Recent Progress in Many-Body Theories: Proceedings of the 13th International Conference Buenos Aires, Argentina

Series on Advances in Quantum Many-Body Theory - Vol. 10

Proceedings of the 13th International Conference

RECENT PROGRESS MANY-BODY THEORIES

Editors

Susana Hernandez Horacio Cataldo World Scientific

RECENT PROGRESS MANY-BODY THEORIES

Series on Advances in Quantum Many-Body Theory Edited by R. F. Bishop, C. E. Campell, J. W. Clark and S. Fantoni (International Advisory Committee for the Series of International Conferences on Recent Progress in Many-Body Theories)

Published Vol. 1:

Proceedings of the Ninth International Conference on Recent Progress in Many-Body Theories Edited by D. Neilson and Ft. F. Bishop

Vol. 3:

Proceedings of the Tenth International Conference on Recent Progress in Many-Body Theories Edited by Ft. F. Bishop, K. A. Gernoth, N. Ft. Walet and Y. Xian

Vol. 4:

Microscopic Approaches to Quantum Liquids in Confined Geometries E. Krotscheck and J. Navarro

Vol. 5:

150 Years of Quantum Many-Body Theory A Festschrift in Honour of the 65th Birthdays of John W Clark, Alpo J Kallio, Manfred L Ristig and Sergio Rosati Raymond F. Bishop, Klaus A. Gernoth and Niels Ft. Walet

Vol. 6:

Proceedings of the Eleventh International Conference on Recent Progress in Many-Body Theories Edited by Raymond F. Bishop, Tobias Brandes, Klaus A. Gernoth, Niels R. Walet and Yang Xian

Vol. 7:

Introduction to Modern Methods of Quantum Many-Body Theory and Their Applications Adelchi Fabrocini, Stefano Fantoni and Eckhard Krotscheck

Vol. 9:

Proceedings of the Twelfth International Conference on Recent Progress in Many-Body Theories Edited by J. A. Carlson and G. Ortiz

Vol. 10: Proceedings of the Thirteenth International Conference on Recent Progress in Many-Body Theories Edited by S. Hernandez and H. Cataldo

Forthcoming Vol. 2:

Microscopic Approaches to the Structure of Light Nuclei Edited by R. F. Bishop and N. R. Walet

Vol. 8:

Pairing in Fermionic Systems: Basic Concepts and Modern Applications Edited by S. Armen, M. Alford and J. W. Clark

Series on Advances in Quantum Many-Body Theory - Vol, 10

Proceedings of the 13th International Conference

RECENT PROGRESS MANY-BODY THEORIES Buenos Aires, Argentina

5 - 9 December 2005

Editors

Susana Hernandez Horacio Cataldo University of Buenos Aires, Argentina

\jjp World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG

• TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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RECENT PROGRESS IN MANY-BODY THEORIES Proceedings of the 13th International Conference Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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CONTENTS Preface

ix

Series Editorial Board and Other Committees

xi

Foreword by the Editors

xiii

F E E N B E R G MEMORIAL MEDAL P R E S E N T A T I O N

1

Raymond Bishop and Hermann Kummel: Feenberg Medalists 2005 The Coupled Cluster Method

3

C. E. Campbell, S. A. Chin and John W. Clark Quantum Theory "Without Measurement"

12

H. G. Kummel Towards a Coupled-Cluster Treatment of SU(iV) Lattice Gauge Field Theory

22

R. F. Bishop, N. Lighterink and N. R. Walet A T R I B U T E TO S E V E N D E C A D E S : PHYSICS AND MUCH MORE A Tribute to John Walter Clark on his 70th Birthday C. E. Campbell Application of Support Vector Machines to Global Prediction of Nuclear Properties

39 41

47

J. W. Clark and H. Li A Tribute to Manfred L. Ristig on his 70th Birthday

62

L. Szybisz The Physics of Liquid Para-Hydrogen T. Lindenau, M. L. Ristig, K. A. Gernoth, J. Dawidowski and F. J. Bermejo Q U A N T U M FLUIDS A N D SOLIDS Quantum Reflection, Evaporation, and Transport Currents at 4 He Surfaces

67

79 81

V. Apaja, E. Krotscheck, A. Rimnac and R. E. Zillich Radial Distribution and Liquid Structure Function for Liquid Para-Hydrogen at Low Temperatures

91

K. A. Gernoth, M. J. Harrison and M. L. Ristig Variational Description of Weakly Interacting Bose Gases in 3 Dimensions F. Mazzanti, M. Saarela and V. Apaja v

95

VI

Stability and Spectra of Small 3 He- 4 He Clusters

105

J. Navarro, S. Fantoni, R. Guardiola and A. Zucker Bose-Einstein Condensation in Bulk and Confined Solid Helium

115

L. Reatto, M. Rossi and D. E. Galli Continuum Theory of Superflow in Supersolid 4 He: A Review

127

W. M. Saslow Analysis of an Interatomic Potential for the Condensed Phases of Helium

138

S. Ujevic and S. A. Vitiello Liquid 4 He Adsorbed Films on very Attractive Substrates /. Urrutia and L. Szybisz Q U A N T U M M O N T E CARLO STUDIES OF M A N Y - B O D Y SYSTEMS A N D Q U A N T U M COMPUTATION Monte Carlo Simulation of Boson Lattices

142

147 149

V. Apaja and 0. F. Syljudsen Thermal Entanglement in Spin Systems

153

N. Canosa and R. Rossignoli Limits on the Power of Some Models of Quantum Computation

158

G. Ortiz, R. Somma, H. Barnum and E. Knill Free Rotation of Dopants in Superfluid Helium Clusters

168

S. Paolini and S. Moroni Projected Entangled States: Properties and Applications F. Verstraete, M. Wolf, D. Perez-Garcia and J. I. Cirac Quantum Monte Carlo Study of Overpressurized Liquid 4 He at Zero Temperature L. V. Markic, J. Boronat, J. Casulleras and C. Cazorla

178

190

COLD ATOMS A N D F E R M I O N S AT T H E B E C - B C S CROSSOVER

201

Spin 1/2 Fermions in the Unitary Regime at Finite Temperature

203

A. Bulgac, J. E. Drut and P. Magierski Deconfinement and Cold Atoms in Optical Lattices M. A. Cazalilla, A. F. Ho and T. Giamarchi

208

Vll

Exact BCS Solution in the BCS-BEC Crossover

218

J. Dukelsky, G. Ortiz and S. M. A. Rombouts Quantum Monte Carlo Study of the Ground-State Properties of a Fermi Gas in the BCS-BEC Crossover S. Giorgini, S. E. Astrakharchik, J. Boronat and J. Casulleras

228

Collapse of K-Rb Fermi-Bose Mixtures in Optical Lattices D. M. Jezek and H. M. Cataldo

238

BCS-BEC Crossover in a Superfluid Fermi Gas Y. Ohashi

243

Bose-Einstein Condensate Superfluid - Mott Insulator Transition in an Optical Lattice

254

A. M. Rey, E. A. Calzetta and B.-L. Hu P H A S E T R A N S I T I O N S A N D LOW DIMENSIONAL SYSTEMS Zero-Temperature Phase Diagram of Dissipative Random Ising Ferromagnetic Chains L. F. Cugliandolo, G. S. Lozano and H. F. Lozza Phase Transitions in Ultra-Cold Two-Dimensional Bose Gases D. A. W. Hutchinson and P. B. Blakie

259

261 266

Quantum Critical Behavior in the Insulating Region of the 2D Metal Insulator Transition D. Neilson and D. J. Wallace Geldart

271

Intermediate Symmetries in Electronic Systems: Dimensional Reduction, Order out of Disorder, Dualities, and Fractionalization Z. Nussinov, C. D. Batista and E. Fradkin

282

Information Geometry and Phase Transitions

293

M. Portesi, A. L. Plastino and F. Pennini Mapping Reaction Paths in Phase-Space

297

J. Tailleur, S. Tanase-Nicola and J. Kurchan Quantum Fluids in Nanopores

307

N. M. Urban and M. W. Cole An Extended Concept of Universality in a Statistical Mechanics Model C. Wexler, C. M. Lapilli and P. Pfeifer

315

Vlll

ELECTRONS A N D F E R M I O N SYSTEMS Theoretical Investigation of 3D Nanostructures on Cu Surfaces: The Influence of the Local Environment S. Frota-Pessoa and A. B. Klautau Infrared-Absorption Spectrum of Electron Bubbles in Liquid Helium M. Pi, M. Barranca, V. Grau and R. Mayol

323

325 335

Fluctuations and Pairing in Fermi Systems: A Crossing-symmetric Approach 345 K. F. Quader Theory of Electron Spectroscopies in Strongly Correlated Semiconductor Quantum Dots

355

M. Rontani Nucleation of Vortices in Thin Superconducting Disks

365

M. B. Sobnack, F. V. Kusmartsev, D. R. Gulevich and J. C. H. Fung N U C L E A R SYSTEMS

369

A New Realistic Many-Body Approach for the Description of High-Energy Scattering Processes off Complex Nuclei M. Alvioli, C. Ciofi degli Atti and H. Morita Variational Description of Few-Nucleon Systems: Bound and Scattering States

371

376

A. Kievsky, M. Viviani, L. E. Marcucci and S. Rosati The Number Self-Consistent Renormalized Random Phase Approximation

380

A. Mariano Coupled-Cluster Theory for Nuclei T. Papenbrock, D. J. Dean, J. R. Gour, G. Hagen, M. P. Piecuch and M. Wloch

385 Hjorth-Jensen,

Correlations in Hot Asymmetric Nuclear Matter A. Polls, A. Rios, A. Ramos and H. Miither

393

AUTHOR INDEX

405

SUBJECT INDEX

407

PREFACE The Thirteenth International Conference on Recent Progress in Many-Body Theories (RPMBT-13) was held at the University of Buenos Aires, Buenos Aires, Argentina, 5-9 December 2005. The present volume contains most of the invited talks plus a selection of excellent poster presentations. This conference series is now firmly established as one of the premier series of international meetings in the field of Many-Body physics. The first official RPMBT meeting was held in Trieste in 1978, in response to several precursor meetings that accentuated the need for a continuing series. The most important of these, which can be regarded as RPMBT-0, is the 1972 conference on The Nuclear Many-Body Problem organized by F. Calogero and C. Cioffi degli Atti in Rome. Additionally, there were two very significant workshops held in 1975 and 1977 at the University of Illinois, Urbana, with Vijay Pandharipande as the chief organizer. The quantum many-body community suffered a profound loss with Pandharipande's untimely death in early 2006. Later conferences in the series have been the 1981 RPMBT-2 meeting in Oaxtepec, Mexico; the 1983 RMPBT-3 meeting in Altenberg, Germany; the 1985 RPMBT-4 meeting in San Francisco, USA; the 1987 RPMBT-5 meeting in Oulu, Finland; the 1989 RPMBT-6 meeting in Arad, Israel; the 1991 RPMBT-7 meeting in Minneapolis, USA; the 1994 RPMBT-8 meeting in Schloss Seggau, Styria, Austria; the 1997 RPMBT-9 meeting in Sydney, Australia; the 1999 RPMBT-10 meeting in Seattle, USA; the 2001 RPMBT-11 meeting in Manchester, UK; the 2004 RPMBT-12 meeting in Santa Fe, USA, and the present 2005 meeting in Buenos Aires. Highlights of past meetings can be found in earlier volumes of this series. This conference series is also responsible for awarding the prestigious Eugene Feenberg Memorial Medal in Many-Body Physics. This medal, first presented in 1985, is designated for work that is firmly established and that can be demonstrated to have significantly advanced the field of many-body physics. Past recipients have included David Pines (1985), John W. Clark (1987), Malvin H. Kalos (1989), Walter Kohn (1991), David M. Ceperley (1994), Lev P. Pitaevskii (1997), Anthony J. Leggett (1999), Philippe Nozieres (2001), and Spartak T. Belyaev and Lev P. Gor'kov (2004). Professors Kohn and Leggett received the Nobel Prize in 1998 and 2003 respectively. We are pleased that the Tenth Feenberg Medal was awarded at this conference to Professors Hermann Kummel and Raymond Bishop for their development and application of the Coupled-Cluster Method to diverse physical problems. During the past half century, the application of this quantum many-body physics approach has contributed substantially to our understanding of challenging problems in condensed matter physics (including electrons in solids, nuclear matter, quantum liquids and gases, and quantum magnetism), atomic and molecular physics, nuclear physics, and subnuclear physics/quantum field theory (see R.F. Bishop and H. G. Kummel, "The Coupled-Cluster Method," Physics Today, March 1987). The Coupled-Cluster Method is one of very few ab initio methods that has an enduring impact in quantum chemistry. The presentation was made by Charles IX

X

E. Campbell, Chair of the Feenberg Medal Selection Committee. The text of the tribute as well as the responses of the Medal recipients are included in this volume. The conference also hosted a special session to celebrate the 70th birthdays of John Clark and Manfred Ristig, whose very substantial contributions to many-body physics were warmly recalled by Charles Campbell and Leszek Szybisz. These reminiscences, as well as Clark's and Ristig's lectures, are collected in this volume. The current conference maintains the tradition of covering the entire spectrum of theoretical tools developed to tackle important and current quantum many-body problems with the aim of fostering the exchange of ideas and techniques among physicists working in diverse subfields of physics such as nuclear and sub-nuclear physics, astrophysics, atomic and molecular physics, quantum chemistry, complex systems, quantum field theory, strongly correlated electronic systems, magnetism, quantum fluids and condensed matter physics. The highlights of the conference included state-of-the-art contributions to the understanding of supersolid Helium, BEC-BCS crossover, fermionic BEC, quantum phase-transition, computing, simulations, as well as the latest results on more traditional topics of liquid helium, droplets, nuclear and electronic systems. The conference continues to demonstrate the vitality and the fundamental importance of many-body theories, techniques, and applications in understanding diverse and novel phenonomena at the cutting-edge of physics. The Local Organizing and Program Committees deserve great thanks in creating a well-run and productive meeting, with an exciting program of talks and poster presentations. We also thank the speakers and participants for making the meeting a lively mutual learning experience. It is a special pleasure to thank the local organizers, Susana Hernandez, Leszek Szybisz, Guillermo Dussel, Ana Maria Llois and Horacio Cataldo, for their tireless efforts in making the conference not only a success, but a memorable occasion. We are all indebted to Susana's resourcefulness, the hospitality and charm of Buenos Aires, and the resilience of the Argentinian economy.

Siu A. Chin Chair, International Advisory Committee for the Series of International Conferences on Recent Progress in Many-Body Theories College Station, USA

SERIES EDITORIAL B O A R D A N D OTHER COMMITTEES

SERIES EDITORIAL B O A R D for the Series on Advances in Quantum Many-Body Theories R. F. Bishop (Chair), UMIST, Manchester, UK C. E. Campbell, University of Minnesota, Minneapolis, USA J. W. Clark, Washington University, St. Louis, USA S. Fantoni, SISSA, Trieste, Italy

I N T E R N A T I O N A L ADVISORY C O M M I T T E E for the Series of International Conferences on Recent Progress in Many-Body Theories R. Bishop, UMIST, Manchester, UK C. Campbell (Treasurer), University of Minnessotta, Minneapolis, USA J. Carlson, Los Alamos National Laboratory, USA S. A. Chin (Chairman), Texas A&M University, College Station, USA J. W. Clark, Washington University, St. Louis, USA A. Fabrocini, Universita di Pisa, Italy P. Fullde, Max-Planck Institut for Physics of Complex Systems, Dresden, Germany Susana Hernandez, Universidad de Buenos Aires, Argentina E. Krotscheck, Johannes-Kepler Universitat, Linz, Austria H. Kiimmel (Honorary President) , Ruhr-Universitat Bochum, Germany C. Lhuillier, Universite Pierre et Marie Curie, Paris, France A. H. MacDonald, University of Texas at Austin, USA E. Manousakis, Florida State University, Tallahassee, USA D. Neilson, Universita di Camerino, Italy A. Polls (Secretary), Universitat de Barcelona, Spain M. Saarela, University of Oulu, Finland

P R O G R A M M E COMMITTEE FOR T H E T H I R T E E N T H CONFERENCE R. F. Bishop (UK) J. Boronat (Spain) C. Campbell (USA) J. W. Clark (USA) G. Dussel (Argentina) E. S. Hernandez (Argentina) XI

Xll

K. Hallberg (Argentina) E. Krotscheck (Austria) D. Neilson (Italy) J. P. Paz (Argentina) A. Polls (Spain) M. Ueda (Japan) M. Weissmann (Argentina)

LOCAL ORGANIZING C O M M I T T E E FOR T H E T H I R T E E N T H CONFERENCE Susana Hernandez (Chair) Leszek Szybisz Guillermo Dussel Ana Maria Llois Horacio Cataldo all at University of Buenos Aires, Argentina

F O R E W O R D B Y T H E EDITORS

The Thirteenth International Conference on Recent Progress in Many-Body theories took place at the University of Buenos Aires, Buenos Aires, Argentina, from December 5 to 9, 2005. The Conference Programme consisted of 43 oral presentations plus near 30 posters. This book contains a majority of those invited talks and a selection of short write-ups corresponding to posters. We warmly thank Ray Bishop and Hermann Kiimmel for their valuable advice in order to collect short contributed papers for publication in this volume. Notwithstanding, the final decision regarding the contents of this book corresponds to the Editors. We regret not being able to include material from several outstanding talks, and warmly thank all authors appearing in this volume for joining us in the effort of offering these proceedings to the Quantum Many-Body international community, for the extreme care in preparing their contributions and for the high quality of the final product. A special word of gratitude is due to Siu Chin, Chair of the International Advisory Committee of the Conference Series, for enriching this volume with an enlightening preface, to Charles Campbell for his vivid remembrance of John Clark's achievements and scientific life in his 70th birthday, and to Leszek Szybisz for his warm recollection of Manfred Ristig's life and career at the turn of his seventh decade. As in previous volumes of this series, the articles are collected into groups that intend to reflect the spirit of the meeting, on the understanding that both the grouping and the headings are somehow arbitrary and that many papers could appear in more than one group, or under a different classification. The present scheme aims at orienting the reader to locate the material delivered at the conference, in a broad sense. As members of the Local Organizing Committee and Editors of this volume, we feel largely gratified for the outcome, that to our belief meets the highest standard of the series.

Susana Hernandez Horacio Cataldo Buenos Aires, Argentina April 30, 2006

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FEENBERG MEMORIAL MEDAL PRESENTATION

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R A Y M O N D BISHOP A N D H E R M A N N KUMMEL: F E E N B E R G MEDALISTS 2005 THE COUPLED CLUSTER M E T H O D

C. E. CAMPBELL School of Physics and Astronomy, University Minneapolis, Mn 55455, U.S.A. campbelWumn.edu

Minnesota

S. A. CHIN Department of Physics, Texas A &M University College Station, TX 77843, USA chin ©physics, tamu. edu J. W. CLARK Department of Physics, Washington University St. Louis, MO 63130, USA jwc@wuphys. wustl. edu

1. The Eugene Feenberg Memorial Medal Raymond Bishop and Hermann Kummel have been named as the recipients of the The Tenth Eugene Feenberg Award in Many-Body Physics, in recognition of their key roles in the development and application of the powerful coupled cluster method of quantum many-body theory. Feenberg Medals were presented to the awardees on December 5, 2005 at the Thirteenth International Conference on Recent Progress in Many-Body Theories, held in Buenos Aires, Argentina. The Eugene Feenberg Memorial Medal was established in 1983 by the quantum many-body physics community to honor Eugene Feenberg, a pioneer in theoretical nuclear physics, for his unique and enduring contributions to the development of the microscopic quantum many-body theory of strongly interacting systems. The award commemorates and celebrates Feenberg's wise stewardship of a field that permeates all branches of physics, his deep physical insights and great formal achievements, his dedicated service as teacher and mentor, and his exemplary integrity in his professional and personal life. The selection of Feenberg Medalists is conducted under the auspices of the International Advisory Committee for the Series of International Conferences on Recent Progress in Many-Body Theories. The previous recipients of the award are David Pines (1985), John W. Clark (1987), Malvin H. Kalos (1989), Walter Kohn (1991),

3

Hermann Kiimmel and Raymond Bishop

5 David M. Ceperley (1994), Lev P. Pitaevskii (1997), Anthony J. Leggett (1999), Philippe Nozieres (2001); and Spartak T. Belyaev and Lev P. Gor'kov (2004). 2. Eugene Feenberg As we near the centenary of his birth, it is especially appropriate to include memories of Eugene Feenberg on this occasion. Born on October 19, 1906 in Fort Smith, Arkansas, Feenberg received a B.A. in physics and M.A. in mathematics in 1929 from the University of Texas, Austin after three years of study. While a Harvard doctoral student he spent a year and a half in Europe as a Parker Traveling Fellow, visiting the institutes of Sommerfeld, Pauli, and Fermi. His time in Europe was cut short by the turmoil accompanying the Nazi ascent to power in Germany. Returning to Harvard, he completed his Ph.D. in 1933 under the direction of E. C. Kemble, founder of one of the earliest significant schools of theoretical physics in the United States. As a second mentor, John van Vleck also had a strong influence on Feenberg's early professional development. His Ph.D. thesis, on quantum scattering theory, contained the first statement and proof of the quantum optical theorem. In the ensuing years, Feenberg held appointments as Instructor or Fellow at Harvard, Wisconsin, and the Institute for Advanced Studies, collaborating with Eugene Wigner, Gregory Breit, John Bardeen, and Melba Phillips, among others. In 1936, back-to-back papers in The Physical Review by Cassen and Condon and Breit and Feenberg presented evidence and arguments for the charge independence of nuclear forces and formally identified the first internal symmetry (isospin) of particle physics.1'2 After eight years on the faculty of New York University, intermixed with four years in defense work at Sperry Gyroscopic during World War II, Feenberg joined the faculty of Washington University in St. Louis as an associate professor. In 1964 he succeeded Edward Condon as the Wayman Crow Professor of Physics, a chair previously held by Arthur H. Compton, and subsequently by Edwin T. Jaynes and currently by John Clark, continuing the very high standards and traditions of this important chair. Much of Feenberg's research during the first phase of his career was concerned with nuclear theory, in work that culminated in publication of Shell Theory of the Nucleus by Princeton University Press 50 years ago. 3 In the mid-1950's, a careerlong interest in perturbation theory stimulated his early entry into the new field of quantum many-body theory, coincident with pioneering developments in Russia and the West driven by advances in field theory. This engagement brought a shift of Feenberg's primary focus from nuclear problems to the theory of quantum fluids, most notably the helium liquids, a subject to which he would contribute very importantly for the next two decades until his death in 1977. Along with his students, he developed the method of correlated basis functions (CBF) in order to cope with the strong, short-range repulsion between helium atoms - a feature that defies mean-field approaches and makes the theory of the helium fluids virtually intractable using ordinary perturbation theory. The early part of this research is

6 the subject of his now-classic monograph Theory of Quantum Fluids.* Eugene Feenberg's courage in confronting the challenge of strong, short-range correlations by conceiving and developing a comprehensive ab initio framework to deal with them quantitatively from first principles, was characteristic of his research style. This body of work is the foundation of much of modern theoretical research on strongly correlated quantum fluids. His personal integrity and his high standards of professional achievement and behavior continue to inspire and guide his former students and colleagues. The esteem in which Eugene Feenberg is held by those who knew and learned from him is eloquently captured in an excerpt from the commencement address "Giants Must Stand on Solid Ground" delivered in 1991 at the Massachusetts Institute of Technology by Walter Massey, President of Morehouse College and former Director of the U.S. National Science Foundation: 5 So, where do young researchers first learn the rules of the game that protect the integrity of their endeavors? For generations, the community has relied on the unique mentor-apprentice relationship that develops during the process of doctoral and postdoctoral research to teach these important lessons. When all goes well, a bond of trust develops between the professor and his or her students which is grounded in intellectual curiosity, a desire to discover new knowledge, and a common commitment to truth. What is conveyed and what is learned is more than simply a body of facts; it is an approach to understanding, an appreciation of standards, and a set of values - in effect, an ethic. I was very fortunate in my own graduate training to have had a research director from whom I learned a great deal of physics and much more besides. Professor Eugene Feenberg (now deceased) was a prominent physicist, well-respected among his peers, a man of integrity and unyielding honesty. All of us who were his students saw these traits in him and profited from working with him. To come to class ill-prepared or not take time to listen to his students and their problems; to publish a result prematurely simply to gain priority; or to put his name on a paper by one of his students when he had not shared in the work himself - all these things were anathema to Gene Feenberg. Those of us who were his students realize how lucky we are-to have had an almost ideal mentor-apprentice relationship. Those who would like to know more about Eugene Feenberg and his work are invited to read George Pake's biographical memoir for the National Academy of Sciences.6 Presentation of Feenberg Medals at successive conferences in the series on Recent Progress in Many-Body Theories serves both to remind us of and to perpetuate this man's extraordinary influence on physics and physicists. Our two laureates, Hermann Kummel and Raymond Bishop, exemplify the steadfast integrity and uncommon commitment to the highest of standards that were so visible in the professional

7 career and personal life of Eugene Feenberg. 3. Hermann Kiimmel The Tenth Feenberg Memorial award cites Hermann Kiimmel for his role in the creation and early development of the coupled cluster method, and pioneering high-accuracy applications of it to problems in nuclear and subnuclear physics. The volume of the World Scientific Series on Advances in Quantum Many-Body Theory devoted to the proceedings of MB 11 contains an excellent biography of Hermann Kiimmel, in a tribute written by Raymond Bishop in celebration of Kiimmel's 80th birthday. 7 From that source we learn that Kiimmel was born in 1922 in Berlin, and that all of his formal education took place in that city. He received the Diplom in 1950 from Humboldt University in the East Zone, and his Ph.D. in theoretical physics in 1952 from the Free University in West Berlin, where he continued with his research until moving to the U.S. to work at the University of Iowa. It was during his two years as a research associate in Iowa City that, in consultation with Fritz Coester, the foundations were laid for the coupled cluster method (CCM) (more commonly known in the early days as the Exponential-5 Theory). Kummel and Coester presented the seminal ideas of CCM in a 1960 paper published in Nuclear Physics.8 After a brief period at the University of Tubingen, Kummel spent several years in Mainz with joint appointments at the Max Planck Institute for Nuclear Chemistry and the University of Mainz. Then came two years back in the U.S. as Professor of Physics at Oklahoma State University. Returning to Mainz as a senior scientist at the Max Planck Institute and adjunct professor at the University, he built a strong research group in nuclear theory. In 1969 Kummel moved with his entire research group to take a Chair in Physics at the newly founded Ruhr University in Bochum (RUB), Germany. His efforts soon established Bochum as one of the world's leading centers in quantum many-body theory. Significantly, it was at this time that he turned his attention in earnest to the theoretical and computational development of CCM. Although nuclear systems remained the primary focus of applications, important results were also obtained for the electron gas, a basic problem in solid-state physics and a testbed for manybody theories. The level of sophistication achieved in Bochum under his leadership, in both formal and computational strength, has rarely been matched. He retired in 1988 at the mandatory age, but has remained active in research at RUB as Professor Emeritus. An authoritative and insightful account of Hermann Kiimmel's pioneering contributions to the formal development of CCM and its subsequent application in a host of problem settings may be found in his paper entitled "A Biography of the Coupled Cluster Method" .9 Kummel was both a co-founder of the CCM and the first to appreciate and exploit its power in the context of nuclear physics. Arguably, the

8

earliest applications of coupled cluster theory by his group to medium-mass atomic nuclei stand to this day as never having been surpassed. Considering the computing power that was available at the time, the results obtained for both closed-shell nuclei and open-shell nuclei comprising one or two valence particles or holes outside closed shells, are breathtaking in their sophistication and accuracy. They were undoubtedly the first fully microscopic calculations for nuclei to be demonstrably converged for a selection of the best available, realistic, two- and three-nucleon interaction potentials, for both ground- and excited-state energies and properties. Based on his successes in applying coupled cluster theory to finite nuclei, Kiimmel next turned his attention to various strongly correlated systems in subnuclear physics and quantum field theory. In particular, his treatment of the deuteron from the field-theoretic standpoint of pions and nucleons interacting via the standard pseudoscalar, isovector coupling, may be recognized as a landmark calculation. It provides a rare example of a fully converged calculation for a non-trivial, strongly interacting quantum field theory. Throughout his long and fruitful career, Hermann Kiimmel has been a pioneer, a leader, an inspiration, and a mentor in the international community of many-body theorists. 4. Raymond Bishop The Tenth Feenberg Memorial Medal award cites Raymond Bishop for his development of the coupled cluster method toward a comprehensive ab initio approach, and innovative applications across the full spectrum of subfields of quantum many-body physics. Raymond Bishop was born in London in 1945. After earning his B.A. degree at the Queen's College, Oxford in 1966 and a Ph.D. in theoretical physics at Stanford University in 1971, he returned to England as Science Research Council Research Fellow. Subsequently, he was Senior Research Associate and Lecturer in the Department of Physics, Manchester University, simultaneously being associated with the Theory Group at the Science Research Council, Daresbury Laboratory. Bishop returned to the U.S. and spent several years at the University of California, Berkeley as Staff Scientist at the Lawrence Berkeley Laboratory and Lecturer in the Department of Physics. In 1979 he joined the faculty of the Department of Mathematics at the University of Manchester Institute of Science and Technology (UMIST). He was appointed Professor of Theoretical Physics at UMIST in 1988 and Head of the Department of Mathematics in 1991. Theoretical Physics moved to the Department of Physics in 1995, where Bishop became Head in 1996. He is currently Professor of Theoretical Physics in the School of Physics and Astronomy of the combined University of Manchester formed in the merger of UMIST with Manchester University in 2004. Bishop initiated his research program in the coupled cluster method during lengthy visits to Bochum in the mid-to-late-1970s. While at Bochum, he began a fruitful collaboration with Karl-Heinz Luhrmann, in which they applied CCM the-

9 ory exhaustively, and with great success, to that touchstone problem of many-body theory, the electron gas. From that point on, Bishop was responsible for a series of pivotal formal developments in coupled cluster theory, often with the close participation of the late Jouko Arponen. He has systematically and energetically developed the CCM into the elegant, consistent, flexible, and accurate tool for microscopic applications that we know today. Indeed, he has been the driving force behind the 25-year evolution of coupled cluster theory from a promising framework for ab initio treatment of quantum many-particle systems into its mature form as a powerful and universal method for quantitative investigation of ground states, excitations, and phase transitions. Applications abound in nuclei and nuclear matter, atoms and molecules, quantum fluids and electronic systems in condensed matter, lattice-spin and lattice-gauge models, quantum optics, and quantum field theory. In more detail, his numerous significant achievements include: (a) incisive analysis of correlations in the electron gas at high, intermediate (metallic), and low densities; (b) extensive development of the extended coupled cluster formalism, its uses to map arbitrary quantum field or quantum many-body theories exactly onto classical Hamiltonian mechanics, and for its applications to the treatment of excited states and the zero-temperature hydrodynamics of strongly-interacting condensed Bose fluids; (c) formulation and implementation of a translationally-invariant coupled cluster theory for light nuclei; (d) pathbreaking studies of spin-lattice models yielding new insights into quantum antiferromagnets; (e) exploration of Hamiltonian lattice-gauge theories in the coupled cluster framework; and (f) publication of comprehensive and authoritative reviews on the important ramifications of coupled cluster theory within both chemistry and physics. At Manchester, Bishop has established one of the world's foremost groups in quantum many-body physics and has directed an intensive computational effort exploiting automated symbolic manipulation. At the same time he has developed and leads what amounts to an informal international school of many-body physics research through his extensive collaborations throughout the world and his development and application of CCM to an enormously broad range of theoretical physics subdisciplines.

10 5. Selected significant publications Hermann Kiimmel F. Coester and H. Kiimmel, Short-range correlations in nuclear wave functions, Nucl. Phys. 17, 477-485 (1960). H. Kiimmel, Theory of many-body wave functions with correlations, Nucl. Phys. A176, 205-218 (1971). H. Kiimmel and K. H. Liihrmann, Equations for linked clusters and the energy variational principle, Nucl. Phys. A191, 525-534 (1972). R. Offermann, W. Ey and H. Kiimmel, Degenerate many-fermion theory in exp(S)form, Nucl. Phys. A273, 349 (1976) H. Kiimmel, K. H. Liirmann and J. G. Zabolitzky, Many fermion theory in the exp S-(or coupled cluster) form, Phys. Rep. 36C, 1-63 (1978) Raymond Bishop R. F. Bishop and K. H. Liihrmann, Electron correlations. II. Ground- state results at low and metallic densities, Phys. Rev. B26, 5523-5557 (1982). J. S. Arponen, R. F. Bishop, and E. Pajanne, Extended coupled cluster method. I. Generalized coherent bosonization as a mapping of quantum theory into classical Hamiltonian mechanics, Phys. Rev. A36, 2519-2538 (1987). R. F. Bishop, M. F. Flynn, M. C. Bosca, E. Buendia, and R. Guardiola, Translationally invariant coupled-cluster theory for simple finite systems, Phys. Rev. C42, 1341-1360 (1990). R. F. Bishop, J. B. Parkinson, and Y. Xian, Coupled-cluster treatments of correlations in quantum antiferromagnets, Phys. Rev. B44, 9425-9443 (1991). R. F. Bishop, An overview of coupled cluster theory and its applications in physics, Theor. Chim. Acta 80, 95-148 (1991) References 1. B. Cassen and E. U. Condon, Phys. Rev. 50, 846 (1936) 2. G. Breit and E. Feenberg, Phys. Rev. 50, 850 (1936) 3. E. Feenberg, Shell Theory of the Nucleus (Princeton University Press, Princeton, NJ 1955) 4. E. Feenberg, Theory of Quantum Fluids (Academic Press, New York, 1969) 5. W. Massey, in The Tech 111, Number 28, p. 2 (June 26, 1991); available from: http://www-tech.mit.edu/Vlll/N28/massey.28n.html. Accessed November 15, 2005. 6. G. Pake, in Biographical Memoirs, (The National Academies Press); available from: http://www.nap.edu/readingroom/books/biomems/efeenberg.html. Accessed November 12, 2005; see also K. A. Brueckner, C. E. Campbell, J. W. Clark, and H. PrimakofF, Nuclear Physics A317, pp. i-vii (1979). 7. R. F. Bishop, in Proceedings of the 11th International Conference on Recent Progress in Many-Body Theories, ed R.F. Bishop, T. Brandes, K. A. Gernoth, N. R. Walet, and

11 Y. Xian (World Scientific, Singapore, 2002) pp. 319-333; R. F. Bishop, International Journal of Modern Physics B17, 5295(2003) 8. F. Coester and H. Kiimmel, Nucl. Phys. 17, 477-485 (1960). 9. H. Kiimmel, in Proceedings of the 11th International Conference on Recent Progress in Many-Body Theories, ed R.F. Bishop, T. Brandes, K. A. Gernoth, N. R. Walet, and Y. Xian (World Scientific, Singapore, 2002) pp. 334-348; , H. Kiimmel, International Journal of Modern Physics BIT, 5311 (2003)

Q U A N T U M THEORY "WITHOUT MEASUREMENT" HERMANN G. KUMMEL Institut fur Theoretische Physik II, Ruhr- Universitat 44780 Bochum, Germany [email protected]

Attempting to describe the "collapse of the wave function" of the orthodox quantum theory as a physical process, a natural assumption is that the measurement apparatus is subject to the same physical laws as the measured object, leading to a theory without rules for measurement. This - together with the loss of classical-quantum correspondence for chaotic systems - enforces the inclusion of the environment. I very shortly describe the present state of the art in this field - a challenge for the quantum many body community. Keywords:

decoherence; environment; measurement

1. Prologue: Why is the Orthodox Quantum Theory Incomplete? This meeting is about applied quantum theory. We all are convinced that there are good reasons to work in this field. Rarely in science exist results as exact as those obtained in quantum theory. Also, we know that there are many beautiful experiments exploring and testing the most sophisticated aspects of quantum physics and that their results are always in agreement with the predictions from this theory. Quantum computing, cryptography, teleportation and so on are the keywords which matter here. Just because we don't have the least doubts that quantum theory in the "orthodox" (i.e. the Copenhagen or Bohr-Born) form correctly describes nature, we rarely worry about the fundamental problems involved. We more or less consider them as a nuisance, something esoteric, even irrelevant, and of interest more for philosophers or for retired and quite old people like myself rather than for active physicists. Yet quite a few of our famous colleagues like Einstein, Schrodinger, Wigner, Feynman, Bell and others never believed that quantum physics is fully understood, or complete, or correct. I quote Feynman as a representative of the younger (post-Einstein) generation: "I think I can safely say that nobody today understands quantum physics". Bell even believed that "Quantum mechanics carries the seeds of its own destruction", he said "It is rotten". Nevertheless all these people fully recognized the utilitarian success of the theory - they had to. I believe, however, that now the situation has changed: a consistent and complete theory is in sight. Thus is may be a good idea to present the new developments to the quantum mechanics practitioner. Let me first write down the well known rules of the orthodox interpretation

13 about which almost everybody agrees that they are correct for applying quantum theory. I leave out some aspects which are irrelevant for the points I wish to make, even though some of them have been hotly debated in the past. From now on "object" means the physical system we want to investigate, and "apparatus" is the measurement apparatus we use to do this. For lack of space I don't go into the rather delicate (almost philosophical) questions of what constitutes these two entities, or what are the boundaries between them, or what an "observable" really means. The rules are: 1. Attach to any observable O a Hermitian operator O = X)°ilVi)(Vi| (f° r instance pairs of generalized momenta and coordinates (Pj,Qj) with the usual commutation relations). This implies the existence of a Hilbert space for the object. 2. Invent a suitable Hermitian Hamilton operator H characteristic for the object at hand. 3. The measurement of an observable O puts an observed object "suddenly" into an eigenstate \tfi) of the Hermitian operator O related to this observable ("collapse or reduction of the wave function"). The value of this observable attached to the object is the eigenvalue o*. The apparatus is macroscopic and the measured value of the observable is imprinted on it as a macroscopic observable. 4. The probability for the occurrence of Oj is given by pi = Ky'ilV')!2 with ip as the state of the object before the measurement ("Borns rule"). 5. Before and after measurements the state of the object moves according to the Schrodinger equation driven by the unitary operator exp(—i/(Ht). To rules 1 and 2: Here one is free to define physical objects (Hamiltonians) and observables and their hermitian operators. Discussing the fundamentals of quantum physics we have to assume that all relevant operators are known some way or another. Rule 4 should be interpreted as follows: pi is approximately the fraction of objects found in state \ipi) of a large finite number of objects investigated. It is standard to introduce a density matrix g representing the ensemble of the quantum objects. For such an ensemble one obtains pi = Tr(g\)(tf>\ => Qafter

= ^ P i l f i ) ^ ,

(1)

where the arrow does not represent a unitary transformation. gafter is interpreted as describing an ensemble with the fraction pt of objects in the state \ipi). According to rule (3) what is called a "measurement" in quantum theory actually is "putting the system into a state". Moreover, after the measurement on a single object we have learned almost nothing about the status of the object as it was before the measurement. Even if we repeat the measurement as often as possible on as many as possible objects - all of them prepared as well as possible in the same way - we merely know (approximately) the set |(vi|V')| 2 and not the wave function before the measurement. Our daily experience and the belief that we can learn something

14 about the history of the universe from its present state is not compatible with this rigor inherent in the orthodox quantum theory. I shall come back to this problem later. On the other hand, after a (perfect) measurement we know everything about the state of the object immediately after this measurement - and forever if we know the Hamiltonian and can use it (rule (5)). After these comments on the established fundamental laws of quantum physics let me come to the central problem: the theory in this form is incomplete since for the sudden change of the wave function there is no equation of motion. In this interpretation the mechanism of the measurement remains unexplained. As we shall see, the search for something very general describing this process, i.e. for some universal features for all kinds of measurement apparatuses, leads to a series of more problems. In any case, as soon as the apparatus is considered to be more than a "black box" one arrives at the following four alternatives for a complete theory: I. Both object and apparatus move according to the Schrodinger equation. II. Object and apparatus are quantum systems not treated on the same footing ("Consistent histories", "many worlds" and other interpretations). III. Everything is classical - although described by waves instead of particles. Quantum phenomena appear only if one does not look - or is not able to look carefully enough. The "hidden parameter" theories are of this kind. Due to Bells famous papers it is known that they necessarily are non-local and as a consequence violate special relativity. IV. A "super"-theory, encompassing everything including general relativity. String theories as theories of everything come into our mind. However, the question how the measurement apparatus is embedded in it to my knowledge never has been considered. Such ideas at present are very far from concrete. I follow a minimalist approach preserving as much of the known quantum theory as possible, especially the quantum equation of motion. This is alternative I of above. In other words: both microphysics and macrophysics including the measurement apparatus will be described as ordinary (if appropriate many body) quantum objects subject to the Schrodinger equation. In the end such a quantum theory would have no rules for the measurement - like in classical physics. Using this approach recently much progress has been made and conclusions have been reached on which very likely a final theory can be built.

2. Object and Apparatus Coupled together Once one has decided to treat also the measurement apparatus as a quantum object one is bound to construct a wave function containing the degrees of freedom of object and apparatus. Thus one necessarily has to deal with product as well as with entangled states: before the measurement the state of object plus apparatus is the product of the wave functions \tp) = 5ZCJ|> =

5>iw>i4>> -»• Scii^)!^),

(2)

where the arrow now is a shortcut for this unitary transformation. Hamiltonians containing an object-apparatus interaction H Q A which perform (2) can easily be constructed, see below. This approach - due to John v. Neumann 1 - seems to be the only one possible as long as one sticks to the idea to have a common entity "object+apparatus" described by the time development of quantum dynamics and separated from the rest of the world, i.e. as a closed system. And one is tempted to be happy by observing that if the apparatus is seen in the state \A,) the object certainly is in the state \?(x)E?(x)]

=

^dtEf

(11)

25 The components of the fields can be obtained via the relation A; = -Ti[A,\a] , ^ = -Tr[B,A a ] . (12) 9 9 The constraints obey the same commutation relations as the generators of the gauge group. Thus, Gauss' law cannot be implemented as a strict operator condition as it leads to contradictions, since the non-commuting constraints cannot all be diagonalised simultaneously. However, within the physical (in this case colourless) subspace defined by S a (x)|Phys> = 0

,

(13)

no such problem arises, since the eigenvalue of the commutators is also 0. The space of states consists of wave functionals, taking values on the SU(iV) group manifold. From Eq. (13) we find N2 — 1 functional conditions on each wave functional, consisting of functions on the group manifold at each space point. As is well known, quantisation of problems involving redundant degrees of freedom (i.e., where some of the equations of motion are constraints) is quite involved. The two main techniques used are Dirac and BRS quantisation, and they require a large amount of additional analysis. For more details one can consult the seminal work by Dirac, 5 ' 6 as well as Refs. [4,7,8,9,10]. If we are able to work within the physical subspace only, we can ignore these formal problems and define the quantisation of the canonical momenta, II? = doAf, by n

?W = ^ = -iSiFw •

(14)

which involves a functional derivative 11,12 with respect to the field variables. Since A$ is not dynamical, we cannot associate a canonical momentum with it. We therefore use the temporal gauge, Ag = 0, which leaves us with a residual gauge freedom ^(x) independent of the time coordinate, such that under the transformation A^ (x) —» ^A^ (x), where ^(x)=^(x)^(X)^-1(x) + i[a^(x)]r1W *F^(x)=^(x)F^(x)^-1(x) ,

,

(15) (16)

with <j> G SU(JV), the Lagrangian is invariant. 2.2. Discretisation

and the

Hamiltonian

Many quantum field theories suffer from singularities, both in the infrared and ultraviolet limits. In many interesting cases, such as QCD, 13 these are renormalisable. Rather than dealing directly with the continuum, we shall regularise the problem by introducing a simple hypercubic lattice in the D-dimensional space, with lattice spacing a. Since we are pursuing a Hamiltonian approach, time will remain continuous. We shall concentrate here only on the pure gauge theory.

26 As is by now well known (and see our full paper 14 for further details), the system is described by a set of gauge fields (or chromo-vector-potentials) N2-l

1

A

>=29^2

X A

" ' >

( 17 )

a= l

that are now defined on the links I of the lattice. They are Hermitian, since Aa is Hermitian. The chromo-electric fields, Ef, are the corresponding canonical momenta obeying the commutation relation, [E?,At!) = -i5u,Saa,

.

(18)

The group elements are the link variables, Ui = exp{iaAi}, which are thus N x N matrices. They correspond to parallel transporters on the lattice. Since Ai is an element of the su(N) Lie algebra, Ui is an S\J(N) matrix. The product of four such group elements around a primitive square on the lattice (usually called a plaquette or Wilson plaquette) defines the corresponding plaquette operator, Up = U\U2U^lU^ . Their traces form the simplest gauge-invariant quantities on the lattice. Ignoring temporarily problems with overcompleteness of the variables, one can derive the Kogut-Susskind Hamiltonian, 9 JV2-1

*KS =laP ^ f e g2 2aD-

£ l W + A5>[2-CV

(HE + \HM)

,

(19)

where D is the number of spatial dimensions, and A = a2D~6/g4. The sum on / in the electric term HE runs over all links, while the sum on p in the magnetic term HM runs over all plaquettes on the lattice. Our physical states are now gauge-invariant in the vacuum sector. So far, the price we have paid for adopting the Hamiltonian approach is twofold. Firstly, explicit Lorentz invariance has been broken and, secondly, the problem remains of determining the physical subspace. 2.3. A first look at handling

the

constraints

As we have seen above, the gauge freedom leads to constrained dynamics. The residual gauge symmetry involves all time-independent local gauge transformations. These gauge freedoms thus generally prove an obstacle to establishing a proper set of variables in which the wave functional may be expressed. Before proceeding let us do some simple counting of the number of independent (i.e., unconstrained) degrees of freedom in our formulation. The primitive variables are the set Z?" defined on the set of Ni links. Hence, the total number of degrees of freedom is (iV2 — l)iV/, where the first factor is simply the number of SU(./V) group generators. However,

27 Table 1. The number of degrees of freedom on a hypercubic lattice of size n X n X • • • n, for different, numbers of space dimensions, D. [The D = " 1 " case comprises n plaquettes on a line.] Dimensionality, D

"1"

number of sites, Ns number of links, 7Vj number of plaquettes, Np

2(n+l) 3n + 1 n

2

3 2

(n + 1) 2n(n+l) n2

D >2 3

(n + 1) 3ra(n + l ) 2 3n2(n+l)

(n + 1)D Dn(n + l ) 0 - 1 | D ( D - l)n2(n + 1)D~2

there are constraints between them due to Gauss' law. Thus, we can readily derive the lattice versions Qf of the continuum generators Qa (x), for each of the Ns lattice sites i. Hence, the number of (unconstrained) independent degrees of freedom is Nu = {N2 - 1){N, - Ns + 1)

,

(20)

where the additional unity term in the second factor in this expression arises due to the overall global gauge degree of freedom that would finally still remain. As shown in Table 1, we see that the total number of plaquettes, Np on the lattice is given by Np = Ni — Ns + 1 for the cases D = 1 and D = 2, but this relation is not true for D > 2. The construction of Gauss' law makes it clear why it is so attractive to work with plaquette variables or, more generally, with traces of products of group operators Ui around closed loops (i.e., Wilson loops), since these variables are automatically gauge-invariant. Hence, for D = 1, 2 the plaquette variables form a complete (i.e., neither over- nor under-complete) set of unconstrained variables. We turn below to the vexed question of what variables to choose for D > 2. 3. Gauge Fixing and Maximal Trees As we have seen above, it is the gauge freedom that leads to the constrained dynamics and the fact that the set of link variables must, in general, be over-complete. However, since the link variables are still one of the most attractive sets to use, we are motivated to fix the gauge as much as possible. We actually choose to fix the gauge fully (apart from an overall global gauge transformation that we cannot fix) by separating all of the links on the spatial lattice into two sets. One set is chosen to be just sufficient to connect any two lattice sites in a unique way. Any such set of links is called a maximal tree. 15 ' 16 Its choice for a given lattice is clearly not unique, but one such choice is shown in Fig. 1 for D = 3. The case shown corresponds to the union of all links on the z-axis for y = z = 0, and all links in the y-direction for z = 0, and all links in the ^-direction, with an obvious labelling for the axes. Thus, all SU(7V) link variables Ui are thereby divided into two non-overlapping sets, {Ui} —• {V; |/ £ tree; W7j |/ £ tree}. The variables {Wi} are now associated with the irrelevant gauge degrees of freedom; they are essentially the irrelevant longitudinal electric fields. Conversely, the variables {V;} form our basis for the relevant degrees of freedom, the magnetic variables. However, these latter link variables are not by themselves invariant under local gauge transformations. Hence the wave

28

Fig. 1. A particular choice of maximal tree for D = 3 is shown on the left; and the path associated with a typical variable X\ for a particular link not on this tree is shown on the right.

function cannot depend directly on them since, gauge in variance would then be violated. Instead, we transform the links not on the tree into closed loops by combining them with (unique) paths on the maximal tree joining each end of the link from and to the origin, as is illustrated in Fig. 1. In this way we make a unique mapping, VI • Xi; V7 0 tree, where Xi simply comprises a product of the corresponding Ui or U[~ operators on the links comprising the closed loop, where the inverse operators appear on links pointing back to the origin, in a convention that orients the links on the maximal tree along the direction away from the origin. Thus, all the variables Xi transform in the same way under local gauge transformations with the gauge transformation at the origin, and are invariant under all other local gauge changes. We know that when we fix the gauge we cannot fix a global gauge transformation, and we are thus led to identify this with the one at the origin. We have shown14 that the electric field operator E" associated with the link / on the maximal tree can be transformed by a body-fixed frame rotation, such that when it acts on one of the X m -variables one of the following relations holds, depending on the position of the link, 1

a

a

1

1 a

*

a

•

Ei Xm = ——A Xm , Et Xm = —XmX , Ei Xm = ~^A Xm + —Xm\ . (21) In this last equation the first result holds if link / is part of the path leading up to link m from the origin, the second result holds if it is part of the path leading back back from link m to the origin, and the third result holds if it is part of both paths. In this way the electric field operator generates long-range interactions between two link variables Xi and Xi>. The links Wi on the maximal tree change under local gauge transformations, and any function in the physical subspace must hence be a function only of the gauge-invariant Xi variables. The gauge is effectively fixed by setting the SU(iV) matrix operators Wi to be the unit operator, so that we have {VI; Wj} —>• {Xi\ 1}. It is now a matter of simple counting to show that the Xi variables are precisely the (unconstrained) canonical variables of our Hamiltonian theory in the colourless sector, as summarised in Table 2. On a finite lattice of length n in D dimensions there are N, = (n + \)D lattice points, and Ni = Dn(n 4- l ) ^ - 1 links. The maximal tree contains Nt = n-|-n(n + l)4-ri(n-|-l) 2 -| hn(n-|-l) z > _ 1 = (n + l)D-l = N,-l

29 Table 2. The number of degrees of freedom in the maximal tree approach on a hypercubic lattice of size nx n X • • -n, for different number of space dimensions, D. Dimensionality, D

2

number of sites, Ns number of links, Ni number of links on tree, Nt number of independent X\ variables, AT; — Nt

3

D > 2

(n+1)2

(n + 1)3

(n + l ) D

2 n ( n + 1) n + n(n + l ) v?

3n(n + l ) 2 (n + l ) 3 - 1 2n 3 + 3n 2

Dn(n+l)fl-1 (n + \)D - 1 Dn(n + 1 ) D _ 1

links, as can easily be seen from (the D-dimensional generalisation of) our explicit choice of maximal tree. Therefore there are iVj — Nt = Dn{n + l ) - 0 - 1 — (n + l)D + 1 remaining links, and the same number of variables Xi. Each of these variables has N2 — 1 degrees of freedom. Hence, the number of degrees of freedom of the Xi variables is precisely equal to Nu, the number of unconstrained canonical degrees of freedom in the theory, as given by Eq. (20). 3.1. The Hamiltonian

in terms

of the Xi

variables

We now start with the Kogut-Susskind Hamiltonian of Eq. (19) and assume that it will act on a function of the variables Xi. For the electric piece of the Hamiltonian, HE, we make use of the result in Eq. (21). In the magnetic part, HM, we can set all of the links on the maximal tree to unity. After some algebra we arrive at a final expression for our Hamiltonian given entirely in terms of our complete set of (unconstrained) canonical variables, Xi. We do not quote the somewhat unedifying expression here. The interested reader is referred to Ref. [14] for further details and the full expression. 4. Colour Neutrality and Further Constraints Although we have solved the major problem above of the constrained dynamics due to the gauge freedom, we still have to face a further problem of overcompleteness related to the number of degrees of freedom in the traces of SU(iV) matrices. This leads to the existence of dependences among the traces of the Wilson loop variables, usually known as Mandelstam constraints. 17 Our wave function in the pure gauge sector studied here should be a colour singlet, and the above complications arise when we impose colour neutrality on the wave function. Thus, the natural choice for us is now to work with traces of products of the variables Xi, as discussed above. A suitable approach would be to construct a basis of eigenstates of the electric part of the Hamiltonian, HE, and calculate matrix elements of the magnetic energy between these states. Such an approach is a quite natural calculational scheme for the Hamiltonian approach. One can also use the method inherent in the Lagrangian calculations, which is based on invariant integration over the full group. 18 However, for a proper Hamiltonian approach this

30 discards many of the advantages of the method. To find eigenstates of the electric operator, one can resort to three general approaches. Firstly, group theory gives us, in principle, a way to construct general eigenstates, the group characters. However, for a large basis, and N > 2, this is extremely involved,19 unless it can be automated, and we see no easy way to do this. A second approach is based on integrating configurations, and constructing orthogonal combinations from them. In this case one must start off with much larger overcomplete sets of configurations, and at increasing orders the integration, based on Creutz's integration method, 20,21 tends to become more and more involved.22 The third approach is based on the action of the electric operator itself, which leads to a block-diagonal matrix which has to be diagonalised to recover the eigenstates. In combination with a symbolic method explained more fully elsewhere,14 this seems to be the most powerful approach, which allows one to tackle any arbitrary SU(iV) group. This is the method that we have developed and that we prefer to use with the maximal tree approach, although we note that it also has wider applicability, for further details the reader is referred to Ref. [14] Our whole approach so far has been designed for applications in mind using a universal method of microscopic quantum many-body theory in the Hamiltonian formulation, such as the coupled cluster method (CCM). 23 We do not have the scope in the present paper to discuss in any detail such many-body applications. Instead, we consider below the simpler limit where all plaquettes appear independently, and in so doing we introduce an extremely useful set of angular variables that considerably aid and simplify the analysis. We show that they serve to map an SU(iV) one-loop problem onto an N-fermion problem on a torus. 5. The One-Plaquette Problem If all of the loop variables are combined in such a way that they form plaquettes, 21 we may consider our wave functional as depending only on the plaquette variables. Since the trace of a matrix is invariant under similarity transformations, we may write N

Ti[Um] = Tr[(VUV-1)m~\

=^e

i m

^:;

m £ [1,2, • • •, N]

,

(22)

->'=i

where V diagonalises the SU(iV) unitary matrix U of the one-plaquette problem. Hence, out of the N2 — 1 degrees of freedom for SU(iV) the trace depends only on its N eigenvalues exp(ij), all of which lie on the unit circle in the complex plane. The angular variables <j>j satisfy the constraint Ylj=i ^i = 0 coming from the SU(N) condition, detU = 1. The wave functions of the one-plaquette problem can be expressed in a basis of group characters 24 labelled by the standard SXJ(N) partitions A = [Ai, A2, • • •, AJV]These are simply proportional to the eigenstates of the electric operator (which

31 is just the quadratic Casimir operator) for the single plaquette. We denote the group characters of SU(iV) as X A ( ^ ) , where we use the shorthand notation

^SSSJ

•

,25)

acts on the antisymmetric wave function $ . When the the term Tr[t/]4-Tr[t/ -1 ] in the magnetic part, HM, of the Hamiltonian in Eq. (19), acts on a wave function, it now reduces to a multiplication 24 of group characters,

[XiW + Xi(-*)]e*= E A'=A + 1

£

*'+ E

£A

' >

(26)

A'=A-1

where the symbols A ± 1 in the sums denote the inclusion of all possible partitions in which one and only one of the A; —>• A,- ± 1 (with 1 < i < TV), while still fulfilling the requirements in Eq. (23). The operators Tr[[/] and Tr[[/ _ 1 ] thus act as raising and lowering operators, which act in a simple way on the Young tableaux corresponding to the group characters or wave functions with specific symmetry properties. 14 We see from the above results that for a single SU(TV) loop comprising L links (where L = 4 for a plaquette), the electric operator, HE, in the Hamiltonian essentially yields L times the difference between the total kinetic energy of the TV particles (now fermions) on a torus (represented by the angular variables i — 0- We need

32 SU(2)

Fig. 2.

SU(3)

SU(4)

SU(5)

The lowest levels of the one-plaquette energy spectra for SU(2) to SU(5).

also to impose the constraint on the centre-of-mass momentum, P = $3,-_i Pj = 0, since 2 , and we resort to the numerical procedure described above to obtain the remaining results shown in Fig. 2. We note that the spectra for N > 2 are much richer than for N = 2. They include not only many avoided crossings but also, very interestingly, what appear to be several real crossings in the cases N = 4 and N = 5. We have verified that the distances between the respective two levels in these latter cases are equal to zero within our numerical accuracy. Such real crossings, of course, are a reflection of some (otherwise hidden) symmetry in these theories. They clearly deserve further investigation. Finally, we note that the region of the coupling constant shown in Fig. 2 is insufficient to observe the asymptotic convergence to the harmonic approximation discussed above. However, we have checked numerically, by going to values g~4 > 50, that our stated results for the degeneracy factors for the equidistant harmonic oscillator spacings in this limit are correct. 5.2. Independent-plaquette

wave

functionals

The results for the one-plaquette problem have more consequences for more general (e.g., variational or CCM) wave functionals than one might at first suspect. Thus, if the trace variables, £ m = m _ 1 Tr([/ m ), of the one-plaquette matrix are used, the wave functional is a function of the group characters only. The specific wave functional that is the sum of one-plaquette functions, =

£

F({>«}f = 1 )

,

(29)

plaquettes a

naturally leads to the sum of one-plaquette problems, leading to total energies which are the sum of one-plaquette energies. However, the corresponding product wave functional

mw)=

F

n

(3°)

({+J}?=I) .

plaquettes a

also leads to the same result as we now show. This absence of correlations between nearest-neighbour plaquettes, follows in our case from the symmetry of (the original bosonic) wave functional, $ \ , in the angular variables, * * ( • • • &

•••*,•

•••)

=

* * ( •

••*,••

• • & • • • )

.

(31)

where we decompose F(^i.-^Jv) = ^cA*A(^i..-0JV) . x Therefore the cross-product term from the electric operator vanishes,

(32)

£ > A T * ^ ••*&)*'*• («*!•••
, (33)

34 where a and /3 are plaquettes containing the link /, and ^ = det(J)^A, where det(J) = |A| 2 is the Jacobian of the transformation from trace variables to angular variables, Jmj = d£m/dj = iexp(im_,). The differential operator DN contains both sets of angular operators, . Therefore the Hamiltonian, when acting on the product wave functional |II), also reduces to the sum of one-plaquette Hamiltonians. 6. Summary and Whither Next? In summary, we have shown how to fix the gauge by using the maximal-tree gauge, which is specific to the lattice versions of gauge field theories. It does not suffer from the typical problems of gauge fixing in the continuum. We then introduced a very natural set of angular variables , simply related to the trace (or Wilson loop) variables for a fundamental plaquette, in terms of which the SU(Vv") one-loop problem can be mapped onto an ./V-fermion problem on a torus. Exact solutions were obtained for a single plaquette in both the weak-coupling (g —>• 0) and strongcoupling (# -» oo) limits. The eigenstates in the strong-coupling limit are the group characters of the corresponding group, and in the weak-coupling limit the harmonic approximation to the fermion problem leads to explicit formulas for the degeneracies of the equidistant energy levels. In terms of the group characters (i.e., the eigenstates of the electric piece of the Hamiltonian), the ground and low-lying energy states of the single plaquette can be determined numerically for arbitrary values of the coupling constant, since the Hamiltonian reduces to a simple linear equation in terms of these group characters. The terms in the magnetic piece of the Hamiltonian are simple raising and lowering operators in this basis, and hence the energy spectra are found by a simple configuration-interaction method technique of diagonalising the resultant block-diagonal Hamiltonian matrix in a restricted basis. The procedure can be automated using the theory of multiplication of group characters, in terms of the Young tableaux of the corresponding group. Another, somewhat unexpected, advantage of our approach is the fact that correlations between spatially distinct trace variables cancel. This leads to the intriguing possibility that spatial correlations in the full lattice problem might be weak in our approach, which would in turn tend to suggest that successive approximations in a fully consistent microscopic many-body approach based on our approach might be expected to converge rapidly. Clearly, the simplicity of the angular variables formulation is promising for more elaborate wave functionals. An obvious extension to include correlations is to employ the coupled cluster method (CCM), 23 which has been extensively applied with great success to a wide variety of quantum many-body systems and quantum field theories. Typical applications of the CCM to strongly interacting continuum quantum field theories include 4fieldtheory 25,26 and a model field theory of pions and nucleons.27 There have been many applications of the CCM to various spin-lattice models in quantum magnetism. 28 Other applications to lattice field theories include the 0(4) nonlinear sigma model as a model of meson field

35 theory with a phase transition due to chiral symmetry breaking. 29 Previous attempts to use the CCM in lattice gauge theory have largely been confined to the ground and low-lying excited states in the pure gauge (gluon) sectors of the U(l) and SU(2) cases in one and two spatial dimensions. 1,2 As we have seen, for D < 3 the plaquette variables Up form a natural complete set. However, they are overcomplete for D > 3. One of our key aims here has been to find a natural complete set of variables for this case. We have shown that the variables Xi fulfil this role. At the heart of the CCM is the parametrisation of the ket-state many-body wave function as an exponential of a correlation operator formed from a linear combination of mutually commuting creation (or excitation) operators with respect to a model or reference state |$), |*> = es|$>

;

S=$>|/>n, from the space of input vectors x into a higher-dimensional feature space, which is "hidden" from input and output (and corresponds to the hidden layer in MLPs). This is done in terms of an inner-product kernel A'(x,Xfc) =ipT(x)ifi(xk) satisfying certain mathematical conditions, notably Mercer's theorem. 20 The second step is to find a hyperplane that separates (approximately, in general) the features identified in the first step. This is accomplished by the optimization procedure sketched above. A self-contained introduction to the SVM technique is beyond the scope of the present contribution. Excellent treatments are available in the original work of Vapnik as expounded in Refs. 6,7 and in Haykin's text 2 (see also Ref. 19). To provide some essential background, let us consider a regression problem corresponding to a map x —> y(x), where x is an input vector with n components x^l\ and suppose that T training examples indexed by k are made available. Then the optimal approximating function takes the form T

2/opt(x) = ^ ( a A - a 4 ) A ' ( x , X f c ) ,

(1)

k= \

Solution of the optimization problem stated above determines the parameters ak and a'k, and the support vectors of the machine are defined by those training patterns for which ak ^ a'k. Different choices of the inner-product kernel appearing in Eq. (1) yield different versions of the Support Vector Machine. Common choices include A'(x,x A ) = (x T x fc + l ) p ,

(2)

corresponding to the polynomial learning machine with user-selected power p; a Gaussian form A-(x,x*) = e x p ( - 7 | | x - x * | | 2 )

(3)

containing a user-selected width parameter 7, which generates a radial-basisfunction (RBF) network; and A-(x,x A ) = tanh(/3 1 x T x A +/3 2 ),

(4)

which realizes a two-layer (one-hidden-layer) perceptron, only one of the parameters j3i, fa being independently adjustable. We also draw attention to a generalization of the RBF kernel (3) introduced recently as a simplified version of what is called ANOVA decomposition, 21 having the form

K(*,xk)=ff>p [-7 (4° - *(°y

d

j '

(5)

The Support Vector Machine may be considered as a feedforward neural network in which the inner-product kernel, through an appropriate set of m elements

52 A'(x,Xfc), defines a layer of hidden units that embody the mapping from the ndimensional input space to the m-dimensional feature space. These hidden units process the input patterns nonlinearly and provide outputs that are weighted linearly and summed by an output unit. As already pointed out, the familiar structures of radial-basis-function networks and two-layer perceptrons can be recaptured as special cases by particular choices of kernel. However, the SVM methodology transcends these limiting cases in a very important way: it automatically determines the number of hidden units suitable for the problem at hand, whatever the choice of kernel, by finding an optimally representative set of support vectors and therewith the dimension of the feature space. In essence, the Support Vector Machine offers a generic and principled way to control model complexity. By contrast, approaches to supervised learning based on MLPs trained by backpropagation or conjugate-gradient algorithms depend heavily on rules of thumb, heuristics, and trial and error in arriving at a network architecture that achieves a good compromise between complexity (ability to fit) and flexibility (ability to generalize). 3. Application to Nuclear Systematics In this section we summarize the findings of recent explorations of the potential of Support Vector Machines for global statistical modeling of nuclear properties. The discussion will focus on the predictive reliability of SVM models relative to that of traditional "theory-thick" models. The properties that are directly modeled in these initial studies, all referring to nuclear ground states, are (i) the nuclear mass excess AM = M — A, where M is the atomic mass, measured in amu, (ii) /3-decay lifetimes of nuclides that decay 100% via the /3~ mode, and (iii) nuclear spins and parities. The requisite experimental data are taken from the on-line repository of the Brookhaven National Nuclear Data Center (NNDC) at http://www.nndc.bnl.gov/. The experimental mass values are those of the AME03 compilation of Audi et al. 22 Extensive preliminary studies have been performed to identify inner-product kernels well suited to global nuclear modeling. Earlier work converged on the ANOVA kernel (5) as a favorable choice, and corresponding results have been published in Ref. 19. More recently, we have introduced a new kernel that yields superior results, formed by the sum of polynomial and ANOVA kernels and named the pA kernel. (Satisfaction of Mercer's theorem is conserved under summation.) The new kernel contains three parameters (p, 7, and d) that may be adjusted by the user. Aside from parameters contained in the inner-product kernel, the SVM procedure involves a constant C giving the user control over the tradeoff between complexity and flexibility, plus an additional control constant e in the regression case, measuring the tolerance permitted in the reproduction of training data. Thus, SVM models developed with the pA kernel contain four or five adjustable parameters (five in all applications reported here). To allow for a meaningful evaluation of predictive performance (whether inter-

53 polation or extrapolation), the existing database for the property being modeled is divided into three subsets, namely the training set, validation set, and test set. These sets are created by random sampling, consistently with approximate realization of chosen numerical proportions among them, e.g. (100-2R):R:R for training, validation, and test sets, respectively, with R < 25. The training set is used to find the support vectors and construct the machine for given values of the adjustable parameters p, 7, d, C, and e. The validation set is used to guide the optimal determination of these parameters, seeking good performance on both the training and validation examples. The test set remains untouched during this process of model development; accordingly, the overall error (or error rate) of the final model on the members of the test set may be taken as a valid measure of predictive performance. When one considers how SVM models might be applied in nuclear data analysis during the ongoing exploration of the nuclear landscape, it seems reasonable that consistent predictive performance for 80:10:10 or 90:5:5 partitions into training, validation, and test sets would be sufficient for the SVM approach to be useful in practice. The SVM approach has been applied to generate a variety of global models of nuclear mass excess, beta-decay lifetimes, and spin/parity, corresponding to different kernels, databases, partitions into training/validation/test sets, and tradeoffs between the relative performance on these three sets. Here we will focus on those models considered to be the best achieved to date. Moreover, due to limited space, we will restrict the discussion to the most salient features of those models and to an assessment of their quality relative to favored traditional global models and to the best available MLP models. Further, more detailed information may be found at the web site http://abacus.wustl.edu/Clark/svmpp.php, which generates SVM estimates of the listed nuclear properties for (Z, N) pairs entered by visitors. This web site will be periodically updated as improved SVM models are developed.

3.1. SVM Models of Atomic Mass

Surfaces

Development and testing of the SVM mass models to be highlighted here are based on the AME03 data for all nuclides with Z, N > 16 and experimental masses having error bars below 4%. This set of nuclides is divided into the four classes: even-Zeven-AT (EE) even-Z-odd-JV (EO), odd-Z-even-JV (OE), and odd-Z-odd-iV ( 0 0 ) . Separate SVM regression models were constructed for each such "even-oddness" class. This does introduce some minimal knowledge about the problem domain into the modeling process; one might therefore say that the models developed are not absolutely theory-free. However, the data itself gives strong evidence for the existence of different mass surfaces depending on whether Z and N are even or odd. Knowledge of the integral character of Z and N may, quite properly, bias the SVM toward inclusion of associated quantum effects.23 Table 1 displays performance measures for models based on an 80:10:10 target partitioning of the full data set among training, validation, and test sets, respec-

54 tively. Inspection of the actual distributions of these sets in the Z — N plane shows that substantial fractions of the validation and test sets lie on the outer fringes of the known nuclei, significantly distant from the line of stable nuclides. Accordingly, performance on the test set measures the capability of the models in extrapolation as well as interpolation. Performance on a given data set is quantified by the corresponding root-mean-square (rms) error ,t + c.c, which is adiabatically turned on with the rate eet, and finally one sets e —> 0 + . The excitation operator 5U(t) is determined by stationarity of the action

S= JdtC(t) = Jdt(9(t)\H + Uext(t)-ih^\*(t))

.

(10)

For weak perturbations we may linearize the equations of motion. 1 6 , 1 7 Truncating the series (2) at the level of 2-body fluctuations, the variations of S with respect to Sui(ri;t) and „(r) and their adjoint states m\ipn) = 6mn. The approximations that lead to working formulas for the dynamic structure function that both capture the right physics and are practical in the inhomogeneous geometry have been discussed in Ref. 16. These approximations allow us to solve the second equation of motion analytically, and lead to the so-called convolution approximation for the density-density response function, X (r,

r »

= ] T v ^ ) ^ ( r ) [Gst(Lj) + Gtt{-uj)] <MO V P F ) •

(17)

St

Here Gst(co) = [H[UJ -UJS+

ie]6st + E ^ u ) ] - 1

(18)

is the Green's function (phonon propagator), with the self energy

•mn

The three-phonon matrix elements Vmn are overlap integrals of three Feynman states, see Ref. 16. By setting T,st(co) = 0 one recovers the random phase approximation (RPA) - which is, equivalently, obtained from time-dependent Hartree theory or time-dependent density functional theory. The three-phonon corrections have the important consequence that the density-density response function can have a non-trivial imaginary part due to phonon decay processes. 2.3. Transport

current

For calculating scattering probabilities as measured in experiments, we need to keep only those contributions that persist in the homogeneous parts of the system (in the vacuum or deep in the bulk liquid) and that have a non-zero spatial average. We shall call this portion the transport current. By a lengthy diagrammatic analysis, 19 the second order current (9) can be written in the suggestive form ;(2)/r..rt_:(2)/

,^;(2).

..

K

jr(r;*)=j^(r;*)+«r;t) = 8m. + n(r) J d3r'Pl(r')Ju