemed now to have failed again.

vin-

in

the thirties.

The Strong Nuclear Interactions

were thought to involve strong couplings - almost by definition.

The value

(in dimensionless "natural" units) of g=/4v

(4.1)

= 14

for pion-nucleon couplings excluded the possibility of a perturbative treat* Note that the receptivity appears to have been very limited.. introduction to the second edition (1930), he wrote, " I t has been rumoured that the "group pest" is Quantum Physics. This is certainly not true..."

42

gradually

In

the

cut out of

ment based on an expansion in powers of the coupling constant, as had been done in QED. For the Fermi Weak Nuclear Interaction, the phenomenologically well established Marshak - Sudarshan - Feynman - Gell-Mann Hamiltonian displayed a coupling GF =

10-5 mN2

(4.2)

With dimensions of an inverse mass squared, thus apparently excluding any prospects of renormalizability. There also appeared to be some more basic d i f f i c u l t y with the formalism i t s e l f , for theories with non-linear interactions, such as the Yang-Mills (YM) model, suggested around that time.

The algebraic elegance of this

theory, extending the Weyl idea of a local gauge invariance to non-Abelian Lie groups, attracted many workers in the f i e l d .

However, i t seemed

impossible here to preserve Unitarity off mass-shell.

This was the result

of a clash between RQFT, based on GL(4,R) - the f i e l d aspect - and the Quantum Mechanical Hilbert space - the particle aspect.

For instance, the

YM potential (the connection) AI~ has four space-time components for every value of the "internal" index i .

On the other hand, we have effectively

known for centuries that l i g h t , i . e. photons, have only two physical polarisations; and the same is true of the YM quanta and as a matter of fact of any non scalar massless eigenstate of the Poincar~ group. On mass shell this is indeed taken care of by the equations of motion, but off-mass-shell the redundant components stay.

This is a result of the clash between the

two dominant group structures in RQFT: the Poincar~ group, whose eigenstates form the Hilbert space of particles, acting on mass shell almost by d e f i n i tion; and GL(4,R), the linear subgroup of the Diffeomorphisms on R4, whose eigenvectors define the classical and Quantum f i e l d s . In QED, i t was possible to deal with these redundant components in a simple way. For the YM case, however, this could not be done as the nonlinear part of the YM hamiltonian coupled physical to unphysica| components inextricably.

The result was an apparent loss of Unitarlty off-mass-shell,

and this put RQFT in bad odour throughout the 1955-1971 period.

We note

that the issue of Unitarity was already solved by Feynman in 1962, and improved upon by De Witt and Faddeev with Popov, but this was disregarded,

43

once RQFTwas "set aside", u n t i l the entire renormalizatlon program was completed. Meanwhile, the "S-matrix" approach was favoured, i . e. dealing with the Hilbert space states only, discarding the f i e l d operators altogether. Unitarity is preserved at every step; in fact i t represents the only constraint on intermediate-states insertions. An S-matrix diagram may correspond to an i n f i n i t e sum of Feynman diagrams. As a result, i t was natural to look for new ways of representing dynamlcs, so that only the Hilbert space states become involved. I d e a l l y , i t ought to be algebraic.

Gell-Mann's suggestions of using the Algebra of the

SU(3) currents and of t h e i r SU(3)xSU(3) Chiral extension, then even the i n f i n i t e algebra of the components of their current-densities, were very successful and revived the Matrix-Mechanics approach, after 40 years of "Schroedinger dominance". The SGG program aimed at further dynamics through sum-rules and other algebraic characterization. I t f u l f i l l e d the on-massshell and S-matrlx conditions, and the a r t i c l e s [11-12] can be considered as the launchin9 of a program [13]. Another feature displayed by the hadrons were the Regge sequences, bands of excitations with JAJJ = 2, resembling the nuclear rotational or vibrational bands. Originally, the assumption had been that the hadron excitations were pionic in nature, and the Regge poles were thought to correspond to Yukawa potentials; in that case there would have been very few recurrences, the trajectory turning back very fast in the complex angular momentum plane.

Experimentally, they seemed closer to the Harmonic Oscillator model,

a fact that f i t t e d an extended structure idea. Both the Quark Model and the Regge excitations thus pointed to hadrons as extended structures.

SU(6) and the ]ike seemed to indicate that the

internal symmetry U could be combined with the Poincar~ group P. This meant that the "comprehensive" group G is non-compact, and thus carries i n f i n i t e unitary representations - again f i t t i n g the experimental bands. The fact that the hadron masses obey the Gell-Mann - Okubo formula implies that the Strong Hamiltonian, a generator of P, does not commute with

44

the Internal group U.

I t was thus possible to visualize an algebraic struc-

ture in which this would be a feature of that comprehensive group G and the manner in which SU(3) would be s i t t i n g in i t [19].

Several schemes of that

nature were suggested, sometimes with "catastrophic" results, such as the non-commutativity between the Poincare and Internal subgroups of G spreading over Isospin and Spin. Rotating or accelerating a neutron might turn i t into a lambda hyperon... Various no-go theorems were proven. The most comprehensive and precise theorem is due to L. O'Raifeartaigh [20]. He proved that there can be no non-trivial discrete mass-spectrum for a f i n i t e G, assuming G is simple and contains P. Y. Dothan [21] identified the role of SGG as symmetries, f i t t i n g the Noether theorem. They represent symmetries with an e x p l i c i t time dependence (or in a r e l a t i v i s t i c framework, coordinate dependence), similar to the Lorentz boosts. The latter are symmetries of the S-matrix, and yet they do not commute with the Hamiltonian: Moi = xi H - t Pi,

d/dt Moi = B/Bt Moi +i [ H , Moil = 0

(4.3)

The SGG generators G (p,q;t) obey the commutators i [ Gi(p,q;t), Gj(p,q;t) ] = i cij k Gk(p,q;t)

(4.4)

ia/Bt Gi

(4.5)

[ H , Gi ] = 0

B/Bt Gj (p,q;t)

=

mkj

Gk (p,q;t)

(4.6)

Equation (4.6) can be rewritten, using (4.5), as [H , Gj ]

= i mjk

(4.7)

Gk

the Hamiltonian maps the algebra onto i t s e l f [21]. The method was generalized to supergroups in the context of

45

the superstring.

5.Nuclear Physics and the Quadrupole Groups We have seen in #3 that E l l i o t t ' s SU(3) involved quadrupole operators in the description of the deformed nuclei.

The SGG according to Lipkin and

Goshen is Sp(3,R) [16]. There i s , however, another p o s s i b i l i t y .

Reference [11] made a different

suggestion for deformed nuclei: the non-compact SL(3,R) has unitary (ladder) representations with IAJI = 2, where angular momentum J is identified with the SO(3) c SL(3,R) , the maximal compact subgroup. The only m u l t i p l i c i t y - f r e e representations of that type are: Jmin = 0 :

j= 0 + 2 + 4 +...

o ER

(5.1)

Jmin = i :

j= I + 3 + 5 + . . .

o ¢ R

(5.2)

where o is the second quantum number defining the representations. More N

recently, i t was shown that the double-coverin9

~

SL(3,R) has a unique

m u l t l p l i c i t y - f r e e unitary representation [22,23]. Jmin = i / 2 :

j= I/2 + 5/2 + 9/2 + 13/2 + .. o=O

(5.3)

The algebra commutation relations can be checked [24] by measurements of the moments of i n e r t i a etc. There are altogether three p o s s i b i l i t i e s with respect to the quadrupoles. The angular momentum subgroup obeys the same commutation relations in all three: (Ji , J j ] :

i s i j k Jk

(5.4)

and the quadrupoles behave as a J=2 representation (a,b:1..5) {Ji ' Qa] = i Xiab Qb

(5.5)

46

For the Q , there are three possible commutators,

a

{Qa ,Qb ] =

i aabj

[Qa ,Qb ] =

0

(5.6)

Jj

(5.7)

[Qa ,Qb ] = - i aab3 Jj

(5.8)

I t is thought [25] that the 5U(3) commutators (5.6) f i t the case where the quadrupole contribution is due to the valence shell ( i . e. a peripheral deformation).

The SL(3,R) commutators (5.8) correspond to "core" contribu-

tions. The Abelian vanishing commutator (5.7) is encountered in cases where both regions of the nucleus contribute, but cancel each other's right hand side in the commutators. All three systems appear together as different subgroups of the Goshen-Lipkin Sp(3,R), and i t is very instructive to check on these commutators experimentally and reconstruct the Sp(3,R) SGG from the measurement of these matrix-elements. An alternative route that has been followed in the study of nuclear structure is due to Arima and lachello [26]. Effectively, i t is as i f they truncate the sequence in (5.1). Note that the ladder representations of the non-compact group generally appear as the limit of the t o t a l l y symmetric representation of the corresponding compact group, resulting from a total symmetrization of the product of n ~ - fundamental representations of that compact version. For SL(3,R), the representation (5.1) is the n ~ ® limit of the SU(3) Young tableaux

I I

I I I

I I I I

I I I I I

...

(5.8)

Arlma and lachello use su(3) rather than sl(3,R) and stop at the 2nd tableau in the sequence, instead of continuing to n ~ -

. They thus find a

6-dimensional j = 0 + 2 unirrep of SU(3). The truncation thus implies going from i n f i n i t e to f i n i t e unirreps, from non-compact to compact group or algebra. Arima and Iachello then further postulate as a symmetry the full

47

U(6) that is carried by the 6-dimensional representation space. The Interacting Boson Model (IBM), as further developed by lachello and collaborators, has been very useful in understanding certain types of nuclei. The U(6) is reduced over several different chains, depending again on the physics of the relevant nuclei; 0(3)

=

su(3)

:

u(6)

0(3)

=

0(5)

=

u(5)

~

u(6)

0(3)

,

o(5)

,

o(6)

:

u(5)

(5.9)

The model has been extended to the Supergroup U(6/1), with a good experimental f i t . More recently, the SGG has been used in the study of the scattering [27].

6.Hadrons For the hadrons, we encounter two d i f f i c u l t i e s : the O'Raifeartaigh theorem [20] positing that the only possible SGG containing the Poincar~ group as a subgroup is an i n f i n i t e group; and the fact that using unitary infinite-dimensional representations of the SGG automatically implies unitary ( i n f i n i t e ) representations for the Poincar~ and Lorentz subgroups. In that case, boosting a proton, for instance, would transform i t into a higher spin excitation, adding at the same time to the mass or potential energy (as M2 ~ J); but we know that the physical proton behaves like an "elementary" particle, always preserving i t s spin.

A physical Lorentz boost Just acce-

lerates the proton with purely kinetic energy. The two original suggestions for a hadron SGG have nevertheless been pursued, using different methods to bypass these d i f f i c u l t i e s . Barut and Bohm [12] have used S0(3,2) and S0(4,2). The dynamics involve constraints, which may be a way of evading the O'Raifeartaigh theorem. S0(3,2) describes a "dumb-bell" and may thus f i t hadrons as extended bodies, at least for the quark-antiquark case. The f i t with experiment is good [28]. As to the boost d i f f i c u l t y , i t is claimed that the composite proton may indeed absorb energy from the boost and t r a n s i t to an excited state; a l t e r n a t i v e l y , one assumes that the Lorentz subgroup here is not the complete Lorentz group.

48

The DGN paper [11] contained two phenomenological suggestions for the hadrons. The U(6) x U(6) of the constituent quarks' spin-unitary spin invariance is embedded in U(6,6) as a SGG. These systematics should represent some version of the "Symmetric Quark Mode]", but this SGG has not been investigated to date. The second conjecture relates to the Regge trajectories, assigned to the ladder representations of 5L(3,R) as in (5.1) and (5.2). The dynamical origin of these excitations is assumed to derive from quadrupole excitations of the hadrons as extended structures.

In fact, the

non-compact operators of SL(3,R) are the pulsation-rates, the time derivatives of the gravitational quadrupoles d/tit

I m r(i

rj)

dv

=

Qij

(6.1)

The original d i f f i c u l t y with SL(3,R) was the (wrong) impression, at the time, that the ]inear groups SL(n,R) had no spinor, i . e. double-valued, representation; i . e . a representation reducing into a sum of double-valued representations of their maximal compact subgroup S-~)(n,R). The representation (5.3) was constructed in 1969 [22] while I was s t i l l working with D. W. Joseph on the S~(3,R) scheme for the hadrons, and showed that SL(3,R), at ]east, had i t s spinors, a mathematical result; however, for the physical apP]Ication we were interested in at the time, the fact that we had also Proven that there was no similar multiplicity-free unirrep starting with Jmin = 3/2

appeared to rule out the entire approach, since the best known

nucleonic excitation is the j : 3/2 , I : 3/2

Fermi resonance, with well-

established Regge recurrences. In 1977, working on Gravity and Supergravity, I realised that the spinor representation (5.3) whose existence I knew by construction also imp]ied that the classical statement in every text book on General Relativity that "there are no spinor or double-valued representations of GL(4,R)", was simply wrong. The statement is given as reason for the introduction of tetrad frames to represent spinor fields. I proved [23] that the doublecovering exists for all GL(n,R). The entire issue has been recently reviewed [29] and we refer the reader to that article for illustrations of the classical errors in this subject.

49

As a result, the DGN program was revived, this time in the form of a field theory. The unirreps [30] of the double-covering G[(4,R) have provided for new types of fields: just as ordinary tensor fields carry f i n i t e , nonunitary representations of GL(4,R), so can the i n f i n i t e "bandor" representations form new tensorial or spinorial infinite-component fields ("manifields"). Both fundamental d i f f i c u l t i e s with SGG are thereby resolved: (I) a field theory involves an i n f i n i t y of degrees of freedom and therefore escapes the O'Raifeartaigh theorem; (2) the Lorentz boost excites the momenta and generates kinetic energy through the use of an inner automorphism [29,30] of GL(n,R), A = exp {(7/4) Too }, where Too is the symmetric tensor-shear generator's time-time component, an angular momentum singlet. The unirreps can then be redefined in a non-unitary way, with the Lorentz SO(1,n-1) represented non-unitarily by f i n i t e representations, just as in ordinary tensor fields (in the unitary representations of GL(n,R) N

~

these would represent the compact subgroup S0(4)). The scheme f i t s experiment nicely as a SGG [31,32]. At the same time, i t has important physical implications. The interaction of hadrons with Gravity, for instance, can be described in two ways, equivalent in principle. Either one stays at the "fundamental" level, i . e. quark fields with the QCD Lagrangian, with all derivatives becoming covariant-derivatives for Gravity - or alternatively, one can work at the "phenomenological" level. In that case, we treat the nucleon directly, as a hadron manifield. The derivatives in this manifield's kinetic energy term become i n f i n i t e covarlant derivatives for Gravity. QCD then does not appear at a l l , since the hadron manifield is already the effective outcome of QCD and the quarks. The interaction with Gravity has thus been studied in particular [33], including the use of the manifield to represent either an anholonomic GL(4,R) in Metric-Affine Gravity or a world-spinor in Riemannian or Affine Gravity . In the latter case, the manifield supports a non-linear realization of the diffeomorph!sm group double-covering DT~(4,R), as functions over the matrices of the linear subgroup GL(4,R). Back in 1965, another attempt to resolve the Hadron spectrum by an i n f i nite SGG, to overcome the O'Raifeartaigh theorem, was based upon the local current algebra [34]. Taking the matrix-elements of this (Kac-Moody like, we

50

would now say) i n f i n i t e algebra between infinite-momentum states answers all axiomatic d i f f i c u l t i e s . Lorentz covariance is guaranteed by an algebraic constraint, the "angular condition". The representations of this algebra were studied and classified [35], but the consistency with the angular condition represents a major computational d i f f i c u l t y and the program has been abandoned. I t is worth noting that the successes of the Algebra of Currents [36] were important in the emergence of the entire SGG program in the Physics of Particles and Fields in the mid-sixties., They revived in practice the Matrix-Mechanics approach. The Adler Weissberger sum rule is in the s p i r i t of the Thomas-Reiche-Kuhn sum-rule of the twenties [37].

7. Dual Models as Hadron SGG Since 1984, i t appears that the 1965 SGG program may indeed have set the right course. The program's evolution, as often happens in science, was somewhat roundabout, and yet the Quantum Superstring of the late eighties is very much the realization of the original SGG idea [11-13,16], and this was consciously so at several stages of the development of the model. I t evades the O'Raifeartaigh theorem through the use of an i n f i n i t e algebra and is thus capable of providing algebraically for the mass-spectrum, for instance. The main surprise, however, is that this is not the spectrum of hadron states we had in mind in the sixties, with SU(3) and i t s extensions.

The only

mass-shell states we identify in this spectrum are the massless Graviton, With spin J=2, and (perhaps) some YM quanta (with J=1). And yet the conjecture is that the algebraic systematics and operators do represent some very fundamental quantities related to either quarks and leptons or to their constituents at an even more basic level of compositeness. The Quantum Superstring is now a strong canditate for a unified theory (TOE - "Theory of Everything,,) including (hopefully f i n i t e ) quantum gravity. Work on this model has grown into a full discipline. The S-matrix amplitudes do appear f i n i t e , even though this has as yet not been proven to all orders.

For

further study of the string, we refer the reader to the various textbooks [38] and reprint volumes [39, 40] and to our review of i t s SGG features [41]. Present work is dedicated to searching for non-perturbative presen-

51

tations. I t should be noted, however, that the truncated massless sector of the theory is thought to correspond to a Relativistic Quantum Field Theory. I t has been pointed out [42] that this would imply the simultaneous existence of a renormalizable RQFT of gravity.

8. A Digression into Quantum Gravity. Work on the renormalization of the RQFT of gravity has yielded [43], in a serendipituous manner, a possible common explanation for (a) the ~ N

emergence of the SL(3,R) of Regge recurrences in hadron systematics (section 6), (b) the relatively good picture of hadron dynamics which was provided by the 1967-1974 Dual Model or string (section 7), and (c) the Arima-Iachello "IBM" and other dynamical symmetries in nuclei (section 5). Accordingly, we review some results relating to the renormalization program for Quantum Gravity, necessary for the derivation of our new results. I t has been established that Einstein's gravity is perturbatively non-renormalizable [44]. The final perturbatively renormalizable r e l a t i v i s t i c quantum field theory (RQFT) of gravity, i f i t exists, should therefore d i f f e r from Einstein's in the quantum region ( i . e . Planck energy), though i t should reduce to that theory in the low-energy (AX~Iplanck) domain. The main obstacle to the renormalizability of Einstein's theory consists in the dimension d = 2 of Newton's constant and the resulting need for new counter-terms in each order of the perturbation series.

Alternatively, the

d i f f i c u l t y can be viewed as deriving from the d : 2 of the Einstein-Hi]bert Lagrangian, linear in the curvature. This should be compared with the dimensionless coupling and d = 4 curvature-quadratic Lagrangian of a Yang-Mills gauge theory, guaranteeing cut-off independence of S-matrix amplitudes and a f i n i t e set of counter-terms. The addition (to the scalar curvature) of d : 4 terms quadratic in the curvatures, for the Einstein quantum Lagrangian, has been shown [45] to lead to a power-counting renormalizable theory. This is due to the 1/p ~ graviton propagators, a result of the Riemannian constraint DogNv = 0 relating the connection Fa~v to the metric gnu' so that the (8F) and (FF) in the cur-

52

vature R become ~ (B2g) and (Bg)2, making the R=-type terms yield 11p4 propagators. Quartic propagators can be shown, however, to contain ghosts, and this renormalizable theory is non-unitary. An analysis of the most general Such additions to Einstein's Lagrangian [46 - 48] has resulted in some combinations that were shown to be unitary; however, power-counting renormalizability is lost. For the sake of completeness we should mention that we have recently suggested a model [49] in which the Lagrangian has d = 4 and the coupling is dimensionless. The theory is based on gauging the linear group GL(4,R); i t reduces in the symmetric l i m i t to some linear combination of the irreduCible components of the SKY [50] Lagrangian, quadratic In GL(4,R) curvatures. A spontaneous symmetry breakdown (SSB) is then triggered by two Goldstone-Higgs fields and the Einstein Lagrangian then also emerges from a d = 4 term, with a "soft" effective Newton constant. The Riemannlan condition is also a low-energy effective result of the SSB. The connection is thus independent of the "pre-metric" gab(x) which is an SL(4,R) gaugedependent symmetric tensor (until i t becomes the LE effective metric, as a result of SSB). This theory has been proved to be renormalizable [51].

9_L-An Effective "StronQ" Gravity from QCD We now terminate our excursion into Quantum Gravity and return to the origin of the hadron and nuclei SSG. The "fundamental" dynamics have to be those of Quantum Chromodynamics (QCD). The non-perturbative features of that theory have however made i t d i f f i c u l t , in almost any situation, to apply i t exactly, except for the (perturbative) asymptotic domain. Even before the rise of QCD and throughout the e a r l i e r stages in the evolution of the theory, an ad hoc "strong gravity" hypothesis [52] was t r i e d , in which the fo meson (with J=2+ and a mass of 1270 MeV) was given a central role as the "strong graviton". In the l i g h t of the f ° ' s quark-antiquark structure, i t s postulated gauge-field nature can now at most be regarded as "phenomenological". Moreover, the results were inconclusive at the time. Using the lessons derived from the work on the renormalization of gravity, we now construct an "effective" stronQ Qravity, derived d i r e c t l y from QCD. I t reproduces most features of the strong-coupling region in QCD: dynamical

53

color confinement, the successes of the hadronlc string, the structure of the hadron spectrum for baryons and mesons, (including Regge trajectories) and scaling. In addition, i t explains the low-energy nuclear physics quadrupole-lnduced spectra and "predicts" the highly successful Interacting Boson Model ("IBM") of Arima and lachel]o [26]. Our assumptions reduce to the SU(3)c saturation properties, i.e. that the color-singlet configurations are the lowest lying ones. Since the hadron lowest ground states are colorless and in the approximation of an external QCD potential (in analogy to the treatment of the hydrogen atom in the $chroedinger equation), the hadron spectrum above these levels w i l l be generated by color-singlet quanta, whether made of dressed 2-gluon configurations, 3-gluons,..No matter what the mechanism responsible for a given flavor (or nuclear) state, the next vibrational, rotational or pulsed excitation corresponds to the "addition" of one such color-singlet multigluon. In the f u l l y r e l a t i v i s t i c QCD theory, these contributions have to come from summations of appropriate Feynman diagrams, in which dressed n-gluon configurations are exchanged. We rearrange the sum by lumping together contributions from n-gluon irreducible parts, n = 2, 3,.. = and with the same Lorentz quantum numbers. The simplest such system will have n = 2. The color singlet external f i e l d can thus be constructed from the QCD gluon field as a sum (qab is the SU(3) metric, dabc is the t o t a l l y symmetric 8 x 8 x 8 ~ I coefficient)

(9.1)

Bag Bbv qab + Bag Bbv BCo dabc + " -

In the above, Bag is the dressed gluon field. We seggregate Nau, the zeromode of the field. With Fagv as the curvature or field-strength, Bag = Nag + Aag ,

F(N) = 0

(9.2)

We now rewrite the 2-gluon configuration as (9.3)

Ggv(x) = Bag Bbv nab which looks very much l l k e a spacetlme,,,metric.

I t is an e f f e c t i v e

spacetime

metric. I f (9.3) is the dominating configuration in the excitation systema-

54

tics and since Lorentz invariance forces this "metric" to obey a Riemannian constraint

DoGpv = O,

(9.4)

where Do is the covariant derivative of the effective gravity (the connection is a Christoffel symbol constructed with the metric (9.3)). The separation of the f l a t part of Bap reproduces the separation of a tetrad eap(x) = 6ap + fap(x) into a f l a t background and the quantum gravitational contribution. As a result, Gpv(X) i t s e l f can be separated similarly. We note two POints: (1) Out of the 10 components of Gpv in (9.3), the 6 that survive after the 4 Riemannian constraints have spin/parity assignments JP = 0+,..2.... +. Thls Suggests a relationship with the IBM model systematics [26], in which the fundamental excitation of isospin-symmetric nucleon pairs has these quantum numbers, an observational result. Note that the absence of dipolar excitations is in i t s e l f an indication that a 9ravity-like force is involved [53]. Algebraically, G~v(x) carries the lO-dimensional irreducble representation of GL(4,R). The non-relativistic subgroup is SL(3,R). Under that group, the 0+ and 2+ states span together the 6-dimensional irreducible representation (non-unitary for sl(3,R), unitary for su(3)) and the couplings of the "effective" gravity to nucleons will be the same (they are GL(4,R)-unlversal in a gravitation-like theory). (2) An effective Riemannian metric induces Einsteinian dynamics. However, our correspondence is between low-energy (IR) QCD, with i t s strong Coupling, and the hlgh-energy (UV) strong coupling region of our effective gravity (and not with the weak coupling Newtonian l i m i t ) . Gravity becomes a St~rong force in the quantum regime (two Planck mass particles attract each Other gravitationally 10~° x 10I° = 103~ times stronger than two nucleons). The quantum gravity lagrangian includes curvature-quadratic counterterms generated by the renormalization procedure and corresponds to the (effective) invariant action, m

lin v = - fd'x I/-G (a R]jV RiJV _ B R2 + Y K-2 R)

55

(9.s)

This is the Stelle [45] action, when used for true gravity. As we saw, i t was shown to be renormalizable, a feature befitting this application, since QCD is renormalizable and any piece of i t should preserve the finiteness feature; true, (9.5) is not unitary, but this is no problem for this application: a "piece" of QCD should not be unitary, QCD being an irreducible theory. Stelle's main result, however, was to show that renormalizability is caused by I/p ~ propagators. But I/p ~ propagators are dynamically eqivalent to confinement [54] and thus f i t well in the present context. Note that the 1/p ~ propagators cause any colored state to be bound and confined; adding a quark or gluon to a color-singlet hadron will polarize the vacuum, creating \ pairs until the configuration becomes color-neutral. In recent years, quadratic lagrangians similar to (9.5) have been investigated classically in the context of the Poincar~ Gauge Theory of grav i t y [55]. The exact solutions display, aside from the Newtonian potential M/r, a component behaving ~

r 2, dominating the strong-field limit and ori-

ginating in the curvature-squared terms as in (9.5). I t has been shown [56] that the String, as a gravitational theory, is equivalent to an action such as (9.5), i . e . with quadratic counterterms. Also, the embedding of the String in curved "target" spacetimes has been interpreted [57] as a series of constraints on the manifold's geometry, preserving the cancellation of the conformal anomaly. Such constraints are regarded as replacing Einstein's equation in fixing the geometry of the target space - and their lowest terms are also those of (9.5). Our ansatz thus explains the successes of hadronlc strings in 1967-74. The external field Ggv(X) transforms under Lorentz transformations as a (reducible) second rank tensor f i e l d , with Abellan components, i . e . [Ggv,Gpo] = O. Algebraically, Gpv and the Lorentz generators form the algebra of TIO(O)SO(1,3), an inhomogeneous Lorentz group with tensor "translations" (the symbol (o) denotes a semi-dlrect product). This is a classical r e l a t i v i s t i c algebra. For the quantum case, the gluon field is expanded in creation and annihilation operators and we can write, Ggv = Tpv + Ugv , where the quadrupolar excltatlon-rate is glven by Tgv = qab SdE [aag+(k) abv+(k) e21kx + aag(k) abv(k) e-21kx]

58

(9.6)

for (infinite) gl(3,R) non-compact excitation bands [11], whereas Upu = qab Sd~ [aap+(k) abv(k) + aap(k) abu+(k)] •

(9.7)

generates f i n i t e u(3) spectral multiplets. We have made use of the canonical transformation: [aap(k)

+ 112 NaB eikx ] *

[aap+(k) + I12 NaB e-ikx]

aap(k) ;

(9.8a)

~ aap+(k)

(9.8b)

Using [aap(k),abv+(k')] = 6ab 6Nv 6(k-k'), one verifies that the operators Tpv and Upv, together with the operators SNv : qab

[dE [aap+(k) abv(k) - aap(k) abv+(k)]

(9.9)

close respectively on the gl(4;R) and u(1,3) algebras. Note that the largest (llnearly realized) algebra with generators quadratic in the ap+, ap operators is sp(1,3;R), where the notation "1,3" implies a definition over Mlnkowski space.

u(1,3)

su(1,3) *

I

I

Sp(1,3;R) ~I~ gl(4;R)* ~ ~ ~ ~ ~ sl(4;R) ~ ~ ~I ~ ~ so(1,3)

I

I

i~ ti0(o)so(1,3)~ ~ ~ tg(o)so(1,3)~I

(9.10)

gl(4;R) is a Spectrum-Generating Algebra for hadrons [31,32]. We now return to the expansion (9.1). The s1(4,R) is generated by the 2-gluon term. For 3-gluon and n-nucleon exchanges, the corresponding algebras do not close and generate the full Ogievetsky algebra of Minkowski space Diffeomorphisms [58]. The maximal linear subalgebra of diff(4,R) is gl(4,R). The remaining generators (i.e. diff(4;R)/gl(4,R)) can be expliCitly realized in terms of the gl(4,R) generators [59], both for tensors and for spinors. In our case, this would involve functions as matrix elements of the representation of our generators Tpu and Spy in (9.6 - 9.8). As in 57

General Relativity, the entire "Gpv-covariance" can be realized in terms of the invariant action given in (9.5). Note however that diff(3,1;R) can also be represented linearly. In that case, i t w i l l involve i n f i n i t e , ever more massive, repetitions of the representations of s1(4,R). In either case, we find that using sI(4,R) takes care of the entire sequence in (9.1). The inhomogeneous versions of the algebras in (9.10) i . e . their seml-direct product by translations t 4, are relevant to the hadron Hilbert space.

In the

case of u(1,3) in (9.10), when selecting a time-like vector (for massive states), the s t a b i l i t y subgroup is the compact u(3) with f i n i t e representations - as against the non-compact gl(3,R) for sl(4,R).

This f i t s with

the situation in nuclei, where symmetries~such as the u(6) of the IBM model [26] are physically realized over pairs of "valency" nucleons outside of closed shells. There is a f i n i t e number of such pairs, and the excitations thus have to f i t within f i n i t e representations. The u(1,3) SGA of (9.10) is related to both the highly-successful IBM u(6) symmetry and to the sp(3,R) systematlcs [16,60] with i t s (Elliot) su(3) [15] and DGN [11] sl(3,R) subgroups for deformed nuclei [61]. Particle physicists who have read through the last section may well be puzzled. First, our assumption that the exchange of n-gluon terms (9.1) is the relevant contribution in QCD seems to run counter the accepted wisdom that regards meson exchange, i . e . quark-antiquark pair exchange, as the main force in nuclear physics. This is how Yukawa conceived of his meson, and present treatments of nuclear forces Just l i s t parametrized contributions of the exchange of the qo ~o ~+ ~- go pO p+ p- ¢o and higher spin mesons. Moreover, to the extent that we build on a 2-gluon configuration, lattice calculations seem to indicate that such a gluon pair appears as a "glueba11" bound state, with a mass of roughly 1.5 GeV. This would imply a range much shorter than that of the ~ etc.. These arguments are however irrelevant. We are dealing with a gluon pair whose propagation is described by 1/p 4 propagators. This is not the exchange of a particle. Such an exchange would have corresponded to the K|ein-Gordon equation and 1/p 2 propagators. At "best", the 1/p ~ propagators describe the difference between two particle exchanges, or more precisely, a particle exchange and a ghost exchange. The element of QCD in i t s IR region whose effects we have studied is not a particle exchange. I t is a gravity-like

5B

effect with I/p ~ confining propagators. I t corresponds to the effects of a massless pseudo-graviton. Massless - because like in gravity, the GNv gauges the diffeomorphisms. At f i r s t sight, the expressions in (9.1) are not gaugeinvariant; this is true in the sense of QCD, but in terms of the d i f feomorphisms, (9.5) is a gauge-invariant Lagrangian.

10. The Effective Strong Gravity in Nuclei Returning to nuclei, we look again at the ansatz defining the IBM model U(6) symmetry: a fundamental 0+ and 2+ quasi-degenerate set of excitations. Such a set would never arise from the exchange of the lighter quarkantiquark pairs (i,e. mesons with spins 0 and 1). This would not generate quadrupole excitations. Skipping the i - dipole is generally intimately connected with the (tensor) gravitational potential [53]. The 2+ mesons such as the f°(1270) are represented by tensor fields and could thus do i t , but their range is too short. Our QCD-induced effective gravity as described by (9.5) appears to supply the correct answer. When applying " e f f e c t i v e g r a v i t y " to nuclei, i t is natural to assume that closed shells assume the role of "vacua", as r i g i d structures. "Graviton" e x c i t a t i o n s should then be searched for in the valence nucleon systematics. In t h i s sector of even-even nuclei the GNv quanta can indeed excite nucleon pairs; the overwhelming preponderance of proton-proton and neutron-neutron over proton-neutron pairs can be f u l l y explained in terms of the Clebsch-Gordan c o e f f i c i e n t s in the d i r e c t channel. Dynamically for one Pair, we assume that the p a i r i n 9 force i t s e l f

is due to the exchange of a

"strong graviton" between the two nucleons. The paired system then displays f u r t h e r excited states with the absorption of additional such quanta. The picture now is of an external f i e l d supplying these quanta, perhaps l i k e the role of the electromagnetic f i e l d in the hydrogen atom in the Schroedinger equation treatment. I t is thus natural that proton pairs and neutron pairs should have the same energy difference between O+ and 2+, since these are due to the same flavor-independent component of QCD - precisely for the same reason that the Eightfold Way (flavor) SU(3) invariance is due to the flavor-independence of QCD.

59

In estimating the amplitude for such pair excitations, we note that the Z-gluon exchange here is dominated by a pole in the d i r e c t channel at k2 = 4 m2, m the nucleon mass. Remembering that "effective strong graviton" excitations are also seen in hadrons, where they cause the AJ = 2 sequences of resonances along Regge t r a j e c t o r i e s , we get a value for the effective 1/K 2 in (9.5). I t is given by the slope of the Regge t r a j e c t o r i e s , roughly 1/K = ~ 1 (GeV) =.

We now return to the Hamiltonian corresponding to (9.5), as translated into our effective dynamics. terms Br + F.F;

The curvature R corresponds in true g r a v i t y to

the Christoffel formula gives F ~ G-~BG and

R ~ B(G-~BG) + (G-~BG)(G-~BG). Here G stands for G~v, b and b_+. represent the destruction and creation of a 6-dimensional "strong graviton" quantum, H = (I/M=)fdk {C1 (k'/K 2) (b+.b) + + C2 (k21K2) (b+.b)(b+.b) + + A1 k 4 (b+.b)(b+.b) + A2 k 4 (b+.b)(b+.b)(b+.b) + + A3 k 4 (b+.b)(b+.b)(b+.b)(b+.b).

(10.1)

The coefficients Ci and AI respectively contain dynamical information relating to the y and 4,B terms in (9.5), an approximation for the n o n l i near effect of the V-G, the reduced matrix element for the coupling to the nucleon pair, etc. M is a mass parameter that takes care of the dimensionality.

We select M to be of the order of the impacted system, i . e . the

valence nucleons, M ~ 20 GeV. For k ~ we use the dispersion r e l a t i o n resu]t mentioned above, i . e .

~

4 (GeV)=. Using our strlng-Regge r e s u l t (I/K 2)

~ 1 (GeV)2, and assuming the C coefficients to be of the order of u n i t y , we get for the f i r s t term ~ .5 MeV . This is of course by far no more than an order of magnitude check, but i t seems the values are roughly in the r i g h t ballpark. The values w i l l decrease for ]arger M. Our (10.1) is of course equivalent to the IBM Hamiltonian with higher order terms. One l a s t comment with respect to the symmetries of nuclei. Some time ago Lipkin conjectured [62] that the sp(3,R) set of algebraic operators and the

60

Arima-lachello IBM generators involve the same creation and annihilation elementary operators, linearly in the su(3) or sl(3,R) and quadratically for the IBM. Lipkin's idea is f u l l y vindicated in our derivation: the sp(3,R) oscillators correspond to gluons, those of the IBM describe "strong gravitons", i.e. they are quadratic in the gluons. References I. y. Ne'eman, Nucl. Phys. 2_66(1961), 222; M. Ge11-Mann, unpublished. 2. H. Goldberg and Y. Ne'eman, Nuo. Cim. 27 (1963), i ; M. Gel1-Mann, Phys. Lett. 8 (1964), 214; G. Zweig, unpublis-hed. 3. y. Ne'eman, Phys. Rev. ~134 (1964), 61355. 4. R.J. Oakes and C.N. Yang, Phys. Rev. Le~tt. 11 (1953), 174. 5. S.L. Adler, Rev. Mod. Phys. 54 (1982), 729.-6. M. Gell-Mann, Phys. Rev. 125--(1962), 1067. 7. G. ' t Hooft, NATOAdv. Stud. Inst. Series B59 (1980), 117. 8. W. Pauli, Z. Phys.36 (1926), 336. 9. M. Gell-Mann and Y. Ne'eman, The Eightfold Way, W. A. Benjamin, New York (1964). 9. M. Gell-Mann and Y. Ne'eman, The Eightfold Way, W. A. Benjamin, New York (1964). IO.F.J. Dyson, S~TnmetryGroups in Nuclear and Particle Physics, W. A. Benjamin, New York (1966). 11.Y. Dothan, M. Gell-Mann and Y. Ne'eman, Phys. Lett. 17 (1965), 1107. 12.A. Bohm and A. O. Barut, Phys. Rev. 139B (1965), 1107"~. 13.Y. Dothan and Y. Ne'eman, in Resonant Particles (Proc. Athens, Ohio Conf. 1965), p. 17; see also I,A.Malkin and V.I. Man'ko, Yad. Fiz., (1966), 372. 14.Y. Ne'eman, Algebraic Theory of Particle Physics, W. A. Benjamin, New York (1967), 287. 15.J.p. E l l i o t t , Proc. Roy. Soc. (London) A245 (1958), 128. 16.S. Goshen and H. J. Lipkin, Ann. Phys. ~ 6 (1959), 301. See also D.J. Rowe, "The shell model theory of nuclear collective states" in Dynamical Groups and Spectrum Generating Algebras, A. Bohm, Y. Ne'eman and A.O. Barut editors, World Scientific, Singapore (1989), pp. 287-316. 17.H. Weyl, Gruppentheorie und Quantenmechanik (1928), translated as The Theory of Groups and QuantumMechanics, Dover Pub. (1931). 18.A.0. Barut, Phys. Rev. 139B (1965), 1433. 19.A.0. Barut, in Symmetry Principles at High Energy (Proc. 1964 Coral Gables Conf.), B. Kursunoglu and A. Perlmutter eds., W.H. Freeman, San Francisco (1964), p. 81. 20.L. O'Raifeartaigh, Phys. Rev. Lett. 14 (1965) 332 and 575. 21.Y. Dothan, Phys. Rev. D2 (1970) 2944~for applications, see for instance I.A. Malkin, V.L. Man'k-'o and D.A. Trifonov, Nuo. Cim. 4A (1971), 773. 22.D.W. Joseph, "Rep. of the Algebra of SL(3,R) with lAJl~--2", Univ. Nebraska preprint, unpub. (1969). 23.Y. Ne'eman, Ann. Inst. Henri Poincar~, Sec. A 28 (1978), 369; Proc. Nat. Acad. Sci. USA 74 (1977), 4157. 24.0.L. Weaver and L.C. Biedenharn, Phys. Lett. 32B (1970), 326; R.Y. Cusson et a l . , Nucl. Phys. A114 (1968), 289. 25.G. Rosensteel and D. J. Rowe, Ann. of Phys. 123 (1979), 36; 126 (1980), 198 and 343. 26.A. Arima and F. lachello, Phys. Rev. Lett. 35 (1975), 1069; Ann. Rev.

61

Nucl. Part. Scl. 31 (1981), 75; F. lachello and I . Talml, Rev. Mod. Phys. 544 (1987), 339. 27.Y. Alhassld, F. Iachello and J. Wu, Phys. Rev. Lett. 56 (1986), 271. 28.A. Bohm, Phys. Rev. 175 (1968), 1767; D3 (1971), 377.-29.Y. Ne eman and Dj. Si~cki, In. J.M.Ph~. A2 (1987),1655. 30.0j. $ija~ki and Y. Ne'eman, J. Math. Phys.~6 (1985),2457. 31.Y. Ne'eman and DJ. ~ija~ki, Phys. Lett. 1575"(1985), 267. 32.Y. Ne'eman and DJ. Sija~ki, Phys. Rev. D ~ 1 9 8 8 ) , 3267. 33.Y. Ne'eman and DJ. SiJa~ki~ Phys. Lett.-1-57B (1985), 275. 34.R.F. Dashen and M. Gell-Mann, Phys. Rev.-Te~t. L7 (1966), 340. 35.s.J. Chang, R.F. Dashen and L. O'Raifeartaigh, Phys. Rev. 18__22(1969), 1805. 36.S.L. Adler and R.F. Dashen, Current Algebra, W. A. Benjamin, New York (1968). 37.J. Mehra and H. R. Rechenberg, The Historical Development of Quantu~ Theory, Vol. 2, p. 246. 38.M. Green, J.H. 5chwarz and E. Wltten, Superstrln 9 Theory, Cambridge U. Press, Cambridge (1987), Vol I (469 pp.), Vo1. I I (596 pp.); see also W. Siegel, Introduction to Strin 9 Field Theory, World Scientific, Singapore (1988), 244 pp. 39.M. Jacob, ed., Dual Theory, Physics Reports reprint book series 1, North Holland Pub., Amsterdam-Oxford (1974), 399 pp. 40.J.H. Schwarz ed., Superstrings - The First 15 Years of Superstring Theory, World Scientiflc, Singapore (1985), (1141 pp,). 41.Y. Ne'eman, Found. of Phys. 18 (1988), 245. 42.A. Casher, Phys. Lett. B195 T1987), 50. 43.Dj. ~iJaEki and Y. Ne'eman, Phys. Lett. B (to be published). 44.M. Goroff and A. Sagnotti, Phys. Lett. B160 (1985), 81. 45.K.S. Stelle, Phys. Rev. D16 (1977), 953-an-dGen. Rel. Gra. 9 (1978), 353. 46.0. Neville, Phys. Rev. D~i-~'-(1978), 3535 and D21 (1980), 567T 47.E. Sezgin and P. van Nie'u-wenhuizen, Phys. Rev-T-D21 (1980), 3269 and D22 (1980), 301. 48.R. Kuhfuss and J. Nitsch, Gen. Rel. Grav. 18 (1986), 1207. 49.Y. Ne'eman and Dj. Sijacki, Phys. Lett. 82~-~ (1988), 489. 50.C. Lee and Y. Ne'eman, Phys. Lett. B242Ti-9-~0), 59. 51.G. Stephenson, Nuo. Cim. 9 (1958), ~ C.W. Kilmister and D.J. Newman, Proc. Cam. Phil. Soc. 5._77T1961) 851; C.N, Yang, Phys. Rev. Lett. 33 (1974) 445. 52.C.J. Isham, A. Salam and J. Strathdee, Phys. Rev. DB (1973), 2600; ibid. D9 (1974), 1702; ibid. Lett. Nuo.Cim. 5 (1972), 96~'~'. 53.~e e.g.C.W. Misner, K.S, Thorne and J.A. Wheeler, Gravitation, W.H. Freeman and Co. Pub., San Francisco (1973), §36.1. 54.See e.g. J, Kiskis, Phys. Rev. Dll (1975) 2178; G.B. West, Phys. Lett. 1155 (1982), 468. 55.P. Baeckler and F.W. Hehl, in From sU(3) to Gravity, E. Gotsman and G. Tauber editors, Cambridge University Press, Cambridge (1985), p. 341; see also F.W. Heh], Y. Ne'eman, J. Nisch and P. v.d. Heyde, Phys. Lett. 78B (1978), 102. 56..~-~--Deser and A.N. Redlich, Phys. Lett, 1765 (1986), 350. 57.E.S. Fradkin and A.A. Tseytlin, Phys. L ~ 5 1 5 8 (1985), 316. 58.V.I. Ogievetsky, Lett. Nuo. Cim. 8 (1973), 9~-~-. 59.DJ. ~lja~ki and Y. Ne'eman, Ann. ~hys. (NY) 120 (1979), 292. 60.G. Rosensteel and D. J. Rowe, Ann. of Phys. 123 (1979), 36; 126 (1980), 198 and 343. 61.L.C. Biedenharn, R.Y. Cusson, M.Y. Han and O.L. Weaver, Phys. Lett. B42 (1972), 257. 62.H.J. Lipkin, Nucl. Phys. A350 (1980), 16.

62

SPECTRUM GENERATING GROUPS - IDEA AND APPLICATION* Arno Bohm Center for Particle Theory The University of Texas Austin, Texas 78712

I. Introduction Motions are best described by groups. One distinguishes two kinds of motions of an extended physical object: 1) The motion of the object as a whole or the center of mass motion, and 2) the intrinsic motions of the extended object like, e.g. the rotations about the center of mass or the vibrations. The center of mass motion is described by the symmetry group. This is the conventional use of group theory in quantum physics. The symmetry group of non-relativistic quantum mechanics is the Galilei group, the symmetry group of relativistic quantum mechanics is the Poincar~ group. The intrinsic motions are described by the spectrum generating groups,[ 1] which are also called dynamical groups. [2] This is the modern use of group theory in quantum physics.[s] Different intrinsic motions are described by different spectrum generating groups and complicated intrinsic motions require a large spectrum generating group. The spectrum generating group approach thus analyzes an extended physical object in terms of its intrinsic motions. Understanding Mways means reduction to the simpler. The classical way of understanding is the reduction to the simpler objects, the constituents of the extended object. This constituent picture also prevailed in quantum physics. The way of understanding, that underlies the spectrum generating group approach, is the reduction to the simpler motion. One does not ask the question: What does it (the extended object) consist off But one asks the question: What does it do? An extended object can perform a simple motion, yet involve a large collection of constituents. In this case the collective models that use this motion picture are always more successful. The mathematical description of these collective models is given by the spectrum generating groups. The motion picture and the constituent picture have been used concurrently in many areas of physics, of which we want to consider here three: molecular physics, nuclear physics and hadron physics. Each of these areas has its standard model based on the constituent picture. In molecular physics it is the ( M a r N) body SchrSdinger equation of M electrons and N nuclei, in nuclear physics it is the microscopic manybody theory of nuclear forces; in hadron physics it is quantum chromodynamics of quarks and gluons. These standard models are in principle solvable but not of great practical value especially when the number of constituents involved is large. The practitioners analyze, low energy spectra and structure [4'5] of the non-relativistic domain in terms of collective motions, principally rotations and vibrations. The relativistic spectrum generating group approach suggests the same for hadron physics only that here the intrinsic collective motions are also relativistic. * Dedicated to Yossef Dothan, my colleague and friend.

63

We shall in the following sections give a review of the spectrum generating group approach in the three areas: molecular physics, nuclear physics and hadron physics. In section II we explain the idea of the spectrum generating groupie] using the (diatomic) molecule as an example. In section III we go over some applications of the idea in nuclear physics,[7] and in section IV we present attempts to extend this idea into the relativistic domain. If. Molecules provide the basic idea of the Spccmml Generating Group Ha. CollectiveMotions as Part of ~c Molecule The collective models for molecules are based on the Born-Oppenheimer procedure.[s] In this procedure the physical system is divided into two kinds of motions, the fast motion and the slow motion. The fast motion is the motion of the electron cloud around the nuclei; the fast (electron) variables are denoted by (p, r). The slow motion is the collective motion of the molecule; the slow m o m e n t u m and position variables are denoted by (P, ~).[9] They can be imagined as the relative variables of the nuclei, in particular if one considers a di-atomic molecule (which we suggest to do), because the electron cloud instantly follows the motion of the nuclei. The complete Hamiltonian is

ar = 7"~ + ~

+ v(~, ~) +7-~

"

(1)

The last term, the kinetic energy of the center of mass (c.m.), with Pi being the generator of the Galilei (symmetry) group is ignored. One first solves the eigenvalue problem of the sub-I~Iamiltonian p~ h ( 0 = ~ + Y(~, r)

(2)

governing the fast variables and considers ~ as a fixed parameter~ because the nuclei are almost stationary compared to the fast moving electrons: h ( 0 In; ~) = ~-(~) I n; 0"

(3a)

n denotes the electronic quantum numbers or the quantum numbers of the fast motion. To consider ~ as a fixed parameter means to assume that it commutes with P and is not the conjugate operator of P. Consequently, h ( 0 commutes with P and so also commutes with H. Thus H and h(~) can be diagonalized simultaneously:

h(0 1t~, n; 0 = ~ ( 0 1 t¢, n; ~), H I iV, n; 0 = ~N,n(~) I N, r~;~),

(3b) ca)

where

IN, n; ~) =-I tO)® t n; ~). With (3) solved for every fixed value of ~, (4) means that

(P' ) ~-~7+~n(0

l i V , - ; 0 = ~ N , ~ l i V , n;0,

64

(5)

where gn(~) is now treated as the potential energy (induced into the slow dynamics by the fast system). N stands for a set of quantum numbers, N = (j, v,...), which characterize the collective motions (j for rotation, v for vibration). Equation (5) determines the energy values Ej,~;n of the collective excitations for every fixed value of n. The Born-Oppenheimer procedure leads thus to the well known spectra of molecules, Fig. 1: The electronic levels ~n axe split into vibrational excitations Ei,~,;n(v = 1, 2, 3,...) which axe in turn split into rotational bands E#,,m(j = 0,1, 2,...). v=0

.

.

.

.

v= 3

V~

.

.

.

£2

'

2

E0,3.1

i|iii

j14 ,ja3

i

E3,1,1

Eo.i.1 ja4 j=3 v=0

ii

..........

£1

Hgure 1. Schematicsof typicalmolecularspectra. We are interested in the slow, collective degrees of freedom and the spectrum given by the solution of (5) for a fixed en(~). We will not discuss the solution of (3), and the fast motion interests us only in as much as it determines the observables [e.g., potential energy e,(~)] of the slow motion. Our observables are the intrinsic, collective variables ~,P, the intrinsic angular momentum S (which may be S = ~'^ P, but only if the angular momentum coming from the fast motion is zero). In the above, old, BornOppenheimer approximation P, ~ S are the usual canonical variables fulfilling the usual commutation relations. That something can go wrong with the old Born-Oppenheimer approximation (when the slow variables change on closed orbits) was first noticed in molecular physics.[I°] This problem was explained by M e a d and Truhlar by introducing a vector potential into the equation of motion ("molecular Bohm-Aharonov effectn). The general occurance of such induced vector potentials was then discussed by Berry.[11] If one does not consider the slowly varying parameters as fixed but as slowly changing observables,

65

a straightforward calculation shows [12] that instead of the Hamiltonian on the r.h.s, of (5) one has to use an effective Hamiltonian for the slow motion given by: H =

~72

+

(6)

where

= P - A

(7)

is a covariant m o m e n t u m and the [gauge potentiM (Berry connection) A, induced by the fast motion, is given by A,(~) = / n , ~ l i 0 These induced gauge potentials modify the canonical structure of the phase space. In particular, zrl will obey anomalous quantum mechanical commutation relations. If the fast motion is the spin ~ around the internuclear axis (, with h(~) = pa.~, " " then the induced gauge potential turns out to be the vector potential of a magnetic monopole. It t,t2] We will return to this picture below. ]:lb. The Idea of the Spectrum Generating Group We will now explain how one uses the spectrum generating group to obtain the spectra of the collective excitations for the (diatomic) molecule and how the above intrinsic variables ~', P and S are related to the generators of the spectrum generating group. We will discuss in detail only the rotational bands of Fig. 1, i.e., the energy levels for a fixed value of n and a fixed value of u. This is the energy spectrum of an extended object which is rigid (in the vibrational ground state u - O) and capable only of rigid body rotations about its center of mass (c.m). [f the object is also elastic, performing e.g. harmonic oscillationsand rotations, a spectrum generating group larger than the rotator group will have to be used. As an example let us consider an extended object [as shown in Fig. 2(a)] with a center of positive charge at Q(+), a center of negative charge at Q(_), and a dipole moment e~*-'-d. It performs rotations about its c.m. generatedby the angular.momentum operator relative to the center of mass Si (Si = Ji - (Q A P)i, where Ji, Qi,ffi are the generators of the symmetry group, which in the case under consideration is the Galilei group). The Si thus fulfill the angular momentum c.r.

[s. si] =

(9)

W h a t are the c.r. of ~i, or what group do they generate? First, it is natural to assume that ~i is an intrinsicvector operator, i.e.,

[Si,~j]-- i~ijk~k.

66

(10)

-e .,.,. A -VC

¢

)

Origin

Origin

(a)

(b)

Figure2. (a)Extendedobjectwithcenterof positivechargeQ(+ ) and centerof negativechargeQ(_). The dipole moment is e~. Due to calslraints, the relative position ~and mornentann 7r of the centers of charge need not be indeixmdent canonical variables. Ca) Vibrating dumbb¢ll consisting of two unconstrained charges. The commutator of ~i and ~i may be anything. Three simple alternatives, in which the commutator closes with the Si into a group, are: [~i,~j] = 0

[~i,~j] -" --ieijkSk [~i,~j] = +i~ij~S~

leading to a group E(3)~,s D SO(3)s, leading to a group S0(3, 1)~,s D SO(3)s, leading to a group SO(4)e,s D SO(3)s.

(lla) (lib) (lle)

The conventional assumption is ( l l a ) , commuting dipole operators, and its justification is based on the following arguments: Let the extended object be a rigid or vibrating dumbbell consisting of two masspoints, as shown in Fig. 2(b), with Q ( + ) , P ( + ) and Q ( _ ) , P ( _ ) which separately fulfill the canonical c.r.. Then, in addition to the center of mass position and total m o m e n t u m operators (which are generators of the Galilei group), Q, = ,.
3-

--

2-

o%-

4*--- 2"---

Th. (nd,O)

(nd,I)

(nd-2,0)

2"--

6 4--3+-.-- 0+..- 2¶/_

o'.-

4"--- 2+_._

or--

I-

O-

2*._

2"..-

or--

o*--.

U(5)

Fig.l. An example of U(5) dynamic symmetry (chain I) in nuclei:t1°Cd. On the left the experimental spectrum. On the right the spectrum predicted by the energy formula (2.9),I.

E (MeV]

156n,4 64 ~ 9 2

(24,0)

Exp.

~o,z)

(J6,4)

_

Th.

(1604)

(re,o)' ' m4,o) (~,2}

:_--

(18,0)

_

o÷-

10%..

5*----

°'-: i: 6~

6t - " 5*---

6.- ~=~'-

0+--

6+--

6 t-

4*.--,2*.-

Su(3) Fig.2.An example of SU(3) dynamic symmetry (chain II) in nuclei:lS6cd. On the left the experimental spectrum.On the right the spectrum predicted by the energy formula (2.9),II.

90

E (MeV)

196

Exp.

78Pt118

Th.

3-,

(6,0)

(6,0)

(4,0}

(6J)

12:'- od 0 ÷-

(6,1]

//\

(2,0)

(4,0}

(2,0)

J~

o*--

3t.. 0%-0"-.--

4"-2"--

2 ~_

2"--

2*.--

O- o*--

.o*.--

0(6)

Fig.3. An example of S0(6) dynamic symmetry (chain III) in nuclei:t~SPt. On the left the experimental spectrum. On the right the spectrum predicted by the energy formula (2.9),III.

Algebraic operators.

methods

can

Particularly

also be

important

transitions between levels.

used

to

evaluate

are operators

matrix

elements

of

inducing electromagnetic

These are also expanded into elements of g,

@ - to +~#t # Gap

(2;10)

The structure of the matrix elements is then

(2.11)

< Irrep of g I Tensor of g I Irrep of S >

and their values can be obtained in

terms

of isoscalar factors for nested

chains g D g' D ~'' D ... It i s structure

also worthwhile to remark that associated with every algebraic there is a geometric structure.

91

This

is the

structure

of the

coset space

[6].

In the case described above the geometric

particularly important, the bosons. extensively

This in

since it provides an alternative

classical,

the

or

geometric,

description

of

is

interpretation of

interpretation

collective

structure

states

has in

been

nuclei

used [7].

Classical variables can be introduced through a coherent state

IN, =. > - (st + 7. ~. dt. )N l o >

The variables u~ (c-numbers)

(2.12)

are in the coset space U(6)/U(5)®U(1).

They

can be associated with the shape of an object described by the surface

R-R 0 (1+~

The three dynamic symmetries

Y2~ (0,4))

(2.13)

(2.5) describe then three special cases of the

ellipsoid (2.13), i.e. (I) spherical shape, (II) axially symmetric deformed shape and (III) non-axlally symmetric deformed shape (7-unstable shape).

3.

DYNAMIC

SUPERSYMM

TRIES

IN NUCLEAR

PHYSICS

The concepts of spectrum generating algebras and dynamic symmetry can be applied to any algebraic structure,

not necessarily

a Lie algebra.

particular, one can make use of graded (or super) Lie algebras. again introduced for applications have

found

their

most

useful

to elementary

application

to

particle date

in

These were

physics nuclear

In

[8] but physics.

Spectrum generating superalgebras and dynamic supersymmetrles can be defined in the same way as in (I.I) and (1.2). (a)

Spectrum generating superalgebra.

This is the superalgebra g* onto which all physical operators are expanded,

H - f(G:)

(b)

G*Q ~ S*

Dynamic supersymmetric.

92

(3.1)

This

is

a

situation

in

which

the

Hamiltonian

contains

only

invariant

(Casimir) operators of a chain of (super) algebras ~* D ~*' D ....

H - f(G:)

Applications

(3.2)

of these

concepts

interacting boson-fermion model interacting boson nucleons

model

to

, also individual

to nuclear physics

[9].

include,

This model in

addition

are based

on the

is an extension

of the

to

correlated

(unpaired) pxoton and neutrons.

needed if one wants to describe odd-even nuclei,

i.e.

pairs

of

This model is

nuclei with an odd

number of protons or neutrons, since in this case at least one particle must be unpaired.

The building blocks of the interacting boson-fermion model are

still six boson operators as before bt~, b e (=-I ..... 6), but in addition one has now a certain number,

O, of fermion operators,

The number ~ depends on the particular orbit unpaired particle (or particles).

a% i , a i (i-I .... ,D).

(or orbits)

occupied by the

The billnear products

S*: olj -

aI aj

F t - b~ a i ~i (3.3)

Fi~ " ~I b

generate the graded algebra $* - U(6/O).

The G~# and Gij's are its Bose

sector, while the F~i and Fi~'s are its Fermi sector. constructed

from

elements

of g*.

For

example,

the

All operators Hamiltonian

are

can

written as

H - H B + H F + VBF

93

,

(3.4)

be

where H B describes the bosons, H F the fermions and VBF their interaction. Each individual piece is written in terms of elements of g* as

1

~'~#%# +2c,#~-~6~#-~6%# c~

HB - EOB+:#I

1

HF -

+

alj + 21j l Vljkl GiJ ckl (3.s)

VBF- =:lJ Z ~ # l j % # c l j + =~lJ E w'=#iJ F=i t Fj#

States are obtained by acting with the operators b1= and al l

on

a

vacuum

state

(3.6)

They

form

totally

supersymmetrlc

rep~esentatlon

of

U(6/~)

with

Young

supertableau

[~} = [] [] ... z

(3.7)

In (3.7) all bosonlc indices are symmetrized and all fermionic indices are antlsymmetrized.

•

N-NB+N F • The procedure

denotes

described

the

total

above

to

number

of

construct

bosons

dynamic

their associated energy formulas can be repeated here.

plus

fermions,

symmetries

and

There are now many

classes of possible symmetries, since there are three classes of symmetries for

the bosons

and

several

(depending

on ~)

for

the

fermions.

A

good

fraction of the symmetry classes of the interacting boson-fermion model have been

investigated

[9].

In

particular,

the

case

f~-4, corresponding

fermlons with J-3/2 has been studied in considerable detail. visualize

the rather

complicated nesting of algebras,

94

to

In order to

it has been found

One of these

convenient to introduce the concept of lattices of algebras. lattices, originating from U(6/4) is U(6/4) uB(6)

uF(4)

®

oB(6)

suF(4) "~

°B(5)

SpinBF(6)

~

~

$

spF(4)

~ SpinBF(5)

oB(3)

suF(2) -~

SpinBF(3)

~q~

(3.8)

SpinBF(2)

Again the importance of dynamic supersymmetries is that of providing a classification quantities.

scheme

with

explicit

expressions

all

observable

This is particularly useful here, in view of the very complex

structure of spectra of odd-even nuclei. since

for

in a

supersymmetric

Another important point is that,

representation,

K,

there

appear

nuclei

with

different numbers of bosons (pairs) and fermions (individual nucleons), this symmetry provides a simultaneous classification of many nuclei,

some even-

even and some even-odd.

in nuclear

Several examples

physics have been found. route

a

of supersymmetries

Fig. 4 shows one such an example, corresponding to

in the lattice

(3.8).

States here

are

characterized

by

representations

U(6/4) DUB(6) ® uF(4) D oB(6) ® suF(4) D SpInBF(6) D 1 NB

NF

Z

(=i,=2,=3)

D SplnBF(5) D SplnBF(3) D SpinBF(2) (~l,~2),vA

J

Mj

with energy elgenvalues given by [9]

95

,

(3.9)

the

El

7~lrl t4

7~0S114

(MeV

Th

.r.13/2+

"~.

(~)1 --

.

~-I

~ 9 /. : .9~+L1 ~2/ , ~ 7/2+

""

(3 1 . . . . . . . 7/2 +

....... ~.~'E)

(MeV

--

..T

""

+

"-.

-

-

3/2÷ - - - - ~

- ._.

771rl 14

-(22)

-4

~

-..

760s114

(~'°) " ' " ' - -

~V2+

Fxp

--

T'S'~"

'I '~'91Z

o,/~,~L.. T

(2 2)

.,.

--....

--.

, "".

~,o~-;... " - - - . . . ~ , ~ ~ . . . . Fig.4. An example of U6/4) supersymmetry in nuclei: excitation spectra predicted by Eq.(3.10) (top part) and experimental spectra for the pair of nuclei 1"°0s-1911r (bottom part).

96

E(X,NB,NF,Z,al,a2,a3,fl,f2,WA,J,Mj) E 0 + EIN B + E2N

-

+ AZ(T.+4) + A' ~l(al+4)+a2(a2+2)+~3

+

4.

-

+

] + cJ(J+l)

(3.1o)

CONCLUSIONS Algebraic

theory,

i.e.

the expansion of all operators

of a physical

system into elements of an algebra S, has become an important tool in the study of problems

in physics

and chemistry.

Use of this tool

structure physics has led to major advances in this field. of the dynamic symmetries (and supersymmetries)

in nuclear

The exploitation

associated with the algebra

g has played a major role in the classification of nuclear spectra. particularly

true for complex spectra for which other methods

This is

of analysis

are unsuited. ACKNOWLEDGEMENTS This work was supported in part by D.O.E. Contract DE-AC-02-76-ER03074. EEFEKENCES [i]

W. Pauli, Z. Physik 36, 336 (126); V. Fock, Z. Physlk 98, 145 (1935).

[2]

M. Gell-Mann, Phys. Rev. 125, 1067 (1962); Y. Ne'eman, Nucl. Phys. 26, 222, (1966).

[3]

F. G~rsey and L. Radicati, Phys. Rev. Left. 13, 173 (1964).

[4]

A.O. Barut and A. BShm, Phys. Rev. 139B, 1107 (1965); Y. Dothan, M. Gell-Mann and Y. Ne'eman, Phys. Lett. 17, 283 (1965).

[5]

F. lachello and A. Arima, "The Interacting Boson Model", Cambridge University Press, Cambridge, 1987.

[6]

R. Gilmore,

"Lie Groups, Lie Algebras and some of their Applications",

Wiley, 1974. [7]

A. Bohr and B.R. Mottelson,

"Nuclear Structure", Vol. II, Benjamin,

Reading, Mass., 1975. [8]

H. Miyazawa, Progr. Theor. Phys. (Kyoto) 36, 1266 (1966); P. Ramond, Phys. Rev. D3, 2415 (1971).

[9]

F. lachello and P. van Isacker, "The Interacting Boson-Fermion Model", Cambridge University Press, Cambridge, 1990.

97

SO(3,2) F O R O S C I L L A T O R A N D H Y D R O G E N L I K E

SYSTEMS*

Mark E. Loewet Arnold Sommerfeld Institut ffir Mathematische Physik Technische Universit~it Clausthal Leibnizstrafle 10, D-3392 Clausthal

R e p r e s e n t a t i o n s o f 8 0 ( 3 , 2 ) and t h e t h r e e - d i m e n s i o n a l oscillator w i t h s p i n

Generators Jba = --J,,b of S0(3,2) job ,.,Sp(4,~) J,b.~ a, bE {0,1, 2, 3, 5}, fulfill [Jab, Jcd] = - - i (7]aclbd "~ ~bd~ac -- ?TadJbc -- r]beJad) ,

where 77o0= -rhx = -~22 = -~3s = ~Ts~= 1, and ~b = 0 for a ~ b. Its maximal compact subgroup is SO(2)roxSO(3)j~ where F0 - Jo5 mad Ji - ~le ijkJik, i , j e {1,2,3). The generators J/, Fi ----Jis, Fi = Joi are components o£ SO(3)ji-vector operators J, r , F. Subgroup chain

Commuting subgroup invaria~ats

Eigenvalues

SO(3,2)ro,r,,n,s, u

c(4)_=(r0J-rxF)~-(J.r)~-(J.F) 2 s(,+l)(Eo-1)(E0-2)

SO(2)ro xSO(3)j, U S0(2)~ a

n + E0 j ( j + 1) js

F0 32 Js

II Majorana and daughter 13 • and daughter O M S0(4,2) dyon ~' J half-SO(4,2) uy""on

5/2

S 3/2 I J,,r ~

I/2

"

-7/2. -3 -5/2 -2 -3t2 -1 -1/2

0

0

0

Eo

1/2 1 3/2 2 5/2 3 7/2 4"~

Figure 1. Combinations of (E0, s) for which unitary irreducible representations (unirreps) Dj,~(E0, s) of S0(3,2) j.~ exist in which E0 is either the lower (if E0 >0) or upper (if Eo < 0) bound of the spectrum of F0 [1,2].

98

•

0

2 2

•

11

o.:oi~ angula~

n

1:

mon~ntum

0 • 0

1

1

3

0 •

2 •

2

3

•

1

j

2

3 ~

1

1

1

Q

tD

0

o * 1/2

Dj.,(E0 > ½,s=0)

3/2

. 512

J

"

7/2 ~

Dj.,(go > 1,~= ½)

Figttre 2. Spectra of nlj in Dj~,(Eo)I,s=O) and Dj,~(Eo> 1,s= 1) [3,41. Each dot stands for a tmirrep Dro,j~(n,j) of SO(2)ro xSO(3).r~ within which j3 e { j , j - l , . . . , -j}. Actions of the raising and lowering operators

V_~=½[~(F~-r~)-F~-r~], L , = N(J,-~J~), m l = ½[F~-r~-~(a+r~)l,

U~= -';u(F~+~r~), u+~ = ½[~- r~ +i(N +r~)], J~, r0, D~=~(a-ir~), D+1 ½[ i ( r l - ~ ) - ~ -r21 =

on basis vectors In j j3) in Djo~(Eo>_½,s=O) and D3,b(Eo:> l , s = } ) are given by

r01n j j3) = j j~) =

(n + EO) js

&In J±lln

j ja)

=

U~ln j j~) =

=t=

In j j3), In j j3),

x/(j:FJ~)(j+j3+I)/2

In j j3:i:l),

-

u+(n,j)

In+l j + l J3)

-

~o(~,j) ~_(n,j)

I~+lj

u+(n,j) uo(n,j) u_(n,j)

In+l j+l/3:f=1) In+l j j3 =t:l) ]n+l j - 1 j~q-1),

v/(J-l-ja+i)(j-j3+l) j~ + 4(J4J3)(Y:Y3)

U±lln j j3) = - ~/(j+j3'+l)(j=t:j~+2)/2 x/(j+j~+l)(j~=j3)/2 X/(j~j3-1)(j=Fj3)/2 -

D3lnJ J3) = - x/(j+ja+l)(j-j3+i) + ........ J~ + ~/(j+j3)(j-j3)

j~)

In+l j-1 ja),

u _ ( n - l , j + l ) In-l j + l Ja) ,u,o(n-!,j) ............ In-1 j j3) u + ( n - l , j - 1 ) In-1 j - 1 j~),

Dq-lln J J3) = - ~/(~j3'-I-1)(j+j3+2)/2 u _ ( n - l , ] + l ) In-1 j + l j3±1) j3el) :F ~(j-t-j3+l)(j:T:j3)/2 uo(n-l,j) In-1 j - ~f(j~:j3-1)(j~:j3)/2 u + ( n - l , j - 1 ) I n - 1 j - l j s ± l ) ,

99

where the reduced matrix elements are given [5] for the a = 0 unirreps by

/(2Eo+n+j)'(n+j+3) and uo(n,j)

u "n '"

/(2No+n-j-I)

(n-j+2)

= 0 and for the s = ½ unirreps by

u+(n,j)=

~i[2Eo+n+J+½(-1)n+.i-½][n+j+3-½(-X)"+.i-½]/2,

=

'

w

Dimensionless position, momentum, and orbital and intrinsic spin angular momentum vector operators are defined in DA~(Eo > ½, s =0) and DAb(Eo > 1, s = l_.)uby rim j j 3 ) -

Pl n j j3) -

lim

E0-*c~ -

1 ~'Fln .7. ja),

1 lira ~ r l n E o ~ V'~0

j j3),

L[n j j:~) =

lira __1r x F I n J Ja), E0~oo E0

Slnjjz ) -

lira E0---~oo12;0

~(r0J-rxF) lnj j~).

It follows that L = r x p and J = r x p + S ; that r, p, S fulfill the Heisenberg and intrinsic spin angular momentum commutation relations of generators of the direct product group HW(3)r,,,,,I ×SO(3)s, [6]:

[ri,pj] = i#ijI,

~i,Pj]

[ri, rj] = 0,

= 0,

[St, $1] =

ieq/cSk;

that F0 is the essentially-self-adjoint energy operator for isotropic harmonic oscillation but with the zero-point energy shifted from -~ to E0:

1"0 = ~ ( r2 + p2) _ 2 + E0,

2 @2 q. p2) tn j ja} =

n+

In j js);

and that squares of the orbital and intrinsic spin angular momenta fulfill L21n j ja) =

t(t + 1)lh j

ja),

S~I'~ j ja) = ,(s + i)ln j j3),

where s is the invariant intrinsic spin quantum number which characterizes the continuous sequence of unirreps Dj,,,(Eo, s) by determining the unirrep of the SO(3)s~ factor and where the spectrum of (n, t) is that of total vibration mad orbital angular momentum as in the usual three-dimensional oscillator unirrep of the HW(3)r~,p~ ,I factor. The spectrum and multiplicity of (n, j ) is that of total vibration and total angular momentuna of the direct product of the three-dimensional oscillator with the intrinsic spin s. For the 8 = 0 sequence L = J and the orbital angular momentum l is a redundant quantum number given by l = j. For the ~ = ½ sequence l is a redundant quantum number

100

given in terms of n and j b y / = j - ½ ( - 1 ) "+j-½. For the 8 >_1 sequences bases In l j j3} again exist but l is no longer a redundant quantum number. A parity operator that anticommutes with F, r , r, p is defined on the bases In I j j3) = In j j3)inDj=,(Eo>_½,s=O) and Dj=b(Z0_>l,s=½) by P In l j j3) - OP(-1) 1 In I j j3} where rip = +1 or UP = - 1 gives the intrinsic parity. P represents the transformation that inverts the space-like 1, 2, 3-axes and leaves the time-like 0, 5-axes unaffected. Dimensionless vector potential, field strength, and covariant m o m e n t u m vector operators similar in form to those of dyons [7] are defined by r A---- ~-~ × S,

i B=~(pxA+Axp),

~-p-A.

Their matrix elements can be calculated with the help of those of the operator 1 / r 2 inverse to r 2, which are given in Djo, (E0 > ½, s =0) and Dj.,(Eo > 1, s = ½) by

1 In l j j3}

(n' l' j' J'31-fi

=

~',1

2

V~--0::(n'+l+D!!'

8j,,j 8j~,j~ 21 +-----i"x

if

_> n,

/(n-~)r~(n'+t+l)'!

V(n,_01~(.+I+,)21,

if n' < n.

The field strength and covariant momentum operators fulfill [7] [Tri, lrj]

=

1r2 = p2 --

i¢ijkBk, + ~

B j2 - ~

=

+

r.S

=

r.J

---fi--r

=

2L. S +

5S 2

r4

r,

When E0 ~ ½ in DJob(Eo > ½, s =0) or E0 ~ 1 in Dj.,(Eo > 1, s = ½) their n = j - s and n > j - 8 subspaces become S0(3,2) Job-invariaat [2,3], but not HW(3)r,,p, j×SO(3)s~invaria~t. The respective n = j - s subspaces give the Majorana [8] unirreps Dj.,(Eo = ½, s = 0 ) and D]ob(EO = 1, s = ½) and the respective n > j - 8 subspa~s give the daughter tmirreps D Job(E0+2 = -~, s = 0 ) , in which n - - 2 e {0,1,2,-.-}, and D jo~( E 0 + l = 2, 8 = ½), in which n - 1 E {0, 1, 2,.-.} and -r/p plays the role of intrinsic parity. When E0 ~ ~ + 1 in an s > 1 continuous sequence DJo, (E0 > s + 1, 8) two subspaces again become SO(3,2)Job-invariant [2,9]; one gives the unirrep DJ.b (E0 = 8+1, s), which is multiplicity-free with respect to SO(2)r0×SO(3)j~ and extends (see below) to either of two dyon unirreps, and the other gives the daughter unirrep DJo, (E0+l = s+2, 8-1), in which n - 1 E {0,1, 2,...}, s - 1 plays the role of intrinsic spin, and - y e plays the role of intrinsic parity. Analogous facts hold for the unirreps Djo,(Eo < 0, 8), in which n E {0, - 1 , - 2 , . . . } and in which the reduced matrix elements of F and r have reflection symmetry compared to those in DJo~(Eo > 0,8) [2,3]. An operator that anticommutes with F0 and r transforms between D Job(Eo, s) and DJ.b (-E0, s) and anticommutes with p [10].

101

The

s = 0 a n d ~ = ½ r e p r e s e n t a t i o n s as factors of h y d r o g e n l i k e s y s t e m s

Orbital structure of hydrogenlike systems is usually described using a # = 0 dyon unirrep of an SO(4,2)]an [7,11]. The generators ]BA = --JAB of S0(4,2)3~n ,.~SU(2,2)2~, A, B e {0,1, 2, 3, 4, 5}, fulfill

where ~/00= -zh~ = -~/22 = -~/sz = -Y44 =~/~s = 1 and ~/AB= 0 for A ~ B . The maximal compact subgroup of S0(4,2)3an is S0(2)~o× SO(4) ~,~, where V~-- J~4. Dyon unirreps D ~ axe the only unirreps in which the generators also fulf~ [12]

JAB, ycB} = _2a6AC

= constant.

They are characterized by a helicity type quantum number # E {0, =t=½,=t=l,. • .}, which determines (spectrum ] . V ) = {/~[/z[+ 1),# (1#1+2),...}, and by ¢-- sgn(spectrum P0). In them 5 = 1 - #2 and i~0~ = J z + ~ 2 + 5. They remain irreducible with respect to S0(4,1)2,a ' a, fl e {0,1,2,3,4}. D g>°'¢ Jaa also reduce into the SO(3,2)~.~ unlrreps D : . , ( / ~ 0 = f f l ~ l + G S - - t ~ l ) and D ~'=°'¢ :a~ reduce into the direct sums of two S0(3,2)~.~ unirreps, D2.,(J~0 = ~, ~ = 0) ~ D3,~(Eo = 2G'~= 0) [12], one with zero-point shifted by down from ~; and the other with zero-point shifted by ½ up from s¢. (See Figure 1.) A # = 0 dyon unirrep is used for orbital states of hydrogenlike systems because .l and are then orbital angular momentum and "tilted" Runge-Lenz vector operators and the spectrum of the S0(4) 3,,~-invariant operator - R / P ~ =-R/(.J2-~-~rb'r2-jf-1) is given by the Balmer formula - R / N ~ (where R is a Rydberg parameter) with N~-fold degeneracy. The four-dimensional space of solutions of the Dirac equation [13] is an .;.,,.,~.., . . . ~"~r~Dirac )an of SO(4,2))an whose generators also fulfill the above relation with &= - ~ . Actions of the generators on a basis of eigenvectors [P~ )s) of P0 and )s, with eigenvalues ps 6 { ½,-½} • and 3s ~ { ~1 , - ~1} , are obtained, using the S0(4,2)3an commutation relations, from [14] F0 lps )s) = ps lpz )z), •

1

~

3

where tb4----)0, and P4----&s. D ~an Di'ac remains irreducible with respect to the S0(3,2))o, and SO(2,1)N,p4,1.oxSO(3)), subgroups and factors into the direct product of a twor~p-~pace of SO(2,1)~,i~4,i~° and a spin-½ unirrep D~7t2 of SO(3)A: dimensional irrep "~,t0,/'4 - - r~p.-~pace D3D , ~ i r-a~c&,~4,t'o ® D~? 12"

Hydrogenlike systems with orbital-, fine-, and hyperfine-structure might be described using the respective direct products DJo~ (E0 > ½, s = O) ~ '~" r~P-~P~c~ ~4,~o D J°b (E0 >

102

3, s = O) ® D Dir~*)AB or DJo,(E0 > 1,s = ½)®.~a,l~4,i~o,nP-~P~¢*and DJo~(Eo > I , s = ½) ® D )AS Dir~*" Bases in these direct products are given by eigenvectors IN l j ] ]3) of the operators 1_2(r2+p2)+~o, L2, j2, ] 2 £ __ where ] = J + J is now the total angular momentttm with respective eigenvalues N, I(I+1), j ( j + l ) , ](]+1), ]3 (see Figure 3); note that in the first three direct products this labeling is redundant since l = j =], l=j, and j =], due to S=O=], S--0, and J--0, respectively.

3 N

0

1

2

3

•

•

•

•

0

1

2

•

•

•

0

1

•

•

1,2 @

2,3 @

3

0,1 ®

1,2 ®

2 •

2

0,1 @

1 •

1

o •

N

o •

1

0,i @

4

3 •

h.

0 Dj.b(Eo

1 > ~, s

2

T

0,1 @

1,2 @

2,3 @

3

0,1 @

1,2 @

2 .

2 1

1/2

=o) ~~' *,"-~°°~ " ~,~'4,~'o

4~

N

3 "

3/2 ~, 5/2 J

7/2 ~

D j . ( E o > ½,~=0) ® n ,~'~° JAB

3

3 •

orbital

0,1 ®

N

1 •

0

0,! 1 @•

1

•

0

•

112

1/2

3/2 ,,, 5/2 J

0

7/2 ~

I / •

mome41ta

0 112#,/2 3]2r5/2

1

2

5/2~7/2 7/2r9

3

J

4r~] "

Dj., (E0 > 1, s = ½) ® TlDir~e)aa

•./.d p - - a p a c e

Dj.b(Eo > 1 , 8 = ½ ) ® ' ~,P,,Po

Figure 3. Spectra of Nl 3. Each dot or circle stands for a (2]+l)-dimensional subspace. An operator whose spectrum is given by the Balmer formula - R / N 2 with N2-fold, 2NZ-fold, 2N2-fold, and 4NZ-fold degeneracies within the respective direct products is -R 2

= lim

-R

103

=

llm

-R

A parity operator is defined in the direct products by

P IN l j .~33) - ~P(-1) ~IN l j .713) where e i t h e r . ~ p - +1 or ,?p = - 1 gives the intrinsic parity. P anticommutes with r , F, r, p, r , F, Fa, F4 and again represents the transformation that inverts the space-like 1, 2, 3, 4-axes and leaves the time-like 0, 5-a.xes unaffected. Dimensionless vector potential, field strength, and covariant m o m e n t u m vector operators can be defined using J or J + S in the same way as with S. These angular momenta can then be weighted by parameters; e.g., weighting S by a parameter A in the previous definition of A leads to

[ri, ~rj] = (2--~)ieijkBk,

~r2 = p2 +

2L. S +

[

S 2 - ~-{

.

Replacing p2 in the Baimer operator by this r 2 produces an operator whose spectrum depends on A and describes splitting of the Balmer levels when A ~ 0. Hmailtonians whose spectra describe level splittings of hydrogenlike systems might be formulated in connection to the factorization method [15], dyons [16], SO(3,2)jo~ gauge and QED theories [17], and their supersymmetric versions. Note that the eigenvalues of ½ (r ~+p~) +n0 that result from doubling = d shifting the zer~point of ½ (r ~+ p~) by the eigenvalues of n0 are the same as the zero-poim values E0 = 1 and ~70 = 2 occuring in the reduction of the dyon unirrep D v=° JAB'+"2 D d.b (/30 = {, S = ½) extends to either dyon unirre.~ apparently, Dsob(Eo 3 ~ _ m ~ n O l r ~ , _ D_ (2, ½ ) - * D j . b ( 1 , ½ ) ~ M D #=~1/2'+' dAB D3ob(2,{ ) and Dj.b(E0 = 5 DDi~,¢_ 0)._+D3.,(2,1).. . D3.~(3,0)]e Dj.b(2 , 0)@Dj.~(3,1) contain Gupta-Bleuler-triplets [17]; and D Dirac is the representation by which so(3,2)$.b extends to the superMgebra osp(1,4). Dependence of a Hamiltonia~l'S spectrum on the continuous parameter E0 goes beyond its dependence on any parameters which appear in the Hamiltonian's expression in terms of the generators Jab and .lAB. The possibility to vary E0 away from particular values connected with other theories, e.g., E0 = ~, might allow better descriptions of level splittings. The eigenvalues of the SO(3,2)job Casimir operators C(2) and C(4) are quadratic functions of J~0 with minima occuring when/~0 = -~. The two values E0 = 1 and J~0 = 2, which might correspond to shifting from E0 = ~, 3 produce degenerate spectra for C(2) and C(4) but two other values of J~0 that differ by 1, which might correspond to similar shifting from another value of E0, would split this degeneracy. Analogous facts, including existence of an analogous Balmer operator with the same spectrum and degeneracies, hold for the direct products obtained by replacing the first factors Dj°~(Eo > ½, s = O) and .Dj,,b(.E 0 :> 1,s = ½) by the corresponding first factors Dj.,(Eo < ½, s = 0 ) and Dd.b(Eo < 1, , = ½) in which the spectrum of F0 is bounded from above. An operator that aaticommutes with F0, r , p, I'0, I', and either F4 or ~ and [18] transforms between corresponding direct products.

-~,s={)® ~.,,,-[Dj.~(3,

104

Acknowlegements, references, and footnotes I thar~ Professors A. Bohm, W. Heidenreich, P. Horv£thy, and J. Tolar for beneficial discussions. I thank the Fulbright Commission, Bonn, for Graduate Fellowship support and Professor H.-D. Doebner and members of the Arnold Sommerfeld Institute for kind hospitality. I also thank the members of ASV Barbara, Clausthal, for kind hospitality. * t [1] [2] [3] [4]

Based in part on an invited lecture, XVIIIth ICGTMP, Moscow, USSR, June, 1990. Permanent address: 12882 Olympia Way, Santa Ann, California, 92705, USA. N. T. Evans, J. Math. Phys. 8, 170 (1967), classified these unirreps. M. E. Loewe, Ph. D. Thesis, University of Texas, Austin (1989), Fig. 4.2. J. B. Ehrman, Ph.D. Thesis, Princeton University (1954), Figs. 7-9 and 7-15. C. Fronsdal, Phys. Rev. D 12, 3819 (1975),

[51 rtef. [31, Eqs. (7-173,185)and (7-204); ref. [21,

(4.91).

[6] A. Bohm, M. Loewe, P. Magnollay, M. Tarlini, R. R. Aldinger, L. C. Biedenharn, H. van Dam, Phys. Rev. D 32, 2828 (1985). See also E. Char~6n, P. O. Hess, M. Moshinsky, J. Math. Phys. 28, 2223 (1987), and references therein. [71 D. Zwanziger, Phys. Rev. 176, 1480 (1968); J. Schwinger, Science 165,757 (1969); A. O. Barut and G. L. Bornzin, Phys. Roy. D 7, 3018 (1973), e.g., Eqs. (65,66). [8] A. Bolun, M. Loewe, L. C. Biedenharn, H. van Darn, Phys. Rev. D 28, 3032 (1983), appendix; note that Eq. (A31) should read Cj = i/2 for 3"--1, ~,~2, ~,-.5 .. [9] M. Lesimple, Left. Math. Phys. 18, 315 (1989), and references therein, indicates a more complicated situation involving indecomposable representations. [10] This operator represents the transformation that inverts the 5-axis and leaves the 0,1, 2, 3-axes unaffected, perhaps interpretable as charge-conjugation C. [11] A. O. Barut and Raj Wilson, Phys. Rev. A 40, 1340 (1989). [12] A. O. Barut and A. Bhhm, J. Math. Phys. 11, 2938 (1970). [13] A Bbhm and B. Mainland, For~schr. Phys. 18, 285 (1970); A. Bhhm and R. B. Teese, J. Math. Phys. 18, 1434 (1977). [14] These actions are equivalent to four-dimensional Dirac matrices %,, a ~ , etc.. [15] A. Stahlhofen and L. C. Biedenharn, Proc. XVIth ICGTMP, Springer Lecture Notes in Physics 313, 261 (1988); A. Stahlhofen, K. Bleuler, Nuovo Cimento 104B, 447 (1989), and references therein. [16] L. Feh~r, P. A. Horv£thy, and L. O'Raifeartaigh, Phys. Rev. D 40, 666 (1989), and references therein. See' also P. A. Horv£thy, these proceedings. [17] H. D. Doebner, W. F. Heidenreich, Symposia Mathematica XXXI, 157 (Istituto Nazionale di Alta Matematica Francesco Severi, 1989 or 1990); C. Fronsdal, Left. Math. Phys. 16, 173 (198,8). [18] If it anticommutes with F4, then it can "represent the transformation that inverts the 5-axis and leaves the 0, 1, 2, 3,4-axes unaffected; if it anticommutes with ~ and ~r, then it can represent the transformation that inverts the 4, 5-axes and leaves the 0,1, 2, 3-axes unaffected.

105

CONFORMAL GROUPS IN THE KEPLER PROBLEM Josl~ F. CARIIqENA*, CARLOS L6PEZ*, M A R I A N O A. DEL O L M O t A N D M A R I A N O S A N T A N D E R t

*Depto. de Ffsica Te6rica, Universidad de Zaragoza, 50009 Zaragoza, Spain *Depto. de F~siea Te6rica, Univ. Complutense de Madrid~ 28040 Madrid, Spain tDepto, de F~sica Te6rica, Universidad de Valladolid, 47011 Valladolid, Spain. Abstract. A group $0(3, 2) of point transformations on the space ft of all (plane) Kepler orbits is introduced. This group has a subgroup of dimension seven induced from point transformations in the configurationplane X = R2 - {0}. In particular 2~ appears as an homogeneousspace of SO(2,1), and the (one-parameter family of) invariant connectionsin this homogeneousspace have the configurationspace Kepler orbits as auto-parallell curves with the anomaly as the canonical parameter. Another SO(3, 2) group is similarly related to the known groups (Moser -Osipov - Belbruno- Milnor) acting on the sets of Kepler orbits of constant energy,whichcan be identifiedto the sets of geodesicsof the Riemannian 2-dimensionsd spaces of constant curvature. ORBITS IN THE KEPLER PROBLEM: CONFIGURATION AND VELOCITY SPACES. In addition to the known dynamical symmetry in phase space, the Kepler problem has interesting symmetries in velocity space [1]. The set 2hrg of all velocity vectors compatible with a fixed energy E can be endowed with a unique Pdemaunian structure of constant curvature ]c = -9.E, whose geodesics are the hodographs of the Kepler problem [2-6], and whose isometry groups are (in the plane case) SO(3), E(2) or SO(2, 1) after E < 0, -- 0, > 0. Either the set ME or the manifold f~g of all (plane) Kepler orbits of energy E appear as homogeneous spaces of these groups, which act as groups of point transformations in both spaces. TRANSFORMATION GROUPS IN THE ORBIT SPACE Here we introduce two groups Go, Gv of point tranformations on the space f~ of all plane Kepler orbits. As abstract groups they are isomorphic to the conformal group ~qO(3, 2) of the (2+1) Minkowski space. For each of these two groups, there are subgroups induced by point transformations in respectively the configuration space • or the velocity space ,A~f. The subgroup of Gv,E mapping the submanifolds f~E onto themselves contain as subgroups the velocity space groups S0(3), E(2), S0(2,1) wlch acts as groups of point transformations both in 2~fE and in f~E New results are obtained from Go. The subset of G¢ induced by point transformations in the configuration space is a seven-dimensional subgroup, isomorphic to the (2 + 1) Poincar6 group extended with dilatations. This group acts by point transformations and transitively in A" and fL The Lorentz subgroup Partial financial support fi'omC~GYTand from Universidadde Valladolid is acknowledged.

106

8 0 ( 2 , 1 ) acts transitively on the submanffold ~c of all Kepler orbits of constant C = i-r, ~s and these transformations are induced by a SO(2, 1) group of point transformations in ~ ; here the action is transitive but not primitive because the isotopy subgroup of the action is a one-parameter parabolic subgroup of `90(2,1) and there is no any invariant Pdemannian metric nor a unique invariant connection in ~'. Instead of this we have a whole family of connections, r (c), and the fanfily of Kepler orbits with a fixed value of C = . ~ , considered as curves in X, are the autoparallel lines of the connection r (c). For each such Kepler orbit O, there is a one-parameter subgroup Go of `90(2, 1) mapping the orbit into itself;the canonical parameter of Go turns out to coincide with the (eccentric / parabolic / hyperbolic) anomaly, which is known from classicalmechanics to be the good parameter, and which also coincides (up to a factor) with the Levi-Civita parameter used in the regularization of singular Kepler orbits. The structure so obtained is a coni~gur,ation space counterpart to the previously known for the velocity space, where for each value of the parameter E there is a ,90(3) / E(2) / SO(2, 1) homogeneous space structure in both the submanifolds ~E and .A4E • The main difference is what these groups act as isometrics of some invariant metrics, whereas this is not so for the actions on 2d. However, the groups of transformations on 2d appear as subgroups of Go, while the groups in the vctocity space are subgroups of Gv. We describe here in detail only Ge; for Gv the construction is quite similar starting from another Minkowskiau metric in the orbit space; a detailed description will appear in a forthcoming paper [7]. THE CONFORMAL GROUP Gem The general equation of a (regular) Kepler orbit is p 1 r ( 8 ) = 1 + ¢ c o s ( 6 -- ¢ )

Q(e) -

-r = t -F x cos 8 + y s i n e

where t = ~ and (x,y) = ~(ecos¢,esin¢) and we adopt the convention p < 0 for orbits with negative angular momentum. The parametrization (t,x, y) is called the ~qlinkowskian pararnetrization for Kepler orbits. RELATION BETWEEN (t, X,y) AND THE KEPLER ORBIT. A simple geometrical construction using some projective geometry.

~

1) Let II be the polar plane of (t, x, y) relative to the quadric t 2 + x = + y= = 1.

107

,,, It~_..._.. (~.,,,..,~)

2) Consider the intersection of II with the cone 42 - x 2 - y2 = 0. It is a conic in II. 3) Project this conic orthogonally onto the plane (x, y), identified with the configuration space with x, y ordinary cartesian coordinates; the projected conic has always the focus at the origin and is the Kepler orbit we are looking for. The unique reference in the literature we have found for this construction is in Arnold [8] and is credited to A.B. Givental. INTRODUCTION OF A MINKOWSKIAN METRIC. From all this it is only natural to introduce a metric do 2 in the orbit space given by do 2 = dt 2 - dx 2 - dy 2. A time-like (resp. null, space-like) separation corresponds to nonintersecting (resp tangent, intersecting) orbits in X. We consider the associated isometry group (a (2 + 1) Poincar6 group) and we call Gc the conformal group 0(3, 2) of this Minkowski space, acting as a (local) group of txansformations in f/. This group Gc maps isotropic cones into themselves, but does not in general map planes into planes, although there is a large subgroup (the (2 + 1) Poincard group extended with dilatations) which does. In view of the previous geometrical description, this subgroup can be expected to be induced by point transformations in the configuration space X, and this is indeed the case. The standard form of the conformal group action on Minkowski space is 0

0

0 0 Kz = -x~-~ - t ~ x , 0

Po=--Or' D=tO

K2 = - y ~ O

P~=--Ox ' 0 O

0

- ~

0 y 0

P2=-~ Oy

Co = - ( ~ + z 2 + V2) ~O - 2 t x ~ x - 2 t y ~ . ~ Cx = 2tx 0 + (t2 + x2 - y2) ~---~+ 2Xy~y 0 0 2 0 C2 = 2ty-~ + 2xy-~x + (t2 - x2 + y )Oy"

CONFORMAL TRANSFORMATIONS INDUCED BY POINT TRANSFORMATIONS IN Is each of these transformations induced by some point transformation in ~d?. For each generator A = at(t,x,y) ° + aX(f, z,y)'b¥° + a V ( t , x , y ) ~v try to find an associate generator g acting on X, Pi = a°(Q, 0) ° + aq(Q, o)~q, and a function A(Q, 0; *, x, y) such that A ~ - M E = AE, where ~(Q, O; t, x, y) = t - x c o s O - y s i n O - Q = Ois the equation of the orbit. The subgroup of G, for which these equations have solutions is

108

.

generated by J, Kz, K2, P1, P2, Po, D and the corresponding generators in configuration space are given by 9 a J = D = Qb0 + sin8~-~

Kt = Q c o s 0 ~ Po = OQ

Pz

=

K2 = QsinO~-~ - cosO~-~O

e__._O cos OQ

0 P2 = sin8 OQ

Hence the (2 + 1) Poincar4 group acts in the standard way in the orbit space, but also acts in the non-standard way as a transitive group of point transformations in the configuration space X of the Kepler problem. From the former expressions, the Lorentz subgroup 80(2, 1) acts transitively on the submanifolds 42 - x 2 - y2 = C, which corresponds to Kepler orbits with a fixed value of C = %-r. 2E X AS AN HOMOGENEOUS SPACE FOR 8 0 ( 2 , 1)

Fix now C and consider the submanifold of all Kepler orbits with that value; as curves in the configuration space they are a two-parameter family of conics. Can this farnily be considered as the geodesic curves of some connection in the configuration space invariant under 80(2,1) ? The most genera/invariaat spray under the S0(2,1) action depends on two parameters and is the following 0 0 3 ~ a b ~ O [~ r,.b = "Q~-0 + " ' ~ + [50 "q + ~ , ~ , 0 - ( + ~)v0 ] 0-7~Q + ~ + ~v~" 2-1~_;,0 C = 00KBITS (PARABOLAE) Take now the two-dimensions/faxnily of Kepler orbits with C = 0 (ie. E = 0, and

hence parabolic orbits). If Q = ~(1 + ez cos 8 + % sin O) is the general equation of a such conic depending on two parameters (as %2 + e~ = 1), it is easy to see that X0 = J - % K i + e~K2

is tangent to the conic; the new local adapted coordinates are sz, sz given by

Xosl

= 1,

Xos2 = 0

whose integration gives 0

Q

81 = - t a n ( ~ ) ,

~2

=

1 +cose

and we get the parametric description of the conic in terms of 81 = h 2 Q(h) = p(1 + h2) ' 8(h) = - 2 a r c t a n h By eliminating p and h the spray F00 is easily recovered. Notice that h coincides with the so-called parabolic anomaly.

109

C > 0 ORBITS (ELLIPSES) C = c~ > O. Here as E < 0 we have e < i. The equation of the conic depends on two independent parameters e=, ey. (Here e=~ + ey2 < I) and is e

Q = ,.~--___~(I + ~cos0 + e, sine) vtl-~) The tangent field is now x+ =

~ [1J

- e~K= + eJ, x~)~2(SN_I) = --/5'-1 d N - l n ~ ( d ) * ~ ( d )

dZ~-lQ

=

1

/"

J~N-I ----(~b,~)Ssh-®,.

(5.5a) +oIQr,+°ml

dN-lq

[qJ2/4p2)N-1 =- /•N-1 dN-lq ~Cq)*~b(q), (1 +

~(d(q)) (5.5b)

3Compare. with the geometrical,,(classical)..expressi°nI$' )8 uslng.the Schr~dinger.quanti~ation map on p~. Keep in mind that operators ~i cannot be selr-adjomt m domains orfunchons whose Fourier transform have compact support, because they are generators of translations that do not leave the domain invariant.

116

where 6 ¢ = ± ( Q ) = e ( ~ ) and ~(q) = (1 + lql214p2)¢N-x)/2e(~(q)), and similarly for • and ~. Under this inner product integral, symmetry transformations are unitary and their infinitesimal generators (5.6b) are seif-adjoint. When we use the customary inner product form (~b,~)~h_~, in (5.5b) for a 'flat' space of measure d N - l q , the so(N) generators ~i i and Mi axe the SchrSdinger quantization of the 'classical functions' in (3.8) and Ki, K N in (4.2I. 4 The functions axe linear in the components ofPl , so there is no ordering ambiguity: any quantization rule gives the same result. The optical fish-eye Hamiltonian (2.2), being neither p2/2 + V(q) nor of the form pf(q) + g(q), I#ould be subject to ordering freedom [21]. As Caslrnlr operator however, it is the well-defined sum of squares of the operators (5.2). The wave Hamiltonian is thus unique and independent of the qua~tization scheme. As in the Hydrogen atom, there are a finite number of independent Maxwell fish-eye states for each allowed color. They may be labelledby the so(N) representation £A, in (5.3),and the row indices of the canonical bask [22],{l~r-1....,~2}, with integer ly's bound by the usual branching rules. For N = 3 (the 2-dlmensional Maxwell fish-eye),we have the spherical harmonics on the surface of the ordinary sphere S z. In this plane optical world, the description of the wave patterns in the Maxwell fish-eye by the usual {l,m}-labels is by projecting ~/t,,~(~) to [20]

T&m(q't) =

",...1 +

[ql2/4p ~"

vibrating with v(£) = (c/nop)vfl-(-~-+ 1), £ = 0,1,2,....

V

hop

c .o)

These functions can be visualized ms

patterns of light with an intensity weighted by an obliqulty factor N ITt,m(q)12~(1 + Iq12/4p2) -2. For m = ± l we have waves travelling around the equator. The nodes of the real part axe moving sphere meridians. This rotating light pattern projects on the fish-eye plane as nodes moving as spokes in a rotating, rigid wheel. When the rotation axis is inclined, the belt of maxima projects on an off-centercircle and the nodal meridians on circular nodes that cross through the two projected rotation poles. W e conjecture that these 'circle-of-light'solutions are the best wave analogues of the geometric light orbits. W e note that since the phase velocity is not linear in l, chromatic disperslon takes place whenever more light has more than one constituent color. The periods are incommensurable. The m = 0 solutions contain a Legendre polynomial Pl(cos fl) and are independent of the longitude angle ~/. They can be described as standing-wave solutions that have their maxima at two conjugate points. They will be in or out of phase according to the parity of L A flash at some point of the fish-eyewill decompose into Pl(cos/~0)'s,subject di#perslon again. Thus signals will loose their shape, even though the optical path between two conjugate points is equal along any circle arc, and wavefronts are well defined [2]. The Maxwell fish-eye is thus not quite a perfect imaging device [3] because it cannot forestallthe chromatic dispersion. Other operators in the hidden, potential, or dynamical algebras may be used to define other bases for the polychromatic, wavlzed fish-eye. W e may choose the two commuting operators = - & ) = - OlOq , i = 1,2. These yield plan wave has , hut since they do com~ute with the driving Hamiltonian, the solutions quickly loose their shape. They are in fact the cross-basisrepresentations between the elliptic(£,m) and parabolic (Pt,P2) subgroups of the SO(3,1) group that we have realized on the R 2 plane [22]. The conformal so(3,2) algebra adds 4i.e. through the replacements qi ~-~qi', (multlpllcation by ql), and pi ~ ~ ..~ --iO/aqi.

117

the to the list the generators H i = qiP, i = 1 , 2 , / I s = )~ - 2pp, a~d H 4 = ~ = p(1 + [q12/4p2) p. T h e y are wv.vized w i t h t h e integral o p e r a t o r t h a t realizes the formal root of the L s p l a c i a n integral operator p = ~ p~.

References. [1] R.K. Luneburg, Ma~h¢maticcJ Thsorv o/Optics (Univexsity of C~li£ornla, Berkeley, 1964), Sect. 28 [2] O. Stswoudis, The Optics o/Rags, Wave/fonts, and Caustics, (Academic Press, New York, 1972). [3] ~VLBorn and E. Wolf, Principles o/Optic= (Pergamon, New York, 1959), Sect. 4.2.2 [4] V. Pock, Zur Theorle des Waeserstoffatoms, Zei~c~r~ ffLr Phy~ik 98, 145-154

(1955)

[5] V. Bergmann, Zur Theorie des Wesserstoff~toms, -Bemerkungen sur gleichnamigen Arbeit yon V. Fock, ZeJtsobrJt'¢ flu" PJ~ysJk99, 576--~;82 (1936) [7] H.A. Buchdahl, Kepler problem mid Maxwell fish-eye, jourAm. J. Phys.46840-84378 18] T. Sekiguchi and K.B. Wolf, The HazailtoniLn formulation of optics, Am. J. Phys. 55, 830-835 (1987). [9] K.B. Wolf, Elements of Euclidean Optics. In L/e Methods in Optics, II Workshop, Lecture Notes in Physics, VoL 352, (Springer Verl~g, Heidelberg, 1989). [10] V.L Man'ko and K.B. Wolf, The mapping between Heieenberg-Weyl and EuclideLn optics is cometic. In Lie Methods in Optics, II Workshop, Lecture Notes in Physics, Vol. 352, (Springer Verlag, Heidelberg, 1989). [11] E. L6pes-Moreno and K.B. Wolf, De la ley de Snell-Descartes ales ecuaciones de Hamilton en el espacio fuse de la 6price geom6trica, Rev. Mex. F~. aS, 291-300 (1989). [12] A.. Frank, F. Leyvras, and K.B. Wolf, Hidden symmetry and potential group of the Maxwell fish-eye. Comunicaciones T~cnicas IIMAS, Inveatig~i6n, # 562 (1990), accepted in Journa/ofMathematka/Physice. [13] S. Steinberg, Lie series, Lie transformations, and their Applications. In L/e Methods in Optlcm,Lecture Notes in Physics, Vol. 250, (Springer Verlag, Heidelberg, 1986). [14] (a): H. Goldstein, Claedcal MechanlcJ (Addison-Wesley, Reading, Mess., 196~), Chapters 7 and 8; (b): E.G.G. Sudarshan and N. Mukunda, Olassical Dynamlc,: A Modern Perspective (Wiley, New York, 1974), Chapter 8;(e): K. Sundermeyer, ConJtrained DynomleJ, Lecture Notes in Physics, Vol. 169 (Springer Verlsg, Heidelberg, 1982), Ghapter IIL [1~] A. Frank and K.B. Wolf, Lie algebras for potential ecutterin 8. Phys. Rev. Left. 52, 17-~7-1739 (1984); A. Frank and K.B. Wo~f,Lie algebras for systems with mixed spectre~ The scattering P6schl-Teiler potential. J. Ma~h. Phys. 26, 975-985 (1985) [16] Y. AIhessid, F. G~rsey, a~d P. Ia~eIlo, Group theory approach to scattering, Ann. Phys. (N.Y.) 148, $4~380 (1985); lb. H. The Euclidean connection, Ann. Phys. (N.Y.) 167, 181-200 (1986); J. Wu, F. IacheUo, and Y. Alh~id, ibid. HI. Realistic Models Ann. Phys. (N.Y.) 173, 68-87 (1987). {17] K.B. Wolf, Dynamic groups for the point rotor and the hydrogen atom, Suppl. Ntwvo (~Lmento 5, 1041-1050 (1967) [18] A.O. Burut and G.L. Born=in SO(4.2)-Formulation of the symmetry breaking in rel~ivisitie Kepler problems with or without magnetic charges. J. l~ath. Phys. lZ, 841-846 (1971); see B.G. Wybourae~ Cla4sieal Groups /or Phvslelt~s (Wiley, New York, 1974), Sect. 21, p. 297. [19] J.M. Sourian, Btruc~urs des Syfl~mes D~namiques (Dunod, Perle, 1970); B. Kostant, Quanti~ation ~nd Uni~ar~Representations, Lecture Notes in Mathematics 170 (Springer Verlag, New York, 1970); J. ~nintycki, Geometric Quanti~ation and Quantum Mechanics (Springer Verleg, New York, 1980); V. Aldaya, J.A. de

118

Asc£rr~ga, and K.B. Wolf, Quantk&tinn, symmetry, and natured polarization, Jr. Math. Phys. 25, 506-512

(1984).

[20] L.C. Biedenhv.rn and J.D. Louck, Angular Momentum i, Quontum Phy~icJ. Theory and Application. EncyclopediL of MLthemLticm,Vol. 8 (Academic Press, New York, 1981). [21] K.B. Wolf, The Heisenberg-Weyl PAngin Quantum Mechanics. In Group Theory and ira Applieationa, Vol. 3, Ed. by E.M. Loebl (Academic Press, New York, 1976). [22] R. Gilmore~ Lie Group~, Lie AIgcbra~, and Some o/their Applleatlon~ (Wiley Intersclence, New York, 1978).

119

Transition to chaos in hadronic systems Piotr Kietanowski Institute of Theoretical Physics, Warsaw University ul. Ho~a 69, 00-681 Warsaw, Poland The interactions of mesons and baryons show that they posses an internal structure. There exist various attempts to describe this structure. One such a.n attempt is to assume that mesons and hadrons are composite and are built out of quarks. A complementary description is to treat hadrons as extended objects (strings) which can perform various motions. These motions manifest themselves in the complicated structure of the excited states of hadrons. The framework for the description of the internal collective motion of a hadron is the relativistic quantum mechanics with the choice of dynamical variables based on the Spectrum Generating Group (SGG) of the system while the Center of Mass motion is described by the Poincar6 group. We will consider here the excited states of a nucleon (nucleon resonances) and of a hyperon E. TABLE I.

N(1440)PI1 N(1680)F15 N(1520)DI3 N(1700)D13 N(1535)Sll N(1710)Pll N(1540)P13 N(1720)P13 N(1650)Sll N ( 1 9 6 0 ) N(1670)D15 N(1990)F17

N(2000)F15 N(2080)DI3 N(2090)Sll N(2100)Pll N(2190)G17 N(2200)H19

N(2250)G19 N(2600)Illl N(2700)Kl13 N(,,,3000)

Excited states of a nucleon. TABLE 2.

~(1193)Pll ~(1385)P13 ~(1480) E(1560)

~(1670)D13 ~(1690) E(1750)S11 E(1770)Pll

~(1915)F15 ~(1940)D13 ~(2000)Sll ~(2030)F17

X](2250) ~(2455) E(2620) E(3000)

E(1580)DI3 ~(1775)D15 ~(2070)F15 ~(3170) ~(1620)SII ~(1840)P13 ~(2080)P13 ~(1660)Pll ~(1880)PlI ~(2100)G17 Excited states of a hyperon E.

120

The following model will be used to describe our system [1] * Spectrum Generating Group is chosen to be

S0(3, 2).

* The physical states form the irreducible representation D( 3, ½) of SGG. • The hamiltonian of the system is chosen to be (1)

=

Here ¢ is the Lagrange multiplier of constraint hamiltonian mechanics, P~, and/5 t, are the linear momentum and the velocity of a hadron and £ , are the generators of the S0(3,2) group. This hamiltonian is a relativistic generalization of an oscillator [2] Within this model the spectra and the radiative transitions of nucleon resonances are well reproduced [1]. We will now study more closely the mass spectra of the two families of nucleon and E resonances. These spectra, described by the same hamiltonian (1), show striking differences. These differences can be well seen in the mass distribution of the nearest-neighborspacing (NNS) of the states in the Fig. 1 and Fig. 2. The NNS spectrum of nucleon resonances is monotonic and decreasing while for E resonances the NNS spectrum shows distinct maximum.

Figure

10-

1

N Nucleon resonances

8' \ \

6'

\ \ \ \

4'

2'

0

0

I

I

I

I

20

40

60

80

A E

121

I

100

|

120

I

140

MeV

Figure 2 8-

N

\

/ /

Sigma resonances

\ \

6

\

/

'\ \

tI

4.

\ \ \

1! I 2-

\ \

!

\

t 1 0 0

\

, 2O

I

4-0

I

60

"l,-1oo

sb

2, E

i:~o MeV

The question arises how so different spectra of the same hamiltonian are possible and what might be the meaning of this effect? The problem of quantum manifestation of classical chaos is not straightforward. In quantum mechanics the notion of trajectory does not exist and thus for quantum systems the criterion for the chaotic versus regular motion has to be decided from the properties of the spectrum. In this respect a general view is that for nonintegrable chaotic systems the NNS distribution of the level corresponds to the Wigner distribution [3]

Pw(=) = ~ e x p ( - ~ =

),

= >_ o.

(2)

For integrable systems with regular motion the NNS distribution corresponds to the Poisson distribution

Pp(=) = ~--,

= _ 0.

(3)

It has been shown that there exist mechanical systems which at different energy ranges perform either chaotic or regular motion. Such a system is e.g. the magnetized hydrogen atom [4]. The quantum behavior of the magnetized hydrogen atom has also been studied [5] and it has been shown that NNS distributions of levels correspond either to the Poisson (regular motion) or Wigner (chaotic motion) distribution. For the case of the Poisson distribution in the discussed energy range there exists an additional approximate integral of motion. In Fig. i the continuous line corresponds to the experimental NNS distribution of levels

122

for the Nucleon resonances. The dashed line corresponds to the Poisson distribution. The experimental and theoretical distributions fit rather well. The internal motion for nucleon resonances is thus regular. In Fig. 2 the continuous line corresponds to the experimental NNS distribution of levels for the Sigma resonances. The dashed line corresponds to the Wigner distribution. The experimental and theoretical distributions again fit well. The internal motion for Sigma resonances is thus chaotic. If this interpretation is correct then one should expect that there exists in the case of Nucleon resonances an additional approximate constant of motion. The microscopic explanation of the effect of the transition from the regular to the chaotic motion may have its origin in the large mass difference between the non-strange and strange quarks. Acknowledgments It is a pleasure to thank the organizers of the Colloquium and especially Professors M.A. Markov and V.I. Man'ko for their kind hospitality and an excellent organization. This research was partially supported by the Research Program CPBP 01.03. References [1] A. Bohm et al., Int. Jour. of Mod. Phys. A 3, 1103 (1988). [2] A. Bohm et al., Phys. t~v. D 32, 2828 (1985). [3] O. Bohigas and M.J. Giannoni, in Mathematical and Computational Methods in Nuclear Physics, edited ny J.S. Dehesa, J.M. Gomez and A. Polls, Lecture Notes in Physics, Vol. 209 (Springer Verlag, Berlin, 1984). [4] M. Robnik, J. Phys A 14, 3195 (1981); J.B. Delos, S.K. Knudson and D.W. Noid, Phys. Rev. A 30, 1208 (1984). [5] D. Wintgen and H. Friedrich, Phys. Rev. Left., 57, 571 (1986); G. Wunner et al., Phys. Rev. Lett., 57, 3261 (1986).

123

THE DYNAMICAL GROUP OF RIEMANN ELLIPSOIDS G. ROSENSTEEL Department of Physics, Tulane University New Orleans, Louisiana 70118 USA

A dynamical group is constructed from a Lie algebra of physically relevant observables which characterize a system. A dynamical group may be exact so that the algebra's observables engird all the degrees of freedom of the physical system. Examples of exact dynamical groups include the Heisenberg group for a spinless non-relativistic particle, the Poincar~ group for a relativistic free particle, and gauge groups such as color SU(2). On the other hand, an approximate dynamical group models a complex system by including only the most important degrees of freedom in the algebra and omitting observables which are adiabatically deeoupled. Examples of approximate dynamical groups in nuclear physics include U(6) of the interacting boson model [1] and Sp(3,R) of the symplectie collective model

[2]. Dynamical groups are realized in quantum mechanics as unitary irreducible representations. Less familiar, but no less significant, dynamical groups arise in classical mechanics as transitive groups of canonical transformations on phase space [3]. In this article, GCM(3), which is an acronym for general collective motion in 3 spatial dimensions, is shown to be the dynamical group for the classical Riemann ellipsoids. The Riemann ellipsoids model rotating galaxies (period ~ 1015 see), stars (10ss), gaseous plasmas (?s), and rapidly-rotating atomic nuclei (10Z~s) [4-6]. The Lie algebra GCM(3) = [R6]GI(3,R) is a semidirect product of GI(3,R),

124

the motion group, with the abclian ideal R ~, spanned by the inertia tensor, which determines the system's spatial configuration. For a fluid with density p, the inertia tensor is given by L

(~ij --- ~ p Xi X.j d3X where X denotes the Cartesian position vector. For an ellipsoid, the eigenvalues of (~L equal (M/5)a~, where M is the total fluid mass and a t denote the seml-axes lengths. The Lie algebra of the motion group is spanned by the shear tensor L

Nij

~'~

J'p

XI

U L d3X j

where U L denotes the Cartesian velocity vectorfield of the fluid. the shear tensor, the vibrational momentum is given by

N..

~--

In terms of

(M/10) d 'a2'

the angular momentum by

and the Kelvin circulation by -1/'2 •

.

112

In order to use GCM(3) as a dynamical group, it is necessary to verify that it is spectrum generating, i.e. that the Hamiltonian H is in the G C M ( 3 ) enveloping algebra. If GI(3,R) is the motion group, then the velocity field is a linear function of position and the kinetic energy is known to be

K.E. = 1/2 tr(~t'(~-l':20 . The potential energy depends upon the specific problem. For a rotating star or galaxy, the potential energy is the gravitational self-energy, which is an elliptic integral functionally dependent upon the semi-axis lengths. It is a good approximation to take M2 P.E. ~ - - 3 5 (a 1+ a2 + a3)/3 For atomic nuclei, the potential is the sum of the Coulomb repulsion between

125

the protons and the surface attraction. These two terms are also just functions of the deformation and are approximately given by P.E. ~ + 3

5 (al+

Q2 . ~'s'4~(ala2 + ala3 + a2a3 )t3 a 2 + a3)/3

where Q is the total charge and ?~ denotes the surface tension. Thus, the potential energy is generally a function of the semi-axis lengths. Since these lengths are just defined by the eigenvalucs of the inertia tensor, the potential energy is a rotationally invariant function of the inertia tensor. Therefore, GCM(3) is a spectrum generating dynamical group. A symplectic manifold on which GCM(3) acts as a transitive group of canonical transformations is an invariant classical phase space for this dynamical group. Such phase spaces play the same role for dynamical groups in classical mechanics as unitary irreducible representations do in quantum mechanics. These phase spaces are constructed explicitly and exhaustively by the group's co-adjoint orbits. The GCM(3) co-adjoint orbits are indexed by the Kelvin circulation ~, whose square is the Casimir invariant of GCM(3). Hence, the Kelvin circulation is constant on each orbit. The orbit spaces are coset manifolds

~GCM(3)/SO(2),

~'~0,

14dim

# M = ~GCM(3)/SO(3),

~=0,

12dim.

The main result is as follows: Theorem: The Chandrasekhar-Lebovitz virial Newtonian equations of motion of a Riemann ellipsoid for which the conserved circulation is .~ are equivalent to the Hamiltonian dynamical system

on the co-adjoint orbit #.~ of GCM(3). The proof is given in Ref [7].

126

Acknowledgment This work was supported in part by the US National Science Foundation (PHY-8711380).

References 1. 2. 3. 4. 5. 6. 7.

F. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987). G. Rosensteel and D.J. Rowe, Ann.Phys. 126, 343 (1980). J.-M. Souriau, Structures des Systemes Dynamiques (Dunod, Paris, 1970). S. Chandrasekhar, EllipsoidaI Figures of Equilibrium (Yale University Press, New Haven, 1969). F.J. Dyson, J.Math.Meeh. 18, 91 (1968). G. Rosensteel, Phys.Rev. C41, RSll (1990). G. Rosensteel, Ann.Phys. 186, 230 (1988).

127

AN ALGEBRAIC MODEL OF CLUSTER STATES IN ODD-MASS NUCLEI G. L~vai and J. Cseh

Institute of Nuclear Research of the Hungarian Academy of Sciences Debrecen, P. O. Box 51, Hungary 4001 Introduction The interplay between collective and single particle degrees of freedom is an essential feature of many odd-mass nuclei. For the description of nuclear collective motion several phenomenologic algebraic nuclear models have been proposed. The first model of this kind was the Interacting Boson Model (IBM)"giving account of quadrupole collectivity in nuclei. The three dynamical symmetries of this model (called the U(5) [1], SU(3) [2] and 0(6) [3] limits) are able to describe various aspects of nuclear collectivity in even-even nuclei. Later the vibron model was introduced as an algebraic approach to dipole type collectivity [4, 5]. The 0(4) limit of this model has been applied to the rotational-vibrational motion of diatomic molecules, while the U(3) limit was used to classify cluster states in even-even light nuclei [6]. In connection with this application the role of the Pauli principle has also been discussed [7]. Later these phenomenologic models of collective motion have been extended to incorporate single particle degrees of freedom as well. The coupling of collective (bosonic) and single particIe (#ermionic) degrees of freedom is formulated in terms of group theory, with a group structure depending on the fermionic states taken into account in these models. While the fermionic extension of the IBM (called IBFM) has been worked out for each dynamical symmetry [8, 9, 10, 11], the similar treatment of the vibron model has only been carried out for the 0(4) limit [12], giving rise to the vibron-electron model of diatomic chemical molecules. We have introduced the missing U(3) limit of the vibron-fermion model [13] as an algebraic approach to the cluster states in light odd-even nuclei. The SU(3) x U(2) limit of the vibron-fermion model In this model the particle-like or hole-like fermions are allowed to occupy the states of an oscillator shell with n oscillator quanta. (This shell is not neccessarily a nuclear shell.) An essential point of this model is the decomposition of the fermionic angular momentum into orbital and spin part, resulting in the fermionic group structure UF(m) O U / ( m / 2 ) × V~(2) D SU[(3) x U#(2) O Of(3) × SU#(2) D SpinF(3) where I and s refer to the orbital and spin momenta of the fermionic states and m = (n + 1)(n + 2) is the number of fcrmionic states taken into account. These fermionic states with j = 1/2, 3/2, ...n + 1/2 are labelled with l = n, n - 2, ...1 or 0 and s = 1/2. The fermionlc group chain is then coupled to the

us(4) uB(3) suB(a) o'(3)

128

group structure of the U(3) limit of the vibron model, leading to the SU(3) × U(2) dynamical symmetry of the vibron-fermion model. (We chose this name in accordance with the corresponding limit of the IBFM [11].) The group structure associated with this limit is uB(4) x UF(m) D u s ( 4 ) × UF(m/2) x U~F(2)

vB(3) × SU[(3) × U#(2)

sus(3) × su#(3) × u~(2) su(3) × u~(2) 0(3) x su,q2)

sp~n(3) The basis states are labelled by the representations of these groups. Depending on the representations of the fermionic groups we can consider the coupling of particle-like and hole-like fermionic structure to the collective motion. Collective (duster) bands are labelled by the (A,/~) quantum numbers, while n~- is used to label states with different bosonic configuration. Similarly to the U (3) limit of the vibron model the Pauli principle is built in approximately by the truncation of the basis from the small n~ side [6,7]. The Hamiltonian associated with the group chain presented above is written as

H = / % +~sC,(UB(3)) +-rBC2(suB(3)) + 7C~(SU(3)) + &C~(O(3)) + etc.(Spin(3)) = H~+asn,r +2(0's + 7)QB" QB + (3(3'B + 0')/4 + 6 + e)LB. LB +6LE .LF -k eJR" JF =k47QB • QF + (37/2 + 26)LB • LF + 2eLB . Jv where QB, LB and QF, L f are the quadrupole and orbital angular momentum operators associated with the bosonic and fermionic structures, respectively, ,TF is the full fermionic angular momentum operator, while n~ is the number of ~--bosons associated with the relative motion of the clusters. Symmetry conserving third order terms of the type n,rC2(G) can also be introduced to consider effects depending on the bosonic structure. Besides this strong (SU(3)) coupling limit a weak (angular momentum) coupling limit of the model can also be formulated. The electromagnetic transition operators can be written as Hermitian combinations of the generators of bosonic, fermionic and coupled groups: T(s:)

T(~M') =

:

_ ~(2)

, ~(2) -{-p=p(2)

t/2~m/~ nu ~ 2 % $ F #

(gSL(B)~ + g,L(~)~-~,s ± " s 0)~ + gq[Q(2) x j o ) ] ~ ) + gp[p(=)x j 0 ) ] 0 ) )

(P(=) is a two-body bosonic operator P(=) describing transitions with

IA-=I =

2.)

T ~ ) = d[~t × ~1~) + d*[~t × ~l~') One nucleon transfer operators linking SU(3) × U(2) basis states with vibron model states can be expressed using the generators of bosonic groups and fermion operators.

129

Relation to other models Our model shows some similarity with the local cluster model of Buck eta/. [14] in several respects. Besides the structure of the energy spectrum, for example s the approximate treatment of the Pauli principle is similar in these two models. Our model is comparable to the nuclear vibron model [15], which also takes into account the excitation (quadrupole, rather than single particle) of one of the clusters. The basic idea of the model is the same as that of the vibron-electron model [12]. In this latter model the 0(4) (rather than the U(3) ) limit of the vibron model is used and the fermionic (electronic) single particle states taken into account are hydrogenic levels with a given principal quantum number. The role of the SU(3) and 0(4) groups as the degeneracy groups of the harmonic oscillator and Coulomb problem in three dimensions helps us to explain why the field of application of the 0(4) and U(3) limits of the vibron (and vibron-fermion) model are so different. From the mathematical point of view the SU(3) x U(2) limit of the vibron-fermion model can be related to the U(5) x U(2) [9] and SU(3) x V(2) [11] limits of the IBFM, while the vibron-electron model (with 0(4) × U(2) group structure) can be considered the generalised analogue of the 0(6) x U(2) [10] limit of the IBFM. (There are some other dynamical symmetries of the IBFM without equivalents in the vibron-fermion model. These are based on the isomorphisms o(5) ~- sp(4) and o(6) ~- su(4) [82 9, 10].) Applications As a possible application of the SU(3) x U(2) limit of the vibron-fermion model first we propose the o~-cluster states of the 192" nucleus for several reasons: /) The a-cluster states of the neighbouring 2°Ne nucleus have already been interpreted in terms of the V(3) limit of the vibron model [6, 7]. //) The mathematical formulation of the model is quite simple for this system. /fl') It is one of the most well-studied cluster systems. Many low lying (Ez ~_ 10 MeV) states of the l ° F nucleus are known to have marked 15N + a or 16O + t cluster character. Several cluster bands have been identified, some of which have equivalents in the 1sO + a system. Microscopic studies showed [16] that satisfactory results can be obtained taking only the 15N + a configuration into account and neglecting the 160 -b t configuration. At the same time the importance of the excited state of the 15N core with J~ -- 3 / 2 - has been emphasized. In terms of our model the fermionic states taken into account in this case are those with j r = 1 / 2 - and 3 / 2 - , so the corresponding oscillator shell is the one with n = 12 and the fermionic angular momentum is decomposed into I~ = 1- and 8 -- 1/2. We fitted a spectrum with SU(3) x U(2) dynamical symmetry (corresponding to strong coupling) to 25 well known cluster states belonging to 6 cluster bands [13]. The result of the fitting procedure is presented in the figure below. In addition to the usual one-- and two-body terms in the Hamiltonian two third order terms were also considered to give a more realistic description of the spin-orbit coupling. Our calculations in the SU(3) x U(2) tim.it showed that this simple phenomenologic model is somewhat less successful than other models in reproducing the E2 andf M1 tran.qition rates~ nevertheless it gives a better approximation of the E1 transition rates.

130

The study of the one nucleon transfer reaction going from the ground state of the 2°Ne nucleus to the cluster states of 19F showed that our results are in good agreement with the available experimental data [13]. Algebraic cluster models are unable at present to describe cluster spectroscopic factors, but work is in progress to cure this problem [17].

10

I

--512-

19 F

- -

712

-

--1112-

--31213/2" =====11/2-

--1312" --312"

--SI2"

-

512" ~3/2-

112~ 2 - 3 1 2 - - -

--312"

?p+ __5/2+--112 .

--712* --1312

--312"

--1/2

+

--712+

--3/2"

--1/2"

--1/2"

6 q =. ([j + m]q![j - re]q!) -½ (a[)S+'~(a~)S-'~10>q. (4.10) One easily verifies the commutation relations, and the eigenvalue relation: Jzq --* m. The value of j is determined by the weight of the highest weight state; it is easily found that: ½ ( ~ + ~ ) -~ j. The realization for SUq(2) defined by (4.9), and the set of eigenkets {lJ, rn)q} given in eq. (4.10) define every unitary irrep of SUq(2), with a (finite dimensional) irrep for every j = 0, ~, 1 1,..., with the vectors in an irrep j labelled by m, running by integer steps over the range j _> m _> - j . Let us next consider mixed s y m m e t r y states in Uq(2). To construct these, recall first the situation for U(2). The mixed symmetry states here are denoted by the \

Cel'fand-Weyl pattern

(m) = (m12 rn22 ~, with m12 > m l l _> rn22. (The state ljm) \

T~11

/

corresponds to putting m22 = 0, 2j = ram, 2m = m n -mz~.). Using the boson calculus we find I rn12mnm22> = M - ½ ( a 1 2 ) m 2 2 a , ~ n - , ~ 2 a ~ = - m ~ t l O ) , (4.11) where M is the norma~zation, am =- a~ b2 - a2bl, and {bl, b2} is a second (commuting) set of boson operators (further details may be found in Ref. 7). What are the q-analogs of these states and, more particularly, what is the q-analog to the antisymmetric combination a12? We can answer this question explicitly using q-boson operators, and to do so we introduce a second (commuting) set of q-bosons {bl, b2} and form the generators by co-multipllcation (we drop the q index now): J+ = q(N~-N~)/dal~ 2 + q-(N~-N~)/dblb2,

(4.12a)

o7_ = q(N~t-N~)/da2~ 1 + q - ( N ~ - n ; ) / a b 2 b l , 1 1 J=--- :Z ( -N -~ - N ~ ) + Z~ - (N~ - N~) , - - - - - -

(4.12b)

153

(4.12c)

where the superscripts a, b refer to the q-boson sets {a), {b) respectively. The generators in (4.12) act on the space V ® V of polynomials in al,a2, bl, b2. Since the finite dimensional representations of Uq(2) are in correspondence with those of U(2), we seek an irreducible subspace of V @ V in which the states carry the labels mii. The state of highest weight is annihilated by J+, j + ma2m12m22> = 0,

(4.13)

and so this state must be a linear combination of the polynomials \ amll--8.mt2--ml•+aLsLrn22--slN 1 ~2 VlV2 IV/~

(4.14)

where 0 _< s _< m22, since these states are eigenfunctions of Na = N~ + N b and N2 = N~ + N~. The specific linear combination 19 is found by solving (4.13) and the general state is found by applying the lowering operator J_. By direct calculation one

gets:

) m12

~22

rn22 r Xs.m(s)

=

M-½ 2

mll

~-/A___

~mll--a,~s'brn12--mll~81~m22--slrl\

,=0 [s]q![m2~ - s]q]'~l

"~-

"1"2

I"t,

(4.15)

where

re(s) = [(s + m12 - m n

-b 1)m22 -- sin12 - 2s1/4,

(4.16)

[m12 -b 1]q![ml2 - mll]g[[mll - m22]g][m22]q[ [m12 - rn~2 + 1]q!

(4.17)

and M =

Remarkably, the unwieldy expression (4.15) can be compactly written in the operator form I m12 m 2 2 / = M-½a~22a~,l-m22a~-mH[OI (4.18) ~11 where a12 is the operator defined by:

a12 = q(N~+N~+l)/4alb2 -- q-(N~+N~+l)/4a2bl.

(4.19)

This operator is the q-analog to an an~isymmetric boson pair operator, and shows no symmetry under the exchange ai ~ bi. The generators also show no symmetry under al ~ bi, since, as noted earlier, the addition of q-angular momenta depends on order. The proper q-symmetry operation here is R: al ~ bl, q ~ q - l , which obeys 1~2 -- 1. Under this operation one sees that the generators are q-symmetric, the monomial operators are q-symmetric, and the q-boson pair operator a12 iS q-antisymmetric. The states (4.18) are q-analogs of the boson states (4.11). An important difference, however, is that the operator a12 in (4.19) does not commute with al, a2, bl, b2; unlike the q = 1 case, the ordering in (4.18) is essential. This property, and the structure of 154

(4.19), which derives ultimately from co-multiplication, requires further investigation, but already one sees a hint of q-analog structures in symmetric function theory.

B. D e f o r m i n g

Func~io~al,.~ 20'21'22'23

The fact that all unitary irreps of SU(2) and SUq(2) have exactly the same dimensionality (for generic q) strongly suggests that the formal deformation involved may be made fully explicit; this is the idea of a 'deforming functional', Q, introduced by Curfright and Zaz~hos2°. Since the generators Jzq and J~ have identically the same spectrum, Jzq is itself not deformed; that is =

Q(s

) =

Jz.

(4.20)

For the generator J_~ one finds:

:$

=

Q(:+)

=

([jop - J, lq[jop + Jz + !]q~ 1/2

(4.21)

with

jq_ = Q(j_) = (j~_)t = (Q(j+))t.

(4.22)

In this result one has introduced the formM operator jop defined as the positive solution of the quadratic Casimir invariant. (In the boson calculus Jop is realizable as a bona fide linear operator.) The mapping Q defined above maps the generators of SU(2) into the generators of SUa(2 ). For generic q, this map is, in fact, invertible, and links the representation structure of SU(2) with that of SUq(2). Of particular interest is the fact that the deforming map Q links the composition law for representations in two systems. The map Q thus induces a non-commutative co-product for SUq(2) from the corresponding commulalive co-product in SU(2), and similarly for the other Hopf algebra operations. This aspect of deforming maps has been studied in detail 2°. The map-induced co-product may, however, be quite difficult to handle in particular cases. When q is not generic, but a root of unity, the deforming map is singular, and may not always be well-defined24. Where the map is well defined, it is of considerable help in understanding the structure of the indecomposable representations. Non-invertible deforming maps have been studied2s for SU(1, 1) and in connection with current algebras. 26

C. Other Realizations of SUq(2): As the deforming map approach might suggest there can be many alternative deformations that realize SUq(2).

155

Let us note that Sklyanin's original definition 27 of SUa(2) gave a trigonometric deformation:

[S0,S3]=0,

[S3,S4-] = (SOS+ + S±SO),

[S+,S_]=4SOS3,

[so, s ~ ] = ± tanh ~ ~ ( S ~ & + S z S ~ ) ,

where

80~ - S~ t ~

(4.23a, b,d)

~ ~ = 4 sinh 2 ~. (4.24~, b)

Woronowicz2s gave a very differently appearing realization: [y0,y+]~, = s~voy+ - ~ v + v o

= v+

[v-,v0],, = v_

1

[V+, V_L/, =_ ~V+ V_ - sV_V+ = Vo.

(4.25a, b) (4.25c)

Rather similar results (two versions) were found by Witten 29. The deformation having the most symmetric appearance is that of Falrlie22:

qXY-q-IYX=Z

qYZ-q-IZY=X

qZX-q-IXZ=Y.

(4.26a, b,c)

The Casimir invariant here is a cubic form.

5. The Fundamental Theorem for q-Tensor Operators Applications of symmetry structures in quantum physics is heavily dependent on the fundamental theorem for tensor operators (the generalized Wigner-Eckart theorem). Such a theorem need not ezis~ for avly given symmetry structure, but depends rather upon the specific way in which the symmetry is realized 7. In the prototypical symmetry structure in quantum physics--the quantal rotation group SU(2)--there are two required properties of the realizationT: gquivariance (the action of the group generators on a tensor operator (a set of operators) realizes a linear representation defined by transformations on this set) and Derivation (the generators act on products as a derivation: Ji(ab) = Ji(a)b + aJi(b).) It is a consequence of these two properties that the Wigner-Ctebsch-Gordan (WCG) coefficients for SU(2) occur in two logically distinct ways: (a) as coupling coe3ficien~z for the addition of angular momenta carried by kinematically independent constituent systems, (the Clebsch-Gordan problem) and, (b) as matrix elements (up to a rotationally invariant scale factor) of physical transition operators, (the Wigner-Eckart problem). Conversely, if equivariance and derivation are not valid for a given realization, then this latter result (b) fails 3°. It is not obvious that one can extend both (a) and (b) to quantum groups, particularly when one realizes that the derivative property corresponds to a commuta$ive co-product, which is invalid for a general quantum group. Moreover, the equivaviance condition is problematic as well. For a Lie group, equivariance is effected by the adjoint "156

action: ad=(y) = [x, y]. This cannot work for a quantum group, since the quantum group irrep corresponding to the adjoint representation is finite dimensional in contrast to the infinite number of linearly independent generators obtained under commutation. The resolution of this problem for quantum groups has been given by Biedenharn a~d Tarlini 31 and by Kirillov and Reshitikhin 3~. We follow Ref. (31) below. First we formalize the Hilbert space on which the operators act to be a model space a3, M, defined to be the direct sum of vectors carrying unitary irreps of the group G, each equivalence class of irreps occurring once and only once. The operators on M are defined to belong to the linear space T with the action: T:

M --+ M.

(5.1)

We introduce now the key concept of an induced acgion; this is a mapping: T ®T

, T,

(5.2)

which we will denote by A ( B ) = C, that is, A acts on B to give C, with A, B, C ET. (One must carefully distinguish this action from the natural product of operators on Hilbert space, which we denote as usual by juxtaposition: AB.) Using these concepts, we can now recognize that the standard setting for the Wigner-Eckart theorem consists first of the induced co-product (A):

A(,]+)(ai ® Ira)) = J+('n) ® l(Im)) + l(al) ® .]+ (Ira)),

(5.3)

followed by a mapping c: c:

T@M

.~ M.

(5.4)

Applying this operation in the standard setting, yields:

,r+(ailm)) = S+(ai)lm)

+

aiJ+lm),

(5.5)

for which one uses the induced action ]+(ai) = [J+,a/], that is, the adjoint action. The novelty in this construction is that the co-product A ( T ) acts, via (5.3), on the tensor product T ® M of differen~ vector spaces and, in addition, there is a further operation (the mapping c) which requires that the induced co-product and the induced action be compatible. The compatibility of these two structures is guaranteed for the standard (Lie group) case since the standard matrix action of operators in Hilbert space induces (by commutation) a commutative co-product (a derivation) and the induced commutator action of the generators realizes, by equivariance, an irrep cas'ried by the tensor operators (this compatibility is the content of the tensor operator theorem (generalized Wigner-Eclmrt theorem)).

157

The compatibility requirement can be expressed most succinctly by using the language of diagrams. Consider the following diagram (where E is a generator): T ®M

Zx(E) T ® M ....

~

cI M

cI s

(5.6)

M.

The requirement of compagibility is that the diagram above be commutative. Assuming a given co-product determines a compatible action (or vice-versa). Thus, for example, for a Lie group one uses the diagonal (commutative) co-product and determines the commutator to be the compatible induced action.

It should now be clear how to extend this structure to tensor operators for quantum groups (q-tensor operators). Let us formalize these considerations: DEFINITION: Let T denote the vector space of operators mapping the model space M of the compact quantum group Gq into i~self." M T~ M. An irreducible q-tensor operator is a set of operators, {t~.,~} 6 T which carries a finite-dimensional irrep E, with vectors ~, of the quantum group Gq. That is:

E~(t~,~) --- ~

(Z, ~'[E,,I=. , ~)t.n,~,,

(5.7)

where E~ is a generator of Gq, E~(t~,~) denotes an action o{ E~ on T , and ( . . . > denotes the matrices of the generators for the irrep =. A q-tensor operator is accordingly a linear combination of irreduciMe q-tensor operators, with coefl]cients invariant under the qgroup action.

THEOREM. /f {t~,~} is a q-tensor operator of the compac~ quantum group Gq such that the co-product of Gq is compatible with ~/ae action E~(tv.,¢), that is, d/agram (5.6) is commutative, then ~he matrix e/emends of {t~,~} ha M are proportional to the q-WCG coet~cients of G a with the constant of proportionality an invariant. Conversely, if {t=_,~} is a q-tensor operator and the ma~r/x elements of {t~,~} are proportional to the q-WCG coett~cients, then d/agram (5.6) is commutative.

6. T h e Algebra of q-Tensor Operators The q-WCG coefficients are the matrix elements of au inver~ible map (denoted W) from M®M into M. Expressing this diagrammatically we have the following commutative diagram: M®M w M A(E~) 1 M®M

1El W-1

- -

158

M.

(6.1)

Now, because of the way the action has been defined on the q-tensor operator {tm,~}, for t,,a acting on the vector I~ ) - - a n element of T ® M - - w e have a well-defined compatible co-product action A(Ei), with T ® M replacing M ® M above. It follows from the diagrams that the mapping c can be identified (to within an invariant factor) with the mapping W, that is, with the q-WCG coefficients. Moreover the co-product A has a well-defined compatible action that extends to T ® T thus yielding a further commuting diagram. It follows that we have as corollaries to the q-tensor operator theorem: COROLLARIES:

(a) There exists an algebra of q-tensor operators: T®T

W

T,

carrying products of irreducible q-tensor operators into irreducible q-tensor operators. (b) There exJsts a product carrying an irreducible q-tensor operator into an invariant (having the properties of an inner product). Thus a norm exists and irreducib]e unit q-tensor operators (T ) axe well-defined whose matr/x elements, by the fundamental tensor operator theorem, are q-WCG coeJtJcients. (c) Denote the mapping in Cor.(a) by T ~ T 6 T, and denote the invarian~ product in Cor. (b) by T . T. Then the (6-j) operators are defined by: ~£.(T ~ T). Similarly (3n-j) operators can be defined. To clarify the meaning of these results let us give some examples. Example 1: Define the operator pair by:

4:=1,,_.

_N_z

q ',

-

4½,_½ -

(6.2a, b)

This pair is a q-tensor operator of SUq(2), with the explicit induced action

J :(0

=

-

(6.3)

(Note that for q --* 1 we recover the commutator action.) One verifies that this induced action on 4x -~z obeys the equivariance condition with j = ~, 1 that is:

• By using q-WCG operator coupling, this result for the spin-½ q-tensor operators extends 4he induced ("adjoir~t') ac4ion to all q-tensor operators in SUq(2).

159

Example 2. It was remarked earlier that the generators-of SUq(2) are not irreducible q-tensor operators. The q-tensor operator in SUg(2) carrying the three-dimensional (adjoint) irrep is denoted Tl,m with m = q-l, 0, and has the explicit matrix elements: 54

(J,M + IITI,+IJ, M ) = ( - 1 ) q -~f'~

([2]q[J -

Mlq[J + M + 1]g) 1/2 ,

(J,M[TI,o]J,M) = (q_(J+½-M)[j + MI q _ q ~ [ j _ M--1

M],),

(J,M - I[T~,_IJ, M ) = q ~ ([2]g[J + M]q[J- M + 1]q) 1/2.

(6.5a)

(6.5b) (6.5c)

The Casimir invariant (scalar product) TI" T1 has the eigenvalue: T1. T1 ~

[2J]g[2J + 2]g.

(6.6)

The unusual form for T1,0 in (6.5b) should be noted. One can verify that under the induced action Tl,m transforms properly as the adjoint q-irrep; for q --* 1 we recover the commutator action and the usual matrix elements of 2J. It is interesting to remark that less systematic approaches yield forms 29'32 for the Casimir invariant different than (6.6) and distinct from each other, forms that introduce fracgiortal q-numbers. (Although q-integers do not form a ring, all operations in SUg(2) do preserve the q-integer form. In particular, fractional q-numbers axe never necessary.) We remark that the induced action allows one to dispense with the Serre relations for the q-tensor operators of generator type (adjoint q-irreps). The algebra of tensor operators, and its q-tensor operator extension, are thoroughly understood for SU(2) and SUq(2) but have not been as fully developed for SU(3) and SUg(3), although an existence theorem has been proved. 34

These opera~or algebras are eharaegerized by a well-defined artd fully ezplicit opera~or product ezpansion. Little is known of their subalgebra structure although it is known that the universal enveloping algebra is a subalgebra and it is believed that some diffeomorphism group commutator algebras axe indeed subalgebras. 7. A p p l i c a t i o n s o f Q u a n t u m

Groups

It is important to indicate that there axe sound physical reasons for the current

interest in quantum groups. Within the.limitations of this overview, it is best simply to cite here recent important applications. (a) Solvable two-dimensional systems, with factorizeable g-matrices, via inverse scatfeting techniques and the Yang-Baxter equation: Faddeev 35, Kulish and Reshetikhin 3s, Sklyanin 3T, Jimbo ss, Ge et al39, Burroughs40 and Takhtajan 3~. (b) Solvable lattice models in statistical mechanics; anisotropic spin chain Harniltoniarm: de Vega~1, Pasquier and Saleur 12, Batchelor et al42. 160

(c) Rational Conform~l Field Theory: Alvarez-Gaum~ 11, Moore and Reshetikhin 4S, G6mez and Sierra 44. (d) Two dimensional gravity: Gervais 45. (e) Three dimensional Chern-Simons theory: Witten 29, Guadagnini 46, Majid4L (f) Knot theory applications: Kauffman 4s. (g) Symmetry interpretation of q-hypergeometric functions, q-strlngs: Romans s°.

Lohe 49,

(h) Non-commutative geometry" Martinsz. (i) Non-standard quantum statistics:Greenberg s2. (j) Applications to nuclear rotational spectra: Smirnov et als3. 8. Acknowledgements: I wish to thank the organizer, Professor Vladimir Man'ko for the opportunity to take part in this colloquium (held at a most interesting time!) and for inviting this review. I wish to thank my colleagues Professors Marco Tarlini and Max Lohe, for their help; our joint work has been the basis for this review. I am indebted to Dr. Cosmas Zachos for the favor of a preprint 24 version of his timely review of this field.

References 1. E. Sklyanin, L. Takhatajan, and L. Faddeev, Teor. Mafh. Phys., 40 (1979) 194. 2. P. Kulish, N. Reshetikhin, Zap. nauch, serninarov LOMI, 101 (1981) 101, ibid. J. Sovief Mafh. 23 (1983) 2435. 3. V. G. Drinfeld, Quanfum Groups, Proc. Int. Congr. ofMa~h., MSRI Berkeley, CA, (1986) 798; Soy. Math. Dokl. 36 (1988) 212. 4. M. Jimbo, Left. Mafh. Phys. 10 (1985) 63. 5. G. Lusztig, Adv. in Math. 70 (1988) 237. 6. M. Rosso, Commun. Math. Phys. 117 (1988) 581. 7. L.C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications, 8, Addison Wesley, 1981; reprinted Cambridge University Press (1989). 8. M. Jimbo, Commun. Mafh. Phys. 102 (1986) 537. 9. G. Lusztig, Conf. Mafh. 82 (1989) 59. 10. P. Roche and D. Arnaudon, Left. Math. Phys. 17 (1989) 295. 11. L. Alvarez-Gaum~, C. Gomez, and G. Sierra, Nud. Phys. B319 (1989) 155. 161

12. V. Pasquier and H. Saleur, Nucl. Phys. B330 (1990) 523. 13. L. C. Biedenharn, J. Phys. A. Math. Gen. 22 (1989) L873. 14. A. J. Macfarlane, J. Phys. A. Math. Gen. 22 (1989) 4581. 15. C.-P. Sun and H.-C Fu, J. Phys. A. Math. Gen. 22 (1989) L983. 16. P. Kulish and E. Damashinsky, J. Phys. A23 (1990) L415. 17. M. Chaichian and P. Kulish, Phys. Left. 234B (1990) 72. 18. M. Chaickian, P. Kulish, and J. Lukierski, Phys. Lett. 237B (1990) 401. 19. L. C. Biedenharn and M. A. Lohe, invi%ed paper at the Rochester Conference (honoring Prof. S. Okubo) 4-5 May 1990 (to appear in the Proceedings, World Scientific, Singapore). 20. T. Curtright and C. Zachos, Phys. Left. 243B (1990) 237. 21. T. Curtright, Miami prepring TH/1/89, to appear in Physics and Geometry, L-L. Chau and W. Nahm, eds., Plenum, 1990. 22. D. Fairlie, J. Phys. A23 (1990) L183. 23. A. Polychronakos, Florida preprint, UFIFT-90-14, 1990, to appear in Proceedings of the Argonne Workshop on Quantum Groups, T. Curtright, D. Fairlie, and C. Zachos (eds.), World Scientific, 1990. 24. C. Zachos, ANL-HEP-PI~-90-61 to appear in Symmetries in Science V, B. Gruber (ed.), Plenum (N.Y.). 25. D. Fairlie, J. Nuyts, and C. Zachos, Phys. LetL 202B, (1988) 320. 26. H. Itoyama and A. Sevrin, Stony Brook preprint, ITP-SB-90-12, February 1990. 27. E. Sklyanin, FuncL Anal. Appl. 16 (1982) 263. 28. S. Woronowicz, Comm. Math. Phys. 111 (1987) 613; 130 (1990) 381. 29. E. Witten, Nud. Phys. B330 (1990) 285. 30. D. Aebersold and L. C. Biedenharn, Phys. Rev. A15, (1977) 441. 31. L. C. Biedenharn and M. Tarlini, LetL Math. Phys. 20, (1990) 271. 32. A. N. Kirillov and N. Reshetikhin, Harvard Preprint 90/B261 (April 1990). 33. I. M. Gel'land and A. V. Zelevinsky, Societ~ Math. de T~ance, Astdrique, hors serie, (1985) 117. 34. D. Flath and L.C. Biedenharn, Can. J. Math. 37 (1985) 710; L.C. Biedenharn and D. Flath, Commun. Math. Phys. 93 (1984) 143.

162

35. L. Faddeev, N. Reshetikhin, and L. Takhtajan, A1g. Anal. i (1988) 129; also in Braid Group, Kno~ Theory and Statistical Mechanics, C. Yang and M. Ge (eds.), World Scientific, 1989. 36. P. Kulish and N. Reshetikhin, Le~. Ma~h. Phys. 18 (1989) 143. 37. E. Sklyanin, Uspekhi, Mat. Nauk. 40 (1985) 214. 38. M. Jimbo, Int. J. Mod. Phys. A4 (1989) 3759. 39. M.-L. Ge, Y.-S. Wu, and K. Xue, Stony Brook preprint, ITP-SB-90-02. 40. N. Burroughs, Comm. Math. Phys. 127 (1990) 109. 41. H. J. de Vega, Int. J. Mod. Phys., 4 (1989) 2371. 42. M. Batchelor, L. Mezincescu, R. Nepomechie, and V. Pdttenberg, J. Phys. A23 (1990) L141. 43. G. Moore and N. Reshetikhin, Nud. Phys. B328 (1989) 557. 44. C. G6mez and G. Sierra, Phys. Left. 240B (1990) 149. 45. 3.-L. Gervais, Comm. Math. Phys. 130 (1990) 257; Phys. Left. 243B (1990) 85. 46. E. Guadagnini et al., Phys. Let~. 235B (1990) 275. 47. S. Majid, In~. J. Mod. Phys. A5 (1990) 1. 48. L. Kauffman, Int. J. Mod. Phys. A5 (1990) 93. 49. L.C. Biedenharn and M.A. Lohe, "Quantum Groups and Basic Hypergeometric ]~'hnctions", to appear in the Proceedings of ~he Argonne Workshop on Quantum Groups, T. Curtright, D. Fairlie, and C. Zachos (eds.), World Scientific, 1990. 50. L. Romans, in Strings '89, R. Arnowitt et al. (eds.), World Scientific, (1990). 51. Y. Manin, Comm. Math. Phys. 123 (1989) 163. 52. O.W. Greenberg, to appear in the Proceedings of the Argonne Workshop on Quantum Groups, T. Curtright, D. Fairlie, and C. Za~hos (eds.), World Scientific, 1990. 53. P.P. Raychev, P~. P. Roussev and Yu F. Smirnov, J. Phys. G: Nucl. ParL Phys. 16 (1990) L137. 54. M. Nomura, J. Ma~h. Phys. 30 (1989) 2397.

163

INTEGRABLE THEORIES, YANG-BAXTER ALGEBRAS AND QUANTUM GROUPS : AN OVERVIEW H. J. de Vega LPTHE, Universit~ Paris VI, Tour 16, ler. etage, 4, Place Jussieu, F-75230,PARIS Cedex 05, FRANCE.

Integrable massive QFT and conformal invariant models follow from lattice integrable models in suitable scaling limits. There are Yang-Baxter algebras (YBA) associated with all these two-dimensional models. These YBA allow one to construct the exact solution (spectrum, S-matrix, form-factors .... ) for this class of theories. Braid groups and quantum groups are derived as limiting cases of YBA when e (spectral parameter) goes to +oo.

Integrable lattice models (on vertex or faces), integrable (massive) quantum field theories and conformal (massless) theories are the subject of this short review[I]. The interrelation between these three subjects is described by the triangle depicted in fig. 1. The two upper sides of the triangle constitue the "royal way" to construct massless and massive QFT : starting from a critical lattice model in a finite volume, the continuum limit can be rigorously constructed letting the lattice spacing a to zero and the volume V to infinity. Moreover the exact solution of the continuum theory follows as the a = 0 limit of the lattice model solution. The light-cone approach[2,3, 4] is probably the more powerful one to derive massive QFT as scaling limits of integrable lattice models. No parameters are loosed in the scaling limit within the light-cone approach. It must be stressed that both the massless limits (Conformal Field Theories, CFT) and the massive limits are u n i v e r s a l . That is, all models in the same universality class (integrable and non-integrable) posses the same scaling behavior. The CFT describe the behavior at criticality : a 2

,

b i bi+ 1 b i = bi+ 1 b i bi+ 1 (20) This is precisely a braid group. In other words, the ultrarelativistic limit of the YB algebras provide braid group representations. This fact has been recently exploited successfully to compute link and knot invariants[20]. Let us study the e = + * limit of the six-vertex YB generators. The six-vertex model being the simplest non-trivial trigonometric/hyperbolic YB solution. We find[ 1,21] T11(e)

= (y_+)N exp(2,fSz)

[1+O(y+-2)]

172

e --) +oo

T22 ( 0 ) = (y±)N exp( +7 Sz) [ 1 + O(y± "2) ] (21)

e ._) +c~ T12( 0 ) = (y±)N-1 sh 7 J-( +7 ) [1+O( y±-2)] 0 ---) +oo T21(e) = (y±)N-1 sh 7 J+(+7) [1 +O(y±'2)] 0 --) +oo where

y+ = + exp[+(e + 7/2)] , S z = (1/2) ~. (O-a)z 1,.

be eon-

(2.6)

The nonmalizing faotor oaloulated by using the relation

J+J_" = J_~J+ + [n]J ~-I[2Jo-~+I ]

(2.7)

Per the finite dimensional IR DJ only :I=0,1/2,1 .... and m=$,$-1, .... -~ are allowed. Thus the structure of SUq(2) IR's is similar to the IR's of usual SU(2) algebra and the dimension of IR's are the same in both of oases is equal to (2J+I). Aoting by generators J_ and J+ on vector (2.6) we obtain

~lJm>

= ,,/[J+m][J-m+l]

IJ,m-~>

,

(2.8)

J+lJm>

= J[J-m][$+m+l]

IJ,m+l> .

(2.9)

Thus the explicit form of D J IR for SUe(2) eoinoides with ooz~espending formulae for SU(2) exoept for the substitution of usual number (J±m) and (J~m+l) by q-numbers in two last rows. In the theoz-J of SUq(2) IR the important role plays the 0asimir operator 2 2 C 2 = J_J+ + [Jo+I/2] = J+J_ + [Jo-I/2] , (2.10)

184

whioh is Hermitian one. The vectors (2.6) are its eigenveetors

C21dm>= r J + l / Z ] Z l E m >

.

(2.11)

3. Projection operators for s ~ ( 2 ) algebra First of all we are interesting in the pr~jeotion operator ~ j , j = ~ having the property

~lJ'm=J> = o~,j, tJJ>,

(PO)

(3.1)

i.e, aoting on an ambitrar~ veotor ]J) of weight m=J

Ira=J) = ~ Bj. IJ'J)

(3.2)

J" >4 the operator PJ projects the oomponent veotor of IR

tUlJ)

IJ J> being the highest weight

= B:FtJJ) •

(3.3)

Similarly to Refs. [22-25] we seek this PO as a power series of gener-ators j+ and J_ OO

r~oC J_~J+~

(3.4)

The exponents of these generators are the same due to oondition

(3.5)

~,Jo J = o. Sinoe

l~lJJ> = IJJ> ,

(3.6)

we obtain because of (2.6) that

O0

=

J+

/~tJ)

(3.7)

t

and CO

= J+

,.~oC,,. J_'J+'tJ)

= o .

By u s i n g o f Eq. ( 2 . 7 ) the f o l l o w i n g r e o u r r e n t r e l a t i o n olents oan be found Or_ 1 + C t " ] [ 2 J + r ' + l ] O

(3.8)

for C

ooeffi(3.9)

= 0 .

Solving it we have [2J+1 ], O = (-1)" r [ r ] , [2J+r'+l ] , Obviously the PO is Her~itian one

(3.10)

( ~ ) + = P'I. (3.11) By H e r m i t i a n o o n j u g a t i o n o f Eq. ( 3 . 8 ) we o b t a i n an i m p o r t a n t p r o p e r t y of PO P J = 0 . (3.12)

185

The operator

19j_ . -J

= ~.OCr. J +'J-"

(3.13)

,.=

with the same ooeffioients C on the lowest weight of IR The most general

as in Eq. (3.10) is an extremal projeotor .

form of projeeting

operator

!~,

-/ [2J],.[J-m],J-J-'~ 19 ÷

~/ [2J],[J-m' ],

will need in further oaloulations. These P0s have the properties

4.

"Vector coupling"

o f q-ap4pzlar momenta

Now we turn to the of "veotor ooupling" of angular momenta in the oase of SUq(2) algebra . The generators of summar~ angulam momentum J ( 1 , 2 ) ave of the form

J~(1,2) = Jo(l) + Jo(2),

(4.1a)

,;o(2) + q-Jo (1) J±(2)

J~+(1,2) = J+_(1) q

In standard notation for Hopf written in the following form

algebras

(4.1b) the relations

(4.1) must

be

J~o = Jo ®I + I@Jo '

= Z±¢q

Jo

+ q

-Jo

cJ, .

However we shall use below notation (4.1) in order to oonserve the maximal possible similarity with usual AMT. It is easy to prove that the operators mutation relations (2.1) and (2.2).

(4.1) are satisfying to oom-

The action of generators (4.~) on basis veotors tensor produot of IRs is given by formulae

~ ( l . 2 ) lJ,m,>lJEaz> = (m~+m2)lJ~ml>lJ~2> J~(l.2) lJlral>lJ#',2>

= q

IJ1m1>lJ2m2> of the

,

<J1'ml±llJ±lJlta1>lJlm~±l>lJ~2>

+ q "-'~I <Jz,mz±llJ±lJzmz>lJlral>lJzm2±l> It should be remarked that the q-analog of the binomial formula is valid

186

(4.2a) +

(4.2b) expansion

r"

~

JO ( 2 )

[r] '

s

=

8

I s ] ,. [ r - s ]

.v

g±(1) J±

-Jo ( 1)

~-B

(2) q

r

SJo(2)-(~-a)Jo(1) .

(4.3)

The generalization of the veotor eoupling prooedure on the ease of SU (2) oonsists in the following. It is n e o e s s ~ to oonstr~let from thqe tensor produot basis veoto~-s IJ1m1>IJ2m2> suoh linear oombinations

I,]1,,f2Z,~m>cl =

~. <Jlml;J27/z21J11z>q tJll'ltl>lJ211z2> m l "~2

(4.4)

Whioh belong to the IR ~ of SUq(2), i.e. they are eigenveetors of the Oasimir operator with eigenvalues

A=[J+½]z:

C~(I,2)

o~(1,2) lJlJz;Jm> = rJ+l/2]ZlJ~J2;Jm>q

(4.5)

The ooeffielent <Jlml;J2tlz21Jm> in linear combinations (4.4) are called as 01ebsoh-Gordan ooeffioients (q-0G0) for quantum algebra. To find them we shall use the P0 appPoaoh. Simultaneously the struotume of Clebsoh-Gomdan series for SUq(2) will be found or mope OorTeotly it will be oonfirmed that the 01ebsoh-Gordan series for SUq(2) ooinoides with the Eq. (4.1).However before to tulna to this point it is pertinent to list the orthormality relations for the q-0G0s

SUq(2)

<J/m,;JznzzlJm ~ <Jl=/;J_jA, zlJ'm' ~ = O.,,,,,O,~, ,, ml,m2

(4.5a)

j~.~ <jl,nl ;jEn~lJm>q q = o,~ ,"~, o '~z,"~, .

(4.5b)

The Q-0G0s form an oPthogonal matrix and the following equation whioh is inverse with 1~espeot to trg/Isformation (4:4) is valid

IJ,ml>lJ?z> ~ ~

<Jlr~,;Jz~zI,J=> tJlJz;Jnz>

(4.6)

5. Q-analogs of Clebsch-Gordan coefficients Using PO we oar, write the veotor (4.4) in a form

IJ,.I2;.I,~ = r % ) - ' ~ : ~ , r l.2)lj,m;( l ) > l j ~ ) c e ) > .

(5.1}

where m'--m~+m2. Thus the q-OGO oan be oaloulated as the matrix element of PO

<J lm~ ,J ~ z l Jm>q =

= (Q'q)-'<j~m, (1)l<jEne(z)I~:~,

' (~ .z)ILm; (~)> I,'E,/2)

>

(5.a)

where Qq is a normalizing faotoP. As for values of re;and m~ in Eqs. (5.1) and (5.2) they oan be ohosen in avbitr~r~j manner but for simplifioation of oaloulations it is oonvenient to take m~=J I and m~=J-J I . 187

Then the Eq.(5.2) oan be rewritten in the form <J~m~;J2m21Jm~ =

=

(~)-~<J~(1)l<J2rnz(2)l~:~(~,Z)lJIJ~(~)>lJz,J-J~(2)>,

(5.3)

<j1jl(l)l<jz,j_jl(z)l~:~(1,z)lj~jl(l)>ljz,j_J~(2)>"

(5.4)

where ~q2

=

Sines IJIJ1(1)> is a highest weight veotor the generators J+(1) in PO PJ'qfl,2) o a n b e omitted and for the normalizing factor Qq we obtain

Qq2

=

<j1jl;jz,j_jlljj>2 = <j2,j_jl(2)l~V)(g)lj2,j_Jl(2)>"

(5.5)

where (-1)rE2J+11 '

1=~,q)(2) = ~ . (J1 r>..o [rJ![2J+r+l]! q

-2~j

"

l j ~(2)j+r(2). -

(5.6)

Let's adopt an usual phase oonvention for Qq being positive (arithmetie) square root of Qq2. As a results all q-OGOs will be real. It is clea~ from Eq.(5.3) that only values of summary angular momentum J satisfying the oonditions -J2lJ2,J-J~ (2)> =

<JIJ1 (1) f<J2,J-Jl (2)IP J'q(1,2) IJIJ1 (1)> I,f2,J-J I (2)> 1/2"

(5.7)

To oaloulate the numerator of this expression it is neoessar7 to express PO P~,jCIJ'q,2) in tez~ns of generators J+(1,2)_, then to do the binomial expansion of thei~ powers in terms of J±(1) and J±(2) using the Eq. (4.3). The last step is a oaleulation of matrix elements of j+s(~) using formulas (2.8), (2.9).Here we give an explioit expression only for denominator:

188

<J1Jl(l)l<J2,J-J1(2)lI:~:~(l,2)lJIJ1(l)>lJ2,J-Jl(2)> = =

-2~J 1

[2J+l]!,[Jt+Jz-J]! [J-J +J ]~

(-l)r[J-Jl+J2+r]! q [r]![2J+r+l]~[J +J -J-r]!

(5.8)

This sum may be o a l o u l a t e d u s i n g one of Vandermonde formulae [27].As a Pesult we find

Qq2 = q(Jl+J2-J)rJ-Jl+J2 +I)

[2Jl]![2J+1]! [J+Jl-Jz]![Jl+Jz+J+l]!

It should be noted that the matPix element (5.8) can be oaloulated also in Peourrent manner [28]. Thus we obtain the following explioit analytioal formula fop q-OGOs

<Jlml;JzrnzlJm~

=oj,~+~2

q- ~(J1+J2-J)($+$~+J2+I)+JWz-J2~

. /[2J+l][J+m]![J2-m2]![J~+Jz+J+l]![J~+Jz-J]![J1-J2 +J]!

[J-m]t[J~-ml]![J~+ml]![J2+mz]~[J-J~+J2]~ (_l)Jl+J2-J-Z

[2J2-z]![Jl+J2-m-z]!q Z(Jl+~ 1 ) z [zl~[J1+Jz-J-z]![Jz-m2-zl![Jl+J2 +J+l-z]~ In a "olassioal" limit

q=l

(5.9)

it ooinoides with the general foz~nula fop

OGOs obtained in Ref.[18]. It is onoe mope vePsion of q-OGOs fozvnulae alternative to ones dePived in R e ~ s . [ 9 - 1 1 , 1 4 - 1 7 ] . Simple analytioal oases [28]

formulae

oan be found fop important paPtioula~

= <Jrn;OOlJ'rn'q>

= 6j,j,

<jm,j'rn" IOO> = %, ,j

, [2 j+I]

~

,,

(5.1o)

(_l )J -~ q~ "

(5.11)

<Jlml;J2rnzlJJ>~ = = ~j,~1+~

* ¢//~j

j I( (-1) 1"~1 q'2 J1+J2-J)(J-Jl+J2+1 )-(J+l)(Jl-r~ I) z

[2J+l]t[Jl+ml]![J2+m2]t[J~+J2-J]~ 1-ml]~[J2-m2]![Jl-J2+J]~[J-J1+Jz]~[Jl+J2+J+l]! 189

,

X

(5.12)

<J ,J ~;J:Iz~lJnz> = O , j ~"~2 q

(J 1 +J2 - J ) ( J - J I +J2 + I ) - J 1 ( J " ~ )

[2J+l][j+m],[2j~],[j_nzz],[j_j~+j2] '

(5.~3)

6. 3J-symbols and their symmetry properties

In Ref. [11 ] the q-analog of 3S-s~mbols was defined as a follows

[~,J a'/~] ,

q =

(-1)J'-Jz-~a' -5 ( % -~2 )

In omdem to i~olude the Regge sym~etr~j properties of 3~-symbols it is oonvenient to introduoe the Regge symbol

;, J2 J3] -=

, ma ma q

[J,+m, IJ~-m,

Ua-J,+J2

J2+~2 J2-m2

Ja+ma ] Ja-ma |

(6.2)

Ja+J,-J2 - J a + J , + J ~ q

that is invamiant with respeot transposition and even pez~nutations of rows and oolumns. At odd permutations of rows and oolumns the phase faotor (-1) J1+J2+J3 appears and the substitution q * q takes plaoe.

7. Tensor operators, Wlgner-Eckart

theorem

As a q-analog of rank /~ tensor operator we shall oonsider a set of 21¢+I operators /~(q) (8e=~,~-I.... ,-/~+I,-~) satisfying the following oormnutation relations with generators of SUq(2) algebra

[Jo,~(q)l= ae T~(q),

(7.1) -Jo

Aoting by generators Jo,± on veotors oount (7.1) and (7.2) we obtain

~&,ae(q)-1~(q)iJm> and

taking ao-

(7.4) 190

From the comparison of these expressions with (4.2) it is olear that veotora ~:~(q) ~ e t ~ f o r m ~ as basis veoto~s o~ the t e ~ s o ~ p ~ o duot ~ ® ~ of IRs of b~/q(2) algebra. Therefore it is possible to exto the IR of pand these veotoms on oomponents mJ'~:J~(q)belong "m" SU q (2)

Multiplying both sides of this equation by veoto~ <J'm' J and taking aooount the orthogonality px~opert~

where ~ , ~ ( q ) is independent on m', m, ~ we obtain the q-analog of well known Wigner-Eokart theorem:

<J'm' IT~a(q)IJm>= <Jm;~IJ'm'> (J" IT~(q)IJ)

(7.7)

or in more standard f o r m

<J'm" t~(q)IJm> =

<Jm;~lJ'm' >_ o. ~(-l )"~<J" igCq)l,,'>,

(7.s)

v/'~'$+1 ] As an example t h e t e n s o r o p e r a t o r t h e f i r s t r a n k El(q) oonstrueted by us from generators Jo,+_ in explicit form:

j1+1(q) =

~I q-SO J+_ ,

j~o(q) =

1 I

~[2]

(ae=O,+l) i s (7.9a)

q-1[2Jo + (q_q-1)j+j_) =

[q-'CZ:o3 +

(7.9b)

It is clear that these expressions are more oomplioate then in 571(2) oase but in the limit q=1 they ooinoide with standard oyolio oomponents of angular momentum. Calculating neoessar~r OGOs we find the following expression for the reduced matrix elements of the tensor (7.9)

q=L-\'u(JlJzJ3;JlzJ23)qIJlJJ3(Je3):Jm>q J3 It is useful to ~troduoe a q-analog of 6j-symboI instead

two

these

(8.1) o f Raoah

ooeffioients

U(abco];e,f)q = ~/[2e+l][2.1'+l] (-1)a+b+c+c~ {: b /[} c q°

(8.2)

If to express it in terms of 3j-symbols

X

then the symmetry properties of 6j-symbols ean be easy found. Namely the 6j-symbols are invariant with respeot to permutations of eolumns

{]1 J2 J3} = f ]2 J1 J3} ~1 ~2 ~3 q

....

(8.4.)

Z2 ~1 L3 q

Also they are invariant with respeot to substitution two arbitraz~j momenta in the first row by oozTesponding momenta from the seoond row

{J, J2 J3} = ~1 ~ J3} ~1 "L2 7"3 q 1 "f2 "t q

....

(8.5)

Finally Raoah ooeffioients U(...) and 6j-s~rmbols are invariant with respeot to substitution q~q as it oan be seen from the general aualytioal formula (8.14) for Raoah ooefficients. The last one may be deri-

192

red usir4~ the projection operators in the manner used in Ref [18] for usual AMT. As a result we have

u(J1J2JJ3;J12J23) q = v/[2J12+l][2J23+l] (_1) J-323.J12+J2

X

*

MJ~J2J12) 5(J2J3J23) h(J12J3J) h(J1J23J) [J~-J2+J12]![-J1+J2+J12]![J2-J3+J23]![-J2+J3+J23] ! [J12+J3+J+l]~[J1+J23+J+l]!

X

,

(8.6)

[Ji-J23+J]![-J12+J3+J]! (_l)~[jl÷j_J23+r]![j3+j_j12+r]![j2_j+j12+J23_r]! r [r]![2J+r+l]~[Ji-J+J23-r]![J3÷J12-J-r]![J2+J-J12-J23 +r]'

X

•

Here

n(abc)

['a+'b-c]![a-b+c]![-a+b+c]!

=

[a+b+c+l]!

In the particular ease of one vanishing angular momentum in the 6jsymbol we obtain

{JoIJ2J32} v/[zJ2+l][zJ3+1] =

J3 J

,.(,-1,)Jl+J2+J3

(8.7)

~

Authors are thankful to A.N. Kirillov, A.U. Kllmyk and Ya. 8oibelman for illuminating discussions. Refereoes 1.

2.

E.K. 8klyanin: Punot. Anal. Appl.16 263 (1982). P.P. Kulish, N.Yu. Reshetikhin: Zapiski° Naueh. 112

3. 4. 5. 6. 7. 8. 9.

Semin. LOMI 101

(1980).

V.G. Drinfeld: DAN 88SR 32 254 (1985), N. Jimbo: Lett. Math. Phys. 12 247 (1986), N.Yu. Reshetikhin: LOMI preprints E-4-87, E-17-87 L.O. B i e d e n h ~ : J. Phys.A 22 I~73 (1989). A.J. Maefarlane: J. Phys.A 22 4581 (1989). M. Rosso: Comm. Nath. Phys. 11~ 581 (1988), T,.L. Vaksmau: DAN 888R 306 269 (1989),

193

(1988).

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

A.N. Kirillov, N.Yu. Reshetikhin: Preprint LOMI E-9-88 (1988). Zhong-Qi-Ma: Preprint IOTP I0/89/162 (1989). A.Oh. Ganohev, V.B. Petkova: Preprint IOTP I0/89/158 (1989). R. Roohe, D. Arnaudon: Lett. Math. Phys. 17 295 (1989). I.I. Kaohtu-ik, A.U. Klimyk: Preprint ITP-89-48E (1989). H. Ruegg: Preprint UGVA-DPT 1989/O8-625 (1989). H.T. Koelink, T.H. KoI~uwinder: Preprint of Mathematioal Institute, University of Leiden, W-88~12 (1988). M. Nomura: J. Math. Phys. 30 2397 (1989). D.T. Sviridov, Yu.F. Smir~ov: r~eor~ of o p ~ o a ~ spectra of ~ra,~s~o~ m~,$a~ ~o,~s, Mosoow, Nauka (1977) (in Russian). D.A. Varshalovioh, A.N. Moskalev, V.K. Khersonsky: Q u ~ Theof-~ of ~ a r N o - - S u m , Leningrad, Nauka (1975) (in Russian). A.P. Yutsis, A.A. Bandzaitis: T;~eor~ of ~ a ~ m o m e n t u m e,~ q~,~~ ~c~os, Vilnjus, Mintis (1965) (in Russian). L.0. Biedenharn, J.D. Louok: X ~ E ~ a ~ M o m ~ $ = m ~ Q u ~ P~s~cs, Addison-Wesley (1981). R.M. Asher~va, Yu.F. Smirnov, V.N. Tolstoy: Theor. Math.Fiz. 8, 255 (1971). R.M. Asherova, Yu.F. Smi~nov, V.N. Tolstoy: Matem. Zametki.36 15 (1979 ). J. Shapiro: J. Math. Phys. 6 1680 (1965). D.P. Zelobenko: DAN SSSR 4 317 (1984). G. Raoah: Phys. Rev. 62 438 (1942). G. Gasper, M. Rahman: S ~ c ~e,'~:eo,.~$~'¢o se,.~es, 0ambridge University Press (1989). Yu.F. Smirnov, V.N. Tolstoy, Yu.I. Kharitonov:Preprint LINP ~1607 (1990).

194

Quantum Dynamical

Algebras Systems

and

Symmetries

of

P. P. KuIish Leningrad Branch Mathematical Institute AN USSR 191 011 Leningrad, Fontanka 27,

LOMI

The development of the quantum inverse problem method [1] and the study of solutions to the Yang-Baxter equation [2] gave rise to the notion of quantum groups and algebras (cf. [3,4] and references therein). These days quantum Lie groups and Lie algebras are popular topics in different branches of theoretical physics and modern mathematic physics as one can conclude from this Proceedings. The structure of this object with respect to many properties is quite similar or richer than the Lie group and Lie algebra one due to appearence of a deformation parameter q (representation theory, Clebsch-Gordon-Wiegner-Racah calculus, q-special functions, non-commutative differential geometry etc.). We will consider in this report the quantum algebras as symmetries of dynamical systems. The simplest quantum algebra V = slq (2) or quasitriangular Hopf algebra [51 depends on a parameter q and is generated by three elements I, X± satisfying relations

[I, x~ ] = +x~, [21]q ---

[x+, x_ ] = [2~q,,

(q2I q-~.I) (q_ q-l)

(1) (2)

_

The Itopf algebra structure on V is defined by maps of antipode (coinverse) S : V ~ V (antihomomorpMsm), counit e : V ~ C and coproduct or comultiplication ,4 : V - . V ® V (algebra homomorphism) which satisfy a set of axioms [3-5] (we'llnot discuss them here).

s(x+)=-a~x+,

s(I)=I,

s(z)=-z,

~(i) = I,

~(I) = d x + ) = 4 x - ) = o,

(3)

A(~) = I®I, ,4(I)=Z®a+I®Z, A X ± = X± ® ql + q-1® X± . For general parameter q (special values are roots of unity:q M = 1, M E IN) the finite dimensional representations of sly(2) have the same structure as the Lie algebra s/(2) ones (ef. [6] ). 195

The contraction limit of spin j representation [7] I=j-N,

a +=

lim X± ~-.oo ~

(4)

gives rise to a new algebra A ( q ) with generators satisfying relations

[N,e~]=~,

[e-,~+]=q

-2~ .

(5)

Introducing the operators /q

a=q-ra-

=Ca+) + ,

A = q J V a - = (A+) +

one can rewrite the relations (5) as follows aa+ _ qaa +, AA + -qAA

= q-N

(o) (z)

4 = 1 .

As far a these transformations are invertible one can pick up any set of generators e.g. (a, a +, N) to define an associative algebra A ( q ) . It is natural to give to these operators the name q-oscillator for in the limit q ---, 1 (5) (7) reproduce the usual harmonic oscillator [7-12]. In the limit j ---* oo one obtains the irreducible representation of A ( q ) which coinsides with the Fock space representation of the harmonic oscillator acpo = 0 ~

¢Pn =

(8)

a+ n ~Po

where we use the notation (2): [n]q - ( q , , _ q - , ) / ( q _ q - Z ) . a, a + in ?gj~ in terms of Bose-operators b, b+ : a + = ~

It is possible to express b +. In particular

5

a + a = [N]q in 7~F. However the algebra A ( q ) is not equivalent to the harmonic oscillator algebra for general q. It has nontrivial centre z = q - N ( [ N ] q -- a+a)

and others irreducible representations which are not equivalent to 7-/.~. Taking as a Hamiltonian in 7~F the operator H = a + a one has U(1) as the symmetry group with the generator N and the q u a n t u m algebra suq(1, 1) as the dynamical symmetry algebra with generators Ko=N+~,

ot

[K0, K+ ] = ± K + ,

K_ = V/IN + e]qa = (K+ )+,

(9)

[ g + , K_ ] = -[2Ko]q •

Let us concider a simple example of two q-oscillators as dynamical systems with the q u a n t u m algebra suq(2) as the symmetry of this system. The Hamiltonian is H = a + a l q N= + q

196

-~' a 2a2 +

•

(1o)

It commutes with the generators of the suq(2) X+ = a + a = ,

X_ = a + a , ,

1

I= ~(~r~-lV~).

(la)

However this statement is correct only in the Fock space representation for the qoscillators o~, i = 1, 2), where H = [N1 ÷ N2]q due to the relations a+al = [Ni]q. As a result the space 7~z ®7~ is decomposed into direct sum of finite dimentional subspaces co

~1@7~2 = ~ V k , k=0

dimVk = k + 1 ,

(12)

where Vk are irreducible representations of the suq(2) with spin j = k/2. In the limit q - , 1 we reproduce the picture of two standaxt oscillators with the Lie algebra su(2) as their symmetry. More of that the corresponding eigenstates ]k - m) ® Ira), m = 0, 1 , . . . , k of the Hamiltonian with the eigenvalue [k] do not change being constructed from the eigenstates of Ni, i = 1, 2 which coincide with the number of operators of standart oscillators. This construction can be easily generalized to the quantum algebra of higher rank e.g. suq (n) as the symmetry algebra of n dimensional q-oscillator. For n = 3 the Hamiltonian is +Ira -I- a+3a3q -Na -Ira H "- at~zlq Na'I'N= -l- a+ 2 a2,,-N= "~

(13)

In the Fock space representation it is equal to [Nz +N~ +Ns] and it commutes with the generators of the suq(3) x + = ~+~ = ( x ; ) + ,

I I1 = ~(2v~- 2v~),

x ~+ = ~+~ = ( x ; ) + ,

i x~ = ~(N2-

N~)

(,4)

•

The generator X ~ corresponding to the highest root is constructed using qcommutator or q-adjoint action x + = ( x ; ) + = [x~+ , x + ], =aa,(x~)x+ = x+x+ -qx+x+ ~_~+~.

(is) It is interesting to note that the Hamiltonians (10), (13) have interactions among different modes and their structure reminds the noneoeommutative comultiplication of the quantum algebras. There are more realistic integtable models wich posses quantum algebras as symmetry or dynamical symmetry algebras. They axe non-affine Todd field theories [13], generalized 3aynes-Cummings model Hamiltonian of quantum optics [14] integrable spin models such as the XXZ model of spin ½ [15] or its generalisation to higher spin [16], relativistic oscillator [17] and models of the eonformal field theory such as Liouville equation and WZNW model [18].

197

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Faddeev, L. D. : Les Houches Lectures 1982 (Amsterdam: Elsevier) (1984) 563. Kulish, P. P. , Sklyanin, E. K. : Lect. Notes Phys. 151 (1981) 61. Jimbo, M. : Int. 3. Mod. Phys. 4 (1089) 3759. Takhtajan, L. A. : Adv. Stud. Pure Math. 19 (1989) 435. Drinfeld, V. G. : Proc. Int. Congr. Math. (Berkeley, CA, MSRI) 798 (1986). Kirillov, A. N. , Reshetikhin, N. Yu. : Adv. Series Math. Phys. (World Scien. ) 7 (]989) 285. Chaichian, M. , Kulish, P. : Phys. Left. B 2 5 4 (1990) 72. Kuryshkin, V. V. : -quantizatiou. Dep. V I N I T I 3936-76 (1976); Ann. Found. L. de Broglie 5111 (1980). Biedenharn, L. C. : J. Phys. A: Math. Gen. 22 (]989) L873. Macfarlane, A. J. : J. Phys. A: Math. Gen. 22 (1989) 4581. Kulish, P. P. , Damaskinsky, E. V. : 3. Phys. A: Math. Gen. 23 (1990) L415. Ui, H. , Aizawa, N. : Mod. Phys. Lett. A 5 (1990) 237. Leznov, A. , Mukhtarov, M. : Theor. Math. Phys. 71 (1987) 46. Chaichian, M. , Ellinas, D. , Kulish, P. : Phys. Rev. Left. 65 (1990) 980. Pasquier, V. , Saleur, H. : Nucl. Phys. 330 (1990) 523. Kulish, P. P. , Sklyanin, E. K. : (unpublished). Kagrarnanov, E. , Mir-Kasimov, R. , Nagiyev, Sh. : Preprint I C T P (~rieste), IC/89/42 (1989). Faddeev, L. D. : Preprint HU-TFT-89-56 (1989).

This article was processed using the I ~ _ ~ macro package with ICM style

198

COULOMB GAS REALIZATION OF SIMPLE QUANTUM GROUPS C. Rarrdrez*, H. Ruegg** and M. Rniz-Altaba** *Fachbereich Physik, UniversitKt Kaiserslautern **D6pt. de Physique Th6orique, Universit$ de Gen~ve

A b s t r a c t . Generalizing a procedure of Gomez and Sierra, we realize simple quantum groups by their action on screened vertex operators of WZNW models in the Coulomb gas representation. 1. I n t r o d u c t i o n Quantum groups or, more precisely, quasi-triangular Hopf algebras [1, 2] have appeared in the study of integrable systems [3]. They also have been shown to be hidden symmetries of two-dimensional conformal field theories [4]. In this latter context, Gomez and Sierra [5], using the Coulomb gas representation for minimal models [6, 7], have been able to construct an explicit realization of the relevant quantum group, with the deformation parameter q a root of unity. The quantum group acts on screened vertex operators [8] by increasing, lowering or counting the number of screenings. We have extended this construction to simple quantum groups [9]. The basis vectors e(z) for irreducible representations are Gomez-Sierra (GS) integrals over monomials in the vertex V(z) and screening operators J(z) of the Wess-Zumino-Novikov-Witten model, in its Coulomb gas realization [10,11]. The main ingredients for the construction are braiding, which follows from the operator product expansion of the V's and J's, and deformation of the contour of GS integrals. For instance, one gets the/~-matrix satisfying the Yang-Baxter equation by braiding the basis vectors e(z). As an example of the general method, we shall consider the enveloping algebra of sl(2)q, its generators, co-products and Clebsch-Gordan series. For the general case we show, in the ChevaUey basis, how the quantum Serre relations are satisfied. For convenience, we recall that to eac~ simple Lie algebra G corresponds a q-deformation Uq(G) of the universal enveloping algebra, the dual of a quantum group. Its generators

199

X H, Hi (i = 1 , . . . I = rank G) satsify the relations [12] [Hi, Hj] = O

;

[Hi,X~]=+(oq,aj)X~

(1.1)

q~rd2 _ q-Hd2

[ X ~ , X ; ] = 6ij q'~/2 q--I/2

(1.2)

and the q-Serre relations [13] l-el#

i ~j

E (-l)n[l-~i'] (,Xi,+l-Cil-nxJ±(,X~ ' ,±'n=0

n=O

(1.3)

qi

where q E C, Cij = 2czi • aj/o~ 2 is the Cartan matrix of G, qi = qa~/2 and

[m] n q

=

{ (qml,

q-,n/,)...(q(*n-*t+X)/, q-(m-n+X)/,) (q,,12_q-,=l,)...(qZ/2_q-Xl2)

1

(1.4)

ifn=Oorm=n

In addition, a Hopf algebra is characterized by a co-product A, an antipode '7 and a co-unit. Of interest is only

=

qH,/4 + q-H,~4 xH

(1.5)

In the limit q --* 1, one recovers the classical U(G). 2. W Z N W

m o d e l in C o u l o m b gas r e p r e s e n t a t i o n

A W Z N W model based on a simple Lie algebra G (dim(G) = d) contains t free real scalar fields ~ with a boundary term (charge at infinity) and a spin-1/3 - "7 system which will play no essential role in the following [11]. This model is invariant under a K a c Moody algebra G, whose generators are complicated functions of the ~,/3, 7 fields. We shall need the vertex operators of primary fields

v Cz) exp[Cil ) iM ' jCz)] =

(2.1)

where v = ~ -l-h*, k is the level of the Kac-Moody algebra, h* =0. (0 + 2p)/O 2 is the dual Coxeter number, 0 is the highest root, 2fiis the sum of all positive roots of G, and Mij = ai " a~. The screening currents of the Coulomb gas associated to the simple roots a i are, up to a polynomial in/~, %

J (z) =

(2.2)

Using the operator product expansion

~i(z)~j(zt) = --~i " ~j log(z -- z t) + regular terms

200

(2.3)

one finds the braiding through e iTr (the fl, 7 fields are inert under braiding, hence our lack of interest in them)

VX(z)VxI(Z' ) = q½~'M-";Vx,(z')Vx(z ) s~C~)Vx(¢ ) =

q-~,~',14vxcz,)s~(~ )

(2.4)

Si( z ) Sj( z t) ---. q~'aJ /2 Jj( zt)Ji( z ) where q = e 21ri/v2

: exp [ k 2~i + h* ~ J

(2.5)

is the quantum deformation parameter of Uq(G), related to the Kac-Moody level k.

3. The q u a n t u m enveloping algebra Uq(s£(2)) In this case, ~ = 2j, the vertex operators are Vj = exp[ij¢/v] and the screening current is J =/3 exp[-i¢/v]. One gets a basis of a representation of Uq(s£(2)) introducing (n+ 1) vectors. Start with e°(z) = Vj(z) and generate a family of screened vertex operators e~?(z) By successive addition of the screening currents on GS contours:

e~(z) =

dtiS(ti)

,5"

]

Vj(z)

(3.1)

The GS contour in the complex plane ~i starts at infinity (where the curvature singularity is concentrated), then down, then counter-clockwise around the point z where the vertex operator is inserted, then Back up to infinity. Using Braiding and contour deformation, one gets

which shows that n < 2j hence dim{eT(z)} = 2j + 1. Now define the quantum group lowering operator as

F

= _]~sdtJ(t)

(s.3)

which acts among these states by Fe~(z) = e~+l(z). The co-multiplication is defined by r~

201

n I

?

where the GS contour goes around z and z t. Deforming this into two GS contours and using braiding one finds n

n'

t

(3.5) It is natural to define

(3.s) so one gets

A F = F ® 1 + q-gin ® F

(3.7)

AH=H®I+I®H The raising operator E in the present realization destroys a GS contour. It can be shown [9] that it is d o s d y related to the zero-mode of the K a c - M o o d y lowering generator of the W Z N W model, which acts as d/dz on J(z). Setting E = q - g / 4 x + , F = q - H / 4 x - ,

one gets the st(2)q generators in (1.1) and (1.2). For the Clebsch-Gordan series and its truncation when q is a root of unity, see ref. [9].

4. ~q(G) T h e general case Uq(G) is very similar to Uq(st(2)), with an i m p o r t a n t difference: the screening operators Ji(z) do not commute, and therefore it is necessary to verify the Serre relations in the Chevalley basis. This can be sketched as follows [9]. Introduce for the time--ordered multiple integral

i....il = T

at~

)

si.(t,)...ai,(tl)vs(:

,]

(3.8)

In a somewhat abusive but d e a r notation,

:(,-- q ~'+ai'aS) iF'V.-.+ qai'aS/2 In general,

°

Fi,, ...FI, Vx(z) = ~ q½~"~,~- O. Four algebra homo,lorphisms of • • are obtained by combining ~(~I) or ~/~2) with ~ 3) q

B_(m,n)

.

"n-1

q

or ~

~1"~'n-i ~3~ra-1

4)

into W (n,m) ~. . A un:ttary

m-1

representation follows from usi,l~ ~¢An:l and ~m:l" This is the only

one

that we shall describe e x p l i c i t l y ( t h e others are similarly constructed) ek

= bkbk+l'

fk = bk+lbk '

hk = Nk - NR-Z'

tb

~1+,, %1

=

~,+,~I' %1

Nn ,

NI+~' 1:1,..,~-1,

%+,

:

em+n

= (-) ~Jm' f ~ n = ~Jm(-)M' hm+n = 2 % - 1.

:

k=l,..,rr-1,

:

"I

-

(20)

where M=~=I Mi. I t is not d i f f i c u l t to check that the defining relations of Bq(m,n) are s a t i s f i e & Note that a Klein operator enters ill the expression of ~+n and fm+n' Let us point out that d i f f e r e n t homomorphisms of Bq(m,n) into Wq(n,m) can be obtained by exchanging the b' s and ~J s. However, one then needs to use a set T with more than one element.

, Bq(O,n).

The representations of Bq(O,n) only require q-bosons, ltomomorphisms of Bq(O,n) into Wq(n,O) can be constructed from e i t h e r 1~,,1)

An-I

or 1~,2)

/In_1"

In the f i r s t case one has

%

-

bt'b', "' fk KKI ?

en= bn ,

= bk+~b k,

fn= bn ,

h k = N~ - ~k+~'

~''""~"

hn = 2Nn + 1.

(21)

This representation is unitary. The other one has (en, fn, hn), k=l,..,n-1 from (15) and (e n, fn' hn)from (21). These bosonic r e a l i z a t i o n s of ospqfl,2n) were given in ltef. [17]. . C_(n+l).q~ We have two homomrl)hisms of Gq(rY~I) in Wq(n,I). The f i r s t uses W~~) and is defined as follows -fl

~ ek+l = bkbk+l' fk+l = bk+Ibk' en+1 (i/2cosh ~2

hk+l :t~-N~+ r

213

k:~, ' " .-~,

This representation is not unitary.

~1 second one uses u (2) : e1=~lb 1,

An_ I e3=ib2b3,. "" and so u n t i l index n+1 is reached. For odd n the generator en+I coincides with t h a t in (22) and f o r even n i t is given by fn+1 from {22). Other generators can e a s i l y be reconstructed from e i' s. e2=iblb2 ,. t t

, 19 (re, n). Two homomorphisms of D ~m,n) into Wq(m,n) are obtained upon cgmbining n(AI) or n(t2) with n(~3)q. The f i r s t produces unitary realization:

"'n-i

"'n-I

e,+i: i i÷l' em+n =

,f .f

e;~dJm;l'

"'m-I

h*.÷.b.I,K K

hk = N~ - Nk+ I, k=l,.:,n-l,

Vllbn ,

h n = Nn + M1 ,

= @~+i@I, hn+ I = N 1 - MI+ I,

fm+n

=

%=

÷

1=I,..,m-I, -

I.

(23)

For I)q(m, 1) the form of this q-oscillator representation had been conjectured in Ref. [16]. h second r e a l i z a t i o n is formed by taking; the f i r s t n-1 g;enerators from (15). Finally, two new homomorphisms cfia be obtained from the described rel)resentations by l e t t i n g b i ~: + M-

+~a e-2

I

(z7)

linear oscillator

iaV~ =

2

e 73-~'~ -

and coin-

j +

(a++a -)

sh

-a-

(18)

2

COS The

deformed

commutator

(ii)

for

the

relatlvistc

oscillator

takes

the form [M-, M+]q = qM-M + + q-IM+M- = 2(q "I - q) = 2sh ~4 Then the hamlltonlan

operator

is written in a factorized

of the relativistic

linear

(19)

oscillator

form

where T=

It follows from

" - - ~ x] cos

e= oq

; 2

(19) and (20) that

217

"i -

q = 2sh~.

(21)

q

-I

q +

o p e r a t o r s H, M~ a r e r a i s i n g

This means t h a t the

and l o w e r i n g o p e r a -

tors for energy levels. Putting

n÷1 ] - 1 / 2 M+On(x), ~n+l(X) = ~( e x p.n÷i ( ~ 2 . 4 sh 2 u~

n = o,1 ,2, ... (23)

we obtain from (22) and (9) the recurrence relations connecting the nelghbouring levels en and en÷ i Their solution gives us the spectral

formula ,

.,.÷l

en = 2 [ e x p ( ~ ) : -

ch

~)

(241

It Is worthwile to notice that the same spectrum corresponds to the

hamiltonian H 2mc

(H

4mc

is the nonrelativistic hamiltonian,

/

dimensional variables are

nr

rostored). The wave functions have the form 2 ~x

~n(X) = Nn e

2

hn(x)

(26)

°

N. =

~/~)~" 2" e- ;7 in],

}.,.

(27~

where by d e f i n i t i o n =

4/~

U

exp,--

4

k=i

The normalization condition is

The polynomials hn(X) = hn(Sln~-) are given by the integral representation 2

e- ~ x

t2 i ( 2 i ~ n I Tn(t) e2iXt e- ~ at hn (x) = ~V~t-g-j')

(30)

where the kernels T (t) are defined by the recurrence relation n

2n÷3 -

Tn+i(t ) = e The

sh~ ch~

following operators

218

~'E rn(t)

sh~ d

(31)

L÷ = - ~1 (M*M-) '/~ "*'

L- = ~, .-

(.*.-)1/~.

,.3 _- ~

~.

C3Z)

where

I

-I/2

-i

-~

(q-~ - q~).

--

(33)

obey the deformed Lie algebra SU (I,I) relalatlons: q

[L÷,L -] z= L 3, q The expressions (33)

[L±,L 3] ±1= ; L ± q

(34) as c ~

= t u r n i n t o the well-known formu-

SU(1,1) of dynamical symmetry o f

lae f o r the generators of the group usual l i n e a r o s c i l l a t o r ( e . g .

(34)

[9]).

Let us i n t r o d u c e the operators a

+

=

(4sh_~ 4 I/2M-exp( - (aN),

a-

=

(4sh4) I/2M + exp(- ~N (3s)

2 In(1 + I

)

+

Operators a- obey the following deformed commutation relations -

aa

+

-

-e

-

~a~

+

mN

£

-

%

(36)

=expL~-J +

Now, i f we d e f i n e t h e q - g e n e r a t o r s

-~I (*-~l/2a~j J +=2v3 a a

J-, j3

by

I ,*-,I/2) J-= ~7-~[a a

a,-

d3= 2N+I,

(37)

the deformed SU (1,1) dynamical symmetry will be described q (cf. [2] and tel. therein)

by the

relations

[j3

j±] = ±j±

J-] =

[J+'

sh(~ j 3 ) sh£4

(38)

References. i. E.D. Kagramanov,

R.M. Mir-Kasimov,

Sh.M. Naglyev, preprlnt

(IC/89/

42, Trleste, 1989).To be published in Journ. Math. Phys. 2. L.C. Biedenharn,

J.Phys. A22 (1989) L873; A.J.Macfarlaine, J.Phys.

A22 (1989) 4581; R. Floreanl, V. Splrldonov, 90/TEP/12, University of California, Phys.

L. Vlnet, preprlnt

1990); T. Hayashl,

Comm. Math.

127 (1990) 129.

3. L.D. Faddeev Integrable Models in (i+I) Dimensional Theory

(UCLA/

(Les Houches XXX]X) ed. J.-B.

Amsterdam,

Quantum Field

Zuber and R. Stora

(Elsevier,

1984).

4. P.P. Kullsh

and N.Y. Reshetlkhln,

P.P.Kullsh and E.K. Sklyanln, (Springer, Berlin,

Lecture

1982) 61. 219

J. Sovlet. Mat. Notes

in

23

(1983)

Physics,

2435;

Vol. 151

5. L.D. Faddeev, N.Y. Reshetlkhln and L.A, Takhtajan, A~gebra i Analis I. (1989) 178. 6. Yu.I.Manln, Annales de l'Instltut Fourier 37. (1987) 191. 7.

M. Chaichlan,

P. Kullsh

and

J.L. Lukierskl,

Phys.

Lett.

B237.

(1990) 401. 8. S.Vokos, B. Zumlno and J.Wess, preprlnt (LAPP-TH-253/89,

Annecy-

le-Viex, 1989). 9. I.M.Malkin, V.I.Manko,Dynamlcal Symme%ries and Coherent States of Quantum Systems,

"Nauka" Publishers, Moscow,

220

(1979).

Some Aspects of the Angular Momentum Coefficients in SUq (2) K.Srinivasa Rao and V.Rajeswari The Institute of Mathematical Sciences, Madras-600 113, India Recently there has been a considerable interest in the study of the deformation or modification of the Lie algebra that involves an indeterminate parameter q. Mathematicallyit is a Hopf algebra which is loosely referred to as a quantum group. The nature, structure and representation of quantum groups have been studied by : Sklyanin[1], Jimbo[2],Drinfeld[3],Woronowicz[4], Fadeev[5] and others. The simplest quantum group which has been extensively studied is the group SUq (2) - viz. the q-deformation of the Lie algebra of SU(2). Several aspects of SUq (2) have been dealt with by Sklyanin[1], Vaksman and Soibelmann[6], Kulish and Reshetikhin[7], Kirillov and Reshetikhin[8], Pasquier[9], Bo-Yu Hou, Bo-Yuan Hou and Zhong-Qi Ma[10], Matsuda et al[11],Nomural121, Macfarlane[13], Biedenharn[14] and Ruegg[15]. The q-analogues of the RacahFock formulae for 3-j symbols were obtained by Vaksman[16]. Other representations of the q-analogues of the 3-j symbols - viz. the vander Waerden and Majumdar formulae, besides the Racah formula - as well as their symmetry properties were found by Kirillov and Reshetikhin[8]. Kirillov and Reshetikhin[8] have also given the q-analogue of the 6-j symbol. They note that the q-analogues of the 3-j and the 6-j symbols correspond to the basic generalized hypergeometric functions 3~2 (q) and ~qia(q) respectively. Bo-Yu Hou, Bo-Yuan Hou and Zhong-QiMa[10] explicitly derive the q-3j and the q-6j coefficients for the quantum SI(2) enveloping algebra. Our aim here is to establish the full connection between the q-3j and the q-6j coefficientson the one hand and the basic generalized hypergeometric functions on the other. In fig.l, we show a schematic connection between different aspects of quantum theory of angular m o m e n t u m and hypergeornetric functions as it has evolved. The starting point for our present work is the q-analogue of the vander Waerden form of the 3-j coefficientsand the q-analogue of the Racah form for the 6-j coefficient,given explicitly by Kulish[7], Kirillov and Reshetikhin[8] and others.

vander Waerden,Wigner,

Angular Momentum algebra :SU(2) q--+l

Quantized 1 Universal Enveloping ! algebra: SU~(2) I i

!

i~ Generalized

I

,,, >

]Hypergeometric I

Racah,Majumdar,KSR,VR.

[Functions :p F, I

E.Heine (1898) t F.H.Jackson (1905-60) q --~ 1 G.N.Watson . I Generalized "~ I basic hypergeometric functions :p ~ Fig.1.

221

The q-factorials: (1 - q" )

[~]! = [n][~ - 1] ... [2][l],where [a]

(1 -

q)

are related to the q-gamma functions by:

[~]~ = r q ( ~ + ~), and the Fq a satisfy the property: r , C~) r , (1 - ~)

r,(~+~)

r,(1-~-~)

= C-z)° q

. . . . .

c°-,/=,

which can be derived from the properties of the"q-gamma functions given by Jackson[17]. With these results, we can show that the q-analogue of the vander Waerden form of the 3-j coefficient, given by Kirillov and Reshetikhin[8] becomes:

£

£

~}

~I

~2

~3

q 2

×(1-I H ~=1

1/2

3

[8, -

,~,]! [8,P/[8~ + 82 + 8~ -

,~1 -

,,~ + 1]! )

j=l

x ( r , ( l - ~ , 1 - ~ , 1+~i, 1+82, 1+~)} -1

(i) where

rq(,, y,...) = rqc,)r~ (y)..., [x], = ( 1 - q ' ) ( 1 - q ' + l ) ( 1 - q ' + 2 ) . . . ( 1 - q ' + " - l ) , al = jl-j3+m2,

a2 = j 2 - j ~ - m l ,

81 = j l - m l ,

82 = ] 2 + m 2 a n d S s = ] 1 + . i 2 - ] 3

•

The above expression (1) is manifestly invariant under the 3~ numerator parameter (81,82,83) permutations and the 2! denominator parameter ( a l , a 2 ) permutations, thereby exhibitLug only 12 symmetries of the q-3-j coefficient. In the limit q -+ 1, (1) will reduce to the ordinary 3-j coefficient corresponding to (p q r) = (1 2 3) given in Srinivasa tLao[181. As in [18], it is obvious that a set of six 3O2(q)s is necessary and sufficient to account for the 72 symmetries of the q-3-j coefficient. However, the set of six s ~ (q) functions split into two sets of three s (I)2(q)8, one corresponding to the even and the other to the odd column permutations of the q-3-j coefficient. It is possible to write the set of three 3~2 (q)s corresponding to the odd permutations as either a set A of 3~2(q)s or a set B of ~ 2 ( q "+~+ . . . . ~)s where the set A and the set B can be obtained from the set of three 3(I)2(q)s corresponding to even permutations by q --+ q - l , or by the reversal of series, respectively. In the limit q --+ 1, these results reduce to the set of six 3 F2 (1)a, as represented schematically in Fig.2.

222

/ -Set

or iX"

3F (1). _

q-+l

q-~l

.-1

r

!

Set A of three L

~l 3~,(q)........ }~ t

1

I

odd p erms.

reversM~

l

; even perms. I

a~2

Ji

odd perms,

i

Fig.2 The q-analogue of the Erdelyi-Weber (or Whipple) transformation[19] :

[a, f l , - n ; q , q )

= q,,~ r q ( ~ + r t -

oq"l)

(a,

5-

fl, -rt;q, ql+~-"~

(2)

can be used to obtain from the vander Waerden form (1) of the q-3-j coefficient, the l%acah, Wigner and Majumdar forms, in a way completely analogous to that given by Rajeswari and Srinivasa Rao[20]. In the case of the Racah coefficient, the Kirillov-Reshetikhin[8] q-6-j coefficient can be shown to be given by:

{abe} dcf

= q{#,{#1-1,-,.{~.-1,-#.1,.-~)+(-#.+,.+#.)1...+...+o~+°,-~#.,}/, q

x r . (/~, +2) {r, (1 - a ~ +fl,, l - a , +fl,. 1 - a z +/~,, 1 - a , +fl,. l + f l , -fla. l + & - f l , ) } - '

x,03\-fl,-1,1+fl,-fl,,l+fl3-fl, ] where

al = a + b + e ,

a2 = e + d + e ,

fll = a + b + c + d , , O 2

a3 = a + c + f, a4 = b + d + f

= a+d+e+f,

,

f13 = b + c + e + f .

This expression (3) is manifestly invariant under the 4! numerator parameter (~1, a2, ct3, t~4) permutations and the 2! denomiantor parameter (f12,f13) permutations, so that, it exhibits only 48 of the 144 symmetries. It can be shown that there exists a set I of three 4 ~ ( q ) s to which (3) belongs, which in the limit q --* 1 become the set I of three 4Fs(1)s given by Srinivasa Rao et.al.[21]. The reversal property for the terminating generalized basic hypergeometric series has been shown[22] to be given by :

223

:,+o,+...+=, )

, 1 + #, + #2 + ...+ #p ~, q r..i.

t Xp+1~p

I

~ -- 0 ~ , 1 -- r~ -- 0 ~ 2 , . . . , 1 -- r~ -- <Xp

(4)

/

Using (4), we can obtain the set II of four 4~s(q) functions to represent the q-6-j coefficient, which is the q-analogue of the set II of four 4F3 (1)s given by Srinivasa Ram and Venkatesh[23]. The details of these results will be presented elsewhere. The authors wish to thank Drs.R.Chakraborti and R.Jagannathan for interesting discussions. REFERENCES 1. E.K.Sklyanin, Usp.Math.Nauk. 40 214 (1985); Func.Anal.Appl. 16 263 (1982); 17 273 (1983). 2. M.Jimbo, Lett.Math.Phys. 10 63 (1985); 11 247 (1986). 3. V.G.Drinfeld,Proc. ICM Berkeley (1986) p798, Ed. A.M.Gleeson (Providence, RI : AMS); Zap. Nauch'n. Sem. LOMI 155 18 (1986). 4. S.Woronowicz, Publ. RIMS (Kyoto Univ.) 23 117 (1987); Commun. Math. Phys. 111 613 (1987). 5. L.D.Fadeev, N.Yu.Reshetikhin and L.A.Takhtajan, Quantizat{on o/Lie Groups and Lie Algebras, Leningrad preprint, LOMI E-14-87 (1987). 6. L.L.Vaksmann and Ya.S.Soibelmann, Funkt.Anal.Pril. 22 1 (1988). 7. P.P.Kulsih and N.Yu.Reshetikhin, J. Soy. Math. 23 2435 (1983). 8. A.N.Kirillov and N.Yu.Reshetikhin, Representations of the algebra Uq(Sl(2)), qorthogonal polpnomials and Invariants of links, Leningrad preprint, LOMI E-9-88 (19s8). 9. V.Pasquier, Commn.Math.Phys. 118 355 (1988); Nucl. Phys. B395 491 (1988). 10. Bo-Yu Hou, Bo-Yuan ttou and Zhong Qi Ma, BIttEP-TI-I-89-7 and 8 (NWU-IMP89-11 and 12), (1989). 11. Y.Matsuda et.al. Repre.sentaHon~ o/quantum groups, Preprint Kyoto RIMS 613 (1988). 12. M.Nomura, J.Math.Phys. 30 2397 (1989); J.Phys.Soc. (Japan) 57 3653 (1988); ibid 58 2694 (1989). 13. A.J.M~cfarlane, J.Phys.A:Math. Gen. 22 4581 (1989). 14. L.C.Biedenham, J.Phys. A:Math. Gen. 22 L873 (1989). 15. II.Ruegg, J.Math.Phys. 31 1095 (1990). 16. L.L.Vaksmann, Func. Anal. Applns. (1988). 17. F.H.Jackson, Q.Jour. of Pure & Appl. Maths XLI 193 (1910). 18. K.Srinivasa Rao, J.Phys. A:Math. Gem 11 L69 (1978). 19. D.B.Sears, Proc. Lond. Math. Soc. 53 158 (1951). 20. V.Rajeswari and K.Srinivasa Rao, J.Phys. A:Math. Gen. 22 4113 (1989). 21. K.Srinivasa Rao, T.S.Santhanam and K.Venkatesh, J.Math.Phys. 16 1528 (1975). 22. K.Srhlivaaa Rao and V.Rajeswari (to be published). 23. K.Srinivasa Ra~ and K.Venkatesh, Vth ICGTMP, Group Theoretical Methods in Phys., F,d. by R.T.Sharp and B.Kolman, Academic Press, N.Y. (1977) p.649.

224

CONTRACTIONS

AND

ANALYTIC

REPRESENTATIONS

CONTINUATIONS

OF THE Nikolaj

Komi

Scientific

Centre,

QUANTUM A.

Physical

i. The purpose cription

of

the

The

[I-3].

the This

particular, tence tum

this

report

of

notion field

is

Jimbo

[3]

of a q - a n a l o g

Academy

to

the

now and

at

constructive

all

s u C2). q method gives

quantum

investigation [7]

of

algebra

and

have

of t h e G e l ' f a n d - T s e t l i n

des-

continuations

problem

group

active

Ueno

a

unitary

inverse

quantum

of t h e

USSR

analytic

of the

under

of Sciences

give

the

quantum

the

of

t h e USSR,

Man+ko

is

and

of

USheR

117333,

representations

development

to

Moscow,

contractions

the irreducible

rise

of

I.

Institute,

USSR,

s u C2~ q

of S c i e n c e s

I•7810,

Vladlmir P.N. L e b e d e v

ALGEBRA

Gromov

Academy

Syktyvkar,

OF THE IRREDUCIBLE

[4-~].

shown

basis

algebra

for

In

the

exlc-

the

quan-

group

G L Cn+l). The representation operators of G L C n + l ) q q obtained from those of G L C n + I ) by replacing of s i m p l e

are

Gel'fand-Tsetlin In

this

report

the

unified

the

analytic

sical

multipliers

groups

we

with

generlize

description

of

continuations

to

the

q-analogs. case

of

Wigner-Inonu

the Cot

their

Weyl

unitary

a quantum

groups

contractions

trick9

of t h e

and clas-

[8-10].

2. TAe ~ n £ £ ~ r 5) C ~ N £ e ~ - K £ e £ n ~[~ebro.$ suC2;ji>.

Let

us d e f i n e

the map W:

Cz

~ CzCj,

9 m

%uzo o

where

Zo.Zi~z,

nates

and

imaginary

= z o,

Wz i

Zo.ZIG4~zCjl)

are

=

the

parameter

Jl n ~ y

unit

to the Clifford

i or

be equal

jizi

CI)

complex to

the

dual

Cartesian

real

unit

unit

t .The i

225

coordi-

or dual

to

the

units

are characterized k~m;

~=0;

The

by the following form

Iz'12+lz~l z of C is t r a n s f o r m e d e £ 2 i n t o t h e f o l l o w i n g q u a d r a t i c f o r m of C z C j i )

(z.z)=IzopZ+j~Iz unitary

group

Cayley-Kleln

of all

keeping

t h e s p a c e CzCj,)

transformation

and

respectively

under

Iz

¢2)

group

SIJC2;jiD

is

defined

as

the

invariant

the form

Ce) t r a n s f o r m a t i o n s

in

w i t h t h e u n i t deterntlnant.

the

algebra

Lk~O ; LkLTn=LmLk~¢O,

Lk/~k=i.

quadratic

t h e m a p CI)

The

properties:

of

the

group

T h e m a p CI)

S"JC~-) i n t o

transformation

of

s u C 2 ; j D. T h e g e n e r a t o r s

the

[8]

the

algebra

3_+,3"s of s u C 2 )

induces

group

S U C ~ ; j i)

suC2D

into

are

the

transformed

as follows:

J = J:C-~),

J + = jsJ~C_ --~},

w h e r e b y ..]':C -'+D.J~C -->D a r e d e n o t e d lar t r a n s f o r m e d parameter,

or t h e a n a l y t i c

unified

unit.

The generators different

JiC--+),J:C-+)

are defined

representations

the

Gel'fand-Tsetlin

are

specified

by

corresponding

continuations. approachs

a

of

of

transformation pattern

give the following

generators

J +_ l l , m > = j i .JC j -* i l +-m ) C j[*l+m+l)"

Eqs. C3)

theory.

In

of

namely

these the

[J.,J+] 3, T h e ce with sions

generators

components

l=jsl",

for

the

the quantum

3__ll,m>

=

the

m=m . Then

m+j 3"ll,m_+l> ,

relations

rJ ,j

] = Jz~Ja':L

generators algebras

l

of

I+

the

with

CS::)

suqC2; j~::).

the multipliers

generators

of

c4.)

9%4aTlt%un Cc~5;Le~;-KLe~n c~LFebr ~

[3] ,[7] w e r e p l a c e

C49

obtaine

= +d+,

for

of suC~.;j i)

...... I1, m-+l>=lCi~jJ;OCl_+j

the commutation

ways

particular,

3 Ii ,m>=mll,m>, which satisfy

t o i,

in a different

suC2)

law

fi*.m*>,

singutending

t o t h e Weyl

Therefore,

in g r o u p

suC2).

representation

the

t h e Gel " f a n d - T s e t l i n

fill

W h e n j, is e q u a l

two different

for

Eqs. C3)

to dual

give the transformartions,

trick

naturally

the Wigner-Inonu

a n d jx p l a y t h e r o l e of z e r o

w h e n j, is equal

t h e n Eqs. C3) unitary

generators

C3)

their

irreducible

In accordan-

in the final q-analogs.

expresThen

representations

we of

suqC~;j i) in t h e f o r m

[l-T-J~.m:l[l--+j~m+j~]

"

ll,m-+l>,

226

J ll,m>

= roll,m>,

C8)

where

the

q-analog

[x]

[Js,J+]

(8)

for

tions

of s u ( e ) . q following

Z

ji=l

of a H o p f

[J+,J_]

the

:

with

[jl][2ji.Ts], the

for

~CJ+)

= c C J s)

A,

algebras

® e

commutation

counit

s u q C 2 ; j i)

~

rela-

and

the

anti-

structure

hJ s + e

s ® j+,

ACJs)

~CJs)

=-Js"

From

Eqs. C7)

~(J-+)

= -e

we find

the

the -

-

sl nhCh) commutator

J_+~l,m>

CO)

i

-

,

sl nh(h)

-

h

unit C1~.)

h

h and

the

Jsll,m>

irreducible

of t h e dual

[:L]

[J÷,J_]=O

(0)

generators

= roll,m>,

representation

s u (2; L ). q i If w e p u t Ji=i, t h e n

Cll)

s = _j+e+h

q-analog

[I ]

= Eli ll,m+l>,

the

® Js'

-hJ sJ+e

L

i

-

i

+ i

CIO)

sinhC L h) ]

= Js ® 1

= O,

hJ

realize

C8)

well-known

coproduct

quantum

-hJ = J+

then

relations

algebra

At J+)

[~

(7)

, h ~ ~.

the commutation

formulae

on

by

2h

coincide

define

x is d e f i n e d

q = e

satisfy

= _+J_+,

which

pode

number

si nhC xh) sinh(h) '

=

The generators

The

of t h e

of

in

l->O, I ~ ,

the

m~=Z

form (13)

quantum c o n t r a c -

the

ted algebra

[i] and

the

sinhCih) sin(h) Sinh(h) = isinh(h ) ,

=

commutation

the following

relations

commutation

[39,3_+] = +J_+, of

the

the algebra

with

we

their

generators

(8)

for

ji=l

q-analogs

C8)

sinZ(h)

for

ji=i

do

-I,

#

si nhZ(h) not

(14)

coincide

with

= -[2J ],

C15)

algebra

the

quantum

s u (I,i). q have another

J + (4),

z= [i]

relations

[J+,J_]

pseudounitary

generators

But

we find

do

not

su CI,1). Therefore q describe the representation

possibility in

the

to

replace

intermediate

the

of

multipliers

expressions

for

the

namely

J+ I1, m> =jiJ [ Jlll ;m] [jlll _+m+i ]

Ii ,re+l> ,

227

J s l l , m > = m J l ,m> ,

CIS)

Then

the generators [Js,J+]

which

for

C16D

satisfy

= -+J+,

[J+,J_]

ji=i

are

exactly

the commutation = J

2

the

of

the

pseudounitary

Ci7)

[~-Js],

commutation

s u CI,ID. If w e p u t ji=i, l=a-i/~-, a ~ q t h e g e n e r a t o r s of t h e p r i n c i p a l s e r i e s resentation

relations

relations

in

Eqs. C 1 6 ) ,

of

the

quantum

we

of

obtaine

irreducible

algebra

rep-

su CI,1) q

in

the form 3+l!,m>

=

1-_+m+ial[i-+m-ia ]

_

J

ll,m>

2

2

= roll,m>,

a~,

ll,m+l>

=

m~.

C189

s

It

will

in the

be

case

noted

of

the

of

taine

the following

I

=

their

since

undefined

in the the

case

representations

[8]

quantum

groups

same

is true

in a higher

are

not

generators

the

defined

for

dual

C18D

con-

on

and

Csee

of

classical

C 1 @~)

z

the generators

is s p l i t

C al g e b r as9

applicable

+ .... 3!

description

of

not

the

of c o n t r a c t i o n s

unified

the

a~e

h +

previously

continuations

CIS9

functions

s i n h C h)

discussed

be used

they

Indeed,

-

Therefore analytic

formulae

ji=Li.

sinhChD

As w e h a v e must

parameter

Li

i

the

contraction

value

'*

that

the

given also

[I~-]).

contractions

groups

Calgebras)

two different their

b y Eqs. C6)

methods

and and for

r epr e s e n t a t i ons. T h e

dimensions.

References. I.

P.P. K u l i s h , E.K. S k l y a n i n : Zap. N a u c h n . Sere. L O M I g 5 I R g c 1 g e o ) C i n R u s s l and ~.. V.G. D r i n f e l d : Proc. I C M - 8 6 , B e r k e l e y , I ~ 8 7 , pp. Tge-eP.o. 3. M. Jimbo: Lett. Math. Phys. I 0 6 3 C l g 8 5 ) ; 11, R 4 7 C l g 8 8 ~ . 4. L.C. B i e d e n h a r n : 3. Phys. A: Math. Gen. R 2 L 8 7 3 C l g e g ) . 5. P.P. K u l i s h , N. Yu. R e s h e t i k h i n : Left. Math. Phys. 1 8 1 4 3 c i g e g ) . 6. L.C. B i e d e n h a r n , M. T a r l i n i : Left. Math. Phys. ~ 0 C l g g O ) . 7. K. Ueno, T. T a k e b a y a s h i , Y. S h i b u k a w a : Left. Math. Phys. 1 8 R l 5 Cig89). 8. N.A. Gromov: Contractions and analytic continuations of t h e c l a s s i c a l groups. U n i f i e d a p p r o u c h , S y k t y v k a r , IggO, 2 2 0 pp. C i n R u s s i and. g. N.A. G r o m o v , V.I. Man'ko: 3. Math. Phys. 31 1 0 4 7 C l g g O ) . lO. N. A. Gromov: P r e p r i n t Komi S c i e n t i f i c C e n t r e , N 2 3 S ClgSOD. ll.E. I n o n u , E.P. Wigner: Proc. Nat. Acad. Sci. U S A 3 g S i O C l g S 3 ) . 12. N.A. G r o m o v , V . I . M a n ' k o : P r e p r l n t PIAN, N S O C l g S O ) C i n R u s s i a n ) .

228

UNIVERSAL R-MATRIX FOR QU~d~%lM SUPERGROUPS S.M. Khoroshkin Institute of New Technologies K~q-ovogragskaya str., 11, Moscow 113587, USSR V.N. Tolstoy Institute of Nuolea~ Physics, Moscow State University Moscow 119899, USSR Pot quantum deformation of classical finite-dimensional Lie superalgebras we give an explicit formula for the universal R-matrix. This formula generalizes the analogous formulae for classical quantum groups obtained by M. Rosso, A.N. Kirillov and N. Reshetikhin, Ya. Soibelman and S. Levendorskii. Our approach is based on careful analysis of rank two algebras, a combinatorial structure of the root systems and algebraic properties of q-exponential functions. We don't use quantum Weyl group. I. I n t r o d u c t i o n

V.G. Drinfeld [I ] and M. Jimbo [2] introduced the notion of quantum group that gives a number of examples for solutions of Yang-Baxter (YB) equation. Later, Drinfeld [3,4] defined the quasitriangular Hopf algebras with the universal solution of YB equation. Namely, quasitriangular Hopf algebra is a Hopf algebra A with an additional element R E A®A such that A" (z) = RA(z)R -I,

z E A ,

(1.1)

(A®~d)R = R~aR 23, (~d@A)R = R~3R ~2. (I .2) This element R satisfies the YB equation and is called "the universal R-matrix". The method of oonstrttotion the quasitriangular Hopf algebx~s is based on the quantum double notion [3]. If A is any Hopf algebra then the quantum double W(A) is a quasitmiangulam Hopf algebra (~ A@A" as a vector space) with the canonical R-matrix R = ~

e¢ ® e ¢ ,

(1.3)

where e¢ and e ~ are dual bases in A and A'. F o r ~ quantum group Uq(g) (the D~infeld-Jimbo defox~m~tion of Kao-Mood¥ algebra g) there exists an epimol~phism to Uq(g) from quantum double of the oo~esponding Bor~l subalgebra: W(Uq(D+)) 4 Uq(E). Thus any quantum group Uq(g) is a quasitriangular Hopf algebra.

229

The problem is to obtain an explioit expression for the universal R-matrix direetly in terms of U(g). General form of sueh an expression was found by Drinfeld [3,4]. M.Rosso [5] obtained the explieit factorized expression of the universal R-matrix for U (s~(~J) by examining the identifioation of U~(s~(~J) with quantum double of U (b+J. This formula was generalized in [6,7] to quantum deformation of semisimple Tie algebras usir~ q-Weyl group. We deduoe the analogous fozmmAla for quantum supergroups (q-deformation of finite-dimensional simple Lie superalgebras). Our proof is different to that of [5-7]. We don't use quantum Weyl group. Our approach is based on oareful analysis of rank two algebras, a combinatorial struoture of root systems and algebraio properties of q-exponen tial funetions. 2. The Oar%an-Weyl basis for quantum supergroups Let g(A,8) be a oontragredient finite-dimensional superalgebra with a symmetrizable Caftan matrix A (i.e. A = DA (e), where A(e)=(a Cs)) is a sy-mmetrio Oartan matrix,and D = D ~ ( d I,... ,d~), ~ # 0 ) and with a parity funotion 8: {simple roots}-~{0,1}. We define the quantum supergroup UqCgCAoe))=UqCg) as the Drinfeld-Jimbo deformation of U(g). The definition differs from that of [3,4] by replaoing the Lie braokets [ , ] with superoom~nutator [G,D] = a b - (-l)sCa)eCb)bC~ and superoommutativity of tensor produot [8]. For the oomultiplioation we use the following formulas:

Aq(e~ = e

® I + q -~¢ ® I ,

A q ( e _ ~ =e_~® q

+ I ® e_~.

(2. 1)

To define the Oartan-Weyl b a s i s i n Uq(g) we ohoose a normal o r d e r ~++ in the reduoed system of positive roots Y+ and define root veotors on induotion as follows [8]: If 7=~I+~, ~l; Ca)q!

wher'e

o.d~q x (y) =Ix, y], ~f'+~x

q

acZZxq (y) ~ [x,[x,y]]q,

( y ) = [ x , aa~ x ( y ) ] . ,

... ,

(Ix, z] ~ = xz -

q

q

q~zx).

(3.9)

U s i r ~ q - a n a l o g o f t h e H'hdamam f o ~ n u t a we show t h a t ( 1 . 1 ) i s j u s t an additive property of q-exponents for q-oommuting variables. For other type of quantum supergroups we prove the equality

(3.10) in induotion on the height of root 7E~+ and then repeat Rosso's quantum double arguments [5 ], [7 ]. Acknowled~ents. We are sinoerely grateful to Professors I.V. Oherednik, A.N. Kirillov, Yu.F. Smir~ov and Ya.S. Soibelman for useful and stimulating disoussions. References

1. 2. 3. 4. 5. 6. 7.

V.G. Drlnfeld: DAN SSSR 283 1060 (1985). M. Jimbo: Lett. Math. Phys. 10 63 (1985). V.G. Drinfeld: I0M Prooeedings, Berkeley 798 (1986). V.G. Drinfeld: Algebra andAnalysis I ~2 30 (1989) (in Russian). H. Rosso: 0 o ~ . Math. Phys.124 307 (1989). A.N. Kirillov, N. Reshetikhin: Preprint HUTMP 90/B261 (1990). S. Levendorskii, Ya. Soibelman: Som8 appl~oa$~ons o f q ~ n ~ m ~ W e y l ~oup. Preprint Rostov-on-Don (1990). 8. V.N. Tolstoy: Ex¢~em~l p~oJec¢o~s yo~ qu~w~$~ze~ Kc~o-Moo~y supe~alg e b r ~ s ~1%d~ s o ~ O ~ %~e(~ ~ p p ~ ( o ~ $ ~ o ~ s , Workshop on Quantum groups, Olausthal (1989) (to appea~ in Leot. Notes in Physlos),(see also these Prooeedlngs),

232

RECENT PROGRESS IN NON-COMMUTATIVE GEOMETRY* by Richard KERNER LP.T.P.E., Universit~ Pierre et Marie Curie, Tour 16 El, 4 Place Jussieu 75005 PARIS (France)

Abstract We expose here some recent results of the common work performed with M. Dubois-Violette and John Madore. The leading idea of a non-commutative geometry is to replace the commutative algebra of smooth functions on a differential manifold by a more general, non-commutative associative algebra with unity. The maximal ideals can be identified as points, and the corresponding "manifolds" reduce usually to a discrete set of points in the non-commutative case. Nevertheless, the notions of vector fields, exterior forms, metric and connection can be easily generalized. A new version of the Kaluza-Klein type of theory can be set up, introducing such a non-commutative structure instead of internal space.

* Talk at the XVlllth ICGTMP Conference in Moscow, June 1990. 233

1.

Introduction

In the modern approach to the differential geometry as it is exposed in the book of Kobayashi and Nomizu~], one can take as a starting point the aloebra of smooth functions on j~.~3~_D~3,J]Lf~d. Traditionally, these functions are first conceived as functions on the topological Hausdorff space representing the set of points that form a manifold; therefore the elements of this algebra are denoted by f(p), p representing a point. The addition and the multiplication in this algebra are defined then by adding and multiplying the values of the functions pointwise :

(flf2)(P)

= ft ( P ) f 2 ( P )

( l a)

( f l + f2)(P) = fl(P) + f2(P)

(1 b)

etc. The algebra so defined has to be rich enough in order to distinguish between any two points : this means that if Pl ~ P2, there exists at least one function belonging to the algebra such that

f ( P l ) ~ f(P2)

(2)

The dimension of a given manifold also finds an appropriate expression in the properties of the algebra of functions; namely, if the dimension is given and equal to n, this means that for any point p there exists an open set Up containing p such'that one can find n elements of the algebra of functions which will be enough to distinguish all the points of Up. The derivations of the algebra of functions are identified with the vector fields on the manifold. A well known theorem concludes that the aloebra of derivations contains all the information about the differential manifold contained in the algebra of functions itself. Finally, let us consider all the functions belonging to the algebra, which vanish at a given point p; this is clearly an ideal of the algebra of functions, because if f(p) = 0, then g(p)f(p) = 0 whatever g we choose. It can be shown that such ideals are also maximal ideals. Therefore, there is even no need to introduce the points first, and the functions later : one can start with the differential algebra, and then identify the background manifold as the set of all maximal ideals of that algebra. Now, the real or complex valued functions on any differential manifold form always a commutative alaebra. The idea of a non-commutative geometry is to reproduce as fairly as possible all the notions of the usual differential geometry starting with a more general algebra, which will be supposed associative differential algebra with unit element, but not necessarily commutative. The very notion of a manifold is drawn into shade now, although one can still speak of "points" identified with the maximal ideals of the algebra. In what follows, we show the simplest example of such a construction, based on the algebra of (n x n) complex matrices Mn(C). 234

2. Non-commutative geometry of matrices Let us denote an abstract associative algebra by ,,1.; for an infinitely smooth differential manifold V we have ,,t = C°°(V); the derivations identified with the vector fields on V are denoted by X ~ D¢r ( A ) = 9 ¢ r (Coo(V)). We shall use the same notations for the algebra Mn(C). This algebra consists of n 2 generators and all their (complex) linear combinations. The canonical basis of Mn(C) is given by the n x n identity matrix "t n, with Tr 1 n = n, and (n2-1) satisfying the following multiplication table :

Ek EL = n g k L 1

SkL Em-iCk~

Em

(3)

The generators iEk span the well known Lie Algebra of Sl. (n, C) : m [iEk, iEL] = 2 CkL fiEm)

(6)

m whose structure constants are to be identified with 2 Ckl" . We have at our disposal n2 "functions" forming an algebra over C. The "manifold" contains only two points : the algebra Mn(C) itself, and 0. The derivations of Mn(G), denoted by ::Dcr (Mn(C)), are linear mappings satisfying the Leibnitz rule :

a ( a l a 2 ) = (dal)a2 + a1(da2) (7) It can be easily shown that all the derivation~ of Mn(C) are internal, i.e. induced by commutators with the elements of Mn(C) : there are only n2-1 independent derivations, which can be chosen as

(8)

~k EL df ad (iEk) EL = i[Ek, EL] = 2 Ck~ m Em and ak 1 n = 0 These derivations form a Lie algebra over C with a usual commutator law

(9) by virtue of Jacobi identity. We have also

(~o)

~l.(Ek Era) = (aL Ek)Em + Ek(dl, Em) (Leibnitz rule). But these derivations can not be multiplied by fields because

235

functions, like usual vector

(Ek dL) Em df Ek(~L Em) j

.

~

( 11 )

of Mn(C). This is the first important difference we encounter here : the

vector fields do not form a module over the algebra of functions, like in the classical case. This is the general feature of a non-commutative geometry, not only of the algebra Mn(C) : The differential forms and their exterior algebra can be defined esily by introducing the basis of n2-1 one-forms, dual to the basis of vector fields ak : we introduce 8k such that

ek (aL) = ~ I

(12)

The 1-forms span a module over the algebra

Mn(C), because we are able to define the

multiplication

(Em Sk)(aL) = Em (8k aL) = Em ~ . I = ~LLEm therefore

Em ek = ek Em •

(13) (14)

The exterior product is defined as usual :

(e k ,~ e m) (~L, aj) =

(ek(c~L)e m (aj) - e m (~)ek(c~j))

(15)

The exterior differential operator d can be defined now on the exterior algebra of forms : for any function, i.e. O-form f, and a vector field X we define

dr(x) = Xf Therefore,

(16)

d l = 0 and dEk = 2C~kL Em eL

(17)

The 1-forms dEk might have served for a basis of 1-forms, but these do not commute with all functions Em dEk ~ dEk Em• One checks easily that

ek= EL Ek dE~

(18)

with EL = gLmEm; gkl. is the Cartan-Killing tensor of SL(n,C). Finally, a very important identity defines the 2-forms dek:

l dok = - CLm k e6^ e m

(19)

The exterior algebra spanned by ek is similar to the algebra of Maurer-Cartan forms on the group SL(n,G). The Lie derivative with respect to a vector field X can be defined as usual. First, we define an anti derivation ix :

ix ~ = O ff ~ is a O-form;

(20)

(ixco)(Xl .... Xp.1)=oJ(X, Xl ..... X p - l ) 236

if oJ is a p-form.

(21)

if co is a p-form. Then we define

£o) df d o i x o ) + i x o X

(22)

do)

Also in a classical case,

~xlix2+ Jx2~x~=o [~.,xl,~xE=i[x~,xE and

(23)

[~"x l' £ ' x j = £"[x l >:2] A natural metric 2-form can be introduced as

(24)

gkL ek ~ eL The unique volume element is then

el^

e2 /, ...^ e n2-1

(25)

It is easy to check that this volume element is invariant, i.e.

~(~T

e I ^ e2 ^ ... ~ e "2-1) = 0

(26)

for any vector field X. The integration of any (n2-I)-form can be defined as follows : any such form is proportional to the volume element, i.e.

# = B ~ - ~ e t ^ e 2 ^ ...^ en2-1

(2 7)

with B ~ Mn(C); then we set

t

(28)

/# =n Tr B This definition should be implemented by the following properties :

(29) (30)

f # = 0 if# -- do) and

fa^7=

(-1)Pq/7^a

if a and 1' are, respectively, a p-form, and a q-form, so that p+q = n2-1. The Hodge duality operator * on the p-forms as usual,

. (;,' ^ j2

... ,, jp"

,

~ i l J l g i2J2

~

. . .

- (n 2- 1 -p) I

J l J2""Jn2- I - p

J 0 + 1 ^ ... ^ J n 2.1-p E

gJPJP

denoting the Levi-Civita symbol with the property.

Jlj2...jp 237

* (,4 8i1,,, .... ^ 8ip) = A * (si1A .... A 8ip)

(32)

co any p-form. The integration defined by (30) enables us to introduce the scalar product for any two forms the same degree (e~, #) = / a A

*#

(33)

as well as the anti-derivation 8 : < 6~ l # > d f < o~l Sj~ >

(34)

or equivalently de~ = (-1) (n2" 1)p+n2 * d * a

(35)

for any p-form ~t. The Laplace-Beltrami operator is defined as zi = d& + &d

(36)

A particular 1-form, defined independently of basis, and which will play an important rSle in this geometry, is the following : 8 df Ek8k=6~kEk

(37)

Ones proves easily the following formulae : de + 8,,, e= o,

(38)

dB = i [8, B] = i 8B - i Be

(39)

and

for any "function" B ~ Mn(C). This canonical 1-form is also invariant : £8=0 ak

(40)

for any vector field X. Before closing this section, and in order to be better accustonned to a non-commutative geometry, we shall work out two examples of its application. 1) Canonical svmDlectic structure. Although there is no underlying manifold in a proper sense, and there is non definite dimension, a canonical symplectic structure can be introduced in the algebra of "functions"

Mn(C). Consider the two-form ~ defined as Q=d8=-848 or

(4f)

~ = de = CkLm Ek 8L " em

(42)

238

It is obviously closed : d£~ = dd0 = 0; it is also non-degenerate in the following sense : if

Q(X,Y) = 0 for all Y,

(43)

then X = 0. We can introduce Hamillonian vector fields corresponding to a given function B e Mn(C) as usual : for any X e Dcr (Mn(C)), we define Ham(B), a hamiltonian vector field of B, as follows :

(X, Ham(f))

d=f X f = df(X)

(4 4 )

Then, the Poisson bracket of any two functions A and B can be defined as follows :

{f,g}

df

~ (Ham(f), Ham(g))

(45)

In our case, we easily compute

~(dk, aL) = cmij E m 8 i ^ ~ (~k, dL) = 2 CmkL E m = i[E k, E L] = dk E L

(4 6)

Therefore, for any two matrices A and B, their Poisson bracket is equal totheir commutator multiplied by i, like in quantum mechanics :

{A,B} = i[A,e]

(4 7)

In this sense there is no difference between classical and quantum mechanics in the non-commutative geometry; they are in one-to-one correspondence. 2. The eioenvalues of the laolacian in M2(C). The simplest algebra of matrices is spanned by our 2x2 complex hermitian matrices 1 and the three Pauli matrices ek = 1,2,3. In this case CkLm = ekL m, gkl. = 2~kL' and SkLm -- 0. The Hodge *-operator acts as follows : 1 _,,k

* I

eL ^ em

1 ekLm e L ^ em . ek = -~ (48) *

^

= SkLm e m

. ~ ^ eL ^ em = ~ # m f ;

dek = - skLm e~ ^ em

The Laplace-Beltrani operator & ,= d5 + 5d has exactly 32 "eigenfunctions" in the space of P-forms. These are the following :

z~ I = O, z~eL = 4e L,

Z~Gk = 8ok z~dcfk= 8dek

(49)

4(crmem ) = 4(emem), 2

2

z~(~keL + el'Ok - -~ ~ 0m ~m) = 16 (ok eL + eLek --~ ~k em era) 239

and the 2-forms and 3-forms which are the Hodge duals of these ones. As we can see, the spectrum of the Laplacian is finite, which will profuce a finite number of masses if we use it as a Laplacian of the internal space in a generalized Kaluza-Klein type theory. For more details and developments, see the list of References below. References

1.

S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, lnterscience, N.Y.

2.

1963. M. Dubois-Violette, C.R. Acad. Sci. Paris, 307 (1), 403 (1988).

3.

A. Connes, Publications I.H.E.S., 62, 257 (1986).

4.

M. Karoubi, C.R. Acad. Sci, Paris 297 (1), 381 (1983).

5.

C. Chevalley, S. Eilenberg, Trans. Am. Math. Soc., 63, 85 (1948).

6.

A. Connes, C.R. Acad. Sci. Paris, 290 (1), 599 (1980).

7.

M. Flato, A. Lichnerowicz, D. Sternheimer, Journ. Math. Phys., 17 (9), 1754

8.

(1976). F. Bayen, M. Flato, Phys. Rev. D., 11 (10), 3049 (1975).

9.

M. Dubois-Violette, R. Kerner, J. Madore, Phys. Lett. B217, 485 (1989).

10.

M. Dubois-Violette, R. Kerner, J. Madore, Class. & Quant. Gravity, 6, 1709 (1989).

1 1.

M. Dubois-Violette, R. Kerner, J. Madore, Journ. Math. Phys. (in press.), (1990).

240

ALGEBRAIC

GAUGE

NONCOMMUTATIVE

THEORY

AND

GEOMETRY

G.Landi 1,3 and G . M a r m o 2,3 z SISSA, S t r a d a Costiera, 11 - 1-34014 Trieste m Italy. 2 D i p a r t i m e n t o di Scienze Fisiche - Universit~ di Napoli M o s t r a d ' O l t r e m a r e , Pad.19 - 1-80125 Napoli - - Italy. s Istituto Nazionale di Fisica Nucleate - Sezione di Napoli M o s t r a d ' O l t r e m a r e , Pad.19 - 1-80125 Napoli - - Italy.

Introduction

The quest for a "purely algebraic description of reality" as Einstein puts it in appendix 2 of The Meaning of Relativity [Ei],goes back to yon N e u m a n n in connection with quantum mechanics [viNe]. More recently, m a n y efforts have been made to deal with a "non commutative differential geometry" [Co], [Ka], [Du] and "quantum groups" (see for instance [Dr],

[WoD. In a series of paper lLaMa1-8] we have tried to "algebrize" classical physics to have a better understanding of algebraic structures involved in this process with the idea that afterwards we might feel more confident on the more difficultroute to quantum physics. Here we give a brief account of the main ideas involved while referring to pubblished work for more details [LaMal-8]. W e also briefly show how to deal with a noncommutative situation in our framework. For a brief account of a noncommutative geometry approach to gauge theories we refer to [Ke], [DKM].

Algebraic Differential Calculus T h e construction of an algebraic differential calculus relies on a t h e o r e m by R. Palais [PaJ asserting t h a t the only n a t u r a l derivation o p e r a t o r s on a manifold M are provided by the exterior derivative d, t h e inner contraction i x associated with a vector field X , and their (graded) c o m m u t a t o r L x = ixd + dix. T h e latter coincides with the Lie derivative on forms. T h e other crucial observation is t h a t a manifold M can be replaced with the ring 5r = C°*(M) of s m o o t h real valued functions on M . T h e transition to 5r is already known as a useful.trick to "linearize" differential calculus on M .

Thus our starting point will be to forget about M and start with a ring 7. W e require ~r to be an associative algebra over R with a unital element I which allow to

241

immerse t t into Y'. As we are interested in explaining the main ideas we do not bother to make explicit all the properties 7 should have to constitute a reasonable replacement for M. Very often the assumptions being made will be clear from the context. Vector fields on M shall be replaced by the set DerY" of all derivations of ~'.Der~r is a Lie algebra over R. When it is considered to be a module over ~" (in such cases we shall also call DerY" an Y'-Lie module), the algebra ~r is obliged to be commutative. The vector space of 1-forms on M is replaced by the Y'-lineax maps from D s r f to f , Lin~(DerY', F) =: (DerT)*. The exterior algebra of forms on M is replaced by the graded differential algebra built out of (Der~r) *. By using A(DerSr,~r) =: ~p AP(DerY", 7), the exterior derivative d is defined by setting

d: AP(Der~ ", ~) ~

A"+'(Der~ ", Y) , "

k

d,(X,,X=,... ,X,+,) =: E(-~) '+' X,,. ~(X,,..4., X,,+,) k k

l

+ E(-ll~+'~,([x~,x,I,X,,.¢..,.¢..,x~÷,)

, (~)

k I Lie denotes each solution which appears for increasing values of n, by

~1(z,~,yl,... ,Y,,,-I), ~.(z,y, y1,... ,y,,,),...,

~(z,y,y~,...,

Y,,,-2+~).

F r o m the recu~sion relation 3(3 / one can prove [5], eq.(2.92): =

+

So, when aa-~== 0 one can choose ~os = ~ a { , . . . , ~

~h

{d~l ~-I , . , ~ = d~2 -j~-,-~-,

v',,+,(,,1) = ~ah-1.

In the general case 4:

d ~~--V-(-~-) - i d~t _~

=

3(8)

3(9)

This means that with the solutions of the system 2(5) for/¢ = 2, one can build up to an arbitrary order the general form of the O D E with a given symmetry algebra. Lie gave m a n y examples; we give here three of them s . : {O:~,zO~,,z~O=,O~,you, ys ~ } ,,~ ,qL~ X ~L~; then r~' = m = 6 and t h e Lie determinant gives the t h i r d order invariant equation: e - m~

3

- ~ ~

= o,

300)

Beside this equation, all ~-inwriant O D E of order n >_ 5 are of the form ~ . = 0, with f~n an arbitrary functions of ~o~, 1 < ~ < n - 4; with the notation ' for the total derivative e: = (~11 - l e ,

¢~1 "~" (4S8" -- ~.~'~1~{- s ,

(~2 = ( 4~'9',, -- 1 8 5 ~ ' 8 " -{- lSS'S)~--9/2;

the other ~k's are computed according to 3(9). = {@ffi,O~,z@f,y~,y~,mO~,zD, yD} with D

Q

=

3(111

mOf+yO~; so Q .~ ~ L s . T h e n

A ~- - 2 y s u s = 0 with u = 9y~y5 - 45~/s~/sy4 + 40y~

3(12)

yields two invariant equations: y" = 0 and a fifth order one: u = 0. T h e expressions of ~ , i = 1,2 given b y Lie would take one page 7 i So we skip them. = {u= ~ } , 1 < ~ < m > 2, with the u= linearly independent, i.e. the Wronskian w(u=) ~ 0. So ~ is isomorphic to a m-dimensional Abelian Lie algebra. T h e n m' = m - 1, which implies that the Lie determinant va,!sh identically. The u~'s are solutions of the linear differentialequation mo = 0, 4

LIE [2]-X,p. 247 obtained heuristically relation 3(9): in his very short proof, one step is incorrect.

S

Their choice will become clear in the two next sections.

3(13)

6 This example is in [2]I,section 2.12, but with the Lie's implicit remark in [2]II-1.9 that the expressions he introduced are successsive total derivatives of m , 7 Using total derivatives, probably they could be greatly simplified.

254

where: |

. . .

;

~.

"', . . .

"

/

s( 4)

~(m-~) ~(-~+0/

So the ODE of order m + k, invanant by ~, are of the form s

= o,

3(15)

As a trivial application of this powerful method, for the one dimensional symmetry algebras generated by 0= (respectively, a~, ~; 0u) we obtain the obvious result: the invariant ODE £ = 0 do not depend explicitly on x(resp, y,.~/), i.e. 0=£ = 0, (resp. 0~£ = 0, 0~, £ = 0. Since an one dimensional subalgebras of vector fields are equivalent, any ODE with a non trivial symmetry algebra can be put under one of these forms. 4 Symmetry algebra of an ODE; examples of LODE. The previous section sunn-narizes Lie's paper [2]I; the three other papers were essentially devoted to the (eventually partial) integration of ODE with given symmetry algebra. The paper [2]-I also gives implicitly a method for finding the symmetry algebra g ~e of a differential equation e = 0. This technique is explained in detail in text books and powerful (interactive) computer programs have been developed for solving this problem, although an explicit expression for ~e often requires solving the equation! Lie knew that the symmetry algebras of the LODE ~/(n) = 0 are respectively, for n = 1, ¢ = {~(~,y) 0~ +~(.V) 0~}, for r~ = 2, ¢ = SLs given z0 in section 3, for n > 2, ¢ = .~,~(1, ~, ~2 . . . , ~(n-t)) ~ ~L~, the center ~ ~ of the general linear algebra acts on A~ by dilatation while its shnple subalgebra SL2 (~z, 2z 0= + ( ~ - 1)y 0~, - z 2 0= - ( n - 1)zy 0v) acts through its (unique up to equivalence) n-dimensional irreducible representation. It seems that a general study of the symmetry of order r~ > 2 LODE has been done only recently [81 [9]. Up to an isomorphism the subalgebra .An~.AI(~O~) belongs to the symmetry algebra of an LODE. This is easy to understand: given an order n linear differential equation £n = 0, we know that its solutions form a n-dimensional vector space, so the set of solutions {z~} (and therefore the equation) is invariant under the translations by the solutions: infinitesimally they are represented by the n-dimensional Abelian Lie algebra of vector fields Jt~ = {u0~}. Since the equation is homogeneous in ~, the symmetry algebra contains also ,41(~a~). We will now characterize the LODE with a larger symmetry algebra. Without loss of generality we write the order r~ linear differential equation in the form 1(3). To belong to its symmetry algebra, a vector field ~ must satisfy the equation: pr( )e

-

+

+

= 0.

40)

As we have seen, this is a polynomial in the independent varlablea ~/~ = ~/(~), I < ~ < s; to be zero, all coei~cients of its independent m o n o m i a ~ must vanish. For instance, from the terras/n 8

N o t e t h a t for all vector fields ~ = 0 a n d t h a t u p to t e r m s in ~oa, the successive d e r i v a t i v e s of too are the

9 lO

T h a t is t h e s t a b i l i z e r i n ~l~ii~ of t h e equation. I n d e e d $ L 3 is the projective linear algebra a c t i n g on the two d i m e n s i o n a l l i n e a r m a n i f o l d of solutions;

here i t acts locally. W e recalled in the i n t r o d u c t i o n t h a t all order 2 L O D E are on t h e s a m e o r b i t of D i f f . .

255

Y~Y.-z and YlY.-t

we obtain ~ = 0 and 7/#~ - n~=v = 0, so ~7~ = 0. W i t h these results we can

write explicitly 11 the non vanishing terms of ~[k]:

I=1

Moreover we know a general form for ~, 7: = f(z),

~ = g(z)y + ~(~).

4(3)

Using the expression 4(2) for ~[k] replacing y. by its value in £n = 0, interchanging the two sums on k,t and separating the term in Y.-t, equation 4(I) becomes: rt--2

n--1 L=0

"-=

"

¢=1

k=£

(:) () k

~=,+,-,)

+ ~-

CkW=,

=

4(4)

0.

/.=0

The term in Y~-I yields:

#,

= -. -;t- f , .

"-~ i.e. g = --f-

f' + K.

4(5)

So the functions f, u in 4(3) and the constant K i n 4(5) are the only unknowns. The equations they satisfy are obtained directly from 4(4): /. = n - 2 :

i < l < .-

3: ( . - I - l)

+ 4c._:f

(=) I

p . + ~ - o + -~- '

+ 2c:_~f

~((._

i)

-- 0

(:)(:), -

I¢"+~-%

k=l+l

+2c~(.- l)f + 2~f = 0

~ - ~ ( . - 1)ckf(k+~) + 2.~of ÷ 2 4 f = O, ~(~) = O.

4(7) 4(S - 9)

k=l

Equation 4(9) corresponds to the Abelian algebra ~4.({u}) and no condition is imposed on the constant K: this corresponds to the algebra .Ai(yO~). The symmetry algebra of the linear equation £. = 0 is strictly larger iff (=if and only if) the n - 1 equations 4(6-7-8) have at least one c o m m o n solution. W e verify that this is the case when the equation £ = 0 has constant coefficients: then f = constant is a solution. To study the general case we can assume that the Laguerre transformation 1(4) has been performed. Then equation 4(6) reduces to f " = 0, so the equations 4(7-8) have a c o m m o n solution with it iff the following simpler system of n - 1 equations

I " ' = o, o < ~ < . -

3, (~ + ~ ) ( n - ~ - ~)c~+~" + ~(,~- Z)cJ' + ~c~f = o

4(zo)

has n o n trivial solutions. The first equation is of order three, that with l = n - 3 is simply:

= o, 3c._~f' + c._~i ' 11 Use 3(1), begin by a partial derivative again on ~.

Oy on

4(11)

~/, and continue with the condition to never use @y on ~ oI

256

and the other are at most of order two (depending on the values of their coefficients 1. So the s y s t e m has three linearly independent solutions (the set o f polynomials in • of degree < 21, iiT all cl vanish:

T h e o r e m The symmetry algebra o[ an order ~ > 2 ~near dif[erentialequation has dimension n ÷ 4 J~ by the Laguerre trans/orrnatJon the equation reduces to ~(") = O. Then the synnnetry algebra is ,A~ >~ Q L2. This theorem has been s t a t e d in [9] in an equivalent form. In [8] we obtained a similar theorem with the condition t h a t the order r~ equation be iterative, t h a t is of the form:

= L"[y] = L ~-t [L[y]] = 0 with L[~] - ~(=)~' + q(~)~.

4(12)

Reference [9] gives a similar structure for the equation, b u t in the form 1(31: (.-1)/2

.

.

d2

n odd : ~

r~

even:

d-~ + (2k)=c'~-2 ~ = 0 ~/2

/

/==1

--

4(13'I

\

i n((%

-1- ( 2 k - 1)2c._2

)) # = 0.

4(13")

I t e r a t i v e equations 4(12 / have a set of linear independent solutions of the form tLr=-l-kv/~, 0 _< k < n - 1; this is a well known form for building the irreducible representations of,SLz b y tensor symmetric power from the two dimensional one with basis u, v. If after the Laguerre transformation the coefficients are not all zero~ let k _< r ~ - 3 be the largest index of the non-vanishing ones; 4(10) shows t h a t the equation for l = k is a first order equation in f whose solutions satisfy / = L~c~/(h-n) where L;= is a constant and, moreover, ~ a polynomial of degree < 2. This shows t h a t the dimension of the s y m m e t r y algebra of a n order n L O D E is either r~~- 4 or r~+ 2 or r~ -F 1, b u t 12 it cannot b e r~ % 3. I t is z=Jr 2 iff, for the equation in t h e Laguerre form, its coefficients satisfy the non trivial equations of 4(10); explicitly:

.f=A= 2+Bz+C,

k < t < n-

3, cz = O,

0 _< ¢ < k, cL = .f~-"(KL + A(¢ + 1)(n - l - 1)

/;

ck = K k / k - " ,

cL+l (t)/'~-~-l(t)dt).

4(14)

We can prove here a simpler Theorem If" the symmetry algebra o£ an order n Jinear o r d i n a r y d J ~ e r e n t i a / e q u a t b n has dimension n-i-2, this equation can be transformed into a Jinear d~lTeren~JaI equation w~th constant coefJ~cients. We have already proven (see 4(3-4)) t h a t the s~n~netry algeSra has the structure: ~ . ( ~ ( = ) 0~) > .

~4=(~, ~1 with ~ = f(z I ~= + ~ f ' y ~ which normalizes the subalgebra ..4.(u(zI @~) where the u's are the solutions of the equation. The linear ordinary differentialequation 1(31 with this symmetry a1~ebra is transformed by the dhTeomorphism of type I(2): x = into another one with s

1"

I(t)-let,

Y = ~r(=), with r = f - ( ' ~ - ~ ' 2

4(lS)

4 (rOr,Ox) where

etry atgebra 0 =

z = ~(X) is the inverse function of X(z) in 4(15). In the symmetry algebra the term in Ox shows 12 This result is g i v e n in [9]. Although the contrary has s o m e t i m e s b e st&ted, a non llnee~rlz~bledifferential equation of order w m~y have ~t symmetry e/gebza of dimension n + -~: this is the case for the equation u = 0 With ~ = ~ in 3(12).

257

that the new linear equation has constant coefficients. In :Diffz there are many non isomorphic subalgebras ~ as defined above: by a suitable basis in ~ the linear map induced on it by a x can be put in the form of a Jordan matrix whose trace can be removed by a suitable combination with the multiple of the identity operator Y a r . So the generic family has n - 1 parameters of isomorphic classes. To summarize: there is an infinity of strata for the LODE of order n > 2, and equivalently an infinity of Lie subalgebras of :Diff~ containing a ,4n algebra. For the Lie algebras Jr,, and :>~~41, this last factor acting by dilation on the first one, the equivalent classes are labelled by r t - 2 arbitrary functions, and this is also true of the strata of the generic LODE: the functions are their coefficients in the Laguerre form. There are n - 1 parameter families of order n LODE strata with a r t + 2 diinensional symmetry algebra. Finally, as we have seen, the order rt LODE with rt + 4 dimensional symmetry algebras form a unique orbit (and stratum) of Diff2; their symmetry algebra class: W , = An :>aGL2 are maximaJ finite dimensional subalgebras of ~iff2. 5 T h e e q u i v a l e n c e classes o f finite dimensiona]~ algebras o f Diff2. As we explained at the end of section 2, the publication by Lie [7] of the list £ of equivalence classes (under the adjoint action of Diff,) of the finite dimensional Lie subalgebras of 2)iff2 completed implicitly the problem of the classification of symmetries of ODE since we have the list £ of strata and, as explained in section 3, for each finite dimensional subalgebra ~ C :Diff2 one can find all g-invarlant ODE. Classification of finite dimensional Lie algebras was just beginning, but Lie used very cleverly the concept of primitive and imprimitive actions (the latter transform a given family of curves into themselves) of equivalence classes of finite subgroups of Diff2 on the plane z,y. Indeed these concepts are very relevant to the problem~ As we have already seen from some examples in the preceding section, this list £ is infinite. Remark that £ is a partially ordered set (by inclusion of subalgebras up to conjugation in Diffz). Of course, with the results we now know on the structure and classification of finite dimensional Lie algebras, this list can be obtained faster Is. This will be done in the companion snmrner school (at Rachov, Ukraine) and will be published in its proceedings. Here we just give the essential results; most of them are given in tables 1,2 and diagram 3.

type [a ^ b] = 0

dim ~t dime(f)

o) (o lo) (o

('0

ad d

[d ^ a] = a [d ^ b] = ~b

2 0

•¢2J1 [d ^ ~1 = a [dAb]=a+b 2 0

,/~2 ,1 [d ^ a] = [d A b] = 0 1 1

[~ ^ ~] = [d ^ b] = 1 1

o

[d ^ =] = 0 [d ^ b] = 0 0 3

T a b l e I Types (=Isomorphism classes) of non simple 3-dimensional Lie algebras. The symbols 8,.~" m e a n s respectively solvable, nilpotent Lie algebras; their first index is t h e dimension of the m a x i m a l Abelian ideal and t h e second is the dimension of t h e corresponding quotient. There is an infinity of types "q~tl ~'~ S1/;~ ~,1 so we a s s u m e here 0 < l~l < 1. A(2,I is the Heisenberg algebra of center a.

13 Also t h e results c~n be given with greater precision. Instead of giving the llst ~, Lie gives examples of subalgebre.s, he is sometimes r e d u n d a n t a n d it is not ~]w~ys obvious how to construct t h e partial ordering of ~:.

258

T h e r e a r e (29 + three one parameter families) equivMence classes of Finite dimensional ~ i f f = subalgebras which do n o t contain a three dimensional Abelian algebras; t h e s e c/asses a r e those o f t h e subalgehras of S Z , x 5 L, and 5 La. T h e two corresponding classes [I L~ X 5 L=] and [SLy] are m a m m a l i n £ . T h e p a r t i a l l y o r d e r e d set of t h e s e e q u i v a l e n c e classes is g i v e n i n d i a g r a m 3. The notations are explained in the previous tables.

[{~]

[,,42]+ [.42]-[SI,z] + [Sl,l]-

[.4~]+* [s~]+

1

[s~,l]+

z

[s~,l 1+

1

~o

b,h.

c,d,e+

centrallzerC

normalizerH

• o~

~(=)o~, ~"(=) # o

~C~(~)o~)

~i~bl~

4~(x(1-~) -' 0~)

.43 x.4;-

AI(~ o~)o

[sf,~ • sc, d

oo

x O~

oo

~ o~

oo

-0

+ ,Xx o= -D

= o~

-,j o~

o

[z~d

o.

-(a= o= +yo~)

o

[s,+~ e s~j

[s~,t-

,

[N,~]-

*

o=

-(= o. +~ o~)

o

Hfr=]-

[S~,l-

~

o,

-(~o=+(~ + ~)o~)

o

[sl,,,,]

oo

,jo=

o=

~,(o,)o

~ >~(so,,) -

-v=o~

~ia(=)

ceg

~tz(= o=)

ce~

i[~,,]

sL,]+

oo

12vo,

too

tsL=]-

oo 12D

sz.2] °

oo

t 2D

-v(~.~o=+~o~)

-2=~jo= -(=~ + ~=) O~

o

T a b l e 2. E( u i v a l e n c e d a s s e s o f t w o a n d t h r e e d i m e n s i o n a l s u b a l g e b r a s o f Di/e.{,( z , tJ). The first column lists the conjugation classes of the three dimensional subalgebras. There is an infinity of equivalence classes for algebras .A3; they are l~belled by -F and the function ~. In lines 2 and 6 there is an infinity °f isoluorphism types; they are labelled by the parameter A ~ 1, 0 < IAI < 1 snd they are defined in t~ble 1. The columns 2,3,4,5 give the incidence of the four equivalence classes of two dimension&l subalgebras into those of dlraenslon three. A typical representative eubalgebra is given by a basis of three vector fields: a or e_ = 0~; the two other generators are given in columns 6,7: D is a short for z 0= +~0~. The last two columns glve the centralizer and the normalizer in :Diff(z~~/) of e~.ch subalgebra (the upper index ¢ indicates thl~t the centralizer iS the center).

259

One can be astonished that Lie did not invent the simple concepts of centralizer and normalizer of a subalgebra. For example, isomorphic subalgebras of ~Diff= cannot be equivalent ff their centralizers and normalizers (respectively denoted by C~ifr=(~) and A/'~ier2(P) ) are not isomorphic. The centralizers are easy to compute and, in general, it is not difficult to compute the norrnalizers (recall that ~ and C~)tfr=(~) are ideals of Af~i~r=(~)). For the subalgebras of GL2 x SL2 and G/Ss, their equivalence classes in ~Piff2are separated by the isomorphism classes of their centralizersand normalizers. All one dimensional sub algebras of~)iff2 are equivalent; this class is denoted by [v41].There are two isomorphic classes of two dimensional algebras, one Abelian: v42 and one non Abelian: Sz,z (S is for solvable), with commutation relation [a, b] = a. Each of these two isomorphic classes has two equivalent classes in ~)iff2 depending on whether the functional dimension of the subalgebra is I (upper index +) or 2 (upper index - ) . We give here an example of a subalgebra

for each of these four classes: Jt2(@f,y@=)E [~L=]+, Ft~(O=,O~)E [Jt2]-, 8t,~(@e,ma,) E [Sl,t] +,

s~,~(o.,=a= +z,o~,)

e [,s~,~]-. Table 1 gives the isomorphy classes of non simple Lie algebras of dimensions 3. The nilpotent algebra A/2,1 is also called the Heisenherg algebra; its commutation relations can also be written: [a,c] = 0 = [b, c], [a, b] = c. There is one simple Lie algebra of dimension 3: SL2; we can write its commutation relations: [h, e±] = +2e±, [e+, e_] = h. Table 2 gives the equivalence classes of the 3 dimensional Lie subalgebras of Diff,, their centralisers and normalizers. It also gives the partial order relations between 2 and 3 dimensional Lie subalgebra eqtdwlent classes. . ...........

s

[srs]

s

4

[ax,= x aL=]

~

,

]

+

[¢L~]+

3 is

/2]-

D i a g r a m 3. Partial ordering of the equivalence'"classes of ~Diff2 subalgebras (of' dimension > 3) which are smaller than the two maximal classes: [SL2 X SL2] and [SLs]. The four direct products in diagram 3 as well as [~L2] + = [.A1 × SL2+], have one factor in 2~iffl(z) and the other in :Diff=(y). Vie introduce a new family of 4-dimensional algebras: 8~,,I ,1 .-.~2,~(~,b,c):~X~(d) :

[a,b] = c, [~, c] = [b, c] = O, [4~] = c, [d,a]= ;~a, [d,b] =

260

(i -

~,)b.

s(~)

1-A are isomorphic but inequivalent; however, for each value of One verifies that 8~,t, z and 82,1,1 A (in the complex plane) there exists s unique equivalence class that we denote by [8~,1,z]. A representative of this class is: (a = a~, b = N0=, c = a=, d = - ( = a=+Aya~)> e [S~,1,1]

5(2)

As we saw in section 3 (before 3(12)) a natural representative of the class [S/;8] is: sL3 = (a=,a,,=a=,ya,,=a~,~a=,~D,

yD) with D = ~a® + y a ~ .

5(3)

with the natural gIz- subalgebra:

eL=- = ~t1(D) × SL2-(~ 0., = 0~, = 0. -~ 0~).

5(4)

The Bore/subalgebra ( = m a ~ m a l solvablesubalgebra) of SLs is denoted by Bs; it is generated by the first five terms of 5(3). The affine algebra .A~ff2 is a maximal subMgebra which belong to two classes:[.Aft2] ~F = [~4~ >~gL2-] corresponding respectively to the classes of stabilizers of points and of lines in the projective plane (indeed SL3 ~ t P L = , the special projective linear algebra in dimension 2). The two classes: [fl2f~]~F are exchanged by the outer automorphisms of 8Ls; their representatives in 5(3) are generated by the first (respectively last) six generators of this equation. 6 The strata of order 2 ODE. The classificationof the symmetries of order 2 O D E has been given by Tresse [10]in 1896 from the study of their differentialinvariants. This even classifiesthe orbits, and Lie in [2]-IIIhad already given a characterizationof all order 2 O D E on the orbit of the linear ones (for a more precise formulation see e.g. [11]). To conclude this lecture we apply the general sections 2,3,5 to order 2 O D E . This yields very fast the complete listof their strata. From 3(15) we know that the symmetry algebras of o~ler 2 O D E do not contain an Fts, so they be2ong to equlva2ent classes of dim 0,1,2 and those of diagram 3. Consider an arbitrary second order differentialequation:

and let Qz be its symmetry algebra. With the prolongation of vector fields,equation 2(5) yields:

O~/E gg ¢#w~= O; t o y E ~Z ~ zwy-l-wy~ =0;

yt)yEQz,~z~w=ywy+ylWyz.

6(2)

With these relations we obtain: ~=(a~, = a~) + e g e * ~, = ~(=),

s,(a~, v or) + e ge = ~2 =

~(=)~ + #(=);

6(2)

i.e. the equation is linear and~ as we have seen, its s~ mmetry algebra is SL3. All those results can be summariezd in the L e m m a Outside S i s , the symmetry algebra o f an ODE o f order 2 cannot belong to an equiva2~nce clas, >_ [~t=]+ or _> [a~,~]+ i~ z. From table 2, we see that the only possible ~ of dimension 3 are

[s~,~]-,~ ¢-0, [s~,]-, [s~]-, [s~,] °.

6(3)

Diagram 3 shows that there are no larger symmetry algebra classes outside [SLs]. Of course "nearly all" second order ODE have no symmetry, but in practical problems we meet mainly equations with symmetry, this property helping to solve them. The equations of

261

the PaJnlev~ Gambier transcendentals are examples of order 2 ODE without symmetry; another family of examples is:

~ = o : .~ # - 3 ,

~ = Ay ~ + / ( ~ ) ,

f # c , # C(~ + K) 2 ~ / ( I - ~ ) .

6(4)

For order 2 ODE, even a one dimensional symmetry algebra allows the integration of the equation; indeed this algebra can be transformed into ~dz(0=). By a B.iecati transformation X = y , Y = Yt (which is not in 7)iff2(~,y)) we can decrease by one unit the order of the equation (it becomes Y Y ' = w(X, r ) ) . For each strata with non trivial symmetry we give examples of equations:

~[sh]-,:~#o,~,l,2: ;~ ~ [SLq °, Ge E [SL2]- :

I-2A

y..=cy~-r:r, ~,. = ~ ( 3

Y~ = (Y~ + C)(2Y) -1,

~FL]-:

~,=c~-,.

- ~ + Cyi-~/')(= - ~)-~, ~e E [SLa] :

linear equations.

6(8,9) 6(10) 6(11.12)

We leave as an exercise to the reader the listing of the strata (with examples of equations for each stratum) for a given order n >_ 3 of ODE! 7 References. [1] LAGUEI~REE., Sur quelques invariants des ~quations diff~rentielles lin6aires. C.R. Acad. Sci. Paris, 88 (1879) 224-227. [2] LIB S., Klassifikation und Integration yon gewShnlichen Differentialgleichungen zwischen ¢, y~ die eine Gruppe yon Transformationen gestatten. I to IV. Arch. ]or Math Kristiana 1883-I884, new edition for I, II in Ma~h. Ann. 1888-1889. The four papers are reproduced in Sophus Lie Gesammelte Abhandlungen, vol 5, X, XI, XIV, XVI. Teubner (Leipzig 1924). [~] OVSIANNIKOVL.V.~ Group analysis of differential equations (translation from Russian). Academic Press, 1982. [4] IBZtAGIMOVN.H.~ Transformation groups applied to mathematical physics (translation from Russian). Reidel 1985. [5] OLVER P.J., Applications of Lie groups to differential equations. Springer, 1986. [6] LIE S., Vorlesungen iiber continulerliche Gruppen mit geometrischen und anderen Anwendungen. Bearbeitet und herausgegeben yon Dr G. Scheffers, Teubner (Leipzig 1893), Chap. 12, p. 298~ "Satz 3". [7] LIB s , Theorie der Transformationsgruppen, drifter abschnitt, Abtheitung I~ Unter I~twirkung yon Pr. Dr F. Engel, Teubner (Leipzig 1893). [8] KRAUSE J., MICHEL L., Equations differentielles lin~dres d'ordre n > 2 ayant une alg~bre de Lie de sym~trie de dimension n + 4, C.R. Aead. Sci. Paris, 307 (1988) 905-910. [9] MAHOMED F.M, LEACH P.G.L., Symmetry Lie algebras of n ~a order ordinary differential equations, to appear in J. Math. Anal. A~pl. (Preprint Dept. Computational and Applied Mathematics, Univ. of the Witwatersrand, Johannesburg, S.A.) [10] TRESSE A.,D~termiuation des invariants ponctuels de l'~quation diffdrentielle ordinaire du second ordre y" = w(z, y, y~). GekrSnte Preisschrfft, HJxzel, Leipzig 1896. [11] ARNOLD V, Chapitres supplementaires de la th~orie des ~quations diff~rentielles ordinaires. I6E exercice, Nauka~ Moscou 1978. J.K. research is supported by FONDECYT (project 0462/89). L.M. has benefited from it; he is also grateful to DIUC, Pontificia Universidad CatSlica de Chile~ for an invitation in the fall of 1989.

262

CONDITIONAL SYMMETR.IES AND CONDITIONAL I N T E G R A B I L I T Y F O R NONLINEAR SYSTEMS

Pavel W i n t e r n i t z Centre de recherches math~matiques, Universit~ de Montreal C.P. 6128-A, Montr6al, Qu6bec, H3C 337, Canada

ABSTRACT. Two methods of obtaining particular solutions of nonlinear differential equations are reviewed. The first makes use of "conditional symmetries", i.e. local Lie point transformations leaving a subset of solutions of an equation invariant. The second consists of adding further equations to the given one, so that the equation, together with the supplementary conditions, figures as compatibility conditions for an algebra of linear operators. 1. INTRODUCTION. Virtually all of the fundamentM equations of physics are nonlinear, as are most of the differential systems describing specific physical phenomena. It is hence both of conceptual and practical interest to develop techniques for solving nonlinear differential equations. In this presentation we shall concentrate on methods of obtaining exact analytical solutions, rather than numerical ones. The motivation is that analytical solutions, even if they are particular, rather than general ones, very often provide a lot of insight into the qualitative behavior of a system. Analytical solutions can also serve as the starting point for further perturbative calculations. They often turn out to be particularly stable and may provide asymptotic formulas for solutions developing from wide classes of initial conditions. Two different teclmiques are extremely useful in constructing solutions of nonlinear partial differential equations (PDE's). One is the method of ~ymme~ry reduction. It consists of a systematic application of Lie group theory to obtain solutions that are invariant under some subgroup of the symmetry group of the considered system. The invariance conditions are used to reduce the system of PDE's to a lower dimensional system, which may be easier to so.lve. The method is applicable if the 263

differential system under consideration has a nontrivial symmetry group in the first place. The price we pay for reducing the number of independent variables is that initial conditions, or boundary conditions, can only be imposed on specific types of surfaces. The method of symmetry reduction goes back to S. Lie and is reviewed in numerous books and articles [1]-[6]. A recent burst of activity in this area is due, among other reasons, to the possibility of performing otherwise tedious calculations on computers, using various symbolic languages [7], [8]. The other systematic technique for solving PDE's analytically is that of the spectral transform and its generalizations [9]-[11]. It is based on the possibility of finding a Lax pair, i.e. a pair of linear differential equations, for which the original nonlinear equations are compatibility conditions. The method, when applicable, provides large classes of solutions, among them solitons, multisolitons, periodic and quasiperiodic solutions. Nonlinear equations for which the spectral transform method is applicable are called "completely integrable", and they are relatively rave. Our purpose here is to review some recent results extending the applicability of both of the above techniques for solving nonlinear PDE's. 2. CONDITIONAL SYMMETRIES AND THE DIMENSIONAL REDUCTION OF PARTIAL DIFFERENTIAL EQUATIONS. The problem that we are addressing can be formulated as follows. We are given a system of partial differential equations (PDE's)

A=(¥,~',~'=., ~'=.= ..... ) = 0 a=l,...,N,

~ E R P, ~ E R q

(2.1)

of any order M for q functions of p variables. How can we reduce it to a system of equations involving k < p independent variables? We wish to do this by finding functions Ui and za such that the substitution

~( ¥) =

ui(-;, ~i ( ~ , ...~k ), ...~( ~i, ...~ k ) )

:j = ~j(7),

+ = I,..., q

(2.2)

will reduce the system (2.1) to a new system of differential equations of the form

£ . ( 7 , ~, 3+~, ;zo z,, ...) = o

a=l,...,N, "~ER k, ~ER q, O stands for the integral f-~oo d= O(1)(z)/(x) O(J)(x). For the discrete spectrum we have oo

=

t

(o(2)()v) i~:~1 o(~)(,~) ~i(~)o(:)(~)}~=~

-~-:-

-

a#

- 2 i i o ( ' ) ( A j ) lo~=X~=l e(:)(~

))1

(8)

and the similar relations for Akt and "~kc. It should be stressed that the equations (7) and (8) are exact ones, but the exact equations cannot be directly applied because their right-hand sides should be calculated for unknown solution of the perturbed nonlinear equation. Hence, an effective procedure is needed for pxactical calculation of the matrices O and 6) in a relevant order of e. Below we describe .a method of obtaining the first and second-order corrections in the case of the perturbed nonlinear Schrbdinger equation (NSE) irt + r== + 2[r{2r = i e R ( r ) , but this algorithm permits the advancement to higher orders, an actual possibility of such advancement being limited only by an amount of calculations. J

1

2

4. We seek a solution of the perturbed NSE in the form r = r + r -t- r, where m .

i

__(~

r is a soliton solution, while the terms r are of the order of e~. Thereby O 1

2

1

2

+ O + O and O = 6 + O + O. Since b(A) = e -2i~= act ]D(1),O(1) ] and b-= 0, we 1

2

1

2

1

get b =b + b, which gives ~ =~o + p and ~ =~ + ~. As a consequence, the basic equations (4) are written in the form (A1 = ~z + i~h) ,

1

2ri

,

~

2i~h

/;,,( oo ~

-

,~ (=, g ) +

o,

1

(=, ~,)

:,.1(

$ (z, A) 4- O (x, A)+ O (x, A) = I - A - ,~;

]b (/z)l =

)(; ,£,

(=, ~)+ e (~, ~)

1 4- ~ /

[b

oo

+ ~'

::= ~ -

a

(~' '~)+ ~, (=' ~')

)(;

x

)

(~,)1=

"

('~' '~)+ ~' (~' '~)

(o) ×

)' (10)

where we employ the matrices

0

]FI(x) = - I ~{-I(z) "~I (z))0 296

(11)

for taking into account the conditions (2) and use the formula a(A1) = (2i7h)-1 exp [(2~ri) -1 .r_~oo( # - A1) -x l n ( 1 - ]b(/~)[~) d . ] up to the second order of e. The subsequent manipulations consist in the following. Collecting the terms in (9) without • we get a system of algebraic equations J

b (x, ~) = z

~1 - ~-1 6 (~, ~ , ) r l ( ~ ) , A-At

~, (~,~) = z

~i - ~i 6 ( ~ , ~ ) r ~ ( ~ ) .

(12)

A-At

Solving this system with respect to ~, we find a non-perturbative soliton ~'. Sol-

_~ 'ring then (8) with O - and 6 --O and substituting the obtained perturbationdependent parameters in ~, we reach the adiabatic approximation [2]. Now collect the terms in (9) of the order of e. This yields

o(z,A)-

A-A1

1

- -1

1

,~ A ~ (~,~) 0 (~,~) (13)

with parameters satisfying the equations of the adiabatic approximation. It should be noted that the integrands in (13) include O and O obtained within the previous step. Thereby, equations (13) are again algebraical and not inte1

gral ones. Solving (13) with respect to O we get a matrix ~ (x) which gives a first-order correction to the soliton shape in the form Q (x) = ~ era, For finding second-order corrections we collect in (9) the term of the order of e2 :

0 (z, A) - , x2in1 -i~ 21ri

(A1) 6 (~, A1)+ o2 (~, A1) F, oo ~

o(z,~)-

(X1) ~ ) ( z , ~ ) + O ( x , i ~ ) +~

9.

'

A

,f_o

~,~

,x

6 0 # 6 0 + e 6, 0 # ( ~ )

1 ~1+ 04)

I

I-hre C (,,) = (2,~) - 1 J ' ~ l ~' ( ~ ) 1 = ( # - ~,)-'d~,. We point out that the ~rst term in the right-hand side of (14) which corresponds to the discrete spectrum 2

part includes G which is determined by the continuous spectrum, thereby, ;in the second-order approximation there is no complete separation between the contributions of both kinds of the spectrum.

297

As an example of a possibility to overcome calculational barriers we give a second-order correction to the NSE soliton in the case of a dissipation R -- - r : ie2 [ 32~1a 2 ( 1 - z t a n h z ) I 2 - 4 v I ~ ( z ) t a n h z -

r2 ( z , ~ ) -

+ 6 tanhz)Io(z) +

(4u-4vztanhz

4((3)(1 - z tanh z)2(2 - sinh ~ z) ×

x (ln2coshz-ztanhz)+T~(1-4ztanhz+3tanhz)+

z ~ tanh z + u z ~ + T5

--u _ _' 2

+

(

z2 +

z3tanhz

(2 + t~nh z) + z~ (z + 3 tanh z + e-" sinh

z2 ztanhz-usinh2zln2coshz+2uzsinh ei'~sechz £5=1 12/ J

z)

2z+-~-

.

Here z = 271 (z - X(t)), ~ = ~ V i ' ~ z + Z~(~), "r~(~) = e x p / ( a ( ~ ) - 2A~X(~)), u = eL

g*

const., Io(z) = fo In(1 + y2)~, Ix(z) = fo Inyln(1 + y2)~, I2(z) = f01 ln2(1 + y~) d-!cy.The leading term of the asymptotics r: (x, t)Iz{~oo indicates the absence of a tail behind the soliton.

(ie2)/(64r}~) z%i'~sechz

References I. 2. 3. 4. 5.

D. 3. Kaup : S I A M J. Appl. Math. 31, 121 (1976).

V. I. Karpmann and E. M. Maslov : Zh. Eksp. Tcor, Fir. 73,537 (1977). D. J. Kaup and A. C. Newel] : Proc. Roy. Soc. London A 361,413 (1978). Yu. S. Kivshar and B. A. Malomed : Rev. Mod. Phys. 61, 763 (1989). V. E. Zakhaxov and A. B. Shabat : Soy. Phys. J E T P 84, 62 (1972). 6. T. K a w a t a : 3. Phys. Soc. 3apan 51, 3381 (1982). 7. L. D. Faddeev and L. A. Takhtadjan : Hamiltonian Method in the Theory of So]irons (Springer-Verlag, Amsterdam, 1987).

This article was processed using the ~ T F X macro package with I C M style

298

E x t e n d e d Chiral C o n f o r m a l M o d e l s w i t h a Quantum Group Symmetry Ivan T. Todorov Institute for Nuclear Research and Nuclear Energy Bulgarian Academy of Sciences~ B G 1784 Sofia, Bulgaria

Introduction to current work of L. K. Hadjiivanov, R. R. Paunov and the author. Topics reviewed: 1. Statistics and symmetry. 2. Gauge invariant 2-dimensional (minimal) p-models. 3. An extended monodromy group acting in the space of conformal blocks. 4. A ehiral p-model with a quantum group symmetry and a 1dimensional monodromy. 5. (Anti)self dual 4-point functions of the Uq-extended chiral theory. 6. Braid invariant inner product in the space of Uq-blocks. Reconstruction of the (unitary) 2-dimensional Green functions. 1 Introduction The statistics of quantum fields restricts the type of their possible internal symmetry. A basic input is the assumption that sla~is~ics operators (which provide the rule for exchanging two fields in a product for space like separation of their arguments) should commute with the symmetry operations. This justifies the application of the celebrated Wey] theorem which says that the commutant of the action of a (say, compact) group G in the n-fold tensor product of an irreducible G-module coincides with the permutation group trn, and moreover, allows to decompose the tensor product into a direct sum of products of irreducible representations of G and o',~. Thus ordinary statistics (including parastatistics) associated with representations of the permutation group are to be coupled with standard symmetry only. In contrast, the symmetry of fields (with multivalued correlation functions) obeying a braid group s~atis~ics has to be described by a "quantum group". Braid group statistics are only consistent with locality in low dimensional space-times: in 1+1 dimension for point like fields, and in 2+1 dimensions for "string fields" (For a precise formulation and further references see [1]). Two dimensional quantum field theory is simpler and has the advantage of providing 299

a wealth of mathematically non-trivial soluble models. Our treatment, based on current work of L. K. ttadjiivanov, R. R. Paunov and the author [2], deals with a class of minimal conformal models [3,4]. The first published attempt to introduce internal quantum group symmetry in a current algebra chiral conformal theory [5] was deficient, since it did not realize that non-unitary and indecomposable representations of both the Virasoro algebra (Vir) and of the quantum universal enveloping (QUE) algebra Uq are needed in order to trivialize the representation of the braid group acting on a Uq-invariant chiral Green Function. A correct (and more conceptual) treatment of (Wess-Zumino-Witten) current algebra models is being worked out by K. Gawedzki [6]. An operator approach to quantum group extended chiral minimal models using the Coulomb gas picture is developed in [7].

2

"Gauge-invariant"

2-dimensional

(2D)

p-models

A minimal 2D eonformal model [3] can be characterized in the language of the algebraic approach to quantum field theory (recently reviewed in [8]) as a theory with a finite number of superselection sectors and a vacuum subspace that splits into a finite sum of irreducible Vir ® Vir modules. In field theoretic language Vir ® Vir is generated by the analytic and anlianaly~ie (right and left movers') components of the stress energy tensor T = ½ (T o + T~), T = ( T O - T~)

w(~) = Z z,,z-"-:, ~¢ 77

c 2 - 1)&~+,~. z;~ = L_., [Z=, Z~] = (= - m ) L . + . n + i-i=(,~

(1)

(The "physical value" of z = ei'~ are confined to the unit circle interpreted as a compactified light ray, 0 = ~r° - al ). The observable algebra is the tensor product a®K of the (right and left movers') ehiral algebras, which are isomorphic to each other. The state space of the theory can be written as a finite direct sum of tensor products

= O"~n~

®~

(~

= 0,1, 2)

(2)

vP

of irreducible positive energy Vir modules. The unitary minimal models [4] are characterized by Virasoro central charge

¢ (= c (p)) = 1

6

p(p-

1)

(3)

and involve at most ~ ( p - 2) (p - 1) different lowest weights of Vir. h pmode~ is a submodel of a unitary minimal model closed under superposition of I also called a thermal submodel - see [7]

300

superselection sectors (or operator product expansions) involving only conformal weights of the set /)

ZX. = ~A.,

} (~, z 2 >z~ > z 4 > - z s .

(16)

In order to define a permuted block we need to specify a homotopy class of paths in ©4 {zlj = 0 for at least one pair i # j} (the space of quadruples with no coinciding pairs) that permute, say, zi with zi+z. We thus arrive at the notion of statistics

opera,or = n

--'

= rt

Lv~-t

]

vi+zj

302

i = 1, 2, s

=

= 0)

(17)

P

ADE label

p

A~_~

(primary fields generating the algebra of observables) v=0

4z+~

D2,÷2

~,= o,4z

4/+4

D2~+a

v=0

12

E~

,, = o, 6

Labels of primary fields intertwining between different supcrselection sectors (v,v) v = 0 , 2 , . . . . p - 2

(2j,2j) ÷, ( 2 i , 4 l - 2 j ) j= 1,2,...,21--1 (2j, 2j) j = l , 2 . . . . . 21+1, (2j + 1,4z - 2i + 1)

(3, 3) (3,7)

(7, 3)

(7, 7)

(4,4) (4,10) (I0,4) (~0,I0) 18

F,~

~,=0,16

(4,4) (4,12) 02,4) 02,12) (6,6) (6,10) (10,6) (10,10)

(~,8) (8, 2) (8,14) 04, 8) 30

E,

~ = o, 10,18, 28

(6, 6)

(6,12)

(6,16)

(6, 22)

(12,6) 02,12) (12,16) (12, 22) (16,6) (16,12) 06,16) (16, 22) (2~, 6) (22,12) (22,16) Table 1. ADE classifikation of local 2D primary fields in a p-model

which exchanges two neighbouring factors, implying analytic continuation in the positive direction, so that (Rj :) zjj+z --* ei~zjj+l

(zjk ~-* zj+l,k for k > j + 1 etc.).

(18)

The statistics operators are represented by d4 × d4 matrices in the space of 4-point blocks subject to the (constant) Yang-Baxter equation R I R i + I R i = tL~+ltlaRI+I, or, more precisely (for i = 1), v2 1/4 t~A

a

//3 1,'4 /~a

vl//4 aA

(19) The matrix Rx is diagonal in the s-channel basis with eigenvalues determined from the 3-point function (10):

(20) The fact that we are dealing with a representation of the braid group on the sphere implies the additional relation [12] v2v~

R t

vsv~

R 2

v4v~

R a

vav~

R s

vlv~

R 2

vlv2

R I

= e-i41rA~l

I.

(21)

W e are investigated in the subgroup of transformations generated by the statistics operators that leaves the order of the "external lines" (P1u2vav4) intact.

303

This "statistics group" is the smallest group Gp = matrices which contains the monodromy operators

Gp(ul,vs, va, v4)

of d4 × d4

mx = t t lu=~ Rl~a~= = eS~i(,a~+,a~,- a.~- a,.~)R~,~RV,~0z z .

(22)

V~3 m2 = R V23 V = R 2

(23)

(and diagonMizes R~ 2v3)

and which has.the property: if A E Gp thdn R~-IARi E Gp (i = 1, 2, 3). One can choose the relative normalization of the S- and U-blocks (i.e. of the

structure constants c~,v in (8) (9) (10) in such a way that F is involutive: F -~ = a .

(2s)

gq.(21)andtherelationR[ol~2]=ei~(~,+a~.-a,'~-a~=)R[~aO4]suggest (and one indeed verifies - - see [13]) that P can be written as eir(a,., - a,.a -,a,,= - a,,, ) F = Rz,,2v,

l(.2v, v~R~,v= -7- R2vxv=Rav, v~ Rz='2='3.

(26)

The p-dependence of Gp is reflected in the fact that m~P = m~P = 1

(m; = m; = 1 for odd P l -

(27)

The following statement shows that Gp can be viewed as a generalization of a monodromy representation of the braid group. P r o p o s i t i o n 1 . Gp(v,v,v,u) is isomorphic to the representation of ~he braid group B4 on conformal blocks of type (v, v, v, v). P r o o f : It follows from (26) that R t I F R 1 = e ~ i a ~ R z R ~ , hence e2'~ia~R2 E Gp; similarly, e2'ria-Rx E Gp. We remark that the monodromy (or "pure braid") group M4 appears as an invariant subgroup of B4 (-~ Gp) and the relation RI = Rs implies in this case that the factor group is isomorphic to the 6-element permutation group .~.

304

4 A QUE Algebra Extended 1-dimensional Mono dromy

p-model

with

We saw that a single 2D correlation function gives rise to a finite dimensional space of conformal blocks. The question arises: can one extract a "chiral square root" of a 2D field theory which can be itself interpreted as a (1-dimensional) quantum field theory with an internal symmetry? We have already noted in the introduction that whenever conformal blocks of non-trivial monodromy are involved such a symmetry cannot be described by an ordinary group. In simple cases like the p-models, studied here, and current algebra models [6] the notion of a (Hopf) QUE algebra [14,15] appears to be adequate (A more general concept - of a quasi-Hopf algebra [16] - is required for treating some orbifold models [17]). A p-model is related to the QUE algebra

uq := uq(sl (2)0,

(2s)

where the value of the deformation parameter q is a pth root of - 1 : [2] := qq- q-1 = 2cos lr. (29) P We summarize the basic facts about Uq, fixing on the way our notation (see for more detail [14,15,13,2]). Uq is an associative algebra generated by 4 elements q:i=J3 and J± subject to the relations

q+J~q~:J~ = 1 ,qJ*J~ = j + q J ~ : l [j+, j_] = [2J3] . -

q2J~ _ q- 2J~ q _ q-1

(30)

It is equipped with a coproduc~ Zi : Uq ~ Uq ® Uq, an algebra homomorphism satisfying A(q:t:j:) = q:~.r~ ® q±j~, A(J+) = J+ ® q-J* + qJ~ ® J± .

(31)

There also exists an algebra homomorphism e : Uq ~ ~], the co-unit, such that =

1,

o .

(32)

The tensor product ring of finite dimensional representations of Uq, generated by the 2-dimensional representation of q-spin 1/2 that is identical with the defining representation of SU(2): J+ = o'+ =

0

'

contains, for q satisfying (29), exactly p unitarizable irreducible highest weight representations ]~,

A(=2I) =0, l,...,p-1,

dimV~ = A + I

(33)

as well as an ideal of indecomposable representations [18,1] (for a mathematical study o f the QUE algebra representations - see [19,20]). The correspondence between Uq and the p-models (on the related SU(2)p_2 models is based on the following observations [21,9,12,22,23,1].

305

(i) The quantum (q-) dimensions of the first p - 1 representation (33) satisfy the BPZ fusion rules (defined in (1)): q~t ~ q - - n

[,k-l- l][/z'-l- I] = ~"~Nx,~,[v-l- I] for [n]-

q_q-1

(34)

v

(ii) The Uq 6j-symbols [24] coincide, essentially, with the fusion matrix F of (24). We shall further fix The choice of q among the two roots of Eq. (29) by demanding that the product of monodromy matrices m l of the p-model and of the Uq-theory is a multiple of the unit matrix. To this end we use the Uq universal R - m a t r i x [14,24,12] R E Uq ® Uq which relates the coproduct (31) and its permuted ~.a(x) = R~(X)R-',

X e U , , ~(X ® Y) = Y ~ X

(3s)

with the result

1•

o j

(3o)

For v~. _> va, I~ >_ 11 we find v ~ - vz +2n h - lz + n --- ( - 1)Vlz+Vlei~V= (v2+~.)q- 2I~ (12+l)ei~rn(v=-v~),q2n (/2-I1) (qei,r,)n (n+l) , n = 0, 1,...,~1 •

(37)

In order to make the last factor independent of n we should set q2 = e x p ( ~ ) ; this, together with (29), gives q= er (38) (We remark that for an S"ff(2)k current algebra model the primary weights are A21 = p - t I (I -I- 1), 21 = 0, 1 , . . . , k = p - 2, and we find q -- e i~/p instead of (38)). The diagonal assignment 2Ii = vi (--- ~ ) also kills the linear dependence in n in the exponential of the square of (37). This observation suggests the following correspondence between diagonal 2D p-models and their extended chiral counterpart. To each local 2D primary field of label (u,P --- v) = u we make correspond a (is+ 1)-component chiral V i r ® U q primary field

where [ : ] is a q-deformed binomial coefficient

= [ ~ ] ! ~ ; : n]!'

[~1~ = [~1[~ - 1]~, [01: = 1,

306

(401

u is a formal variable substituting the "magnetic quantum number" m (= n - ~) (see [22]). • is an analytic (malt±valued) primary field with respect to V i r of weight Ziv (see [3]), while its Uq transformation properties are summarized by q.rS~,~ = ~,mqJ~+m,

(f~m = ~},-n(z; 2I), m = - 1 , 1 - I , . . . , I)

J±~,n = ([I ~ m][I :i: m + 1])½d~m+lq -J: + qmqI'mJ± .

(41)

The vacuum of the theory will be assumed both MSbius and Up invariant, L,~10) = 0

for n = O , 1,2,...

(~

L_~}0)

= 0),

X]0) =¢(X)10 ) for all X E U q (i.e. J±lo)

= o = (q±J~ - 1)to) ) •

(42)

The 2- and 3-point functions of ~ are then computable and their z- and udependence factorize; we choose the normalization of • so that {01~5(zl, Ut; v)~(z~, u2; t~)10) - z ~ ' a " [ q - ½ u z ( - ) q ½ u 2 ] where

v

(43)

Z 1 2 ~- Z 1 - - Z 2 , v

[q-½ux(-)q12u2] v : ]_.~q ~-ntlY~Itn* tnJ lk-u2)~u-n ~m0

= (q - "~

- q ~ u ,.)[u ~(-)~] " - ~

;

(44)

the 3-point function is a product of (10) with the (q-deformed) "3j-symbol" ( A /~ v )

, + =[q--r-u~(-)q-W'u~j*+= [= 1½(~+v-X) rq- ---r"÷= u~ ( - ) q ~.+~ ' ,½(~+~-V)u2j

I t I I12 ~t3

× [q-~uz (-)q~ua] ½(*+v-l*)

(45)

(which vanishes unless all three exponents are non-negative integers). It turns out that the overall structure constants of the extended theory can be defined in such a way that the 4-(and higher) point Green functions transform under a 1-dimensional unitary representation of the statistic group Gp (introduced in See. 3). 5 Fusion Invariant chiral theory

4-point

functions

of the

Uq-extended

We start, for the sake of definiteness, with a diagonal (Ap_l-type) model and consider the 4-point Green function (for 1 < tt _ p - 1 , however, we find that the fusion rules ]~wA of the extended theory have to differ from the BPZ rules (N~vA _< l~l~vA). Indeed, in the BPZ approach, each function fA and g2n satisfies two differential equations, of orders # + 1 and ( p - 2) ( p - 1 - # ) , stemming from null vector conditions in the Verma module of lowest weight z~f,. The second equation kills # + v + 2 - p among the solutions of the first ( thus reducing the dimensions of the space of conformal blocks to d4 - see (7)). In the extended theory only the first of these equations is satisfied. For instance, in the Ising model (p = 4) the null vector condition

L,~(rL2_I-L_2)#(z=O,u;v=P=2)JO)

=0

for n = 1,2 . . . .

(52)

does not imply the vanishing of the level 2 null vector in the extended state space of the theory (what happens is that whenever the BPZ structure constants C~v~(r) tend to zero - in the unitary normalization - for r ---+ 1 - ~, the corresponding Uq structure constants diverge to infinity in such a way that their product with the C's tend to a finite non-zero limit). The (anti)self duality (more generally, the Gv-covariance ) of Mv~ is thus achieved at a price: we need non-unitary Vir (and Uq-) modules in the extended state space of the theory.

308

We shall illustrate the situation by two examples, corresponding to v = p - 2 andv= land2. For/~ = 1 and any v (> 1) the spaces of conformal and Uq blocks are (each) 2-dimensional and the corresponding fusion matrices are computed to be [2]

(~v #- 0 is a free parameter reflecting the remaining freedom in the normalization of f~-l+2n and g2, after imposing (25)). For [v + 2] = [p] = 0 the matrix F becomes lower triangular, 5r, upper triangular, and the 4-point function (49) (50) is only (anti)selfdual if the irreducible n o n - u n i t a r y representation of V i r of lowest weight P AP-I : ""4p 1(p~ _ 2 p - I) contribution to the sum in intermediate states. The second example (/~ = 2, v = p - 2) involves, to begin with, indecomposable representations of V l r and Uq. The property normalized Dotsenko-Fateev (Coulomb gas picture) integral representations of fp-2 and fp [25] coincide and so do the finite Uq blocks 9p and 9p-2. However, ~-](fp-2 - fp) has a limit for v ---* 1 - 1_ ([p] ..., 0) and is linearly independent offp. Similarly, one can redefine P the s- and u-channel bases of Uq blocks ending up with the following triangular fusion matrices [2]:

: 2

1 Y-

= *~

(54)

where z and y depend on the normalization. Non-diagonal primary fields, corresponding to the Dzj+a series of 2D models with 2Iv = Valso admit a fusion invariant 4-point function - see [2] (they involve, typically, a smaller number of non-unitary contributions than the diagonal ones). The remaining non-diagonal 2D models (listed in table 2) involve an extended algebra of observables. In the first (D4-) example of this type, the chiral Potts model corresponding to p = 6, a is generated by the field of spin A 4 : 3; this is the simplest example of a W - a l g e b r a , introduced by Zamalodchikov [26]. It is conveniently described by the coset construction [27]

su(3)l × su(a)l su(a)=

, c-2+2

8×2

4

a+ 2

5

- - -

(55)

The superselection sectors can be labeled by the level 2 SU(3) weights 0 = (0,0), 1 = (1,0), 1" = (0,1), !1" = (1,1), _2 = (2,0), _2" = (0,2).The corresponding lowest weights of V i r are I 2 ~Q = o, ~ ! = ] - g = ~ ! " , ~ ! ! " = g , ~

309

2 = g = Aa'-

(56)

The fusion rules of the model, displayed in [281 (see Eq. (19)), are those satisfied by the positive quantum dimensions of the irreducible representations of the QUE algebra Uq(sl(3)) for q + q - l ( =

[2])=2cos ~' = -1+v/'5 T(=

[3]).

(57)

The intersection of the set of weights (56) with the "thermal" series of the p = 6 model consists of 0, 2 and _2" and obeys the (closed) set of fusion rules

2 x 2 = 2 " , _2" x_2* = 2 , 2 x 2" =_2" x _ 2 = 0

.

(58)

We intend to treat this and other examples of non-diagonal chiral minimal elsewhere. 6 Braid Invariant Inner Product i n t h e S p a c e o f Uq Blocks. Reconstructing the 2D Green functions The presence of an unitafizable representation of the statisticsgroup Gp in the space of Uq-blocks suggests the existence of a Gp-invariant inner product in this space. W e shall construct such an inner product starting from a factorizablc sesquilinear form defined on the tensor product ring generated by the irreducible Uq modules ~)x (33) (factorizable meaning (vl ® va, v~ ® v~) = (vl, v~)~l(v2, v~)x=

for v~l) E 1)xl, i = 1) 2). The form is fixed by giving its values in the polynomial basis for an arbitrary irreducible representation 1;2::

(-~I-~ uI+ra=ei,-r [ 21 ] -1 '

I + m

&,m

•

(59)

We note that this form is not hermitian; for 1 = i1 + n it takes independent of m pure imaginary values. However it becomes hermitian and positive semidefinite when restricted to Uq-invariant blocks; for instance

ra)~=--I

X_tl -m-~t_--..l-~I+'~ I,~ i "~ 2

4, ) "1

I+m.,~ I2- - m ~1

I

=

\ ut

u~

~

q-~"' = [2x + 1] ,

(60)

ua /

[z~ + I~ - x~]! [z,. + x. - x~]! [x~ + I~ - I:]! [z~ + I2 + Ia + 11! • (81) [211]![212]![213]!

310

The norm square of any 4-point block A

(

rh ~2 t~s tJ4) ~P~ ul u2; us u4

t

,

;

~tl u 2 - m

)(:

;

,

u3 ~4

)

,

(62)

where we have used the expansion (39) (40) giving

Ul

~$2

n=O

~I

U2

;

?I --

1

can be computed from the factorization relation

[q-~u(-)q~u']~ Ul

~

~2

ul u2; u3 u4 ,gt

U3

U4

(63)

One easily verifies that it vanishes whenever either one of the triples (ux, u2, A) and (A, va, v4) violates the BPZ fusion rules. The Green functions of any of the (ADE) local 2D theories is restored by taking the inner product of a chiral Green function of Yir ® Uq primary fields of type (r,~, 2I~ = P~) with the respective diagonal one (of fields of type (~i,~)). Its monodromy invariance and selfduality then follows from the Gp-invariance of the inner product in the space of Uq-blocks and from the Gr-eovariance of the (extended) chiral Green functions (of type studied in Sec. 5).

It is a pleasure to thank Academician M. A. Markov and Professor V, L Man'ko for their kind hospitality in Moscow during the 18 International Colloquium and Group Theoretical Methods in Physics.

311

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312

24. A. N. KiziIlov, N. Yu. Reshetikhin: Representations of the algebra Uq(s/(2)), qorthogonal polynomials and invariants of links~ in: Infinite Dimensional Lie Algebras and Groups~ Proceed. of 1988 CIRM, Luminy Conference~ Ed. V. G. Kac (World Scientific, Singapore 1989)pp. 285-339 25. VI. S. Dotsenko, V. A. Fateev: Nucl. Phys. B240 [FS12] 312 (1984) 26. A. B. Zamelodchikov: Theor. Math. Phys. 65 1205 (1986) 27. P. A. Bais, P. Bcuwknegt, K. Schoutens, Surridge, M.: Nucl. Phys. B$04 371

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This article was processed using the ~ X

Inacro package with ICM style

313

C o n f o r m a l R e d u c t i o n of W Z N W and W-Algebras

Theories

L. O'Ra fea taigh Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland.

A b s t r a c t : It is shown that Toda field theories can be regarded as reduced WZNW theories and that the reduction generalizes to yield families of conformal and non-conformal integrable field theories. The advantages of regarding the conformM theories as reduced WZNW theories are outlined, and include the natural appearance of two-dimensional gravity~ the easy derivation of the general solutions from the standard WZNW solution~ and, for the Toda theories, an intuitive understanding and relatively simple construction of the W-algebras.

1 Introduction. Two of the most celebrated classes of conformally-invariant two-dimensional field theories are the Wess-Zumino-Novikov-Witten (WZNW) theories[I] and the Toda field theories[2]. Recently it has been shown that these theories are not independent and that, in fact, the Toda theories can be obtained from the W Z N W theories by placing conformally invariant constraints on the currents[3]. More recently still, it has been shown that a similiar reduction leads to a whole series of coaformal integrable field theories, which interpolate between the WZNW and Toda theories [4], and that there even exists a non-conformal version of the constraints that leads to non-eonformal integrable systems, in particular to the a~ine Toda systems. In the present talk I should like to describe these reductions of W Z N W theory and to outline the various features and advantages that emerge by regarding the reduced conformal theories as constrained W Z N W theories. Perhaps the most remarl~ble feature that emerges is the appearance in all cases of a two-dimensional gravitational field~ in non-trivial interaction with itself and with the other fields. 314

The emergence of this gravitational field is is in some sense the converse of the Polyakov's embedding [5] of the string-induced two-dimensional gravity in the WZNW group SU(1,1), but it is present for all WZNW groups. One of the greatest practical advantages that accrues from regarding the conformal theories as constrained WZNW theories is that their general solution can be obtained in a rather simple manner from the general WZNW solution, which is well-known to be quite trivial. For reasons of space the derivation will not be given in this talk but the general method will be indicated (with references for details) and the end-result, which is quite simple, will be presented. One of the remarkable features of the Toda'.theories in particular is that they realize [6] the polynomial algebras (so-called W-algebras) defined [7] abstractly by Zamolochikov. Within the confines of Toda theories it is not immediately obvious why these algebras should exist, and one of the great advantages of regarding Toda theory as a reduced WZNW theory is that in the broader WZNW context their existence becomes quite natural and understandable. Indeed in the reduced WZNW context the W-algebras have a simple intuitive interpretation as the algebras of gauge-invariant polynomials of the constrained currents (and their derivatives), the gauge group in question being that one generated by the constraints. This identification not only provides an intuitive understanding of W-algebras, but also provides a relatively simple algorithm for their computation. This is because of the existence of a gauge in which the gauge-invariant polynomials reduce to the currents themselves. In this gauge the W-algebras manifest themselves as the Dirac star algebras of the gauge-fixed constrained currents and, because all constraints are linear in the currents, the W-algebras can then be computed relatively easily from the Kac-Moody algebras of the associated WZNW theories. For those not completely familiar with two-dimensional conformal field theory we begin by recalling the features of theory which are relevant to our discussion, in particular the WZNW and Toda theories and the Zamolochikov W-algebras. With these aspects of the theory in hand the reduction will be seen to bc quite staightforward.

315

2 Recall of Conformal Field T h e o r y and W - A l g e b r a s . We begin by recalling the situation for conformal inva~iance in more than two dimensions (D > 2). Let L(¢(z)) be the Lagrangian density for any set of tensor fields ¢(x) and T ~ ( ¢ ( z ) ) the corresponding energy-momentum tensor density. If L(¢(x)) is conformally invariant then according to Noether's theorem the generators L~v, P~, S~, D of the cor~ormal group are moments of T~v. For example, for the dilation D one has D = f z ~ T ~ o d ° - l z . In all the D > 2 cases the conforreal group is finite-dimensional ((D + 1)(D + 2)/2-dimensional actually), and thus involves only a finite number of moments of T~v. It is also semi-simple and thus admits no central extensions. In two dimensions the situation is different. If x = (xl, x2) are the usual Cartesian coordinates, then the conformal group consists of all transformations of the form z --+ f ( z ) and w --+ ](w), where z , w = xl :l: x2, or z , w = xl :l= ix2, according as the metric is Minkowskian or Euclidean, and f ( z ) and g(w) are arbitrary analytic functions. Thus it is an infinite-dimensional group and is a direct product of a left and a right part. F~zrthermore, it is well-known that each part aclmits one central extension [1]. For conformally invariant Lagrangians the (three component) energy-momentum tensor density T~z, -- [Tz~,Tzz, Tww] reduces to [Tzw = O, T~z -- L(z) and Tww = Z(w)], and the Noether generators of the conformal group consist of all its moments i.e. consist of the quantities Ln=fznL(z)dz

and

Zn=Jw"]~(w).

(1)

From the structure of the conformal group it follows that the L ( z ) and f,(w) commute with each other and that each satisfies a Virasoro algebra i.e. an algebra of the form {L(z), L(z')] = 2L(z)O,6(z - z') + OzL(z)6(z - z') + - c~ O ,36 ( z - z'),

(2)

where the last term is the central extension and c is a constant that depends on the original Lagrangiau. The tensors with respect to the conformal group are called primary fields and have the transformation properties

¢(x) - .

(3)

where the quantities j and ~ are called conformal weights and arc often integers. 316

With these properties recalled let us turn to the definition of W-algebras. According to Zamolochikov, who first introduced them [7], a W-algebra is an extension of a Virasoro algebra by primary fields, such that the Poisson bracket (or commutators) of any two primary fields is a polynomial in the fields and their derivatives (both primary and Virasoro), the order of the polynomial being less than the combined order of the two primary fields. In other words a W-algebra consists of the Virasoro algebra, the transformation law (3) (with one of the coordinates (w,say) dormant) and a set of Poisson bracket (or commutation relations) of the form

[~(")(Z), ~(b)(zt)] -~- P(a'b)(~(Z), L(z), ~(z - zl)),

(4)

where p(,,b) is polynomial of lower order than (a+b) in L(z), q~(z), ~(z - z') and their derivatives. In counting the order the delta function and the derivative are each given unit weight. 3 Standard

Examples

of 2-D conformal

Field Theories.

Two standard examples of 2-D cor~ormal field theories are the Wess-ZumlnoNovlkov-Witten (WZNW) theory and the Todd theory. The WZNW Lagranglan takes the form k

= J

2+

j

3,

where the three-dimensional integral is over a space whose boundary is the twodimensional one of the first (kinetic) integral. As a result of the addition of the three-dimensional integral, whose variation is purely topological, the field equations of the theory take the form

O j ( = ) = 0 and 0,](=) = 0,

(6)

where

J(=)=g-'(=)O,g(=)

,~d

](=)=(O~g(=))g-'(=).

The field equations mean, of course, that the currents J(=) and J(=) are functions only of z and w rcspectively and from the symmetry of L w z with respect to (rigid) left and right group multlplication (g ~ hg and g ---*gh), and the Noether theorem, it follows that they satisfy Kac-Moody (KM) algebras with centres 1~. Thus ](z), for example, satisfies the KM algebra

317

[J=(z), Jb(~')]

= / h J o ( z ) 6 ( ~ - ~') + k6=b0~6(z -- z').

(7)

The Toda Lagrangian, on the other hand, takes the form

LToda

/ d'zx[CiiO¢~(x)OeJ(x)+exp(gij¢i(x)¢J(x))],

(8)

where C and K are the Coxeter and Cartan matrices for any semi-simple simple Lie group. Thus to every Dynkin diagram there is associated a Toda field theory. Recently it has been shown that every Toda field theory admits a W-algebra, the W's being coefficients in an equation called the Gelfand Dickey equation[8]. This equation is a linear differential (or pseuda-differentiM) equation of the same order as the dimension of the defining representation of G, and which is satisfied by certain left-and right-moving functionals of the Toda fields. Its role in our discussion will be to help identify the W-algebras.

4 R e d u c t i o n of W Z N W Theories. What we want to show is that the Toda theories can be obtained by putting conformally-invariant constraints on the WZNW theories and that by generalizing the constraints one obtains not only the Toda theories but a whole class of theories that interpolate between the WZNW theories and the Toda theories. These theories may be regarded as interacting WZNW theories or as generalizations of the Toda theories in which the individual Toda fields are replaced by WZNW fields,the usual Toda theories being the extreme case in which a~ the subgroups are abelian. A remarkable feature of the reduction is the emergence of an abelian field that plays the role of two-dimensional gravity. Some advantages of deriving the Toda theories in this way are: (i) the emergence of the two-dimensional gravitational theory just discussed (ii) the emergence of a new set of conformally invariant integrable field theories (iii) the derivation of the general solutions of these theories from the (trivial)

general ~olutio~ g(~) = g(z)~(~) of the WZNW theories (iv) the emergence of a simple intuitive explanation of the W-algebras of Toda theory and of a relatively easy algorithm for their computation (v) the fact that the whole procedure can be generalized so as to obtain a series of non-conformal field theories including the affine Toda field theory. One also obtains a formula relating the KM and Virasoro centres for the quantized theory [3][4] but this will not be discussed here. 318

The reduction of the WZNW theories requires setting some of the WZNW currents equal to non-zero constants and since these currents, being space-time vectors, have conformal weights (1,0) or (0,1) with respect to the usual conformal group, the problem is how to set them equal to constants without breaking conforreal invariance. By the usual conformal group is meant here the group generated by the Noether currents L(z) and L(w) belonging to the energy-momentum tensor of the WZNW theory and the way that is used to circumvent this dii~iculty is to note that the this conformal group is not unique. In fact, there is a two-parameter family of conformal groups equivalent to it and the procedure will be to choose a member of this family with respect to which some of the currents are no longer vectors but scalars i.e. have conformal weights (0,0). However, to make the appropriate choice of member requires some Lie-algebraic technicalities and these will be discussed in the next section.

5 Lie Algebraic Technicalities. The simple WZNW groups G which are used for our reduction will be the (maximally non-compact) ones generated by the reallinear span of the Cartan generators i.e. by the generators [Hi, E~] in conventional notation. For the A and D algebras, for example, these are the groups SL(n,r) and SO(n,n). Within the Carman algebra there always exists an element H such that each of the simple roots Ea, is an eigenvector of H with eigenvalue unity or zero.

[H, Eo,,]=hEa~ where h = 0 , 1 ,

i=1,2...l,

(8)

and I is the rank. (To see this note that H can be written as w.H, where w is a sum over any subset of the I fundamental coweights). Then H provides an integer grading of the whole Lie algebra, [H,E=]=hE=

where

heZ.

(9)

In particular the elements of the algebra of the little group of H, which we shall call B, will have zero grade. It is not dif~cult to see that the set of little groups B for all possible choices of H are just the non-compact versions of the set of little groups in the adjoint representation of the compact form of G. In particular for w = s, where s is the sum over all the simple coweights (=half the sum of the positive coroots), the little algebra is the generic one, namely the Cartau algebra itself. (It will be seen later that this case corresponds to the Toda reduction). 319

Finally we note that G admits a local Gauss decomposition G = ABC, where B is the little group and A and C are the (nilpotent) groups generated by the root vectors Ea with weights which are strictly positive and negative with respect t o / / . (This decomposition may not be global, but the parameter space may be divided into a finite number of patches on each of which the decomposition is valid up to left- or right-multiplication with a constant group element). At the KM level we have, correspondingly, [H(z), J ' ( z ' ) ] = 0

except

[H(z),H(z')] = k0=5(z - z ' ) t r n ~,

(10)

=

(11)

and [H(z),

-

6 P r e s e r v a t i o n of C o n f o r m a l Invariance in t h e R e d u c t i o n . We come now to the crucial point. Let L(z) denote the Virasoro operator which is the component T~(z) of the energy-momentum tensor of the W Z N W theory, and with respect to which all the KM currents J(z) are have conformal weights unity, or, more precisely, (1,0). Then we replace L(z) by A(z), where

A(z) = L(z) + O,H(z),

(12)

It is to be noted that A(z) is again a Virasoro operator i.e. satisfies a Virasoro algebra of the form (2). The only difference is that the centre c changes to c 12ktrH 2. It will turn out that A(z) is actually the improved (i.c.traceless) energym o m e n t u m tensor of the reduced theory. Once the crucial change (12) has been made the rest is almost automatic. With respect to the conformal group generated by A(z) the KM currents J(z) are no longer vectors of conformal weight (1,0) but have the following transformation properties: (i) Except for H(z) the currents JB(z) belonging to the little group B are still vectors i.e. have conformal weights (1,0). (it) the field H(z) now transforms not as a spin-one vector but as a spin-one

connection. (iii) The currents Ja(z) transform as conformal tensors (primary fields) of conformal weight (1 + h). Thus, in particular, the current~ of grade h = -1 trans/orm as conformal

scalars. 320

With this information in hand we are ready to impose the constraints, namely, J_~l(z) = J_~l(0) # 0,

and

J~(z) =

0,

h < -1.

(13)

Here the set of constraints with non-zero right-hand-side do not break the conforreal invariance generated by the new Virasoro operator A(z) since they are scalars with respect to this operator, and the set of constraints with zero right-hand-side are added so that the complete system of constraints is first class. For the righthanded currents ,T(w) similiar constraints are imposed, but with h < 0 replaced by h > 0. In order to obtain an intuitive feeling for the meaning of the constraints (13) let us consider the case of G=SL(n,R), in which case the constrained current J(z) takes the form

Joo"''.(z)

=

'J,l(z) S21(0) 0 o o ,,°°°

J12(z) J=(z) ]~3(0) o o ° , H ° ° ° ° °

o

S,3(z) J23(~) J~(z) J~,(o) o ° ° ° , , ° ° , ,

o

o

............ ............ ............ ............ ............ H

*

~

*

~

*

H

J.._,(o)

S,.(z)' J=.(,) J~.(~) :,.(z) Js.(z) t o o I H I o o

J.,,(z),

where the Jab(z) denote submatrices of currents which in general are not single entry or even square. Note that the constraints can also be expressed ms J~,a = M

and

]vos = N,

(14)

where M and N are constants matrices of grade minus one and plus one respectively, and rteg and pos refer to the sign of h. The constraints (13) are not invariant with respect to general KM transformations, J(z) --o U(z)-lJ(z)U(z)+ U(z)-lOzU(z) but there exists a residual group of KM transformations with respect to which they are invariant. These are the KM transformations for which U(z) lies in the group A of the Gauss decomposition which is generated by the root vectors with negative grade (E~, for h < 0). Thus they are just the transformations that would be generated by the constraints themselves. The idea is to regard these residual KM transformations as gauge transformations and regard only those functions, or functionals, of J(z) which are invariant with respect to this gauge group as physical. Thus finally we have (dimG-dimB)/2 constraints and (dimG-cUmB)/2 gauge degrees of freedom, leaving just climb physical fields altogether. It is possible to choose the gauge (at least locally) so that the physical currents are just the ones JB(z) belonging to the little group B.

321

7 Field Equations. It is easy to see that the constraints (13) are consistent with the W Z N W field equations (6), indeed are special solutions of some of them, and hence the WZNW field equations can be reduced to field equations for the unconstrained components of the currentJ(z). After some simple Mgebra one finds that the reduced field equations take the following form

O~,JB(~.) = [b(z)Nb -1 (~), M],

(15)

and =

]°Cz ) =

=

(0~c(z))c(z) -1 =

(16)

b-l(¢)Nb(z),

where M and N denote the constant matrices defined in (14). Note that, in c o n t r a s t to the WZNW currents, the currents JA(z), JB(z), iv(z) and ] B ( z ) are not functions of z and w alone. The most interesting feature (15) is that the equations for JB(z) do not involve the fields Jl~(z) for h # 0 and thus are self contained. Furthermore, it is easy to verify that they can be derived from the effective Lagrangian

Leff(b(z)) = Lwz1~w(b(~))

+

/tr(b(z)Mb -l(z)N), d

where

b(z) e B.

(17)

This Lagrangian can be interpreted in two ways. First, it can be regarded as the generalization of the WZNW Lagrangian for fields belonging to the group B, but where, because B is reducible, there are interactions between the simple and abelian parts of B. Note, however, that since the constant matrices M and N have grades ±1 there is a non-zero interaction only between the components of B which differ by one grade (nearest neighbour components). Second, by noting that the Lagrangian (17) reduces to the Toda Lagrangian when B is abelian (i.e. when b(z) = exp(Hi¢i(~)) and M = r n i E = , , where the m's are constants and tr(HiHj) = C~j), one sees that it can be regarded as a generalization of the Toda Lagrangian to the case in which the nearest-neighbour interacting fields are no longer abehan fields but WZNW fields belonging to the irreducible components of B. Thus (17) may be regarded as describing either interacting WZNW fields or generalized Toda fields.

322

8 T w o - D i m e n s i o n a l Gravity. As mentioned in the Introduction, the Lagrangian (17) incorporates also a twodimensional gravitational field. This comes about as follows: Since B is defined as the little group of H it follows that the one-parameter abelian group R(1) generated by H is in the centre of B. Hence, locally at least, B may be written as the direct product R(1) ® Bo, where Bo denotes the rest of B. If we let h(z) be the WZNW field belonging to R(1) then the Lagrangian (17) can be re-written in the form

(is) But we have already seen that, unlike the rest of the components of the current which transform like primary fields, the components in the direction H transform like spin-one connections, and it is not difficult to deduce from this that the field h(x) transforms like x/g where g ~ is a two-dimensional metric. Accordingly, if one define8 a metric as gt~u = h(x)rlt, u, where rh~u is any fiat (constant) nonsingular mettle, introduces general coordinate transformations, and extends the tensor properties of the currents to be the same with respect to general coordinate transformations, one finds that the Lagrangian (18) may be written as

~(g)tr (boMb'~' N) ,

(19)

where R(x) is the Gauss curvature a n d / k the two-dimensional d'Alembertian operator. It is clear that this Lagrangian describes a theory in which a two-dimensional gravitational field h(z) and the WZNW fields bo(z) interact with themselves and with each other. The purely gravitational part of the interaction (which is obtained by setting bo(z,) = 1) is just the Liouville gravitational interaction which is induced by string theory in less than 26 dimensions [9]. This Liouville theory was embedded in an SU(1,1) Kac-Moody theory by Polyakov [5] in order to facilitate its quantization, so our procedure may be regarded as the converse of Polykov's for SU(1,1), and a generalization of the converse for the other groups.

323

9 Solutions

of the Field Equations.

The general solution of the field equations (151(16) for the fields b(x) are obtained from the general solution for the WZNW equations for the group G, namely, g(m) = g(z)~(w), where g(z) e G and ~(w) e G are any arbitrary functions of the coordinates z and w respectively. I do not have time to describe the procedure by which the solution of the reduced system is obtained from this solution, but it is not difficult and is given in [4]. Here we shall simply present the result, which is that the general solution takes the form

b(x) = b(z)D(z, w)b~,(w),

(20)

where b(z) e B and b(w) e B are again arbitrary functions of z and w respectively, and D(z,w) is the B part in the Gauss decomposition of c(z)a(w), where a(w) and c(z) are the solutions of the remaining equations in (14) and its rlght-handed counterpart, namely,

O,a(z) = a(z)(b(z)Mb-l(z))

and Owa(w) = a(w)(b-l(w)Nb(w)),

(21)

with initial values c(0) = a(0) = 1. It might be thought, of course, that this solution is not complete because it leaves the differential equations (21) still to be solved. However, because of the nilpotency of the groups A and C these equations can be solved by simple iteration. Indeed if c(z), for example, is decomposed into its H grades ch(z) then the solution is co( ) ---- 1.

ch(z) =

(22)

Note the resemblance between the general solution (20) and the general solution b(z)b(w) for non-interacting WZNW fields belonging to the little group B. Indeed (20) reduces to this solution in the non-interacting case, for which M = N = 0 and hence, from (21), D(z, w) -= 1. 10 T h e W - A l g e b r a s

of Toda

Theory.

In this section we show that the W-algebras that have emerged in the Toda theory become much more understandable and tractable in the reduced WZNW context. First we identify them by means of the equation 0zg(z) = J(z)G(z) connecting the WZNW fields g(z) with their currents J(z). These equations can be regarded as 324

first-order differential equations for g(z), given J(z), and, it turns out that, in the constrained case, they can easily be reduced to higher order differential (or pseudodifferential) equations for those components of g(z) that are gauge-invariant with respect to the residual gauge group discussed earlier. Since the coefficients of the powers of Ox in these higher-order equations are gange-invariant with respect to the residual gauge group by construction, and are polynomials in the constrained currents and their derivatives because the group A is nilpotent~ we see that they are gauge-invariant polynomials in the constrained currents and their derivatives. The crucial point now is that the higher-order equations obtained in this way are just the Gelfand-Dicke equations. Since the coefficients of the latter equations are just the base elements of the W-algebra of the Toda theory this immediately gives us an identification of the W-algebra as the algebra of local gauge-invarlant polynomials in the constrained currents. Although the identification of the W-algebra of Toda theory as the algebra of gauge-invariant polynomials of the constrained WZNW theory is very natural and intuitive it is not of great help for practical computations in arbitrary gauges. However, there exists a set of gauges in which it is very useful and practical, and in which we obtain an alternative interpretation of the W-algebras as Dirac star algebras. These are the (DS) gauges introduced [10] by Drinfeld and Sokolow. In these gauges the local gauge-invariant polynomials in the constrained currents reduce to the currents themselves,

o,"

os = J(1) (z),

(23)

where the J(1)(z), of which there are I, form a basis for the W-algebra. The gaugefixing is complete in these gauges and the system of constraints obtained by combining the original constraints and the gauge fixing form a second class system of constraints. Hence their Poisson-bracket algebra (which, from (23), is just the W-Poisson-brac~et algebra) is not their no 1real Kac-Moody algebra but the corresponding Dirac star algebra, [P(i),P(k)]

ms TDSl*

r'rDS 7DS1

= [Ji0 ,'(k) l = t'(i) , "(k) l -

DS ms [Jii) 'Ca][Ca, C#]-I[C#, J(k) 1"

(24)

We thus obtain an alternative identification of the W-algebra as the Dirac star algebra of the constrained currents in the DS gauge. This identification is very useful for practical purposes because in this gauge the gauge-fixing as well as the original constraints impose linear conditions on the currents. This means that the the constraints Ca in (24) can be replaced by components J= of the currents

325

themselves, in which case the right-hand-side of (24) can be expressed completely in terms of KM commutators. Furthermore, because of the nilpoteney of the gauge group it turns out that the inverse constraint matrix [jDS, j ~ s ] - i is easy to compute and is a polynomial in the currents. Again we shall not give the details of the computation here but refer to the literature [3] in which, as examples, the W-algebras for the groups G = A2, B2 and G2 are computed. It is well-known that that the W-algebra for G2, which involves the Poisson bracket of two sixth-order polynomials, is very difficult to compute by direct methods. Indeed, as far as we know it has not yet been computed this way. 11 R e d u c t i o n

to Affine Toda

Theory.

The reduction described up to now has been conformally invariant, but there exists a natural non-conformally-invariant generalization which leads, inter alia, to the affine Toda theories. The generalization is obtained by noting that in the equation (15) for the reduced field equations no use was made of the fact that the group B was the little group of H. Thus, in principle, one could use any subgroup B and any two cosets A and C in the Gauss decomposition (so long as they were complete in the sense that every group element g could be written as g = abc) and then impose the constraints (14). The constraints would still be special solutions of the original WZNW field equations. The only difference would that there would be no reason for the constraints to be conformally invariaut, or expressible linearly and/or locally in terms of the original currents J(z), and, in general, they would not be so. However, this would not in itself make them uninteresting and to illustrate the kind of theory that one would obtain we show now indicate how the affme Toda theories can be obtained by such a reduction. The reduction consists of simply replacing the conditions M = Z

o,E o,

and

¢t~

(25)

= Ol~

where the ~i denote the l simple roots, by a similiar sum in which i denotes not only the simple roots but also the most negative root so, say. Thus i = 0,1,2, ..i instead of just 1,2,...l. It is not difficult to see that in this case the effective Lagrangian (18) reduces to the affine Toda one. In particular, for SL(2,R), the Lagrangian (17) reduces to the sinh-Gordon Lagrangian.

326

References. .

2. 3. 4,

5. 6.

. .

.

10.

P. Goddard and D. Olive, Int. J. Mod.Phys. A1 (1986) 303. A.N. Leznov and M.V. Savaliev, Comm. Math. Phys. 74 (1980) 111. J. Balog, L. Feher, P. Forgacz, L. O'l~ifeartaigh and A.Wipf, Physics Lett. B227 (1990) 214; B244 (1990) 435 ; Annals of Physics 202 (1990). L. O'Raifeartaigh and A. Wipf, Preprint DIAS-STP-90-19, ETtt-Ttt/90-20. A. Polyakov, Mod. Phys. Lett..4.2 (1987) 893. A. Bilal and J-L. Gervais, Phys. Lett. 206B (1988) 412; Nucl, Phys. B314 (1989) 646; B318 (1989) 579; O. Babelon, Phys. Lett. 215B (1988) 523. A.B. Zamolodchikov, Theor. Math. Phys. 65 (1986) 1205; V.A. Fateev and A.B. Zamolodchikov, Nucl. Phys. B280 (1987) 644. V.A. Fateev and S.L. Lukyanov, Int. J. Mod. Phys. A3 (1988) 507; S.L. Lukyanov, Funct. Anal. Appl. 22(1989) 255, K. Yamagishi, Phys. Lett. 205B (1988) 466; P. Mathieu, Phys. Lett. 208B (1988) 101; I Bakas, Phys. Lett. 213B (1988) 313. M. Green, J. Schwarz and E. Witten, Superstring Theory, Cambridge University Press, 1987. V. Drinfeld and V. Sokolov, J. Soy. Math. 30 (1984) 1975.

327

THE RESURRECTION

OF A FORGOTTEN

DE BROGLIE'S

SYMMETRY:

SYMMETRY

Henri Bacry Marseilles, France

The aim of the present talk is not historical, even if I am calling for historical facts in order to perform an assessment of the present situation in particle theory. What I want to show you is that a symmetry which played an important role in the birth of quantum theory has been neglected, forgotten and buried, without a ceremony or tears, at the age of three. Paradoxically, to-day, the majority of the physicists think that this symmetry is still alive and textbooks are mentioning it without any announcement of death. In listening to that, you must have the feeling that I am referring to some accident in the history of physics and, consequently, that the situation I want to talk about is not very dramatic; after all, it is the kind of things historians or philosophers are interested in, and we, physicists, we can ignore it. This is not true. This lack of symmetry had very serious consequences for the present status of quantum theory and we are faced with inconsistencies which have to be cured. To-day, I intend to tell you what these inconsistencies are and I will show you how to restore the initial symmetry and put back the coherence which has been lost.

1. Symmetry with or without group theory. The Einstein-Faraday symmetry. There are essentially two ways in introducing symmetries in physics, even if the frontier between the two is not easy to draw. The frost one consists in examining the equations of some theory and in looking for its invariance group of transformations. This is, for instance, what was made by Henri Poincar~, when he examined MaxwelI's equations and the transformations introduced by Hendrik Antoon Lorentz. He arrived at the so-called inhomogeneous Lorentz group, which is now known as the Poincar~ group 1. Later, it was shown independently by the British physicists Harry Bateman 2 and Ebenezer Cunningham3 that Maxwelrs equations in the vacuum were invariant under a fifteen dimensional Lie

* Contribution to the XVIIIth International Colloquium on Group Theoretical Methods in Physics, Moscow,Juno 1990. 1This appellation was proposedby EugeneWigner around 1961. 2H. Bateman, Proc. London Math. Soc., 8, 223 (1910). 3E. Cunningham, Proc. London Math. Soc., 8, 77 (1910)

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group, the so-called conformal group 1. Eugene Wigner was the one who completed the Poinear6 group in adding parity and time reversal2. It is certainly true that, in his symmetry investigation of Maxwelrs equations, Poincar6 was very close to the discovery of special relativity. What is not very well known is that Albert Einstein's derivation of this theory was induced by another kind of symmetry approach3. Einstein was worrying about the fact that there were two different interpretations of the electromagnetic induction effect, depending on the way the objects were moving. If a circular wire is moving near a fixed magnet, the Laplace force obliges the electric charges of the wire to move and create a current. But, if the circular wire is now fixed and the magnet is moving, we have an induced electric field and the Laplace force is useless. This dissymetry between the two explanations was not accepted by Einstein once he realized that the phenomenon only depends on a symmetric quantity, namely the relative velocity. It was a single effect and he thought that a given phenomenon cannot suffer two completely opposite explanations. Any symmetry in a given effect must correspond to some symmetric element in the theory. According to Holton, this is the analysis which led Einstein to special remativity.

2. The de Broglie symmetry (1923). Another "added symmetry" was proposed by Louis de Broglie in 1923, one year after the explanation of the Compton effect. It is interesting to underline that it was the time where physicists began to accept the light quanta of Einstein and, therefore, the so-called wave-corpuscle dualism for light. The symmetry idea of de Broglie was to propose the extension of the wave-corpuscle dualism to matter particles and, especially, to electrons. It must be emphasized that, at that time, nobody was able to understand really what was the deep meaning of this dualism. Since 1905, almost all physicists were aware that there was a difficult problem to solve: the necessity of reconciling the (discontinuous) theories of the Compton and the photoelectric effects with the (continous) Maxwell theory. It is this de Broglie symmetry - the "democracy" of all particles - I am interested in. I want to show how it has been given up three years later, without saying it exphcifly. Surprisingly, this death was unnoticed.

3. De Broglle's symmetry buried (1926). The de Broglie symmetry consisted in extending the 1905 Einstein waveparticle dualism of light to all kinds of particles, However, the relationship between the wave aspect and the particle aspect was not very clear, The experiment of Clinton Ioseph Davisson and Lester Germer in 1927 was considered with just reason as a success of de Broglie's idea, although it was not suggested by it. The first particle interpretation of the SchrOdinger wave function is due to Max Born (1926). This statistical interpretation broke the de Broglie symmetry, since Maxwell's field could not be given such a meaning. The reason why the Born suggestion is incompatible with the de Broglie symmetry is that it cannot be applied to the photon. This can be shown in many ways. Let me give you two non sophisticated manners of proving this fact. The first one is based on dimensional analysis. If one of the electromagnetic vectors (A, E, or B) IFor historicalfacts,see Jo~ M. Sanchez-Ron, The roleplayed by symmetries in the introductionof relativi~ in Great Britain, in M. G. Doncel,A. Hermann,L. Michel and A. Pais. Editors,Symmetries in physics, publishedby Serveide Publicacions,UniversitatAutbnomade Barcelona,Spain (1987). 2E. P. Wigner, Annals of Mathematics, 40, 149 (1939), reprintedin Y. S. Kim and W. W. Zachary, Proceedings of the International Symposium on Spacetime Symmetries, North Holland, Amsterdam

~G,.~{-Iolton,Thematic origins of scientific thought, HarvardUniversityPress (1973).

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described the probability amplitude density to find a photon at some place in space, we must have a dimensionless integral of the form f I~.AI2 d3x, where ~, is some factor which can be expressed with the aid of the fundamental constants c, e, ft. Unfortunately, dimensional analysis shows that this problem has no solution. Therefore the Born interpretation o f the wave function (the counterpart o f the electromagnetic wave) cannot apply to the electromagnetic wave. The other elementary way of showing the difficulty concerning the photon consists in proving that the vector operator i~--pcannot be diagonalized. Indeed, define F=B

-iE

and suppose that we have the following eigenvalue equation

I~-~ ' ~ F j(p) = xiFj(p)(i,j = I, 2, 3) The Maxwell equation p.F(p) = 0 (transversality condition) gives

i•pi

PJ ~F. ~Pi

[P'F(P)] = i~p'~l ~ [ Z p J Fj(p)] = iFi(p) + i Z J

= iFi(p)+ ixi[p.F(p)] = 0 and, therefore, F(p) = 0.There is no localized state for the photon. You could protest and say that iff-ffpis perhaps not a good position operator for the photon. After all, the photon is a relativistic particle and the Schr0dinger position operator is a non relativistic one. This is a good point, but I have two answers: first, my argument based on dimensional analysis is still valid; second, everybody knows that, in 1949, T.D. Newton and Eugene Wigner t (followed by Arthur Wightman 2, in 1962) have shown, from relativistic considerations, that there were no localized states for the photon. The reason is not due to the massless character of that particle but to the fact that it is a spinning and massless particle. In other words, it is due to i) the transverse character of the electromagnetic field (spin), ii) the impossibility of the value p = 0 for a massless particle 3. The N.W. (Newton and Wigner) result is well known but it is usually considered as a curiosity or a technical point and many physicists do not mind because they say that the photon is not a matter particle. But this does not solve the problem. To-day, particle physicists believe in a theory where the photon is a brother o f the intermediate boson which itself has localized states in the N.W. sense. Nobody is

IT.D.Newton and E.P.Wigner,Rcv.Mod. Phys.21,400 (1949). 2A.S. Wighlman, Rev. Mod. Phys. 34, 845 (1962). 3It is perhaps important to underline that the energy and the momentum of a masslcss particle obey the relation E ffi Ipl and that to be at rest for a particle means really p = 0, in contradistinction with what is said in almost all books on quantum mechanics, where the meaning of this expression is related with the speed. Everybody knows how the speed is defined in quantum mechanics: it is the group velocity of the wave associated with the given particle. According to the superposition principle, a photon is not necessarily in an energy-momentum state (another generally accepted idea!). Since the group velocity can have any value (less than or equal to c), a photon has states of any velocity, even the zero vclocityt

333

worrying about the way a broken symmetry is accompanied by a broken de Broglie symmetry.

4. Restoring the de Broglie symmetry. I guess that many people of the audience will object that, to-day, the waveparticle dualism is no longer the one I am describing but refers to the modem quantum field theory, where fields obey differential equations. Before examining this objection, I will tell you how we can restore the de Broglie symmetry in the context of quantum mechanics. For that purpose, I have to recall you that the Born statistical interpretation of the wave function is no longer considered as an axiom of quantum mechanics. It became a consequence of one of the axioms of the theory of quantum measurement. More precisely, it concerns the measurement o f the position operator X. Using the Dirac formalism, Iw(x)12 is just the modulus squared matrix element I<xlXlt>#, where Ix> describes a localized state for which three commuting observables have been measured. This means that we could say that the Born idea, in its generalized formulation, is simply a probalistic statement about the measurement of a complete set of commuting observables for a system in a given state IV>. The interesting fact is that, if we do not require the commutativity of the coordinate operators X, Y, Z as one of the Wightman axioms, the photon can be given a position operator. From consistency arguments, it also follows that the spinning particles with mass must also have non commuting coordinate operators. For the moment, I only want to mention that the de Broglie symmetry can be restored at such a price and that for that price one gets some extra advantages.

5. The Bohr complementarity principle and the Einstein-Faraday symmetry. I described at the beginning of this talk how Einstein worded about the double 1 explanation of the induction effect and how profitable was his investigation. To-day, there is still an experiment which is suffering two explanations. It is the electron slit experiment when a light source is used to detect through which slit the electrons go. If the source is intense, the interference pattern disappears completely. If we make the intensity of the source decreasing progressively, interference fringes reappear. The two distinct explanations I am referring to are the following ones: a) If the source is such that the flux of photons per second is decreasing, there are more and more electrons which are not scattered by photons; they are not detected and, therefore, contribute to the interference pattern. This is a particle argument. b) If the frequency of the source is decreasing, the image of the electron is not a point, but a spot of increasing radius which does not always permit to say through which slit the electron went. This is a wave argument. Since we are free to choose a light source as we want, the reapparence of the interference pattern needs both wave and particle arguments and Bohr's complementary principle according to which an experiment is either of wave type or of particle type is not allowable. To have recourse to Bohr's principle is equivalent, from Einstein's point of view, to explain the induction effect with the help of some complementary principle where wave and particle are replaced respectively by induced electricfield and Laplace force (or vice versa). iThe word "double" is not the right one. The induction phenomenonoccurs whateverare the speeds of

the wire and of the magnet. At the time of Einstein's investigations, we needed a continuous set of

explanations.

334

Fortunately, when one chooses the non commutative position operator I proposed for the photon t, we get a single explanation for the slit electron experiment we have just discussed. This is for the following reason. When a photon is almost in an eigenstate of the momentum operator, say in direction z, we have the aproximative commutation relation 10.

[x,y] - i ~-~ which gives the uncertaintyrelation Ax Ay ~ ~,2 the particlecounterpartof the wave spot argument. This relationprovides us with a nice and satisfactoryexplanation of the fact thata photon is more and more localizablewhen itsfrequency is largerand larger. It is now time to give a rdsumd of the advantages of the new position operator. They are five in number. Up to now, we have examined the firstthree ones. I. De Broglie's symmetry is restored.All particles(bosons and fcrmions) have a position operator built with a unique procedure. The only change is that one cannot measure simultaneously two components of the position operator for spinning particles. 2. Obviously, there is no localized state for a photon in the N.W. sense but now, without calling upon the complementary principle of Bohr, we have the following quantum situation: a photon is more and more localizable when its frequency is larger and larger. 3. The Bohr principle was not able to describe experiments which were intermediate between pure wave type and pure particle type. The typical experiment is the two slit experiment for an electron beam, when a light source of variable intensity is used to localize some electrons. We now have a unique quantum explanation of this experiment. 4. When the spin of an electron is neglected, the new position operator reduces to the ordinary Schr6dinger one. In the non.relativistic approximation (small momenta2), the potential in the S c h r ~ n g e r equation is modified in such a way that the spin-orbit coupling rises automatically, with the correct factor. 5. The interference of light waves in the slit experiment can be given a particle interpretation. After all, each photon is known to face the two slits. Everybody will agree that such a situation corresponds to a quantum question concerning a position. Given a photon in the momentum state p = (0,0,pz) at time zero, the presence of the two slits is equivalent to the question: "is x equal to J: a?". The answer is "yes". After the measurement, the photon is in a reduced state. This new state Iv(t)> permits to OO

know the probability density l<xlNt(t)>l2 at any time t and the integral r/I<xl~(t)>12 dt ~tJ

must give the function of x expected by the interference pattern.

1H. Batty, Localizability and space in quantum physics, Lecture Notes in Physics, vol. 308, SpringerVerlag (1988). See also H. Batty, The position operator revisited, Ann. Inst. Henri Poincar6, 49,245 (1988) andThe notions of localizability and space,from Eugene Wigner to Alain Connes, in Y.S. KJm and W. W. Zachary, op. cit. 2an approximationwhleh is distinct from the Galilean approximation(c going to infinity).

335

Before examining the problem of quantum field theory, I have the following comments to add.

6. Comments on the new position operator. 1. I have shown in my book how the construction of the new position operator is related with the SchrCkiinger zitterbewegung for the electron. The motivation of Schrtdinger is to-day forgotten. What he wanted to do is to propose a new position operator f o r the Dirac electron. As far as the Dirac equation governs the Hilbert space of the one electron states, the S c h r ~ g e r procedure is similar to mine. When I read the Schr~linger paper, I had the feeling that there was a kind of zitterbewegung for the photon behind my position operator. Recently, in his attempt towards a point description of a massless particle with helicity, Plyushchay 1 found a proof of it. 2. It can be shown that the difficulty about the lack of localized states for spinning massless particles is not only quantal but has its counterpart in classical theories. I refer you to my book and to two more recent articles 2 which concern the problem of the position of a classical massless particle. In one of these articles, Grigore wrote the rigorous classical counterpart of the Wightman axioms for localized states and is led to coordinates which are not (Poisson) commuting. 3. I remind you that all the difficulties related with the problem o f localized states wcrc due to the existence of the spin (for spinless particles, the N.W. conclusions are satisfactory). If we note that the unit of spin is the Planck constant, we can bring together this fact and the discreteness of the electric charge and conclude that in quantum physics, we do not have to "measure" angular momenta nor charges; we only to ' count them. In that sense, the experimental values of the two fundamental constants /z and e belong to classical physics. This is opposite to the ordinary statement made in all textbooks where /z is considered to characterize quantum physics. We can deduce from that that position measurement of particles have a mysterious discrete character and, if position measurement has something to do with space, we probably have to fred the real microstructure of our space before constructing a new quantum theory of fields. 4. As I explained in my book, I knew for more than twenty years that the difficulty about the lack of localized states of the photon was due to the Poincar6 group itself 3. To-day I can say more: the microstructure of the Minkowski space is unacceptable 4, even in the context of classical physics. What I mean is not that experiments oblige us to give up the Minkowski space, but that special relativity cannot be built with the aid of a continuous spacetime.

7. Why Minkowski's space is unacceptable. It woud be too long to give a detailed discussion about the unacceptability of the Minkowski space in the classical theory of special relativity. Let me say in a few words about the main argument. The fact that Einstein used light rays (and clocks) to explain how to measure distances is in itself contradictory. Indeed, measuring distances between points implies that these points have a small expanse. It is easy that this expanse is of a few wavelengths of the light rays if we want the light rays to propagate IM.$. Plyushchay, Massless point particle with rigidity, Mod. Phys. Lctt. A4, 837 (I989). 2D.R. Grigore, Localizability and covariance in analytical mechanics, Jour. Math. Phys. 30, 2646 (1989). C. Dural, I. Elhadad and G.M. Tuynman, Puk ~nsky's condition and symplectic induction, ~Sreprint(1990). ee J.-M. Soutiau, Structure des systdmes dynamiques, Dunod (1970). 4H. Bacry, A contradiction in Special Relativity, preprint (1989).

336

at group velocity c. Since the wavelength is not Lorentz invariant (Doppler effect), we cannot say that the points we are considering are small There exist observers for which they are as large as we want. This means that Minkowski's space is good provided we restrict ourselves to small boosts and expanded points. It also has an important corollary: the xtz coordinates in Maxwell classical theory cannot have a sharp space-time interpretan'on.

8. Quantum field theory and the de Broglie symmetry. If we examine the evolution of theoretical physics between 1905 and 1934, we can say the foUowing things: 1. There were two known fields, the Maxwell field and the gravitational field, both relativistic. 2. Among these two fields, only one has waves. 3. Since 1905, electromagnetic waves haye also panicle characters. 4. Since 1923, it is stated that electrons and protons (the only matter particles known) have also wave properties. This is the birth of de Broglie's symmetry. 5. Schr6dingcr writes the wave equation for these particles. It is a nonrelativistic equation. The Born interpretation of the wave function destroys the de Broglie symmetry. 6. In 1928, the Dirac equation replaces the Schrtdinger one. 7. In introducing the meson in 1934, Yukawa introduces a new kind of democracy: the field-particle duality. The de Broglie symmetry seems to have be restored by Yukawa. This is an illusion because the introduction of a field does not erase the dissymetry related with the measurement of the position for one particle states. Moreover, the notion of field has an ingredient which is unsatisfactory, namely the Minkowski space. Fortunately, Fock told us how to build fields without it. For each kind of field, we start with the one particle Hilbert space of states, a carrier space of an irreducible representation of the Poinear6 group 1 for which the energy-momentum and angular momentum are natural observables. The direct sum of the symmetrized (or antisymmetrized) tensor products of this Hilbert space provides us with the Hilbert Fock space for this kind of field. Once the tensor product of all Fock spaces is taken, we have constructed in principle the space of all systems. The only thing which is left is a procedure to determine the Hamiltonian of the system we choose to study. Unfortunately, no procedure is known, except the Lagrangian one, but a Lagrangian needs some parameters and these parameters are the x u, the so-called coordinates of the Minkowski space, which do not have a space-time interpretation. For me, the solution of this difficulty must be found in the non commutative position operator. This means probably that we need a non commutative space.

9. Are we living in a non commutative space? First, let me try to answer a question asked to Professor Manin to-day. Somebody in the audience wanted to understand what was a non commutative space. His handwaving expressed his confusion. The problem of space is an old one. Greek thinkers were aware of the relationship between space and numbers. Everybody knows the difficulties they encountered with irrational numbers. The existence of a bijection between real numbers and the points of a straightline seemed more and more natural to philosophers and scientists. With Descartes, the isomorphism between our space and N 3 b e c a m e 1Wigner's work on the representations of the Poincar~ group is an illustration of his belief in the de Broglie symmetry.

337

obvious, even if the concept of real numbers was not completely understood at that time. However, on the one hand, nobody is able to show you a point in space, on the other hand, I can speak of a lot of real numbers. The points are out of reach but we are idealizing them with the aid of real numbers, a mind construction. We are so accustomed to this surprising association, that we are not bringing that into question. However, as physicists, we cannot be satisfied by that: there are absolute scales (the size of an electron, for instance) in space, there are not in the set of real numbers, a set which looks the same at any scale! The most trivial fractalt However, in order to investigate a space, the mathematicians made a new step, gave up the set of real numbers and used one of the many (commutative) algebras of functions on the space. They discovered continuous spaces, differential manifolds, etc. They replaced an abstract construction by another abstract construction. Nowadays, they are not more abstract than before. We are still not able to understand with our body the intimate structure of space but they are able to generalize geometrical computations in replacing the commutative algebra of functions by any non commutative algebra. We are not yet accustomed to this idea,.., not yet... I said that the real line does not possess an intrinsic scale, neither does a commutative space, but a non commutative one? We could expect that our space will explain why the angular momentum is discrete. The non commutative geometry was founded by Alain Connes. As claimed by his inventor, this new geometry was inspired by quantization itself. If the operators x, y, z do not commute, it is natural to imagine that we are living in a non commutative space. But Connes has other arguments in favour of such an idea. He made some attempts in the use of noncommutative spaces in quantum field theory t and very encouraging facts are an indication that we have to explore this kind of solution. The main argument is that Connes have shown that replacing the ordinary spacetime by a non commutative space is equivalent to introducing a cutoff: the ultraviolet divergences vanish... If you want more details about the arguments I have mentioned in my talk, you can refer to my book and the general introduction in Connes' book. My hope is that I already convinced you that we are riving in a non commutative space. If it is the case, I invite you to read the whole book ofAlain Connes.

1A. Connes, G~omdtrie non commutative, InterEditions paris, 1990). A. Connes, Essay on physics and non-commutative 8eometry, preprint ((1989). A. Connes and J. Lott, Particle models and noncommutative geometry, preprint (1989).

338

Relativistic invariance of a many body system with a Dirac oscillator interaction M . M o s h i n s k y * , G. Loyola, C. VillegasT Ins~i~u~o de F(sica, UNAM Apdo. Postal 20-36.t, M~zico, D.F., 01000 M~:ico

A b s t r a c t . In a recent publication Nfoshinsky and Szczepaniak considered a one Particle Diraz equation linear in bo*h momenta and coordinates. As in the nonrelativistic limit it gave a Haxniltonian containing an ordinary oscillator, the equation was referred to as that of a Dirac oscillator. In this paper we extend the concept of Dirac oscillator interactions to a system of r~ particles showing that it can be derived in a relativistically invariant way. Thus the eigenstates for the mass operators of Dirac oscillators with 2 or 3 particles, that were discussed in previous publications, are basis for irreducible representations of the Poincar6 group, and they can be used to derive a relativistic mass formula for baryons.

In a recent publication[1] , Moshinsky and Szczepaniak discussed a single particle Dirac equation that was linea~r:ln bo*h momenta and coordinates. As the large component in this equation turned out to be an eigenfunction of an ordinary oscillator plus a strong spin orbit force, they gave to the problem the name of Dirac oscillator. The authors mentioned [1] extended their analysis to many body systems with Dir~c oscillator type of interactions, but their discussion of the relativistic invariance of the problem was not considered complete. Nevertheless they obtained [2] in an exact and aaMytic form the eigenstates and eigenvMues for the two particle Dirac oscillator. For three particles the problem could still be solved exactly, but required the determination of roots of secular equations associated with finite matrices, for which numericM computation were needed. This problem was discussed [3] and further extended to determine a relativistic mass formula for baryons [4]. In this note our wish though is to prove that the Dirac oscillator problem for a many body system can be obtained from a relativistically invariant equation, i.e. one that commutes with the generators of the Poincar~ group. Thus the eigenfunctions mentioned in the previous paragraphes, that were obtained in the frame of reference where the center of mass was at rest, are basis for irreducible representations of the Poincar~ group. * Member of E1 Coleglo NacionM Present address: IMAS, UNAM

339

W e shall start by briefly reviewing the mass operator for an n - b o d y system with Dira~ oscillator interaction and later proceed to rewrite it in a relativistically invariant form.

For a single free particle the Dirac Hamiltonian takes the form [5] (1)

Ho = a . p + m f l ,

(0

where

p=

-iV, a =

a

'

o)

0

'

with a being the vector of Panli spin matrices and in units where h = c -- 1. For n-free particle of the same mass m, the Hamiltonian becomes n

(3)

z¢0 = E ( ~ . 'p. + m#.), g=l

where the matrices are direct products such as fl, = I ® . . . I

® fl ® I . . . ®

I,

(4)

with fl in the position s, and similarly for the as. Introducing the total momentum

(5)

P=P1 +Pz+"'+Pn, the Hamiltonian can be rewritten as

/-/o = n - l ( a I + oe2 + - . . + an)" P +/-/oI,

(6)

where R

/-/to=

E(~..p'.+m/~,) ;p',=p.-n-Ip,

(Ta, b)

s=l

In the frame of reference where the center of mass of this system of n-particles is at rest i.e. when P = 0, the Hamiltonima becomes the H i of (7a) and can be interpreted as the mass of the n-free particle system with relative motions between them. The Dirac oscillator n-body mass operator, which we shall designate by JP[ was obtained [1,3] when in (7a) we made the replacement

p'.

(8)

, p'. - i~'.B,

where

x', = x , - X , X = n - l ( x l

+ x2 + . - .

B=#®#®#

..... ® #

+ x,),

(ga, b) (9c)

Thus we have n

(10)

$=1

340

whose spectra and eigenfunctions for n = 2 and 3 were determined in reference [2] and [3]. Clearly the previous analysis does not seem invariant under transformations of the Poincar~ group, but we shall show below the .hd can also be derived from an equation that is explicitly relativistically invariant. We shall start this part of our discussion by returning to the one body problem and defining the 7 ~ matrices, where # = 0,1, 2, 3, as

= 1,2,3,

7° = fl,7 i = flai,i

(lla, b)

where our metric will be g~v = 0, # ¢ u; gn = g22 = g33 = -g00 = 1,

(12)

and the 7 ~ matrices satisfy the anticommutation relation 7~7 v + 7v7 ~ = -2g ~v.

(13)

As usuM [6] the spin part of the generators of the Lorentz group is given by s "v =

v

-

(14)

so that from (11) and (12) we obtain is.v,

= i(g.

Tv _ g w

(15)

On the other hand the orbital part of the generators of the Lorentz group is given by [5]

L ~' = x " p v - xVp ~,

(16)

so that from [z~,p v] = ig ~'v we obtain

[L"~, xr]

--- i ( g " r x ~" - g~r x~'),

(17)

and similarly for the momentum pr. The fu~ generator of the Lorentz group for the one body problem is then [5] J ~ = L ~ + S "~,

(18)

and from (15) and (17) we see that 7 r , x r , p r, transform in the same way under the j~v so we can interpret all three of them as four vectors. For an n - b o d y problem 7 ~ , s = 1, 2 , . . . ,

n are direct products as for example

7 °` = I ® I . . . I ® f l ® I . . . ® I ,

34~

(19)

were we made use of ( l l a ) and the fl is in the s ~h position. The 7~, x~, p~ remain though four vectors as their commutation relation with n

s "~ = ~ s:~,

(20)

8=1

continue to be of the type (15) or (17), where j~v are defined by (14), (16), (18), adding an index s to all the variables involved. The total momentum four vector P~, defined as in (5), will play an important role in the following discussion, as well as d0es a unit time like four vector which we shall designate by % . As a final point in notation we define the scalars

rs = ( z t u . ) - l r ,

r = II "~.

(21~, b)

r=l

where repeated indices are summed over # = 0,1, 2, 3. Note that (7:u~) -1 in (21a) just eliminates the corresponding term in the F of (21b) so Fs is still in product form. We now turn our attention to papers of Bazut and Komy [7] and Barut and Strobel [8]. In them they derive, from an appropriate variational procedure applied to a field theoretical action, a single covariant equation for an n - b o d y system. For non-interacting particles, and in the notation introduced above, this equation takes the

form [7,8] rs (-r:p., + ,~) ¢ -- o,

(22)

where so far the u~_ appearing in the I'~ of (21) is arbitrary, except for the requirement that it should be time like, of unit length, and transform as a four vector. We first show that (22) is the covavia~t form of the equation obtained when we apply the operator (3) to ¢, i.e. for the system of n non-interacting particles. For this purpose we choose the frame of reference in which (u~) = (1000) where (22) takes the form o # r,(~,p~, + , ~ ) ¢ = 0,

r° Z:po, + s=l

(23)

a=l

with n

r ° = II ~ = B, r~ - ( 7 D - l r °.

(24a, b)

r=l

Multiplying (23) by F ° mud making use of (2b,c) and (11) we obtain - ~ ' ° + ~ ( ~ 8 - p8 + , ~ s ) s-----1

342

¢ = e,

(25)

where we put the time like component of the four momentum in its covariant form using the metric (12). Comparing (25) with (3) we see that we achieve the objective indicated at the begimaihg of the paragraph if we interpret po as Ho, as is usually done. Equation (22) can also be written in the form

n -~

r.(~P,)

+ ~ r.(~p,.

8=1

+ m) ¢ = 0.

(26)

8~I

! where P~ and P~s are the total and relative four momenta defined respectively by (5) and (Tb) when we put an index I~ = O, 1,2,3 on all the variables.

We now give to the u~ appearing in the I's of (26) a dynamical meaning by writing

u . ='(P~/P), P = ( _ • p . ) 1 / 2 ,

(27a, b)

which implies that the unit time like four vector u~ takes its form (100O) in the frame of reference in which the center o~ mass is at rest. By an analysis entirely similar to the one leading from (22) to (25) we see that in this frame (26) becomes

[- v o + ~": ( ~ ,

]

• p' + m#,) ¢ = 0,

(2s)

,~1

which is exactly what we obtain if we apply (7a) to ¢ and identify po with H~. Thus we see that, with the choice (27) for uu, we get from the covariant equation (26) the total energy for a system of n non-interacting particles in the frame of reference in which their center of mass is at rest. We can immediately generalize this analysis to the case when there is a Dirac oscillator interaction between the particles if we make the replacement

' ~ P~s

' - i~,r Pgs

(29)

I t in (26), where P~s, X~s are given respectively by (7a), (9a) when we put an index g -- 0,1,2,3 on all the variables and r was defined in (21b). The covariant equation for the n - b o d y Dirac oscillator becomes then

.-~

r,(vfv,)+~r,[~;(pg,-i~,r)+m

¢=0,

(30)

s=l

and it reduces to the equation that is obtained when we apply (10) to ¢ when we pass to the frame of reference in which the center of mass is at rest, where now we identify po with M . Note that ug appearing in the I', r~ of (30) is now given by (27) and is not the unit time like four vector required by Barut [7,8] in his discussion of the single covariant a body equation (22) in an arbitra~ frame of reference.

343

Equation (30) commutes with P~, j~v and thus is an invariant of the Poincar~ group whose Casimir operators [6] are 1

~ ~

p2 = _ p ~ p ~ , W 2 = W u W ~, W~ = ~e~v~rP J

,

(31a, b, c)

which in the center of mass frame of reference reduce to [6] p2 = (po)2, W 2 = (p0)212,

(32a, b)

where po is now the total mass Ad and j2 the total angular momentum of the system of n particles. In the solution of the equation J ~ ¢ = #¢,

(33)

where 2¢/is given by (10), for systems of 2 and 3 particles, [2,3] we not only considered the eigenstates corresponding to eigenvalues # of the operator 2,4, but required also that they should be eigenstates of the total angular momentum J 2 with eigenvalues j ( j + 1). Thus our states [2,3] for two and three particles are basis for irreducible representations , of the Pomcare .• . group characterized . .• by it 2 and it 2 .3 (3• + 1). Fur. thermore, these states, and thmr corresponding elgenvalues, were used m the deternnnatmn of a relativistic mass formula [4], whose predictions were compared with experiment. We wish to acknowledge fruitful discussions on this subject with C. Quesne. References

1. 2.

8.

4.

5.

Moshinsky, M. and Szczepaniak, A. J. Phys. A: Math. Gen. 22, L817 (1989). Moshinsky, M., Loyola, G., Szczepaniak, A. The two body Dirac oscillator (Anniversary Volume in Honor of J.J. Giambiagi, World Scientific Press, Singapore, 1990). Moshinsky, M., Loyola, G., Szczepaniak, A., Villegas, C., Aquino, N.,Thc Dirac oscillator and its contribution to the baryon mass formula (Proceedings of the Rio de Janeiro International Workshop on Relativistic Aspects of Nuclear Physics, 1989, World Scientific Press, Singapore, 1990) pp. 271-307. M. Moshinsky, G. Loyola y C. Villegas, Relativistic Mass Formula for Baryons, Proceedings of the 13th Oa~xtepec Symposium on Nuclear Physics, Notas de Fisica, Vol 13, No. 1 pp. 187-195, (1990). Schiff, L.I. Quantum Mechanics (New York, McGraw-Hill) Third Edition, Chapter 13, (1968).

344

6.

Kim, Y.S. and Noz, E.M., 1986 Theory and Applications of the Poincard Group (Dordrecht, Reidel Publishing Co, 1986) Chapter III.

7.

Barut, A.O. and Komy, S.,Fortsch. Phys. 33, 6, (1985).

8.

B~rut, A.O. aad Strobel, G.L., Few-Body systems, 1, 167, (1986).

345

DERIVATION

OF THE GEOMETRICAL

BERRY PHASE

Arno BOHM, Luis J. BOYA, and Brian KENDRICK Center for Particle Theory, Department of Physics The University of Texas at Austin, Austin, Texas 78712

I. I n t r o d u c t i o n The state vector of a quantum system which undergoes cyclic evolution develops not only the usual dynamical phase but also a geometrical phase [1],[2]. Cyclic evolution means that the physical state of a quantum system returns to the same physical state after some time period T. Since state vectors which differ by a phase represent the same physical state, the fmal state vector can differfrom the initialstate vector by a phase. The dynamical part of the phase depends explicitly on the Hamiltonian. The geometrical part of the phase is produced by the non-trivial geometry of the space of physical states. This geometry can be described using the mathematical theory of fiber bundles [3],[4]. We will begin by describing the geometry of Hilbert space in terms of a fiber bundle. We will then introduce the geometrical ideas of a connection and horizontal lift, and see that the scalar product defines the connection. The resulting geometric phase will be expressed in terms of the connection one-form.

II. T h e G e o m e t r y o f H i l b e r t S p a c e A state vector will be denoted by [ ¢(~)) which is an element of a N-I-1 dimensional or infinite dimensional complex vector space denoted by C N+I - {0} or 7~ - {0). This vector space is endowed with the usual scalar product or Hermite~u metric. We want to consider normalized state vectors undergoing unitazy evolution, namely all [ ¢) such that (¢(t) [ ¢ ( t ) ) = 1 for all time. Normalized state vectors are elements of the sphere S ~N+I or S °°, which is a submanifold of C N+I - {0} or ~ - {0). In quantum mechanics a physical state is not represented by a normalized state vector [ ¢(~)) but by a ray. A ray is the one-dimensional subspace to which this vector belongs. Two normalized vectors are equivalent I ¢)' "~[ ¢) if they belong to the same ray, i.e. if [ ¢)' = e i° 1¢) where e i° E U(1). This equivalence relation forms equivalence classes on S N+I or S ~ . The set of all equivalence classes S°°/U(1) forms the space

346

of physical states (rays) which we denote by P(7-/) = S ° ° / U ( 1 ) = ~-{0} s ¢~ or by c-{0} = "-£C P N ( N dimensional complex projective space) when N is finite. We can express the above ideas in terms of fiber bundles. A fiber bundle consists of a topological space E called the total space, a topological space M called the base space, a fiber space F , a group G acting on the fibers (called the structure group) and a projection map Ir which projects the fibers above M to points in M. In our case the fiber bundle consists of a total space E which is the normalized state vectors in C N+I - {0) or 7 - / - {0), the base space M is the complex projective space C P N or P(T/) whose elements are the rays (one-dimensional subspaces of 7-/- {0}), a fiber consists of all unit vectors from the same ray, the group G is U(1) and the association of the unit vector ] ¢(~)) to the operator [ ¢(t))(¢($) l is the projection map 7r. This fiber bundle is a particular type of fiber bundle called a principal fiber bundle over C P g or P(7%) with group U(1) [41.

III. T h e C o n n e c t i o n a n d H o r i z o n t a l Lift The geometry of the fiber bundle is given once a connection is chosen. Intuitively a connection provides a way to compare fibers at different points on the space M. Mathematically a connection is specified by defining a horizontal subspace H of the tangent space T E to E. Complementary to the horizontal subspace is a vertical subspace V such that T E = I - I ~ V . Consider a point u in E, the vertical subspace at u is defined to consist of those tangent vectors in T E which are tangent to the fiber passing through u, i.e. whose projections to the tangent space M are zero. While the vertical subspace is defined by the fibers, the horizontal subspace (connection) is a matter of choice. Once a connection is specified, the notion of a horizontal lift can be introduced. A horizontal lift is defined by lifting the tangent vectors of a curve in M to tangent vectors of a curve in E such that they are horizontal. The horizontal lift of a closed curve is in general open. Starting at a given point in the fiber, the horizontal lift will return to a different point on the same fiber. This difference is called holonomy, and in our case it is a phase. In this way, the horizontal lift with respect to a given connection defines the geometrical phase. The total phase can then be decomposed into a geometrical part aad a remaining part called dynamical. Before choosing a horizontal subspace (connection) we will identify the vertical subspace (or vertical direction). The action of the group U(1) on S 21v+1 or S °° generates the fibers. Each element of a fiber points in the same direction (they just differ by a phase). This direction generated by the U(1) action is called the vertical direction. The scalar product provides a natural choice for the horizontal subspace. To see this consider I ¢(t)), the tmugent vectors to the curve 1 ¢(t)> in E. These tangent vectors are in T E and can be decomposed into vertical and horizontal parts via the scalar product, t ¢ ( t ) > = ( ¢ ( 0 1 ¢ ( t ) ) l ¢ ( t ) > + l h~(t)>. (1) From the above discussion we know that [ ¢(t)) points in the vertical direction (so does [ ¢'(t)) = e i° [ ¢(t))). Thus, (¢(t) [ ¢(t)) is the vertical part o f [ ¢(t)). We note that

@47

the above decomposition is independent of the particular fiber element we choose to represent the vertical direction. The horizontal component satisfies (¢(t) i h,(t)) = O.

(2)

This equation defines the horizontal subspace as being orthogonal to the vertical subspace providing a natural connection on the fiber bundle. Vertical tangent vectors axe proportional to [ ¢(t)), and horizontal vectors are proportional to [ he(t)). We will now evaluate the holonomy produced by the horizontal llft of a closed curve in M with respect to the connection given above. We will denote the horizontal lift by I f3(t)). By definition, the tangent vectors to the curve 1¢(t)) must be horizontal. From equation (1) this means

in E in terms of a closed path I ¢(t)) in E

I ¢(t)) = J m ) JC(t)>

(4)

where I ¢(T)> = e i(I(T)-I(°)) I ¢(0)) and I ¢(T)> =1 ¢(0)). The path I ¢(t)) represents a section which is a continuous mapping of a patch U on M into the region of E above U. A section maps a closed path in M onto a closed path in E. Choosing a different patch on M corresponds to choosing a different section or dosed path in E. Different sections axe related by the structure group,

I ¢'(t)) = d °(t) I ¢(t)).

(5)

In order for 1 ¢'(t)) to be a dosed path in E, O(t) must satisfy O(T) = 0(0) + 2rrn (n an integer). Equation (5) is called a gauge transformation. Defining fl = I ( T ) - f(O), substituting equation (4) into (3) and integrating yields

#

(¢(t) 1 ¢(t))dt.

= i

The tangent vector I ¢) is given by d

.0

_>~, 0

d--/I ¢(t)> = o N I ¢) + x 82-; I ¢> where 0 is the fiber coordinate and the X ~ are the coordinates of M. Contracting this equation with i(¢ I from the left and integrating yields

= i J0T 0(¢ I ~0 I ¢)dt + i f0T X " ( ¢ I ~ -0~ 1 ¢)dt.

(6)

By considering a U(1) action, it can be shown 01 ¢) _ i 1 ¢>.

00

348

(7)

Using equation (7) and (¢ I ¢) = t, equation (6) becomes

/3 = -

//

Odt + i

//

(¢ I

1¢)2"dt.

(S)

We define the connection form 0

~i = i(¢1 b - ~ I ¢)dX".

(9)

Using equation (9), we can express the second term in equation (8) as

i

I -6-27 l ¢ ) 2 " d t =

.4.

The first integral in equation (8) yields ~o T Od~ = O(T) - 0(0) = 2~rn.

This contribution to the phase represents the gauge freedom as discussed above. The holonomy (or geometric phase) e i# is independent of the choice of gauge e i# .~. e i ~c/te--i2~rn ei # ~_ ei ~ / i

With this understanding we can choose a gauge and write /3 = ~ A.

(10)

The phase angle/3 is the standard geometric phase angle. Equation (10) expresses fl as a line integral of the connection form A over a closed path C in M. We note that for unitary evolution R e ( ¢ [ ¢) = 0 which implies that equation (9) can be written as = -Im(¢[d[¢)

where d is the exterior derivative with respect to the coordinates X ~' on M. curvature two-form of M is

= -Im(d(¢

]) A (d]

¢))

and by using Stoke's theorem we can express ,8 as [5],[6]

/3=jfs~ where S is the two-dimensional surface enclosed by the path C in M.

349

The

IV. S u m m a r y We have seen that the equivalence of state vectors which differ by a phase, along with the scalar product, define the geometry of Hilbert space (i.e. the fiber bundle and connection). The geometry is non-trivial. It induces a U(1) holonomy in a normalized state vector which undergoes cyclic evolution. This induced phase is called the geometric phase. It depends only on the path in the space of physical states, not on the Hamiltonian which generates this path. Physical effects of non-trivial geometries appear in molecular physics [7]. These effects ave described by the introduction of a vector potential (connection) into the molecular equations of motion [8]. The relative momenta coordinates t5 go into/5 _ .~. This change alters the canonical commutation relations [9], and may be of interest in a spectrum generating group approach [10].

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

M.V. Berry, Proc. Roy. Soc. London 392, 45 (1984). Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987). T. Eguchi, P.R. Gilkey and A.J. Hanson, Phys. Rep. 66, 213 (1980). S. Kobayashi and S. Nomizu, F o u n d a t i o n s of Differential G e o m e t r y , Interscience, New York, 1969, Vols. I and II. D. Page, Phys. Rev. A36, 3479 (1987). J. Samuel and R. Bhandari, Phys. Rev. Left. 60, 2339 (1988). G. Herzberg and H.C. Longuet-Higgins, Discuss. Faraday Soc. 35, 77 (1963). C. Mead and D. Truhlar, J. Chem. Phys. 70, 2284 (1978). R. Jackiw, Int. J. Mod. Phys. A3, 285 (1988). A. Bohm,"Berry's Connection, the Charge-Monopole System and the Group Theory of the Diatomic Molecule", in S y m m e t r i e s in Science III, p.85 B. Gruber and F. Iachello, eds. Plenium Press, New York, 1989.

350

GAUGE TRANSLATIONS AND THE BERRY PHASE R. R. Aldinger Department of Physics, Gettysburg College Gettysburg, Pennsylvania

17325,

U.S.A.

ABSTRACT Quantization of an

SO(4,1) gauge theory is carried out leading

to a Hamiltonian which describes the kinetic energy of an object moving on a space endowed with a connection.

In the

nonrelativistic limit, an association is made between the translational

gauge

component

and

Berry's

connection.

Take the vacuum state of space-time to be a 4+1 dim dS space with curvature of radius R and possessing a dS SO(4,1) as its global symmetry group of motions. The manifold can be parametrized by five coordinates constrained by: ~ A o B ~IAB = - R2 A,B = 0,1,2,3,5 where the dS metric: "qAB = diag (1,-1,-1,-1,-1). The algebra is generated by :

JAB = - JBA where:

[ JAB ' JCD ] = - i (TIAD JBC " TIBCJAD + TIBD JAC " rlAC JBD )"

(1)

Given a point of coordinates, ~A,the SO(4,1) generators can be decomposed as [1] : JAB = Jij + ~ A gj + ~B rci where: Jij = JAB - R-2 ( ~A ~C JCB - ~B ~ C JAC ) ~j = R-2 c A JAB

and

(2a) (2b)

which leads to the usual 4-dim realization of SO(4,1). Now associate to each point of the underlying space-time M 4 (no longer in the vacuum state) an internal dS space

Z 4 (local copy of the vacuum). The dS-structured connection: Fl't(x) = 2 FIxAB(x) JAB

(3)

specifies the transport of an SO(4,1) vector and the covariant derivative: Vttc~A = ~Ix~A + I-.Ac ~C leads to the dS curvature: With

coefficients:

351

lAB [ VIx, Vv] = i 2 ~Ixv JAB

RBvAB : DIt FvAB- 3v FBAB + FBAC FvCB" FvAC FIaCB .

(4)

The S0(4,1) Lie-algebraic decomposition allows the connection on the principal dS (frame) bundle to be pulled back to the Lorentz frame subbundle: FIX= ~ - 0 I X = 1 o~JJij- 0~ ni where

co~J---FO.ij

(5)

is the Lorentz gauge connection and

pseudo-translational piece.

0~ - R F 5 i is the

The field strength also decomposes according to:

1RIS AB 'AB=2QIXvJ J i j - s l x i r q

where:

Qlxij = Rtxvij + R -2 (0t~ 0 j - 0 ~ ) 0~)

(6a)

RIXij = ~IXr~vJ-0v ~ J + ~o~k C0v~- C0vik COlXj

(6b)

Consider spontaneous symmetry breakdown which allows the passage from a linear to a nonlinear realization of SO(4,1) on

G/H = SO(4,1)/SO(3,1) = Z 4.

For any g E G = SO(4,1), there exists the decomposition: g = gF gH {gF $ ~4 and gH E SO(3,1)}.

In the exponential parametrization scheme: gF = exp (- i ~. ~)

and the original linear gauge fields:

[FI.t AB } = {c01xiJ , 01t i }

are used to induce

the nonlinear fields on the associated bundle such that [2]: i F~t--i21-~JJi j - i OA~i = exp (i ~,~)[ ~l.t + i2Lco~JJij- i o~ini] exp (-i ~.n) The induced fields: {

~

""

B}= {~-~j, ~ }

(7)

are functions of the linear gauge fields,

coset parameters and their derivatives and may be written as: ~ j = co,J+ o i • V~j+coixij and ~i ° i V i • o and 0o ix= OIX+ Oix+ Ol~ where co

denote those

parts of the forms which depend upon the coset parameters and their V V derivatives while co and 0 are proportional to the linear gauge fields only. Under local S0(4.1). the induced fields transform according to: iL 2 ~ , IX qJij = gill( i 2L ~ Ix IJJij )g-IH1 + gill( ~IX )g-lttl - i OIx i 7ri = gill( - i 0-tl i ~i

where

)g-lltl

gill is a nonlinear function of

gi and

(8b)

SO(4,1).

In order to make the

reduction to a Lorentz gauge subtheory complete, one may

352

(8a)

eliminate 0l~ b y

identifying it with the space-time vierbein field: 01~1 =-hp. 1 [3]

(the soldering

condition). So, one may interpret the nonlinear gauge fields as the true spacetime spin and vierbein fields. Lorentz

4-vector

The usual notion of parallel transport of a

vl't= hil'tv i is generated using the covariant derivative: Dt.t = ~bt + i 1 ~'~t IJJij

(9)

and transport about a closed curve results in the Riemann curvature: -R ~ "'~ = ~ v

ij

.

~

~v~-~ ij +

. J ~ i k 7 7Wvk - ~v

ik

~ - ~ ."

(10)

In order to obtain the complete geometrical picture I follow the work of Stelle and West in [2] and define another differential operator (along with its associated notion of parallel transport called development [4]): Al-t- Ot-t+ i 2 °)l.tiJ'liJ - i 0 ~ i where transport about a closed curve on

(11)

:1'1.4 results in the curvature fields:

[AB Av] = i 1--Q~tiJJij- i Sp.j~ i '

2

"

The complete geometric structure of of

(12)

M 4 is obtained by considering the action

(11) on an arbitrary nonlinear vector field according to: Al.t~i = (~l.t + i 1 ~ g iJJij + i h-; ~i)V' = (DI.t + i h-~ ~i)V'

(13)

Where one first parallel transports ~'i across M 4 using (9)followed by parallel transport in r,4 with help from the dS boosts. The curvature terms 1,n AB _1i'j - i become: ~-~,-IXV JAB-~-O4tx; J i j + S t l v n i where the rotational sector: -O,vij = -R,~ 'j"" + R.2 ( r i ~ j

- hv,~-j)

(14a)

- "

gives the net S0(3,1) rotation and is the difference between the Lorentz curvature tensors on M 4 and y4. The torsion tensor (translational sector):

-s~vi = ~ - v i ~v -ih~ + -i=~ k ~-vk- = i k h%k indicates the amount by which an image curve on .,

to be as flat-as-possible, I choose: { ~ J , -

i

Constraint: Sp.v = 0 which gives:

•

E 4 fails to close.

(14b) For :1%4

.

h~t1} = {0, 61~} and impose the

[AI~ , Av] = i R-2 Jl.tv .

To quantize the dS gauge theory [5] , I return to the purely gauge-theoretic process of development, (11), which in the unitary gauge (where the Higgs

353

field CA(x) = 85A R such that: class of horizontal

~-gij= ottij

and

O-~ = 011i)

and for the special

Lorentz cross-section becomes: Atz E ~tt - i X J51x

(15)

where Z, = R "1 specifies the strength of the symmetry-breaking In the Schr'6dinger picture: Ptt = - i bit and Bgt = - i Alx so that: By identifying

B~t = P t t - k J 5 t t . J5tt with the dimensionless form of the quantum

analogue of the Finkelstein

[6]

mechanism. (16) mechanical

relativistic displacement operator: (1 7)

(where btt specifies how far the center-of-mass trajectory is displaced from the origin) one obtains another representation of a dS S0(4,1) generated by Jttv and Btt.

This representation turns out to play the central role in the

construction

of a model for relativistically

extended objects (which has been

quite successful in describing the rotational

aspects of single hadronic bound

states [7] ) in that the nonlocal system is characterized by the eigenvalues of Z,2 CII = B~tBIx- X 2 J ~tvJ ttv irrep X2 c~2 . 2 one obtains the relativistic Hamiltonian:

the 2nd-order Casimir of S0(4,1): Using constraint mechanics,

H = - 2 ~ [BttBIx- ~2J2 ttvJ IXV _ X2 ct2]

(1 8)

which describes the kinetic energy of an object moving on a space endowed with a connection of nonzero curvature:

[Xb'g,~.bvl=i~,2Jttv.

Now consider, in general, a Hamiltonian in the nonrelativistic case: H = p2 p2 + + V(R,r) 2M 2tt where {P,R} = slow (nuclear variables) and {p,r}

(19)

= fast (electronic variables).

The fast subHamiltonian depends parametrically on the slow coordinates: h ( R ) = p 2 + V(R,r) (20) 21~ and generates the eigenvalue equation: h(R) I n ; R > = en(R) I n ; R > w h e r e the instantaneous states can give rise to Berry's [8] connection defined by: An(R) = < n ; R

354

liVRIn;R>

(21)

The

effective Hamiltonian for the slow motion in the Born-Oppenheimer

approximation

(where one ignores the off-diagonals) reads: H e f f = I--L- (P - A n ( R ) ) 2 + en(R) 2M

(22)

Therefore the fast system induces into the slow dynamics a potential energy arising from Berry's connection which enters through the curvature.

Also, a

velocity dependent force changes the velocity commutator leading to the anomalous

commutation

relation:

[ M l~i, M l~j ] = i eij k Jk

where MR = I~ - A.

Now consider the nonrelativistic limit of the generalized momenta, (16). Since Bit.-->oo in the limit, I will consider ~'Bit- Btt~c " For the space components: center-of-mass

its dimensionless

generalization:

Bi ---~-MQi - Mdi - - MYi where

the

Yi = Qi (charge position)+ d i (displacement from the charge

position to the center-of-mass).

Nonrelativistically, one has the generator of

pure Galilean boosts: G i = M Y i where: [ G i , P j ] = i M S i j is the anomalous commutation relation that characterizes the nontrivial extension of the Galilei group.

It is well-known that in the Hamiltonian formalism anomalies manifest

themselves through central extensions of the Lie algebra of the group of gauge transformations.

Therefore

the group is only projectively realized and

the phases that arise are not removable by a mere redefinition of the states which is equivalent to the existence of a Berry connection [9]. which satisfies: [Z.b'g,Z. b v ] = i ~2Jitv is,

Therefore

L bg

in a certain sense, the relativistic

analogue of the Berry connection for a geometrical SO(4,1) gauge theory.

1. A. D'Adda: Theory of Fundamental Interactions North-Holland, Amsterdam 268 (1982). 2. K. S. Stelle, P. C. West: J. Phys. A 8 L205 (1979); Phys. Rev. D21 1466 (1980). 3. R. J. Mckellar: J. Math. Phys. 25 161 (1984). 4. S. Kobayashi, K. Nomizu: Foundations of Differential Geometry_ 'col, I Interseience, New York (1963). 5. R. R. Aldinger: J. Phys. A 23 1885 (1990). 6. R. J. Finkelstein: Phys. Rev. 75 1079 (1949). 7. R. R. Aldinger: Int. J. Theor. Phys. 25 527 (1986); Phys. Rev. D32 1503 (1985); R. R. Aldinger et al: Phys. Rev. D29 2828 (1984); ibid: D28 3020 (1983). 8. M. V. Berry: Proc. Roy. Soc. London A392 45 (1984). 9. p. Nelson,L. Alvarez-Gaum6: Commun. Math. Phys. 99 103 (1985); L . D . Faddeev, S. H. Shatashvili: ibid. 167B 225 (i986).

355

REGULAR

GEOMETRIC QUANTIZATION" REPRESENTATIONS AND MODULAR

ALGEBRAS

GERARD G. EMCH Department of Mathematics, University of Florida, Gainesvilte, FL 32611, USA Abstract. A link is established between the geometric quantization programme and the decomposition theory of the regular representation of the Weyl group for homogeneous manifolds of constant curvature K __.0. 1. M o t i v a t i o n a n d results. The purpose of this review is to look at geometric quantization in a novel light, namely the general contexts of the decomposition theory of regular representations and of the Tomita-Takesaki theory of modular algebras [30]. In this Section, we give our motivation and state the main results~ the proof of which is sketched in Section 2. With this apparatus in hand, we present in Section 3 an application of these results to the passage from prequantization to quantization, as well as some general comments on the scheme outlined in Section 1 . Recall first that, traditionally~ three mathematical categories are involved in the discussion of quantization: (a) unitary representations of Lie groups; (b) symplectic geometry of classical mechanics; (c) non-commutative operator algebras of quantum theories. While it should be clear that the philosophical primacy of one of these categories over the others is largely a matter of personal taste, each of the possible choices of one of them, as the starting point of a quantum theory of dynamical systems, has its own history and disciples. The elusive passage from (b) to (c) is known as the Dirac problem; in its most restritive formulations this problem was proven to have no solution (see e.g. [1] and references quoted there; a version of this proof can also be found in [8]). The formulation of (c) in terms of (a), but independently of (b), is examplified by the system of imprimitivity approach of Mackey [21]. The passage from (c} to (b) is known as the classical limit; for the homogeneous manifolds, discussed later in this section, this part of the programme was discussed in [8] using techniques from non-commutative harmonic analysis. Finally, the geometric quantization programme [1~13,16,26,27,28], at its most schematic level, is mostly motivated by (a); it starts by classifying the co-adjoint orbits of the group considered; as these are homogenous manifolds, naturally equipped with a symplectic form [16], a link with (b) is established; this structure is then used to construct a reducible (and hence partial) solution to the Dirac problem: this is known as the prequantization stage of the theory; the third step of the geometric quantization programme is the reduction of this representation into irreducible representations by the so-called polarization method. The motivation for the approach taken in this lecture comes from the following remark [29], the origins of which may be traced back to [15,24].

356

While the prequantization map [1,13,16,26,27,28] is a solution of the Dirac problem that produces a reducible representation of the CCR, acting in the space )l of square integrable functions on phase-space: (1-1)

P = -ihaq

,

Q = +ihap + q ,

the Hilbert space )4 also harbors an anti-representation of the CCR, namely (1-2)

,

Q'=-iaop

,

which commutes with the representation (1-1). The contact between this observation and the theory of modular algebras is established by the following result. SCHOLIUM [7,10]. The prequantization representation (i-1)generates a von Neumann algebra )¢ which is a factor of type I, while the antirepresentation (1-2) generates the commutant

(i.3)

J¢'= { A e B()I) [ [A,N] = 0 Y N e )¢ }

of )4; moreover, there exists an involutive anti-unitary operator J that establishes an isomorphism between ]4 and J¢', i.e. (i-4)

.,V" = J)4J

As usual, a technical remark should be made here, namely that all the operators appearing in (1-1) and (1-2) are evidently unbounded; they can nevertheless all be defined in such a manner that they axe self-adjoint, and thus generate unitary groups, (1-5)

{U(a) = e x p ( - i a P ) l a e R }

;

{V(b) = e x p ( - i b Q ) I b e R }

(1-6)

{U'(a)=exp(-iaP') laeR}

;

{V'(b) =exp(-ibQ') I b e R }

s~tisfying

v(b) = exp(-iab) v(b) a~d

(1-8)

U~(a) V' (b) = exp(+iab) V' (b) U ~(a)

When we say that the yon Neumann algebra J¢ [resp. N'] is "generated" by (1-1) [resp. (I-2)], we mean that ,~/ [resp. )/'] is the algebra of operators obtained as the weakoperator closure of all finite linear combinations of the unitary operators appearing in (1-5) [resp. (1-6)].

357

The commutation relations, written in the Weyl form (1-7), suggest the introduction of the "Weyl group "

(1-o)

~o = {w = (a,b,0)

{a

e R n , b e Rn,O E R}

(a,b,O)(at,b',O')=(a+a',b+b',O+O'-~(al -b'-a t-b))

While in the physics literature, this group usually shows up in connection with quantum mechanics, it must be noted that it arises in classical mechanics as well when the momentum map is introduced (1]. It is important for our purpose to note that this group is a central extension

(1-1o)

O--~ R ~ ~o ~ R:--~ O

of the group of translations R 2 which acts transitively and freely on the phase space T ' R . The generalization to T*R ~ is straightforward. The new question we want to address here is whether the above Scholium is itself a consequence of a deeper result. In answering this question, we were led to an alternate derivation of the prequantization representation, summarized in the following statement. THEOREM. The prequantization algebra )4 appears as a / a c t o r in the central decomposition of ~he regular representation of ~he Weyl group. Together with the Scholinm, this Theorem shows that: (i) the prequantization representation of the CCR can be obtained directly from the central decomposition of the regular representation of the Weyl group; and (ii) it generates a yon Neumann algebra spatially anti-isomorphic to its commutant. This a~uti-isomorphism is at the heart of the Tomita-Takesaki theory of modular Hilbert algebras [30]. REMARK: This theorem is not a peculiarity of the flat configuration space R ~ and it extends, in particular, to the Weyl group ~ for simply-connected n - dimensional homogeneous manifolds )/~ of constant negative curvature K = _~;2 < 0. For illustrative purposes, in the first case of interest, namely r~ = 2, ~ can be described explicitly as follows. For K = - 1 , ~ is the Poincare half-plane, which is isometrically isomorphic to the unit space-like hyperboloid in the (2 -k 1)- dimensional Minkowski space, where the Riemann metric om ~/~ is induced from the Minkowski metric. In order to be able to interpolate readily between the results relative to the curved configuration manifold )/12 and those obtained for the corresponding flat manifold R 2, it is convenient to consider the Minkowski metric d82 = dx 2 + dy~ - c2gt 2 for which the corresponding unit spacelike hyperboloid ~ has curvature K = _~2 with tc = 1/c. The considerations to be presented explicitly below for ~ extend, for instance, to the three-dimensional unit space-like hyperboloid X~ in (3 + 1) - Minkowski space. We showed elsewhere [11] (see also [8,25,31]) that the natural generalization of the Weyl group ~o to the case where the configuration space is ~/~ is a 5 - dimensional, simply-connected, exponential and tame Lie group ~g~ on which a coordinate system

358

can be chosen in such a manner that the only non-vanishing brackets in its Lie algebra are:

(1-11) Clearly this group contracts, in the flat case limit ~ ~ 0 to the Weyl group Y~o for the configuration manifold R 2 . It is a central extension

(1-12)

o

R

o

of a 4 - dimensional group G~ that acts transitively and freely on phase space, i.e. on the cotangent bundle (1-13)

T'N~ ~- ~ / Z o

where Zo ----R is the center of SP~. In (1-12), the group G~ is itself an extension 0 --+ R 2 --+ G,~ - * H,¢ --~ 0

(1-14)

where H,~ is the group of lower-diagonal 2 × 2 matrices, that occur in the Iwasawa decomposition (see e.g. [14]) K - ~ , of SL(2, R) (where K is the compact isotropy group of rotations). 2. S k e t c h o f t h e p r o o f for t h e m a i n t h e o r e m . For the sake of simplicity, the proof of the main theorem is conducted in this Section for the flat case R'*; see the concluding comments in Section 3 for an indication of some among the technical precautions that have to be taken when dealing with the non-flat manifolds )t~. The proof proceeds in two steps: the first is an explicit application of the central decomposition theory of the regular representation for the Weyl group ~o; the second step establishes the link between the primary representations so obtained and those obtained by prequantization. Let ~/# (with ~ standing for either R or L) be the yon Neumann algebra generated by the (right- or left-) regular representation U # of the Weyl group ~o , i.e.

and let J be the involutive anti-unitary operator

(2-2)

=

Then (see for instance [5,6,16]) :

(2-3)

JUR(w)J

=

ur'(w)

,

JJC~J----)VL

= {X/n]'

Because of (1-10), we have immediately that the central decomposition ([5,6])

359

(2-4)

•

#

v#(w) = f]~ dAu~ (w)

is implemented, in the sense of Gelfand triplets I12], by the Fourier transform (2-5)

~(ao,bo)-

v ~1 fRdOe_~oogl(ao, bo,Oo )

namely (with # standing again for either R or L, and e/~ = +1, eL = -1):

(2-6)

[V~# (a, b, 0)#A](ao, bo) -= IV# (a, b, 0.)~]~ (ao, bo) 1 = exp{--iA[--e#O + ~ (ao-b - a-bo)]} ~,~ (ao + e#a, bo + e#b)

[JA#:~](ao, bo) ---[Jq~]~,(ao, bo) = # ~ ( - a o , - b o ) * We read directly from these expressions:

(2-~a)

j , v#(,~)j~ = v2 (w)

and, with (2-7b)

~/~ - {U~(a,b) } (a,b) e R 2 " y ' ,

C2-7c) (2-7d)

~ # n ~ 2 = C X.

Since the R 2" part of ~o acts transitively and freely on the cotangent bundle T*R'* " ]Vo/Zo (where Zo ~ R is the center of ~o), we introduce the coordinate identification (2-8)

(a,b) e R 2'~ +-+ ( - q , p ) e T*R '~

corresponding to the classical action (2-9)

e x p { - ( X p - a + X q - b ) } : (q,p)~-~ ( q - a , p

+b)

where X~ is the Hamiltonian vector field for the Hamiltonian function H(p, q) (i.e. XH ] w = - d H ) , and w is the symplectic form dp A dq. Upon defining the generators of the representation U # ('IPo) by

(2-10)

V#(a,0,0) = e x p ( - i a - P # / h ) , V # ( 0 , b , 0 ) - exp(-ib.Q~#/~) V # (0, 0, 0) _=exp(-i0 0 r Ih),

360

we obtain ,

(2-11)

=

o ~ = - (~h)++i [p~ q+~ = + ~(~)~#~ A'

AJ

In particular, for the choice

(2-12)

(~h) = -1

{P~,Q~} is a representation of the CCR, while { P ~' , q ~' } ~ { P ~L , Q ~L} is an antirepresentation of the CCR. Moreover (2-11/12) generate two yon Neumann algebras J¢~ -~ ~/~ and [ ~ ] ' = )¢~ that are the commutant of one another, and satisfy

(2-13)

j~z~ j~ = [~]'.

Finally, the apparent discrepency between (2-11) and (1-1/2) is only due to a residual ambiguity [16,18] in the usual geometric quantization proceedure, which can be "gauged away rq'by the unitary multiplication operator U× = expix/h with X E C°°(T*R'~), corresponding to the fact that, in the definition of the prequantization map, the oneform ~ appearing in the connection

(2-14)

Vn;x

=

X - ilt-l~(X)

is uniquely determined only up to an additive term dx. This completes the proof, for the flat configuration space R ~, of the results announced in Section 1. 3. A p p l i c a t i o n a n d C o m m e n t s . The results of Section 1 (namely that there exists an anti-unitary isomorphism between: (i) the yon Neumann algebra )¢.~ generated by the primary representation of the CCI~ usually obtained by prequantization; and (ii) its commutant [)CA]' ) have several

~onsequences [2,7,10]. The most important one is that (1-4) [i.e. (2-7c}] allows one to identify in terms of observables quantities (i.e. in terms of elements in J]~) the maximal abelian algebra ~ in [X/~]' corresponding [5,6,16] to a decomposition of the primary representation U~ ( ~ ) as a direct integral of irreducible representations. For instance, if the decomposition is made with respect to

(3-1) where ~ is the maximal abelian subalgebra of M~ generated by the position operators on ~/", the decomposition of U~(~,~) precisely produces a direct integral of irreducible systems of imprimitivity [8,21] for the group H~* acting on the configuration space ~ . In general, the choice of a decomposition with respect to a maximal abelian subalgebra (i.e. a complete set of commuting observables) ~A C [JVx]' = JAJC.~JAis the algebraic

361

equivalent of the geometric reduction procedure {19] associated with the choice of polarization when one carries out explicitly the geometric quantization programme for the group ~,~. Physically, this choice can be interpreted in terms of the choice of a measuring process, which in turn is reflected in the correspondance one obtains between classical and quantum observables, i.e. the choice of an ordering. For instance, in the quantization of a one-dimensional harmonic oscillator, we can choose for .g the maximal abelian algebra generated by the Hamiltonian H = ½P~ + ½Q~; the correspondance one obtains from this choice is then given by the so-called antl-normal ordering. This result, discussed in [2], follows from fact that the decomposition of the prequantization representation into irreducible representations can be implemented by a reproducing kernel K that is easily interpreted in terms of coherent states. In fact, the prequantization representation itself, namely

(3-2)

P~

= --i~Oq + ~ p

,

Q~ =

+ihOp +

lq

obtained in Section 2 as a factor in the central decomposition of the right-regular representation of the Weyl group ~ o , coincides with a representation of the CCR that also shows up in [29], where Streater points out its relation with the Bargmann formalism [4] (and related works in flat-space quantum field theory [24]). To be more explicit about the way the quantum ordering (or correspondance "principle ") enters into our theory, we recall that the Hilbert space )/ (of square integrable functions on phase-space), recovered through (2-5), not only carries the representation (1-1) and the antirepresentation (1-2), recovered in (2-11), but can also be viewed as the Hilbert space on which the Koopman formalism [17] for classical mechanics operates: every classical observable f , being a real valued function on phase-space, can be viewed as the multiplication operator M f , acting in )/, defined by

(s-s)

[MSk~](p,q) f (p, q) ffg(p,q) =

(we ignore here the domain questions as these can be taken care of by the usual techniques). The correspondance between classical and quantum observables is then given

by (3-4)

f ~-~ K M / K

where K is the reproducing kernel introduced above; this gives directly the specific ordering corresponding to K and thus to the maximal abelian alegebra ~. An explicit example of the construction of such an ordering is given in [2]. From the point of view of the mathematical structure of the theory presented here as compared to the usual derivation of the prequantization representation, it should be remarked that the choice (2-12) that selects one particular primary representation in the central decomposition (2-4) matches exactly the choice one makes in the geometric quantization programme for the value of the constant h ¢ 0 that enters in the definition of the prequantization map when one imposes that the curvature ~ , of the connection (2-14) is proportional to a specific multiple of the symplectic form w, namely - i h - l w . 362

Finally, we elected to present the proofs in Section 2 for the particular case where the configuration space is R '~ . The analysis of the central decomposition of the rightadd left-regular representations of ~,~, with ~; ~ 0, can essentially be conducted as in section 2, with however two technical differences: (i) Y/~ is not unimodular, and neither is G~ nor even ~/~ ; (ii) the non-abelianness of the extension G~ of H~ also implies that the first-order differential operators appearing in {P~.~, /~ Q~.~} are different from those appearing in {pL.~, QL.~}) (a distinction that is blurred in the fiat case); they are now the teft-(resp. right-) invariant Hamfltonian vector fields of the classical description. With these precautions, one can then indeed generalize the argument presented in section 2, and prove that the results stated in section 1 remain true in the case where the flat configuration space R '~ is replaced by the curved homogeneous manifold N'*. As for the short list of open questions, with: which a review should end, we would like to mention two directions of possible extension for the approach to quantization presented here. The first line of investigation would be concerned with non-tame Lie groups, and with group actions that appear in classical dynamical systems and lead to non-type I representations [20,23,32]. The second line would be to consider the special infinite-dimensional Lie groups that occur when one considers systems with infinitely raany degrees of freedom [3,9]. A c k n o w l e d g e m e n t s . The author wish to thank Dr. S.T. All for discussions on stochastic quantization and the correspondance principle; Dr. A.J.Rica da Silva for discussions on the global structure of the Weyl group ~ ; Dr. L.Faddeev for reminding him again of the pioneering importance of the work of van Hove as a precursor of the motivating Scholium stated in Section 1; and Dr. A.M.Vershik for pointing out to him possible connections with the work of Krieger. This research was supported in part by NSF Grants DMS-8801749 and DMS-8802672. REFERENCES I, K, Abraham and J. E. Maraden, "Foundations of Mechanics," Be~amin/Cummings, Reading, Mass, 1978. 2. 8. T. All and G. G. Emch, Geometric quaatization: Modular rvduc~ion theo~ at~ O o h a ~ gates, Journ. Math. Phys. 2T (1986), 2936-2943. 3. H. Araki and E. J. Woods, R.*presentation~ of the Garmrdcal Oommu~a2ion Relatiotas describing a Non I~lath, ktic Infird~ F~e~Bose Gasj Journ. Math. Pb-ys. 4 (1963}, 637-662. 4. V, Bargmann~ Or, a Hitber~ Space o~/ AnalbCic Ftme~ion, add an A$$ocia~ed Integral Tin,form, Commun.Pure and Appl.Math. I4 (1961), 187-214. 5. J. Dixmier, "Lea alg~bres d'op4rateurs dana l'espace Hilbertien/' Gauthier-Villars, Paris, 1957. 6. J. Dixmier, "C*-algbbres et leurs reprdsentations," Gauthier-Villars, Paris, 1964. 7. G. G. Emch, Pr~quantiza~ion arid K M S Strueture~, Intern'l Journ. Theor. Phys. 20 (1981), 891-904. 8. G. G. Emch, Qttanfura add Classical Mechaaic~ on Hornog~neot~ Pdemannian Ma~Jfolds, Journ. Math.

Phys. 2a (1982), i~85-1791. 9. G. G. Emch, "Mathematical and Conceptual Foundations of 20th-Century Physics," North-Holland, Amsterdam, 1984. 10. G. G, Emchj KM,.q 5~rv,c ~ r ~ ir~ Georngf~c Q t ~ o n ~ in "Contemporary Mathematics," vo].62 (P. E. T, J~rgensen and P. S. Muhly eds.)p AMS, Providence, RI, 1987, pp. 175-186. 11. G. G. Emch and A.J.FL. da Silva, Th~ Wsyl Group for Carved Mardfold, in Classical and Quantum Thsor/~st preprinL University of Florida (1989).

363

12. I. M. Gelfand and N. Ya. Vilenkin, "Generali,.ed Functions IV," Academic Pressp N e w York and London, 1968. 13. V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics/' Cambridge University Press, Cambridge, 1984. 14. S. Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces," Academic Press, N e w York and London~ 1978. 15. L. v a n Hove~ Stir Is prob~me de8 rsla~ion~ er~rs Is8 tran~formation~ unltairel de fa m~canique qsantiqu~ ~t l~trar~lormatioras ca~omqua8 de la rndcardque cla~eique,Mere. Acad. Sci. Belgique 3~" (1951), 610-620. 16. A. Kirillovj "Elements de la th~ioriedes reprdsentations," Mir, Moscou, 1974. 17. B. O. Koopman, Harrdltordan By~tsm~ and Tra~formatior~ in HilbertSpaces, Prec. Natl. Acad. Sci. 17

(1931), 318-318. 18. B. Kostant, "Quantisation and Unitary Representations, Lecture notes in Mathematics vol.170/~ Springer, N e w York, 1970. 19. B. Kostantj Bgmplectic Spinors, Syrup.Math. 14 (1974)j 139-152. 20. W. Kriegerl On conztructing non-*isomorphic hyper]inits factor~ of type III~ Journ. Funct. AnMysis 6 (19~0), 91-109 21. G. W. Mackey, "Mathematical Foundations of Q u a n t u m Mechanics," Benjamin, New York, 1963. 22. R. T. Powers, Rapr~sen~ation~of Uniformly H~perfinite Algebras and their Associated yon Neumann R~nos, Ann. M a t h . 86 (1967), 138-171. 23. K. Schmidt, Algebraic Topi¢~ is Ergodic Theory, Lecture notes preprint (1989), 1-67. 24. I. E. Segal~ Quantizat~on olNor,-Linear Systems, Journ. Math. Phys. I (1960)t 468-488. 25. A.J.R. d a Silva, OWasaicalDynamics is G, rvad Space, Dynamical Groups~ and Geomdri¢ Quantization, Ph.D. thesis, University of Florida (1988). 26. D. J. Simms and N. M. J. Woodhouse, "Lectures on Geometric Quanti~ation/' Springer, Berlin and N e w York, 1976. 27. J. Sniatycki, "Geometric Quanti~ation and Q u a n t u m Mechanics," Springer, Berlin and N e w York, 1980, 28. J. M. Souriau, "Structure des syst~mes dynamiques/' Dunod~ Paris, 1970. 29. R. F. Streater, Oanonical Quan~i~ion, Commun. m a t h . Phys. 2 (1966), 354-374. 30. M. Takesaki, "Tomita's Theory of M o d u l a r Hilbert Algebras, LNM 128," Springer, New York, 1970. 31. G. M. T u y m a n and W. A. J. J. Wiegerinck, General E~ten~don~ and PhyMc~ Journ. Geom. and Physics 4 (1987), 207-258. 32. A. M. Vershik~ Nos M~a~urableDscompo~i$ion~, Orbit Theory, AIg~bra~ of Operator~ Dokl. Akad. Nauk SSSR 199 (1971), 1004-1007,

364

G E O M E T R O - S T O C H A S T I C QUANTIZATION AND QUANTUM G E O M E T R Y Eduard Prugove~ki Department of Mathematics, University of Toronto Toronto, Canada M5S 1A1

Abstraet. The conceots of first and second quantized bundle, as well as of propagator for parallel transport in such bundles, are briefly reviewed. It is shown that such propagation between points in the oase manifold of these bundles can be described by path integrals resulting from the parallel transport of quantum frames along the stochastic paths connecting those points. The Itt-Dynkin concept of stochastic parallel transport is used to formulate solutions of Klein-Gordon equations within quantum bundles. A recently developed [1-9] method of geometro-stochastic quantization, presented in a self-contained and systematic manner in [10], has led to a framework for quantizing gravity [6,7] in which all basic concepts, including that of the states for particles and fields, are provided in geometrically local terms. Hence, in complete analogy with the practice in general relativistic classical gravity [11], they describe physical features that are independent of any assumptions about the global nature of spacetime, or of the many-world interpretations underlying recent developments in quantum gravity and cosmology [12,13]. In the classical regime, the geometrically local aspect of the general relativistic framework enables the assignment of a central role to the equivalence principle. This results in an intimate interrelationship between the special and the general relativistic frameworks: it is arrived at by transferring the arena of the former, from Minkowski space to the tangent spaces TxM over every single point x e M of a given curved classical spacetime manifold M for the latter, and henceforth to related tensor spaces (T:)xM. This fundamental feature of classical general relativity suggests that general relativistic quantum theory should bear a similar relationship to its special relativistic quantum counterpart. However, the conventional notion of states of quantum point particles envisages their representation.by wave functions defined over Cauchy surfaces a which foliate a globally hyperbolic stationary spacetime M (cf. [14], p. 277), rather than by wave functions defined either over tangent spaces TxM, or over related tensor spaces (T:)xM, in generic spacetimes. On the other hand, by abandoning the notion of quantum point particles in favour of that of stochastically extended geometro-stochastic (GS) excitons (which exhibit stringlike features in the mass zero case [7,15]), the geometro-stochastic framework can consistently deal with a geometrically local notion of quantum state represented by a wave function over TxM x Vx c TxM ~ T x * M , where Vx denotes the forward 4-velocity hyperboloid at a given point x e M. The quantum states of such GS exeitons belong to the fibres of bundles over a 4-dimensional differential manifold M, in which the interaction between the quantum gravitational field g and other fields gives rise to various mean Lorentzian metrics ff = g#vdxvdxv. so that x generically represents the base location of the origin of quantum Lorentz frames constructed from excitons. If we characterize these excitons in their role of quantum test bodies by the same absence of self-field characterizing classical test bodies [16], then under the influence of a

365

quantum gravitational field they propagate in a geometro-stochastic manner [1-4], which in the semiclassical approximation is entirely governed by ~ . In this note we shall show that this mode of quantum propagation, formulated in the present form in See. 5 of [4], is related to the Itt-Dynkin [17-19] concept of stochastic parallel transport underlying the development of stochastic differential geometry [21,22]. For the sake of simplicity we shall restrict ourselves to quantum test bodies of zero spin. The local states ku of such test bodies can be prepared and subsequently measured by observers at the base locations x e M, who are equipped with local quantum frames {¢x,¢} constructed [1-3,10] from such quantum objects. For a choice of tetrads {ei(x) = Z~(x)al, I i=0 ..... 3}, orthonormal with respect to the mean metric ff = g~vdxudxV, and constituting a smooth section s of the Lorentz frame bundle L(M) [22], { ~x,g} is given at each x by generalized coherent states [22] labelled by the coordinates ~ = (q,u) that correspond to the general element ~" of TxM x Vx . These coherent states are constructed by means of irreducible unitary representations U(a,A) of the Poincar6 group ISO0(3,1), which act in the fibres Fx over all x e M, as well as in the typical fibre F. This typical fibre is a Hilbert space wave functions W(~"), that carries a phase space irreducible spin-zero system of covariance [1], and the inner product [1,10,23] dZ(~) = 2 uidcri(q)~iu 2 - 1 ) d4v ,

(~r~ ~,) = S ~*(~') W'(~)dT.(~'),

(1)

where the integration in q can be performed over any spacelike hypersurface in Minkowski space. The quantum frames {~x,~ } provide a continuous resolution of the identity in the fibres Fx, so that any ~UeFx can be expressed in the form

=fd.~)~F(OC~x,¢

,

WeF,

YJ(0= (~x,~-[~/-~,

(2)

which obviously representsa counterpartof the expansion qiei(x) for any element q e T x M of the vector bundle T M . The analogy actuallyruns deeper, sinceE is equal to the G-product La(M)×GF, where G --ISO0(3,I),and where LA(M) is the principalframe bundle of affineLorentz frames [10,24].Therefore E is associatedwith the principalbundle LA(IVI),so thatwe can introducein itthe quantum connection [2,5,8]

V = d - iOJPy + (i/2)a&tM kl

,

d = d.x~a~

,

(3)

defined by means of the same connection forms OY and o&t that appear in the affine extension [10,24] of the Levi-Civita connection I7 for parallel transport in TIM. On the other hand, the presence in (3) of the infinitesimal generators P./and M ~ of the representation U(a,A ): ~U(q,v) ~ Y/(A-'(q--a),A-tv), imposes quantum features upon the parallel transport in E determined by this connection. Indeed, for any choice of smooth path ~, joining any two points x' and x" in M , this parallel transport gives rise [2] to a unitary mapping ~l,(x", x ' ) of Fx, onto Fx., which can be uniquely described by means of a propagator

K~,(x",~"; x',~') = (~x"d" I'#r(x ' , x')(19x',~') ,

(4)

for that parallel transport, and which displays the following basic properties:

(~r(x", x')~e)(~") = fKr(x",~"; x',~') ~.'( ~')ol.,~ ~') , Kr(x",~"; x2 ~') = ~Kr(x",~"; x , ~ KT(x,~'; x "

366

~gd2~)

(5) ,

x eT

.

(6)

In the case that M is fiat, the parallel transport described by (3) and (4) is path-independent. Furthermore, in that ease we can work with a section s of LA(M) that corresponds to a global Minkowski frame {ei [i = 0,...,31, so that all TxM can be identified with M itself, and therefore all the fibres F= can be identified with standard fibre F, which carries a representation of the Poincar6 group for changes of such global Lorentz frames. Hence it turns out (of. [2], Sec. 4; or [10], See. 5.4) that if x" is in the chronological future l+(x3 of x' [111, then the propagation described by cr is equivalent to the quantum propagation described in See. 2.4 of [1], which is governed by the Klein-Gordon equation in R*. This suggested [2-4] that in case of curved M the quantum propagation should be described by a path integral corresponding to an averaged parallel transport along broken paths connecting two base locations x' and x". In order to arrive at a physically acceptable formulatibn of such propagation, we have to embed the relativistic causality principle into the geometro-stochastic framework, so that propagation from any x" ~ M would take place causatly to points x" in the chronological future I+(x9 o f x ' . The solution to this problem, proposed in [4,51, consisted of defining a geometro-stoehastic propagator which was the outcome of averaging the propagators in (11) over broken paths F a , built from arcs of timelike geodesics. These piecewise smooth paths Ft, corresponded to foliations of a globally hyperbolic Lorentzian manifold M by means of maximal spacelike hypersurfaces o(t). Thus, for various choices of subdivisions A = (t' = to < tl < - • • < t~ = t " ) , points x(tn-1) ~ o(tn-1) and x(tn) ~ ct(tn) on successsive hypersuffaces in a given foliation were joined with timelike geodesics ~(n) for n = 1.....N , and then (4) was applied to each such geodesic arc. Hence, by iterating (6), averaging over all causal broken paths connecting x' and x", and then taking the limit IAI = maxn (tn -tn-1) --> 0 as N --~ oo, the following expression for the geometro-stochasticpropagator between those two points was obtained,

K(x",~"; x',~') = lim ~dZ( ~n_~)K(x",~";x(t~_~),(N-1) ~dX( ~M-2)K(x(tN_t),(~-6 x(tn_=),~n_2) N--~

x . . . x Sd2~(,) K(x(t2),C 2 ;x(t,),C,)K(x(t,),(,;

x',(~)

,

(7)

where [((x(t.),(n; X(tn_l),(n_l) denotes the propagator (4) for parallel transport along 7An) renormalized by division with the area of ¢s(tn)c~l+(x(tn-1))rV-(x"). The formal similarities between the ensuing treatment of relativistic geometro-stochastic propagation presented in See. 5 of [4], and It6's treatment of stochastic parallel transport in [17,191 are unmistakable. In view of the central role which diffusion processes play in other stochastic quantization schemes [21,221, it is of interest to compare geome~ro-stochasdc propagation with propagation treated by means of stochastic processes ~F = {~F~(co)I ¢0 ~ M [0,°°), lr ~ [0p*)} , associated with diffusion processes with continuous stochastic paths a~ = {x(c) I .r e [0,,,o)} in M. Such diffusions are based on four independent Brownian processes qi(~;) in the typical fibre of TM, so that the following stochastic differential equations [17- 19], ax,(~) = p , [xCz)]dz + X J [x(O]aq"(~) .

p, = -0/2)~"

P~,.

xK0) = x , .

(8)

are satisfied after the substitution eo --> e4, and that the generator of the diffusion process in M is [17,18]: gs#v ~

367

,~-,~a=l"a

"-a

-

(9)

The parameter ~ ~ [0,oo) for the ensuing stochastic paths co could be treated in a more geometric manner by embedding M into a 5-dimensional manifold, in which a suitable fibration would be subsequently carried out (cf. [3], Appendix 2). However, for the sake of simplicity we shall follow the formal procedure originally used by Feynman in his path integration approach to the propagation of Klein-Gordon point particles (cf. [27], Appendix A), and which Guerra and Ruggiero [28] tried to apply to relativistic Markov processes. It should be noted, however, that the metric in (8) and (9) is a Riemannian metric, rather than the original Lorentzian metric of M - albeit the connection coefficients axe those of the Levi-Civita connection corresponding to the Lomntzian metric in M. Hence, in contradistinction to the role which diffusion plays in Nelson's stochastic mechanics [25,28,29], in the present approach the diffusion probabilities play only an auxiUiary role as technical tools for defining limits for parallel transport along paths which arc not smooth, and, as such, they are not related to any of the quantum probabilities [4,5] for the observation of test particles propagating in a geometro-stochastic manner. We shall therefore view the Itb-Dynldn method of construction of stochastic parallel transport as merely one of the mathematical steps in constructing solutions of the following general relativistic Klein-Gordon equation in the quantum bundle E:

(g,vv

vv +,,,2)

= o.

=

.

(lO)

This construction will provide a quantum diffusion process within the relativistically covariant bundle E, which is related by analytic continuations to diffusion processes in auxilliary bundles that display Euclidean rather that Poincar6 gauge invariance. The first step in this construction is to build, for each choice of section

s(x) = {(a(x),ei(x)) ei(x ) = '~i U(x)au, i = 0,1,2,3, x ~ Ms c M } of the Lorentz affine frame bundle LA(M), an auxilliary Euclidean quantum bundle

(11)

Es=p(MS,G) xG F s.

This auxiliary bundle is associated with the principal subbundle P(MS,G) of LAOVI), that has the structure group G=E3, containing the subgroup SO(3) regarded as a subgroup of SO0(3,1) that leaves invariant the timelike elements of the tetrads in (11) - which in this context will be denoted by e40c). The standard fibre F s consists of all solutions of the diffusion equation in ~:E R ÷ = [0,0o),

(o3.+½dada)f.(q~,v)=O,

qoER',

Oa=O/Oqa=-O a , a = 1 , 2 , 3 , 4 ,

(12)

that have analytic continuations to all pure imaginary values for ~ and for the 4-th component o f q o , so that 0

~

(Pm(qb'q°'v) = S0

•

-

exp(-Lm2t/2) f'a(q"'tq~'v)dt '

O

q;' ~ R+'

(13)

is well-defined. By using standard techniques, we then arrive at solutions of the Klein-Gordon equation,

(0~,0" +mZ)cp.(qv,v)=O ,

,O,=O/Oq~o = r/u,,0 ~,

#,v=0,1,2,3

.

(14)

The system of equations (12) and (13) represent a Euclidean counterpart of Feynman's system of equations in Appendix A of [27]. In fact, the solutions of (14) can be described as being the analytic continuations of those solutions of the heat equation in (12), which are subject to a subsidiary condition involving the rest mass m in such a manner that m~/2 plays the role of eigenvalue for a formal "proper time operator" lot :

368

(-i 4 + ½Cgaaa)fit.m(qu,v) = 0 ,

2iatA.m = m2A.m .

(15)

After these solutions of the Klein-Gordon equation in (14) are obtained, they can be cast into equivalence classes, modulo Lorentz boosts Ao to 4-velocities o , by setting:

q~m(Aoq,o) = ¢pm(q,0) ,

Vq = qo e R + x R s,

Vv e V + c R 4 .

(16)

As a consequence [10], the typical fibre F of E can be embedded into the standard fibre F s of E s . We can now reinterpret the diffusion equation for 0 = (1,0,0,0) E R4in (12) as a Fokker-Planek equation for Brownian motion in R 4. Its solutions for initial conditionsfo(q) at ~ = 0 , that are continuous and Lebesgue-integrable in R 4, can be then expressed in the form

f~(q'°) = E°,q [f°(q~,a)]:= fR' f°(q')P(f'q;O'q')d4q" '

t~ = (I,0,0,0) ,

(17)

wherep denotes the transition density of a Wiener process in R 4 :

p(~,q;~',q') = (2~r(~- ~'))-%xp[-lq-q']2/2(~ ¢ > ar,a R1

'

q,q, E R 4

- ¢')] ,

iq_q,[~ =~a=l( q 4

'

a

-q

(18a) ,a 2

)

•

(18b)

The equivalence relation imposed in (16) then results in the analytically continued Lorentz covariant families of functions

fit(q,v) = Eo.q~[f0(q~,o)] ,

t,q ° ~ R+ ,

(19)

which are the stochastic quantum mechanics [1] counterparts of the functions containing a fictitious-time parameter in Feynman's path integral approach to the propagation of Klein-Gordon particles [27]. However, the analytically continued expectation values in (19), and their subsequent counterparts incorporating connection-dependent terms, represent the outcome of mathematically well-defined functional integrations, which can be performed by It6 or by Feynman-Kac methods, rather than by Feynman's formal method of path integration used in [27]. Given now an initial wave function fo(qv,v) = ~(qu,v) ,

~ e F',

qv4 = iq ° e R + ,

(20)

the above construction leads to a Euclidean-space wave function ~,(q,v), which can be related to that initial wave function in (20) by the following direct-integral decomposition:

= ;;* ~iit.rn*din2 '

?'~.,n' (q°'v) = exp(-im2z/2) 5Y(q°'v) ' q°° E R ÷ .

(21)

Thus, we see that in the special relativistic regime we can convert SO(4) covariance into SO(3,1) covariance by relating Wiener processes running simultaneously in a Euclidean fictional time into equivalence classes, determined by the action of all Lorentz boosts Av to 4-velocities v acting on the canonical frame in R 4. Naturally, the key to recovering relativistic invariance lies in the choice of initial conditions, which belong to the Hilbert spaces introduced in [29,30], that carry relativistic systems of convariance, and to the analytic continuation method, which converts Euclidean into Minkowski metric relationships in R 4.

369

We can now extend these considerations to the curved Lorentzian manifold M by using (8), in any coordinate chart for M, to convert Brownian motions in R 4into diffusion processes in M. These diffusion processes give rise to a stochastic parallel transport that is compatible with the Lorentzian metric in M if we choose the connection coefficients in (8) to be those of a connection compatible with that metric - in particular, if we set them equal to the Christoffel symbols for the Levi-Civita connection in the Lorentzian manifold M - as we did earlier. Indeed, according to the Itt-Dynkin method, such a stochastic parallel transport is obtained from the ordinary parallel transport, by approximating stochastic paths with the earlier described piecewise smooth curves 7'/, which are constructed from geodetic arcs 7(x(c;n-,),x(zn)) of the (in the present case) Levi-Civita connection in the Lorentz manifold M, carrying out the purely geometric type of parallel transport along all such curves 7'z~, and then, as 1At ~ + 0 , taking a limit-in-probability with respect to the probability measure of a diffusion process defined by (8). Thus, the stochastic aspect of the definition manifests itself only in taking this last limit. The role of the diffusion process is merely to enable the extension of the geometric concept of parallel transport, which requires curves that are piecewise smooth, to paths which are continuous, but do not possess a tangent at any of their points. We can therefore allow the elements of the Euclidean bundle E s to propagate in accordance with a ItfDynkin [17-19] type of stochastic equation in the typical fibre F s ,

d~',.,.o = ( G ~',.,,o)ax'( ~) + ( ( a,r, ) ~',.,.o + r,r, ~,,.,.o )a:( ,)a~"( ~).

(22)

for Euclidean state vector coordinate functions. The operators F, = a~ -

V~

=

i,t,J, ej;. -"2i a}jk - (0~) M u:k ,

(23)

are obtained taking the covariant derivative, based on (3), in the On direction, and are well-defined if we start from initial conditions ~g0,x,o = Un(O,Ao) Wooc ~ Fx that are in their domain at x. We can then construct by the Itt-Dynldn method [17-19] the operators

r,,r(x,x(~)):

F~ ~

F~(,) ,

(24)

for parallel transport from x to x(z ) along stochastic paths 7'. Consequently, geometro-stochastic wave functions can be defined in the Euclidean regime by the expectation values

tF(~,x, o) = Eo.x[ T~s,~(x,x,.v) W(x(z))] .

(25)

These geometro-stochastic wave functions are therefore obtained by an It5 type of functional integration over stochastic paths joining x andx(c), and hence satisfy the heat equation in M s [17-19]:

a, ~P('tr,x,o) = ½~ ' V ~,V v ~('t:,x,v) .

(26)

Up to this point, the above procedure is only E~, rather than ISOo(3,I), gauge invariant. We therefore execute now in all the fibres of the auxitliary Euclidean quantum bundle E s direct integral decompositions that are the counterparts of those in (21),

~iit(v'x'v) = fR~+ ~a,m'('~'x'v)dm2 '

~it.x,m==exp(-im=v/2)~x,m=

(27)

The transition to the relativistic regime can be then carried out [10] by analytic continuations which convert

370

into it, q4 into iq ° , and 2~~'into i2o~, so that they are extended via the vierbein fields to the metric in (26), and therefore (27) in combination with (27) yields (10). The geometro-stochastic propagators K(x",~"; x',~') defined by (7),. and conceptually related by the above considerations to diffusion processes, govern the propagation of local state vectors ~ E Fx' to all the points x" is in the chronological future I+(x') of x'. The physical outcome of this propagation is prorider by the following probabilistic interpretation [2,4,10]: if a GS exeiton were prepared in a state ~ at the base location x', then, for any given secdon s of the Lorentz frame bundle L(M), the GS wave function ~(x"; f") = ~ K(x",~"; x',~')f(x',~3 d~:(~')

(28)

Supplies, upon setting above ~'"= (0, o), the relative probability amplitude for observing that exciton at the base locations x" along a reference hypersurface o(t) c/+(x'), and of displaying there the stochastic local 4-velocities v" = o"i ei(x ") ~ Vx" in relation to the markers of the origins of the quantum Lorentz frames

{~x,,¢,] correponding to the tedrads {ei(x')} ~ s . All the above considerations can be transferred to the case of the second-quantized bundles E , described in [2,5,10], by extending all the preceding basic derivations from the one-body quantum frames {~x.q}, to the many-body quantum frames {~x.t I f ~ Fx } in their Fock fibres 5rx . The elements ~ E 5rx of these fibres can be described by means of functionals ~(t), f ~ Fx, that are associated, via the first quantized bundle E, to the principal affme Lorentz frame bundle LA(M). Using the techniques of functional integration expounded by Berezin in [31], we can extrapolate (2) into the expansion = S d f a f * tP(x,f) O=.r ,

~(x,f)= (~=,t[~u )

,

(29)

in terms of the coherent state vectors ~x,te Y:x, constituting many-body quantum frames. These frames can be obtained by applying the creation operators ~+)(x,f), related to quantum frame fields O(x;O,

¢(x;O = ¢÷~(x;O + ,t,¢-~(x;~), ¢~+)(x,0 = [f(x,~') ~+>(x,~3dZ,(~3

¢÷~(x;O = [¢-~(x;~3]*,

,

f ~ l~,,

f~ F.

(30) (31)

to the local vacuum state ~/0~ at the considered base locadon x s M : ~x.r = exp[- (1/2)(f If ) + ~+)(x,lDlW0~,

,

f e Fx .

(32)

The extension of the quantum connection in (3) to the second-quantized bundle E can be obtained by inserting in (3) the corresponding infinitesimal generators Pj and M kt of the representation U(a,A), induced, by the original representation U(a,A) in F, in the Fock space Y" constructed from F. The parallel transport along any smooth path ?" joining any two points x' and x" ~n M, to which this connection gives rise, determines a unitary mapping ~r~,(x",x" )of 5rx• onto 9rx-. It is therefore compatible with the quantum metric in E [10], and can be described by means of a propagator for that parallel transport, Kr(x",f"; x',f') = (~x,',f,,l~r(x", x')~x,r,) ,

(33)

M'rich represents the counterpart of the one in (4). The relations (5) and (6), for the one-body problem, then generalize into respective many-body relations as follows:

371

(34)

('ry(x", x ')~r/)(x",f") = ~Kl,(x",f"; x',f')~(x',f') dfdf* , Kl,(x",f"; x',f') = [Kr(x",f"; x,f) Kr(x,f; x',f') dfdf* ,

X~T •

(35)

By using (35) the propagator for parallel transport defined in (33) can be related to an action principle [2,5]. The geometro-stochastic propagator K(~(x"); ~ x ' ) ) , obtained by corresponding types of averaging procedures over stochastic paths as in the one-body case, yields the GS probability amplitudes ~(~(x")) = J' K(~(x"); O(x3)~((~(x')) D~b(x5,

(36)

to which a physical meaning can be assigned by extending in an obvious manner the probabilistic interpretation of the one-body probability amplitudes in (28). The adaptation of these considerations to the case of other massive GS quantum fields is straightforward [2,7,8,10], but the treatment of GS quantum gauge fields and of GS quantum gravity requires superfibre bundle techniques [4,7,8], which will be presented in a systematic manner in [10]. References.

1. E. Prugove~ki: Stochastic Quantum Mechanics and Quantum Spacetime (Reidel, Dordrecht, 1986). 2. E. Prugove~ki: Nuovo Cimento A 97, 597, 837 (1987). 3. E. Prugove~ki: Class. Quantum Grav. 4, 1659 (1987). 4. E. Prugove~ki: Nuovo Cimento A 100, 827 (1988). 5. E. Prugove~ki: Found. Phys. Lett. 2, 81, 163, 403 (1989). 6. E. Prugoveeki: Nuovo Cimento A 101, 881 (1989). 7. E. Prugove~ki and S. Warlow: Found. Phys. Lett. 2, 409 1989). 8. E. Prugove~ki and S. Warlow: Rep. Math. Phys. 28, 107 (1989). 9. W. Drechsler and E. Prugove~ki: "Geometro-stochastic quantization of a theory of extended elementary objects" (to appear). 10. E. Prugove~ki: Quantum Geometry (Kluwer, Dordrecht) - to appear. 11. C.W. Misner, K. S. Thome and J. A. Wheeler: Gravitation (Freeman, San Francisco, 1973). 12. C.L Isham: in General Relativity and Gravitation, ed. M.A.H. MacCallum (Cambridge University Press, Cambridge, 1987). 13. M. A. Markov: in Quantum Gravity, eds. M. A. Markov, V. A. Berezin and V. P. Frolov (World Scientific, Singapore, 1988). 14. J. V. Narlikar and T. Padmanabhan: Gravity, Gauge Theories and Quantum Cosmology (Reidel, Dordrecht, 1986). 15. E. Prugove~ki: Nuovo Cimento A 100, 289 (1988). 16. J. Ehlers: in General Relativity and Gravitation, ed. M. A. H. MaeCallum (Cambridge University Press, Cambridge, 1987). 17. K. It6: Proc. Int. Congress Math. Stockholm 1962, pp. 536-539. 18. E. B. Dynkin: Sov. Math. Dokl. 179, 532 (1968). 19. K. It6: Springer Lecture Notes in Mathematics 451, 1 (1975). 20. Yu. L. Daletskii: Russian Math. Surveys 38, 97 (1983). 21. Ya. I. Belopolskaya and Yu. L. Dalecky: Stochastic Equations and Differential Geometry (Kluwer, Dordrecht, 1990). 22. S.T. All and E. Prugove~ki, Acta. Appl. Math. 6, 47 (1988). 23. E. Prugove~ki, Phys. Rev. Lett. 49, 1065 (1982). 24. W. Drechsler: Fortschr. Phys. 32,449 (1984). 25. F. Guerra: Phys. Reports 77, 263 (1981). 26. G. Parisi and Y. S. Wu: Sci. Sinica 24, 483 (1981). 27. R. P. Feynman: Phys. Rev. 80, 440 (1950). 28. F. Guerra and P. Ruggiero: Lett. Nuovo Cimento 23, 529 (1978). 29. E. Prugove~ki: J. Math. Phys. 19, 2260 (1978). 30. E. Prugove~ki: Phys. Rev. D18, 3655 (1978). 31. F. A. Berezin: The Method of Second Quantization (Academic Press, New York, 1964).

372

S O ( 4 , 1 ) - C o h e r e n t S t a t e s and t h e G e o m e t r o - S t o c h a s t i c Q u a n t i z a t i o n of a G a u g e T h e o r y for E x t e n d e d O b j e c t s

WOLFGANG DRECIISLEK Max-Planck-lnstitut f/ir Physik und Astrophysik - Werner-Heisenberg-Institut f/Jr PhysikP.O.Box 40 12 12, Munich (Fed. Rep. Germany)

ABSTRACT Coherent states associated with the principal series of unitary irreducible representations of the (4,1)e Sitter group are considered for the use in a stochastically qusntized gauge theory defined on a Hilbert undle over curved space-time carrying a phase space representation of S0(4,t). The bundle geometry m characterized by a fundamental length parameter R taken to be of a size typical for hadron physics. The conventional local operator quantum field theory over Minkowski space-time yields a theoretical formalism describing pointlike particles. The opinion has often been expressed in the past that the physically questionable idealization of representing interac~lons in high energy physics in terms of pointlike elementaryparticles and corresponding ~ a l Minkowski operator fields may be the origin of the infinities plaguing the theory. ae aim of this note is to give a short account of an endeavour to set up a quantized gauge theory for elementary objects possessing an intrinsic geometric length scale of extension of the order of R ~ 10-13 cm typical for hadrons [ Leptons would be represented 'n. a similar manner by taking a formal limit, R ~ oo, corresponding to a contraction ot the gauge group from SO(4, 1) [ see below ] to the Poincare group ISO(3, 1) ]. The procedure will be to start from a classical gauge theory of the (4,1)-de Sitter group forraulated on a certain soldered fiber bundle [1-3] associated to the de Sitter frame bundle, P(B, G - 80(4, 1), over curved space-time B, and then go on to quantize such a theor in a geometro-stochastic " " • manner [4-6] using phase space representatwns of the (4,1)-deY Sitter group and coherent states related to the principal series of unitary irreducible repHresentati°ns (.UI~.) . .of. $0(2,11. . . The Lorentz gauge theory defined throu..gh the subgroup . --- S0(3, 1) of G = S0(4, 1) is supposed to be related to gravltatmn in a vierbein Zormulation involving torsioia (c3mpare ~7], [8]). The transition to a geometro-stochastically quantized theory is carried out by introducing a soldered Hilbert bundle 7"/= 7-/(B, 5c = 7-/~p), 0(A#)) over a curved space-time base manifold 1) B possessing a standard fiber 7-/(p) which carries

a

phase spa~e representation of the (4,1)-de Sitter group denoted by Cr(P)(Ag) = U(Ag), with Ag E S0(4,1), Yielding thereby a quantum description of spinless particles in terms of wave motions in a (4,1)-de Sitter space of radius R. At an intermediate stage this requires the introduction of a soldered bundle/~, associated to P , possessing de Sitter phase space (see below) as - $

1)B = U4 in the Riemann-Cartan case, or B = V4 if torsion is neglected (long-range GR-limit).

373

standard fiber. 7"/~p) is a resolution kernel Hilbert space with resolution generator (proper state vector), ~ = ~(P), related to the principal series of UIR of S0(4,1) which, in the spinless case, is determined by the number ~ = - ~ + ip, 0 _< p < c~; and U(Ag) acts as the structur~ group in ~ . Scalar de Sitter gauge field transforming under U(Ag) are then later introduced as representatives of quantized spinless matter pre~ent in the geometry and appear as fields, ~ (~, (), defined on 7~ where x E B denotes a point in the space-time base and (~, ~) are de Sitter phase space variables to be defined below playing the role of local stochastic variables. Here we like to focus the attention on the properties of the one-particle Hilbert space 7/,~-p)providing the standard fiber of the bundle 7-/, i.e. we look at the general formalism ¢ for a fixed wlue of x E B. We shall construct a coherent state basis for 7~(~p) in terms of ~ and de Sitter horospheric~ waves [9] [10] the latter being analogous to the plain waves in Minkowski space-time. The (4,1)-de Sitter space m denoted by V~ - - is a space-time of constant curvature having curvature radius/~. As mentioned above, R is taken to be a fixed scale of length typical for hadron physics which appears here as a parameter characterizing the standard fiber of a bundle providing the arena for hadron physics including gravitation2)~ and describing extension of elementary objects in physics as a gauge phenomenon related to the de Sitter degrees of freedom. This is expressed in terms of the variables (~, () which play the r61e of local stochastic variables in the theory defined on 7-/. For a detailed discussion of the gauge aspects of the theory, the definition of the (~, ( ) - and x-dependent fields ¢(P)(~, ¢) and par_an,el transport on n as well as questions related to a roans particle, i.e. "second quantized formalism we refer the rea~er to a forthcoming paper [11]. Let us, after these remarks, return to the space 7~(p), which is a space of squareintegrable functions defined over de Sitter phase spa~e Af~== V~ x C:t=. Here V~ is (4,1)-de Sitter space [ isomorphic to the noncompact coset space3) G / H = S0(4,1)/S0(3,1) ] represented by a one-shell hyperboloid ~a~b~ab = --R 2 ;~ab = c l i n g ( l , - 1 , - 1 , - 1 , - 1 ) , ~a; a = 0,1,2,3,5, in a Lorentzian embedding space //4,1, and C± = C ± N F , where C :~ is the cone, ~a~b~ab -~ O, in R4,1 with 4- = sign( ° and F is the surface (5 = ]~. The points at infinity in (4,1)-de Sitter space are determined by the vectors ( lying on the cone in R4,1. The origin of the homogeneous space G / H ~-- V~ will be denoted O

O

by ~ [where ~a = ( 0 , 0 , 0 , 0 , - R ) ] with H = S0(3,1) being the stability subgroup of O

G = SO(4,1) leaving the point ~ invariant. Following [9] [10] we identify antipodal points on the hyperboloid V~ (yielding an imaginary Lobachevski space in the notation of [9]) and introduce the notation [~, ¢] = ~a~b~ab, i.e. =

c*

:

[¢,¢1 =

o,

(1)

2) For simplicity we disregard an additional U(I) x SU(2) fiber associated with the electroweak interaction and focus the attention solely on the S0(4~ 1) properties [ with subgroup S0(3,1) ] in the underlying principal bundle. 3)The homogeneousspace SO(4~1)/S0(3,1) is, actually, a symmetric space.

374

with (a = i, 5; i = 0,1, 2, 3)

One has C(±)i((±)k~li k = (.( = RI--~,where the dot denotes the scalar product'in Minkowski space. It was shown in [i0] that a horosphere in (4,1)-de Sitter space parallelto a horosphere through ~ [ the latter characterized by ((±)i and called H~ ] is given by the equation

H~: I[~,C]l=c; ~ e ~ , ¢ e e

+,

(3)

where Rlog c measures the distance of H~ from H~. The analogue of a plane wave solution of the wave equation in Minkowski space is provided by the horospherical wave solution in de Sitter space which is, in the spinless

ca~e, given by

¢~P)(~) = I[~,¢]I-~+IP; 0 < p < 0% ~ e Vd, ¢ e C±,

(4)

These are eigenfunctions of the Laplace-Beltrami operator 4) on VJ, 2"1 = 2-~RLab(~)Lab(~)

~,,

with eigenvalue .

(S)

The second term at the r.-h. side of (7) is again due to the shift of the origin for genera/Ag E S0(4, 1). Eq. (7) can be used to express the transformed horosp~erical wave ][Ag~, ¢][-~+iP in the following manner:

e(-~+iP)

=

e(-~+ip) e(-~÷iP).

(9)

After these remarks we can now write down the coherent states related to the symmetric sp~ce S0(4,1)/S0(3, 1) of the (4,1)-de Sitter group which are a~sociated with the principal series of UIR of S0(4,1): ~(P)'.~'~ = (~r(A(~, & ) ) O ) C )

=

e(-~+iplc~

•

(18)

This overlap integral is nonzero due to the nonorthogonal nature of the coherent states (10). The free propagator thus defined satisfies the following relations: =

¢,¢'),

(19)

fr'a~d

[R~P)(~',¢')]g = K~ - (p) (Ag~,Ag¢ , , ;Ag~,Ay¢).

(21)

R E F E R E N C E S

[1] W. Drechsler, Fortschr. Phys. 38, 63 (1990). [2] W. Drechsler, Found. Phys. 7, 629 (1977). [3] W. Drechsler and E.M. Mayer: Fiber Bundle Techniques in Gauge Theories, Lecture Notes in Physics, Vol. 67, Springer Verlag, Heidelberg 1977. [4] E. Prugove~ki, Stochastic Quantum Mechanics and Quantum Spacetime, Reidel, Dordrecht, 1984; corr. printing 1986. [5] E. Prugove~ki, Nuovo Cimen~o A 97, 597, 837 (1987); A 100, 827 (1988); A 101,853 (1989); A 102,881 (1989). [6] E. Prugove~ki, Class. Quantum Gray. 4, 1659 (1987). [7] W. Drechsler, Fortschr. Phys. 32,449 (1984). [8] W. Drechsler and W. Thacker, Class. Quantum Gray. 4, 291 (1987). [9] LM. Gel'fand, M.I. Grace and N.Ya. Vilenkin, Generalized ~'hnctions, Vol. 5, Chapter V, Academic Press, London 1966. [10] W. Drechsler and R. Sasaki, Nuovo Cimento A 46, 527 (1978). [11] W. Drechsler and E. Prugove~ki, Geometro-Stochastic Quantization of a Theory for Extended Elementary Objects, Preprint MPI-PAE/PTh 18/90. [12] S. Helgason, Lie Groups and Symmetric Spaces, in Battelle Recontres, Editors: C.M. De Witt and J.A. Wheeler, Benjamin Inc. New York 1968. [13] S. Helgason, Topics in Harmonic Analysis on Homogeneous Spaces, Birkh£user, Basel 1981. [14] A. Perelomov, Generalized Coherent States and Their Applications, Springer Verlag, Heidelberg 1986.

378

COvariant Q u a n t u m Mechanics and the Symmetries of its Radiation Fields M.C. Land and L.P. Horwitz School of Physics and Astronomy Raymond and Beverly Saekler Faculty of Exact Sciences Tel Aviv University, I~ma~ Aviv, Israel

Abatrc~c~. The requirement of gauge invariauce for the StSekelberg equation for the relativistieally covariant wave function of a system evolving according a universal world (or historical) time r implies the existence of a five dimensional pre-Maxwell field on the manifold of spacetime and r. The MaxwelI theory is contained in this theory; integration of the field equations over ~ restores the Maxwell equations with the usual interpretation of the sources. The energy-momentum-mass tensor of this field has the form of a four dimensional invariant harmonic oscillator at each point in four-momentum space. The known solutions of the two body bound state problem indicate that the electromagnetic field strengths must lie in a restricted part of the spaeelike region of Minkowski space in order for the spectrum and the wave functions to have the correct structure for the quantized radiation field. The wave functions form irreducible representations of 0(2,1); the 0(3,1) transformation properties of the field correspond to an induced representation. Stiickelberg was motivated to introduce the parametric description of the evolution of the classical world lines, and the quantum mechanical evolution (generating approximate world lines through the Ehrenfest motion of the wave packet), by the observation that pair annihilation can be described by a world line which reaches some region in finite spaeetime from t = - o o , and returns to t = - o o moving in the reverse direction of time. He remarked that the variable t is clearly not adequate to parametrize the motion in such a configuration, and therefore introduced the invariant parameter r. The equation

• a,/,,- (:~) = ZC(p, z),b,- (~)

:

aT

(1)

was assumed by Stilckelberg [1] to describe the evolution of a wave function in the invariant parameter T in a manifestly covl~riant quantum theory. Horwitz and Piton[2] extended the ideas of St~ckelberg (which can be traced back to Foek[3]) to the formulation of a theory applicable to mauy particles with electrornagnetie interaction and action at a distance (in spacetime) potentials. It has been shown, for example, that the Newton-Wigner position operator is naturally contained in the structure of the theory [2], and the uncertainty relation of Landau and Peierls [4] (interpreted by Aharonov and Albert [5] as a causal requirement) follows from a COmmutation relation.

379

The m a s s spectrum for the two body problem (with Lorentz invariant potential functions) was obtained in ref. [6]. It was shown that the energy spectrum then goes smoothly to the total mass plus the Schrbdinger energy spectrum in the nonrelativistic limit. It was essential in obtaining this result that the support of the two body wave functions, in relative coordinates, is restricted to a subspace of the Minkowski measure space which satisfies x l 2 + x22 - t 2 > O. This restricted (spacelike) region, called the lZMS, has an orientation along the 3 axis ( this axis, and the corresponding lZMS, is left invariant under the corresponding O(2,1) of O(3,1)). The eigenfunctions obtained are irreducible representations of (the double covering) O(2,1) in the discrete series. The transformation properties of this system under O(3,1) are then obtained as an induced representation over a spacelike vector (of the form (0, 0, 0,1) in the special frame of the analysis above) [7]. The solution of the two body relativistic bound state problem, as we show below, plays an important role in the quantization of a new electromagnetic theory, which we shall call the p r e - M a x w c l l theory, required for the consistency of the manifestly covariant formalism. The Maxwell field, which is independent of r, has for its source the integral over r of the current associated with Eq. (1), taking for the generator of motion a form quadratic in the energy-momentum vector [2]. This current satisfies the condition O,j ~ +

(2)

= 0,

where p = [~b[2. Integrating over r, the second term vanishes under appropriate boundary conditions; hence the integral of the r-dependent current j~ , on - o o to oo, has a vanishing four-divergence. The dynamical problem of the evolution of events is then, however, not well-conditioned. The motion of each of the events would depend on the entire world line of the other, which is responsible for the field in which it moves. The solution of the problem then requires an iteration to find a self-consistent field, a procedure which may not be stable. A well-conditioned procedure can be formulated by introducing r-dependent fields, with the five-dimensionally conserved r-dependent current as its source [8]. Such a theory emerges naturally from the gauge invariant form of Eq. (1). The requirement of covariance in structure under the transformation

¢, =

(3)

can be satisfied by introducing five gauge compensation fields, a~, for a = O, 1,2, (which we call pre-Maxwell fields), and write the equation as (iOr + eoa4)~br(x) = K ( p -

eoa, x)¢r(x),

a,4 (4)

where, under gauge transformation, a~

-* aa + OaA

preserves the form of the equation (see also refs.[9]). Second order field equations can be obtained (along with the equation of motion (4) for the wave function), by assuming

380

a Lagraagian containing the term ( })f,~f,~e (where lap - aaap - apa~), of the form IS] ,9~f "e = ej'L (5) where e = e0/A. Note that we have formally written the ~ = 4 index as a subscript or superscript to conform with the covariant and contrava~qant forms of the Lorentz indices. This theory leads to a consistent, well-posed, dynamical problem, in which the particle world lines devdop through the motion of the events under the influence of fields generated by the motion of other events in spacetime. Furthermore, integration of the equations (5) for the a = # components (with appropriate asymptotic conditions), reproduces the usual Maxwell equations with the r-integrated currents as sources. We remark that then e0 and A have the dimension of length, and e is dimensionless. Hence, a pos~eriori, the solution of the problem satisfies the requirements of the macroscopic Maxwell theory. The Lorentz forces [10] are, however, different; they approxin~te the Maxwell Lorentz forces only if the field f~4 is small [10]. The antisymmetric derivative of the field equations (5) results in [8]

(o~o~ + o~o~)f ~ = - 4 o ~ J e

-

o~J~).

(6)

In the five dimensional Lorentz gauge O~a a = 0 gauge, Eq. (5) becomes 0e0e~

= -ej ~.

(7)

The Green's functions for Eq. (6) have been worked out in ref. [11] The spacetime Fourier transform of a source-free equation of the type (6) or (7), say, OzO~f = 0, is ( & 0 ~ - k . k . ) f ( k , ~) = 0; (S) if kJ' is spacelike (timelike), in order that the solutions f(k, r) to be bounded for all r, the second order r-derivative should occur with a minus (plus) sign. The 0(3,2) and 0(4,1) choices of the signature of the r-derivative therefore correspond to stable solutions of the source free field equations for spacelike and timelike four momenta of the off-shell fields respectively. We now turn to the problem of second quantization for the radiation field. The energy-momentum-mass tensor for the pre-Maxwell field (we assume the 0(3,2) case, for the part of the spectrum for which kzk z > 0, for defini{eness, in the following)

T;a c ~d _- A { -

4

g~13fu.x f.ya + f~'r f~ }

(9)

provides the generator of evolution for the off-shell radiation field,

1:'~l*'e4 + ¼f"'g,}. J~

T44 = Mt 2 ¢

(10)

The #v components can be eliminated, in the absence of sources, with the help of the relations (6) and (5), i.e., for the Fourier transform of the fields,

381

and

a.P"O0

= o,

to obtain the four-dimensional oscillator form (this structure is maintained in the O(4,1)

case)

where ~2 = k~ks,. As we have seen above in our discussion of the two body bound state, the solution of the invariant four-dimensionM oscillator has an acceptable spectrum and eigenfunction structure if we restrict the support of these functions to the 1%MS for the amplitudes ]4~. In this way we obtain a quantization of the pre-Maxwell field as a 4-dimensional oscillator of relative mo~ion (displacement field), providing a photon Fock space with no ghosts (it can easily be shown [8] that writing Kra~ as a bilinear form in the positivenegative frequency complex components of the fields is non-negative when the fields are spacelike). The fields ~4"(k) = Z(n)~faV(k) (12), where L(n)~, a transformation in the coset Space O(3,1)/O(2,1), is defined to bring the spacelike vector n ~ labeling the orbit of the O(2,1) little group to the standard form (0,0,0,1), transform with 0(2,1) under the action of the Lorentz group. It is convenient to write the wave functions in terms of the variables 6~(k); the eigenfunctions, in this case solutions of the harmonic oscillator problem in the standard frame, are irreducible representations of 0(2,1) in the discrete series. They form induced representations of 0(3,1) under the action of the Lorentz group over an orbit labeled by n~; as in the two body bound state problem, they belong to the principal series. The photons of the preMaxwell theory therefore transform under the non-compact little group O(2,1), with three polarization states, which deform to 2 @ I under integration on T. References 1. E.C.G. Stueckelberg, HelP. Phys. Acta 14, 322 (1941); 14, 588 (1941); 15, 23 (1942). 2. L.P. Horwitz and C. Piron, Helv. Phys. Acta 48, 316 (1973). 3. V.A. Fock, Physik Z. Sowjetunion 12, 404 (1937). 4. L.D. Landau and R. Peierls, Zeits. f. Physik 69, 56 (1931). 5. Y. Aharonov and D. Albert, Phys. P~ev D 24, 359 (1981). 6. 1%.I. Arshansky and L.P. Horwitz, Jour. Math. Phys. 30 66 (1989). 7. 1%.I. Arshansky and L.P. Horwitz, Jour. Math. Phys. 30, 380 (1989). 8. D. Saad, L.P. Horwitz and 1%.1. Arshansky, Found. of Phys. 19, 1125 (1989). See also, L. Hostler, Jour. Math. Phys. 21,2461 (1979); 22, 2307 (1980), and 1%. Kubo, Nuovo Cimento 85A, 4(1984). 9. F.H. Molzahn, T.A. Osborn and S.A. Fulling, Ann. of Phys. , to appear, and references therein; 1%.A. Corns and T.A. Osborn, to be published. 10. M.C. Land and L.P. Horwitz, Tel Aviv University preprint 1824-90, to be published. 11. M.C. Land and L.P. Horwitz, Tel Aviv University preprint 1798-90, to be published.

382

DOUBLE S T R A T O N O V I C H FOR

A

Joseph Dept.

HUBBARD TRICK

-

SYSTEM

OF

L. Birman

of Physics,

AND NOVEL PATH

INTERACTING

and

INTEGRAL

FERMIONS

Alexander V. G o r o k h o v

Clty College,

CUNY,

New York,

N.Y.

10031

i. Introduction The c a l c u l a t i o n

technlques

approach have proven to be

a powerful

quantum mechanical

applications.

and

lald

Hubbard

[2]

calculating With

average

the

the p a r t i t i o n

two-body over

by

Feynman

path

Integrals.

Partition

functlon

Hamlltonlans

the

and

S-H

method

average"

methods

the

integral

Physics

the S t r a t o n o v l c h distinguished

structure

from

of

for

the

realistic

super-Weyl

Address

for statistical

operators,

While

[3], solid

are of considerable

"Gausslan

state

importance,

functional

averaging

It is based on a G r a s s m a n l a n operator

group

:

of the

method.

(plus the usual one)

evolution

for c o m p u t a t i o n

Permanent

the

with These

[~,5]

but

the

incomplete.

system.

the S-H t r a n s f o r m a t i o n relevant

be

Gaussian

fields.

representation

In thls paper we glve a new

methods

therefore

of

system

a

interacting

classical

a [I]

used method

Z( ~ ) as

is not known for e l i m i n a t i o n

[6,7] physics,

for a m a n y - f e r m l o n of

wldely

should

In

one must c a r r y out a v e r a g i n g of time-dependent operators. A general

to the Feynman path

aethods are at present

ling

tools

is very complicated:

and universal

and nuclear

integral

Stratonovlch

systems

Unfortunately,

in

of n o n - c o m m u t i n g

functional

for

integrals

the exponents in contrast

a

fluctuating

f l u c t u a t i n g flelds are fictitious functional

effective

expressing

functions

external

(S-H)

for

path

function Z( ~ ) of a m a n y - p a r t i c l e

partition

HUbbard

and

Thirty years ago

Dasls

interactions,

"tlme-dependent"

that come under the

SW(N),

and

utilizing and

on

a

properties

fermlonlc

of

, K u l b y s h e v University,

General

and

Kulbyshev

383

Theoretical

4~3011,

USSR

of

disentangof

coherent

of the trace.

Dept.

procedure version

the state

2.

Fermlonlc Hamltonians ,

Consider

,

and Double S-H Trick

,,,,

the H a m l l t o n l a n

,

,

H for a system of fermlons

interacting

via a t w o - b o d y potential V

. :

61cicl

÷

~

Where

,

lJkl

i

{Ck#}- are the usual fermlonlc

both m o m e n t u m and spin indlces Is the o n e - p a r t l c l e

k

operators;

R

(k,

the Index k includes

a );

8 k : (

q 2/2mo - ~ )

e n e r g y , molS the mass of the fermion and

~

-

the chemical potential. Usually,

the methods

with such

Hamiltonlans

mean-field

approxlmatlon:

of d e t e r m i n a t i o n

of some values connected

involve the l l n e a r l z a t i o n

scheme based on

a

AB _ A + B , where A and B are an a p p r o p r i a t e and

quadratic

, are the e x p e c t a t i o n

resulting

approximate

able because Lle algebra over,

it

becomes

(Dynamlcal

the p a r t i t i o n

values

(mean-fleld) linear

Algebra

6 i,

in

the

generators

function and the m a n y - f e r m l o n

determined by a Dynamical

[8].

Algebra

for the calculation

operators,

from

the

A

The

Hm_ f Is d i a g o n a l l z -

of the m e a n - f l e l d up

Cj

In some ground state.

Hamiltonlan

the reduced problem can be built suggestion

in

of any

compact

Hamlltonlan).

More-

Green functions

of

factors

completely

convenient

approach

of the m e a n - f l e l d

theory c o r r e c t i o n s

Is the main purpose of thls paper. For simplicity we now restrict a t t e n t i o n to wlth H an o p e r a t o r - v a l u e d Lie

a

function of the g e n e r a t o r s

generic X ab(~

model

)

of

a

algebra

v ab

k

a'b

ab (

Havlng appltcatlons of

In

solid

state

a quaslmomentum

f a ( ~ ) f b ( ~ 1. The notation quasimomenta

(l,

k,~ (sum over all repeated

explicit use

)Xab

Indlces) physics

label

{flIpermits

( ± k , k ~ Q , etc.)

and,

384

k us

e.g.

In

mind,

and write to

we

make

ab(k

) :

distinguish

various

in the BCS case

permits

a t

correspondence ~

wlth

familiar

(£i(k). (2(k~),(cC+.~, lines unifying,

creatlon/annlhllation

along

these

superconductivity and charge and spln density

waves

discussed elsewhere

c_{{)etc.

A~

su(8~ model

operators

[9].

Here the operators

Rab(k)

[ Rab(~ ) . Ra.b(~")] of the Lle algebra

N

(the

in formula (I),

fermlonlc

relations

ab. X a,b(k

if a, b . . . . . . : I,

depends upon the model considered) excitation energy of the

commutation

the

(~iba.Rab(k'* )-~

={)~'t

gI(N,R),

obey

oscillation

)) number

~ (k) of

the

(2) N

is

the

kind

"a"

Which may differ in sign from the corresponding one-partlcle energy. The matrix element of the Interaction expanded i n

2

operator

V

potentlal

(e

can

be

m I)

and

a series

aba'b"

ab

e

Consider the simplest represent the right restriction is n o t

"

case slde

"

of a separable of

eqn.

(3)

as

V a b ( ~ )Va,b.(~').

required for the following,

but

in

the

(This

general

case the calculations are more unwieldy). Then,

Hamlltonlan has the structure

our

M = ~m-f * ~ ' ' wh, re

=

(4)

8 aC

) R aa

k and

V: (

Vab(~

)Xab(~ )) (~ Ga,b(~')Ra*b (2"))

k (Here we consider a case diagonal form in comblne

our

the

approach

when

the

Dynamical with

a

mean-fleld

Algebra

~

A B .

Hamlltonlan

generators;

Bogollubov-llke

we

has have

a to

dlagonallzatlon

Procedure In more general cases). The Grand Partition Function for the Hamlltonlan of (d) can wrltten 0

385

be

where T d - the time ordering operator, and V( ~ ) m exp( ~ H m _ f ) V exp (- ~Hm_f) £s V in the interaction representation. We now follow a familiar procedure : In eqn. (5) divide the interval [ O, ~] into m equal segments as [0, ~ i]...[ ~ m-l' ~ m : ~] l: ( ~ l/m); use the S-H transformation for each factor in the product.

exp(- ( ~/m)V(¢1))

z

exp(-(~/m) A( ¢ i) B( ~i )) =

d Re z( ~ i ) d Im z( ~ i ) ( % m / ~

)-1 e x p ( - ( ~ / m )

z( ~ i ) z( ~ i )) x

(6) (valid to terms of order that

operators

Z( ~ ):N llm Tr k x

labelled

( ~/m)2); by different

[Td~...~r~m

m "~

then using k

d Re z(

(6)

commute,

and

recalling

we o b t a i n

l)d Im z( ~ l ) ( ~ m / ~

)-Ix

i:I

exp(-( ~/m)z(~i)~-~i)) exp( -( ~/m)Sa(~ ) Xaa(~ )) x

x exp[( ~/m)(~( ~l)~ab(~ )

z( ~l)Vab(~ ))Xab(~, q;l)] ] . (7)

Note in the product (7) the "time dependent" operators £ab(~, %1) are

some linear combinations

£ab ( ~ )

of the

which preserve the Lle

simple case of (4)

"tlme-lndependent"

algebraic

£ab ( ~ '~ 1 ): Xab(k

Below, we shall omit the explicit

)exp

k

no confusion. Taking the formal limits integration and trace, we get

z( ~ )= FI j" D,(~., .) exp(-J"

~.( '~ ) . (

k

,386

structure

operators

(2)

1( 8a (~)-8

for

b (k))

the .

dependence where It can cause m -"~,

and

~ ) d'C ) Tr [exp(-~8

interchanging

£aa)"

" g < Xac{. # >aAuss k

Where

m

D(z,

and ~ G l s

z) = llm m-~

~ d Re z( 9;1 ) d Im z ( g l ) I=i

the character of the

(reduclble)

(gm/~)-i

representation

o£

Lie

group G mGL(N, Rj, or of some particular subgroup depending on specific form of H. These methods are well known [3-7]. Unfortunately we cannot obtain the characters ~ G explicitly since the z( 9; ) and z( % ) are general fluctuating

functlons of

g.

Often a

"static"

Path approximation is used: z( ~ ) []const, which is equlvalent to mean-fleld theory. To overcome this limitation we return to eqn. (7) and introduce the Grassmanlan version of the S-H transformation. Namely, If Fl and F2 are antlcommuting operators {F I, F2 } = 0, then

Here

~ and { *

are Grassmanlan

antlcommlting

variables: *

•

= 0,

To apply this transformation definition of operators F1 Fl = ~

{~,

~i,z}

= {~

in our case, and F2 :

(~( 9;l)Vac - z( g l )vac)

In this case the antlcommutator

we

, ~i,2 } = o.

use

an

assymmetrlc

exp( 9;~a ) f+a "

{F I, F 2}

"

( ~/m)

and

can

De

neglected in the limit m ~ . Consider the eqn. ( 7 ) and r e p r e s e n t the last exponent, as a Product, and got each factor, use the Grassmanian S-H trick to obtain: Z( ~ ) = r~ ' ( ~m/#

lira

Tr

exp(-~ea

a

a) d

""

1-I ~ d { ab ( 9; I )d ~ ab { 9; 1 1( ~/m)-lexp( -( ~ /m)~( 9; 1 )z( • 1 ) ) ' ab

exp(_(~/m)

~ab( ~ i ~ ab( ~lllexp(-( ~ / m ) ( £ ; ~ a (

a87

9;l)-~'a (9;l)fa)

(9)

( Under

the time-orderlng

operation

Td

all

operators

commute.

)

In eqn. (9) we introduced notation

~ a ( ~ i ) = exp( ~ l S a

) ~

(Z(~l)Vab

b

- z ( ~ l ) V a b ) ~ab( ~ I) ,

,

C a ( ~ i ) : e x p ( - ~ l S a ) ~ ~ba( ~ i ) b Hence, In the l i m i t m-K~, Z( ~ ) has been represented as a double Gaussian average over the fluctations af the complex fields z( ~ ), Z( q; ) and the Grassmanlan

fields

~ ab ( ~ ) and

~ ab ( I[ ) :

r

Tr / exp(-~8 f+f ),~ D(Z.Z; ~ *

Z(.~) = n

L

k

+

a a a

~ abt~ab ) dl;

exp

~

ab

o] o o,o ]

(ZZ +

~ )exp [ - f '

( f a+ ~

a

- C

"1;

,

(lO)

a 0

where m

D(Z,Z; ~ *, ~ ) : D(Z,

-I

n R d{ab( I : I ab

Z) lim m-~

3. Super-Well Group and Novel,Path Integral Structure The Dyson-Exponent Schrodinger-llke

in the right side of (I0)

satisfies

the

equation

(11)

d ~ dexp[,., ] = ~ { ~ T dexp[...] d~

here the Hamiltonian tors fa" fa

with

Dyson-exponent

~ (~ )

is

a linear combination

tlme-dependent

violates

Grassmanlan

hermlcity,

but

it

of the opera-

coefficients, is

restored

integration over both Grassmanian and complex variables.) to present a factorized form of exponent entangled"

solution of eqn.(ll)

el. Observe that the set

In

(Thls after order

in eqn.(ll) we seek a "dis-

as:

a

a"

fa' fa (a=l ..... N) plus

SaB

the

unit

operator

generate

the super-Weyl

the intermediate

Group SW(N) wnlch

Hanlltonlan

~

) of

is the dynamical

eqn.(ll).

(12) into (ll) we get a system of equatlons a and

~a

Now

group

of

substituting

for the coefflclents

~,

These can be solved to glve

"

~a( ~ ) =I Ca c ~ ') d ~ "

.

~a( ~ , = - f

Ca( ~ ", d ~"

0

,

0

X(~) = ~ I I

eC~'-~")¢a(~'lCa

C~'') d ~ ' d ~ "

,

00 e ( ~ ) is

Where

e(0)

the usual s t e p - f u n c t i o n with the boundary

condition

=o.

In order to s i m p l i f y the expression duce the system of fermlonlc coherent

• .

el,.

"+ ,eN > = e x p C - e ifi).

. .

for

states

Z( p ), let us intro(FCS)

[iO]

e x p ( - e N f N" + ) ~ 0 .....

:

O.

~ 0 ..... O> is the ForM-vacuum vector) wlth the scalar product

(here

< e 1. . . . . eN [ e l . . . . . e N > = exp( Note that our d e f i n l t l o n Ref. lO .

of

the

e[e

FCS

Is

For the trace of any operator

Tr ~ = I < e i . . . . . e N i

1 +

.

.

+ eNe

.

sllghtly R(f+,

).

different

f) we have

from

:

~ l e x. . . . . e N > e ~ p c e l e i . . . . ÷ e # N ) d e i d e i . . . deNd8

Moreover,

(13)

as in the case of Glauber coherent

states

It

Is

N

(II)

easy

to

Prove that N

sI

.+.

N exp( Safaf a)l e l . . . . . eN > = I a=l Using the trace formula

(14),

(13) and (15),

the Integrals

8N,

e N*

are

Gausslan,

and

we can calculate

over the external

sN

e 1. . . . .

e

(iS]

eN >

computing the matrix element

"evolutlon" operator between FCS, formulas

e

taming

into

of

the

account

the

the trace exactly,

Grassmanlan

variables

e

and get the following path Integral

389

because

e I, 1 ..... representa-

tlon for the partition function:

k

0

ab

O0 a

O0

where ZO( ~ . ~ ) = ~ [l + e x p ( - ~ S a ( k ))] is the partition of the system of a nonlnteractlng fermlonlc oscillators;

function and na=

[l+exp(- ~ S a ( ~ ) ) ] - l l s

fermlons

Of kind "a".

the average n u m b e r of nonlnteractlng

Examining eqn.(16) we may

the complex fields

z( ~ ), z( • ).

;D(z,z)exp[- £f z( ~ ) ~ ( %-%

first

integrate

over

all

This integrals have the structure

")z( % ")d'~d ~; "+~(Jz+J~-)d'~ ].

O0

0 -p

where

"the currents"

tlons of

~ ab

o(k, ~)=

ab

and

J(k , • )

and

J(k , '~ ) are blllnear combina-

~ ab

[8(~-~

l)-na(k

)]e

ab (

ca ( ~ l ) d ~

i,

0

:-G

~[

c ab

(

8a(k )( ~ - ~ i ) l ) - n a ( ~ )]e

.

~ ab ( ~ ) ~ ca ( ~ l ) d ~

i"

0

Calculating those integral, we obtain a path integral nlan variables only, namely:

Z( ~ ) : D Zo( ~ , ~ ) ; D ( ~ *, ~ )exp[~

over

Grassma-

ab ( ~ )~ ( ~ - ~ ")~ ab ( ~ ' ) d ~ d ~ O0

O0

It is essential to stress that in the derivation of this formula do not make any approximation,

once we were glven

H. Since the path

integral (17) Is not Gaussian we cannot compute it exactly,

ago

we

but

it

"]

Is useful

for p e r t u r b a t i o n

Utilize

a power

permute

the-.path and

integrals method. matic

series

(for each

treatments.

expansion

"tlme term

For

this

purpose

of the last exponent

" integrations,

In series)

and

using

in

thus

the

consider

integration

In the functional

integral

(16).

the general

The

case.

field

integrations

where

~

integral

Thls

over

the

using

diagramelsewhere.

Grassmanlan

is a more d i f f i c u l t

which

these

function

and these calculations will De reported

Alternatively

involves

can and,

obtain

generating

to formulate this calculations

It is natural

techniques,

we

(17)

only

fields

problem

the

in

Grassman

we may write as

O0 is(in general)

components elements

~aband

depends

structure

~:I

of which M

I

zero trace.

expression

of the matrix

of the g r o u p Computed

Is unity matrix,

G.

Calculation

formal

exactly

n[(-l)n/n]

Is c o n v e n i e n t

multiplied

In the simple because

trMn

4.

a simple

matrix

K

matrix

has

a

elements

the

integral

and (18)

O0

z ~ z d ~ d ~ "1

we observe

Z 0 is equal (small

number

of the power

(19)

that

the deter-

to the

character

N)

detK

series

can

det~

function of the

representation

1iN

expansion

behavlour of the system

In the

be = It

in

the

for

the

llmlt N " ~

calculations). example

As a brief consider

whose

for n > nma x (some finite number).

to use the p a r t i t i o n

Slmple

(8),

by

cases

last form (19) for the d e r i v a t i o n

study of the t h e r m o d y n a m i c

the

of

)exp[-2

the members

m a y vanish

(the s a d d l e - p o l n t

matrix

the compo-

relation

(19) wlth eqn. ~

with

~ ( ~ - ~ ') on the maln diagonal,

D(~,Z)Zo( ~ , ~ ) d e t ~ ( z . z ; k

Comparing mlnant

N 2 x N 2 functional

~-functlons

with

column

z. It Is e a s y to show that the

where

us to the f o l l o w i n g

Z( ~ )= N I k

N 2- dimensional

is the

on zand

+M,

are equal

is a matrix

leads

K

the

and c oncluslons

illustration

o£ an applications

(but non-trlvlal)

391

"Lipkln-llke"

of model

our

method,

H = 2~

where

"2 4%Jx

Jz-

Jx' Jy= i[Jx,

describe

a system

states

[6].

space:

I)

and

Hamiltonlan

-2 8 J z + 8 ~ introduce

of N types

If N >> 1 there

<Jx>Jx.

be c a l c u l a t e d applying

Jy]' Jz generate

~b < 8 / 2 N ,

mean[fleld

for

N8 2 ~/2

two types

full

Here <Jx >

( ~

to the m e a n - f l e l d and d e n s i t y

theory

for

can

approach,

~ a(a:l,2)

and

I):

Thls

] , ...]N

(21)

coincides

with

this

to c o m p u t i n g model

the SU(8)

and the A n d e r s o n

using

the double super-Weyl

a new factorized

path

Z of m a n y - f e r m l o n and systematic

such a p p r o a c h

the

-

usual

and unified model

the

for

corrections

several

model

superconductivity wlll

be

discussed

Is

S-H

trick

and

group

SW(N)

developed

integral

systems

which

corrections

form of the Grand offers

to the

effective

at

disentangling

for

we

Partition

the p o s s i b i l i t y

mean-fleld

least

here

theory.

of We

certain

model

IREX

Board

Hamiltonlans.

This work was AVG)

which

[ll].

four-fermlon

(to

-~).

The

in 2) Hm_ f = our

~ a'

[6].

E = -(d/d ~ ) In Z( ~ ), E = - N k / 2

model,

for the d y n a m i c a l

alternate

different

in ref.

use

can

In p a r a m e t e r

parameter

To

variables

of our a p p r o a c h

- BCS SU(2)

Summarizing:

believe

procedure.

energy

wave model,

elsewhere

obtained

~ O is the order

In region

O; N

two

regimes

as d e s c r i b e d

model

result.

Hamiltonlans

Function

occupying

)l~[~+~.13/(t,chS[3

system

group, This

Hm_ f -- - 2 e J z, while

of G r a s s m a n i a n

The a p p l i c a t i o n

method

~>8/2N,

In I) is:

we flnd

+ O( ~ 2)

pertubatlve

fermlons

via a variational

eqn.(17),

the

o£

SU(2)

are two s i g n i f i c a n t

2)

z( 13 ) = [2(1,chSp and

(20)

supported

In part

and P S C - B G E - F R A P - C U N Y

Fund.

392

by a grant

from

References:

i. R.L.

Stratonovich:

Translated 2. J Hubbard: 3. F.W.

Phys.

Dokl.

2

416

Rev.

Lett.

3

77

Phys.

Path

Integrals

and

Solid

J.T.

Devreese

5. Dal,

Xlan-xl

(USA) 8. J.L. 80 9. A.I. I0.

ii. A.V.

and their

(Plenum

ed.

Press,

115 (1957)

57

(1975).

3

by

522

(1962);

Phys.

B. Laurltzen

and

in Quantum,

G.J.

an

London,

Rev. G.

2B

in

Statistical,

Papadopoulos

New York and T1ng:

:

(1958).

Applications

and Chln-Sin

309

and

16C

SSSR

(1959).

Phys.

Physics,

G. Bertsch,

Birman

1978).

5243

Puddu:

and (1984).

Ann.

Phys.

(1988).

P. Saracco: and

A.I.

Nuovo Cimento

Solomon:

Prog.

lID

Theor.

303

Phys.

(1989). (Kyoto),Suppl.

(198~).

Solomon

J.W.F.

Reports J. Math.

State

183

62

Nauk.

Soy.

Wlegel:

7. R. Cenni

Akad.

Phys.

4. B. Muhlschlegel:

6. P. Arve,

Dokl.

Valle:

Gorokhov

and

J.L.

J. Math. and

J.L.

Birman: Phys. B1rman

393

J. Math. 22

1521

Phys.

28

(1981).

(In preparation).

1526

(1987).

NONRELATIVISTIC

LIMIT OF SUPERFIELD

Jos4 A. de Azc£rraga §t

and

THEORIES*

Dami~n Ginestar §$

§Departamento de Fisica Te6rica, Universidad de Valencia, 46100-Burjasot (Valencia) Spain. t IFIC (CSIC), Centro Mixto Universidad de Valencia-CSIC $ Departamento de Matem~.tica Aplicada (EUITI), Universidad Polit~cnica de Valencia, 46071-Valencia, Spain.

1

Introduction A crucial point in any discussion of 'non-relativistic' (NI~) quantum theories (for a

review, see [1]) is their relation to their parent relativistic ones, i.e., the e --. oo limit. This is a rather subtle transition, as exemplified by the well known fact that the KleinGordon and the Dirac lagrangians do not possess a limit unless the corresponding fields are previously redefined. The contraction process only relates Lie groups with the same number of parameters; it obviously cannot relate the Poincar4 group P to the elevenparameter U(1)-extended Galilei group [2] ~ which is the relevant group in NK quantum mechanics. Thus [3] the c --* oo limit has to transform a U(1)-extension 2-coboundary of P into a non trivial 2-cocycle on the Galilei group ~. The close analysis of this fact is sufficient (and, in fact, necessary) to explain the subtleties of the non-relativistic limit. Supersymmetric theories (see, e.g., [4]) have occasionally been considered in a Galilean framework [5]. Here, we wish to sketch the procedure which provides the Galilean limit of the superfield formulation of the massive Wess-Zumino (WZ) model.

This requires

the use of the notion of pseudoeztension groups [3] and will expose the role of the mass, which necessarily appears in the expansions of Galilean chiral superfields in order that the component fields have canonical dimensions.

*Partially supported by the CICYT (Spain) research grant AEN-90-0039

394

The simplest supersymmetric models (see, e.g. [4]) are based on the N=I superPoincar6 group ST" whose group law is given by x"~' = x'~' + (A'x) ~' + iA~'7~S(A')8

=_ x '~ + (A'x) ~' + A~'(b', b)

(1.1) 8 "~ = 0 'a + S(A')apOP

A(A")~ = A(A')~A(A).P,,

,

For A = 1 the translation-valued ~'(O', 0) is [6] the 2-cocycle [2] defining the extension aTr (parameters x, O) of the group of the supertranslations (parameters O) by the ordinary translations Tr; non-equivalent 2-cocycles ~ can be obtained by varying ;~. The N = I superPoincard group ST" (as T" itself) cannot be extended by a central charge. As for T', however, it is possible to define a U(1) 2-coboundary on S P with a non-trivial contraction limit. If ~cob is generated by the one-cochain on S$ ° 6(g) = - m c x °, the pseudoextension S~o of ST" by U(1) is defined by (1.1) plus

¢ " = ~*"ex~p X¢¢ob,'" {cob =

-me

{h'.° ~ +

i;~#,.rOS(A,)e

_ xo} ,

(1.2)

where ~ _-__exp-~a is the central variable. The natural way of introducing a U(1)-extended (quantum [7]) superGalilei group ~-~, which is to the N=I ST' group

(1.1) what

~ is to T',

is to define S ~ as the NlZ limit o f ~ / ( 1 . 1 ) , (1.2)]. The existence of such a limit requires that [A] = c-Z, n > 1, and its non triviality that ~ = 1/c : the existence of a non-trivial N R limit of tlle N = I superPoincar4 group J~es [8] = I~T-1/2.

The complete group law of ~dC is obtained by performing the c --* oo limit of (1.1), (1.2) with ~ = 1/c. Using the Weyl realisation for the four-spinor 8, it is given by t" = t' + t,

~" = ~' + fft + t ~ x ,

ff' = ~' + l ~ v ,

Y ' = 8' + D1/2(R')~?

(1.3)

~ " = ~'¢exp ( - i ~ {{g2t + ~tRTx - i [O'tD(R')O + O'-gD(R')*O*] }) in terms of the Pauli two-spinor contents of the Majorana spinor. We note that (1.3) does not include the supergroup underlying Witten's supersymmetric quantum mechanics [8], Which hence cannot appear as the Galilean limit of N=I supersymmetry. The same can be said of the 15-parameter group [9] which includes G and Witten's group as subgroups and which contains an (odd) vector translation.

395

To conclude this section, let us mention that for N=2 a central charge z may arise because a Dirac (not self-conjugate) spinor, zh allows us to define a non-trivial 2-cocycle ~(~/',~) = -~z(~/'~?- ~ f ) . Because 77 now contains two unrelated two-spinors, more possibilities to derive a Nit-limit arise, but still the coboundary generated by mcx ° is needed together with the previous cocycle and the condition m = z to obtain a limit. This situation recalls the behaviour of certain massive superparticle models [10] where the mass appears both as a central charge accompanying the Wess-Zumino term in the action (which is associated with the above non-trivial cocycle), and also as a constant multiplying the kinetic term. This last one defines the bosonic constraint p~ = rn2c 2, so that m is related to a representation of the original Poincard symmetry. In principle, it could be unrelated to the central charge z which characterizes the eztended supergroup ( N = 2 superPoincard with a central charge). However, when they are equal the theory presents the appropriate reduction of degrees of freedom; the saturation of the bound z < m identifies two parameters of different group origin into one single one, the superparticle mass. We shall restrict ourselves here to the simplest N = I 57) case.

2

T h e G a l i l e a n limit o f t h e W e s s - Z u m i n o m o d e l To incorporate the full Dirac spinor of the basic matter supermultiplet, two N = I chiral

scalar superfields q ± are needed. Their component expansion and the form of the WZ a~tion is given, e.g., in [4]. Since the Berezin integral measure behaves as a derivative, [dS] = L - 1 T 1/2. Thus, [¢±] = M1/2T -1/2 and accordingly we find for their (independent) components [A+] = M l l 2 T -112 (as for a KG field) and [¢±] = Ml12L-112T-1/2 (as for a Dirac field). In natural units, of course, [#±] = [A±] = M; [¢±] = M s/2. As in the Klein Gordon and Dirac cases, we cannot take the c ~ oo in the WZ lagrangian (or in the corresponding Euler-Lagrange equations). We may, however, introduce the redefinitions dictated by the fact that 6(g) = mcx ° ~+(x,O,~) = exp

-i

~ + ( x , 0 , 6 ) , ¢*_(x,O,~) = exp

396

- i ~

(z,O,0). (2.1)

In terms of the new ~4-, the spacetime Lagrangian density is given by

=

".~,

~*~_] __+~c2

-.-.

Before looking for the Nit limit of (2.2), we notice that because the ~± were chiral superfields, the redefinition (2.1) implies that ~4- now satisfy

( ~ + -~0~(..).a~ + ~e.(.o).a) ~+ = 0 ,

(2.a)

with - m for (~_. Eqs. (2.3) are, in fact, chirality conditions: they are defined by the new eovaria~t derivatives given by the left invariant vector fields of the 8:P group [(1.1), (1.2)] for A = 1/c and A = I ; the expression (2.3) makes use of the dimensional reduction of ~4provided by the U(1)-equivariance condition E ~ . ( x , 0, 0, () = =[:i~_(x, O, 0, ¢'), where ~ is the central generator [11] ). The expansion of ~)+ given by

~+(~,e,~)

= exp-~[eo(.~a~ -

2 o~ i-~,°La~aJ}+(=,e) = A+ + ~-(-~e ¢+~

+~oeoeo-~(~o)o~¢g + ~ooejaOeDA+

+

i~o~'eAa~aOoA + + ~o~oooaOaa + ; (2.4)

that of ~_ is obtained by substituting ( - m ) for m above.The natural appearance of m (and hence of h) in the expansion just reflects the fact that the new chirality condition (2.3) was defined on superspace extendedby a central variable and that the U(1) extension has a quantum character. Using (2.2), and eliminating as usual the auxiliary fields F4- = ( - ~ A ~ : ) , F:~, we get that the component contents of the WZ lagrangian adapted to the Nit limit is given by £.wz =

-O#A~O~'A+ + i ~

(A~. .OoA+ )

- O~,A_O~A _• + ~--~.~c A-OoA*_

+i (~7"0~,~ + -~-¢"*c - (1 - i7 °) ¢)

;

(2.5)

Where ¢ w = (¢+a,¢_~). ~ w z is, as it should, the sum of two KG (for the fields A+ and A*_) and one Dirac lagrangians already adapted to the NR limit. If we call ~+,~-,X1,X2

397

the Galilean fields associated with A+,A*_ and with the spinor components of the (DiracPauli) spinor CDP, the Galilean limit Z~c of ~wz is given, after eliminating X2, by

(2.6) We shall mention here that (2.6) may be directly obtained in terms of Galilea~l chiral superfields by using the Gaiilean covariant derivatives; these are given by the left invariant vector fields associated with (1.3) or, equivalently, by the NI~ limit of (2.3). We conclude by saying that the same procedure can been applied to the NR limit of supersymmetric QED; in this way one directly incorporates the contribution of the intrinsic magnetic moment of a Galilean spin 1/2 particle [12], but we shall omit the details.

References [1] J.-M. L~vy-Leblond, in Group Theory and its Applications, vol II, E.M. Loebl Ed., Academic Press (1971), p.221. [2] V. Bargmann, Ann. Math. 59, 1 (1954). [3] V. Aldaya and J.A. de Azc£rraga:, Int. J. Theor. Phys., 24, 141 (1985); see also ]~. Saletan, J. Math. Phys. 2, 1 (1961). [4] J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton Press (1983). [5] P~. Puzaiowski, Acta Phys. Aust. 50, 45 (1978); G. de Franceschi and F. Palumbo, Nucl. Phys. B162, 478 (1980); S. Ferrara and F. Palumbo, Galilean approximation of massIess supersymmetric theories, in the 2 nd Europhysics Conference on the Unification of the Fundamental Interactions, Erlce (1981), CEI~N preprint Th 3225 (1982). [6] V. Aldaya and J. A. de Azc£rraga, J. Math. Phys. 26, 1818 (1985). [7] V. Aldaya and J.A. de Azckrraga, J. Phys. A16, 2633 (1983). [8] E. Witten, Nucl. Phys. B188, 513 (1981). [9] J.A. de Azc£rraga and L. Lusanna, unpublished (1987). [10] J. A. de Azc£rraga and J. Lukierski, Phys Lett. Bl13, 170 (1982) [11] See in this context J. Scherk, H.J- Schwarz. Nucl. Phys. 153. 61 (1979); see also L..~lvarez-Gaum~ and D. J. Freedman, Commun. Math Phys. 91. 87 (1981). This condition has also an easy interpretation in geometric quantisation schemes. [12] J.-M. L~vy-Leblond, Commun. Math. Phys. 6, 286 (1967).

398

REMARKS ON THE SUPERSYMMETRIC WKB QUANTIZATION FORMULA A. Inomata and A. Suparml Department of Physics, State Unlversity of New York at Albany Albany, New York 12222, USA

S. Kurth Physikallsches Instltut der Unlverslt~t Wth'zburg 8700 WGrzbur8, FRG

The supersymmetrlc ~

quantlzatlon formula proposed by Comtet,

Bandrauk and Campbell Is shown

to be seen

as a supersymmetrIc

counterpart of the WKB quantizatlon formula.

I. Introduction In recent years,

supersymmetrlc quantum mechanics [1] has attracted

considerable attention.

In particular,

the supersymmetrlc WKB formula

proposed by Comtet, Bandrauk and Campbell [2] is thing worth attending to.

a sl/rprlse

a/%d some-

The formula of Comtet, Bandrauk and Campbell

(the CBC formula in short)

provides

exact enerEy quantlzatlon

for

a

variety of nonrelatlvlstlc systems [2,3]. The standard semlclasslcal WKB quantization formula, x

b

4

f x { 2 M [ E - V ( x ) ] } t / 2 dx = (n+~)=5 where E = V(Xa ) = V(Xb)'

i s known

(n=O,1,2

)

(1)

to y i e l d exact energy s p e c t r a f o r a

c e r t a i n c l a s s o f systems p r o v i d e d t h a t ad hoc m o d i f i c a t i o n s the p o t e n t i a l

term [4].

For instance,

hydrogen atom can be obtained

the exact energy spectrum of the

from the formula (1)

Langer replacement, ~(~+I) + (~+~)z

are made to

i s applled.

if

the well-known

In contrast,

the CBC

formula [2], b

f~

{2M[E-¢2(x)]} I/2 dx = n~h a

399

(n=0,1,2 .... )

%

(2)

where E=E+Eo, reproduce

VCx)-Eo=¢2(x)-(h2/2Vr~)d¢/dx ,

with no ad hoc modification

been obtained

(2) works magically well. blguousl y verified.

In fact,

Furthermore,

E=¢2(xa)=#2(xb) ,

can

all the exact results that have

from the WEB formula (I)

term. Despite its surprising success,

and

modified

with the Langer-llke

it is unclear why the CBC formula

the CBC formula has never been ~ -

there is no qualitative explanation of

the observed exactness of the formula C2). It is certainly important to understand formula.

the real nature of the CBC formula

in relation

to the

The aim of this paper is not to verify the CBC formula

report our observations

but to

which might shed light on the understanding of

the CBC formula. 2. Transformations

of Vsu-iables

We start with the observed fact

that

the CBC formula (2)

provides

exact energy spectra at least for the following three systems; a) the one-dlmensional

harmonic oscillator,

b) the three-dlmensional

harmonic oscillator,

and

c) the PSschl-Teller oscillator. For convenience,

let us call these three the elementary systems.

Our first observation

is

that

any of the examples

that have been

reported exactly soluble via the CBC formula is reducible to one of the three elementary systems by a change of variable. if the variable x is changed into y=fCx),

The CBC action (2),

is written as

W = ~vb {2HIE - # a C x ) ] / [ f ' C x ) ] Z } t / a

dy.

C3)

a

If

the s u p e r p o t e n t i a l

i s a c o n s t a n t and

# ( x ) has the form,

CaCx)/f'aCx)=C + Z ( Y ) ,

Z(Y) i s a w e l l - b e h a v e d f u n c t i o n o f y,

then

where C t h e CBC

formula can be put into the form W = ~vb {2M[~ - 2(y)]}I/2 dy,

(4)

a

where

~ =C-C,

~2(y)=xCy)+C-E/f'z(y)

What we are claiming is that

and

C is an adjustable constant.

by an appropriate change of variable

transformed squared superpotential

~2Cy) of (4)

can be put into one of

the following forms;

Ca)

Z C y ) = ~ y2

C-= ~ y < ®)

(b)

ZCy)

= ~ y2 + ~y-Z + •

(0 ~ y < =)

(c)

ZCy)

= ~C~-I )sec 2 Cay)+~C~-l)csc 2 Cay).

CO ~ y ~ ~la).

400

the

For instance,

the modified Morse potential,

V(x)=Ae -zax - Be -ax, has

the superpotentlal of the form, #2(x)=Ae-2ax - [B-ah(A/2M)I/2]e -ax - Eo, where E ° is the ground state energy e-aX = y2 (0 s y < ~),

of the Morse system.

then we have

If

we

let

~ = C + 4a'2[B - ah(2m) "I/2]

and

~z=(4A/a2)y2 - (4E/a2)y -2 + C. Thus the CBC action for the Morse system is transformed into the CBC action

for the radial harmonic oscillator.

Since the solution of the radial harmonic oscillator is already the energy spectrum of the Morse system is determined the constants

involved.

Similarly,

the charged particle

the Dirac-Coulomb problem

among others

to the action of the radial harmonic oscillator. Rosen-Morse potential, potential

limit),

the

Hulth~n potential

(including

the Kepler problem in a hypersphere

these examples are

soluble

in a uniform

can be reduced

The actions

are reducible to that of the PSschl-Teller potential. to mention that

by comparison of

the actions for the Kepler problem

(in two as well as three dimensions), magnetic field,

and

for

the

the

Yukawa

many others

It is instructive

not only

by the standard

Schr~dinger methods but also by the algebraic methods [S] and integral methods

known,

the path

[6].

3. Conversion of ~

into CBC

Since the three elementary systems can be solved exactly and many of the Integrable examples are

reducible

to the elementary systems,

exactness of the CBC formula for a wide class of examples a mystery.

For the one-dlmensional harmonic oscillator,

is

no longer

in particular,

the WKB formula can be reduced to the form of the CBC formula. cation of this is rather simple and omitted here. clarified is w h y

the CBC formula is exact

the

What remains

Verifito

be

for the last two elementary

systems. Our next observation provides no solution for the remainlnE mystery, but presents

a curious fact

that the WKB formula with the LanEer-llke

modification can be converted to the CBC formula by (a) the replacement,

n + n - I/2, in the case of the one-dimenslonal

harmonic oscillator, (b) the combined replacements,

n + n - 1/2

and

case of the radial harmonic oscillator,

and

(c) the replacement, meters,

say

~+

~ + ~ + I/2, in the

n ÷ n - I/2, combined with the changes of para~ + I/2

and

PSschl-Teller oscillator. 401

I + A + I/2,

in the case of the

Let us examine

consequences of the above replacements

by rewriting

the WKB formula in the form, t

~[ [ ( ~ 2 ) ~ where Veff(x)

2 -

v°~f(x)

+ E.]dt = (~ + I/2)~h,

is the effective potential modified with the Langer-llke

term. Upon replacement,

the equation (S) changes into

t

[~ [c~2)i" Ja where En = En_I/2

and

-

%.(~) +

~o]dt

= n.h,

C8)

Veff(^x) is the transformed effective potential

The time integral over a half period

integral

(s)

can be

converted

into the space

o v e r t h e r a n g e (Xa, Xb) between t h e two t u r n i n g p o i n t s , N

~b

(7)

{2M[~n_Oeff(x ) ]}t/2 dx = n~h. a ^

$eff ( X ) = • e ( X )

+ ~0'

N

^

E =E +E

Now we shall show that for the three elementary systems,

n

n

0

and

that

SO

En-Veff(x)=En

- @2(x),

(8)

with which (7) coincides with the CBC formula (2). (a) The One-Dimensional Harmonic Oscillator: In this case, Vefr(X)=V(x)=@2(x)=M~2/2 and En=(n+I/2)h~. n

=E

n-I/2

=r~u and E 0 =0.

Consequently,

we have E =E n

n

By n ÷ n

- 1/2,

and VeffCx)=@2(X),

arrivln E at the CBC formula for the one-dimenslonal oscillator. (b) The Radial Harmonic Oscillator.: The radial potential of the harmonic oscillator in three dimensions

is

V(r)=(M~2/2)P 2 + ~(~+l)h2/(2Mr2).

modification,

it

The superpotential

cor-

becomes

With the

Langer

Veff(r)=(M~2/2)r2 + (~+I/2)2h2/(2Mp2).

responding to the radial potential V(r) has the form (£+l)2hm/(2Mr 2) - (~+l)h~. E =(2n+~+3/2)h~, n

The energy spectrum for this oscillator

and E =2nh~. n

#2(r)=(M~2/2)r2 +

In this case, the replacement n ÷ n - I/2

alone does not lead us to any significant result,

but

if we associate

with the above replacement another replacement, ~ ÷ ~+I/2, n=(2n + ~ + l)h~,

is

Eo=(~ + l)h~,

and hence

En = ~n -~0"

$eff(r) = (M~2/2)r 2 + (~+l)2h2/(2Mr 2) = @2(r)+E O. desired relation (8).

402

Thus,

then we get Also we have we obtain the

( c ) The P ~ s c h l - T e l l e r

Oscillator:

The P~schl-Teller potential, V(x) = Vo{~;(~-l)csc2(ax) + X(A-l)sec2(ax)} where V =~2h2/(2M), ~:>I, k>l, and O-<x- Iql+l the multiplicity is 2(p2-q2) and for the E = 0 ground state it is 2lql. For the scattered motions the symmetry is o(3,1)@o(3) which we used 4 to derive, following Zwanziger 5, the S - matrix. The situation is reminding to the one encountered by McIntosh and Cisneros 6, and by Zwanziger 5 who studied a spin 0 particle described by H o and found that it has an o(4) dynamical symmetry and its spectrum is (1.5) except for the O-energy ground state and a degeneracy (p2_q2), half of the one in the spinning case. This raises the questions: (i) What is the physical interpretation of the 'dyon' system ? (ii) How to explain the origin and specific form of K and f~ ? We show below that H 1 and H o are the component Hamihonians of a supersyrametric system, namely of a massless Dirac particle in the long-range field of a self-dual monopole. The conserved quantities K and I'~ are essentially the images of those vectors conserved for H o. _2, THE MIC-ZWANZIGER SYSTEM About 20 years ago, Mclntosh and Cisneros 6, and Zwanziger 5 suggested to consider a spin 0 system in a combined monopole + scalar potential field, described by 1 2-q2 +(2-~+ H° = 2"~ r

q2 ) 2r 2 "

(2,1)

In addition to the orbital angular momentum, Lo = r x n - @ ,

(2.2)

this system also has a conserved 'Runge-Lenz' vector, 1

Ko= ~ (~xLo-Lox-i~)-q2~

.

405

(2.3)

L o and K o generate an 0(4) dynamical symmetry for bound motions and an o(3,1) for the scattered motions. Applied to Ho, the Pauli method 3 yields 5-7 once more the bound state spectrum. 3, SUSY IN THE FIELD OF A SELF-DUAL MONOPQLE Let us now consider a self-dual SU(2) monopole 9 i.e. a static, magnetic (A o = 0) solution (O, A) of the

Bogomolny equation DO

= B (Bi = 1 ~ ijk Fjk) which satisfies the

boundary condition IOI ---> 1 as r ~ -0. Setting A 4 = O, a 3 dimensional Yang-Mills-Higgs field can also be viewed as a pure Yang-Mills field A~t on the (1+3) dimensional euclidean 1

space whereas the Bogomolny equation corresponds to the self - duality, Fij = ~ e ijkl Fkl. The asymptotic values of the Higgs field define a mapping from the 'sphere at infinity' ~ 2 into the orbit SU(2)/U(1) = ~ 2 and thus a winding number m = [O] E 7r~2(~2) z Z . m is called the

topological charge.

Let us now study the Dirac operator ~b = ~D~t,

o "Tk, Q It

=

•

12 + i c . ~

is straightforward to see that 2QtQ

o.(DO

- B)). For a

o,.,o.)

=

self-dual monopole

QtQ= 1(O9-- D2)-¢r.B = H 1

(02

0 - D 2 - o.(DO

+ B))

and 2QQt

= (02

- D 2 -

DO and thus

B =

and

Q Q t = 1 ( O 2 - D 2) = H o"

(3.2)

1

H 1 and H o describe spin ~ particles with anomalous gyromagnetic ratios g = 4 and g = 0, respectively. The Hamiltonian

.o

(3.3)

acts on ff~ z C4®L2 which we decompose into upper and lower components, which are the z¢1 eigenspaces of the 'chirality operator' F = i ~ =

1

"

The unitary operators U

406

O = Q ~1

U.1 = U* = ~ 1

and

Q*.

(3.5)

act between the upper and lower components. They intertwine Ho and H1, UtHoU = H1 and UH1Ut = H o . If lit 1 is an eigenvector of H l then ~o = Q ~ I is an eigenvector of H o with the same eigenvalue as long as Q~I # 0. On the other hand,

Q gl = 0

¢~

Ib

= O.

(3.6)

The number of solutions is the Atiyah - Singer (Witten) index. It only depends on the topology and not the gauge field. For a self - dual monopole the A-S index has been calculated 11 : for isospin i"I it is m, the magnetic charge. If K o is a constant of the motion for the H o dynamics, [H o, Ko] = 0, then K 1 = UtKoU

(3.7)

is conserved for H1, [HI, Ki] = 0. For example, the lower component has g = 0 and its spin 1

So = ~-o

(3.8)

is hence trivially conserved. A short calculation shows that U t o U = - 2H and thus QtSoQ

-_ 1-L.ni.e. -

2H1

Sl =

-2Yl( I~ ( ~ 2

_ ¢2)

(~ + ~ (X x (Y) - (O'.~)/l;

)

(3.9)

is conserved for H I . Notice that this result is true in the entire space. 4. THE BPS-1 e A S E

The simplest solutionof the Bogomolny eqn. (3.1)is the one found by Prasad and Sommerfield 9, -7

th r -

,

= ~=k

407

~-

1 -

.

(4.1)

Due to manifest spherical symmetry the total angular momentum, J in eqn. (1.2), is conserved for both components. Remarkably, U J U t = J. For large distances the fields behave as --->

x

- 7-

(1)

1

-

'

l

xa

"-> ~ aik ~ "

(4.2)

which represent an imbedded Dirac monopole of unit charge plus a long range scalar field. The electromagnetic properties can be defined for large r .In the BPS case the electric charge operator is xa

Qem = - Ta ~" • 1

(4.3) 1

For isospin ~, e = emin = ~ • The A-S index is thus 12 21ql. It is straightforward to show that, as r --->o% H1 ~

Hl(D'Hoker-Vinet),

Ho --->Ho(MIC-Zwanziger).

(4.4)

In this limit, the spin 0 Runge-Lenz vector K o is conserved for H o. Its super-partner K 1 = U ( K o 12)Ut is found to be K 1 = K o 12 + ~ x ~ +

° -

o - (o. B) r - q ~HI,

(4.5)

Notice that the Runge-Lenz vector K (1.3) of D'Hoker and Vinet is rather U(Ko 12 f2 qo/2)Ut = K1 - q 2-"~'1" By construction, L1 = U t L o U = J - Sl,

and

K1 q:2-ffl

(4.6) Ko

generate the same 0(4) symmetry 5-7 as Lo and ~ _ ~ ,

to which S 1 adds an extra 0(3).

This explains the D'Hoker-Vinet result. REFERENCES [1]

D'Hoker E and Vinet L 1985 Phys. Rev. I.~tt. ,~6., 1043; 1986 in Field Theory,

Quantum Gravity and Strings, Springer LNP 280, 156; D'Hoker E, Kostelecky V A and Vinet L 1986 in Dynamical Groups and Spectrum Generating Algebras, World Scientific: Singapore

408

[2] D'Hoker E and Vinet L 1986 Lett. Math. Phys. 12, 71; [3] Pauli W 1926 Z. Phys. 36, 33; Bargmann V 1936 Z. Phys. 99, 570; [4] Fehtr L Gy and Horv~thy P A 1988 Mod. Phys. Lett. A3, 1451; [5] Zwanziger D 1968 Phys. Rev. 176, 1480; the idea of calculating the S-matrix from the dynamical symmetry appears in Biedenharn L C and Brussard P J 1965 Coulomb Excitations, (Oxford University Press, N. Y. ) and has later applied to nuclear physics, see Alhassid Y, Gtirsey F and IacheUo F 1983 Ann. Phys. N.Y. 148, 353 [6] McIntosh H V and Cisneros A 1970 J. Math. Phys. 11, 896;

[7] Barut A O and Bomzin G L 1971 Joum. Math. Phys. 4, 141; Sch6nfeld J F 1980 Joum. Math. Phys. 21, 2528; Fehtr L 1986 J. Phys. A19, 1259; 1987 Proc. Sitfok Conf. Non-Perturbative methods in Quantum Field Theory, Horv~th, Palla, Patkts, (eds), World Scientific; [8] The introduction of the quantities x and and y are motivated by the treatment of the relativistic Kepler problem in Biedenham L C 1962 Phys. Rev. 126, 845; [9] On monopoles see, e.g., Goddard P and Olive D 1978 Rep. Prog. Phys. 41, 1357; [10] Jackiw R and Rebbi C 1976 Phys. Rev. DI3, 3398; Mottola E 1978 Phys. Lett. 79B. 242; Weinberg E 1979 Phys. Rev. D20. 936; Bais F A and Troost W 1981 Nucl. Phys. B178, 125; Din A M and Roy S M 1983 Phys. Lett. 12913, 201;

409

SUPERSYMMETRY

AND

ELECTRON

Semenov Institute

of

Spectroscopy

Troltsk First theory

, we

of

an

one

can

the ~2 3

pPoblem

anEular

introduce

and

of

be of

of

operator

findinE

Jz m a y

eiEenfuctlon

some

momentum

the

the

and

Jz

[I]

by

p~

=

One

quent Then

-i~0~-

can

.

P8

simplify

that

3" z Furthem

=

S

fop

p~

the K"

*

simplification

-

1 sin8

J

UIP~

reached

K*=(sinS)I/ZK"(sin8) -I/z

.

K

and

~

and

,

of

The

obey

K

Oz(Pe

-

the

out

common

equation

=k~

. two

conse

•

becomes i~

czce)

.

tPansfommation

~*=(slne)*/z~ ''

the

cos 2 + i zsin~

T

and

uslnE

case,

therefore.

3z"

inte¢em

caPmyinE

z

this

+ o=)p e + ~ ) ~ is

iOssln

Pauli's

eiEenvalues

the

m

.

the

and

coomdinates)

expressions

opePators

is

fop

and

cos

+ ~

and

ayetzesin~

-iho~-

these

transfoPmations expmessions

=

Sciences

. USSR

: K=(O,~)

sphemical

-(oot=eoos~ + whePe

, i~2092

eiEenfunctlons

(in

of

results concerninE 1 spin ~ particle. In

a

K

substituted K

Academy

Re¢ion

briefly

MOMENTUM

V.V.

, USSR

, Moscow

remind

ANGULAR

Finally.

:

we

have

(la)

K

$

~$

Equation for

the

~

(la)

funatlons

(-

1

a,p~

~HO

is

easil~

f(e)

and

+

0

,PO)~ $

inteE~ated

= ~k~ ~

afte~

(lb) the

Eq.

followlnE

:

which

(ib)

b(e)

~8 = exp(i(m + ~)~)I Ib(8) f(8) •

d

The particular

sine-(m

ae

(

(

1

-

system , it

d ae (2) was

becomes

1

+ ~))f(e)

1

as

= kb(e)

(2)

1

sinS-(m * ~))b(8) = ke(e) for studied

the in

spherical paper

410

spinors [2]

is

However,

well

known,

another

way

in of

flndlnE (2)

the

as

metric the of

solutions

the

the

function

fore

sector

this

the

case

methods

malisable

and

introduce

the new

SSQM

components

to

. However

of

Unfomtunately

easily

, this

used

find

may

be

supersymmetry

sin~

one

can

. There

-

the

-

nor

overcome

by

So

us

,

let

i F(e)

, which

-

to

the

sector

, as

broken

difficulty

broken

supersym

represent

"bosonic"

spontaneously

be

Eqs.

8

cos~

~8-

the

to

the

treat

:

8

(

may

f(~)

represent

is not

may

. one

consider

function

spontaneously

functions

f(e)

can

b(e)

weve

solutions

"~ecoverinE"

, we

supersymmetry of

. namely

between

So

supersymmetric

, in

possible

transformations

wave

"fermionic"

see

is

(m

+

)etze)F(8)

=

(k

÷

obey

the

m)B(8)

equations

:

,

(3)

d (

-

A~aln

, we

nents

of

charEes

Q

1

~8-

-

(m

consider

the

÷

Q+

Eqs.

have

m

=

(3)

supersymmetric

Q-and

(k

~)

as

-

the

wave

the

form

=

A*

0

0

relations

second

order

between

function.and

the

two

compo

nilpotent

-

super-

:

0 .

.

I)F(e) -

1001 0 i0 1 .

The

etze)B(e)

differential

.w h e r e

equations

A for

±

+

F(8)

d NO and

1

-

(m+~)ct¢O

B(e)

.

ame

2

+ AoAoB Where

=

W

=

The

With

For Other

÷

-

2 d dS-

÷ W

(m

,

equations:

= (k

-

~8-

"Eround"

followinE

Fo(8)

(

-

C, ÷

( sln8

m)(k m

~

0

w

-

w'

)F

=

(k

+

m)(k

-

m

-

I)F

,

(aa)

~ W'

)B

=

(k

÷

m)(k

- m

-

I)B

.

(ab)

by

one

2

I ~)ctE8

.

state of this system + AoF(8)=0 . A~B(8)=0

)m÷,/z

- m

-

the

normalisable

normallsable

i)

=

Bo(e )

•

0

solutions

=

c2

is

defined

: their

( sin8

solutions

)-m-l/z

of

the

are

p

. solution F

and

411

B

may

is

Fo

be

received

and

Bo

otherwise by

construc

-

fine

the

Eqs.

hierarchy

(~)

in

a

2 d de-

( -

of

form

operators

1 ~

(m

÷

~

(m

÷

A~

A~

[3]

So

,

let

us

rewrite

: )(m

i ~)(sine)

-

-2

)F

=

(k

-

I z ~)

F

})(slne) -2

)B

=

(M

-

i z 8)

B

.

(5a)

2 (

d de-

-

Consider FO by

the

Note m÷l

A:A

case

that

B=(K

÷

m

we

(M of

one

the

may

as

=

operators

by

i

of

the ,

membem

of

as

case

accordinE each

-

, with

another

(k

~

m

2)

=

~

0

+

AoA~Az"

and

is

-

-

G

Bo

=

-

with _+ A L

=

''A(n-i)(

(

for

new

)m+z/z, F

st&re and

opePatom

AO

B A:

for

the

iteration ,

any A?

the

shiftinE

"Emound" A +£ m a y

operator

)m+n+x/2

hierarchy

be A:

m

state found + Ao

ame

.

~ +-

ae

of

-

(m+i÷

neEative

)trEe, m

to

Eqs.

(3)

other

and

so

may

the we

(k+m+n)(k-m-n-l)=0, be

AoA~A z .

(k+m+n)(k-m-n-l)=0

treated

quite

normallsatlons have

for

m

0

F

and

B

are

.Further, connected

:

F (e) 1 ÷

• + Ao

m

Fn(@)

(sinS)m÷n+A/z A(n_i > .

m ~ 0 . .n = l. . R,. . analoEously

of ~

.

,

.

n=l.2

A~

±

=

d

÷

.

....

412

~and

fop

m

,

0

each

sine

obeys

sine

Thus

of

)ctEe

"Eround"

the

ope~atoms

states

is m

, which

between

2)

3 ~

÷

Then

Cs

continued

the

(m

obtained

m

state

replacinE

2)Q

F i of

....

B(8)

Fn

m

state

"excited"

d de

+

relations

of

(Sa)

as

the

operators

F(8)

where

-

. HavinE

i)(k

hierarchy

-

l)(k

obviously

As

the

"ground"

Eq.

+ -=

A

(Sb)

superpartner

÷

uslnE

m

the

from

mepmesented

B

be

all Fn

be

2)B

and

, A

the

, when

" E z ~ o u n d '' s t a t e

may

of

may

=

(k

Az

m

obtained

"excited"

, with

A

+

to

first

definition

A£-=

The

- m - + AIA i

consequently,

Eiven

with

I)(M

procedure

of

in

the

the

A:B o

be

-

to

Dair

opematoP

This

m

+ -AIAIG

+

find

Fi

and

m

positive

may

-

match

:

+

the

l)(k

can

supersymmetric

Bo

of

Eq. (5b)

÷

equation

with

3 +

Eq. (Sb)

. Therefore

Now the

1 ~)(m

~8

-

(m~i~

/ 0 B

normalized

=

m

+

1

) ,

=

0. and

.

1

)-m-I/2 are

can

the

be

the

V.A.

Fock

2.

L.

Infeld

3.

C.V.

B

that

. wi~h easily

consider and

if

then

also

,

T.D. :

we

m

and Ib >. In Fig. (5), we are showing the case of a GOE. tn this case it is clearly a m a t t e r of indifference which vectors we choose. The d a t a are for dimension N equal to 40. It is seen t h a t the variance is still quite significant, i.e. the constant mentioned above is fairly large. Note that its average is three, as expected. On the other hand, Fig. (6a,b,c) show the case of a band matrix with ~ = 1 and N = 40. The distributions are shown for three different choices of the vectors la > and Ib >. These depend quite strongly on the pair of vectors chosen. The variance of the distribution function for a given pair is empirically found to be well correlated with the mean of the distribution function.

In

tables 1 and 2, this connection is indicated for the case of the band m a t r i x model and of the Porter-Rosenzweig model respectively. Fig. (6a,b) show cases where the variance is smaller than in the GOE. This need not always be the case, however. In particular, the variance of distributions with exceptionally high mean m a y be larger t h a n the GOE counterpart. Nevertheless, there is no possibility of confusion, since the average and the shape will be altogether different from the GOE case. Such a case is shown in Fig. (6c). For yet smaller values of a, the variance would show a tendency to decrease and the average would vary over a broader range. These findings have indeed been confirmed, b o t h for the band matrix and the Porter-P~osenzweig models.

IV: C o n c l u s i o n s

The spectral properties of quantum systems the classical analoga of which are in a transition from order to chaos are quite well understood. The surprising conclusion that classical order is associated to r a n d o m spectra, while chaos implies small fluctuations is almost universally accepted. The two deviations, namely the harmonic oscillator and the saturatioun effect leading to long-range stiffness are b o t h well understood.

On the other hand, the

discussion of the properties of wave functions is still in its beginnings. The subject of scars

430

Table 1 N

a

Average

Variance

80

2

2.466

0.183

2.561

0.210

3.005

0.273

3.620

0.440

3.743

0.573

4.154

0.581

2.187

0.172

2.692

0.455

3.175

0.756

3.758

0.609

4.415

1.660

5.027

1.510

40

1.

Table 1: This gives the average and the variance of the distribution of Qoo for various choices of vectors for the band matrix model with width a.

The correlation between

average and variance is apparent. has called considerable attention in the last years, and we have seen some outstanding features of this kind. In the last section we presented a new approach to describe differences between functions associated with Poisson, GOE and intermediate ensembles. We have seen that for finite matrix ensembles the wave functions show very clear and basis independent signature of order and chaos i.e. for the transition from Poisson to GOE-like behaviour. This behaviour does not depend on the transition model we choose and can thus be assumed to be universal, i.e. valid for a wide class of transition models.

431

Table 2 N

40

.01

Average

Variance

2.258

0.206

2.497

0.243

3.211

0.491

3.620

0.741

4.499

0.759

Table 2: This gives the average and the variance of the distribution of Qoo for various choices of vectors for the Porter-Rosenzweig model with parameter a.

The problem is how to apply this in the analysis of numerical or actual experiments. In the classical limit the method is useless as we would have large numbers of states involved. This does not i m p l y w f o r t u n a t e l y - - t h a t it may not be used way up in the spectrum as long as we restrict ourselves to a small number of states. The logical way seems to be to form a wave packet or some other superposition of eigenstates that cover a limited range of energy and to shift it over the energy range we have experimentally or numerically available. This sounds quite satisfactory if we consider the assumed ergodicity of our problems but we must keep in mind that we really change our state as we move up in energy; thus we are not doing exactly the same thing as in the ensemble treated in section III. Nevertheless we believe it should be tried, but we need scale invariant systems or at least systems that do not significantly change over the energy range under consideration. As far as the connection to group theory is concerned, we have seen that the invariance properties of the ensemble we wish to analyze and of the space in which we perform our numerical experiments are closely related and thus make the detection of the former very difficult. A deeper group-theoretical background of the transition from order to chaos resides in the structure of phase space in the two cases. For chaotic systems time evolution

432

for almost all points is given by the translation group along the infinite chaotic orbit, while for integrable systems a direct product of 0(2) groups characterizes the motion. An appropriate deformation might shed some light into the problem. The Authors would like to thank Pier Mello for valuable discussions. Iteferenees 1. E. Heller, Phys. Rev. Lett. 53, 1515 (1984); in Proeeeddings of the International Confer-

ence of Cuernavaea, T. H. Seligman and H. Nishioka eds., Lecture Notes in Physics 263, Springer Verlag (1986) 2. M. V. Berry, Ann. Phys. 131, 1 (1981) 3. O. Bohigas M. J. Giannoni and C. Schmidt, Phys. Rev. Lett. 52, 1 (1984) 4. T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey and S. S. M. Wong, 1%ev.

Mod. Phys. 53, 385 (1981) S. 1~. Balian, Nuovo Cimento 57, 183 (1968) 6. O. Bohigas and M. J. Giannoni, in Mathematical and Computational Methods in Nuclear

Phltsies, J. S. Dehesa, J. M. Gomes and A. Polls eds. Lecture Notes in Physics 209, Springer Verlag, New-York (1984) 7. T. H. Seligman and J. J. M. Verbaarschot, Phys. Lett. A 18, 183 (1985) 8. T. H. Seligman, J. J. M. Verbaarsehot and M. 1%. Zirnbauer, J. Phys. A 18, 2751 (1986) 9. N. 1%osenzweig and C. E. Porter, Phys. 1%ev. 120, 1698 (1960) 10. M. V. Berry, Proe. R. Soc. London, A400, 229 (1985); T. H. Seligman, J. J. M. Verbaarsehot and M. 1%. Zirnbauer, Phys. 1%ev. Lett. 53,215 (1984) 11. M. V. Berry, J. Phys. A 19, 2201 (1986) 12. O. Bohigas, S. Tomsovie and D. Ullmo, Phys. R.ev. Lett. 64, 1479 (1990) 13. N. Ullah and C. E. Porter, Phys. Lett. 6, 30 (1963)

433

QUANTUM

ERGODICITY

AND EIGENVALUE

PROBLEMS

FOR PLANE POLYGONS

M. Pauri and A. Scotti ul Diparfimento di Fisica - Sezione di Fisica Teorica Universit£ eli Parma, 43100 Parma, Italy and LN.F.N., Sezlone di Milano, Gruppo Oollegato di Parma.

ABSTRACT With reference to our previous characterization of quantum ergodicity and recent results on classical billiards, we conjecture a special form for Poisson's summation formula in the case of rational polygons. This special form allows us to give a precise formulation of an algorithm leading to the expression of the corresponding eigenvalues. i. INTRODUCTION The research presented here is a follow up of a recent one [I]. in which

the problem of finding eigenvalues and eigenfunctions for plane convex polygonal billiards was tackled and solved in the case corresponding to the existence of eigenfunctions which can be expressed in terms of a finit~ superposition of plane waves. The convex polygons admitting such a solution turn out to be not only "rational" (with respect to ~r) but those having very special angles, namely ~r/2, n/3, ~r/6, and 7r/4, that is rectangles, squares, equilateral triangles, and half equilateral triangles and squares. To put our interest in the determination of these eigenvalues in the proper perspective, we need to recall our main conjecture (denoted in following by C J) [2-5], on the connection between classical and quantum ergodic properties: substantiating this conjecture is in fact our final goal. Precisely, we advanced the following conjecture: "Given a quantum system with a pure discrete spectrum and having a classical limit, its eigenvaJues are linearly independent on the rationals if and only if the corresponding classical system is metrically transitive on each of its energy hypersurfaces" . The crucial part of the proof required for such

a substantiation is, of course, that which refers to ergodic systems. To our knowledge, however, the energy spectrum for ergodic systems has never been expressed in analytical form, suitable for the analysis of interest here, namely the study of the linear independence of the eigenvalues. The only alternative left would be to tackle the problem numerically; as a matter of fact, various authors have considered models of this type [6-8]. The eigenvalues were computed numerically, and the nature of the |1 also GNSM-CISM.

434

system characterized using the standard technique of studying the statistical distribution of the energy levels spacings. From our point of view the problem is much more difficult, one would nearly say impossible. In fact, studying linear independence of real numbers on a computer can be done, within a certain approximation, but it is an exacting task [9]. This being the situation, we believe that the study of non-ergodic systems, which is interesting in itself, is also of relevance for our purpose: in fact the existence of even a single one of them having a discrete spectrum constituted of real numbers independent on the rationals would contradict CJ. A typical class of such systems are plane polygonal billiards with "ratiohal" angles [10-11], (from now on denoted PRB). Recent studies on the Connection between eigenvalues and periodic orbits (closed geodesics) for general types of enclosures [12-16], have brought to a generalized Poisson Summation formula. In these papers, however, the assumptions made on the nature of the boundary of the enclosure defining the billiard and on the properties of the periodic orbits (closed geodesics) are too restrictive and do not encompass the simple polygonal billiards of interest to us here. 2. THE C O N J E C T U R E ON THE POISSON FORMULA From the following very crude and elementary considerations we are lead to make a conjecture on the particular form of the connection between the eigenvalues of PRB and the "lengths" of their periodic orbits. Let us consider the distributional function of t defined as follows :

=

E

(2.I

k~ESpecA

and ask ourselves which values of t are likely to belong to its singular support. Clearly, denoting T the values we are looking for, we should have, for an infinite subset of values of i, say I i

[A~/2T] =

¢ + 21rk,

Vj e Ij,

k

integer.

(2.2)

At this point, one immediately sees that a set of values T, which might do the job, is easily found in all those cases in which the eigenvalues ~i are proportional to an integer valued function of i, say ~i = a I n ( i ) , where In(i) is such that for any given value of i there exists an infinite subset Ij with the property In(i) In(j) = the square of an integer, i fixed, Yj 6 Ij.. In fact, if that is the case, eq.(2.2) can be satisfied by taking ¢ = 0 , and

T

2rk In(j) 1/2 -

(2.3)

al/2

In all the cases in which this simple trick works, the connection between "lenghts" of periodic orbits and eigenvalues is direct: the eigenvalues are 435

simply proportional to the square of the "lenghts" of the periodic orbits. Notice, however, that whereas in the direct problem, (given ~, find T), the T values always belong to the singular support of (2.1), the converse is not true in generM, since there can be values of T satisfying eq. (2.2) not given by (2.3). In the case of the square and of the equilateral triangle (as well as half-square and half-equilateral triangle), as is well known, the eigenvalues are proportional to (n~+ m2), and to (3n~" + rn2), respectively, (n, m integers), and they have the required property, being homogeneous functions of second degree in n and m . It is then easy to verify, using the geometrical construction described in the next chapter, that indeed the corresponding square of the "lengths" of the periodic orbits of these billiards are exactly proportional to the same quantities. To substantiate the consistency and correctness of our reasoning we have integrated formally, twice, the distributional valued function in eq.(2.2) to obtain, apart from an irrelevant change of sign, (2.4) A~6SpecA

The series expansion so obtained for H(t) is now pointwise convergent and can be evaluated numericMly. The results indicate quite clearly that the discontinuities of the second derivative of H(,), both in the case of the square and the equilateral triangle (half-square and hMf-equilateral triangle) occur at the expected values of t. We are therefore lead to conjecture that for PRB the eigenvalues are given by an expression of the type considered above and the problem of their evaluation is reduced to that of evaluating the "lenghts" of their periodic orbits. 3. THE GEOMETRIC CONSTRUCTION The basic idea of the construction is taken from Ref.[10]. It consists in following the trajectory of the billiard in a straight line along a given direction by reflecting the polygon (instead of reflecting the trajectory) around a successive side, precisely the one cut by the given straight line. This "straightening-out" procedure can be used to compute the "lengths" of the periodic orbits in the following way. Let a given trajectory be chosen and fixed: "Straighten it out " along the straight line to which it belongs. Considering the replicas of the polygons successively obtained by the iterated reflections there can be only two possibilities : either, at a certain reflection, the current polygon can be obtained from the original one by means of a pure translation, or not. In the latter case the trajectory, originally chosen, not only is not periodic, but does not belong to a "good" corridor of directions close enough to periodic directions. In the former case, the chosen trajectory does belong to a "good" corridor and the corresponding translation is a periodic orbit direction and the 436

magnitude of this translation gives the "primitive length" of the family of periodic orbits singled out in this way. Except for the cases of PRB corresponding to a tiling of the plane (see Ref.[1]), the explicit analytical form of the "lengths" of the periodic orbits for a generic PRB, has, up to now, defeated our ability of evaluating it, even in the simplest cases of triangles. Work is in progress in this direction. We do have, of course, evaluated these "lengths" for various polygons, by means of a computer program [17]. Comparison of the eigenvalues, so obtained via our conJecture, with the "real ones", obtained by standard numericM methods is also in progress. We conclude by noticing that our present conjecture on Poisson's formula for PRB is consistent with CJ.

REFERENCES [1] V. Amax, M. Pauri, and A. Scotti, Schroedinger Equation for convex plane Polygons: an Elementary Derivation of EigenvMues and Eigenfunctions, to be published. [2] A. Scotti: Dynamic Days, Texas, January 5-8, (1988). [3] V. Amar, M. Pauri and A. Scotti: Dynamic Days, Texas, January 4-7, (1989). [4] V. Amar, M. Pauri and A. Scotti: 17 Statphys, Pdo de Janeiro, August 3-10, (1989). [5] A. Scotti: Una nuova congettura per gli zeri della funzione zeta di Riernann , to appear in Rivista di Matematica, Parma, Italy (1989). [6] N. L. Balazs, A. Voros: Ann.Phys.(NY), 190, 1, (1989). [7] M. V. Berry: Ann.Phys.(NY), 131,163, (1981). [8] M. Saraceno: NSF-ITP-89-129i . [9] P. Bellomo, A. Scotti and F. Zanzucchi: Quantum Ergodicity: A Numerical Test of a Recent Conjecture, to be published. [10] A. N. Zemlyakov and A. B. Katok: Mathematicheskie Zametki, 18, 291, (1975), (English translation: Math.Notes, 18, 760, (1976)). [11] G. A. Galperin: Comm.Math.Phys., 91,187, (1983). [12] V. Petkov and L. Stojanov: Am.J.Mathematics, 109, 619, (1987). [13] J. Chazarain: C.R.Acad.Sc., Ser.A, 276, 1213, (1973); Invent.Math., 224, 65, (1974). [14] Y. Colin de Verdiere: Sur les 1ongeurs des trajectolres pdriodiques d'un billiard, in "G~ometrie Symplectique et de Contact: Autour du Theor6me de Poincar6Birkhoff", Hermann, Paris, 1984. [15] J. J. Duistermaat and V. Guillemin: Invent.Math., 29, 39, (1975). [16] V. Guillemin and R. Melrose: Advances in Math., 32,204, (1979). [17] P. Bellomo and A. Scotti, to be published.

437

"CLASSICAL"

IN TERMS OF GENERAL STATISTICAL MODELS K.- E. Hellwig,

M. Singer

Institut f~r Theoretische Physik, HardenbergstraBe

From

the

classical

statistical

if

there

point

a r e no

TU Berlin

36, D-1000 Berlin 12

of

view

uncertainly

a

physical

relations.

system In

the

is

called

language

of

ordinary quantum mechanics where observables correspond to self-adjoint operators,

this means that for a classical

tal self-adjoint by

a

function

quantum

operator

of

it.

mechanics

measure space

such that

Hence

reduces

for

to

system there is a fundamen-

any observable

a classical

classical

can be

physical

probability

represented

system

theory,

is given by the spectrum of the fundamental

ordinary where

the

operator

and

the Borel sets on it. Both, are

classical

special

introduction Gudder

probability

cases

in the

and overview

[2]. A general

with some structure: procedures, obvious

is

and

of

~

ordinary

general

see e . g . G .

Ludwig

statistical model with

to

statistically

experiment.

The

a

convex

events.

~

is

Such

procedures.

The

experiment

obviously

by

the

probabilities

statistical algebra. gative,

experiment

Here,

propositional which

are the values

by a probability

finitely

additive

the value one on the convex

procedure

can

statistical

be

S.

and R ,

identity.

structure. combined

experiments.

be

calculus forms

various

the

for the

events

Boolean

on ~

is assumed

with

Hence,

each any

all

resulting

a real,

algebra in

a

that

any fixed

preparation

~(Z)-

~ ( ~

) has

registration

procedure ~

takes

~4(~)

procedure

registration

non ne-

which

let

. Obviously,

assigns a Boolean algebra but also a function M: y

438

of

arise

algebra

For a Boolean algebra ~

It

statistical set

a Boolean

is understood

on the

contents

the

a prepar-

of a probability content on this

content

function

denote the set of probability a natural

[3],

expresses

Combining

is called

to

statistical

the

For

one gets a setup to pro-

setup

interpreted

events and

Holevo

which

mixtures.

registration in a

mechanics

models.

of two sets ~

structure

form statistical

arising

set

[4], A.S.

consists

ation procedure with a registration procedure duce

quantum

statistical

, interpreted to be the set of all preparation

endowed

possibility

theory

category

to

give

not

only

This

function M, evidently, R ~ ~

is

formally

has to be affine.

characterized

by two

In o t h e r words,

an e l e m e n t

ingredients:

a Boolean

algebra

Boolean algebra~

. One m a y

look at

and an a f f i n e v e c t o r v a l u e d f u n c t i o n M. C o n s i d e r an e l e m e n t R ~ ~

with

the c o r r e s p o n d i n g M as at a real, ments,

one v a r y i n g

in ~

R a

It is c o m m o n l a n g u a g e to

if it is r e p r e s e n t e d by the a f f i n e m a p

s ~ X(s,.)

and to call

f u n c t i o n w i t h two argu-

and the o t h e r in ~P

call R a ~ - m e a s u r e m e n t

~

[0.i] v a l u e d

,

~)~(~),

C-observable

if it is r e p r e s e n t e d

by t h e v e c t o r

va-

lued m e a s u r e

~

~ E ~ M

(., E).

In the f o l l o w i n g we a s s u m e ~ t o ~

and

two

preparation

(s',.), t h e n s = s'. statistical It may

happen

(s,.)

that

nics and let

E ~

s,

s'~ Y

the

identify

distribution is

Let

to

called

E): =

~ (F) and

plete r e g i s t r a t i o n

, i.e.

there

if for all R

holds

F denote

of

r p jE

the

only

preparation one

Ro

ables w h i c h lently,

by

can

~ ~

of c l a s s i c a l

p is a d e n s i t y

as well.

function,

In H i l b e r t

valued

an

sume

that

the

existence

s

~

of

by

an

i m p l i e s the p h y s i c a l

The is

self-adjoint are

relations.

operators

not

informationally

observequiva-

informationally

H e n c e one

com-

funda-

informa-

or,

com-

is t e m p t e d to as-

complete

s y s t e m to be a c l a s s i c a l

registration one.

But this

a s s u m p t i o n is w r o n g in general!

In H i l b e r t tical model,

space

quantum mechanics

positive

considered

operator valued measures

as a g e n e r a l

do r e p r e s e n t

stati-

registra-

tion p r o c e d u r e s as well. A m o n g t h e m t h e r e are i n f o r m a t i o n a l l y c o m p l e t e

439

A M

Mecha-

informationally

is defined.

space q u a n t u m m e c h a n i c s

measures,

P l e t e if t h e r e are u n c e r t a i n t y

Procedure

if

By

(Laplacian demon)

represented

projection

suffices:

the p h a s e

space

M in

d c,

procedure

be

=

procedure.

complete

mental o b s e r v a b l e m e n t i o n e d at the b e g i n n i n g of t h i s n o t e tionally complete

M(s,.)

informationally

~ (F) d e n o t e its Borel sets.

(p, E) ~ M(p, where

allows

procedure

is injective.

procedures

~

This m e a n s t h a t k n o w l e d g e of all d i s t r i b u t i o n s

experiments

registration

be s e p a r a t e d by

ones

as

has

recently vables

been

shown

in general

introduced

procedure

to

physical

only

be

valued

algebra~ ~B

but

state

Ri~ ~

not

In

[6],

Stulpe

of such obser-

space,

which

complete

can

be

sufficient

and injective homomorphisms

Ei)"

ji:~

distinct

statistical

as follows:

Ei ~ Mi(''

to

in quantum mechanics

In general

algebras

If there

i ~

models

Let be given

(i = 1 ... n) with Boolean

~i ~

registration

is lacking?

for commensurability operators.

commensurability

measures

E ~ M(., E),

The

What requirement

criterion

stration procedures vector

[i].

informationally

necessary

for self-adjoint

one has to introduce

Prugovecki

for the existence

models:

of an

systems.

The commutation

and

has to be separable.

existence

seems

classical

Ali

condition

statistical

canonically,

In general,

works

by

gave a sufficient

is

regi-

~i

and

a Boolean

and a vector measure

M(., E) being an affine function on

~

such that

Mi(., Ei) = M(., J(Ei)), then (Ri}i= 1 ...n forms a commensurable set. For projection valued measures this clearly reduces to the commutation criterion.

One may call a general

at least any finite subset of ~ The lacking requirement precisely,

whether

Let R ~ ~ w i t h s ~ ~and

statistical

concerns

conditional

there are respective

M(S E : E M (S E ,

.)

A

or

or, more

procedures

M. Let M(S,

If there is some S E c ~

if ~

set.

probabilities

preparation

respective ~ - m e a s u r e m e n t

E ~.

model classical

forms a commensurable

in~.

E) ~ 0 for some

fulfilling

(.))

=

M(s, E) then S E is called the conditional Now

a

criterion

classical

one

procedure

R ~~

tionally

may

for be

(~,~)

complete

and

is classical.

this assumption

a general

stated

with the

there is a conditional

preparation

if

as

under hypothesis

statistical follows:

If

model there

( ~ , ~ )to is

a

respective ~ - m e a s u r e m e n t

M

for each

with

preparation

(S,

E)

~ ~

x~

S E under hypothesis

- It has been shown by M. Singer

any finite subset o f ~

440

E. be

a

preparation

being

informa-

M(S,

E)

~

0

E in ~ t h e n [5], that under

forms a commensurable

set.

[i] [2] [3] [4] [5] [6]

S.T. All, E. Prugovecki: J. Math. Phys. 18, 219-232 (1977). S.P. Gudder; Stochastic Methods in Quantum Mechanics, Elsevier, North Holland, New York (1979). A.S. Holevo: Probabilistic and statistical aspects of quantum theory, Elsevier, North Holland, Amsterdam (1982). G. Ludwig: An axiomatic basis for quantum mechanics I. Springer, Berlin (1985). M. Singer: Zur Informationstheorie statistischer Experimente, Dissertation, TU Berlin (1989). W. Stulpe: private communication (1990).

441

Temporal

Decoherence

in Lorentz-squeezed

Hadrons

Y. S. Kim Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742 Marilyn E. Noz Department of Radiology, New York University, New York, New York, 10016 The concept of temporal decoherence in relativistic quantum mechanics can be formulated in terms of thermal noise arising from the two-mode squeezed state of photons. The noise due to the temporal decoherence is calculated for Lorentz-squeezed hadrons within the framework of the covariant harmonic oscillator formalism. This noise is then translated into the temperature of hadrons. The hadronic temperature is measured through (v/c) 2 = e x p ( - ? u ~ / k T ) . As the temperature rises, the hadron goes through a transition from the confinement phase to a plasma phase. The concept of the Bohr radius is important in quantum mechanics. It clearly indicates that there is a space-like separation between two objects. For a moving observer, this space-like separation is linearly combined with a time-like separation [1]. However, the present form of quantum measurement theory is not capable of interpreting this time-like variable. Therefore, the only way to take this variable into account is to treat it as unmeasurable. The covariant harmonic oscillator formalism is a case in point. This formalism has a long history [2] and can be interpreted as a representation of the Poincar6 group describing the internal space-time symmetry of relativistic extended hadrons [3]. The formalism is capable of explaining the basic hadronic features observed in high-energy laboratories [3]. Furthermore, it has recently been pointed out that the Lorentz-squeeze property of the harmonic oscillator wave functions is the same as the squeeze property in the twomode system of photons [4,5]. The longitudinal and time-like oscillations in the covariant oscillator formalism are like two separate photons in the two-mode squeezed states. It is interesting to note that the failure to observe one of the photons in the two-mode squeezed state leads to thermal noise [6]. Likewise, the failure to make a measurement of the time-like variable leads to noise in the longitudinal variable [5,7]. This noise then leads to the temperature of Lorentz-squeezed hadrons. As is well known, the mathematics of photon-number states is based on the onedimensional harmonic oscillator. Let us start with the one-dimensional harmonic oscillator described by

¢.(x) =

+ 1/2)¢.(x).

(1)

Under the scale transformation x = ( V r ' ~ X , this equation becomes independent of w:

_

= (. +

442

(2)

Thus the frequency dependence is solely in the scale transformation from X to x. For the case of photons, ¢,,(x) represents the n-photon state. With this understanding, we can construct a product of ground states of two oscillators with different frequencies Wl and w2, and thus two different coordinates zl and x2:

(3) In the language of photons, the above wave function corresponds to the zero-photon state in the two-photon mode [5-7]. This ground state is invariant under rotations around the origin of the two-dimensional space of xl and x2. In particular, we can consider the variables Yl = (xl + x 2 ) / V ~ and Y2 = (xl - x2)/V~. Then the squeezed vacuum corresponds to the contraction and expansion along the yl and y~ axes respectively [5,8,9]:

(4) In terms of Cn(x), this form can be expanded as [3,4] ¢~(xl, x2) = (1/cosh ~) Z ( t a n h ~)"¢,(xl)¢n (x2).

(5)

r~

In the above expression, Ca(z1) and Cn(x~) represent the n-photon states for the first and second modes respectively. It is now possible to construct the density matrix for this squeezed state. If both photons are observed, the density matrix is that of a pure state:

=i, 4 ) =

4).

(6)

If, on the other hand, the second photon is not observed, we have to take the trace of the above density matrix over the x2 variable. The resulting density matrix is

pT(Zl, Xl) : =

f ¢,(xl, x2)¢;(z~, x2)dx2 (1/cosh /)2

(7)

Let us examine the difference between p(zl, z2; z~, x~) of Eq. (6) and PT(z l, x~) of Eq. (7). The trace of the density matrix is 1 for both cases. However, Tr(p 2) = 1 for p of Eq. (6), but is less than 1 for p of Eq. (7). The loss of coherence due to the failure to observe the second photon leads to a non-pure state for which Tr(p 2) is not equal to Tr(p). We call this loss of coherence the "decoherence." We can then compare the expression of Eq. (7) with the density matrix for the thermally excited oscillator state:

pT(Xl,xl)

= (1 --

exp(-w/T)) E (exp(-w/T))" ¢,(xl)¢,~(x~).

(8)

n

This form of the density matrix for the one-dimensional harmonic oscillator is readily available in the literature [5,8,9,10]. 443

As Yurke and Potasek observed in 1987 [6], the failure to observe the second photon leads to thermal noise corresponding to

e x p ( - w / T ) = (tanh 7/)2,

(9)

where w is the energy of the first photon. The temperature can then be measured from

[6] T = - ( w / 2 ) / l n ( t a n h r/).

(10)

Indeed, the decoherence due to the failure to observe the second photon leads to the thermal noise whose temperature is determined from the squeeze parameter. It is remarkable that the relativistic quark model can also be framed into the formalism of the two-mode squeezed state [5,7]. Let us consider a hadron consisting of two quarks bound together by a harmonic oscillator force. If the space-time position of two quarks are specified by Xa and xb respectively, the system can be described by the variables: X = (x, + xb)/2, x = (.xa - xb)/2v/2. (11) The four-vector X specifies where the hadron is located in space and time, while the variable x measures the space-time separation between the quarks. In the convention of Feynman el al. [11], the internal motion of the quarks can be described by the Lorentzinvariant oscillator equation 1

{

2 2

7 a x. -

~

¢(x) = (a~)¢(~),

(12)

where we use the space-favored metric: x~ = (x, y, z,t). The four- dimensional covariant oscillator wave functions are Hermite polynomials multiplied by a Gaussian factor, which dictates the localization property of the wave function. As Dirac suggested [2], the Gaussian factor takes the form

This expression is not Lorentz-invariant, and its localization undergoes a Lorentz deformation [3,7]. Since the x and y components are invaxiant under Lorentz boosts along the z direction, and since the oscillator wave functions are separable in the Cartesian coordinate system, we can drop the z and y variables from the above expression, and restore them whenever necessary. The Lorentz boost along the z direction takes a simple form in the light-cone coordinate system, in which the variables (z + / ) / V r 2 and (z - t ) / v ~ are transformed to e (z + t ) / V ~ and e - (z - t ) / V ~ respectively, where ~ is the boost parameter and is t a n h - 1(v/c). Then the ground-state wave function will be Lorentz-squeezed as [3,7]

'¢,~(z,t) = (-~)l/2exp {-(.~-)(e~~'(z+'t) 2 +e-2~'(z "t)2)}.

(14)

This squeeze property is illustrated in Figure 1. The above wave function can be expanded

as [3]

¢~ (z, t) = (1/cosh ~) ~(tanh ~)2¢. (V-gz)¢,, (./St). f~

444

(t5)

Indeed, this expression is the same as that for the two-mode squeezed state given in Eq. (5), if )t is identified as the squeeze parameter 7. It is thus possible to relate the boost parameter to the Boltzmann factor by (tanh ~) = exp(-w/~F), with w = ~2/rn where m is the reduced mass of the quark. T h e hadronic temperature T can therefore be defined as (,#c) 2 = exp(-~/mT.),

or

(fi/m)/ln ((1 + (M/P)2),

r =

(lO)

where M and P are the hadronic mass and its magnitude of m o m e n t u m respectively. If the hadron is at rest with P - O, T vanishes. The temperature rises as the hadronic momentum increases. As the momentum becomes very large, T increases as (12/mM2)p ~.

QUARKS

PARTONS

EzI

E I

)-

o

DEFORMAT ON

I constant ; I Quarks become (almost) free

q ' z

]>(I//

////~

I fl\i MOMENTUM-ENERGY DEFORMATION

I

I

I ~ l",z

f

,.o ~.~

I--

I

._~ (FortOn moment.m~L_ I ~ distribution I becomes wider

Ii I

Figure 1: Lorentz-squeezed hadron. The upper half of this figure describes the space-time squeeze, while the lower half is for the squeeze in the momentum-energy plane, where qz and q0 measure the longitudinal and time-like m o m e n t u m separations respectively. This squeeze property, together with the present figure, has been repeatedly discussed hi the literature [3,7]. When the hadron is at rest, it appears as a bound state of quarks. When it moves with its velocity close to that of light, it appears as a collection of free partons. It is interesting to note that this transition mechanism is the same as that of thermally excited states of light from two-mode squeezed states. In the present case, the concept of hadronic temperature is derived Crom the Lorentzsqueezed wave function, and the temperature is a measure of squeeze as in the case of 445

two-mode squeezed states of light [5,7]. The Lorentz-squeeze of hadronic distribution is observed experimentally in high energy laboratories in the form of the patton model [12]. If we regard z and t as the first and second coordinates respectively, then the hadronic distribution can be derived from the density matrix:

p(z, ¢) = =

vr t)dt

f

(l/coshA) ~En(tanh)~)='~¢n(vv'ffz)¢*(v/'ffz'),

(17)

whose diagonal elements become the distribution of the quarks: p(z) = p(z, z) = ~,Tr(c~sh A)/

exp (-(f~/cosh A ) 2 ) .

(18)

The failure to observe the ~ variable leads to" the "temporal decoherence," which results in a wide-spread distribution along the z axis as (v/c) 2 becomes close to 1 [3,12] corresponding to a higher hadronic temperature [7]. As is indicated in Figure 1, the momentum distribution undergoes a similar deformation. The simultaneous expansion in both spatial and momentum distributions leads to the transition from the quark model to Feynman's original form of the patton model, in which the hadron appears as a collection of an infinite number of free partons with a wide-spread momentum distribution [3,12]. This means that the rapidly-moving hadron is in a plasma phase [13], as is indicated in Figure 2. W-~-o0

CONFINEMENT

PLASMA

exp ( - hm/kT)-"-I

PHASE TRANSITION )

QUARKS

v -C

"~

1

PARTONS

Figure 2: Transition from the confinement phase to a plasma phase. This figure is an interpretation of Figure 1. The Boltzmnnn factor e x p ( - w / T ) in statistical mechanics and (v/e) 2 in special relativity are two of the most fundamental quantities in physics. It is interesting to note that the concept of hadronic temperature is derivable from the temporM decoherence property of Lorentz squeezed hadrons. The degree of squeeze is determined from the hadronic speed. We are then led to the question of how to determine the critical temperature at which the system undergoes the phase transition. The patton model does not specify the critical speed at which the hadron becomes a collection of plasma-hke partons. This transition is known to be a gradual process. On the other hand, as is illustrated in Figure 3, the (v/c) 2 factor as a function of T has an abrupt change in slope

446

1.0

I

"-

1

"r"-Plasma Phase

0.8 0.6 0.4 Confinement Phase 0.2 0.0

1

o

20

t

40

i

so

t

so

i00

T Figure 3: The (v/c) 2 factor as a function of the temperature T measured in units of ~/m. There is an abrupt change in slope within the interval 5 < T < 10. The critical temperature is within this interval. in the interval between T -- 5f~/m and 15f~/m. The critical temperature is within this interval. The decoherence problem can also be formulated within the framework of the Wigner phase-space picture of quantum mechanics [14]. The Wigner phase-space distribution function is derivable from the density matrix p(x, x') as [5,14]: (10) The generalization to p(xi, x2; z~, z~) is straight-forward. It is thus possible to construct the Wigner function W(xl, z2, pl, pz) of Eq. (6) [5]. The decoherence is achieved through

WT (Zl, Pl) = f W(;gl, ~g2,Pl, p2)dxzdp2.

(2o)

This decoherenced Wigner function then becomes that of the thermally excited oscillator state [5,15]:

(21) As in Eq. (10), the temperature T is derivable from the squeeze parameter. Finally, let us exaa-nine the group theoretical significance of the above discussion. Group theory is a beautiful subject in mathematics. It can exist without any reference to physics, but it often serves as the basic scientific language in many branches of physics. Occasionally, one mathematical method can be useful in several different branches of physics. The case most familiar to us the second-order differential equation which can be used both for the LCR circuit in electronics and for the driven harmonic oscillator. 447

\\\\NN\\\XX\\\\

L

'

AC

II C

mg

Modern Optics

Special Relativity

Figure 4: The role of the Lorentz group in special relativity and quantum optics. The situation is analogous to the case where one second-order differential equation is applicable both to the forced harmonic oscillator and to the driven LCR circuit. Likewise, the Lorentz group is the basic scientific language for both special relativity and squeezed states of photons. This is. illustrated in Figure 4. It was pointed out recently [5] that the underlying symmetry for the two-mode squeezed state is that of the (3 + 2)-dimensional Lorentz group, whose representation relevant to the two-mode state was worked out by Dirac in 1963 [16]. The covariant harmonic oscillator formalism is known as one of the representations of the Poincar~ group [17]. The little group which governs the space-time symmetry of the oscillator wave function is also a subgroup of the (3 + 2)-dimensional Lorentz group applicable to the two-mode squeezed state. The authors would like to thank Professor E. P. Wigner for helpful discussions on the density matrix formulation of quantum mechanics and its possible role in relativistic quantum mechanics. We thank also Professor J. B ttartle for a helpful discussion on the word "decoherence." REFERENCES 1. P. A. M. Dirac, Proc. Roy. Soc. (London) A l l 4 (1927) 243; P. E. Hussar, Y. S. Kim, and M. E. Noz, Am. J. Phys. 53 (1985) 142. 2. P. A. M. Dirac, Proc. Roy. Soc. (London) A183 (1945) 284; H. Yukawa, Phys. Rev. 91 (1953) 416; M. Markov, Nuovo Cimento Suppl. 3 (1956) 760; V. L. Ginzburgh and V. I. Man'ko, Nucl. Phys. 74,577 (1965).

448

3. Y. S. Kim and M. E. Noz, Theory and Applications of the Poincard Group (Reidel, Dordrecht, 1986). 4. C. M. Caves, Phys. Rev. D 26 (1982) 1817; B. L. Schumaker, Phys. Rep. 135 (1986) 317; B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A 33 (1986) 4033. 5. D. Hart, Y. S. Kim, and M. E. Noz, Phys. Rev. A 41 (1990) 6233. 6. B. Yurke and M. Potasek, Phys. Rev. A 36 (1987) 3464. 7. D. Han, Y. S. Kim, and M. E. Noz, Phys. Lett. A 144 (1990) 111. 8. 1%.P. Feynman, Stalis~icalMechanics (Benjamin/Cummings, Reading, MA, 1972). 9. H. Umezawa, H. Matsumoto, and M. Tachiki, Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam, 1982). 10. A. K. Ekert and P. L. Knight, Am. J. Phys. 57 (1989) 692; A. Mann and M. 1%evzen, Phys. Lett. A 134 (1989) 273; Y. S. Kim and M. Li, Phys. Lett. A 139 (1989) 445. 11. 1%.p. Feynman, M. Kislinger, and F. Ravndal, Phys. Rev. D 3 (1971) 2706. 12. R. P. Feynman, in High Energy Collisions, Proceedings of the Third International Conference, Stony Brook, New York, edited by C. N. Yang et al. (Gordon and Breach, New York, 1969); J. D. Bjorken and E. A. Paschos, Phys. Rev. 185 (1969) 1975; P. E. Hussar, Phys. Rev. D 23, 2781 (1981); Y. S. Kim, Phys. Rev. Lett. 63 (1989) 348. 13. R.. Anishetty, P, Koehler and L. McLerran, Phys. Rev. D 22 (1980) 2793; B. Svettitsky and L. Yaffe, Nucl. Phys. 13210 (1982) 423; J. D. Bjorken, Phys. Rev. D 27 (1983) 140. 14. E. P. Wigner, Phys. 1%ev. 40 (1932) 749; M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106 (1984) 121. 15. W. R. Davies and K. T. R. Davies, Ann. Phys. (NY) 89 (1975) 261. 16. P. A. M. Dirac, J. Math. Phys. 4 (1963) 901; 17. E. P. Wigner, Ann. Math. 40 (1939) 149; Y. S. Kim, M. E. Noz, and S. H. Oh, :I. Math. Phys. 20 (1979) 1341.

449

Physical Significance of Correlated and Squeezed States V. V. Dodonov, A . B. K l i m o v , V. I. M a n ' k o Moscow Physics-Technlcal Institute

Lebedev Physics Institute

During last two decades the so called squeezed states were the subject of increasing flow of papers in various fields of q u a n t u m physics. It seems rather difficult to cite the first paper, since the history of these states goes back to Schrbdinger [1], who first investigated Gaussian wave packets in quantum mechanics. Another important step was made by Glauber [2] who introduced the concept of coherent states. A lot of papers were devoted to various generalizations of Glauber's states. We mention only a few of them [3]-[8]. More complete lists of references can be found, e.g., in [9]-[12]. Different authors invented different names for new types of quantum states: m i n i m u m uncerlainty states, two-photon stales, etc. Now these states are usually called squeezed stales. They are in fact nothing but Gaussian wave packets in corresponding representations, and different names describe different characteristic features of these states. In ref. [13] the concept of correlated states was introduced. These states correspond to the minimal possible value of the lefthand side of the RobertsonSchr6dinger uncertainty relation [14,15]

O-p O"q( 1 - r 2 ) > h2 /4 .

(1)

Here trp and trq are variances of the momentum and coordinate operators: ~q = (~2) _ (~)2 , while r is the dimensionless correlation coefficient: r =--

1 ~rpq)~/=, ~ q = ~ ( ~ + F4) -- @(~), Irl < 1 ( O'pO'q

(2)

In the special case of r : 0 (1) turns into the more familiar Heisenberg-Weyl 'uncertainty relation. The explicit form of the correlated state wave function in the coordinate representation is as follows,

~'0(xlr,n)=N~exp - ~

1 j3=2_~ +-~-

(3)

Here N~ is the coordinate-independent part of the wave function (including the normalization factor), T/= ~ q , ~ is a complex parameter. Function (3) satisfies the equation

450

with g=~

1

~+in~

(5)

Beginning with eq. (3) we use dimensionless variables and assume h = 1. If we define the basic annihilation operator as = ~ + i~

(6)

v~ then the relation between two systems of operators is as follows,

g = ~ + v ~ +, I~l'-Ivl==l, [~,a+]=[g,g+]=l,

1 (1

(u)

ir

)

(7)

r/

3ust relations like (5) or (7) were the bases for introducing the new types of coherent states (new, gerteralized, mi~im~tm ~ncergain~y, ~wo-pho~on, squeezed, etc.) in refs. [4]-[12]. Therefore the mathematical grounds of all investigations in this direction are the same - - linear canonical transformations and their properties. Here we would like to elucidate the physical significance of correlated states and to give in this connection a brief review of our recent papers. (The mathematical interrelations between different generalizations of Glauber's coherent states were given in ref. [lfi].) The main field of applications of both usual and generalized coherent states is quantum optics. But in quantum optics experiments people measure usualy not variances of quadrature components ¢rq and % but distribution functions of quanta. If quantum number eigenstates (Fock's states) are defined in terms of the basic operators (6): ~+~1-> = n I n > , (9) then the distribution function of quanta in a state ~ is nothing but Wn = ]Cn [S, where Cn, n = 0, 1 , 2 , ... are coefficients of the expansion

• =~c.

ln>

(,0)

In the case of the Glauber state k~a satisfying the equation

we have the simple Poisson distribution function Wn -

I°'1=" n! e -t°t=

02)

This distribution possesses the single peak at the value n,~,~x ~ (n) = la[ 2. Quite different situation takes place for correlated or squeezed states satisfying eqs. (4), (5), (7), (8). In this case [17]-[19] =

v

/~

[ 451

1

H, (x) being the Hermite polynomial. In spite of a simple analytic expression distribution (13) as a function of the integral variable n shows rather irregular behaviour. For certain combinations of parameters u, v,/3 (or 7, r,/3) its graph has many peaks separated by intervals in the n-axis with almost or even exactly zero values of the probabilities. Moreover, the situations are possible when the probability to detect the number of quanta which is equal to their average value exactly equals zero. Besides, the distribution function usually highly oscillates with the period An = 2, and both situations when the probability is large for even values of n and for odd values of n are possible. For the details see refs. [17]-[19]. Such a complicated structure of the distribution function can be used for detecting the system in the correlated or squeezed states. One of the possible ways to create the co~related ox squeezed state is to use some parametric process. Indeed, if we consider the Schr6dinger equation with a time-dependent Hamiltonian like

~ ( t ) = ~ 1 ~2 + ~l n 2 ( t ) q^ 2 ,

04)

then its general solution (first obtained by Husimi [20]) is as follows, ~P( x , t) -~ exp ( i ~~

2 +... )

,

(15)

where e (t) is a solution of the classieM equation of motion

~'(t) + a ~ ( t ) ~ ( t ) = 0

(16)

satisfying the subsidiary condition re*

-

t * e =_

2i

(17)

Comparing this expression with eq. (3) wc see that solution (15) is just the correlated state with the correlation coefficient r = -- 1 -I,~1

-=

(18)

and the squeezing coefficient

k = ( ~ . / ~ p ) l / ~ = }*/~1

(19)

Using eqs. (16), (18) we can find such a dependence [2 (t) for which, e.g., the correlation coefficient does not depend on time [21]: ( t ) = (2rt + const)-I

(20)

The corresponding quantum state is in fact unsqueezed, because kn = $2(t) (eq/crp)l/2 - 1, aq = (2/2)-1 (1 - r2)-1/~ .

(21)

Note that just the modified squeezing coefficient kn describes the squeezing properly, since the ground-state variances of the oscillator with the unit mass and frequency 12 are as follows: ~rq = h / 2~2, Crp = S2h / 2 . This example shows

452

distinctly the difference between the correlated and squeezed states: the state may be correlated but unsqueezed. If we shall look for the cases when k = const, then two dependences /2 (t) are possible. The first one is $2 = k-Z: then we have unsqueezed (kn = 1) and noncorrelated (r = 0) states, i.e., usual coherent states. The second case corresponds to the unstable system with ~2 = _ k 2, when 1 trq = ] k cosh ( 2 t / k ) ,

~p = ( 2 k ) -z c o s h ( 2 t / k ) ,

r= tanh(2t/k)

(22)

Eq. (16) itself can be considered as the one-dimensional Helmholtz equation describing the propagation of some wave through the po~e~ttial barrier ~2 (z). If the corresponding energy reflection coefficient from this barrier R is known (it is supposed that 1"2(z) = const when z ~ -4-oo), then relations (17)-(19) lead to the following limitations on the possible values of the correlation and squeezing coefficients [17,21]:

Ir[< 2 ' / ~ -z+R'

- - 1 -vr-R < k < v1 ~-+. l+v~-

-z

v~'

kmax = ~ 1 rmax__ + ~ rm=

(23)

Suppose that the final frequency S)f = / 2 (t ---* +oo) = 1 (this means t h a t we normalize all quantities by the value of ~r). Introducing the energy of fluctuations in the final states Ef = ~z (% + ~q) and taking into account the relations

[22]

1

O'q: ~1 Ic(t)[= , ~p = ~ l~(t)l =

(24)

one can obtain from eq. (18) the following inequality:

r are h o w e v e r

Moreover

up to a p h a s e

resolution

for

Here

with

both variances

uncertainty =

The

z-classes

of

coherent

there

observabl~s

Stoler

state

since

(SMUS) [2]

stochastic famous

¢

related

virsa.

states

evolution

H x)SU(I,I).

SS a n d C o r S

and

with

stable

was

quasiprobability

the

invariant

known

states

SS

follows

measure

from

d~(g)for

H x)SU(I,I).

The

stable

notations)

in

case and

a

being

construction

[9].

the group

as

properties

the

of

of the

($/~)$dm~Iz;~>

compmleteness

result

the the

SMUS

form

to

the n o t i o n s

but

Another

form

and

the

theoretical

structure

this

the H e i z e b e r g

case

in

is

uncertainty

distributions.

neral

in

in

correlated

squeezed

in the

resolution used

(SS)[I]

latters

in

between

correlated

viewing

written

the

group

example

greater

particular

paper

between

For

are

being

this

not

c

i.

they P

applications

minimum

covariation

real

states

decreasing

the s y m p l e c t i c

differance

-

be

treating

(CS),

[O,P]

to

[i-4,7].

The

SS

not

several

(i.e.

the

dynamics

for

Poisson

of

papers the

set

brackets

SS

was

[5-8]. of

Here

SMUS)

for

457

considered

the

we

the

(in

construct invariant

stable

diferent

in

the

ge-

symplectic

evolution

equations.

Some g e n e r a l well

as

considered

2.

aspects

several in

o~

the

stable

examples

greater

dynamics of

(but

detail

in

not

the

group related

case

of

SMUS t h e

SMUS i ~

it

is

~ollowing

[8].

proposition

a groud state

for

holds:

an o p e r a t o r

a

M,u,~

~ ~

operators,

,

lHJ

[ a , a ÷]

m e t h o d o~ p a p e r

uncertainty

relation

~l~,~,~;O>

=

0

o~ t h e

The S t o l e r

H ×)9U61,1) linear

V~[(M + u)Q + i(M

luJ

= 1 and a and i a = V~(O + i P )

,

can be e a s i l y [2] .

in

the

type

proved

SMUS a s

u)P]

-

a÷ are

following

derivation

Benoting

states and

order

=

of

the

IM,u,~;O>

is

+ ~

,

(i)

the

boson

~or

example

Schroedinger then

we

have

.

exp(-i~)sinhl~l In

-

= 1

The a b o v e p r o p o s i t i o n

+ ~

state

given

£

A = Ha + u a t

the

were

Symmetry and C o m p l e t e n e s s p r o p e r t i e s

For

where

CS a s

SMUS)

to

I~;~>

establish

we n o t e

transform

of

correspond

~ = -~

that the

,

where

the

the

to

~ =

~ =

connection

operator

(I)

boson operator

A = T(e~,y,oJ)aT÷(c~,7,co)

coshJ~l

,

u

=

l~lexp(i~) with

can

a

the

group

be r e p r e s e n t e d

(~,~ e ~

,

~ ~ ~

as a )

,

,

(2)

where 2

T(~,F,~) H

=

= e x p ( ~ a~ -

coshY

-

i~Y-lsinhY

= _2~Y-lsinhY , = ~

[Y-lsinhY

~ a + 7 a÷

-

y =

tion

of

the

T(~,~)

(nonnilpotent,

T h u s SMUS c a n b e c o n s i d e r e d

~ a

+ i~a~a)

,

(3) (3a)

(41~j2 _ 2)~/~

i~Y-Z(coshY-

- ~2FY-Z(coshY The o p e r a t o r s

-

,

(3b)

1)]

I)

(3c)

realize

an i r r e d u c i b l e

nonsemisimple) a s CS r e l a t e d

group to

representaH ×)SU(I,I).

this

group, (4)

where

JO> i s

The g r o u p invariant

the

ground state

H ×)SU(I,I)

measure being

d/~(c%~',c~)

=

for

a

turned

calculated

Y-2sinh2Y

out as

~x

458

,

aIO> = 0 to

be

. unimodular~

(Y~ = 4 1 ~ [ 2 -

d ~ ' dco

the

~)

(5)

But t h e r e p r e s e n t a t i o n against this d~

(3)

was f o u n d t o

m e a s u r e , Then we h a v e t o

on t h e c o s e t s p a c e

Subgroup o f

the

GIK

X =

fiducial

fixing

square

K

the

integrable

i n d u c e d measure

is

the

stationary

[0>.

T(~,Z,~)

t h e o b t a i n e d phase f a c t o r

Crossection)

where

,

vector

Representing the operators

be n o t

look for

in

a n o r m a l y o r d e r e d f o r m and

(i.e.

choosing

the

we g e t

X = ~ £x C , where D £ is the unit b y t h e p o i t s o f X.~ 2 ISMUS> = N ( ~ , D ) e x p ( ~ a * + n a * ) I O > = I ~ , n >

apropriate

disk

in

C

SMUS t h e n can be l a b e l e d

w h e r e (~ ~ ~ i '

N(~,~)

~ ~ e)

(I

=

The i n v a r i a n t unity

-

= ~

-

In for is

When

SU(I,I)-CS

~ ~ 0

later

of

in

[9]

in

except for

Stable evolution

the

[93

one

÷or

of

be

operator.

operator at

should

the canonical

and e q u . ( 2 ) quadratic

Ci@/~ 2 hla~+ h~a ÷ h a* + h : a ~ +

has

Note that

replace error.

t h e known

of

been

unity

since T(~,~)

me a su r e

CS a r e

results

du(~)

reproduced.

SS and C o t s

and ~ ( ~ ) .

The e q u a t i o n s f o r

the requirement

the resolution

the proposition

most

of

harmonic

be a t e c h n i c a l

SU(1,1)-invariant

t i m e w o u l d r e m a i n SMUS i f f

evolution

as

the above integral.

s w h a t seems t o

against

A ÷ o r some H ( L ) , ~ ( L )

should

proof

the results

According to

resolution

I~ 12)-iRe(~"n2)d2~d2n - (7)

considered

calculation

by R e ( ~ e ) .

integrable

3.

were

the

Re~eDz)]

takes the form

t h e a b o v e f o r m u l a s when ~2 ~ 0 we o b t a i n

the not

provides

+

m a x i m a l s y m m e t r y and t h e m e a s u r e (7)

t h e f o m u l a added i n

R e ( i ~ "2)

that

= I

l~IZ)-i(Inl ~

-

I~ 12)-Sl2exp r (1 -

i~,~>

CS w i t h

o b t a i n e d by d i r e c t in

X ,

~(~,n)

(I

The s t a t e s oscilator

l~12)I/aexp[-i(l

m e a s u r e on

I[~,~> and ~ i s

Y being given previously. is

,
and ~ = are

~ ,

of

variables

the covariation)

: (9) d/~t

In t h e p a r t i c u l a r cal

equations

=

~HI~

c a s e when

were

~
/~t

,

Imh

established

=

= 0 and

2

in

paper

-~HI~

.

h -2Reh = i ~

2

such c a n o n i -

[12]

R e f e r e n c e s I.

O.N.Holenhorst:

2.

V.Dodonov~

V. Man'ko,

J.Klauder:

in

3.

Phys.

~.Lenczevski, 4.

Rev.

~19

1669

EoKurmyshev:

Symmetry i n

Science,If,

Plenum P r e s s ,

N.Y.

eds.

A26

150

(1980).

B.Gruber

and

and L o n d o n , 1987.

F. H a a k e , M . W i l k e n s : i n P h o t o n s and Quantum F l u c t u a t i o n s . E . R . P i k e and H . W a l t h e r , ~ r i s t o l

5.

D. S t o l e r :

6.

D.A. T r i f o n o v :

7

H . Y . Y u e n : Phys.

8.

~.Nikolov,

9.

3.~eckers,

Phys.

E2-E1-79E

10.

(1979).

Phys. Lett.

I.Malkin~

Rev. DI 3217

Phys.

Lett.

A48 165

Rev. A I 3 2226

~.Trifonov:

and P h y l a d e l p h i a

(1970);

D11 3033

,

1990.

(1975).

( 1 9 7 4 ) ; A64 269

(1977).

(1976).

Commun. o f

JINR

(Dubna), E2-81--797,.

(1981). N.Debergh: J.

Math.

Phys.

V.Man'ko~ D . T r i f o n o v :

J.

30 1739

(1989).

Math. Phys.

14 576

(1973). 11.

E.Onofri:

J.

Math. Phys.

i2.

A.Rajagopal~ J.Marshal:

16 1087 Phys.

460

eds.

(1975).

Rev. A26 2977

(1982).

Correlated States of Q u a n t u m Chain O. V. Man'ko

Institute of Nuclear Research Academy of Science of USSR

The aim of the work is to discuss the integrals of the motion and correlated states properties of the t i m e - d e p e n d e n t discrete string consisting of q u a n t u m interacting parametric oscillators. Some solutions for the stationary string may be found in Ref. [1]. We follow to the procedure suggested for nonstationary chain of oscillators given in Ref. [2]. Let us consider a chain consisting of N harmonic oscillators. The distances between neighbours are equal to a. When the distance between neighbours approach a zero, and number N tends to infinity the chain turns into the string. All oscillators vibrate with the frequency D0 and linearly interact with the neighbouts, the frequency D0 depends on time. The Hamiltonian of this system is

1~

[P~ + m ~ = ( ~ ) ( q , ~ - q , ~ + t ) 2 + m f ~ o 2 ( ~ ) q ~ ] ,

(1)

m n=l

where q,~ is a shift from equilibrium point of an n - t h oscillator, p,~ is a m o m e n t u m of the oscillator, m is a mass of oscillators. The chain is closed, i.e.,

qi+N---- qi ,

i = 1, N .

In order to take into consideration the most simple case N must be an odd number

N:2p+] The equations of motion corresponding to Hamiltonian (i) are [I] /i,~ = / 2 2 ( ~ ) (q,~+l + q,~-I - 2q,~) - 2t2g (~)q,~ .

(2)

The case when the frequency D0 (g) is equal to zero has been considered in [2]. Let us go over to new variables which reduce the system of N interacting harmonic oscillators to a set of N free oscillators. The new coordinates are

Zs --

qm rs~=l 461

COS

-

-

,

XN = m=l

q,~ sin m=l

(3)

1, P .

S~

T h e new m o m e n t a are Pm

COS

pm

,

m=l

N

Pz~ = m=l

Pv,

Pm sin

=

,

m--1

T h e new variables oscillate i n d e p e n d e n t l y with the frequencies

(4) a n d satisfy the e q u a t i o n s of m o t i o n

~,+n,=(,)~,=o,

~+no~(t)~=o.

9,+n,=(~)v,=o,

I f one i n t r o d u c e s N - v e c t o r s q-=(ql,--.,qN), tt = (:rl,

...,

y l , -.-,

zp,

P = ( p l , . . . , PN)

y r , , ~rN) ,

and

P# = ( P = , , - - . , Pz~,, Py, . . . .

, P~p, P~t~ ) ,

(5) the t r a n s f o r m a t i o n (3) c a n be w r i t t e n in m a t r i x form q=S#,

p=S-lp#,

whereS=K'T

T h e m a t r i x K is as follows k k2

k2

.-.

(k~)2...

kN = 1

]

. , ,

K

_

v~

k = (k~):

...

1

, . .

1

where

k=exp

(i,~m):

1

.

462

...

1

-1 .

(6)

The N

x

N - m a t r i x K is matrix of a canonical transformation QH : K q .

This transformation is connected with the irreducible representations of the permutation subgroup of oscillators in the chain. This subgroup is the group of symmetry of equations of motion (2). This group consists of N elements

cN, c h , ... c~ = m , where CN is a rotation by angle equal 27r/N. Other elements are degrees of this rotation. The m a t r i x T is a matrix of transformation from complex variables Qn to real variables #. #:TQ

H

The matrix T is equal to T1 T : i T~T40

T=

'

wheIe 2p - dimensional matrices T ~ , T 2 , T a , T4 are E T s = T1 : - 0 ~0

T2 = - i T 4 =

-.-

Ol I0

1

I0 Ol

-..

0 0

,E=

"¢2 01 10

0 ...

0

0 0

-..

10 01

The Hamiltonian (1), rewritten in the new variables, as follows,

[ =

P~'

P~-"

mZ'~ (~) (=, + y, ) + --y-=,~j

(7)

The Hamiltonian (7) is the sum of the Hamiltonians of 2p (where p = - ~ ) independent oscillators with frequencies ~ , (t) (4) and one oscillator with frequency $9o (t) . The solutions for harmonic oscillators are well known. Using these solutions we can write some results in our case.

463

Let us construct "annihilation" operators for variable-frequency chain i

N

X,C~l=~Zcos

(2Nm)

(

(2~m)

(

--

m='l

i

N

K(~)=~si~

--

q~) ~(~)~-~'(~)Tqm) lspm ~(~)--fi--~(~)~ Isp,-n

,

'

wt----1

XN(~) = ~

~

~o(~)--~-io(~) To

,

(8)

where functions eo (t), es (t) satisfy the equations 02 ~

2 (t) e~ = 0, 02 ~o

at--~ + ~s

- ~ - + Oo~ (~),o = 0

(9)

with the additional conditions 0¢. ¢; _ Oe~

0"-¥

•

&o

~ r e~=21,

.

~eo-eo~-r

Oe~

=2i .

(10)

The dimensionless times *o = a0 (0) t, ~ = a, (o) t are used, when one differentiates with respect to time in formulae (8) and (10), in the next formulae the dot means the same differentiation. The numbers lo, ls are amplitudes of the oscillations in ground oscillation states. They are equal to m ~20 (0)

, Is =

m I2s (0)

The commutation relations of the boson creation and annihilation operators take place for the operators (8) [Xs, A+ +] ^+

Ar]=6~r,[K,fi

=~r,[XN,AN]=I,

[Xs, K I n [X~, Br ]= [Xs,XN]= [X~,AN] ^+

= [ K , X ~ ] = [K, AN]=0. It can be shown that the operators .&~ (t), % (t), AN (t) and their hermitian conjugate A S (~), B : (~), A N (~) satisfy the equations dX5

OXs

i [~,

d~-- o-~-+~

X~] = 0

dB~ - 0

'--a-v-

dAN - O,

' at

where H is the Hamiltonian (1), and, consequently, they are integrals of motion. II The ground state of the parametric chain can be obtained from the equations

[3] X, I~o)=0, K / ~ o / = 0, Xr¢ I ¢ o ) = 0 464

and the condition

(~01~o) = 1. The ground state in the coordinate representation is equal to ~'o (ql, ..-, qN, t) --

v~

2%(')zg

l-I,=~ ~,

+x--" i~, (t)

.,=~

~

s=1 2es (g)

m=l

27rsm qm COS f

+(~=~ (N)½q'~sin (-~))2]

} • (11)

The coordinate distribution function has the form Wo = ~PO~'0 = ~

7r-P~p

exp

-

I¢ol I-l,=~ I~.l =

,=~

x

2qm qm,

'=~

l'rt I 71rL

-2tg - - -I~ol - ~ + cos

N

N

~

.

Using the characteristics of Gaussian distribution [4], the determinant of the matrix of coordinate dispersions and elements of the inverse matrix of coordinate dispersions can be obtained. They are P

det

O'q - =

}~01 FI,=I I~,1~ 2N/2 (12)

(~,-~)q,,,q,, = Nlo21~oj ~ + ~ Nl2 iE,[~------~ cos s=l T h e ground state in the m o m e n t u m

representation e

m

N

is

4

~'o (Pl,--., PN, t) = ~rN/4~1/2

P

Hs=I is 2

x exp

2~o

h2

=

-~

"

p,~ cos

(2~ms)) : (la)

465

T h e m o m e n t u m distribution function for the chain is

1

P ~ I~ol-~ 1~ I~,1-= exp [ --

Wo = ~

s=l

N ~

Pm Pro'

~a, r n t =1

and one can obtain, that the determinant of the matrix of m o m e n t a dispersions and elements of the inverse m a t r i x of momenta dispersions are P

net ~p = It0l ]-[ It, I= 2 -N/= ,

41~a=Nit, l=]

(o --~ )p,,.p..,, = ~tt=Nltol + ~ s=l

cos

~2~'s(,~ _ m')

(15)

One should remember that the dimensionless times are used when one differentiates with respect to time in formnlae (S), 0 0 ) , (11), (1~) - (15). One can see, that by varying in a definite manner the functions es (t), ~0 (~) (9)~ the q u a n t u m dispersions of coordinates and m o m e n t a of the parametric chain can be controlled. The dispersions of coordinates can be decreased at the cost of increasing the dispersion of momenta, and vice versa. So if we can change the frequency of interaction I2 (g) or the frequency of all oscillators in our chain $20 (g) then we can have squeezings and correlations in the oscillator system of the chain. Acting by varying the frequency on oscillators we can generate correlated states of the chain. The correlated states off" the parametric chain can be found from the ground state (11) with the help of the displacement operator [4,5,2] P s-,----.1

P

where exs,/3s, ~ , are complex numbers. So, the entire family of correlated states of parametric chain is of the form

P

+x-" s----1

[

[

___ ~ ~= %* + _ '¢~

I~1=

~,(q~,...,qN,t)=~oexp

2

I,~,1=

I~,1=

2

2

2

2 ~o

.

~, ~;

2 es

2 ~s

o,, q

Eo Io

=

2

---E-

466

q~

The correlated coherent states obey the eigenvalue equations

Using the property of correlated states to be a generating function for discrete Fock's states [4]

k~a (ql, . . . , qN, t) : e x p

--2 ~

(1~1=+ i~,1 =) _ I =

s=l

× fi ~=o

~ I (Oq)'~"(/3s)n='(~)n°@no,nl~ s

(nl~!n2~!no!)

(ql ......

~"~"

......

'~

qN t) '

"'"

'

'

'

one can obtain, that the Fock's states are T, o

~P,~o,n,1..... n,p,~2 ...... n : r ( q l , ' ' ' , q N , t ) x

II

×H,~°

( no !

~=1

qm

ni~!n2s ! ) 1/2

x H~,.

xHn2.(fi (2) I/~ q~ m_-i

=@0(ql,..-,qN,t)

q~

sin ( 2 7 r s m ) )

I~,lZ---~

-Y-

] '

where H,,, (z) - Hermitian polynomials. Thus, we have constructed explicitly the wave functions of Fock and correlated states for nonstationary quantum string. It should be noted that the dynamical symmetry of this string is described by inhomogeneous sympleetie group ISp ( 2N, R) in accordance with the statement for arbitrary quadratic quantum system [3]. It would be interesting to calculate the Wigner function and density matrix for the string in thermodynamic equilibrium state and the corresponding density matrix taking into account the influence of the string nonstationary. I would like to thank V. V. Dodonov and V. I. Man'ko for the useful discussions. References 1. E. M. Henley, W. Thirring: Elementary Quantum Field Theory. N. Y.: McGrawHill Book Company, inc. 1962 2. V. V. Dodonov, V. I. Man'ko, O. V. Man'ko: Trudy FIAN1 1989, v.200 3. I. A. Malkin, V. I. Man'ko: Dynamicheskye S~jmmetrii i Kogerentnye Sostoyanya

Kvantovykh System (Dynamical Symmetries and Coherent States of Quantum Systems) Moscow, Nauka, 1979

467

4. V. V. Dodonov, V. I. Man'ko: Trudy FIAN, 1987, v.183, p 140-145. Translated into English by Nova Science, 1989, N. Y.: lnvarlants and Evolution of Nonstationary Quantum Systems, ed. by M. A. Markov. 5. V. V. Dodonov~ V. I. Man'ko, O. ~/. Man'ko: Trudy FIAN, 1989, v.191, p 185-224. Translated into English by Nova Science, 1989, N. Y.: Theory of Nonstationary Quantum Oscillator, cd. by M. A. Markov

This articlewas processed using the D T E X macro package with I C M style

468

Interaction of Weak Coherent System of Two - Level Atoms Cavity

Light with a in a Lossless

M. Kozierowski 1, A. A. Mamedov 2 and S. M. Chumakov 3 1 Institute of Physics, A. Mickiewicz University, 60-780, Poznan, Poland Institute of Physics, Academy of Science of the Azerbaijan SSR Baku, 370 143, Prospect Narimanova 33, USSR Central Bureau of Unique Device Designing, Moscow, 117 432, Butlerova 15, USSI:t; P. N. Lebedev Physical Institute, Moscow, ]17 924, Leninsky Prospect 53, USSR

1 Introduction The 3aynes-Cummings model (JCM) of light-matter interaction despite its simplicity demonstrates a number of interesting phenomena such as collapses and revivals [1], sub-Poissonian photon statistics [2] and squeezing [3]. Early studies showed the appearance of the so-called Cummings collapse [4] at coherent quantum pumping. Eberly et al. [1] have later found a revival of the collapsed oscillations, in fact an infinite sequence of collapses and revivals with Gaussian decrease of the revival maxima. The origin of collapses and revivals in the 3CM is connected with the photon number distribution which produces spread in Rabi frequencies. The Rabi oscillations, initially all in phase, periodically dephase and rephase which leads to collapses and revivals, respectively. Barnett and Knight [5] studied numerically collective collapses and revivals for a group of two-level atoms. The atoms were assumed as initially unexcited or excited (a maser case). In general, there are two sources of spread in Rabi frequencies: the photon number distribution (as previously) and the collective atomic evolution. The origin of the collective collapses and revivals is related with a non-equidistant spectrum of the eigenfrequencies of the system. Recently, a new solution to the problem of interaction of a system of N twolevel atoms with a single quantized field mode has been proposed [6]. Strictly speaking, cooperative spontaneous emission of a small number s of initially excited atoms in the presence of a large number of N - s unexcited atoms (8 where an is an additional label that takes care of the remaining degeneracies. For simplicity we denote the combination o f quantum numbers Anenan by On. ~(1) stands for the individual h.o. radial and angular quantum numbers Ni and Li corresponding to the i'th Jacobi coordinate. The two particle internal wavefunction 9an 15'ewritten as IF2 A2 e2 ; ~2 > =

Ir2 02 ; y~ > = ~N2L2(Y2) = 17¢2);Y2 >

(2)

where e2 = 2N2 + L2, A2 is the internal angular momentum and L2 = A2. F~ is determined by L 2 : r 2 = [2] for even L2 and r2 = [15] for odd L2. The value of the z component of the angular momentum is suppressed. Let us assume that the (n - 1)-particle wave functions, symmetry adapted to Sn-1, have already been constructed. The general expression for the n-particle internal wave functions symmetry adapted to Sn, can than be written in the form IYnOn; Y2Y3. . .Yn

>= ~n_l rj(n) (en = en-1 + 2Nn + Ln)

[Y.-lOn-lV(n)Anl)rnon] IYn-10n-xn("lA.;ffzy3...y.

>

(3)

where Yn = Y,~-xr,~ and the coefficients [ I) ] are the h.o. coefficients of fractional parentage (hocfps). The hocfps defined in Eq. (3) satisfy orthogonality and completeness relations[2] similar to those satisfied by the single shell cfps for arbitrary permutational symmetry, defined in ref. [4]. On the other hand, here, the n ' t h particle state I~l(n);ffn > = INnLn;ffn > is not unique. We have to sum over all the different single particle states consistent with the angular momentum coupling/~n.--1 + ~n = /~n and the energy relation en = en-1 + 2Nn + Ln, since the elements of Sn couple all those states. This is the price paid for using the :Iacobi rather than single particle coordinates. The hocfps are evaiuated recursively by diagonaiizing the transposition class operator (the sum of all the different transpositions) n

c 2 [s.] = ~ ( i , ¢)

0)

i

485

(8)

where the coefficients are C •Y"~"~"~" = ~/(2A, + 1 ) ( 2 ~ + 1) ,~-z d n - 1 ) £ n - 1 h('q

E

Z(2 + 1)

~(,~)

L(~)

A

L~ A~

]~n

L (~-1) £n-1

4. The result is the crossover. D,h. Two sequences C x + --+ C x - , X = L, N give finite rotation. Other sequences C~ + -.., Off ~ C ~ - , n = 3, 4, a n d e S - ~ C~ + ~ C ~ - , n > 4 r e s u l t in crossover plus finite rotation. Td. C~ + ~ C~ ~ C~ ""+ C ~ - gives c r o s s o v e r . Oh. All sequences result in crossover. C~ + -* C~ --~ C~ --* C ~ - , C~ + -+ C~ --4 C~ -~ C ~ - , C~ + --* C~ --* [C~-, Cff-]. Two bifurcations in brackets take place on the same zero-dimensional stratum. Ih. The only possible sequence C~ + -+ Cff -* [C~-, C y - ] results in crossover. There is a number of experimentally studied spherical top molecules (CH4, Sill4, CD4, SnH4, CF4) which demonstrate clearly the organization of bifurcations, appropriate for the effective rotational Hamiltonian with Oh symmetry group [6,9]. Other types of organization have not yet been studied experimentally. 1. I.M.Pavlichenkov, B.I.Zhilinskii: Annals Phys.(N.Y) 184 1 (1988) 2. D.A.Sadovskii, B.I.Zhilinskii: Mol.Phys. 65 109 (1988) 3. V.B.Pavlov-Verevkin, D.A.Sadovskii, B.I.Zhilinskii: Europhys. Left. 6 573 (1988) 4. B.I.Zhilinskii: Chem.Phys. 137 1 (1989) 5. I.M.Pavlichenkov: ZhETF 96 404 (1989) 6. G.Pierre, D.A.Sadovskii, B.I.Zhilinskii: Europhys.Lett. 10 409 (1989) V.M.Krivtsun, D.A.Sadovskii, B.I.Zhilinskii: J.Mol.Spectrosc. 139 126 (1990) 8. D.A.Sadovskii, B.I.Zhilinskii, J.P.Champion, G.Pierre: J.Chem. Phys. 92 1523 (1990) 9. O.I.Davarashvili, B.I.Zhilinskii, V.M.Krivtsun, D.A.Sadovskii, E.P.Snegirev: Pis'ma ZhETF 51 17 (1990) .

489

ALGEBRAS

OF

THE

SU(n)

VECTOR

INVARIANTS

AND

SOME

OF

THEIR

APPLICATIONS

Valery P. Karassiov P,N. Lebedev Physical

Institute

of the USSR Academy of Sciences,

Leninsky prospect 53, Moscow 117924, USSR.

ABSTRACT:

We

generated

by

examine the

new

SU[n)

algebraic vector

structures

Invarlants

oscillator creation and annihilation operators. bosonlc

oscillator

systems

with

internal

introduce

both

infinite-dimensional

nonstandard

polynomial

deformations

dimensional

oscillator

Lie

to

the

SU[n)

invariant

algebras

symmetry

A spectral

the rock spaces of initial oscillators

of

algebras

and mutations

algebras.

are

For analyzing

SU(n) Lie

which

consisted

we and

of finiteanalysis

of

is given with respect

under

consideration.

Some

physical applications in composite models of many-body systems are pointed out.

l. Introductlon. The symmetry approach based on the use of mathematical formalism of represantation

theory of Lie groups

and Lie algebras

is

widely and successfully used in quantum theory of many-body systems (see, e.g.,

[i-6] and references therein). Specifically,

ana-

lysis of many-body problems within the second quantlzation method introduces

in a natural way a symmetry formalism associated with

oscillators of bosonic and fermionic types: of oscillator

Lie algebras

represantatlon

and superalgebras

in the Fock

theory spaces

[1,4-S]. Such an approach is especially

fruitful

in examlnin E compo-

site models with an internal symmetry since it allows to display some hidden consideration

symmetries

and other peculiarities

of

systems

under

[8-II]. Besides, within this analysis we obtain some

new alEebraie structures which differ from usual Lie algebras and and

groups

mutations

and

repreesent

their

specific

deformations

and

(cf. [12-15]).

Indeed, [ of bosonic

let us consider many-body quantum oscillator systems or fermlonlc

types)

which

are associated --~

with

the

~+

creation and annihilation operators x~i and xi=(x i) , respectively (~=i,2 ..... n; i=I,2 ..... m<m, the superscript "+" denotes the Her-

493

mitlan

conjugation).

Here

the

components of one-partlcle with the vector

"~"

superscript

labels

"internal"

states that transform in a accordance

(fundamental)

irreducible

representation

(irrep)

DI(G) of a classical group G:

x", 9

x?

÷,

, D'(G).

where from here on the summation superscripts.

(i I)

is implied over repeated Greek

The operator x ~ x~ satisfy the standard commutation I' J

relations (CR)

(i.2)

ix i,xj] (~)=xtx]+Ax]xt=0=[xl,xj]~CX),

[x~.x~]cr()O=~

j. o-(;~)=sgr~,

where A=-I and I for bosonlc and fermlonic systems,

respectively.

The Hilbert space for these systems are the Fock spaces L F spanned by the basic vectors

I{n~}>=N({n~})~. G (x11)

where

I0>

nll

g2 n22

(x 2 )

is the vacuum vector:

constant;

g

n m

. . . ( x m) m I0>, x=lO>=O |

the range of the exponents

(1.3)

~,i, N is a normalization

{n7}

depends on the type of

the oscillator statistics. All physical operators includin E Hamiltonian H are polynomials in variables x ~, x~, e.g. i ] H=.Z.~..x.x.+~(c.x.+c. ,,j

,j

,

j

,

i

,

,

x.)+higher powers, ,

(1.4)

where the asterisk ~ denotes the complex conjugation. Now we suppose that Hamiltonlan H is Invarlant with respect to the action (i.I) of the "internal" symmetry group G. Then, according to the vector invariant theory [16], H depends polynomially only on some elementary G-invarlants I ({x?,~.}) constructed in r

i

j

terms of G-vectors xi=(x ~) and xi=(xT). Further, of H provokes

a possibility

this G-lnvarlance

of picking out the G-invarlant

sub-

spaces in L F that one may interpret as a existence of kinematically coupled

("confined")

G-invarlant dynamics.

in internal variable subsystems with the

In order to examine such composite subsys-

tems within the general

symmetry approach

[3,4] we need

in con-

m

structing

C -algebras

[17]

of

the G-invarlant

observables

and the G-Invariant dynamic symmetry algebras k(h)(G) m

494

k (G) m in terms of

{Ir({X?,X~})}

as

well

as

studying

representations

of

these

al-

gebras in the spaces L F. Efficient tools for solving these problems are the vector invariant theory [16] and the conception of complementary groups and algebras

[10,18].

Specifically,

the complementarity

theory allows

us to decompose the space L F into direct sum (with a simple spectrum)

LF= ~ LF,

(1.5)

where the subspaces L_ are irreducible wlth respect to an action of the

algebra

gek~)(G)-

("g" being

the Lie algebra of G) and

furthermore the label "~" determines simultaneously both an irrep of k(A)[G). D~[g) of g and an dual irrep D~(kLA)(G))m m physical point of view the decomposition superselection rules

From

the

(i.5) gives rise to some

[19] since the single spaces L F with diffe-

rent "~" do not "mix" under the time-evolution governed by a Hamiltonian H E k(A)(G). Thus the "internal" symmetry algebra g "inm duces" the "hidden" dynamic symmetry algebra k(A)(G). m This program is simply and fruitfully realized in many-body physics for the groups C=O(n) and Sp(n)

since in these cases the

basic invariants Ir({...}) are bilinear combinations of the operators x ~ and x~

i

and therefore algebras k(A)(G) are well-known fi-

J'

m

nlte-dlmenslonal

Lie-algebras

(see,

rences

However

the

therein).

for

situation is more complicated. k(-1)(SU(n)) m dimensional

and k(-1)(SO(n)) m Lie algebras

deformations

of the universal

oscillator algebras

e.g.,

groups

[18,20-22] G=SU(n)

Specifically, belong

refe-

SO(n)

the

for nm3 the algebras

to new classes

[8,9,23]

and

and

of

associated

enveloping algebras

infinite-

with

some

of generalized

[Ii]. A theory of these structures has as yet

been developed not quite enough. The main aim of the present paper is to examine the situation in more detail [pl,...,pn ]

for the case G=SU(n),

is

D(pl ..... pn_1)and

the

highest

DI(G)=D(IOn_2) , A=-I,

weight

of

the

SU(n)

where irrep

dot as a superscript over "a" in "~ " means the r

repetition of "a" r times. Sec. 2

we

investigate

k(-1)(SU(n))=k(-)(n) m

The paper is organized as follows. some

and associated

properties structures.

of

In

algebras

In See. 3 we

study

m

their representations

in the spaces L F. See. 4 is devoted to cer-

tain physical applications of the algebras under consideration. See. 5 some problems and generalizations are discussed.

495

In

2. Bosonic algebras of the SU(n} vector invarlants. So,

specialize

our

further

analysis

for

bosonic

systems

(A=-I). As is well known [8,16] the set of the basic vector invariants

Ir({Xl,Xj})

for the group SU(n)

consists of the following

constructions:

Etj~(XlXj)=X?X~=(Ej~)

m[Xl . . . X 1 ] = E 1

Xl . . . I 1

i,j=l

+,

n

1

n Xl

n

.....

m,

(2.1a}

1. . . X ! n ~ i 1 :J,M=const} M

(3.5)

M

of the disjoint spaces L(J,M) which are equivalent with respect to the action of the algebra k{-)(2)=so~(4). Further, the action of ~2 the subalgebra su(2)invC so (4) does not change the quantum number T while so~(4)

the operators

X

]2

and X

12

of

the

subalgebra

raise and lower its value by one respectively.

space L(J,M)

is a conjuction of the disjoint

subspaces L(J;M;T)=Span(IJ;M;{Tt}>:J,M,T~J

su(l,l)

c

Thus each

sUlnv(2)-equivalent

=const} which are "in-

tertwined" by the operators XI2,X12. Such the action of the algem bra k(')(2)=so (4) on the space L(J,M) resembles that of usual 2 oscillator algebra on the Fock space (cf.[2,4]) and allows us to m obtain the space-carrier of the so (4) irrep DJ(soQ(4)) starting from any vector of the "vacuum space"

Lv(J).

Similiarly,

show that all spaces L(J,M) are the carrier-spaces

one can

of equivalent

irreps of the algebra I(-)(2). 2 The above analysis provides a sample for realizing spectral analysis of the spaces L F in the case of arbitrary

"m" and "n"

[II]. Therefore we outline its logical scheme and point out some peculiarities in the general case. For arbitrary "m" and "n" the algorithm consist of determinin E

"vacuum

spaces"

su(m)-equivalent means

(su(m)

action

of

operators

of

lowering

susnt(n)=Span{EiJsr=1~ x|xJ'r r SUlnv(m)=Span{Eij, mon highest vectors

X

next

constructing

their

c k~-)(n), I(-)(n)m' k(')(n))m by

on the vectors iv> I n LuCPl ..... Pn_1). In turn the spaces Lu(...) are generate~ by means of

action

L (pl...pn_ I) and

replicas

! ...i

operators i~j,

i~j, ~II=EII-EI+l,I+1}

of

the

algebras

£11=E11-Ei÷1'l**}

and

c kC-){n)m

on the com-

~pl...pn_1;max>ml>satisfyinE

the following

equations

500

a) Xi ...1

I>=O'

1

(3.6a}

n

b) ~ il>=pil>=E111>,

c) EtJt>=O=Eijl>,

(3.6b)

i=l, . . ,n-1 . .

(3.6c)

i<j.

As a result we obtain at final step of the algorithm the following specialization of eq.[l.5)

[II]:

LF=L()=,~e,,~,,, L(;~';{~';~}), where

(3.7)

L(;~';{p'';~})=Span{f;~';{~'';T}>}

are

carrier-

spaces of the SUlnt(n) irreps D() and of associated D(]) and

9''

SUint(n)

irreps are and

the

Gel'land

k(-)(n)'Im - ) ( n } m

-Tsetlln

respectively;

SUinv(m),

distinguishing vectors

of the algebras

vectors

within

[Spt>;~';{~'';~}>

(dual to

and k(')(n);m ~'

patterns

for

the

algebras

~ is an extra label for

irreps of k(-)(n) etc m

[IO]. Basic

resemble in their appearance the struc-

ture of the vectors some polynomials

(3.2) but instead of monomlals X s we obtain 12 in variables X i ...t " An algorithm for obtaining 1

an explicit

(quasimonomial

n

in vector

invarlants consisted of x i and some intermediate boson vectors u i, uj) form has been developed in our papers [10,29].

4. Some physical applications. A natural area of applications of the above results is in developing

composite

models

with

internal

SU(n)-symmetries

within

both quantum mechanics and quantum field theory [2,11,30-32]. Such models are governed by SU(n)-invarlant Hamiltonian H Inv formulated in terms of elements of the algebras k (n}: m

Htnv=CI+ ~ (~iEi t + i Z c tJ E iJ + ,J

~

di

1

,..i

n

Xi

1

...i

+ n

o

+

Z

d

I

1

...i

X n

Specifically,

%

1

...1

+higher powers.

(4. I)

n

some effective Hamiltonlans

in quantum polari-

zation optics have this form [11,32]. The quantities

X i ...i I

operators

of creation

Invarlant

clusters.

and Xt ...I n

I

and annihilation,

But,

unlike

usual

501

may

be

interpreted

as

n

respectively,

of SU(n)-

quantum particles

(bosons

and fermions) these clusters have unusual statistics as it follows fromthe CRs (2.5). In particular,

in the case n=2 we obtain from

(2.5) trlllnear CRs

[Xr,[Xlj,Xkl]]=(~jl~rk-~jk~rl)Xls+(~jl~sk-~jk6sl)Xr1+ . . . .

(4.2]

which generalize the Green's trlllnear CR for paraflelds and paraparticles

[2]. The CRs (2.5] imply

also the general form of the

number operator Ncl of such clusters [11]

N

=(I/n)ZE

cl

I

II

-C({E~})=(I/n)ZE I

il

-C({E

}),

lj

where C(...] are some SU(n)-invariant

(4.3]

nonlinear functions of the

SU(n] generators E °~ which are multiple to the identity operator I on each subspace L()

from

(3.7]. Specifically,

for m=n=2 we

have

C({EC~8})=-I/Z+(I/2)(l+2(EI2E21+E21EIe)+(EII-E22)2) I/2. Thus,

(4.4)

taking also into account (2.2], we see that internal SU(n)-

symmetry yields us a scheme of a generalized paraquantlzatlon with constraints nontrlvlal develope

[cf.[30,33]) dimensions

models with

on

the

spaces

LF=eL().

of the "vacuum subspaces" spontaneously

Because

Lv[)

broken and hidden

of

we can

symmetries

(cf. [31]] within above formalism. Another interesting llne of InvestlEatlons here is in examlnlng posslb111tles of constructing canonical bases of observables Ya' Yb ([Yb'Y,]=~ab]

in terms of elements of algebras km(n).

way seems to be perspective [34], we obtained

since,

followln E the general

In [11] explicit expressions

Thls

scheme

for Y, Y in the

case m=n:

Y=

Z

Cj(X12...n)J+l(x12...n]J,

Y=(Y]+,

(4.5]

JZO

where the coefficients C

r are determined from a set of reccurence

relations depending on signatures of subspacles L(). Such developments can be useful in analyzlnE composite models of many-body quantum systems of arbitrary physical tons, phonons etc.).

nature

(pho-

Some examples of solving certain problems in

polarlsatlon quantum optics have been considered wlthln this approach in [11].

502

5.Conclusion.

In conclusion we point out

some problems and generalizations

of the above developments. The results obtained provide a mathematical

tool for analy-

zing composite models with internal SU(n)-symmetry only at alEebtalc level. However, miltonians

for examlnln E time evolution governed by ha-

(4.1) we need

of the theory,

in developing group-theoretical

in particular,

generalized

coherent

aspects

states of al-

Eebras k(-)(n) etc. m

It is also of interest to extend our analysis by common considering both internal and the space-tlme Polncare symmetries. "Grassmann nature"

The

of the SU(n)-clusters

X! ...i Elves hope that I n we can obtain alone this llne certain results which are useful for some developments in strlnE theory (cf.[25,26]) and for analyzing nonlinear phenomena and coherent structures in stronEly interacting many-body systems [35]. Finally we note that formal aspects of the above analysis may be extended completely for the case G=SO(n). tlonls

obtained

by

involving

in

Another Eenerallsa-

consideration

other

than

DI(G)

irreps of "internal" groups G.

References.

1.H. Weyl. The theory of Eroups and quantum mechanlcs.(Dover Publ.,New York, 1950). 2. Y. Ohnukl and S. Kamefuchl. Quantum field theory and parastatlstlcs.(Unlv. Press,Tokyo, 1982). 3. I.A. Malkin, V.I.Man'ko. Dynamics symmetries and coherent states of quantum systems.(Nauka, Moscow, 1979). 4. A.M. Perelomov. Generalized coherent states and their applications.(Nauka, Moscow, 1987). 5.P.Jordan: Zeits.f. Phys. 94 531 (1935). 6. B.R. Judd. Second quantization and spectroscopy.(Maryland Univ. Press,Baltimore, 1967). 7. I.Bars and M. Gunaydin: Commun. Math. Phys. 91 31 (1983). 8. V.P. Karasslov. In: TopoloEical phases in quantum theory.(World Sci., Singapore, 1989), p.400. 9.V.P. Karasslov: Sov. Phys.-P.N. Lebedev Phys. lnst. Reports No.9 3 (1988). lO.V.P. Karasslov. In:Group-theoretlcal methods in fundamental and applied physlcs.(Nauka,Moscow, 1988),p.54. 11.V.P. Karasslov. P.N. Lebedev Phys. Inst.preprlnt No137 (FIAN,Moscow, 1990); J.Sov. Laser Res. 12 No.2 (1991). 12. E.K. Sklyanln: Funkt. Anal. Prll. 16 27 (1982); 16 263 (1982). 13. L.C. Biedenharn: J.Phys. A22 L873 (1989). 14. A.J.Macfarlane: J.Phys. A22 4581 (1989). 15. H.C. MyunE and A. Sagle: Hadronlc J. 10 35 (1987). 16.H. Weyl. The classical groups.(Unlv. Press, Prlnceton, 1939). 17. G.G. Emch. Algebraic methods in statistical mechanics and quantum field theory.(Wiley-Intersclence, New York, 1972). 18. M. Moshlnsky and C. Quesne: J. Math. Phys. 12 1772 (1971).

503

19.A.O. Barut, R. Racka. Theory of group represantatlons and appllcatlons.(PWN-Pollsh Sci. Publ.,Warszawa, 1977). 20. P. Kramer: Ann. Phys.(N.Y.) 141 254 (1982). 21.E. Chaco~, P. Hess, M. Moshlnsky. J.Math. Phys. 30 970 (1989). 22. S.I.All~auskas: Sov. J. Part. Nucl. 14 563 (1983). 23. G. Couvreur. J.Deenen and C.Quesne: J.Math. Phys. 24 779 (1983). 24. Yu.A. Bakhturln. Identities in Lle algebras.(Nauka, Moscow, 1985). 25. A. Pressley and G. Segal. Loop groups. (Clarendon, Oxford, 1986). 26. Y. Matsuo. Preprlnt UT-523(Unlv. Press,Tokyo, 1986). 27. H. Le Blanc, D.J.Rowe: J.Math. Phys. 28 1231 (1987). 28. A.S. Mishchenko. Vector bundles and their applications. (Nauka, Moscow, 1984). 29. V.P. Karasslov: J.Phys. A20 5061 (1987). 30. C. Itzykson, J.-B. Zuber. Quantum field theory.(McGraw Hill, New York, 1980). 31.L. Mlchel:Rev. Mod. Phys. 52 617 (1980). 32. V.P. Karasev(Karassiov) and V.I.Puzyrevskli: J.Sov. Laser Res. I0 229 (1989). 33. P.A.M. Dirac. Lectures on quantum field theory.(Yeshlva Unlv. Press, New York, 1967). 34. R.A. Brandt, O.W. Greenberg: J. Math. Phys. 10 1168 (1969). 35. J.A. Tuszyn'skl and J.M. Dixon: J.Phys. A22 4877,4895 [1989).

504

P R O D U C T FORMULAS FOR. Q-REPRESENTATIVES

P.Kasperkovitz Institut £dr Theoretische Physik, Technische Universit£t Wiedner Hauptstr. 8-10, A-1040 Wien, Austria

Introduction. Let Q be a Lie group whose elements - up to a set of measure zero - are uniquely labelled by global coordinates 7/= (r/l, T/2,... ). Furthermore, let g(T/) ~ gr(~/) be a representation of ~ by unitary operators in a Hilbert space whose elements are identified with the states of the quantum system under consideration. If one of the states is selected by a convention then the states I ' > = O(~)l fixed >

(1)

span a linear space that is, in general, a proper subspace of the state space. If this space coincides with the original Hilbert space, which is especially true if the representation is irreducible, the states (1) are called group-related coherent states [1-3]. For historical reasons the expectation value of an operator A with respect to these coherent states, A(~) = < , l d l , > ,

(2)

is then called the Q-representative (or Q-symbol) of A. Equation (2) assigns to each operator a function of the group parameters. Under certain conditions this mapping is invertible. In the following it is assumed that ~ is compact and that the fixed state from which the coherent states are generated is an eigenvector belonging to the highest weight A of an irreducible representation [3]. In this case the Hilbert space is finite-dimensional and the mapping A ~ A is one-to-one [4,5]. As the functions A, B , . . . contain the same information as the operators ,4, J~,... it is possible to perform all quantum mechanical calculations with functions instead of operators provided that the counterparts of the following operations are known: linear combinations, products, transition to the adjoint operator, and trace operation. Except for the product operation,

i#=O,

, AoB=C,

(3)

the corresponding operations for Q-representatives have been known for a long time. In the present contribution formulas are given for the RHS of (3).

The general product formula. Under the assumptions stated above, i.e. ~ compact and I fixed > = ] A >, the binary relation that corresponds to the product of two operators reads as follows.

(4)

A o B = P

505

In this equation the symbols p represent Gel'fand-Tsetlein patterns that label the basis vectors of the irreducible representation characterized by A. To each label p belongs a sequence of shift operators that transforms the vector [ A > -- [ 0 > into the basis vector [ p > (up to a complex factor). The sum over all p's, including the trivial pattern p = 0, reflects the matrix product of the two operators. However, the number of nonvanishing terms, starting with the product of the values of the two functions at position rl, depends on the functional form of A and B. For given A and p the real number (Alp) can be calculated from the commutation relations of the Lie algebra. The differential operator 0p and its complex conjugate 8~ are uniquely determined by the pattern p and the group parameters ~?. The highest order of derivatives occurring in 8p is equal to length of the pattern, i.e. to the number of shift operators contained in p. A detailed derivation of (4) is given in [6]. It rests on the fact that every Qrepresentaive may be expressed as a linear combination of products of matrix elements of the irreducible representation and its complex conjugate. These matrix elements are solutions of partial differential equations and related by recurrence relations; this follows from the special functions approach to group representations where the elements of the Lie algebras related to the left and right regular representation of ~ are represented by differentiM operators.

Cohcrcn~ Spin Sta~cs. Coherent states related to SU(2) have been considered in [7-9]. For a spin of magnitude 8 the Q-representatives comprise all linear combinations of the spherical harmonics YLM with 0 _< L 0 for i E I, and strongly dominant and in P + + if (A, a~' ) > 0 for i e I. The Weyl group, W, of ~ is generated by the Weyl reflections whose action on 7-/* are defined by ri(A) = A - ()~,a~)a~ for i E I. The orbit of A under the Weyl group action is the set W~ = {wA [ w E W}. For each A in P there exists a unique dominant weight A+ in P+ such t h a t A+ = wx A for some wx E W. If A+ is in P + + then all the elements wA with w E W are distinct, and in particular wx is unique. The null depth of A is then defined to be d()~) = D(A) - D(A+). By exploiting the Weyl group invariance, which implies (),+, ,k+) = (w~ ~, w~ ,k) = (~,)~), it is possible to show that d(A) = (1/2L(A)) E t,,j=x (a,~ ()~,)~j -~+)~+)), where a = S_I w i t h So' = c,(c'[)-lA,i for i,j E I + = { 1 , 2 , . . . ,t}. Each irreducible highest weight integrable module V ~ of ~ is labelled by a dominant integral weight A. Such a module has a weight space decomposition V ~ = @,eu" V~ and the character of this module is formally given by ch V ~ = ~ ,e P m ,A e , where the weight multiplicity m , is the dimension of V,x. Kac [1] has established the character formula:

chV

Z: w~W

O)

w~W t.

where p ~ ~* is defined by p = ~ ~=0 w~ so that (p, a~ ) = 1 for i E I. The tensor product V" ® V ~ of two irreducible integrable highest weight modules of G is fully reducible into a direct sum of such modules. If the multiplicity of occurrence of modules V ~ in this tensor product is denoted by g~, then ch V" ch V ~ =

508

~¢ P+ g~v ch V ~ . Two problems which immediately present themselves are the explicit evaluation of the weight multiplicities rn,~ and the explicit evaluation of the tensor product multiplicities g~v x • In what follows it is demonstrated not only that these problems may be solved algorithmically but also that they are intimately connected. Illustrations ave confined for simplicity to the case ~ = A~1). The formal definition of chV" and the character formula (1) for c h V ~ and c h V ~ imply

E: re:e" E ~EP

yEW

E]

(2)

,~EP+

wEW

For any A E P+ it follows that A + p E P++. Moreover w(A + p) E P + + if and only if w is the identity element of W. Setting ~ = vg, using the fact that m~¢ -- rn~ and picking out those terms on both sides of (2) involving e~ with r1 E P + + leads to the identity

Z:

= Z:

fEP

(3)

)~GP+

(,. -F ~,+ p)+ E P + +

This identity provides a geometric procedure for determining the tensor product multiplicities g ~ from the weight multiplicities rn~ of just one of the constituent irreducible modules in the product. Its use is a straightforward generalisation of a very well known technique [2,3,4] developed in the context of finite dimensional Lie algebras. The identity (3) actually provides an explicit formula [3] for tensor product multiplicities:

g.t =

Z:

= Z:

(4)

wEW

wE~' ( r + . + p ) + =~+A,

where the sum over a has been replaced in the second expression by a sum over w E W since the set of elements a + u + p such that (a + v + p)+ = A + p is precisely W(,~ + p). Use has also been made the fact that m~ = r n ~ . This formula not only allows the explicit calculation of tensor product multiplicities from a knowledge of weight multiplicities but also the converse. Indeed setting u = 0, so that V ~ = V 0 is the trivial one-dimensional module and g ~ = g~0 = 6~, and taking ~ ~ / ~ in (4) gives Racah's familiar recurrence relation [3] for weight multiplicities: ~.wew e(w)m~+p-wp = 0. However, (4) can be used as it stands as a tool for determining weight multiplicities from tensor product multiplicities. By way of illustration, in the case of g = A~1) we have A = (_2 - 2 ) , so that S = (2), G = (½) and d(/~) = ~ 7 ~ ( # i 2 - #i+2). For the simplest non-trivial module, V(i,°;°) with highest weight # = w0, the weights are all of the form a = (1 - 2re)w0 + 2row1 - p6 with m E Z and p E Z+. The top ten rows of the infinite weight diagram are shown below. Ignoring for the moment the foot of the table, the Weyl reflection planes are the vertical lines through 80 = (0,1) and 8i = (1, 0). All weights (r = (1 - 2m, 2m; p) on any vertical string in a column labelled by m are Weyl equivalent to those on the one string in the dominant sector labelled by rn = 0. In fact or+ = (1, 0; k) with k = p - m 2, since 2 = m 2 . The weight multiplicities themselves are given by m~ = ap_,~ :I(al2 --Pl) where a, -- p(n), the number of partitions of n, as can be shown through the use of Racah's recurrence relation [5] or otherwise [1].

~09

m

--3

o°o

(7,-6) 9

, , ,

p=0 p=l p=2 p=3 p=4 p=5 p=6 p=7 p=8 p=9

d(C) (Co,C1)

1

--2

(5,-4) 4

1 1 2 3 5 7

--I

0

(3,-2) 1

1

(1,0) ( - 1 , 2 ) 0 1 61 60

I 1 2

1 1 2 3

I ! 2

3 5 7 11 15 22

5 7 11 15 22 30

3 5 7 11 15 22

2

-

0

(8,-3)

-. •

...

1

T ¢0

0

(6, - 1 )

(-5,6) 9

1 1 2 3 5 7

T ¢1 (10, - 5 )

3

(-3,4) 4

0

0

(4,1)

(2,3)

-

2

(0,5) ( - 2 , 7 )

...

The above weight diagram may then be used to calculate, for example, the tensor product multiplicities for V(1,0;0)® V(2,°;°). The procedure based directly on (3) involves shifting the weight diagram of V (1,°;°) through u+p = (3, 1; 0) so that a = ( 1 - 2 m , 2m; p) goes to ¢ = (4 - 2m, 1 + 2re;p) with L(¢) = 5. This can be effected by the relabelling given at the foot of the above diagram. The reflection planes are now at the positions ¢0 = (0, 5) and ¢1 = (5, 0). All weights ~ on any vertical string either lie on a Weyl reflection plane or are such that ~+ = (4, 1; k) or (2,3; k) for some k, with d(~) = ]rn(rn + 1) or ](rn 2 + m -- 2), respectively. Carrying out the Weyl reflections for each vertical string, taking signatures into account and subtracting p = (1, 1; O) gives the .(~o,~,;k) . following tensor product multiplicities u(1,0;0)(2,0;0)"

= (3, 0) k=O k=l k=2 k=3 k=4 k=5 k=6 k= 7

1 1-1=0 2-1=1 3-2=1 5-3=2 7-5=2 11-7=4 1 5 - 11 = 4

= (1, 2) 1 1 2 3-1=2 5-1=4 7-2=5 11 - 3 -

1= 7

Since tensor products are commutative, the same multiplicities must arise if the problem is approached in the same way but starting from the weight diagram of the module V(2.°;°). This takes the following form in which reflections in the planes signified by ¢0 = (0, 2) and ¢1 = (2, 0) have been used to parametrise the weight multiplicities

510

in t e r m s of those in the dominaalt sector. T h e weights are all of the f o r m T = (2 -2 m , 2 m ; p ) , w i t h L ( r ) = 2 a n d d(7) = ½m 2 or ½(rn 2 - 1), a c c o r d i n g as m is even or odd. m = -4 (~o,~t) "." d(r) 8

p=0 p = p = p = p = p = p = p = p =

1 2 3 4 5 6 7 8

ao

-3 (8,-6) 4

-2 (6,-4) 2

bl b2 b3 b4

ao al a2 a3 a4 a5 a6

-1 (4,-2) 0

O (2,0) 0

1 (0,2) 0

¢1 l

¢o 1

ao at a2 a3 a4 a5 a6 a7 a8

bl b2 b3 b4 b5 b6 b7 b8

ao at a2 a3 a4 a5 as

0 (2,3)

¢0 (0,5)

bl b2 b2 b4 b5 b6 b7 b8

T 2 ...

0 (10 , - 5 ) ( 8 , - 3 )

o (6,-1)

3 (-4,6) 4

4 (-6,8) 8

bl b2 b3 b4

ao

2 (-2,7)

4 (-4,9)

T

¢1 d(¢) (¢o,¢t)

2 (-2,4) 2

0 (4,1)

P r o c e e d i n g as in the p r e v i o u s e x a m p l e on the basis of (3) with # a n d v i n t e r c h a n g e d , n o w involves a d d i n g # + p = (2, 1; 0) to t h e weights ~- to give ~. Of course the reflection planes signified b y ¢0 a n d ¢1 a n d d(~) are exactly as before. C a r r y i n g out the reflections, t a k i n g s i g n a t u r e s into a c c o u n t a n d s u b t r a c t i n g p leads to expressions for the tensor p r o d u c t multiplicities which m a y be solved recursively for the weight multiplicities ak a n d bk of the d o m i n a n t weights (2, 0; k) mad (0, 2; k) of V (2,°;°) as s h o w n below.

k = k=l k = k = k = k =

0 2 3 4 5

(~0, ~1) = (3, o)

(~o, ~t) = (1, 2)

1 = ao 0=at-bt 1 = a2 1 = a3 2 = a4 2 = as-

1=bl 1 --- b2 2 = b3 2 = b4 4 = bs-

b2 b3 b4 - a0 b s - az

b0 bl b2 b3-

a0 at a2 a3

a0 = al=l a2 = a3 = a4 = a5 =

1 3 5 10 16

bl=l b2 = b3 = b4 = b5 =

2 4 7 13

REFERENCES [1] V . G . K a c , Infinite dimensional Zie algebras (Boston, Mass.: B i r k h a u s e r , 1983) [2] A.U. K l i m y k , Amer. Math. Soc. Transl. Series 2 Vol 76 ( P r o v i d e n c e , RI.: A m e r . M a t h . Sot., 1968) [3] G. Ra~zah, in Group theoretical concepts and methods in elementary particle physics, Ed. F. G u r s e y (New York, NY.: G o r d o n a n d Breach, 1964) p p l - 3 6 [4] D. Speiser, in Group theoretical concepts and methods in elementary particle physics, Ed. F. G u r s e y (New York, NY.: G o r d o n a n d Breach, 1964) pp237-246 [5] A.J. Feingold a n d J. Lepowsky, Ado. Math. 29, 271-309 (1978)

511

ATYPICAL MODULES OF THE LIE SUPERALGEBRA gl(m/a) J. Van der Jeugt ~ (University of Ghent, Belgium), J.W.B. Hughes (Queen Mary and Westfield College, U.K.), R.C. King (University of Southampton, U.K.) and J. Thierry-Mieg (University of Montpellier, France) ~ Let G --- G o S G ~ be the genera/l/near Lie superalgebra gl(rn/n) [2] consisting of complex matrices (vA BD) of size (rn -b n) 2. The even subspace G O of G consists of the matrices (oA0D) and the odd subspace G i consists of the matrices (o s° ) . The bracket between homogeneous elements is defined by [a,b] = a b - (-1)=Zba for a e G=,b e G~ (a,~ e (0, i} ---- ~2). Thus the even subalgebra is isomorphic to gl(rn) ~ gl(n). G admits a consistent ~-grading G = G-1 ~BGo (9 G+I where Go = Go, G+x is the space of matrices of the form (0°0 s ) and G-1 is the space of matrices of the form ( o 0 ) . The special linear Lie superalgebra sl(m/n) is the subalgebra of gl(m/n) consisting of matrices with vanishing supertrace. In what follows we put G = gl(m/n), but all of the results can be reformulated for ~l(m/n) as well. The Cartan subalgebra H of G consists of the subspace of diagonal matrices. The roo$ or weight space H* is the dual space of H and is spanned by the forms ei (i 1 , . . . , m ) and 6j (3" -- 1 , . . . , n ) . The inner product on the weight space H* is given by [5] (ei[(i) = 61j, (ei[6i) : 0, (6~[6y) = -6ii, where 61i is the usual Kronecker symbol. In this e6-basis the even roots of G are of the form e l - ei or ~ i - 6i, and the odd roots are of the form q-(el - 6j). Let A denote the set of all roots, Ao the set of even roots and Ax the set of odd roots. As a system of simple roots one takes the so-called distinguished set [3] ez-e~, e 2 - e s , . . . , e=-61, 61-62, ..., Q - 1 - Q . Then the set A + of posi~ive roots consists of the elements e, - ej (i < j), 61 - ~ (i < j) and e, - ~j. Now the notations A+ and A + are obvious; in particular : A + = {~,.~. = c,. - 6~.,

i = I , . . . , m,

j = i,...,,~).

(i)

All simple modules (i.e. irreducible representations) of the classical simple Lie su-

peralgebr were classi ed by Kac [S]. Kac's result speci ed to sICm/ ) implies that every finite-dimensional simple G-module V is a highest weight module V(A) specified by an integral dominant weight A. A weight A E H* is said to be integral dominant if and only if its so-called Kac-Dynkin labels A = [al, a 2 , . . . , a.~-l; a.~; am+l,..., a.~+.-i] are such that al E ]N for i ~ m whereas am can be any complex number. For our purpose it is sufficient to consider only those A for which a,~ E 2~. If A is expressed in terms of the eS-basis as A = ~ ~iei + ~ uj6j., then the Kac-Dynkin labels of A are given by a~ = /~i-/~i+1 (i < m), am = /~,~+uz, a~+y = u y - u y + 1 (3" < n). Note that the coordinates in the e6-basls represent a unique weight of gl(m/n) whereas the Kac-Dynkin labels represent a unique weight of sl(rn/n) rather than gl(m/n). Often it will be useful to represent a weight A by a composite Young diagram, consisting of 1Research Associate of the NFWO (National Funds for Scientific Research of Belgium} 2Talk presented by J. Van dcr Jeugt

512

the diagrams of {g} and {~} in appropriate positions [5]. For example, for gtC4/6 ) and A = (7, 6, 6, 3[i, i, 3, 3, 5, 5) in the e~-bazis (where k stands for - k ) , the composite Young diagram is shown in (5). In this case, for example, the Kac-Dynkin labels of A are [1,0,3;2;0,2,0,2,0]. The basic problem we are concerned with is the determination of the weights and weight multiplicities of V(A). Such information is contained in the so-called character of V(A), which is by definition equal to chV(A) = E~(dimV~) e", where dimV, is the multiplicity of the weight ~ appearing in the weight space decomposition V(A) = ~ , V,. Recall that for a (reductive) Lie algebra Go (in the present case we can think of Go as the even part gl(m) (9 gl(n) of gl(m/n)) the character formula of a G0-module V0(A) with highest weight A is given by Weyl's character formula :

chVoCA) = Lo E

Lo = H

,.uEW

C2)

aEA+

where W is the Weyl group of Go, ~(w) is the signature of w E W and Po = ~ ~.~e,,+ o~. A very important finite-dimensional highest weight module V(A), the so-called Kacmodule, was introduced in [3]. For given integral dominant weight A, the G0-module V0(A) is uniquely determined (up to isomorphism), and can be extended to a Go (9 G+Imodule by putting G+xV0(£) = 0. Then one defines the induced module V(A) = IndaaoOa+, Vo(A) ~. U(G-z) ® Vo(A).

(3)

It follows from the structure of U(G_t) that

car(A) = x CA) = Lo

II (1 + ~ew

(4)

ae~+

When is V(A) a simple module? The answer to this question was given by Kac [3] : only if (A + Plfl) #- 0 for all fl in A +. Herein p = P0 - P,, where P0 has been defined previously and p, = ½E~e~tfl" In the case that all (A + pl/~) # 0, A and V(A) = V(A) are said to be typical, otherwise h and V(A) # V(A) are said to be atypical If A is atypical, V(A) contains a unique maximal submodule M(A) and V(A) -~ V(A)/M(A). Our main aim is to determine characters for such atypical modules. One of the useful tools in studying atypical modules is the so-called atypicality matrLx A(A) consisting of the rnn integers A(A)~i = (A + P]fl~i) [4J, where fl;i has been defined in (1). In terms of its components in the e6-basis, A(A)~i is given by #i + vi + rn - i - j + 1. The m x n atypicality matrix fits nicely into the compositie Young diagram, as is illustrated here for our example, A = (7,6,6,3]i,i,:],:~,5,5) for g/(416) :

V(M is simple if and

III

]

513

This A is atypical of type fix,6, and is singly atypicM. Various character formulae for atypical V(A) have been proposed, most of which are of the following form (see [4,5] and references therein):

= 4'

,C,,,)',, (, "+'° I I

flEA(A)

wEW

+,-")).

(6)

where A(A) is some subset of A +. In particular, Bernstein and Leites [1] proposed A(A) = { f l e A + ](A + p]fl) ~ 0), in which case Xa(A)(A) in (6) is replaced by XL(A). However, counterexamples were found to their formula. Similarly, counterexamples were found to other formulae of the type (6) [5], and in particular we were able to prove that for G = gl(3/4) and A = [1, 1;0; 0, 1,0] no set A(A) exists yielding the correct character of V(A). Hence no formula of the type (6) can give correctly the characters of all simple modules V(A) of glCm/n). There is, however, the important class of singly atypical modules where the problem of finding character formulae has been solved [4]. When there is only one ~ in A + with ( A + p l " / ) = 0 (and (A+plfl> # 0 for all fl ~ ~), A is singly atypical. In this case, we proved that the maximal submodule M(A) is itself a simple G-module, and that M(A) ~ V ( ¢ ) , where ¢ = w . (A - k~) = w ( A - k~ + p) - p and A - k~ is the first element of the sequence A - % A - 2 % ... that can be mapped into an integral dominant weight ~ by means of a w. action. In terms of the composite Young diagram, with the zero in the atypicality matrix at position (i,j), we move to the end of row i in the ~ p a r t of the diagram and to the end of column j in the v-part of the diagram, and perform a strip removal of length k in both parts of the diagram, removing one box at a time until the composite diagram is standard [5]. In our example (5) this leads to the following strip removal :

I I xx 9 8 5 4

N[ol

7 6 3 2[i]~ 6 5 [2l[11[2] 2 1~ 3 6 ?

IIxx

xxl

Note that only after 6 box removals, the remaining Young diagrams are standard. Thus ~I, = w . (A - 6fll,6) for some w E W, and it follows that @ = A - (fix,6 + fll,s + fl2,5 + fls,~ + fl3,4 + &,s) = (5, 5, 3, 31i , i, 2, 2, 2, 7i). Both strip removals (indicated by X's) are necessarily of the same shape, and the positions of the brackets [ ] in the atypicality matrix (which constitute the same shape again) determine the flij one has to subtract from A in order to obtain @. Also, ~ is atypical of type fiB,a, which corresponds to the "tail" of the removal strip. Making use of these properties, of combinatorial properties of the atypicality matrix, and of recursion, one is able to prove [4] that for the singly atypical case chV(A) = XL(A). Then, making a formal expansion, one can rewrite the

514

character as an infinite alternating series of Kac-characters XK(~.) : oo

chV(A) =

x

x (A-

(A) =

(8)

t=0

Let us now return to the more general case of mul~iply atypical modules. For reasons of presentation we shall illustrate here the case of doubly atypical modules. Thus CA Jr" P[fll) = 0, C-h- -~- P[fl2) = 0, and CA + plfl) ~ o for every f l ¢ / ~ l , & . Similarly as in (8) one can formally expand the Bernstein-Leites formula as an infinite alternating series of Kac-characters :

(9) tl,t2=O

CA

where CA = {,~ = A-tiff1 - t2/?~} is the "cone" with vertex A and ( - 1 ) IA-~l = ( - 1 ) t'+$2. Let ~1 = e l - 6 j andfl2 = e k - ~ w i t h i > k a n d j < I. Then there is a u n i q u e w l ~ in W which permutes the components i and k, m + j and m + l, and leaves all the other components of a weight in the e~-basis invariant. Let H12 = {~? e H*lw12 • (71) = r/}. Clearly, such a hyperplane splits the weight space H ' into two half-spaces. The truncated cone C + is defined to be the set of weights of CA that are in the same half-space as A. Then we conjecture : chV(h) = XL(A) = ~c~(--1)[A-~IxK(A) if A is not critical, and chV(A) = ~c+(-1)IA-~l XK(A) if A is critical, where A is c r i t i c a / i f and only if the entry A(A)kj in the atypicality matrix is equal to the "hook length" connecting the two zeros (at positions (i,j) and (k,l)) in the atypicality matrix, i.e. equal to i - k + l - j - 1 [5]. The ways in which this conjecture has been tested, and how it works for atypical modules with degree of atypicality > 2 is described in [5]. Let us emphasize that the given formulae are expansions of chV(A) in terms of the formal characters XK(A), which are characters of Kac-modules when A is dominant integral. One may also consider the inverse problem : given the Kac-module V(A), how can chV(A) be expressed as a Cnecessarily finite) sum of characters of simple modules chV(a)? In other words, what are the non-zero multiplicities n= in the expression chV(A) = ~= nachV(a)? This is known as the problem of the determination of the composition series of V(A). Recently, we have made a lot of progress in solving this question. Our results concerning the determination of the composition factors of the Kac-module V(A) were presented at this Colloquium by R.C. King, who reports on it elsewhere in this Volume. REFERENCES [1] I.N. Bernstein and D.A. Leites, C.R. Acad. Bulg. Sci. 33, 1049-51 (1980) [2] V.G. Kac, Adv. Ma~h. 26, 8-96 (1977) [3] V.G. Kac, Lecture Notes in Mathematics 676, 579-626 (1977) [4] J. Van der Jeugt, J.W.B. Hughes, R.C. King and J. Thierry-Mieg, "A character formula for singly atypical modules of the Lie superalgebra $l(m/n),", Commun. Algebra, in press (1990) [5] - - , "Character formulae for irreducible modules of the Lie superalgebra sl(m/n)," J. Ma~h. Phys., in press (1990)

515

Vector coherent state theory of the non-compact orthosymplectic superalgebras C. Q u e s n e *

Physique Nucl~aire Th~orique et Physique math~matique Universitd Libre de Bruxelles C a m p u s Plaine, C.P. 229 - B1050 Bruxelles - Belgium

Abstract The vector coherent state and K-matrix combined theory is applied to construct matrix realizations of the positive discrete series irreps of the orthosymplectic superalgebras osp(P/2N,R) (P = 2M or 2 M + l ) in osp(P/2S,R) D so(P) @ sp(2N,R) D so(P) ~ u(N) bases. As an example, the case of osp(4/2,R) is treated in detail.

1

Introduction Vector cohe.rent states (VCS), also called partially coherent states, were independently intro-

duced by Rowe [1], and by Deenen and quesne [2] as a natural extension of generalized coherent states [3,41. At the same time, it was noted that coherent states provide a very powerful method for constructing matrix realizations of Lie algebra ladder irreps in bases symmetry-adapted to some maximal rank subalgebra [5,6]. Such a construction is carried out by the so-called K-matrix technique [7,8]. Since then, the VCS and K-matrix combined theory has been applied to a lot of algebrasubalgebra chains (Refs. [7,8] and references quoted therein}. Recent extensions have allowed the method to be used for non-semisimple Lie algebras [9] and for Lie superalgebras [10]. In the present communication, we report on a new application to the positive discrete series irreps of the non-compact orthosymplectic superalgebras osp(P/2N,R), where P - 2M or 2 M + l . In *Directeur de recherches FNRS

516

Refs. [11] and [12], a general method is provided for determining the conditions for the existence of star irreps (and of grade star irreps in the osp(2/2N,R) case), the branching rule for their decomposition into a direct sum of so(P) ~ sp(2N,R) irreps, and the matrix elements of the odd generators in osp(P/2N,R) D so(P) @ sp(2N,R) D so(P) ~ u(N) bases. The cases explicitely worked out include the most general irreps of osp(1/2N,R), osp(2/2,R), osp(3/2,R), osp(4/2,R), osp(2/4,R), and the most degenerate irreps of osp(2/2N,R). We shall review here the osp(4/2,R) example.

2

The positive discrete series irreps of o s p ( 4 / 2 , R ) The osp(4/2,R) superalgebra is spanned by the so(4) generators A~2, A 12, C b, a, b = 1, 2,

the sp(2,R) generators Dr, D, E, and the odd generators G =, //=~ I~, Jo, a - I, 2. We choose to enumerate the weight generators in the order E, C1, C~. Then the lowering generators are A 12, C~, D, G =, and Ja, and the raising ones A~, C~, D t, H ° and I=. The adjoint operation in so(4) (B sp(2,R) can be extended to an adjoint operation in osp(4/2,R) in two ways differing in a sign choice : (Ga)t = ±I~, (Ja) t -- ± H a. On the contrary, it cannot be extended to a grade adjoint operation. Hence, osp(4/2,R) may have star, but no grade star irreps

[13]. The positive discrete series irreps of osp(4/2,R) can be induced from a lowest-weight so(4) sp(2,R) irrep [~1::~] (B (l~) or, equivalently, from a lowest-weight so(4) (~ u(1) irrep [~IE=] (~ {~'1}. They will be denoted by [~xE~n/. Here, ~1,~2, and fl are some integers subject to the conditions ~1 > IE~I~and n > 1. To construct a basis of the [~lE~n) carrier space, symmetry-adapted to the chain osp(4/2,R) D so(4) ~ sp(2,R) D so(4) E~ u(1), one may start from a basis (I[~l~2]{n}c~)} of the lowest-weight so(4) (B u(1) irrep. Since the raising generators H a and Ia (Dr) are the components of an so(4) (B u(1) irreducible tensor ~ (Dr), transforming under the irrep [10] ~ {1} ([00] E~ {2}), one can construct polynomials in ~ (Dr), transforming under an so(4) @ u(1) irrep [A1A2] (B {p} ([00] {v}). Here, v runs over all even integers~/~ over the set ~0,1 .... ,4}, and [~,1A2] over those so(4) irreps contained in the u(4) irrep {1"6}. By acting with these two sets of polynomials on the states I[~r~.=]{N}cx) and by performing so(4) couplings, one can form the set of states

=

[PI°°J~(Dt) × [Q[~,~,J

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