Proceedings of the Julian Schwinger Centennial Conference : 7-12 February 2018, National University of Singapore 9789811213144, 9811213143

Table of contents :
3. Remarks on the Abraham-Minkowski problem, in relation to recent radiation pressure experiments --
1. Introduction --
2. Dielectric model --
3. Conductive model --
4. Statics and dynamics of the circular plate --
4.1. Statics --
4.2. Dynamics --
5. Conclusion --
Appendix A. Relationship to the Casimir effect --
Acknowledgments --
References --
4. Manifest non-locality in quantum mechanics, quantum field theory and quantum gravity --
1. Introduction --
2. Quantum Theory and Quantum Spacetime --
2.1. Modular variables --
2.2. Modular space and geometry of quantum theory --
2.3. Born geometry 3. From Modular Spacetime to Quantum Gravity --
3.1. Intrinsic non-commutativity in metastring theory --
4. Effective Quantum Fields and Manifest Non-locality --
4.1. Non-local excitations: Metaparticles --
5. Conclusions --
Acknowledgments --
References --
5. Unruh acceleration radiation revisited --
Ia. Introduction: Dedication --
Ib. Introduction: Overview --
II. Quantum Optics Route to Obtaining Unruh Radiation in Minkowski Coordinates --
IIa. Accelerating atom in a vacuum --
IIb. Excitation of a Static Atom by the Rindler Vacuum III. Acceleration radiation and the equivalence principle using Unruh-Minkowski modes --
IIIa. Accelerating atom --
IIIb. Accelerating mirror --
IV. Acceleration Radiation and the Equivalence Principle --
V. The "Bogoliubov" Path to Unruh Radiation --
VI. Periodicity Trick for Unruh Temperature --
VII. Conclusions --
Acknowledgments --
References --
6. Non-thermal fixed points: Universal dynamics far from equilibrium --
1. Introduction --
2. Non-thermal fixed points --
3. Kinetic theory of non-thermal fixed points --
3.1. Properties of the scattering integral and the T-matrix 3.2. Scaling analysis of the kinetic equation --
4. Low-energy effective field theory --
4.1. Spatio-temporal scaling --
4.2. Scaling solution --
4.3. The case of a single-component gas, N = 1 --
4.4. Relation to predictions of the non-perturbative kinetic theory --
5. Wave-turbulent transport --
6. Topological defects vs. the role of fluctuations --
6.1. Defect dominated fixed points --
6.2. Fluctuation dominated fixed points --
7. Prescaling --
8. Outlook --
Acknowledgments --
References --
7. Baryon isospin mass splittings --
1. Introduction --
2. Baryon Basis and Interactions --
3. Baryon Masses.

Citation preview

11602_9789811212130_TP.indd 1

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World Scientific

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9/9/19 10:15 AM

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

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Cover image: Informal portrait of Julian Schwinger (AIP Emilio Segrè Visual Archives, Segre Collection)

JULIAN  SCHWINGER  CENTENNIAL  CONFERENCE Proceedings of the Julian Schwinger Centennial Conference Copyright © 2020 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.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-121-213-0

For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11602#t=suppl

Printed in Singapore

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Dedicated to the memory of Julian Schwinger, physicist and gentleman

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Contents Dedication

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Preface

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1. The g factor of an electron in hydrogenlike carbon and the precision determination of the electron mass

1

Jonathan Sapirstein 2. The magnetoelectric coupling in electrodynamics

9

A. Mart´ın-Ruiz, M. Cambiaso and L. F. Urrutia 3. Remarks on the Abraham–Minkowski problem, in relation to recent radiation pressure experiments

31

Iver Brevik 4. Manifest non-locality in quantum mechanics, quantum field theory and quantum gravity

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Laurent Freidel, Robert G. Leigh and Djordje Minic 5. Unruh acceleration radiation revisited

61

J. S. Ben-Benjamin, M. O. Scully, S. A. Fulling, D. M. Lee, D. N. Page, A. A. Svidzinsky, M. S. Zubairy, M. J. Duff, R. Glauber, W. P. Schleich and W. G. Unruh 6. Non-thermal fixed points: Universal dynamics far from equilibrium

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Christian-Marcel Schmied, Aleksandr N. Mikheev and Thomas Gasenzer 7. Baryon isospin mass splittings Lai-Him Chan

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8. Multiquark states in the Thomas−Fermi quark model and on the lattice

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Walter Wilcox and Suman Baral 9. Are dyons the preons of the knot model?

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Robert J. Finkelstein 10. The Schwinger−DeWitt proper time algorithm: A history

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Steven M. Christensen 11. From Julian to Jupiter: Unanticipated outcomes

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Margaret Kivelson 12. Julian Schwinger — Recollections from many decades

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Stanley Deser 13. Schwingerians

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Charles Sommerfield 14. Drell−Yan mechanism and its implications

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Tung-Mow Yan 15. My years with Julian Schwinger: From source theory through sonoluminescence

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Kimball A. Milton 16. The statistical atom

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Julian Schwinger and Berthold-Georg Englert 17. Julian Schwinger and the semiclassical atom Berthold-Georg Englert

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18. Fond memories of Julian and Clarice, especially involving Moshe Flato and Noriko Sakurai

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Daniel Sternheimer 19. Speeches by V. F. Weisskopf, J. H. Van Vleck, I. I. Rabi, M. Hamermesh, B. T. Feld, R. P. Feynman, and D. Saxon, given in honor of Julian Schwinger at his 60th birthday Berthold-Georg Englert and Kimball A. Milton

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Julian Schwinger (12 February 1918–16 July 1994) is best known for his work on the theory of quantum electrodynamics (QED), in particular for computing the anomalous magnetic moment of the electron, for developing a relativistically invariant perturbation theory, and for renormalizing QED. For this seminal work, he was awarded the 1965 Nobel Prize in Physics jointly with Richard Feynman and Sin-Itiro Tomonaga. Schwinger is widely recognized as one of the greatest physicists of the twentieth century and one of the greatest teachers of physics. Indeed, his accomplishments are numerous: he began his career as a leader in nuclear physics and discovered tensor forces; he pioneered the theory of the spin-3/2 field; he introduced powerful variational methods in classical electrodynamics and contributed to the theory of synchrotron radiation; he is responsible for much of modern quantum field theory, including a quantum version of the action principle, and the equations for the Green’s function in field theory; he made major contributions to angular momentum and to a reformulation of quantum mechanics; he effectively laid the foundations for non-equilibrium quantum statistical mechanics and for quantum gravity; he developed the first electroweak model, a SU(2) gauge group spontaneously broken to electromagnetic U(1) at long distances; he inferred that there must be multiple neutrinos; he explored the first example of confinement in the Schwinger model, quantum electrodynamics in 1 + 1 dimensions; he discovered the Schwinger terms of current and stress-tensor commutators and the consequent anomalies in quantized fields; he advanced the theory of magnetic charge, monopoles and dyons; and there are many other topics. From 7 to 12 February 2018, the Julian Schwinger Foundation for Physics Research (JSF) together with the Institute of Advanced Studies (IAS) at Nanyang Technological University and the Centre for Quantum Technologies (CQT) in Singapore organized the Julian Schwinger Centennial Conference. More than 100 participants attended the event; all presentations were by invited speakers. Most of the speakers were related to Julian Schwinger in some way, either as students, post-doctoral fellows, collaborators, or friends. On the first three days (7–9 February), the speakers covered a large variety of topics, all linked to Schwinger’s rich scientific legacy, whereas the final day — Monday, 12 February 2018, Schwinger’s 100th birthday — was devoted to personal recollections. In total, there were 38 talks, including five pre-recorded videos by speakers who could not travel to Singapore, and 17 of these talks are now available in this proceedings volume. Also included are an essay of 1985, co-authored by Schwinger but not published previously, and the transcripts of speeches by distinguished colleagues at the 1978 gathering when Schwinger’s 60th birthday was celebrated. Many thanks are owed to the speakers at the Schwinger Centennial and all the other participants for creating such a memorable event. The staff of IAS and CQT — Chris Ong, Toh-Miang Ng, Louis Lim, Lay Hua Tan, Valerie Lim, Aki Honda, Kelly Giam, Resmi Poovathumkval Raju, Geraldine Teo, and others — deserve much praise for working so tirelessly before, during, and after the event, making sure that everything was running smoothly. Berge Englert (JSF & CQT) Chair, Organizing Committee

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Participants in the Schwinger Centennial Conference (Singapore, 7–12 February 2018)

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[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

Walter Wilcox Gordon Baym Manas Mukherjee Michael Lieber Shau-Jin Chang Yu-Tin Huang Martin-Isbj¨orn Trappe Daniel Sternheimer Herbert Fried Klaus Blaum Lars Brink Kris van Houcke Peter Tsang Lester L. DeRaad, Jr.

[29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

Hwa-Tung Nieh Song He Roland Hablutzel Shang Shan Chong Kimball A. Milton Tung-Mow Yan York-Peng Yao Zvi Bern Tobias Haug Choon-Lin Ho Stephen Adler Luis F. Urrutia Rios

[41] [42] [43] [44] [45] [46] [47] [48] [49]

Luyao Ye Ulrike Bornheimer Tserensodnom Gantsog Daniel Greenberger Y. Jack Ng Hui Khoon Ng Seth Putterman Shrobona Bagchi Steven Christensen

[50] [51] [52] [53] [54] [55] [56] [57] [58]

Jing Hao Chai Max Seah Yink Loong Len Doron Gazit Djordje Minic Rui Han Iver H˚akon Brevik Andrzej Czarnecki Christian Miniatura

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Leong Chuan Kwek Thomas Gasenzer M. A. A. Ahmed Paul Indelicato Devin Ky Alexander Hue Lai Him Chan Long Wen Zhou Weijun Li Jun Yang Sim Gordon Semenoff Berge Englert Dario Poletti Yanwu Gu

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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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Sponsors The Julian Schwinger Centennial Conference was jointly organized and sponsored by – the Julian Schwinger Foundation for Physics Research; – the Institute of Advanced Studies, Nanyang Technological University, Singapore; – the Centre for Quantum Technologies, Singapore.

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Participants in the Julian Schwinger Centennial Conference Dmitry A BANIN University of Geneva, Switzerland Stephen A DLER Institute of Advanced Studies, USA M.A.A. A HMED Universiti Putra Malaysia, Malaysia Shrobona BAGCHI Centre for Quantum Technologies, Singapore Gordon BAYM University of Illinois, USA Zvi B ERN University of California, USA Kishor B HARTI Centre for Quantum Technologies, Singapore Klaus B LAUM Max-Planck-Institut f¨ur Kernphysik, Germany Ulrike B ORNHEIMER Centre for Quantum Technologies, Singapore Iver H˚akon B REVIK Norwegian University of Science and Technology, Norway Lars B RINK Chalmers Institute of Technology, Sweden Jing Hao C HAI Centre for Quantum Technologies, Singapore Lai Him C HAN Louisiana State University, USA Shau-Jin C HANG University of Illinois, USA Thanh Tri C HAU Centre for Quantum Technologies, Singapore Jun Hong C HEW National University of Singapore, Singapore Li Ming C HONG Centre for Quantum Technologies, Singapore

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Shang Shan C HONG Signapore Steven C HRISTENSEN University of North Carolina, USA Andrzej C ZARNECKI University of Alberta, Canada Lester L. D E R AAD, Jr. Julian Schwinger Foundation, USA Stanley D ESER Brandeis University, USA Kai D IECKMANN Centre for Quantum Technologies, Singapore Ruiqi D ING National University of Singapore, Singapore Gerald D UNNE University of Connecticut, USA Berge E NGLERT Julian Schwinger Foundation, USA; Centre for Quantum Technologies, Singapore Robert J. F INKELSTEIN University of California, USA See Kit F OONG Nagoya University, Japan Herbert M. F RIED Brown University, USA Xiaozhen F U National University of Singapore, Singapore Gerald G ABRIELSE Harvard University, USA Tserensodnom G ANTSOG National University of Mongolia, Mongolia Thomas G ASENZER University of Heidelberg, Germany Doron G AZIT Hebrew University of Jerusalem, Israel Edward G ERJUOY University of Pittsburgh, USA

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Paweł G LIWA Ericsson, Sweden Daniel G REENBERGER City University of New York, USA Benoˆıt G R E´ MAUD MajuLab, Singapore Yanwu G U National University of Singapore, Singapore Roland H ABLUTZEL Centre for Quantum Technologies, Singapore Rui H AN Centre for Quantum Technologies, Singapore Tobias H AUG Centre for Quantum Technologies, Singapore Song H E Chinese Academy of Sciences, China Hermanni H EIMONEN Centre for Quantum Technologies, Singapore Choon-Lin H O Tamkang University, Taiwan Yu-tin H UANG National Taiwan University, Taiwan Jun Hao H UE Centre for Quantum Technologies, Singapore Paul I NDELICATO Laboratoire Kastler Brossel, France Khiam Aik K HOR Nanyang Technological University, Singapore Margaret K IVELSON University of California, USA Leong Chuan K WEK Institute of Advanced Studies and Centre for Quantum Technologies, Singapore Devin K Y University of Malaya, Malaysia Hock-Chong L EE National University of Singapore, Singapore

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Yink Loong L EN Centre for Quantum Technologies, Singapore Weijun L I Centre for Quantum Technologies, Singapore Michael L IEBER University of Arkansas, USA Clarence L IU National University of Singapore, Singapore Zheng L IU Centre for Quantum Technologies, Singapore Yiping L U Centre for Quantum Technologies, Singapore Sivakumar M. M ANIAM National Institute of Education, Singapore Kimball A. M ILTON University of Oklahoma, USA Christian M INIATURA MajuLab, Singapore Djordje M INIC Virginia Tech, USA Manas M UKHERJEE Centre for Quantum Technologies, Singapore Hui Khoon N G Yale-NUS College, Singapore Kian Fong N G Centre for Quantum Technologies, Singapore Y. Jack N G Julian Schwinger Foundation, USA Travis N ICHOLSON Centre for Quantum Technologies, Singapore Hwa-Tung N IEH Tsinghua Education Foundation, USA Stefan N IMMRICHTER Centre for Quantum Technologies, Singapore Choo Hiap O H Centre for Quantum Technologies, Singapore

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Yen Chin O NG Yangzhou University, China Krzysztof PACHUCKI University of Warsaw, Poland Kok Khoo P HUA Institute of Advanced Studies, Singapore Dario P OLETTI Singapore University of Technology and Design, Singapore Ignatius William P RIMAATMAJA Centre for Quantum Technologies, Singapore Seth P UTTERMAN Julian Schwinger Foundation, USA Nedumaran R AMASAMY Singapore Shabnam S AFAEI Centre for Quantum Technologies, Singapore Jonathan S APIRSTEIN University of Notre Dame, USA Marlan O. S CULLY Texas A&M University, USA Max S EAH Centre for Quantum Technologies, Singapore Gordon S EMENOFF University of British Columbia, Canada Jun Yan S IM Centre for Quantum Technologies, Singapore Kuldip S INGH Centre for Quantum Technologies, Singapore Charles S OMMERFIELD Yale University, USA Chorng Haur S OW National University of Singapore, Singapore Daniel S TERNHEIMER Rikkyo University, Japan Tiong Gie TAN National University of Singapore, Singapore

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Martin-Isbj¨orn T RAPPE Centre for Quantum Technologies, Singapore Peter H. T SANG Brown University, USA Ko-Wei T SENG Centre for Quantum Technologies, Singapore Luis F. U RRUTIA R IOS Universidad Nacional Aut´onoma de M´exico, Mexico Kris VAN H OUCKE Ecole Normale Sup´erieure, France Walter W ILCOX Baylor University, USA Tung-Mow YAN Cornell University, USA Anbang YANG Centre for Quantum Technologies, Singapore York-Peng YAO University of Michigan, USA Luyao Y E Centre for Quantum Technologies, Singapore Gin Hao Y UEN Centre for Quantum Technologies, Singapore Yicong Z HENG Centre for Quantum Technologies, Singapore Long Wen Z HOU National University of Singapore, Singapore and others.

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Speakers at the Julian Schwinger Centennial Conference The speakers are listed in alphabetical order; their affiliations and the titles of the talks are as stated in the conference programme.

Dmitry A BANIN University of Geneva Ergodicity, entanglement, and many-body localization Stephen A DLER Institute of Advanced Studies, Princeton Investigations on gauged Rarita–Schwinger theory Gordon BAYM University of Illinois With Julian looking over my shoulder through the years: From Martin–Schwinger many-particle theory to transport theory in strongly interacting systems Zvi B ERN Department of Physics and Astronomy, University of California at Los Angeles The methodological unification of gravitons and gluons Klaus B LAUM Max-Planck-Institut f¨ur Kernphysik, Heidelberg Precision tests of QED with stored and cooled highly charged ions Iver B REVIK Norwegian University of Science and Technology Remarks on the Abraham-Minkowski problem Lai Him C HAN Louisiana State University, Baton Rouge Baryon Isospin Mass Splittings Shau-Jin C HANG Emeritus Professor of University of Illinois Working with Professor Schwinger Steven C HRISTENSEN Department of Physics and Astronomy, University of North Carolina at Chapel Hill The Schwinger–DeWitt Proper Time Technique: A History Andrzej C ZARNECKI University of Alberta Precise Quantum Electrodynamics of Bound States

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Stanley D ESER Brandeis University; California Institute of Technology Julian Schwinger — Recollections from many decades (pre-recorded video) Gerald D UNNE University of Connecticut The Search for the Schwinger Effect: Non-perturbative Pair Production from Vacuum Berge E NGLERT Centre for Quantum Technologies and Department of Physics National University of Singapore Julian Schwinger and the Semiclassical Atom Robert J. F INKELSTEIN Department of Physics and Astronomy, University of California at Los Angeles Are Dyons the Preons of the Knot Model? (pre-recorded video) Herbert M. F RIED Brown University Non-Abelian QCD: No longer in the Shadow of Abelian QED Gerald G ABRIELSE Leverett Professor of Physics, Harvard University Director of the Center for Fundamental Physics at Northwestern Greatest Triumph of the Standard Model (by video conference) Thomas G ASENZER University of Heidelberg Universal dynamics and non-thermal fixed points Doron G AZIT Hebrew University of Jerusalem Plasma microscopic phenomena in extreme conditions: New approach to the solar abundance problem Edward G ERJUOY∗ Department of Physics and Astronomy, University of Pittsburgh Memories of Julian Schwinger (pre-recorded video) Daniel G REENBERGER City College, City University of New York Schwinger homework problems as the source of research projects ∗ Very

sadly, Professor Gerjuoy passed away a few days before the Schwinger Centennial.

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Song H E Chinese Academy of Sciences and Perimeter Institute Scattering from Geometries Yu-tin H UANG National Taiwan University Geometric unification of symmetries and unitarity Paul I NDELICATO Laboratoire Kastler Brossel (CNRS, ENS, Sorbonne University) The Seven Year Itch: The proton size puzzle 7 years later Margaret K IVELSON Earth, Planetary and Space Sciences, UCLA, Los Angeles Climate and Atmospheres and Space Physics, University of Michigan, Ann Arbor From Julian to Jupiter: Unanticipated Outcomes (pre-recorded video) Michael L IEBER University of Arkansas Off on a Tangent Kimball A. M ILTON University of Oklahoma Reminiscences of Julian Schwinger — Late Harvard, Early UCLA Years Djordje M INIC Virginia Tech Manifest quantum non-locality in quantum mechanics, quantum field theory and quantum gravity Krzysztof PACHUCKI Faculty of Physics, University of Warsaw Quantum electrodynamics of atomic and molecular systems Jonathan S APIRSTEIN University of Notre Dame The g factor of an electron in hydrogenlike carbon and the precision determination of the electron mass Marlan O. S CULLY Texas A&M University; Princeton University; Baylor University Black hole acceleration radiation from quantum optical perspective Gordon W. S EMENOFF University of British Columbia Entanglement and the Infrared

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Charles S OMMERFIELD Yale University Schwingerians (pre-recorded video) Daniel S TERNHEIMER Rikkyo University, Tokyo, and Universit´e de Bourgogne Fond memories of Julian and Clarice, especially involving Moshe Flato and Noriko Sakurai Luis F. U RRUTIA R IOS Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico The magneto-electric effect in axion-electrodynamics Kris VAN H OUCKE Ecole Normale Sup´erieure de Paris Summation of diagrammatic series for a strongly correlated fermionic theory with zero convergence radius Walter W ILCOX Baylor University Multiquark States in the Thomas–Fermi Quark Model Tung-Mow YAN Cornell University, Ithaca Drell–Yan mechanism and Its Implications York-Peng YAO Leinweber Center for Theoretical Physics, University of Michigan, Ann Arbor Kavli Institute for Theoretical Physics, University of California, Santa Barbara Some Comments on Solving the Scattering Equations

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The g factor of an electron in hydrogenlike carbon and the precision determination of the electron mass Jonathan Sapirstein Department of Physics, University of Notre Dame, Notre Dame, IN 46656, USA [email protected]

The role of the bound electron Green function in the recent high precision determination of the electron mass is discussed. Emphasis is placed on the connection to Schwinger’s use of such Green functions in his early work establishing the modern form of QED, his calculation of leading binding corrections, and his work on synchrotron radiation.

1. Introduction Anyone interested in the contributions to science of Julian Schwinger should be aware of the remarkable work of Sylvan Schweber, “QED and the Men who Made It.”1 Seventy years after the eventful year 1948 it is easy to be unaware of the impact Schwinger’s work had on the birth of the modern form of quantum field theory, but the time is vividly brought back to life by Schweber. The electron propagator in an external field played a central role at that time in Schwinger’s calculations. It continued to do so in his work on hydrogen spectroscopy, and also in his calculations of the properties of synchrotron radiation. It in fact remains central to modern advances in bound state calculations that include radiative corrections. While formal expressions can be written down for Green functions in external fields, using them to carry out calculations with the same accuracy as the ever improving experimental results is a challenging problem for QED theorists. A particular example of this involves determining the electron mass to high precision. Extraordinarily precise experiments on hydrogenic carbon in a Penning trap2 along with highly sophisticated theoretical calculations of the bound electron g factor3 have allowed the determination me = 0.000 548 579 909 067(16) u,

(1)

a result accurate to 0.03 part per billion (ppb). As will be discussed below, one component of the original research that showed the electron has an anomalous magnetic moment was the study of g factors of bound electrons. The effect of binding is relatively small, and the largest contribution is the g factor of a free electron, g = 2(1 + ae ), where the anomaly, ae , is ge − 2 . (2) 2 We will see that by gaining a theoretical understanding of the corrections to the g factor caused by the bound state, the results of the experiment can be used to determine the electron mass with the remarkable accuracy shown above. ae ≡

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2. Historical Context The electron mass determination mentioned above depends on an extremely accurate understanding of the bound g factor of an electron. The most important effect of the g factor of a bound electron is simply its role in the Lande g-factor. This depends only on the angular momentum quantum numbers of the state and two g factors, the orbital factor gL and the spin factor gS . In the Dirac theory gS = 2. Gabrielse and collaborators,4 also using a Penning trap, in this case to confine a “free” electron (the small effects of its being bound in the Penning trap will be discussed later), produced another extraordinarily accurate result, ae = 0.001 159 652 180 73(28).

(3)

One of the first indications that the electron had an anomalous magnetic moment depended on g factor determinations. Zeeman effect experiments on the relatively complicated atom gallium and the simpler, but still complex atom sodium were carried out by Foley and Kusch.5 Despite the nontrivial atomic structure, the g factors were expected to behave according to the standard formulas, with ratios, in the case of gallium, of the 2p3/2 and 2p1/2 equal to 2. The experiment5 found 2.00344 ± 0.000012, a clear deviation from the prediction. If the Lande formula is modified so that the spin part is enhanced through gS = 2(1 + 0.0011(2)),

(4)

agreement resulted. At the same time experiments on ground state hyperfine splitting in hydrogen were also indicating a discrepancy.6 The Fermi formula by itself predicted a splitting of about 0.1 percent smaller than that observed, and the discrepancy could again be removed with the same modification of gS . In the same time period, the Lamb–Retherford experiment7 confirmed earlier indications that the spectrum of hydrogen deviated from the Dirac equation (a fascinating history of how those early indications were buried is given in the chapter “The Shift” in Ref. 8). The newer experiment made it clear the effect was real, and early calculations by Bethe9 indicated that the effect was a radiative correction, though his method was not completely relativistic. Schwinger was able to carry out relativistic calculations based on the same basic approach to these radiative corrections for both hyperfine splitting and the apparently very different Lamb shift, explaining both corrections in terms of the same physics. Schweber emphasizes the enormous effect this had on the theoretical physics community in his elegant summary: “The importance of Schwinger’s calculation cannot be underestimated. In the course of theoretical developments there sometimes occur important calculations that alter the way the community thinks about particular approaches. Schwinger’s calculation is one such instance. By indicating, as Feynman had noted, that “the discrepancy in the hyperfine structure of

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the hydrogen atom . . . could be explained on the same basis as that of the electromagnetic self-energy, as can the line shift of Lamb,” Schwinger had transformed the perception of quantum electrodynamics. He had made it into an effective, coherent, and consistent computational scheme to order e2 .” The actual approach used by Schwinger is described by Schweber, but it is equivalent to the following modern formulation. Restricting our attention to the self-energy of the electron, the basic unrenormalized and unregulated expression for the self-energy of an electron in an external field is   d4 q ¯ (2) 2 dx dy ESE = −4πiαc ψv (x)γμ G(v − cq0 , x, y)γ0 γ μ ψv (y). (5) q 2 + i The Green function of the electron in an external field, which plays a central role in QED calculations on hydrogen and hydrogenic ions satisfies    (6) E − H0 + V (x) G(E, x, y) = δ 3 (x − y), where H0 = −icα · ∇ + mc2 β.

(7)

A formal representation that will be used in the following is Gγ0 =

1 , γΠ − m

(8)

where γΠ = γμ (pμ − eAμ )

(9)

and Aμ is an external potential, in this case having only a μ = 0 component, the electric potential of the proton. 3. Electron Mass The g factor of an unbound electron can be treated either by its direct measurement in a Penning trap or by carrying out QED calculations to evaluate it. If heavier charged particles are ignored, the task for theory is to evaluate the coefficients cn in the expansion ∞  α n  ae = cn . (10) π n=1 The first two contributions are  α 2 α − 0.328 478 440 00 . . . . (11) ae = 2π π The term beginning with −0.328 is the numerical value of the analytic result, first calculated by his student Sommerfield:10 Schwinger spent some effort trying to simplify that already complicated calculation, but, in the voice of one of his characters,

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Harold, expressed some dissatisfaction with the effort in his source theory textbook.11 Thanks to both advances in computers and enormous efforts, particularly by Kinoshita and his group,12 the calculations now extend to five loops. The reason we show the first terms in the theory are that they “live” inside the more complicated bound state terms, forming the largest part of them, with the small corrections needed for the electron mass. One way of organizing the calculation of the g factor involves using the expansion of ae . Specifically, one can introduce a set of functions C (2n) (Zα), with the g factor given as the series g=

∞ 

C (2n) (Zα) ·

 α n

1

π

.

(12)

We concentrate on   1 1 + (Zα)2 + O (Zα)3 . (13) 2 12 The first term is the Schwinger correction and the second was first calculated by Grotch.13 All three terms play an important role in the electron mass determination. The electron mass was determined by a Penning trap experiment on five times ionized carbon.2 Denoting the ion mass M , one can show that the ratio Γ of the Larmor, νL , and the cyclotron, νC frequencies is related to only the electron mass and its bound state g factor through C (2) (Zα) =

me =

g νL M . 2 νC 5

(14)

This is the form used to determine the electron mass, and the experiment2 was able to produce another strikingly accurate result, νL = 4376.210 500 89(11)(7). (15) Γ≡ νC We now show how the electron mass gets more and more accurate as the theory (n) of the g factor is refined, and will give values of the inferred electron mass as me with n starting at zero and increasing as additional theory is added. We recall the end result me = 0.000 548 579 909 067(16) u. If the g had the Dirac value 2 one would find m(0) e = 0.000 548 930 189 u,

(16)

which is off by 0.064 percent, a smaller deviation than one would expect from the known value of the anomaly. This is because the ion has charge 6, and Zα corrections are enhanced. The leading one arises from the Dirac wave functions differing from the Schr¨ odinger wave functions. These were calculated by Breit,14 and change g, before radiative corrections, to

2 gD = 2 · (17) 1 + 2 1 − (Zα)2 . 3

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The g factor of an electron . . .

5

Using this value instead of 2 gives m(1) e = 0.000 547 948 206 9 u,

(18)

now close to the expected result of being 0.116 percent small. In the next step, we use the experimental value of the free anomaly. This picks up the first term in all of the C (2n) (Zα) coefficients, and at this point by almost any standard outside of precision QED a more than adequate value is obtained, me (2) = 0.000 548 579 244 942 u.

(19)

We will stop showing changing values of the mass at this one part per million level, and instead describe part of the theoretical effort required to get the sub-ppb result that has been achieved. The correction of order (Zα)2 to C (2) (Zα) contributes at the level 0.37 ppm, so clearly that term must be more carefully considered. This can be done by explicitly evaluating higher order corrections, but because Z = 6, many terms are required. An alternative approach that avoids expansion is a direct numerical evaluation of the one-loop diagram that gives rise to C (2) (Zα). The first such calculation was carried out in Ref. 15, and is now briefly described. The expression to evaluate has already been given in Eq. (5) if one includes in Eq. (9) both the Coulomb potential and the vector potential for a constant magnetic field. This leads to a very complicated electron propagator, but usually the magnetic field can be treated as a perturbation. That is the case for the magnetic fields of the experiment. There are three contributions that correct the self-energy when a external magnetic field B(x) is present. We give the formula for the correction to the propagator,   d4 q ¯ (2) ψv (x)γμ G(v − cq0 , x, z)V (z) ESE = −4πiαc2 dx dy dz 2 q + i G(v − cq0 , z, y)γ0 γ μ ψv (y). (20) The other two contributions involve the change in the external wave functions and the change in the energy eigenvalue. In the numerical calculation all three terms are important, and the factor 1/2 from the Schwinger correction does not emerge until all three are taken together. When the calculation was first done for the Zeeman effect,15 the factor (Zα)2 /12 could be seen from evaluating a range of Z values and fitting the function, but the higher order terms could not be accurately determined. Subsequent work, however, has done so.3 References to the heroic efforts made to get to under the ppb level, which include including recoil effects and evaluating two loop diagrams can be found in that reference. While the external field just discussed was a constant magnetic field, Schwinger was involved in the same kind of calculation for the case of the nuclear magnetic field.16 His group and another17 found for the function analogous to C (2) (Zα), α D(2) (Zα) = − 2.5Zα. (21) 2π

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The dependence on Z is much stronger than for the Zeeman effect, and we note the above formula changes sign already at Z = 10. For larger values of Z the exact approach is needed, though because hyperfine splitting is sensitive to nuclear physics effects, the change of sign is difficult to check because of uncertainties in the distribution of nuclear magnetism. Another important binding correction was to the self-energy. Despite the great success of explaining the Lamb shift with QED, as experimental accuracy improved a small discrepancy was seen. Calculations by Schwinger’s group18 and an independent group19 of the correction removed the discrepancy, and for most QED theorists is was clear the theory was going to just more and more accurately describe nature, as in fact it did, with one example being the electron mass discussed here. An amusing dissent by Dyson, who expected the whole theory would be soon replaced with a better one, can be found in footnote a.

4. Constant Magnetic Field Another problem requiring use of Green functions is the case where there is no Coulomb field, but only a constant magnetic field. The Green function of an electron in a constant magnetic (or electric) field is remarkable for the fact that it can be represented as a single integral using the proper time technique introduced by Schwinger in his classic paper on vacuum polarization.20 By representing the electron propagator as 1 =i γΠ + m



∞

ds e−is(γΠ+m)

(22)

0

the ultraviolet divergences are associated with the s → 0 limit, and it is particularly easy to isolate them. Using other techniques devised by Schwinger, a remarkable application to the self-energy was made by Tsai and Yildiz.21 The applications of the calculation are quite diverse. For an electron in the ground state or slightly above, the energy shift can be calculated, giving not only the Schwinger correction, but also a kind of binding correction to it, with the binding in this case coming from the magnetic field. These corrections are quite small because for a 1 Tesla magnetic field eB = 2.26 · 10−10 . m2 c 2

(23)

a “It was just patched together out of bits and pieces, in order to explain some experiments,” says Freeman Dyson, one of the theory’s architects, now at the Institute for Advanced Study in Princeton, New Jersey. “We didn’t expect it to last,” he adds. “Every time there was a new experiment, we all expected that the theory would be proved wrong in some interesting way. Instead, each experiment still agrees with the theory. That’s sort of a disappointment.”

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Given the high accuracies experiment regularly achieves, this kind of binding correction might be detectable, as, so for example, the ground state energy is 

2   13 eB eB 4 m2 c 2 2 2 α − 2 2+ ln − Eg = mc + mc . (24) 2π 2m c m2 c 2 3 2eB 18 However, the expansion parameter enters quadratically for spin flip transitions, which are what experiment detects. A very different facet of the mass operator arises from the fact that the energy acquires an imaginary part for states above the ground state. This of course describes decays to lower energy states, but for very highly excited states one is studying an electron orbiting in a magnetic field, emitting radiation. This allows a study of synchrotron radiation, a subject Schwinger contributed to many times over the years. This aspect of his research deserves a separate paper. In conclusion, what I would like to stress is that the basic object, the mass operator, part of the expression given above for the self-energy correction,  1 d4 k μ 1 γμ γ (25) M = ie2 (2π)4 k 2 m + γ(Π − k) is the start of many fundamental QED calculations to which Schwinger contributed. If the reader has the chance, I close by urging him or her to read the section on Schwinger in Schweber’s book,1 and appreciate in this 100th year since Schwinger’s birth the incredible excitement of those early years when quantum field theory first was “born”, though even the most optimistic of its creators could scarcely have imagined the enormously successful structure it has grown into. References 1. S. S. Schweber, QED and the Men Who Made It (Princeton University Press, Princeton, New Jersey, 1994). 2. S. Sturm, F. K¨ ohler, J. Zatorski, A. Wagner, Z. Harman, G. Werth, W. Quint, C. H. Keitel and K. Blaum, Nature (London) 506, 467 (2014). 3. V. A. Yerokhin and Z. Harman, Phys. Rev. A 88, 042502 (2013). 4. D. Hanneke, S. Fogwell and G. Gabrielse, Phys. Rev. Lett. 100, 120801 (2008). 5. H. M. Foley and P. Kusch, Phys. Rev. 73, 412 (1948). 6. J. E. Nafe, E. B. Nelson and I. I. Rabi, Phys. Rev. 71, 914 (1947). 7. W. E. Lamb and R. C. Retherford, Phys. Rev. 72, 241 (1947). 8. C. C. Mann and R. P. Crease, The Second Revolution (Macmillan, New York, 1986). 9. H. A. Bethe, Phys. Rev. 72, 241 (1947). 10. C. Sommerfield, Phys. Rev. 107, 328 (1957). 11. J. Schwinger, Particles, Sources and Fields (Addison-Wesley, Reading, 1970). 12. T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, arXiv:1412.8284 (2014). 13. H. Grotch, Phys. Rev. Lett. 24, 39 (1970). 14. G. Breit, Nature (London), 649 (1928). 15. S. A. Blundell, K. T. Cheng and J. Sapirstein, Phys. Rev. A 100, 100 (2001). 16. R. Karplus, A. Klein and J. Schwinger, Phys. Rev. 85, 972 (1952). 17. N. M. Kroll and F. Pollock, Phys. Rev. 84, 594 (1951).

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R. Karplus, A. Klein and J. Schwinger, Phys. Rev. 86, 288 (1952). M. Baranger, H. A. Bethe and R. P. Feynman, Phys. Rev. 92, 482 (1953). J. Schwinger, Phys. Rev. 82, 664 (1951). W.-Y. Tsai and A. Yildiz, Phys. Rev. D 8, 3446 (1973).

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The magnetoelectric coupling in electrodynamics A. Mart´ın-Ruiz Instituto de Ciencia de Materiales de Madrid, CSIC Cantoblanco, 28049 Madrid, Spain Centro de Ciencias de la Complejidad, Universidad Nacional Aut´ onoma de M´ exico 04510 M´ exico, Ciudad de M´ exico, M´ exico [email protected] M. Cambiaso Universidad Andres Bello, Departamento de Ciencias F´ısicas Facultad de Ciencias Exactas, Avenida Rep´ ublica 220, Santiago, Chile [email protected] L. F. Urrutia Instituto de Ciencias Nucleares, Universidad Nacional Aut´ onoma de M´ exico 04510 M´ exico, Ciudad de M´ exico, M´ exico [email protected] We explore a model akin to axion electrodynamics in which the axion field θ(t, x) rather than being dynamical is a piecewise constant effective parameter θ encoding the microscopic properties of the medium inasmuch as its permittivity or permeability, defining what we call a θ-medium. This model describes a large class of phenomena, among which we highlight the electromagnetic response of materials with topological order, like topological insulators for example. We pursue a Green’s function formulation of what amounts to typical boundary-value problems of θ-media, when external sources or boundary conditions are given. As an illustration of our methods, which we have also extended to ponderable media, we interpret the constant θ as a novel topological property of vacuum, a so called θ-vacuum, and restrict our discussion to the cases where the permittivity and the permeability of the media is one. In this way we concentrate upon the effects of the additional θ coupling which induce remarkable magnetoelectric effects. The issue of boundary conditions for electromagnetic radiation is crucial for the occurrence of the Casimir effect, therefore we apply the methods described above as an alternative way to approach the modifications to the Casimir effect by the inclusion of topological insulators. Keywords: Magnetoelectric effect; θ-electrodynamics; topological insulators; Casimir effect.

1. Introduction Electrodynamics, both the classical 1 and the quantum 2–4 theories, encompass all our understanding of the interaction between matter and radiation. Although the foundations for the classical theory were laid more than a century ago, still today it is a fruitful research discipline and an excellent arena with potential for new discoveries. Specially when precision measurements are at hand and also when new materials come into play whose novel properties, of ultimate quantum origin, result

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in new possible forms of interaction between light and such materials. That is the case with topological insulators, as well as other materials with topological order. Interestingly enough, the interaction between matter characterized by topological order, topological insulators among them, and external electromagnetic fields can be described by an extension of Maxwell’s theory. In fact, in electrodynamics there is the possibility of writing two quadratic gauge and Lorentz invariant terms: the first one is the usual electromagnetic density LEM = (E2 −B2 )/8π which yields Maxwell’s equations, and the second one is the magnetoelectric term Lθ = θ E · B, where θ is a coupling field usually termed the axion angle. Many of the interesting properties of the latter can be recognized from its covariant form Lθ = −(θ/8)μνρλ Fμν Fρλ , where μνρλ is the Levi-Civita symbol and Fμν is the electromagnetic field strength. When θ is globally constant, the θ-term is a total derivative and has no effect on Maxwell’s equations. These properties qualify P = −(1/8)μνρλ Fμν Fρλ to be a topological invariant. Actually, P is the simplest example of a Pontryagin density, 5 corresponding to the abelian group U (1). This structure together with its generalization to nonabelian groups, has been relevant in diverse topics in high energy physics such as anomalies, 6 the strong CP problem, 7 topological field theories 8 and axions, 9 for example. Recently, an additional application of the Pontryagin extended electrodynamics (defined by the full action LEM + Lθ ) has been highlighted in condensed matter physics, where a piecewise constant axion angle θ provides an effective field theory describing the electromagnetic response of a topological insulator (θ = π) in contact with a trivial insulator (θ = 0). 10 A constant θ can be thought as an additional parameter characterizing the material in a way analogous to the dielectric permittivity ε and the magnetic permeability μ, which nevertheless manifest only in the presence of a boundary where its value suddenly changes. In this contribution we discuss some general features arising from adding to Maxwell’s electrodynamics the coupling of the Pontryagin density to the scalar field θ, leading to a theory that we call θ-electrodynamics (θ-ED), retaining the name of axion-electrodynamics for the case where the axion field θ becomes dynamical. We call the piecewise constant parameter θ the magnetoelectric polarizability (MEP). The resulting field equations have a wide range of applications in physics. For example, they describe: (i) the electrodynamics of magnetoelectric media, 11 (ii) the electrodynamics of metamaterials when θ is a purely complex function, 12 (iii) the electromagnetic response of topological insulators (TIs) when θ = (2n + 1)π, with n integer 10 and (iv) the electromagnetic response of Weyl semimetals which can be described by choosing θ(x, t) = 2b · x − 2b0 t. 13 Recently, the study of topological insulating and Weyl semimetal phases either from a theoretical or an experimental perspective has been actively pursued. 14,15 One of the most remarkable consequences θ-ED is the appearance of the magnetoelectric effect whereby electric fields induce magnetic fields and vice versa, even for static fields. This effect was predicted in Ref. 16 (1959) and subsequently observed in Ref. 17 (1960). For an updated review of this effect see for example the

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Ref. 18. A universal topological magnetoelectric effect has recently been measured in TIs. 19 Many additional interesting magnetoelectric effects arising from θ-ED have been highlighted using different approaches. For example, electric charges close to the interface between two θ-media induce image magnetic monopoles (and vice versa). 20–23 Also, the propagation of electromagnetic waves across a θ-boundary have been studied finding that a nontrivial Faraday rotation of the polarizations appears. 21,22,24,25 The shifting of the spectral lines in hydrogen-like ions placed in front of a planar TI, as well as the modifications to the Casimir Polder potential in the non-retarded approximation were studied in Ref. 26. The classical dynamics of a Rydberg hydrogen atom near a planar TI has also been investigated. 27 The paper is organized as follows. In Sec. 2 we present a brief review of electrodynamics in media characterized by a parameter θ (to be called a θ-medium), recalling their most important properties. Section 3 contains a summary of our generalized Green’s function method to construct the corresponding electromagnetic fields produced by charges, currents and boundary conditions in systems subjected to the following coordinate conditions: (i) the coordinates can be chosen in such a way that the interface between two media with different values θ is defined by setting constant only one of them and (ii) the Laplacian is separable in such coordinates. The particularly simple case of planar symmetry is discussed subsequently in Sec. 4, where the reader is also referred to the analogous extensions to cylindrical and spherical coordinates. As a specific application of our methods to the case of a planar interface, the Casimir effect between two metallic plates with a topological insulator between them is considered in Sec. 5. Our conventions are taken from Ref. 28, where Fμν = ∂ν Aν − ∂ν Aμ , F˜ μν = μναβ Fαβ /2 F i0 = E i , F ij = −ijk B k and F˜ i0 = B i , F˜ ij = ijk E k . Also V = (V i ) = (Vx , Vy , Vz ) for any vector V. The metric is (+, −, −, −) and 0123 = +1 = 123 .

2. Electrodynamics in a θ-medium Electromagnetic phenomena in material media are described by the Maxwell’s field equations, ∇ · D = 4πρ,

∇ · B = 0,

∇×E+

1 ∂B = 0, c ∂t

∇×H−

1 ∂D 4π = J, c ∂t c

(1)

together with constitutive relations giving the displacement D and the magnetic field H in terms of the electric E and magnetic induction B fields, plus the Lorentz force. 28 These depend on the nature of the material, and they are generally of the form D = D(E, B) and H = H(E, B). For instance, for linear media they are D = εE and H = B/μ, where ε is the dielectric permittivity and μ is the magnetic permeability. For isotropic materials ε and μ are constants, while for anisotropic materials they are tensorial in nature and may depend on the spacetime coordinates.

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In this paper we are concerned with a particular class of materials described by the following constitutive relations D = εE −

θα B, π

H=

1 θα B+ E, μ π

(2)

where α  1/137 is the fine structure constant and the MEP θ is an additional parameter of the medium, which can be considered on the same footing as the permittivity ε or the permeability μ. In the general situation these parameters may be functions of the spacetime coordinates. The constitutive relations (2) yield the following inhomogeneous Maxwell’s equations 1 ∂(εE) 4π α 1 α ∂θ α = J− ∇θ×E− B. (3) ∇·(εE) = 4πρ+ ∇θ·B, ∇×(B/μ)− π c ∂t c π c π ∂t In fact, the modified Maxwell’s equations (3) can be derived from the usual electromagnetic action supplemented with the coupling of the Abelian Pontryagin density P via the MEP θ,  

 1 1 2 1 α 2 3 S[Φ, A] = dt d x εE − B − 2 θ(x) E · B − ρΦ + J · A . (4) 8π μ 4π c The electromagnetic fields E and B are written in term of the electromagnetic potentials Φ and A as usual, providing a solution of the homogeneous equations in Eq. (1), which are summarized in the Bianchi identity ∂μ F˜ μν = 0. An important consequence of the modified Maxwell’s equations (3) is the appearance of additional field-dependent effective charge and current densities given by α cα α ∂θ (5) ∇θ · B, Jθ = − 2 ∇θ × E − 2 B. 4π 2 4π 4π ∂t Current conservation ∇ · Jθ + ∂ρθ /∂t = 0 can be directly verified as a consequence of the homogeneous equations in (1). Note that these expressions depend only on spacetime gradients of the MEP θ. This is because the Pontryagin density P is a total derivative in such way that the coupling in (4) does not affect the equations of motion when θ is globally a constant. Even though the constitutive relations depend upon the constant θ, their contribution to the equations of motion turns out to be null due to the homogeneous Maxwell’s equations. This can be directly verified from the constitutive relations (2), yielding ρθ =

θα (∇ · B) , ∇ · D = ∇ · (εE) −

π

1 1 ∂ θα 1 ∂D 1 ∂ = ∇× B − (εE) + B . ∇ × H− ∇×E+ c ∂t μ c ∂t π c ∂t

(6)

Physically, the effective charge and current densities (5) encode one of the most remarkable properties of θ-ED, which is the magnetoelectric effect. A large class of interesting phenomena can be described by θ-ED if one considers the adjacency of different media with constant θ. In the simplest case where the (3 + 1)-dimensional spacetime is M = U × R, with U being a three-dimensional

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Fig. 1.

13

Region over which the electromagnetic field theory is defined.

manifold and R corresponding to the temporal axis, we make a partition of space in two regions: U1 and U2 , in such a way that manifolds U1 and U2 intersect along a common two-dimensional boundary Σ, to be called the θ-boundary, so that U = U1 ∪ U2 and Σ = U1 ∩ U2 , as shown in Fig. 1. We also assume that the MEP θ is piecewise constant in such way that it takes the value θ = θ1 in the region U1 and the value θ = θ2 in the region U2 . This situation is expressed in the characteristic function  θ1 , x ∈ U1 . (7) θ (x) = θ2 , x ∈ U2 The two-dimensional surface Σ is parametrized by some function FΣ (x) = 0, such that nμ = (0, n ˆ ) = ∂ μ FΣ (x),

(8)

is the outward unit normal to Σ with respect to the region U1 . In this scenario the θ-term in the action fails to be a global total derivative because it is defined over a region with the boundary Σ. Consequently the modified Maxwell’s equations acquire additional effective charge and current densities with support only at the boundary (in the following we set c = 1) ˜ (FΣ (x)) B · n ˆ + 4πρ, (9) ∇ · E = θδ ∂E ˜ (FΣ (x)) E × n = θδ ˆ + 4πJ, (10) ∇×B− ∂t which reproduce Eqs. (3) in this setting. The homogeneous equations are included in the Bianchi identity. Here n ˆ is the unit normal to Σ defined in Eq. (8), shown in ˜ Fig. 1 and θ = α (θ1 − θ2 ) /π, which enforces the invariance of the classical action under the shifts of θ by any constant, θ → θ + C. As we see from Eqs. (9) and (10) the behavior of θ-ED in the bulk regions U1 and U2 is the same as in standard electrodynamics. Assuming that the time derivatives of the fields are finite in the vicinity of the surface Σ, the field equations (9) and (10) imply that the normal component of

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E, and the tangential components of B, acquire discontinuities additional to those produced by superficial free charges and currents, while the normal component of B, and the tangential components of E, are continuous at the boundary. For vanishing external sources on Σ the boundary conditions read:     ˜ n  , ΔB  = −θE ˜  , (11) ΔEn Σ = θB Σ Σ Σ   (12) ΔBn Σ = 0, ΔE Σ = 0.  The notation ΔVi Σ refers to the discontinuity of the ith component of the vector  V across the interface Σ, while Vj Σ indicates the continuous value of the jth component evaluated at Σ. The continuity conditions, (12), imply that the right hand sides of equations (11) are well defined and they represent surface charge and current densities, respectively. An immediate consequence of the boundary conditions (11) and (12) is that the presence of a magnetic field crossing the surface Σ is sufficient to generate an electric surface charge density there, even in the absence of free electric charges. 3. The Green’s Function Method in a θ-vacuum In this section we review the Green’s function (GF) method to solve a class of static boundary-value problems in θ-ED in terms of the electromagnetic potential Aμ . Certainly one could solve for the electric and magnetic fields from the modified Maxwell equations together with the boundary conditions (11) and (12), however, just as in ordinary electrodynamics, there might be occasions where information about the sources is unknown and rather we are provided with information of the 4-potential at some given boundaries. In these cases, the GF method provides the general solution to such boundary-value problem (Dirichlet or Neumann) for arbitrary sources. Nevertheless, in the following we restrict ourselves to contributions of free sources only outside the θ-boundary with no additional boundary conditions (BCs) besides those required at Σ. Also we consider the simplest media having θ1 = θ2 , but with ε = 1 and μ = 1, which we call the θ-vacuum. In this case, the inhomogeneous Maxwell’s equations can be written as ˜ (FΣ (x)) nμ F˜ μν + 4πj ν . ∂μ F μν = θδ

(13)

Current conservation can be verified directly by taking the divergence on both sides of Eq. (13) and realizing that  ˜ (FΣ (x)) nμ ∂ν F˜ μν ˜ (FΣ (x)) nμ F˜ μν = θδ ˜  (FΣ (x)) nν nμ F˜ μν + θδ (14) ∂ν θδ is zero by symmetry properties together with the Bianchi identity. Since the homogeneous Maxwell equations are not modified, the electrostatic and magnetostatic fields can be written in terms of the 4-potential Aμ = (φ, A) according to E = −∇φ and B = ∇ × A as usual. In the Coulomb gauge ∇ · A = 0, the 4-potential satisfies the equation of motion   ˜ (FΣ (x)) nρ ρμα ∂α Aν = 4πj μ , (15) −η μν ∇2 − θδ ν

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together with the boundary conditions    ˜ 3μα (∂α Aν )  . ΔAμ Σ = 0, Δ (∂z Aμ ) Σ = −θ ν Σ

15

(16)

One can further check that these boundary conditions for the 4-potential correspond to those written in Eqs. (11) and (12). To obtain a general solution for the potentials φ and A in the presence of arbitrary external sources j μ (x), we introduce the GF Gν σ (x, x ) solving Eq. (15) for a point-like source,   ˜ (FΣ (x)) nρ ρμα ∂α Gν (x, x ) = 4πη μ δ 3 (x − x ) , (17) −η μν ∇2 − θδ ν σ σ together with the boundary conditions (16), in such a way that the general solution for the 4-potential in the Coulomb gauge is  (18) Aμ (x) = d3 x Gμν (x, x ) j ν (x ) . According to Eq. (17) the diagonal entries of the GF matrix are related with the electric and magnetic fields arising from the charge and current density sources, respectively, although they acquire a θ-dependence. However, the nondiagonal terms encode the magnetoelectric effect, i.e. the charge (current) density contributing to the magnetic (electric) field. As we will show in the following, a further simplification in θ-ED arises when the system satisfies the following two coordinate conditions: (i) the coordinate system can be chosen so that the interface Σ is defined by setting constant only one of them and (ii) the Laplacian is separable in such coordinates in such a way that a complete orthonormal set of eigenfunctions can be defined in the subspace orthogonal to the coordinate defining the interface. Three cases show up immediately: (i) a plane interface at fixed z, (ii) a spherical interface at constant r and (iii) a cylindrical interface at constant ρ. In all this cases the characteristic function θ(x) defined in Eq. (7) can be written in terms of the Heaviside function H of one coordinate in terms of H(z − a), H(r − a) and H(ρ − a), with the associated unit vectors n ˆξ ˆ ˆ given by k, r and ρˆ, respectively, in each of the adapted coordinate systems. Then Eq. (17) reduces to   ˜ (ξ − ξ0 ) ξμα ∂α Gν (x, x ) = 4πη μ δ 3 (x − x ) , (19) −η μν ∇2 − θδ σ σ ν where ξ denotes the coordinate defining the interface at ξ = ξ0 and the coupling of the θ-term is given by a one-dimensional delta function with support only in the coordinate that defines the interface. Also, the unit vector n ˆ ξ will have a component only in the direction ξ. Let us consider the coordinates partitioned according to ξ plus two additional ones which we denote by σ and τ . Also assume that the Laplacian can be separated in the form ∇2 = L1 (ξ) + f (ξ)L2 (σ, τ ),

(20)

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where the operator L2 (σ, τ ) has eigenfunctions ΨM (σ, τ ) which form a complete orthonormal set in the subspace of the coordinates σ, τ (which we denote collectively by Π) L2 (σ, τ )ΨM (σ, τ ) = λM ΨM (σ, τ ),

(21)

where M denote a set of discrete or continuous labels. The basic properties of ΨM (σ, τ ) are   ΨM (σ, τ )Ψ∗M (σ  , τ  ) = δ 2 (Π − Π ), dμ(σ, τ ) Ψ∗M (σ, τ )ΨM  (σ, τ ) = δM,M  , M





dμ(σ, τ )δ (Π − Π ) = 1 , 2

(22)

where dμ denotes the integration measure in each subspace and d3 x = dμ(σ, τ ) dμ(ξ). Also we have  δ 3 (x − x ) = δ 2 (Π − Π )δ(ξ − ξ  ), dμ(ξ)δ(ξ − ξ  ) = 1. Next we introduce the reduced Green’s function (g νσ )M,M  (ξ, ξ  ) in the following way  ΨM (σ, τ )Ψ∗M  (σ  , τ  ) (g νσ )M,M  (ξ, ξ  ) . (23) Gν σ (x, x ) = 4π M,M 

When substituting Eq. (20) in Eq. (19) we obtain    ΨM (σ, τ )Ψ∗M  (σ  , τ  ) −η μν L1 (ξ) (g νσ )M,M  (ξ, ξ  ) M,M 

+



M,M 

+



  ΨM (σ, τ )Ψ∗M  (σ  , τ  ) −η μν f (ξ)λM (g νσ )M,M  (ξ, ξ  )    ˜ (ξ − ξ0 ) (g ν ) Oμν (σ, τ )ΨM (σ, τ )Ψ∗M  (σ  , τ  ) −θδ σ M,M  (ξ, ξ )

M,M 

= η μσ



ΨN (σ, τ )Ψ∗N  (σ  , τ  ))δN,N  δ(ξ − ξ  ),

(24)

N,N 

since the operator ξμαν ∂α ≡ Oμν (σ, τ )

(25)

contains only derivatives with respect to σ, τ so that it acts upon the functions ΨM (σ, τ ). Multiplication to the right by ΨT (σ  , τ  ) and integration over dμ(σ  , τ  ), followed by multiplication to the left by Ψ∗P (σ, τ ) and integration over dμ(σ, τ ) yields     ˜ (ξ − ξ0 ) Oμν P M (g νσ )M,T (ξ, ξ  ) [(L1 (ξ) + λP f (ξ))] g μσ P,T (ξ, ξ  ) +θδ M

= −η μσ δP,T δ(ξ − ξ  ),

(26)

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where we have introduced the following matrix element   μ  O ν P M ≡ dμ(σ, τ )Ψ∗P (σ, τ )Oμν (σ, τ )ΨM (σ, τ )

17

(27)

which is independent of ξ and ξ  . In this way we transform the calculation of the reduced GF into a onedimensional problem with a delta interaction. The above equation  can be  (26) directly integrated with the knowledge of an additional reduced GF gμσ P,T (ξ, ξ  ), corresponding to the θ˜ = 0 limit, which satisfies   (28) [(L1 (ξ) + λP f (ξ))] gμσ P,T (ξ, ξ  ) = −η μσ δP,T δ(ξ − ξ  ), plus boundary conditions.   The introduction of gμσ P,T (ξ, ξ  ) derives from the existence of a full Green’s function  ΨM (σ, τ )Ψ∗M  (σ  , τ  ) (gν σ )M,M  (ξ, ξ  ) , (29) Gν σ (x, x ) = M,M 

which must respect the coordinate conditions of the problem in a setting where the θ-medium is absent. We refer to them as the free GF’s, emphasizing that they correspond to the θ˜ = 0 case. These GF’s can be taken directly from the vast literature in standard electrodynamics and are the basis for finding the response of an identical system now in the presence of a θ-medium, the interface of which defines the corresponding coordinate conditions. As an illustration take the case of a planar θ-medium that can be embedded in two different ways: (i) either in vacuum, just by choosing the free GF Gν σ (x, x ) with standard BCs at infinity, or (ii) between a pair of conducting plates of infinite extension which are parallel to the interface just by requiring Gν σ (x, x ) to satisfy the appropriate BCs at the plates, which can be found in Ref. 29, for example. This approach was used in Ref. 30 when calculating the Casimir effect between parallel metallic plates in the presence of a planar θ-medium and will  reviewed in Sec. 5.  be In terms of the free reduced GF gμσ P,T (ξ, ξ  ) we obtain   μ   g σ P,T (ξ, ξ  ) = gμσ P,T (ξ, ξ  ) +

  θ˜   μ  g ρ P,N (ξ, ξ0 ) Oρν N M (g νσ )M,T (ξ0 , ξ  ) . 4π

(30)

M,N

This result can be explicitly verified by applying the operator [(L1 (ξ) + λP f (ξ))] to Eq. (30) and using Eq. (28).       It is convenient to think of g μσ P,T , gμσ P,T and Oρν N M as generalized matrix elements of the operators g, g and O, respectively. This allows us to rewrite Eq. (30) in the compact form g (ξ, ξ  ) = g (ξ, ξ  ) +

θ˜ g (ξ, ξ0 ) Og (ξ0 , ξ  ) . 4π

(31)

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This set of equations constitute a coupled system of algebraic equations which can be disentangled according to the following steps. First we set ξ = ξ0 in Eq. (31) g (ξ0 , ξ  ) = g (ξ0 , ξ  ) +

θ˜ g (ξ0 , ξ0 ) Og (ξ0 , ξ  ) 4π

(32)

and solve for g (ξ0 , ξ  ) as g (ξ0 , ξ  ) = 

1 1−

θ˜ 4π g (ξ0 , ξ0 ) O

g (ξ0 , ξ  ) .

(33)

Then we substitute the above result in Eq. (31) obtaining g (ξ, ξ  ) = g (ξ, ξ  ) +

θ˜ g (ξ, ξ0 ) O  4π 1−

1 θ˜ 4π g (ξ0 , ξ0 ) O

g (ξ0 , ξ  ) ,

(34)

    which expresses the reduced GF g μσ P,T in terms of the free GF gμσ P,T . The full GF is reconstructed then from Eq. (23). In the specific cases considered in Refs. 31–33 the solutions of Eqs. (33) and (34) are explicitly constructed in a step by step fashion to be illustrated in the next section. 4. The Case of a Planar Interface The simplest example of the construction previously discussed is when the interface Σ is the plane z = a. Here the MEP θ(x) is θ(z) = θ1 H(a − z) + θ2 H(z − a),

(35)

ˆz where H(z) is the Heaviside function. Then ∇θ = (θ2 − θ1 )δ(z − a)ˆ ez , and e is the unit vector in the direction z. In this way, the dynamical modifications in Eqs. (3) arise only at the boundary z = a, which is the only place where the effective sources (5) are nonzero. That is to say, the θ-vacuum has conducting properties at the boundary Σ, even though its bulk behaves as ordinary vacuum. The general eigenfunctions in Eq. (21) take the form Ψp (x, y) =

1 ip ·x e , (2π)

(36)

where the index M is now the momentum p = (px , py ) parallel to the plane Σ and adds up to realize the Eq. (23) by introducing the reduced GF x  μ= (x, y). This g ν p,p (z, z  ) as  1 ip ·(x−x )  μ  μ    g (37) G ν (x, x ) = 4π d2 p d2 p ν p,p (z, z ) . 2e (2π) In this case the operator in Eq. (25) is Oμν = zμαν ∂α and its matrix elements of Eq. (27) simplify to  μ  O ν p,p = 3μαν ipα δ 2 (p −p ), (38)

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  where pα = (0, px , py , 0) = (0, p ). Since Oμν p,p is diagonal in momentum space,  μ  Eq. (26) indicates that we can also take g ν p,p (z, z  ) to be diagonal, so that we write    μ  (39) g ν p,p (z, z  ) = δ 2 (p −p )g μν z, z  , p . In this way, the final representation for the GF of Eq. (37) turns out to be given in terms of the Fourier transform in the directions x, y parallel to the plane 29 Σ  2  d p ip ·(x−x ) μ   μ  g (40) G ν (x, x ) = 4π ν z, z , p , 2e (2π) as expected. Due to the antisymmetry of the Levi-Civita symbol, the partial derivative appearing in the second term of the GF Eq. (17) does not introduce derivatives with respect to z, but only in the transverse directions. This allows us to write the full reduced GF equation as   ˜ (z − a) 3μα pα g ν (z, z  , p, ) = η μ δ (z − z  ) , (41) ∂ 2 η μν + iθδ σ σ ν where ∂ 2 = p2 − ∂z2 , pα pα = −p 2 and we denote |p | = p. The solution of Eq. (41) is obtained with the introduction of a reduced free GF having the form Gμν (z, z  ) = g (z, z  ) η μν , associated with the operator ∂ 2 previously defined, that solves ∂ 2 Gμν (z, z  ) = η μν δ (z − z  ) ,

(42)

plus BC’s. In the case of standard BC’s at infinity, the choice is 29 g(z, z  ) =

1 −p|z−z | e . 2p

(43)

Note that Eq. (42) demands the derivative of g to be discontinuous at z = z  , i.e., z=z+ ∂z g (z, z  )  − = −1, together with the continuity of g at z = z  . z=z

Now we observe that Eq. (41) can be directly integrated by using the free GF in Eq. (42) together with the properties of the Dirac delta-function, thus reducing the problem to a set of coupled algebraic equations, ˜ 3μα pα g (z, a) g ν (a, z  ) . g μσ (z, z  ) = η μσ g (z, z  ) − iθ σ ν

(44)

Note that the continuity of g at z = z  implies the continuity of g μσ , but the discontinuity of ∂z g at the same point yields z=a+  + ˜ 3μα pα ∂z g (z, a) z=a− g ν (a, z  ) = iθ ˜ 3μα pα g ν (a, z  ) , ∂z g μσ (z, z  ) z=a− = −iθ σ σ ν ν z=a (45) from which the boundary conditions for the 4-potential in Eq. (16) are recovered. In this way the solution in Eq. (44) guarantees that the boundary conditions at the θ-interface are satisfied.

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In this case the formal solution for Eq. (34) for g μσ (z, z  ) can be explicitly obtained in successive steps. To this end we split Eq. (44) into μ = 0 and μ = j = 1, 2, 3 components; ˜ 30i pi g (z, a) g j (a, z  ) , g 0σ (z, z  ) = η 0σ g (z, z  ) − iθ j σ 3ji j j   0 ˜ pi g (z, a) g (a, z  ) . g (z, z ) = η g (z, z ) − iθ σ

σ

0

σ

(46) (47)

Now we set z = a in Eq. (47) and then substitute into Eq. (46) yielding ˜ 30i pi η j g (z, a) g (a, z  ) − θ˜2 p2 g (z, a) g (a, a) g 0 (a, z  ) , g 0σ (z, z  ) = η 0σ g (z, z  ) − iθ j σ σ (48) 30i 3jk 2 0  where we use the result  j  0 pk pi = p . Solving for g σ (a, z ) by setting z = a in Eq. (48) and inserting the result back in Eq. (48), we obtain   ˜ 2 g (a, a) A (z, z  ) + i30i pi A (z, z  ) , g 0σ (z, z  ) = η 0σ g (z, z  ) + θp (49) σ where A (z, z  ) = −θ˜

g (z, a) g (a, z  ) . 1 + p2 θ˜2 g2 (a, a)

(50)

The remaining components can be obtained by substituting g 0σ (a, z  ) in Eq. (47). The result is   ˜ 30i pi g (a, a) A (z, z  ) . (51) g jσ (z, z  ) = η jσ g (z, z  ) + i3jk0 pk η 0σ − iθ σ Equations (49) and (51) allow us to write the general solution as       ˜ (a, a) pμ pν + η μ + nμ nν p2 + iμ α3 pα , g μν (z, z  ) = η μν g (z, z  )+A (z, z  ) θg ν ν (52) where nμ = (0, 0, 0, 1) is the normal to Σ. The reciprocity between the position of the unit charge and the position at which the GF is evaluated Gμν (x, x ) = Gνμ (x , x) is one of its most remarkable properties of the GF. From Eq. (40) this condition demands gμν (z, z  , p) = gνμ (z  , z, −p),

(53)

∗ (z, z  ) = which we verify directly from Eq. (52). The symmetry gμν (z, z  ) = gνμ †  gμν (z, z ) is also manifest. The various components of the static GF matrix in coordinate representation are obtained by computing the Fourier transform defined in Eq. (40), with the reduced GF given by Eq. (43). The details are presented in Ref. 31. The final results are

1 θ˜2 1 √ − ,  2 2 ˜ |x − x | 4 + θ R + Z2

2θ˜ 0ij3 Rj Z √ 1 − G0i (x, x ) = − , R2 + Z 2 4 + θ˜2 R2 i θ˜2 ∂ i Kj (x, x ) , Gij (x, x ) = η ij G00 (x, x ) − 2 4 + θ˜2

G00 (x, x ) =

(54) (55) (56)

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where Z = |z − a| + |z  − a|, Rj = (x − x ) = (x − x , y − y  ), R = | (x − x ) | and √ R2 + Z 2 − Z j j  R . (57) K (x, x ) = 2i R2 Finally, we observe that Eqs. (54)–(56) contain all the required elements of the GF matrix, according to the choices of z and z  in the function Z. Similar results for the cases of spherical and cylindrical interfaces incorporating also piecewise continuous ponderable media have been reported in Refs. 32 and 33. j

5. The Casimir Effect The Casimir effect (CE) 34 is one of the most remarkable consequences of the nonzero vacuum energy predicted by quantum field theory which has been confirmed by experiments. 35 In general, the CE can be defined as the stress (force per unit area) on bounding surfaces when a quantum field is confined in a finite volume of space. The boundaries can be material media, interfaces between two phases of vacuum, or topologies of space. For a review see, for example, Refs. 36 and 37. The experimental accessibility to micrometer-size physics together with the recent discovery of three dimensional TIs 38 provides an additional arena where the CE can be studied. In the scattering approach to the Casimir effect, i.e. using the Fresnel coefficients for the reflection matrices at the interfaces of the TIs, the Casimir force between TIs was computed in Ref. 39. The authors found the most notable feature that, due to the magnetoelectric effect, which now has a topological origin, the strength and sign of the Casimir stress between two planar TIs can be tuned. When the surface of the TI is included in the description, θ-ED is a fair description of both the bulk and the surface only when a time reversal symmetry breaking perturbation is induced on the surface to gap the surface states, thereby converting it into a full insulator. In this situation, which we consider here, the MEP θ can be shown to be quantized in odd integer values of π: θ = (2n + 1)π, where n ∈ Z is determined by the nature of the time reversal symmetry breaking perturbation, which could be controlled experimentally by covering the TI with a thin magnetic layer. 39 For a review of the effective θ-ED describing the electromagnetic response of TI’s see Refs. 10 and 14 for example. The Casimir system we consider is formed by two perfectly reflecting planar surfaces (labeled P1 and P2 ) separated by a distance L, with a nontrivial TI placed between them, but perfectly joined to the plate P2 , as shown in Fig. 2. The surface Σ of the TI, located at z = a, is assumed to be covered by a thin magnetic layer which breaks time reversal symmetry there. We calculate the Casimir stress restricting ourselves only to the contribution of the MEP which now has a topological origin, i.e. we set ε = μ = 1. We follow an approach similar to that in Ref. 40 which starts from the calculation of the appropriate GF, to subsequently compute the renormalized vacuum stress–energy tensor in the region between the plates yielding

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Fig. 2.

Schematic of the Casimir effect in θ-ED.

finally the Casimir stress that the plates exert on the surface Σ of the TI. We also consider the limit where the plate P2 is sent to infinity (L → ∞) to obtain the Casimir stress between a conducting plate and a nontrivial semiinfinite TI. The BCs for the perfectly reflecting metallic plates P1 and P2 are the standard ones nμ F˜ μν |P1,2 = 0, where nμ = (0, 0, 0, 1). The effects of the MEP are incorporated by choosing θ(z) = θH(z − a)H(L − z),

θ˜ = −αθ/π.

(58)

Assuming the absence of free sources on Σ, the required equation for the GF matrix is given by Eq. (17), together with the BCs arising from Eq. (16). The calculation proceeds along the same lines discussed in Sec. 4 for the static case, but keeping the time dependence now. Making explicit the coordinate choice in the transverse x and y directions we can write  2  d p ip ·x dω −iω(t−t ) μ μ  e g ν (z, z  ) , (59) G ν (x, x ) = 4π 2e 2π (2π) where we have omitted the dependence of the reduced GF g μν on ω and p . In the Lorenz gauge the equation for the reduced GF g νσ (z, z  ) is   ˜ (z − a) 3μα pα g ν (z, z  ) = η μ δ (z − z  ) , (60) η μν ∂ 2 + iθδ σ σ ν   where now ∂ 2 = p 2 − ω 2 − ∂z2 and pα = ω, p , 0 . The boundary term (at z = L), missing in Eq. (60), identically vanishes in the distributional sense, due to the BCs on the plate P2 . In this way, Eq. (60) implies that the only topologically magnetoelectric effect present in our Casimir system is the one produced at Σ. Here the free GF we use to integrate Eq. (60) is the reduced GF for two parallel conducting surfaces placed at z = 0 and z = L, which is the solution of ∂ 2 g (z, z  ) = δ (z − z  ) satisfying the BCs g (0, z  ) = g (L, z  ) = 0, namely 29 gc (z, z  ) =

sin [pz< ] sin [p (L − z> )] , p sin [pL]

(61)

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where z> (z< ) is the greater (lesser) of z and z  , and p = problem is reduced to a set of coupled algebraic equations,

23

ω 2 − p2 . Now the

˜ 3μα pα gc (z, a) g ν (a, z  ) . g μσ (z, z  ) = η μσ gc (z, z  ) − iθ σ ν

(62)

We write the general solution to Eq. (62) as the sum of two terms μ (z, z  ) . g μν (z, z  ) = η μν gc (z, z  ) + gCν

(63)

The first term provides the propagation in the absence of the TI between the parallel plates. The second, to be called the reduced θ-GF, which can be shown to be     μ ˜ c (a, a) pμ pν − η μ + nμ nν p2 Ac (z, z  ) + i μ α3 pα Ac (z, z  ) , (z, z  ) = θg gCν ν ν (64) encodes the magnetoelectric effect due to the topological MEP θ. Here gc (z, a) gc (a, z  ) , Ac (z, z  ) = −θ˜ 1 − p2 θ˜2 g2c (a, a)

(65)

has the same form as the previous Eq. (50) with g (z, z  ) → gc (z, z  ). In the static limit (ω = 0), our result in Eq. (64) reduces to the one reported in Ref. 31. As the Eq. (63) suggests, the full GF matrix Gμν (x, x ) can also be written as the sum of two terms, Gμν (x, x ) = η μν G (x, x ) + GμCν (x, x ), each one arising from the respective term in the Eq. (63). We call GμCν (x, x ) the θ-GF. Since the MEP modifies the behavior of the fields only at the interface, we expect that stress energy tensor (SET) in the bulk retains its original Maxwell’s form. In fact, in Ref. 32 we explicitly computed the SET and verified that

1 1 (66) −F μλ F νλ + η μν Fαβ F αβ . T μν = 4π 4 Clearly this tensor is traceless and its divergence is ˜ (Σ) nμ F νλ F˜ μλ . ∂μ T μν = −F νλ j λ − (θ/4π)δ

(67)

As expected, T μν it is not conserved at Σ because the MEP induces effective charge and current densities there. Now we address the calculation of the vacuum expectation value of the SET, to which we will refer simply as the vacuum stress (VS). The local approach to compute the VS was initiated by Brown and Maclay who calculated the renormalized stress tensor by means of GF techniques. 40,41 Using the standard point splitting procedure and taking the vacuum expectation value of the SET in (66) we find  i lim − ∂ μ ∂ ν Gλλ + ∂ μ ∂λ Gλν + ∂ λ ∂ ν Gμλ T μν = 4π x→x   1 − ∂ λ ∂λ Gμν + η μν ∂ α ∂α Gλλ − ∂ α ∂β Gβ α , (68) 2

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where we have omitted the dependence of Gμν on x and x . This result can be further simplified as follows. Since the GF is written as the sum of two terms, the VS can also be written in the same way, i.e. T μν = tμν + TCμν .

(69)



1 1 μν λ  μ ν lim 2∂ ∂ − η ∂ ∂λ G (x, x ) , t = 4πi x→x 2

(70)

The first term, μν

is the VS in the absence of the TI. In obtaining Eq. (70) we use that the GF is diagonal when the TI is absent, i.e. it is equal to η μν G (x, x ). The second term TCμν , to which we will refer as the θ vacuum stress (θ-VS), can be simplified since the θ-GF satisfies the Lorenz gauge condition ∂μ Gμν C = 0. With the previous results the θ-VS can be written as 

 1 1 μν μν μν μ ν λ lim ∂ ∂ GC + ∂ ∂λ GC − η GC TC = , (71) 4πi x→x 2 where GC = GμCμ is the trace of the θ-GF. This result exhibits the vanishing of the trace at quantum level, i.e., ημν TCμν = 0. Next we consider the problem of calculating the renormalized VS T μν ren . We proceed along the lines of Refs. 40 and 42. From Eq. (71), together with the symmetry of the problem we find that the θ-VS can be written as  2     d p dω  μ ν μν ˜ p p + nμ nν p2 gc (a, a) lim p2 + ∂z ∂z Ac (z, z  ) . TC = iθ z→z (2π)2 2π (72) In deriving this result we used the Fourier representation of the GF in Eq. (40) together with the solution for the reduced θ-GF given by Eq. (64). From Eq. (72) we calculate the renormalized θ-VS, which is given by TCμν ren = TCμν − TCμν vac , where the first (second) term is the θ-VS in the presence (absence) of the plates. 42 When the plates are absent, the reduced GF we have to use to compute the θ-VS in the region [0, L] is that of the free-vacuum g0 (z, z  ) = (i/2p)exp(ip|z − z  |), from which we find that limz→z ∂z ∂z A0 (z, z  ) = −p2 limz→z A0 (z, z  ), thus implying that the integrand in Eq. (72) vanishes. The function A0 is given by Eq. (50) using the free-vacuum reduced GF g0 (z, z  ). Therefore we conclude that TCμν vac = 0. Next we compute TCμν ren = TCμν starting from Eq. (72). From  symmetry    11the of the problem, the components of the stress along the plates, TC and TC22 , are equal.  In addition,  from the mathematical structure of Eq. (72) we find the relation TC00 = − TC11 . These results, together with the traceless nature of the SET, allow us to write the renormalized θ-VS in the form TCμν ren = (η μν + 4nμ nν ) τ (θ, z) ,

(73)

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where

 τ (θ, z) = iθ˜

d2 p  (2π)2



  dω 2 ω gc (a, a) lim p2 + ∂z ∂z Ac (z, z  ) . z→z 2π

25

(74)

Our θ-VS exhibits the same tensor structure as the result obtained by Brown and Maclay, 40 but now a z-dependent VS arises since the SET is not conserved at Σ. Using Eq. (61) we compute the limit of the integrand in Eq. (74) obtaining   lim p2 + ∂z ∂z P (z, z  ) = −

z→z

 ×

θ˜ 1 − θ˜2 p2 g2c (a, a)

 sin2 [p (L − a)] sin2 [pa] H (a − z) + H (z − a) . sin2 [pL] sin2 [pL]

(75)

To evaluate the integral in Eq. (24) we first write the momentum element as d2 p = |p |d|p |dϑ and integrate ϑ. Next, we perform a Wick rotation such that ω → iζ, then replace ζ and |p | by plane polar coordinates ζ = ξ cos ϕ, |p | = ξ sin ϕ and finally integrate ϕ. The renormalized θ-VS in Eq. (73) then becomes TCμν ren = −

π2 (η μν + 4nμ nν ) [u(θ, χ)H (a − z) + u(θ, 1 − χ)H (z − a)] , 720L4 (76)

where u(θ, χ) =

120 π4



∞ 0

θ˜2 ξ 3 sh [ξχ] sh3 [ξ (1 − χ)] sh−3 [ξ] dξ, 1 + θ˜2 sh2 [ξχ] sh2 [ξ (1 − χ)] sh−2 [ξ]

(77)

with sh(x) = sinh(x) and χ = a/L with 0 < χ < 1. Physically, we interpret the function u(θ, χ) as the ratio between the renormalized θ-energy density in the vacuum region [0, a) and that of the renormalized energy density in the absence of the TI. The function u(θ, 1 − χ) has an analogous interpretation for the bulk region of the TI (a, L]. This shows that the energy density is constant in the  00 bulk regions, however a simple discontinuity arises at Σ, i.e., ∂z TC ren ∝ δ(Σ). The Casimir energy E = EL + Eθ is defined as the energy per unit area stored in the electromagnetic field between the plates. To obtain it we must integrate the contribution from the θ-energy density  L   dz TC00 ren = EL [χu(θ, χ) + (1 − χ)u(θ, 1 − χ)] . (78) Eθ = 0

The first term corresponds to the energy stored in the electromagnetic field between P1 and Σ, while the second term is the energy stored in the bulk of the TI. The ratio Eθ /EL as a function of χ for different values of θ (appropriate for TIs 39 ) is plotted in Fig. 3. 30 Let us recall that EL = −π 2 /(720L3 ) is the Casimir energy in the absence of the TI. The setup known in the literature as the Casimir piston consists of a rectangular box of length L divided by a movable mirror (piston) at a distance a from one of the plates. 43 The net result is that the Casimir energy in each region generates a

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Fig. 3. of θ.

The ratio Eθ /EL as a function of the dimensionless distance χ = a/L, for different values

Fig. 4. The Casimir stress on the θ-piston in units of FL as a function of χ = a/L, for different values of θ.

force on the piston pulling it towards the nearest end of the box. Here we have considered a similar setup, which we call the θ-piston, in which the piston is the TI. The Casimir stress acting upon Σ can be obtained as Fθp = −dEθ /da. The result is Fθp 1 d [χu(θ, χ) + (1 − χ)u(θ, 1 − χ)] , (79) =− FL 3 dχ where FL − −π 2 /(240L4 ) is the Casimir stress between the two perfectly reflecting plates in the absence of the TI. Figure 4 30 shows the Casimir stress on Σ in units of FL as a function of χ for different values of θ. We observe that this force pulls the boundary Σ towards the closer of the two fixed walls P1 or P2 , similarly to the conclusion in Ref. 43. Now let us consider the limit where the plate P2 is sent to infinity, i.e., L → ∞. This configuration corresponds to a perfectly conducting plate P1 in vacuum, and

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a semiinfinite TI located at a distance a. Here the plate and the TI exert a force upon each other. The Casimir energy in Eq. (78) in the limit L → ∞ takes the form EθL→∞ = Ea R(θ), with Ea = −π 2 /720a3 , and the function  θ˜2 120 ∞ 3 e−3ξ sinh ξdξ, ξ (80) R(θ) = 4 2 e−2ξ sinh2 ξ ˜ π 1 + θ 0 is a-independent and bounded by its θ → ±∞ limit, i.e.,  120 ∞ 3 e−ξ dξ = 1. ξ R(θ) ≤ 4 π sinh ξ 0

(81)

Thus, for this case, the energy stored in the electromagnetic field is bounded by the Casimir energy between two parallel conducting plates at a distance a, i.e., EθL→∞ ≤ Ea . Physically this implies that in the θ → ∞ limit the surface of the TI mimics a conducting plate, which is analogous to Schwinger’s prescription for describing a conducting plate as the ε → ∞ limit of material media. 29 These results, which stem from our Eqs. (64) and (50), agree with those obtained in the global energy approach which uses the reflection matrices containing the Fresnel coefficients as in Ref. 39, when the appropriate limits to describe an ideal conductor at P1 and a purely topological surface at Σ are taken into account. Taking the derivative with respect to a we find that the plate and the TI exert a force (in units of Fa = −π 2 /240a4 ) of attraction upon each other given by fθ = FθL→∞ /Fa = R(θ). Numerical results for fθ for different values of θ are presented in Table 1. Table 1. Normalized force fθ = FθL→∞ /Fa = R(θ) for different values of θ. θ

±7π

±15π

±23π

±31π

±39π

fθ

0.0005

0.0025

0.0060

0.0109

0.0172

A general feature of our analysis is that the TI induces a θ-dependence on the Casimir stress, which could be used to measure θ. Since the Casimir stress has been measured for separation distances in the 0.5–3.0 μm range, 35 these measurements require TIs of width lesser than 0.5 μm and an increase of the experimental precision of two to three orders of magnitude. In practice the ability to measure fθ depends on the value of the topological MEP, which is quantized as θ = (2n + 1)π, n ∈ Z. The particular values θ = ±7π, ±15π are appropriate for the TIs such as Bi1−x Sex , 44 where we have f±7π ≈ 0.0005 and f±15π ≈ 0.0025, which are not yet feasible with the present experimental precision. This effect could also be explored in TIs described by a higher coupling θ, such as Cr2 O3 . However, this material induces more general magnetoelectric couplings not considered in our model. 39 Although the reported θ-effects of our Casimir systems cannot be observed in the laboratory yet, we have aimed to establish the Green’s function method as an alternative theoretical framework for dealing with the topological magnetoelectric

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effect of TIs and also as yet another application of the GF method we developed in Refs. 31–33. Acknowledgments LFU takes the opportunity to thank the organizers of the Julian Schwinger Centennial Conference and Workshop for a wonderful meeting honoring the great physicist and scholar. LFU also acknowledges support from the CONACyT (M´exico) project No. 237503. AM was supported by the CONACyT postdoctoral Grant No. 234774. References 1. J. C. Maxwell, A dynamical theory of the electromagnetic field, Phil. Trans. R. Soc. London 155, 459 (1865). 2. J. S. Schwinger, On quantum-electrodynamics and the magnetic moment of the electron, Phys. Rev. 73, 416 (1948). 3. J. S. Schwinger, Quantum electrodynamics I. A covariant formulation, Phys. Rev. 74, 1439 (1948). 4. J. S. Schwinger, Quantum electrodynamics II. Vacuum polarization and self energy, Phys. Rev. 75, 651 (1949). 5. C. Nash and S. Sen, Topology and Geometry for Physicists (Academic Press Inc, 1983). 6. K. Fujikawa and H. Suzuki, Path Integrals and Quantum Anomalies (Clarendon Press, 2004). 7. M. Dine, TASI lectures on the strong CP problem, arXiv:0011376 [hep-ph], 2000. 8. D. Birmingham, M. Blau, M. Radowski and G. Thompson, Topological field theory Phys. Rep. 209, 129 (1991). 9. M. Kuster, G. Raffelt and B. Beltr´ an (eds.), Axions: Theory, Cosmology, and Experimental Searches (Lecture Notes in Physics Vol. 741) (Springer-Verlag, 2008). 10. X. L. Qi, T. L. Hughes and S. C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008). 11. T. H. O’Dell, The Electrodynamics of Magneto-Electric Media (North-Holland, 1970); L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media (Course of Theoretical Physics Vol. 8) (Pergamon Press, 1984). 12. E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis and N. I. Zheludev, Metamaterial with negative index due to chirality, Phys. Rev. B 79, 035407 (2009). 13. M. M. Vazifeh and M. Franz, Electromagnetic response of Weyl semimetals, Phys. Rev. Lett. 111, 027201 (2013). 14. X. L. Qi, Field-theory foundations of topological insulators, in Topological Insulators (Contemporary Concepts of Condensed Matter Science), eds. M. Franz and L. Molenkamp, Vol. 6 (Elsevier, 2013). 15. N. P. Armitage, E. J. Mele and A. Vishwanath, Weyl and Dirac semimetals in threedimensional solids, Rev. Mod. Phys. 90, 015001 (2018). 16. I. E. Dzyaloshinskii, On the magneto-electrical effect in antiferromagnets, JETP 37, 881 (1959). 17. D. N. Astrov, The magneto-electrical effect in antiferromagnets, JETP 38, 984 (1960). 18. M. Fiebig, Revival of the magnetoelectric effect, J. Phys. D: Appl. Phys. 38, R123 (2005). 19. V. Diziom et al., Observation of the universal magnetoelectric effect in a 3D topological insulator, Nature Communications 8, 15297 (2017).

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20. X. L. Qi, R. Li, J. Zang and S. C. Zhang, Inducing a magnetic monopole with topological surface states, Science 323, 1184 (2009). 21. C. Kim, E. Koh and K. Lee, Janus and multifaced supersymmetric theories, Journal of High Energy Physics, 0806, 040 (2008). 22. C. Kim, E. Koh and K. Lee, Janus and multifaced supersymmetric theories. II, Phys. Rev. D 79, 126013 (2009). 23. F. Wilczek, Two applications of axion electrodynamics, Phys. Rev. Lett. 58, 1799 (1987). 24. L. Huerta and J. Zanelli, Optical properties of a θ vacuum, Phys. Rev. D 85, 085024 (2012). 25. Y. N. Obukhov and F. W. Hehl, Measuring a piecewise constant axion field in classical electrodynamics, Phys. Lett. A 341, 357 (2005). 26. A. Mart´ın-Ruiz and L. F. Urrutia, Interaction of a hydrogenlike ion with a planar topological insulator, Phys. Rev. A 97, 022502 (2018). 27. A. Mart´ın-Ruiz and E. Chan-L´ opez, Dynamics of a Rydberg hydrogen atom near a topologically insulating surface, Eur. Phys. Lett. 119, 53001 (2017) 28. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999). 29. J. Schwinger, L. DeRaad, K. Milton and W. Tsai, Classical Electrodynamics, Advanced Book Program (Perseus Books, 1998). 30. A. Mart´ın-Ruiz, M. Cambiaso and L. F. Urrutia, A Green’s function approach to the Casimir effect on topological insulators with planar symmetry, Eur. Phys. Lett. 113, 60005 (2016). 31. A. Mart´ın-Ruiz, M. Cambiaso and L. F. Urrutia, A Green’s function approach to Chern-Simons extended electrodynamics: An effective theory describing topological insulators, Phys. Rev. D 92, 125015 (2015). 32. A. Mart´ın-Ruiz, M. Cambiaso and L. F. Urrutia, Electro- and magnetostatics of topological insulators as modeled by planar, spherical, and cylindrical θ boundaries: Green’s function approach, Phys. Rev. D 93, 045022 (2016). 33. A. Mart´ın-Ruiz, M. Cambiaso and L. F. Urrutia, Electromagnetic description of threedimensional time-reversal invariant ponderable topological insulators, Phys. Rev. D 94, 085019 (2016). 34. H. B. G. Casimir, On the attraction between two perfectly conducting plates, Proc. K. Ned. Akad. Wet. 51, 793 (1948). 35. G. Bressi et al., Measurement of the Casimir Force between parallel metallic surfaces, Phys. Rev. Lett. 88, 041804 (2002). 36. K. A. Milton, The Casimir effect: Physical Manifestation of Zero-Point Energy (World Scientific, 2001). 37. M. Bordag, G. L. Klimchitskaya, U. Mohideen and V. M. Mostepanenko, Advances in Casimir Effect (Oxford University Press, 2009). 38. L. Fu, C. L. Kane and E. J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett. 98, 106803 (2007); D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava and M. Z. Hasan, A topological Dirac insulator in a quantum spin Hall phase, Nature 452, 970 (2008). 39. A. G. Grushin and A. Cortijo, Tunable Casimir repulsion with three-dimensional topological insulators, Phys. Rev. Lett. 106, 020403 (2011); A. G. Grushin, P. RodriguezLopez and A. Cortijo, Effect of finite temperature and uniaxial anisotropy on the Casimir effect with three-dimensional topological insulators, Phys. Rev. B 84, 045119 (2011). 40. L. S. Brown and G. J. Maclay, Vacuum stress between conducting plates: An image solution, Phys. Rev. 184, 1272 (1969).

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41. J. Schwinger, L. DeRaad and K. Milton, Casimir effect in dielectrics, Ann. Phys. (N.Y.) 115, 1 (1978). 42. D. Deutsch and P. Candelas, Boundary effects in quantum field theory, Phys. Rev. D 20, 3063 (1979). 43. R. M. Cavalcanti, Casimir force on a piston, Phys. Rev. D 69, 065015 (2004). 44. X. Zhou et al., Photonic spin Hall effect in topological insulators, Phys. Rev. A 88, 053840 (2013).

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Remarks on the Abraham–Minkowski problem, in relation to recent radiation pressure experiments Iver Brevik Department of Energy and Process Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway

The classic electromagnetic energy–momentum problem in matter (usually called the Abraham–Minkowski problem) has attracted increased interest, as is natural in relation to the several impressive radiation pressure experiments that have appeared recently. Our intention with the present note is to focus attention on some of these results, and also to give a warning against premature interpretations of the observations. One sees often in the literature that the observable deflections of dielectric surfaces are interpreted so as to mean that the so-called Abraham term is a chief ingredient. Usually this is not so, however. Most of the experimental results are actually explainable by the surface forces at the dielectric surfaces, eventually augmented by Lorentz forces in the interior, and do not involve the Abraham momentum as such. For concreteness we focus mainly on a simplified version of the experiment of Kundu et al. (2017), but extend the analysis somewhat by including time-dependent resonance phenomena. In a short appendix we discuss also the connection with the Casimir effect.

1. Introduction What is the correct — or more appropriately — the most convenient, expression for electromagnetic momentum density in a continuous medium? This is the problem usually called the Abraham–Minkowski problem, and has been under debate at more or less intensity since about 1910. 1,2 The present author has been involved in this discussion for some time, both classically and quantum mechanically, with particular emphasis on radiation pressure phenomena. Some recent papers can be found in Refs. 3–5. There are actually quite a lot of papers in this area; some of them are listed in Refs. 6–26. Let us begin by writing down some essential formulas. In the case of an isotropic nonmagnetic medium with permittivity ε = n2 the electromagnetic force density is 27,28 (the electrostriction term is omitted) 1 n2 − 1 ∂ (E × H). f = ρE + μ0 J × H − ε0 E 2 ∇n2 + 2 c2 ∂t

(1)

The constitutive relations are D = ε0 εE, B = μ0 H. The two first terms above are usually without importance for optical phenomena. The third term which is proportional to the gradient of the permittivity, is usually the most important term. It is common for the Abraham and Minkowski theories and may thus be called the Abraham–Minkowski term. It turns up at dielectric boundaries. The surface force generally acts towards the optical thinner medium. This kind of force was demonstrated already in the classic experiment of Ashkin and Dziedzic, 6 and has been

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repeated several times, for instance when to immiscible liquids are superimposed in the vicinity of the critical point where a large enhancement of the surface deflection can be observed. 29,30 Among other impressible experiments we may mention the one of Astrath et al.; 7 their method was essentially the same as that of Ashkin and Dziedzic, namely to observe the deflection of a water surface when subject to laser illumination from above. Typically, about 30 nm deflections were observed. As another example we may mention the recent radiation pressure experiment of Choi et al., 31 implying use of a liquid-core optical fiber waveguide. Here the radiation force from the laser gave rise to an axial displacement of the interface in the hollow fiber. (A more detailed overview can be found in Ref. 5.) Now proceed to the final term in Eq. (1), fA =

n2 − 1 ∂S n2 − 1 ∂ (E × H) = , 2 c ∂t c2 ∂t

(2)

with S the Poynting vector. This term is called the Abraham term. One may ask: under what circumstances will it be measurable? It is obvious that for a plane wave, under stationary conditions, the Abraham term simply fluctuates out. We will analyze mainly one typical example in the following, namely the recent radiation pressure experiment of Kundu et al. 32 This experiment gives a nice demonstration of how a graphene oxide plate becomes deflected in the downward direction when illuminated by a weak laser beam from above. But does it show the appearance of the Abraham force? As we will show, the answer is no, although this contradicts the main interpretation of the authors. We will extend the analysis somewhat, so as to take into account time-dependent deflections also, and we will finally give a brief account of the connection with the Casimir effect. 2. Dielectric model Assume for simplicity that the incident wave is a plane wave in air (vacuum), propagating from above in the vertical z direction, polarized in the x direction so that Ex = E0 ei(k0 z−ωt) is the only nonvanishing component. Here k0 = ω/c. The Poynting vector of the incident wave is thus SI =

1 ε0 cE02 . 2

(3)

We let the z axis point downwards. The upper surface of the plane graphene sheet lies at z = 0; the lower surface lies at z = d, where d is the thickness. There are in principle two different kinds of forces acting on the plate: (i) Gradient forces acting at the boundaries z = 0 and z = d (the third term in Eq. (1)). At z = 0 the force acts upwards, at z = d it acts downwards (n > 1 assumed). (ii) Lorentz forces acting on the currents in the interior because of the conductivity σ of graphene oxide. The refractive index of this material is complex; calling it

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n ˜ , we have as the mean value at wavelength 532 nm: 33 n ˜ = 2.4 + 1.0 i

(4)

(the plus sign occurs due to our convention e−iωt ). This means that σ is quite appreciable. We might approach the problem using the full formalism for metals. 34 This would however be complicated, and is hardly justified here since we are not aiming for great accuracy in the radiation force. We therefore will follow a simpler approach, namely to calculate the gradient force as if the refractive index were real, and thereafter consider the Lorentz force contribution separately. Thus assume in this section a purely dielectric model, implying that the refrac√ tive index n = ε is taken to be real. Consider first the upper surface z = 0. Making use of a complex representation, the z component of the force density in the boundary layer becomes ∂n2 1 . (5) fz = − ε0 |E |2 4 ∂z Here E , the total electric field parallel to the surface, is the sum of the incident field EI and the reflected field ER . As E is continuous at the surface, we easily obtain the corresponding surface pressure σz by integrating (5) across the boundary layer, 1 (6) σz,z=0 = − ε0 |E0 + ER |2 (n2 − 1). 4 For n > 1 this force always acts outward. From Ref. 34, Secs. 9.10–11, we have for the reflected component   r12 1 − e2ikd ER = (7) 2 e2ikd . E0 1 − r12 Here k = nω/c = nk0 is the wave number in the plate, d is the plate thickness, and r12 =

1−n 1+n

(8)

is the conventional reflection coefficient in the limit d → ∞. For definiteness we need to fix the value of d. We adopt the value actually used in the experiment 32 (communication from K. S. Hazra), d = 300 nm.

(9)

This value of d is actually quite large in a graphene context (a single monolayer is known to be about 3.5 ˚ A thick). With an incident wavelength of λ0 = 532 nm we 7 have k0 = 1.18 × 10 rad/m, and so 2kd = 7.09 n. For simplicity we take in this section the real refractive index to be n = 2.0.

(10)

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The expression (6) for the surface pressure can be rewritten as σz,z=0 = (ε0 E02 r12 )

2 1 − 2r12 cos 2kd + r12 2 4 . 1 − 2r12 cos 2kd + r12

(11)

With the data given above we get r12 = −1/3, 2kd = 14.17, cos 2kd = −0.0328, and so σz,z=0 = −ε0 E02 × 0.35.

(12)

This force acts upwards. Its magnitude is about 35% of that encountered in the case of perfect reflection. Consider next the lower surface z = d, 1 (13) σz,z=d = ε0 E02 T (n2 − 1), 4 where T means the transmission coefficient, T = |ET /E0 |2 , ET being the transmitted field amplitude. When n > 1 this force acts downwards. From Ref. 34, T =

(1 − R12 )2 2 , 1 + 2R12 cos(2kd) + R12

(14)

2 with R12 = r12 . We now calculate T = 0.786, and so

σz,z=d = ε0 E02 × 0.59.

(15)

Then the total surface pressure on the sheet becomes σz = σz,z=0 + σz,z=d = ε0 E02 × 0.24,

(16)

that means, a downward directed force. The direction of this force is not selfevident, since there exists no focusing mechanism here (no lens effect) as it would be for a dielectric sphere, for instance. 3. Conductive model In view of the appreciable conductivity of graphene oxide, it might seem more natural after all to calculate the pressure on the plate via the Lorentz force density J × B in the interior region (we assume the electric charge density ρ = 0, as usual). We will now consider this approach in some detail. It was actually the option followed also in Ref. 32, as they took the radiation force Fz on the plate to be 2P , (17) Fz = c with P the incident radiation power. This corresponds to the surface pressure σz = ε0 E02 , i.e. perfect reflection at the surface z = 0. We shall for simplicity restrict ourselves to the case of a large σ. Formally, this corresponds to the inequality k0 /α  1, where k0 is the incident wave number as before, and α is defined as

(18) α = μ0 ωσ/2.

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As above, we assume that the wave falls normally on the plate at the surface z = 0, the plate now for convenience taken to be infinitely thick. The approximate expressions for the fields in the two regions are:   √ (19) Ex = E0 ei(k0 z−ωt) − R e−iδ e−i(k0 z+ωt) , (z < 0), Hy =

k0 E0  i(k0 z−ωt) √ −iδ −i(k0 z+ωt)  + Re e , (z < 0), e μ0 ω

(20)

k 0 E0 (1 − i)e−αz ei(αz−ωt) , (z > 0), α   k 0 E0 k0 −αx i(αz−ωt) Hy = e , (z > 0), 2 − (i − 1) e μ0 ω α Ex =

(21)

(22)

with R = 1 − 2k0 /α,

tan δ = −k0 /α.

(23)

These expressions satisfy the boundary conditions at z = 0 to the first order in k0 /α. The surface force on unit area of the plate is found by integrating the Lorentz force from z = 0 to z = d,  d 1 Ex Hy∗ dz. (24) σz = μ0 σ 2 0 Insertion of the above expressions gives



k0 σz = ε0 E02 (1 − e−2αd ) 1 − . α

With λ0 = 532 nm, ω = 3.54 × 1015 rad/s we get α = 4.72 × 104 × √ k0 /α = 250/ σ. As an example, we may choose σ = 1.0 × 106 S/m,

(25) √

σ, or (26)

which is about the same conductivity as for manganese steel. Then, k0 /α = 0.25, and the above condition is roughly satisfied. With d = 300 nm the term e−2αd  1, and we obtain σz = ε0 E02 × 0.75.

(27)

One may ask if our choice (26) for the conductivity is reasonable. It corresponds to a two-dimensional sheet conductivity equal to σ 2D = 3 S/m, when d = 300 nm. This is a quantity that is in principle accessible experimentally. To get some more insight at this point, let us go back to Eq. (4) for the complex refractive index and calculate the complex permittivity, ε=n ˜ 2 = (2.4 + 1.0 i)2 = 4.76 + 4.8 i.

(28)

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In conventional notation ε = ε + iε ; thus ε = 4.76, ε = 4.8. Now comparing with the formula iσ , (29) ε = ε + ε0 ω we obtain the value σ = 1.5 × 105 S/m. Although there are considerable uncertainties when associating two-dimensional sheets with three-dimensional quantities, this indicates that our calculation has overestimated σ a bit. We might have used a lower value of σ (giving a weaker surface force), but at the expense of violating the condition k0 /α  1. We will not enter into further detail here, but conclude that the expression (27) should give a reasonable value for the Lorentz force on the plate. When augmented by the surface force (16) at the boundaries we get a value for σz that becomes roughly the same as for total reflection, σz = ε0 E02 .

(30)

We will for definiteness use this expression as the driving force in the following. As mentioned, it agrees with the assumption made in Ref. 32. 4. Statics and dynamics of the circular plate We will apply elasticity theory to estimate the influence from the force (30) on the graphene oxide plate. Now, a practical complication in the experiment 32 was that the plate was residing on a Si substrate. It is natural to assume that there was not a direct mechanical contact between plate and substrate; otherwise there would be no deflection at all. Moreover, if transmission properties in the plate were allowed for, it would be necessary to include the optical properties of the substrate also. We will henceforth avoid these possible complications by assuming that the plate is surrounded by air (n = 1) both on the upper and the lower side. Such a simplified model is yet able to demonstrate the essence of the effect. The plate of thickness d is circular, of radius a, and we assume for simplicity that the pressure σz is constant over the initially flat cross section πa2 . The total radiation force is thus 2P , (31) Fz = c where the incident power is P = SI πa2 =

1 ε0 cE02 × πa2 . 2

(32)

4.1. Statics We will first evaluate the form of the plate in its equilibrium state when acted upon by the cw laser beam. Adopt cylindrical coordinates with the origin lying at the center of the undisturbed sheet and let, as mentioned, the z axis be pointing

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downwards. The stationary deflection is ζ = ζ(r). The governing equation for large deflections is in general quite complicated, of the fourth order in ζ. 35 One may approach the problem in one of the following two ways: First, one may model the graphene sheet as a elastic plate subject to the conditions that both the elevation and the slope of the plate are zero at r = a (i.e., a clamped edge situation). The governing equation is D∇4 ζ = σz + ρgd,

(33)

where D=

Ed3 12(1 − ν 2 )

(34)

is the flexural rigidity. Here E is Young’s modulus, and ν is Poisson’s ratio. For graphene, ν ≈ 0.16. 36 Equation (33) is quite general, holding even if the deflection ζ is large compared with d. With a = 0.9μm, P = 1.4 mW we find σz = 3.67 Pa, while the gravitational pressure is much less, ρgd = 6.65 mPa. Thus the term ρgd can be omitted, and the equation reduces to  

 d 1 d dζ σz 1 d , (35) r r = r dr dr r dr dr D which by integration yields ζ=

σz a 4 64D

1−

r2 a2

2 .

(36)

The slope at r = a is thus zero. Using the maximum deflection from the experiment, ζmax = σz a4 /(64D) = 80 nm, we can estimate the effective value of the flexural rigidity. We find D = 4.7 × 10−19 Nm, corresponding to a very low value E = 200 Pa.

(37)

Second, in view of the large softness it could appear more natural to apply a membrane model in which the zero-slope condition at the edge is relaxed. For a membrane the governing equation is much simpler as it is only of second order, 35 − T ∇2 ζ = σz ,

(38)

where T is the magnitude of the stretching force per unit length of the sheet. The solution is

σz a2 r2 ζ= (39) 1− 2 . 4T a Inserting the same data for ζmax , σz and a as above, we can from this expression calculate an effective value for the stress T . Approximatively, we find T = 10−5 N/m.

(40)

With d = 300 nm this corresponds to a 3D mechanical stress of 62 Pa. It is quite low.

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We can thus conclude so far: the downward bending in the experiment 32 is fully described in terms of the Lorentz force alone, eventually augmented by the surface forces at the boundaries of the plate. The Abraham force does not come into play; it averages out at the optical frequencies. 4.2. Dynamics We will now move on to consider the eigenfrequencies of the plate when there is no radiation, σz = 0, although this is not a topic directly associated with the experiment. 32 For simplicity we assume membrane theory in the following. The driving force on the plate is then only the elastic tension force T ∇2 ζ. The governing equation is ¨ T ∇2 ζ = ρdζ,

(41)

ζ(r, t) ∝ J0 (κr) cos ω0 t,

(42)

with the harmonic solution

where J0 is the zeroth order Bessel function. Here κ2 = ω02 ρd/T,

(43)

from which we calculate the value of ω0 when observing that the first zero point of the Bessel function lies at κa = 2.40. Using the values above, including that of T from Eq. (40), we get ω0 = 0.281/a. Thus with a = 0.9 μm, ω0 = 3.12 × 105 rad/s. We can evaluate how these vibrations can be significantly increased in magnitude by illuminating the plate with a laser power whose intensity varies with a frequency ω lying close to ω0 . Then we have to include the damping, due to air resistance. First, we see that the maximum velocity derived from the formalism above is ζ˙max = ω0 ζmax . Inserting tentatively the value ζmax ∼ 10 nm we find the velocity to be low, ζ˙max ∼ 0.003 m/s. Taking into account the kinematic viscosity for air, νair = 1.50 × 10−5 m2 /s, we see that the Reynolds number becomes also low, Re= ζ˙max a/νair = 2 × 10−4  1. A Stokes hydrodynamical approach, implying a linear relationship between drag force and velocity, is thus justified. Accordingly, we take the drag ˙ where γ is a constant with dimension Pa.s/m. force per unit surface area to be γ ζ, The governing equation becomes − T ∇2 ζ + ρdζ¨ + γ ζ˙ = σz (t).

(44)

We employ a complex harmonic representation σz (t) = σ0 eiωt with trial solution ζ(r, t) = R(r)eiωt ,

(45)

and assume that near the symmetry axis, the radial function R(r) is proportional to the Bessel solution given above. This means R(r) = bJ0 (κr) → b when r → 0, with b a constant. Inserting Eq. (A.3) we first get  γ ρd  2 σ0 1 R + R + . (46) ω − 2iλω R = − , λ = r T T 2ρd

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When the external frequency lies close to resonance, ω = ω0 +  with  small, ω 2 − ω02 ≈ 2ω0 . Moreover, using the approximation λω ≈ λω0 , we see that the governing equation at the symmetry axis reduces to 2ρdω0 ( − iλ)b = −σ0 .

(47)

The physical solution at the symmetry axis, called ζ0 (t), then becomes ζ0 (t) = −

σ0 cos(ωt + δ) √ , 2ρdω0  2 + λ2

tan δ =

λ . 

(48)

This expression is very sensitive with respect to the input parameters, due to the smallness of d. A nontrivial point here is to estimate the value of the resistance coefficient γ. We cannot make use of common drag formulas for rectangular plates moving transversely in a fluid here since the flow is in the laminar regime, but we can get useful information from a comparison with the Stokes drag formula F = 6πρair νair aU for a sphere of radius a moving slowly in air with velocity U . Dividing with the cross section πa2 this yields a resistance coefficient, with the same dimension as γ, equal to 6ρair νair /a ∼ 102 when a = 0.9 μm. The Stokes formula certainly overestimates the drag compared to our case, but it indicates that a realistic value of γ is less by only a moderate factor. For definiteness we choose γ to be one order of magnitude smaller, thus γ = 10 Pa.s/m. With ρ = 2.26 g/cm3 this yields λ = 7.4 × 103 s−1 . Since λ is so large, we will for simplicity ignore the contribution from  in Eq. (48). Inserting ω0 = 3.12 × 105 s−1 and σ0 = 3.67 Pa as above we find the maximum deflection to be σ0 ≈ 1 μm. (49) ζ0max = 2ρdω0 λ This magnitude seems physically reasonable. It is two orders of magnitude larger than the deflection ζmax ∼ 10 nm estimated above for free vibrations. Evidently, also in this time-dependent situation does the rapidly fluctuating Abraham term f A not play any role. 5. Conclusion The recent radiation pressure experiments, on flat dielectric surfaces as well as on curved ones, are of interest both from a fundamental and a practical viewpoint. We have mainly focused on a simplified variant of the recent experiment of Kundu et al., 32 which shows in a detailed manner how a graphene oxide sheet becomes deflected by a weak radiation pressure. A point worth emphasizing is however that some care is needed in interpreting phenomena of this type: they have very little to do with the Abraham momentum as such (contrary to the main conclusion in Ref. 32). The observed results are actually explainable via the surface forces at the boundaries, plus the influence from the Lorentz forces in the interior. In usual cases in optics, the Lorentz forces are evidently absent. When the radiation force on the sheet is known, the deflection of the sheet can be found by making use of elasticity theory. We took the opportunity to give an

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approximate theory of that kind in Sec. 4. Also, the dynamic case was considered, including the possibility of obtaining a considerable enhancement of the deflections at mechanical resonance. In the latter case, it was necessary to take into consideration viscous drag from the air. Appendix A. Relationship to the Casimir effect It is of some interest to consider if this experiment has some relationship to the Casimir effect — this point was actually touched upon already in. 32 Now, the experiment itself belongs to classical electromagnetic theory and so obviously need not electromagnetic vacuum fluctuation in order to be explained. However, let us imagine that there are two graphene sheets, parallel to each other and at a close separation L. Then, the two sheets will attract each other by a Casimir force. We do not intend to go into any detail on this subject, but will give the basic formula allowing one to calculate the force F per unit area between the two sheets, assuming the material to be isotropic. Assume that the sheets have the same width d. As L is naturally much smaller than the relevant electromagnetic wavelengths, we can employ T = 0 Casimir theory (the important parameter is kB T L/(c), which is only about 10−3 at room temperature if we choose L = 10 nm). As usual, one can evaluate the force as the derivative of the free energy with respect to L, with the opposite sign (cf., for instance, Ref. 37). The expression can be written as 

∞

F = −2c 0

dk 2π



∞ 0

 2 −2κ0 L  2 rTE e dζ e−2κ0 L rTM κ0 2 e−2κ0 L + 1 − r 2 e−2κ0 L . 2π 1 − rTE TM

(A.1)

Here k is the magnitude of the wave vector component

parallel to the surface, ζ is

the imaginary frequency, κ0 and κ are defined as κ0 = k 2 + ζ 2 , κ = k 2 + εζ 2 , and rTE , rTM are the generalized reflection coefficients rTE =

rTM =

κ0 − κ κ0 + κ

εκ0 − κ εκ0 + κ

1 − e−2κd , 2  −2κd 1 − κκ00 −κ e +κ

(A.2)

1 − e−2κd . 2  εκ0 −κ −2κd 1 − εκ0 +κ e

(A.3)

Acknowledgments Thanks go to L. C. Malacarne for alerting me to the experiment, 32 and they go also to K. S. Hazra for informing me about the actual thickness of the graphene oxide plate used. This work was supported by the Research Council of Norway, Project 250346.

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References 1. M. Abraham, Zur Elektrodynamik bewegter K¨ orper, Rend. Circ. Matem. Palermo 28, 1 (1909). 2. H. Minkowski, Die Grundgleichungen f¨ ur die elektromagnetischen Vorg¨ ange in bewegten K¨ orpern, Math. Ann. 68, 472 (1910). 3. I. Brevik and S. ˚ A. Ellingsen, Possibility of measuring the Abraham force using whispering gallery modes, Phys. Rev. A 81, 063830 (2010). 4. I. Brevik, Minkowski momentum resulting from a vacuum-medium mapping procedure, and a brief review of Minkowski momentum experiments, Ann. Physics 377, 10 (2017). 5. I. Brevik, Radiation forces and the Abraham-Minkowski problem, Mod. Phys. Lett. A 33, 1830006 (2018). 6. Ashkin, A. and Dziedzic, J. M. Radiation pressure on a free liquid surface. Phys. Rev. Lett. 30, 139 (1973). 7. N. G. Astrath, L. C. Malacarne, M. L. Baesso, G. V. Lukasievicz and S. E. Bialkowski, Unravelling the effects of radiation forces in water, Nature Commun. 5, 4363 (2014). 8. L. Zhang, W. She, N. Peng and U. Leonhardt, Experimental evidence for Abraham pressure of light, New J. Phys. 17, 053035 (2015). 9. G. Verma and K. P. Singh, Universal long-range nanometric bending of water by light, Phys. Rev. Lett. 115, 143902 (2015). 10. M. N. Schneider and V. V. Semak, Ponderomotive convection in water induced by a CW laser, J. Appl. Phys. 120, 244902 (2016). 11. M. Partanen, T. H¨ ayrynen, J. Oksanen and J. Tulkki, Photon mass drag and the momentum of light in a medium. Phys. Rev. A 95, 063850 (2017). 12. V. V. Nesterenko and A. V. Nesterenko, Symmetric energy-momentum tensor: The Abraham form and the explicitly covariant formula, J. Math. Phys. 57, 032901 (2016). 13. S. M. Barnett and R. Loudon, Theory of the radiation pressure on magneto-dielectric materials, New J. Phys. 17, 063027 (2015). 14. M. Bethune-Waddell and K. J. Chau, Simulations of radiation pressure experiments narrow down the energy and momentum of light in matter, Rep. Prog. Phys. 78, 122401 (2015). 15. B. A. Kemp, Macroscopic theory of optical momentum, Prog. Optics 60, 437 (2015). 16. C. Conti and R. Boyd, Nonlinear optomechanical pressure, Phys. Rev. A 89, 033834 (2014). 17. B. A. Kemp, Resolution of the Abraham-Minkowski debate: Implications for the electromagnetic wave theory of light in matter, J. Appl. Phys. 109, 111101 (2011). 18. U. Leonhardt, Abraham and Minkowski momenta in the optically induced motion of fluids, Phys. Rev. A 90, 033801 (2014). 19. S. M. Barnett and R. Loudon, The enigma of optical momentum in a medium, Philos. Trans. Royal Soc. A 368, 927 (2010). 20. C. Baxter and R. Loudon, Radiation pressure and the photon momentum in dielectrics, J. Mod. Opt. 57, 830 (2010). 21. S. M. Barnett, Resolution of the Abraham-Minkowski dilemma, Phys. Rev. Lett. 104, 070401 (2010). 22. E. A. Hinds and S. M. Barnett, Momentum exchange between light and a single atom: Abraham or Minkowski?, Phys. Rev. Lett. 102, 050403 (2009). 23. M. Mansuripur, Resolution of the Abraham-Minkowski controversy, Optic Commun. 283, 1997 (2010).

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24. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, Colloquium: Momentum of an electromagnetic wave in dielectric media, Rev. Mod. Phys. 79, 1197 (2007). 25. R. Loudon, S. M. Barnett and C. Baxter, Radiation pressure and momentum transfer in dielectrics: The photon drag effect, Phys. Rev. A 71, 063802 (2005). 26. R. Loudon, Theory of the radiation pressure on dielectric surfaces, J. Mod. Opt. 49, 821 (2002). 27. I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor. Phys. Rep. 52, 133 (1979). 28. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd edn. (Pergamon Press, Oxford, 1984). 29. A. Casner and J. P. Delville, Laser-induced hydrodynamic instability of fluid interfaces, Phys. Rev. Lett. 90, 144503 (2003). 30. R. Wunenburger, B. Issenmann, E. Brasselet, C. Loussert, V. Houstane and J. P. Delville, Fluid flows driven by light scattering, J. Fluid Mech. 666, 273 (2011). 31. H. Choi, M. Park, D. S. Elliot and K. Oh, Optomechanical measurement of the Abraham force in an adiabatic liquid-core optical-fiber waveguide, Phys. Rev. A 95, 053817 (2017). 32. A. Kundu, R. Rani and K. S. Hazra, Graphene oxide demonstrates experimental confirmation of Abraham pressure on solid surface. Sci. Rep. 7, 42538 (2017); doi: 10.1038/srep42538. 33. X. Wang, Y. P. Chen and D. D. Nolte, Strong anomalous optical dispersion of graphene: Complex refractive index measured by picometrology. Optics Expr. 16, 22105 (2008). 34. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). 35. L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 2nd edn. (Pergamon Press, Oxford, 1986), Sec. 14. 36. C. Lee, X. Wei, J. W. Kysar and J. Hone, Measurement of the elastic properties and intrinsic strength of monolayer graphene, Science 321, 385 (2008). 37. M. Bordag, G. L. Klimchitskaya, U. Mohideen and V. M. Mostepanenko, Advances in the Casimir Effect (Oxford Science Publications, Oxford, 2009).

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Manifest non-locality in quantum mechanics, quantum field theory and quantum gravity Laurent Freidel Perimeter Institute for Theoretical Physics, ON N2L 2Y5, Waterloo, Canada [email protected] Robert G. Leigh Department of Physics, University of Illinois, Urbana, IL, 61801, USA [email protected] Djordje Minic Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA [email protected]

We summarize our recent work on the foundational aspects of string theory as a quantum theory of gravity. We emphasize the hidden quantum geometry (modular spacetime) behind the generic representation of quantum theory and then stress that the same geometric structure underlies a manifestly T-duality covariant formulation of string theory, that we call metastring theory. We also discuss an effective non-commutative description of closed strings implied by intrinsic non-commutativity of closed string theory. This fundamental non-commutativity is explicit in the metastring formulation of quantum gravity. Finally we comment on the new concept of metaparticles inherent to such an effective non-commutative description in terms of bi-local quantum fields. Keywords: Non-locality; non-commutativity; modular spacetime; metastring theory.

1. Introduction Julian Schwinger (1918–1994) was one of the giants of theoretical physics, 1,2 and it is an honor to present this work to the celebration of his centennial. In particular, given Schwinger’s outstanding and seminal contributions to quantum theory 3 and quantum field theory, 4 as well as his pioneering work in quantum theory of gravity, 5 the subject of our contribution seems rather appropriate. Consequently, in this talka we outline the essence of our recent work 6–13 on the foundational aspects of quantum theory, that shed new light on quantum field theory and quantum gravity in the form of string theory. In the first section we describe the hidden quantum spacetime geometry underlying the generic representation of quantum theory (which renders it manifestly non-local) and then in second section we find that the same (and, in general, a Presented

by D. M. at the Schwinger Centennial Conference in Singapore, February 9, 2018.

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dynamical) geometric structure underlies metastring theory, a manifestly T-duality covariant formulation of string theory. Thus quantum gravity “gravitizes” the quantum spacetime geometry. Finally, in the last section we outline an effective description of closed strings at long distance that leads to a non-commutative effective field theory, as implied by intrinsic non-commutativity of closed string theory, and we discuss some of its consequences, such as the new concept of metaparticles. 2. Quantum Theory and Quantum Spacetime We start our discussion by revealing the hidden quantum spacetime geometry of quantization, 10 which will, surprisingly, take us all the way to quantum gravity. We focus on the Heisenberg (or Weyl–Heisenberg) group, which is generated, on the level of the corresponding algebra, by familiar position qˆa and momentum pˆb operators: [ˆ q a , pˆb ] = iδba .

(1)

It will be convenient to introduce a length scale λ and a momentum scale , with ˆ˜a ≡ pˆa /, with λ = . Then, let us introduce the following notation x ˆa ≡ qˆa /λ, x a ˆ a ˜b ] = iδb . Even more compactly let us suggestively write [ˆ x ,x XA ≡ (xa , x ˜ a )T ,

ˆ a, X ˆ b ] = iω AB , [X

(2)

with 12 ωAB dX A dX B = 1 dpa ∧ dq a , where ωAB = −ωBA is the canonical symplectic form on phase space P. The Heisenberg group HP is generated by Weyl operators 14 ˜ k) and ω(K, K ) = k · k˜ − k˜ · k  ) (K stands for the pair (k, WK ≡ e2πiω(K,X) .

(3)

These form a central extension of the translation algebra 

WK WK = e2πiω(K,K ) WK+K .

(4)

The projection π : HP → P (where π : WK → K) defines a line bundle over P (in principle a covariant phase space of quantum probes). In this formulation, states are sections of degree one 

WK Φ(K) = e2πiω(K,K ) Φ(K + K ).

(5)

In this language, geometric quantization means to take a Lagrangian L ∈ P, so that states descend to square integrable functions on L. A Lagrangian submanifold L is a maximally isotropic subspace L with ω|L = 0, and thus {∂/∂q a } ∈ T P defines a Lagrangian submanifold, or “space”. (Indeed, ω(∂/∂q a , ∂/∂q b ) = 0.) This can be understood as a classical characterization of space (and in the covariant context, of spacetime), as a “slice” of phase space. How about a purely quantum characterization of space? We claim that quantum theory reveals a new notion of quantum space (and, more covariantly, a new notion of quantum spacetime).

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Note that for space-like separations the operators of a local quantum field theory commute. Thus in order to understand the meaning of quantum spacetime (quantum Lagrangian), we need to look at a maximally commuting subalgebra of the Heisenberg algebra and the representation that diagonalizes it. Thus, borrowing from notions of non-commutative algebra and non-commutative geometry 15 (such as the theorem of Gelfand–Naimark 16 ), we can say that a Lagrangian submanifold is a maximally commutative subgroup of the Heisenberg group. If we accept this notion of a Lagrangian, then the quantum regime is very different from the classical regime. In particular the vanishing Poisson bracket {f (q), g(p)} requires either f or g to be constant. However, the vanishing commutator [f (ˆ q ), g(ˆ p)] = 0 requires only that the functions be commensurately periodic eiαpˆeiβ qˆ = eiαβ eiβ qˆeiαpˆ,

αβ = 2π/.

(6)

What is interesting here is that similar considerations led Aharonov to introduce modular variables to describe purely quantum phenomena, such as interference 17 (see also the prescient work of Schwinger 3,18 ). 2.1. Modular variables Modular variables are described in great detail in the very insightful book by Aharonov and Rohrlich, 17 where one can find detailed bibliography on this subject.b The fundamental question posed there was as follows: how does one capture interference effects (due to the fundamental linearity of quantum theory) in terms of Heisenberg operators? For example, what are the quantum observables that can measure the relative phase responsible for interference in a double-slit experiment? No polynomial functions of the operators qˆ and pˆ can detect such phases, but opˆ , do. Thus the modular variables erators that translate in space, such as eiRp/ denoted [ˆ q ] and [ˆ p], which are defined modulo a length scale R (the slit spacing being a natural choice), play a central role, where [p]h/R = p mod (h/R) ,

[q]R = q mod (2πR) ,

(7)

ˆ ˆ = eiR[p]/ shifts the position of a partiand h = 2π. The shift operator eiRp/ cle state (say an electron in the double-slit experiment) by a distance R and is a function of the modular momenta. These modular variables (the main examples being the Aharonov–Bohm and Aharonov–Casher phases 17 ) satisfy non-local operˆ = pˆ2 /2m + V (ˆ q ), ator equations of motion. For example, given the Hamiltonian, H the Heisenberg equation of motion for the shift operator is,

q + R) − V (ˆ q) d iRp/ iR V (ˆ ˆ e ˆ =− . (8) e−iRp/ dt  R

Modular variables are fundamentally non-local in a non-classical sense, since we see here that their evolution depends on the value of the potential at distinct locations. b See

also Refs. 3, 14, 18–20.

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Remarkably, thanks to the uncertainty principle, this dynamical non-locality does not lead to a violation of causality. 17 One of the characteristic features of these variables is that they do not have classical analogues; indeed, the limit  → 0 of [p]h/R is ill-defined. Also modular variables capture entanglement of continuous q, p variables. Note that modular variables are, in general, covariant and, also, contextual.c In other words, they carry specific experimental information, such as the length R between the two-slits. In some sense, quantum theory realizes relativity of contextuality. However, in the context of quantum gravity such scales are automatically built in, and the contextuality is in principle fixed by the fundamental scales. The observed contextuality of quantum theory emerges from that underlying realm of quantum gravity. Also, the fundamental dynamical equations for modular variables are non-local in quantum gravity because of the presence of the fundamental length. When exponentiated (i.e. when understood as particular Weyl operators), the i a ˜b ] = 2π δb , the modular variables naturally commute. In other words, given [xa , x 10 following commutator of modular operators vanishes [e2πix , e2πi˜x ] = 0.

(9)

Thus a quantum algebra of modular variables possesses more commutative directions than a classical Poisson algebra, because the Poisson bracket of modular vari 0. ables does not vanish, {e2πix , e2πi˜x } = Here we make a historical note: 21 The above non-local equations of motion were essentially written by Max Born, in the very first paper which used the phrase “Quantum Mechanics” in its title, in 1924, one year before the Heisenberg breakthrough paper. Actually, Heisenberg crucially used Born’s prescription of replacing classical equations by the corresponding difference equations, in order to derive what we now call the canonical commutation relations (properly written by Born and Jordan) from the Bohr–Sommerfeld quantization conditions. 2.2. Modular space and geometry of quantum theory Returning to the subject of quantum Lagrangians, note that the quantum Lagrangian is analogous to a Brillouin cell in condensed matter physics. The volume and shape of the cell are given by λ and  (i.e.  and GN (α ≡ λ )). The uncertainty principle is implemented in a subtle way: we can specify a point in modular cell, but if so, we cannot say which cell we are in. This means that there is a more general notion of quantization, 10 beyond that of geometric quantization. Instead of selecting a classical polarization L (the arguments of the wave function, or the arguments of a local quantum field) we can choose c Aharonov

and collaborators have pushed the logic associated with modular variables to argue for weak measurements of such non-local variables that capture the superposition principle of quantum theory. Similarly, Aharonov and collaborators argue for a time symmetric formulation of quantum theory. 17

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a modular polarization. In terms of the Heisenberg group all that is happening is that in order to have a commutative algebra, we need only ω(K, K ) ∈ 2Z,



WK WK = e2πiω(K,K ) WK+K = WK WK .

(10)

This defines a lattice Λ in phase space P. Finally, we specify a “lift” of the lattice from the phase space P to the Heisenberg group HP . ˆ of the Heisenberg group correspond to latMaximally commuting subgroups Λ tices that are integral and self-dual with respect to ω. 22,23 Given Wλ where λ ∈ Λ ˆ which defines “modular polarization” there is a lift to Λ Uλ = α(λ)Wλ ,

(11)

where α(λ) satisfies the co-cycle condition α(λ)α(μ)eπiω(λ,μ) = α(λ + μ),

λ, α ∈ Λ.

(12)

One can parametrize a solution to the co-cycle condition by introducing a symmetric bilinear from η and setting (with η(K, K ) = k · k˜ + k˜ · k  ,) π

αη (λ) ≡ ei 2 η(λ,λ) .

(13)

Finally, when we choose a classical Lagrangian L, there is a special state that we associate with the vacuum: it is translation invariant (which in our context can be interpreted as “empty space”). In modular quantization there is no such translation invariant state (because of the lattice structure). The best we can do is to choose a state that minimizes an “energy”, which requires the introduction of another symmetric bilinear form, that we call, again suggestively, H. This means, first, that we are looking for operators such that ˆ A , Φ] = [P

i ∂A Φ, 2π

ˆ + λ) = Φ(X), ˆ Φ(X

(14)

ˆ + λ) = Φ(X) ˆ are generated by the lattice obwhere the modular observables Φ(X ˆ = 0. servables Uλ with λ ∈ Λ. Translation invariance would be the condition P|0 Since this is not possible, the next natural choice is to minimize the translational energy. Therefore we pick a positive definite metric HAB on P, and we define 10 ˆAP ˆB , ˆH ≡ H AB P E

(15)

ˆH . This is indeed the most natural and demand that |0 H be the ground state of E choice and it shows that we cannot fully disentangle the kinematics (i.e., the definition of translation generators) from the dynamics. In the Schr¨odinger case, since ˆ the translation generators commute, the vacuum state E|0 = 0 is also the translation invariant state and it carries no memory of the metric H needed to define the energy. In our context, due to the non-commutativity of translations, the operators ˆH  do not commute. As a result the vacuum state depends on H, in other ˆH and E E words |0 H = |0 H  , and it also possesses a non-vanishing zero point energy.

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2.3. Born geometry Thus, modular quantization involves the introduction of three quadratic forms (ω, η, H), i.e. what we call Born geometry, 6,7 which underlies the geometry of modular variables. As we will see, in the context of metastring theory, a choice of polarization is a choice of a spacetime within P but the most general choice is a modular polarization that we have discussed above. From the foundational quantum viewpoint Born geometry (ω, η, H) arises as a parametrization of such quantizations, which results in a notion of quantum spacetime, that we call modular spacetime. Finally, large spacetimes of canonical string theory and general relativity result as a “many-body” phenomenon, through a process of tensoring of unit modular cells, that we refer to as “extensification”. 10 In particular, the symplectic structure ω (ds2ω = 12 ωAB dXA dXB = 1 dpa ∧ dq a ), is encoded in the canonical Heisenberg commutator between q a and pa . The generalized, quantum, metric H comes from the Born rule in quantum theory a + GN dpa dpa ). For weak gravity, this metric reds2H = HAB dXA dXB = 1 ( dqGa dq N duces to the spacetime metric (where spacetime can be viewed as a slice of phase space). Due to gravity’s extreme weakness, we only see spacetime metric at low energies. (The ratio /λ defines a tension; if this is identified with c3 /GN , it is enormous, ∼ 1032 kg/sec.) Therefore, in this formulation the usual dynamical spacetime metric is the low energy leftover of the quantum metric. Finally, the polarization (or locality metric) η encodes the distinction between spacetime-like and energy-momentum-like aspects of phase space (and in this sense it defines an analog of the “causal” structure in phase space) ds2η = ηAB dXA dXB = 2 dpa dq a . This new metric captures the essence of relative locality — when η is constant we have absolute locality. Curving η also means “gravitizing the quantum”. In general all three elements of Born geometry, ω, η and H are dynamical and curved in metastring theory, as we will discuss in what follows. Also, we have that the Lorentz group (in D spacetime dimensions) lies at the intersection of the symplectic (ω), neutral (η) and doubly orthogonal groups (H), 10 O(1, d − 1) = Sp(2d) ∩ O(d, d) ∩ O(2, 2(d − 1)),

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which sheds new light on the origin of quantum theory through compatibility of the causal (Lorentz) structure and the non-locality captured by the discreteness of the quantum spacetime. This also captures the role of relative (observer-dependent) locality 24 needed to resolve the apparent contradiction between discreteness of quantum spacetime and Lorentz symmetry. Let us end this section with a few comments regarding the Stone–von Neumann theorem 25–27 which asserts that all representations of the Heisenberg group are unitarily equivalent. Normally, we think of this as a choice of basis in phase space (a choice of polarization or classical Lagrangian), and all such choices are related by Fourier transform. Similarly, one can pass from a classical polarization (such as the Schr¨odinger representation) to a modular polarization via the Zak transform. 28

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Note that, there is a connection on the line bundle over phase space that has unit flux through a modular cell. (This is very similar to integer quantum Hall effect systems.) A modular wave function is quasi-periodic ˜), Ψ(x + a, x ˜) = e2iπa˜x Ψ(x, x

Ψ(x, x ˜+a ˜) = Ψ(x, x ˜).

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The quasi-periods correspond to the tails of an Aharonov–Bohm 29 potential attached to a unit flux. In particular, vacuum states must have at least one zero in a cell, which leads to theta functions (the Zak transforms of Gaussians). Note that from the point of modular polarization, the familiar Schr¨odinger polarization is just a singular limit. 3. From Modular Spacetime to Quantum Gravity The unexpected outcome of our research is that this fundamental quantum geometry of quantum theory can be realized in the context of metastring theory, where this quantum, Born geometry, is “gravitized” (i.e. made dynamical). At the classical level, metastring theory 6–13 can be thought of as a formulation of string theory in which the target space is doubled in such a way that T-duality acts linearly on the coordinates. This doubling means that momentum and winding modes appear on an equal footing. We refer to the target space as a phase space since the metastring action requires the presence of a background symplectic form ω. The metastring formulation also requires the presence of geometrical structures that generalize to phase space the spacetime metric and the B-field (where the B-field originates from the symplectic structure ω). In fact, in the metastring we have not one but two notions of a metric. The first metric η is a neutral metric that defines a bi-Lagrangian structure and allows to define the classical spacetime as a Lagrangian sub-manifoldd — more precisely, the classical spacetime is defined as a null subspace for η which is also Lagrangian for ω. The second metric H is a metric of signature (2, 2(D − 1)) that encodes the geometry along the classical spacetime (of dimension D) as well as the transverse energy-momentum space geometry. In this formulation, T-duality exchanges the Lagrangian sub-manifold with its image under J = η −1 H. Classical metastring theory is defined by the following action 8 (see also the pioneering papers 30–32 )   1 ˆ (18) S= d2 σ ∂τ XA (ηAB + ωAB )(X)∂σ XB − ∂σ XA HAB (X)∂σ XB , 4π Σ where XA are dimensionless coordinates on “phase space” and the fields η, H, ω are all dynamical (i.e., in general dependent on X) backgrounds. Thus, in general, Born d We remind the reader that in symplectic geometry, a Lagrangian subspace is a half-dimensional submanifold of phase space upon which the symplectic form pulls back to zero. In simple terms, a Lagrangian submanifold might be the subspace coordinatized by the q’s within the phase space coordinatized by q’s and p’s.

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geometry becomes dynamical in string theory, viewed as a theory of quantum gravity. The requirement of conformal invariance determines the equations of motion for the elements of Born geometry: ω(X), η(X) and H(X). In the context of a flat metastring we have constant ηAB , HAB and ωAB

0 δ h 0 0 δ ηAB ≡ ≡ = , ω , H , (19) AB AB δT 0 0 h−1 −δ T 0 where δνμ is the d-dimensional identity matrix and hμν is the d-dimensional Lorentzian metric, T denoting transpose. In this formulation 8 it is convenient, as suggested by the double field formalism, 33,34 to introduce dimensionless coordinates XA ≡ (X μ /λ, Pμ /ε)T on phase ˜a )T , where λ and  represent the fundamental space, or equivalently, XA ≡ (xa , x spacetime and energy–momentum scales. As already stated,  = λ and α = λ . Given a pair (H, η) it is natural to consider the operator J ≡ η −1 H. The consistency of string theory requires J to be a chiral structure, that is, a real structure (J 2 = 1) compatible with η, implying that J is an O(D, D) transformation (realizing generalized T-duality in target space). These three structures, the symplectic Sp(2D) ω, the O(D, D) η and the SO(2, 2(D − 1)) H, define the new concept of Born geometry 6–10 (see also Refs. 35 and 36) which unifies the complex geometry of quantum theory with the metrical geometry of general relativity and the symplectic geometry of canonical Hamiltonian dynamics. 37–41 Note that in the phase space formulation the local phase space coordinates X are quasiperiodic XA (σ + 2π) = XA (σ) + ΔA , where ΔA is the corresponding quasiperiod (which either vanishes for the canonical Polyakov string or is given by the winding number in the usual treatment of T-duality on compact spaces). The worldsheet formulation of the metastring is chiral. Thus, even though the fields are doubled the central charges (left and right) are cL = cR = D and we still have D = 26 for criticality. The metastring is not manifestly invariant under the worldsheet Lorentz transformations and it contains monodromies XA (σ + 2π) = XA (σ)+ΔA . The usual Polyakov string can be obtained by integrating out the dual ˜ for constant η and H backgrounds, and by supposing that the monodromies are X, in the kernel of (η − ω). T-duality is implemented in target space by the action of the chiral J operator (J ≡ η −1 H, J 2 = 1): X → J(X). The target space of the metastring is not spacetime, but, to first order, a chiral phase space P equipped by the symplectic structure ω, and the bilagrangian structure, and in particular, the polarization metric η which relates to the symplectic connection of the Fedosov deformation quantization 42 and thus leads to the star product of deformation quantization, and finally, the quantum H metric which relates to the complex structure in the context of geometric quantization, 43 leading to the concept of Hilbert spaces. This classical Born geometry implements the ideas of Born duality in string theory. 6,7 The classical equations of motion of the metastring ∂τ XA −(J∂σ X)A = 0, implies the relation between momenta and monodromies 2πP = J(Δ). There is soldering

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between worldsheet null coordinates σ ± ≡ σ±τ and the chiral target space structure ∂± XA − (P± X)A = 0, where the chiral projector is defined as 2P± = (1 ± J). This allows us to liberate the left geometry from the right geometry (which is reminiscent of twistor theory). The careful analysis of the metastring action 8 shows that its  1 δXA ηAB ∇σ δXB , where ∇ is the generalized Fedosov symplectic form is Ω = 4π connection found in the Fedosov deformation quantization approach. 42 Also, the operator product expansion of the metastring vertex operators Vk = k eiKX , (i.e. modular variables) lead to the restriction of K on a double Lorentzian integral lattice Γ, that by modular invariance, must be self-dual. These exist in D = 2 mod(8), and are unique. Criticality gives a very unique lattice Γ = Π1,25 × Π1,25 . This fact, in turn, leads to a large symmetry structure (of the fully “compactified” bosonic string) found by Borcherds in the study of the monstrous moonshine. 44,45,e As already noted, the metastring is chiral. This requires the introduction of a preferred worldsheet time coordinate which is fundamentally Lorentzian. 8 How can this be consistent with modular invariance? The answer is given by employing the Giddings–Wolpert–Krichever–Novikov construction: 47,48 given a Riemann surface, provided a choice of local coordinates around punctures is labeled by one scalar, there exists a unique Abelian differential e with imaginary periods. The real part of this Abelian differential is the modular invariant time τ = Re(e). The zeros of e represent interaction points where the worldsheet Lorentzian cones double. Cutting the Riemann surface along the real trajectory of e we obtain a string decomposition of the surface. The Nakamura graphs 49 encode this decomposition and give a very effective cell decomposition of moduli space. Thus Nakamura graphs are the natural Feynman diagrams for closed strings. 50 Finally, the metastring formulation points to an unexpected fundamental noncommutativity of closed string theory, that we address in what follows. 3.1. Intrinsic non-commutativity in metastring theory It is well established that the structure of the zero mode algebra of the compactified closed string depends on a lattice of momenta (Λ, 2η) which is integral and self-dual with respect to a neutral metric: a so-called Narain lattice. 51 In our recent work 6–13 we have refined this structure and we have shown that in fact the kinematical structure of the string zero modes depends on a para-hermitian lattice: a triple (Λ, η, ω), where Λ is a subgroup of R2d that describes the lattice of wave-covectors λK, with λ the string length, η is a neutral metric, a symmetric bilinear form of signature (d, d), and ω is an invertible two-form. This structure needs to satisfy two compatibility conditions: first, the lattice Λ is assumed to be integral with respect to the para-hermitian structure, i.e., (η±ω)(λK, λK ) ∈ Z, for λK, λK ∈ Λ. Second, the metric η and the 2-form ω must be compatible, in the sense that η −1 ω := K is a product structure, that satisfies the condition K 2 = 1. e See also Ref. 46. Modular spacetime interpretation of metastring theory 8 sheds new light on this work.

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These two conditions are a consequence of mutual locality on the worldsheet (i.e. worldsheet causality). It is clear that if (Λ, η, ω) is a para-Hermitian lattice, then (Λ, 2η) is a Narain lattice, so the kinematical structure that we highlight is a refinement of the usual one. The extra information is contained in the 2-form ω. This form does not enter expressions for the spectrum or the partition function and this why it is usually ignored. It does enter however crucially in the definition of the vertex operator algebra and parameterizes what is usually referred to as a cocycle. The role of ω is to promote the zero mode double space P  R2d dual to R[Λ] to the status of phase space: P should be viewed as a symplectic manifold. At the quantum level, both geometrical structures η and ω enter in the commutation relations of string operators. ω controls the non-commutativity of the zero-modes while η controls the non-commutativity of the string oscillator modes. This can be seen if one introduces a double notation for the string coordinate X(σ) ˜ The string commutation relations, that includes the string map X and its dual X. 12,13 were derived in   [XA (σ), XB (σ  )] = 2iλ2 πω AB − η AB θ(σ − σ  ) , (20) where θ(σ) is the staircase distribution, i.e., a solution of θ (σ) = 2πδ(σ); it is odd and quasi-periodic with period 2π. Following standard practice, all indices are raised and lowered using η and η −1 . 1 B The momentum density operator is given by PA (σ) = 2πα  ηAB ∂σ X (σ) and the A previous commutation relation implies that it is conjugate to X (σ). The twoform ω appears when one integrates this canonical commutation relation to include the zero-modes, the integration constant being uniquely determined by worldsheet ˆ P) ˆ the zero mode components of the string operators causality. Denoting by (X, X(σ) and P(σ) we simply have that ˆ B ] = 0, [X ˆ A, P ˆ B ] = iδ A B , [X ˆ A, X ˆ B ] = 2πiλ2 ω AB . ˆA, P [P

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This is a deformation of the doubled Heisenberg algebra involving the string length λ as a deformation parameter. So far we have assumed that the background is trivial, with the fields (η, ω) constant and given by η(K, K ) = k · k˜ + k˜ · k  , and ω(K, K ) = k · k˜ − k˜ · k  . As shown in Ref. 12, we can turn on non-trivial backgrounds encoded into ω by changing the O(d, d) frame X → OX. This change of frame preserves η but transforms ω. Any constant ω can be obtained this way. Since ω has an interpretation as the symplectic form on the space of X’s, modifying ω affects the commutation relationsf ˆ A, X ˆ B ] = 2πiλ2 ΠAB , with ΠAB ωBC = δ A C , where we have introduced the Poisson [X tensor Π = ω −1 . f The algebraic structure that we are working with here has an analogy in electromagnetism in the presence of monopoles. In that analogy, the string length becomes the magnetic length, and the form ω becomes the magnetic field. Another analogy occurs in quantum Hall liquids, the algebra being the magnetic algebra of the lowest Landau level.

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For instance, under a constant B-field transformation X = (xa , x ˜a ) → (xa , x ˜a + b   ˜  ˜ Bab x ), the trivial symplectic form ω(K, K ) = k·k −k·k is mapped onto ω(K, K ) = ka k˜a − ka k˜a − 2Bab k˜a k˜b , and the commutators readg [ˆ xa , x ˆb ] = 0,

ˆ [ˆ xa , x ˜b ] = 2πiλ2 δ a b ,

ˆ˜a , x ˆ˜b ] = −4πiλ2 Bab . [x

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We see that the effect of the B-field is to render the dual coordinates noncommutative (and that the B-field originates from the symplectic structure ω). More generally, we can parameterize an O(d,d) transformation as g =   arbitrary ˆ ˆ ˆ βˆ ˆ ˆ , where Aˆ ∈ GL(d) and eB = B1 10 and eβ = 10 β1 are nilpotent. eB is eB Ae the B-field transformation discussed above, and is associated with the usual B˜a ) field deformation in string theory. We note that the transformation of (xa , x given above does not modify xa , and thus fields that depend only on xa are unmodified. The β-transformation on the other hand corresponds to the map ˜a ) → (xa + β ab x ˜b , x ˜a ). Equivalently, it has the effect of mapping the sym(xa , x plectic structure to ω(K, K ) = ka k˜a − ka k˜a + 2β ab ka kb , and yields commutation relations ˆb ] = 4πiλ2 β ab , [ˆ xa , x

ˆ˜b ] = 2πiλ2 δ a b , [ˆ xa , x

ˆ˜a , x ˆ˜b ] = 0. [x

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Dramatically, the coordinates that are usually thought of as the spacetime coordinates have become themselves non-commutative. Since this is the result of an O(d, d) transformation, we know that it can be thought of in similar terms as the B-field; these are related by T-duality. We are familiar with the B-field background because we have, in the non-compact case, a fixed notion of locality in the target space theory. However, in the non-geometric β-field background, we do not have such a notion of locality but we can access it through T-duality. We note that this intrinsic non-commutativity of string theory can be also explicitly illustrated via a simple closed string product, equivalent to the splitting-joining interaction of the pants diagram, that respects this non-commutativity and is covariant with respect to T-duality. 13 This offers new insights on the relationship between closed and open strings, and the non-perturbative formulation of closed string theory in terms of open strings and even more fundamental (and non-commutative) partonic degrees of freedom. Given the mechanism of tachyon condensation in the open string sector, 53 and this fundamental relation between open and closed strings, we expect that a similar solution of the “tachyon problem” should exist in the closed string sector as well. 4. Effective Quantum Fields and Manifest Non-locality What is the effective description of closed strings that incorporates the above intrinsic non-commutativity? For a closed sting on a circle of radius R (where the g See

also Ref. 52.

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˜ is defined as RR ˜ = 2λ2 and the respective winding integers are n dual radius R, and w) this effective description is captured by the generalized field 12,13  ˜ Φ(x, x ˜) ≡ Φw (x)eiw˜x/R . (24) w

This meshes well with the observation 12,13 that the string product is essentially a representation of the Heisenberg group, which suggests that one should consider the “quantization” map  ˆ ˜ ˆ= Φ(x, x ˜) → Φ Φw (ˆ x)eiwx˜/R , (25) w

from generalized fields to non-commutative fields.h Under this map the T-duality ˆ˜. transformation becomes “localized” and is expressed as the exchange of x ˆ with x 12,13 The T-dual expression is given by   ˆ= ˜ = ˆ˜ − πnR) ˆ˜)einˆx/R , Φ einˆx/R Φn (x Φn ( x (26) n

n

ˆ˜ which has a similar form to (25). We see that the non-commutativity of x ˆ with x allows one to reabsorb all the shifts in terms of a simple reordering that exchanges x ˆ ˆ with x ˜ and is the expression of T-duality. The “quantized” field is simply expanded in terms of modes as  ˆ ˜ ˆ≡ einˆx/R Φ(n, w)eiwx˜/R . (27) Φ w,n

It is useful at this point to generalize the construction to higher dimensional tori. This can be done in a straightforward manner by introducing the modes ˜ n/R). The integrality condition for the lattice KA = (k˜a , ka ), generalizing (w/R,  Λ of admissible modes K, K ∈ Λ reads in this notation asi (η ± ω)(λK, λK ) ∈ Z. ˆ−K , where We now write Φ(K) = K|Φ with the ordering chosen as K| = 0|U ˜x ˆ ik·ˆ x ik· ˜ ˆ UK ≡ e e . This ordering can be seen to be related to the choice of an O(d, d) frame, where we place the operator associated with x on the left and the operator associated with the dual space x ˜ on the right. The key point is that this choice of frame is entirely encoded into the choice of symplectic potential ω and the vertex ˜ k) and X = (x, x operator can be covariantly written in terms of K = (k, ˜) as ˆ i (η−ω)(K,X) ˆ ˆK = e 2i (η+ω)(K,X) e2 . U

Given this notation we can write the string product covariantly as 12,13    (Φ ◦ Ψ)(K) = Φ(K )eiπ(η−ω)(λK ,λK ) Ψ(K ).

(28)

(29)

K +K =K

h Here, we have chosen a specific operator ordering. Given this ordering, the mapping is welldefined and consistent with the string product. i In the one-dimensional case where K = (w/R, ˜ n/R) this follows directly from (η + ω)(λK, λK ) = nw and similarly (η − ω)(λK, λK ) = wn , given that n, n , w, w ∈ Z.

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The non-commutativity of the string product is encoded in terms of a π-flux due to ω. As it turns out the phase factor is exactly the same as the cocycle factor  (K, K ) = eiπ(η−ω)(λK,λK ) that appears in the definition of the vertex operator product. 12,13 4.1. Non-local excitations: Metaparticles To summarize, the above manifestly T-duality covariant formulation of closed strings (i.e. the metastring) implies intrinsic non-commutativity of zero-modes. It is thus instructive to formulate a particle-like limit of the metastring that we call the metaparticle. Given the form for the symplectic structure of the zero modes derived in Sec. 4 of Ref. 12 (Eq. (67) of that  paper, without the contribution coming from string oscillators), the action S ≡ dτ L of the metaparticle is governed by the following Lagrangian L    e pμ pμ + p˜μ p˜μ − m2 + e˜ pμ p˜μ − μ2 , (30) ˜˙ μ − πα pμ p˜˙ μ − L = pμ x˙ μ + p˜μ x 2 where e and e˜ are the Lagrange multipliers for the two constraints that follow from the Hamiltonian (H ≡ ∂σ XA HAB ∂σ XB = 0) and diffeomorphism constraints (D ≡ ∂σ XA ηAB ∂σ XB = 0) of the metastring. 8,11 Note that the usual particle limit is obtained, at least classically, by taking μ → 0 and p˜ → 0. The theory of metaparticles can be viewed as the theory of the zero modes of the closed string, which fully takes into account its intrinsic non-commutativity. Given the form of the above Lagrangian, the metaparticle looks like two particles that are entangled through a Berry phase-like pμ p˜˙ μ factor. The metaparticle is fundamentally non-local, and thus it should not be associated with effective local field theory. In particular, by looking at the metaparticle constraints p2 + p˜2 = m2 and p˜ p = μ2 , we note that the momenta p and p˜ can be, in principle, widely separated. For example, if m is of the order of the Planck energy, and μ of the order of one T eV (which could be understood as a characteristic particle physics scale), then the momentum p can be of the order of the Planck energy, and the momentum p˜ of the vacuum energy scale! Thus the metaparticle theory is able to naturally relate widely separated scales, which transcends the usual reasoning based on Wilsonian effective field theory (and should be relevant for the naturalness and hierarchy problems). We expect that the correct field theoretic description of the metaparticle is in terms of the above non-commutative (modular) field theory Φ(x, x ˜) limit of the metastring. Like their particle cousins, metaparticles should be detectable, and they might even present good candidates for dark matter quanta. 54–59 Such effective non-commutative field theory (similar in spirit to Refs. 60–62) can be also useful in illuminating the vacuum energy problem, via the sequester mechanism 63 in which the dimensional parameters in the effective action of the matter sector depend on the four-volume element of the universe. In some sense, the mataparticle represents a “point-like” implementation of the concept of Born geometry that underlies the metastring. It can be also viewed

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as a quantum of an intrinsically non-commutative quantum field Φ(x, x ˜), that is consistent with Lorentz covariance as well as with the existence of the fundamental length scale. The reconciliation of these two apparently incompatible concepts comes about through observer dependent (or relative) locality, which is the physical principle realized in metastring theory. Also such non-commutative bi-local quantum fields, Φ(x, x ˜), 12,13 represent the correct effective description of T-duality covariant string theory. The metaparticle concept should be of interest not only to string theorists and physicists working in quantum gravity and relative locality, but also particle physicists searching for new fundamental excitations, and (in the non-relativistic limit) to condensed matter physicists interested in going beyond the familiar concept of quasi-particles. Finally, we note that the concept of metaparticles might be argued from the compatibility of the quantum spacetime that underlies the generic representations of quantum theory, as discussed in Ref. 10, and thus the metaparticle might be as ubiquitous as the concept of antiparticles which is demanded by the compatibility of relativity and quantum theory. The details of the theory of metaparticles will be published elsewhere. 64

5. Conclusions In this talk we have discussed intrinsic non-commutativity in quantum gravity 12,13 related to a new concept of quantum spacetime, called modular spacetime 9,10 that also appears as a habitat for metastring theory 8,11 and that is deeply rooted in the foundations of quantum theory (and, especially, in the concept of modular variables that goes back to the work of Weyl, Schwinger and Aharonov). Note that this concept stems from a quantization of spacetime, and not from quantization of gravitational field/metric. Even the flat space is quantized according to our approach to quantum gravity. This allows for superposition and entanglement of spacetimes. Also, this formulation provides for an explicit construction of spacetime quanta or qubits (the fully “compactified” bosonic string 8 ), and a new non-perturbative definition of quantum gravity as “gravitization of the quantum”. 6,7 Such a fully dynamical (or “curved”) construction of Born geometry can be approached from the point of view of “teleparallel gravity” in which one utilizes the flat (zero-curvature) connection and crucially introduces non-zero torsion. 65 This viewpoint is natural for the rigid structure of Born geometry and it allows for “curving” of T-duality. In some sense, by going from our new formulation of quantum mechanics in terms of modular, or quantum spacetime, with hidden but fixed Born geometry, and its application to quantum field theory, to an explicit formulation of quantum gravity that involves dynamical Born geometry, as is the case in metastring theory, we are retracing (in a purely quantum context) the line of development that led from special relativity (and fixed Minkowski geometry) and its application to classical relativistic field theory, to general theory of relativity with a dynamical spacetime geometry. We can only hope that Julian Schwinger would approve of these developments.

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Acknowledgments RGL and DM thank Perimeter Institute for hospitality. LF, RGL and DM thank the Banff Center for providing an inspiring environment for work and the Julian Schwinger Foundation for support. DM also thanks Jerzy Kowalski-Glikman and the quantum gravity group of the Institute for Theoretical Physics at the University of Wroclaw for hospitality. RGL is supported in part by the U.S. Department of Energy contract DE-SC0015655 and DM by the U.S. Department of Energy under contract DE-FG02-13ER41917. Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. DM thanks the organizers of the Schwinger Centennial Conference in Singapore, and especially Berge Englert and the Julian Schwinger Foundation, for the kind invitation and hospitality.

References 1. J. Mehra and K. A. Milton, Climbing the Mountain: The Scientific Biography of Julian Schwinger (Oxford University Press, 2003). 2. M. Flato, C. Fronsdal and K. A. Milton (eds.), Selected Papers (1937–1976) of Julian Schwinger (Springer, 1979). 3. J. Schwinger, Quantum Kinematics and Dynamics (Westview Press, 2000). 4. J. Schwinger, Particles, Sources and Fields, Volumes 1, 2 and 3 (Westview Press, 1998). 5. J. Schwinger, Quantized gravitational field, Phys. Rev. 130, 1253 (1963); Quantized gravitational field. II, Phys. Rev. 132, 1317 (1963). 6. L. Freidel, R. G. Leigh and D. Minic, Phys. Lett. B 730, 302 (2014), arXiv:1307.7080 [hep-th]. 7. L. Freidel, R. G. Leigh and D. Minic, Int. J. Mod. Phys. D 23, 1442006 (2014), arXiv:1405.3949 [hep-th]. 8. L. Freidel, R. G. Leigh and D. Minic, JHEP 1506, 006 (2015), arXiv:1502.08005 [hep-th]. 9. L. Freidel, R. G. Leigh and D. Minic, Int. J. Mod. Phys. D 24, 1544028 (2015). 10. L. Freidel, R. G. Leigh and D. Minic, Phys. Rev. D 94, 104052 (2016), arXiv:1606.01829 [hep-th]. 11. L. Freidel, R. G. Leigh and D. Minic, J. Phys. Conf. Ser. 804, 012032 (2017), doi:10.1088/1742-6596/804/1/012032. 12. L. Freidel, R. G. Leigh and D. Minic, JHEP 1709, 060 (2017), arXiv:1706.03305 [hep-th]. 13. L. Freidel, R. G. Leigh and D. Minic, Phys. Rev. D 96, 066003 (2017), arXiv:1707.00312 [hep-th]. 14. H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, 1931). 15. A. Connes, Noncommutative Geometry (Academic Press, 1994). 16. I. M. Gelfand and M. A. Naimark, Mat. Sbornik 12, 197 (1943). 17. Y. Aharonov and D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed (Wiley, 2005). 18. J. Schwinger, PNAS 46, 570 (1960). 19. Y. Aharonov, A. Petersen and H. Pendleton, Int. J. Theor. Phys. 2, 213 (1969).

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20. G. ’t Hooft, The cellular automaton interpretation of quantum mechanics. A view on the quantum nature of our universe, compulsory or impossible?, arXiv:1405.1548 [quant-ph]. 21. B. L. van der Waerden, Sources of Quantum Mechanics (Dover, 1968). 22. G. Mackey, Acta Math. 99, 265 (1958). 23. G. Mackey, Annals Math. 55, 101 (1952). 24. G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, Phys. Rev. D 84, 084010 (2011), arXiv:1101.0931 [hep-th]. 25. M. Stone, Proc. Nat. Acad. Sci. 16, 172 (1930). 26. J. von Neuman, Math. Ann. 104, 570 (1931). 27. G. Mackey, Proc. Nat. Acad. Sci. 35, 537 (1949). 28. J. Zak, Phys. Rev. Lett. 19, 1385 (1967). 29. Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). 30. A. A. Tseytlin, Nucl. Phys. B 350, 395 (1991). 31. A. A. Tseytlin, Phys. Lett. B 242, 163 (1990). 32. R. Floreanini and R. Jackiw, Phys. Rev. Lett. 59, 1873 (1987). 33. W. Siegel, Phys. Rev. D 47, 5453 (1993). 34. C. Hull and B. Zwiebach, JHEP 0909, 099 (2009). 35. L. Freidel, F. J. Rudolph and D. Svoboda, JHEP 1711, 175 (2017), arXiv:1706.07089 [hep-th]. 36. L. Freidel, F. J. Rudolph and D. Svoboda, A unique connection for Born geometry, arXiv:1806.05992 [hep-th]. 37. G. W. Gibbons, J. Geom. Phys. 8, 147 (1992). 38. A. Ashtekar and T. Schilling, arXiv:gr-qc/9706069. 39. D. Minic and H. C. Tze, Phys. Rev. D 68, 061501 (2003). 40. D. Minic and H. C. Tze, Phys. Lett. B 581, 111 (2004). 41. V. Jejjala, M. Kavic and D. Minic, Int. J. Mod. Phys. A 22, 3317 (2007). 42. B. Fedosov, Deformation Quantization and Index Theory (Akademie-Verlag, 1996). 43. N. Woodhouse, Geometric Quantization (Clarendon Press, 1991). 44. R. E. Borcherds, Proc. Nat. Acad. Sci. 83, 3068 (1986). 45. R. E. Borcherds, Inventiones Mathematicae 109, 405 (1992). 46. G. W. Moore, Finite in all directions, arXiv:hep-th/9305139. 47. S. B. Giddings and S. A. Wolpert, Commun. Math. Phys. 109, 177 (1987). 48. I. M. Krichever and S. P. Novikov, Funct. Anal. Appl. 21, 294 (1987). 49. S. Nakamura, Tokyo J. Math. 23, 87 (2000). 50. L. Freidel, D. Garner and S. Ramgoolam, Phys. Rev. D 91, 126001 (2015). 51. K. S. Narain, Phys. Lett. 169B, 41 (1986), doi:10.1016/0370-2693(86)90682-9. 52. D. Polyakov, P. Wang, H. Wu and H. Yang, JHEP 1503, 011 (2015), arXiv:1501.01550 [hep-th]. 53. A. Sen, JHEP 9808, 012 (1998), arXiv:hep-th/9805170. 54. C. M. Ho, D. Minic and Y. J. Ng, Phys. Lett. B 693, 567 (2010), arXiv:1005.3537 [hep-th]. 55. C. M. Ho, D. Minic and Y. J. Ng, Gen. Rel. Grav. 43, 2567 (2011), Int. J. Mod. Phys. D 20, 2887 (2011), arXiv:1105.2916 [gr-qc]. 56. C. M. Ho, D. Minic and Y. J. Ng, Phys. Rev. D 85, 104033 (2012), arXiv:1201.2365 [hep-th]. 57. D. Edmonds, D. Farrah, C. M. Ho, D. Minic, Y. J. Ng and T. Takeuchi, Astrophys. J. 793, 41 (2014), arXiv:1308.3252 [astro-ph.CO]. 58. D. Edmonds, D. Farrah, C. M. Ho, D. Minic, Y. J. Ng and T. Takeuchi, Int. J. Mod. Phys. A 32, 1750108 (2017), arXiv:1601.00662 [astro-ph.CO].

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59. D. Edmonds, D. Farrah, D. Minic, Y. J. Ng and T. Takeuchi, Int. J. Mod. Phys. D 27, 1830001 (2017), arXiv:1709.04388 [astro-ph.CO]. 60. M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001), arXiv:hepth/0106048. 61. R. J. Szabo, Phys. Rept. 378, 207 (2003), arXiv:hep-th/0109162. 62. H. Grosse and R. Wulkenhaar, Commun. Math. Phys. 256, 305 (2005), arXiv:hepth/0401128. 63. N. Kaloper and A. Padilla, Phys. Rev. Lett. 112, 091304 (2014), arXiv:1309.6562 [hep-th]. 64. L. Freidel, J. Kowalski-Glikman, R. G. Leigh and D. Minic, The Theory of Metaparticles, to appear. 65. A. Einstein, Riemann-Geometrie mit Aufrechterhaltung des Begriffes des Fernparallelismus, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Phys.math. Klasse, pp. 217–221 (1928).

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Unruh acceleration radiation revisited J. S. Ben-Benjamin,1 M. O. Scully,1,2 S. A. Fulling, D. M. Lee, D. N. Page,3 A. A. Svidzinsky1 and M. S. Zubairy Institute for Quantum Science and Engineering, Texas A&M University, College Station, TX 77843, USA M. J. Duff,4,5 R. Glauber,6 W. P. Schleich2,7 and W. G. Unruh8 Hagler Institute for Advanced Studies, Texas A&M University, College Station, TX 77843, USA 1 2 3 4 5 6 7 8

Baylor University, Waco, TX 76706, USA Princeton University, Princeton, NJ 08544, USA University of Alberta, Edmonton, T6G 2R3, Canada Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom Mathematical Institute, Andrew Wiles Building, University of Oxford, Oxford OX2 6GG, United Kingdom Harvard University, Cambridge, MA 02138, USA Universit¨ at Ulm, D-89069 Ulm, Germany University of British Columbia, Vancouver, V6T 2A6, Canada When ground-state atoms are accelerated and the field with which they interact is in its normal vacuum state, the atoms detect Unruh radiation. We show that atoms falling into a black hole emit acceleration radiation which, under appropriate initial conditions (Boulware vacuum), has an energy spectrum which looks much like Hawking radiation. This analysis also provides insight into the Einstein principle of equivalence between acceleration and gravity. The Unruh temperature can also be obtained by using the Kubo–Martin–Schwinger (KMS) periodicity of the two-point thermal correlation function, for a system undergoing uniform acceleration; as with much of the material in this paper, this known result is obtained with a twist.

Ia. Introduction: Dedication Julian Schwinger, that towering figure of 20th century physics, taught us how to tame the infinities of quantum field theory and much more. For example, he and his students taught us how to profitably apply the formalism of quantum field theory to the problem of nonequilibrium quantum statistical mechanics; 1,2 yielding, among other things, the famous KMS condition, which we use herein. Indeed, modern quantum optics owes much to Schwinger’s Green’s function-correlation function approach. In particular, we have found that the tools of quantum optics provide another window into the problem of Unruh–Hawking radiation. It is therefore fitting that we summarize and extend our work on acceleration radiation in this Schwinger centennial collection.

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Fig. 1.

Julian Schwinger rides a hay wagon at the New Mexico Scully ranch in 1987.

Ib. Introduction: Overview The existence of black holes (BHs), regions of spacetime that nothing — not even light — can escape from, is one of predictions of Einstein’s general relativity. Hawking’s 3 demonstration that a non-rotating, uncharged BH of mass M emits thermal radiation at temperature 4 TBH =

c3 8πGM kB

(1)

is mathematically based on quantum field theory in curved spacetime. This remarkable result is intriguing and beautiful but also a bit subtle and mysterious. From a different point of view, our group of quantum optics and general relativity aficionados have teamed up to show 5,6 that atoms freely falling into a BH with the field in the Boulware vacuum (the state of the field in which no Hawking radiation is emitted by the black hole) emit radiation which has a thermal energy spectrum (but has phase correlations between the energy states making the emitted radiation a pure state rather than a thermal density matrix) which to a distant observer has aspects that look like (but also aspects that differ from) Hawking radiation. We call it Horizon Brightened Acceleration Radiation (HBAR). 5 It is produced solely by emission from the atom while outside the BH. This work was inspired by quantum optics in flat spacetime, which predicts that atoms moving with a uniform acceleration emit thermal radiation with Unruh 7 temperature. Although freely falling (having geodesic motion), the atoms seem to a distant observer to be accelerating in their fall into the black hole, and thus seem to that observer to be

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Fig. 2. A detector accelerating through a spacetime region in its field vacuum state detects Unruh radiation. This happens if in the frame relative-to-which the vacuum modes are defined, the atom is accelerating; whether or not the atom is actually accelerating. This could even happen if the atom is inertial, and the metric is flat, or if there are mirrors modifying the boundary conditions of the spacetime modes.

accelerated detectors in the Boulware vacuum (which for a distant observer is one with no particles). However, rather than being excited as though in a thermal bath, they emit radiation whose energy spectrum as seen by the distant observer looks thermal with a temperature TU proportional to their acceleration α, TU =

α . 2πckB

(2)

As is explained in the following section, this “acceleration radiation” arises from processes in which the atom jumps from the ground state to an excited state, together with the emission of a photon. 8,9 In quantum optics, such processes are usually discarded because they violate conservation of energy, and the virtual photons must be quickly reabsorbed in order to maintain the overall energy conservation. However, if the atom is accelerated away from the original point of virtual emission, there is a small probability that the virtual photon will “get away” before it is re-absorbed. Alternatively, the Doppler shift of the accelerated atom takes the otherwise reabsorbed photon out of the atom’s bandwidth. Atom acceleration converts virtual photons into real ones at the expense of the energy supplied by the external force field driving the center-of-mass motion of the atom (in Unruh’s original case, the acceleration results from an external force, while in our case, the seeming acceleration is due to gravity). In an alternate point of view, one can trace the excitation of the atom to a vacuum fluctuation, which in the usual case is canceled by a succeeding, correlated fluctuation. However, in the accelerated case, the velocity of the atom is different by the time that correlated fluctuation hits it, giving a Doppler shift which now means that the fluctuation has the wrong frequency for de-exciting the atom.

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Near the event horizon, at radii close to rg = 2M G/c2 , the Schwarzschild metric is well-approximated by the constant-acceleration Rindler metric, 10 in which an atom would have a gravitational acceleration of α = c2 /2rg (even though to itself it has zero acceleration). The vacuum state through which it falls is one in which observers at rest in that frame see no particles. While in the usual Unruh effect, the atom is excited, in this case, the atom emits photons whose energy spectrum (as seen by distant stationary observers) appears to be thermal. As a result, the temperature, the HBAR temperature, can be obtained from the Unruh temperature by plugging α = c2 /2rg into Eq. (2) to find THBAR =

c2  = TBH . 2πckB 2rg

(3)

THBAR is equal to the temperature of Hawking radiation (1). This radiation differs from Hawking radiation in that, although the probability of emission of the various possible energies is proportional to a thermal spectrum, the emission from any one atom is a pure state, with definite phase relations between the energies. Of course if one has many atoms with incoherent times of fall into the black hole, or if one took into account the recoil of the atom, some of that phase coherence could be destroyed, making the emission look closer to Hawking radiation. However, the physics is very different from that of the Hawking effect. Here we have radiation coming from the atoms, whereas Hawking radiation requires no extra matter (e.g., atoms) and arises just from the BH geometry. There are several features of this finding that some have found surprising. For example one objection could be that the atom is freely falling with proper acceleration of zero. Where then does the radiation come from? However this neglects that the state of the field is assumed to be the Boulware vacuum state in which the particle content near infinity is zero, but near the horizon is full of particles (the energy density actually diverges at the horizon). It is those particles that the atom is interacting with. And from far away, the atom looks as though it is accelerated as it falls into the black hole. In the following section (Sec. II), we first follow a quantum optics path to Unruh radiation and compare it to the more usual treatment based on quantum fields in curved spacetime. In Sec. III, we use two scenarios where, surprisingly, acceleration radiation is emitted by inertial detectors, for discussing the equivalence principle of Einstein (in one case, we have a stationary atom interacting with a moving mirror, and in the other case, we have an atom freely-falling into a black hole). In Sec. IV, we discuss how Unruh radiation occurs because of the difference between mode definitions in different frames — a point of view in which it is not surprising that an inertial observer would detect acceleration radiation. In Sec. V, we present a KMS-inspired method for obtaining the Unruh temperature, an approach pioneered by Christensen and Duff. 11 There, we use the KMS periodicity approach to get the Unruh temperature from both a field and an atom perspective. We summarize in Sec. VI.

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Fig. 3. Feynman’s blackboard as he left it. On the bottom-right corner he inscribed “accel. Temp” under “TO LEARN:”. This is a strong indication of the subtlety and interest in this problem (courtesy of the Archives, California Institute of Technology).

II. Quantum Optics Route to Obtaining Unruh Radiation in Minkowski Coordinates In this section, we provide a simple first principles calculation of the radiation emitted by an accelerating atom. This calculation bears similarities to that of Unruh and Wald. 12 It answers, in part, the implied question of Feynman and Milonni, as in Fig. 3. Milonni wrote: [A] uniformly accelerated detector [i.e., atom] in the vacuum responds as it would if it were at rest in a thermal bath at temperature T = a/2πckB . It is hardly obvious why this should be [emphasis added] — it took half a century after the birth of the quantum theory of radiation for the thermal effect of uniform acceleration to be discovered. IIa. Accelerating atom in a vacuum We consider a two-level atom (a is the excited level and b is the ground state) with transition frequency ω moving along the z-axis in a 1 + 1-dimensional spacetime with a uniform acceleration α. The atom trajectory is given by ct¯ ct¯ ct(t¯) =  sinh , z(t¯) =  cosh , (4)   where t is the lab time and t¯ is the proper time for the accelerated atom, 13 and where  = c2 /α

(5)

is the length-scale in the problem. The interaction Hamiltonian between the atom and an outward-propagating photon with wave number k reads   ¯ ˆ e−iωt + H.a. , (6) Vˆ (t¯) = g a ˆk e−iνt(τ )+ikz(τ ) + H.a. σ

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where operator a ˆk is the photon annihilation operator, σ ˆ is the atomic lowering operator, and g is the atom-field coupling constant which depends on the atomic dipole moment and on the electric field in the frame of the atom. Initially the atom is in the ground state and there are no photons. If the interaction is weak enough, the state vector of the atom-field system at the atomic proper time t¯ can be found using first-order time-dependent perturbation theory, i |ψ(t¯) = |ψ(τ0 ) − 

τ

dt¯ Vˆ (t¯ ) |ψ(τ0 ) .

(7)

t¯0

The probability of excitation of the atom (frequency ω) with simultaneous emission of a photon with frequency ν is due to a counter-rotating term a ˆ+ ˆ + in the kσ interaction Hamiltonian. The probability of this event is 2  ∞  ∞ 2          1 ¯ dt¯ 1k , a| Vˆ (t¯ ) |0, b  = g 2  dt¯ eiνt(t )−ikz(τ ) eiωτ  , (8) P = 2       −∞

−∞

where |b and |a are the ground and excited state of the atom respectively, and t(t¯ ) and z(τ  ) are obtained from Eqs. (4), and using that k = ν/c and changing the −ct¯ / , and taking into account that variable of integration to x = ν c e ∞ dx e

−ix −i ω c −1

x

=e

− 12



πω c

iω Γ − c

,

0

where Γ(x) is the gamma function, and the property |Γ(−ix)|2 = π/[x sinh(πx)], we finally obtain that the probability is P =

1 2πcg 2  ω  . αω exp 2π c − 1

(9)

    We find that P is proportional to the Planck factor 1/ exp 2πωc − 1 which α is the probability that the atom is excited and a photon is emitted. The Planck factor corresponds to excitation probability with a temperature that is proportional to the acceleration α, TU =

α . 2πckB

This can be understood as was discussed in the previous section, as generating a photon by breaking adiabaticity due to the acceleration of the atom. Another physical picture involved the promotion of vacuum fluctuations. In any case, the a†k (t, z) tells us that the (Minkowski) photon is emitted and operator product σ ˆ † (t¯)ˆ the atom is excited.

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Fig. 4. (a) An atom is fixed in Minkowski spacetime at coordinate z0 and the field is in the Rindler vacuum (created by an accelerated mirror). (b) An atom moves in the vicinity of the BH event horizon in the Boulware vacuum, emitting acceleration radiation. These two cases are equivalent to each other, given that the acceleration of the mirror is related to the BH mass by Eq. (60).

IIb. Excitation of a Static Atom by the Rindler Vacuum Having seen that an atom accelerating through the Minkowski vacuum emits (Minkowski) photons, we consider the “inverse” problem of a stationary atom in an accelerating Rindler vacuum. To put this in perspective, Sec. IIa represents the Cavity QED problem of an atom passing through a stationary cavity. In this section (IIb), we are essentially dealing with an accelerating mirror 14 (with the state of the field being a Rindler-like vacuum) and stationary atom, as in Fig. 4b. This is the physics behind the present Rindler coordinate analysis. We proceed by assuming that an atom is fixed in the inertial reference frame (t, z) at position z = z0 (see Fig. 4a). We make a coordinate transformation into a uniformly accelerating reference frame, ct¯ ct¯ z¯/ z¯/ , z = e cosh , (10) ct = e sinh   where  is defined in the same way as in Eq. (4), which gives that the proper acceleration at z¯ = 0 is α. See Fig. 5.

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Fig. 5. Minkowski space divided into four wedges. Of particular relevance are the right and left wedges, which are called “the right Rindler wedge” and “the left Rindler wedge,” respectively.

The coordinate transformation (10) covers only the part of the Minkowski spacetime with z > c|t| (right Rindler wedge). It converts the Minkowski spacetime line element ds2 = c2 dt2 − dz 2 to the Rindler line element, 10,15  2  (11) ds2 = e2a¯z/c c2 dt¯2 − d¯ z2 . An observer moving along the trajectory z¯ = 0 in the Rindler space is uniformlyaccelerating in the Minkowski space along the trajectory (4), which is a special case (¯ z = 0) of Eq. (10). Normal modes of scalar photons in the conformal metric (11) take the same form as the usual positive frequency normal modes in the Minkowski metric, e.g., one can take them as traveling waves, 1 ¯ φν (t¯, z¯) = √ e−iν t+ik¯z , ν

(12)

where ν is the photon angular frequency in the reference frame of the Rindler space and k = ±ν/c. However, the modes (12) are a mixture of positive and negative frequency modes with respect to the physical Minkowski spacetime. Therefore, the vacuum state of these modes is not the Minkowski vacuum but rather the Rindler vacuum, which is what we assume for those modes. From Eq. (10) we obtain t¯ and z¯ in terms of t and z,

 2  z + ct z − c 2 t2   , z¯(t, z) = ln . (13) ct¯(t, z) = ln 2 z − ct 2 2 The atomic trajectory is obtained from Eq. (13) by setting the Minkowski space

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position to z = z0 . In the Rindler space, the atomic velocity is c2 t d z¯ =− . V¯ = dt¯ z0

(14)

From the perspective of the atom, it passes through the right Rindler wedge within the proper time interval −

z0 z0 0 have a positive value for the norm, while those for ω < 0 have a negative norm. We however, use a different complete set of modes, Eq. (32) below, which are similar to Eq. (23), for expanding solutions of Eq. (29). Instead of the solutions (30) and (31), we elect to use a complete set of modes for the field by e−πΩ/2 lim (u − iλ)iΩ , φˆΩ+ =

8πΩ sinh(πΩ) λ→0+

(32)

where we normalized Eq. (23) and use a different variable, u. These are a complete set of positive norm (often called the positive frequency Unruh–Minkowski modes, 7,12 ) even though Ω takes all values positive and negative. The negativenorm modes are just the complex-conjugate of these (due to the sign of iλ, or ultimately, the definition of the branch-cut).

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IIIa. Accelerating atom We are now going to place a mirror at position z = 0. We will take the boundary conditions on the solutions φ that they be zero at the mirror. The solutions of Eq. (29) then are of the form φ(u, v) = g(u) − g(v) ,

(33)

for some function g. Since at z = 0, the null coordinates are both u = v = ct/, then we see that φ(z = 0, t) = g(t) − g(t) = 0, which satisfies the boundary conditions. Using Eq. (33) and the modes (32), we have that the modes satisfying the boundary conditions are   e−πΩ/2 (34) lim (u − iλ)iΩ − (v − iλ)iΩ . φΩ (u, v) =

4Ω sinh(πΩ) λ→0+ For the two-level atom, let us define the two states |b as the ground state of the atom and |a as the excited state, with proper energy ω, and the atomic raising operator σ ˆ † , which takes σ † |b = |a , having time dependence eiωt¯ in the interaction picture, where t¯ is the proper time of the atom. ˆ in terms of the null coordinates u and v We can write the quantum field Φ   ˆ ˆ†Ω φ∗Ω (u, v) . (35) Φ(u, v) = dΩ a ˆΩ φΩ (u, v) + a In terms of the null coordinates (27), the path of the particle (4) is u(t¯) = −e−ct/ , ¯

¯

v(t¯) = ect/ .

The interaction between the atom and the field will be taken to be  ¯ ¯ ˆI = g σ ˆ, H ˆ e−iωt + σ ˆ † eiωt wμ ∂μ Φ

(36)

(37)

where wμ is the four velocity of the atom, and t¯ is the proper time along the path of the detector. In the frame of the atom, it is stationary, thus we have ˆ = ∂t¯Φ ˆ, w μ ∂μ Φ

(38)

where the derivative is evaluated along the path of the atom. This interaction is ˆ an ohmic-coupled bath for the detector, in the chosen because it makes the field Φ nomenclature of Caldera and Leggett. 18 See Fig. 7. Since the atom begins in its ground state, and the quantum field in the Minkowski vacuum state, in the atom-field interaction, the only term that contributes to the probability amplitude that the atoms becomes excited is the “counter-rotating” term, in the language of quantum optics. I.e., we need terms ˆ† . If the atom is not accelerated, such counter-rotating terms will that look like σ ˆ†a give zero when integrated over time. However, using the above definition of the field, and the fact that the time-dependence of the atomic raising operator σ ˆ † is

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Fig. 7. An atom accelerates in the presence of a stationary mirror. Initially, the atom is in its ground state |b, and the field is in the Minkowski vacuum |0M . There is some amplitude |Aex  for the atom to become excited. We note that since the mirror destroys the Lorentz invariance (in contrast to the true, Lorentz-invariant vacuum case in Sec. V), the state seen by the accelerated atom is not a static thermal bath, but this aspect does not matter for our conclusions.

eiωt¯, we get an excitation amplitude of     ¯ ¯ ˆ − e−ct¯/ , ect¯/ |b, 0M |Aex = a| g dt¯ σ ˆ † eiˆωt + σ ˆ e−iˆωt ∂t¯Φ  T ¯ dt¯eiωt =g −T



×

∞

dΩ −∞

(39)

 e−πΩ/2 iΩc  ¯ ¯ a ˆ†Ω |0R , (−e−ct/ )−iΩ + (ect/ )−iΩ

 8πΩ sinh(πΩ) (40)

since ∂t¯φ∗Ω (u, v) is ∂t¯φ∗Ω (u, v) =

  e−πΩ/2 iΩc

lim (u + iλ)−iΩ + (v + iλ)−iΩ ,  8πΩ sinh(πΩ) λ→0+

(41)

where we used Eq. (34). I.e., the first-order excitation is due to the σ ˆ†a ˆ† term, a product of the counter-rotating terms in the quantum optics nomenclature. If Ω > 0, then the second term in the square brackets will be zero after integration over t¯, while if Ω < 0, it is the first term that will be zero. Now (−x + iλ)iΩ = xiΩ e−πΩ for positive x since one must take the contour around the upper −x complex values so that (−1)iΩ = (e+iπ )iΩ = e−πΩ . See Fig. 6. The integral in Eq. (40) thus becomes (in the limit that T → ∞)  e−πω/2a a ˆ†ω/a − a ˆ†−ω/a |0M , |Aex ≈ 2T

8πω sinh(πω/a)/a

(42)

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where a ˆ†ω and a ˆ†−ω are the creation operators for the right- and left-moving modes, Eqs. (30) and (31), respectively. The probability of atomic excitation is Pex = Aex |Aex ,

(43)

which is proportional to the thermal factor 1/(e2πω/a − 1). We note that this is interesting in that there is really no horizon hiding the partner particles from the quantum field from the detector. There is entanglement between the incoming field in the right Rindler wedge and that behind the incoming horizon. But the latter gets reflected out by the mirror. Thus the entanglement in the Minkowski vacuum occurs between the ingoing modes in the right Rindler wedge and the outgoing modes in that same wedge, instead of being hidden behind the horizon. We can ask whether or not the system is truly thermal by comparing the probability of emission of radiation by an excited accelerated atom with the absorption of the counter-rotating term by the unexcited atom. IIIb. Accelerating mirror In the second case, we consider an accelerated mirror, with a stationary detector whose surface is at uv = −1, and the field initially in the Rindler vacuum (as defined by Fulling 19 ). With the mirror accelerated, the field is expanded in terms of the positive-norm Rindler modes, 10   −iΩ , u > 0 u 0, u>0 1 1 φ¯Ω++ = √ φ¯Ω+− = √ iΩ 4πΩ 0, 4πΩ (−u) , u < 0 u 0 v 0, u > 0 1 1 φ¯Ω−+ = √ φ¯Ω−− = √ 4πΩ 0, 4πΩ (−v)iΩ , v < 0 u 0.

(46)

In these regions, v < 0 and u > 0, there is no mirror. We have the positivenorm “2-modes,” which interact with the mirror. The region of the spacetime with negative u and positive v contains the mirror, which lies on the surface uv = −1.

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Fig. 8. A stationary atom with a moving mirror. While usually the spatial left- and right-moving modes are independent, in this scenario, we have three families of modes which interact with the atom. The first (labeled ‘2’), consists of right- and left-moving components, with relative phase (−1) between them, so that they vanish at the mirror. Also in this spacetime are left-moving modes and right-moving modes (labeled ‘1’ and ‘3’, respectively) which do not interact with the mirror. Those have a random phase relationship to one another. The mirror follows a trajectory of constant acceleration, and the atom is at rest. The three cases depicted are (1) the positive-norm “1-modes” that originate before the past right null asymptote of the mirror trajectory and travel to the left; (2) the positive-norm “2-modes” that originate from the left before the past extension of the future null asymptote for the mirror, bounce off the mirror, and continue traveling to the left after the past null asymptote of the mirror; (3) the positive-norm “3-modes” that originate from the left after the extension of the future null asymptote of the mirror and travel to the right. The field is in the Rindler vacuum state, which means that each of the three types of modes above (and the “4-modes” (not depicted) that are to the right of the mirror and do not interact with the atom) are independent of (unentangled and uncorrelated with) any of the other modes and have no particles detectable by accelerated observers in either Rindler wedge. The atom is at rest (moving along an inertial static world line) at distance  from the closest approach of the mirror.

These modes are a superposition of the positive-norm left- and right-moving Rindler modes, Eq. (44), which vanish at the mirror. They are  1  iΩ (47) (u) − (−v)−iΩ , φ¯Ω2 (u, v) = √ 4πΩ and zero elsewhere. These are a bit subtly-defined, because the right-moving piece is defined for u < 0, but the left-moving part is in v > 0, see Fig. 8. We also have the “4-modes” (not shown in the figure). These are confined to the region u < 0 and v > 0 (the right wedge) and vanish at the mirror, but do not interact with the atom, so we ignore them. In terms of the positive-norm mode families which interact with the atom, ˆ is Eqs. (45), (46), and (47), the field Φ

3  ∞  ˆ dΩ ˆbΩi φΩi + H.a. , (48) Φ(u, v) = i=1

0

where the summation over i is to include all three mode types. The atom travels

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along the path u = v = ct/. The state of the field, the Rindler vacuum, |0R , is defined by ˆbΩi |0R = 0 for all values of Ω, and the only terms in the amplitude which survive if the detector is initially in its ground state |b are  T  ∞ dt dΩ eiωt ∂t φ∗Ωi (t, t)ˆb†Ωi |0R , (49) |Aex = −T

0

ˆ and Eq. (37) where 2T is the interaction time, and we used Eq. (48) for the field Φ for the interaction Hamiltonian. To calculate (49) for infinite interaction time T , we first compute WΩ± , where

±iΩ  ±∞ ct iωt WΩ± = ± dt e ∂t ± . (50)  0 To compute WΩ+ , we rotate the contour of integration from the real t-axis to the imaginary t-axis, with tI = Im {t} and where the branch-cut is not in the first quadrant of the complex t-plane. iΩ  ∞ ctI −ωtI dtI e ∂ tI e−πΩ/2 . (51) WΩ+ =  0 Changing integration variables from tI to x = ωtI we get  ∞  c iΩ  c iΩ −πΩ/2 e dx e−x ∂x xiΩ = (iΩ) e−πΩ/2 Γ(iΩ) WΩ = ω ω 0 √ πΩ e−πΩ/2 iϕ(Ω)  c iΩ e =i

, ω sinh(πΩ)

(52)

where ϕ(Ω) is the slowly-varying phase of the complex argument gamma function Γ(iΩ), which  starts at −π/2 for Ω = 0 and reaches 0 only once Ω ≈ 3, by which

time e−πΩ

Ω 2 sinh(πΩ)

will have dropped by a factor of about 103 . I.e., the phase of

Γ(iΩ) is essentially constant over the range in which the Γ(iΩ) is non-zero. Similarly, one can rotate the contour in Eq. (50) the other way and evaluate iΩ   0  0 c iΩ ct iωt dt e ∂t i = dx e−πΩ/2 e−x ∂x xiΩ . (53) − WΩ− =  ω ∞ ∞ We thus find that WΩ = WΩ+ = (WΩ− )∗ , and therefore, the excitation amplitude per Ω is   g ˆ† (54) |AΩ = √ bΩ1 WΩ∗ + ˆb†Ω3 WΩ + ˆb†Ω2 WΩ − WΩ∗ , 4πΩ where, using (49), the full amplitude is |Aex =

 dΩ |AΩ .

(55)

Integrating the amplitude |AΩ in Eq. (54) over Ω gives some constant which is independent of the frequency of the atom, and certainly not thermal. However, the

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probability of emitting a mode with frequency Ω is proportional to a thermal factor PΩ = | 0R | ˆbΩi |AΩ |2 ∝

1 1 − exp[2πΩ]

(56)

which was also found in Ref. 6. Thus, an accelerated atom above a stationary mirror with the field in the Minkowski vacuum (no particles detected by the mirror as striking the stationary mirror) is excited with a probability proportional to the thermal factor, while an accelerated mirror above a stationary atom, with the field in the Rindler vacuum (i.e., no particles detected by the mirror as striking the mirror) emits Rindler modes with a probability proportional to the thermal factor. We must distinguish this statement from stating that the atom emits particles into a thermal state. The atom emits modes with correlations between the modes, given by the phase factor i(a/ω)iΩ eϕ(Ω) , as in Eq. (52). I.e., what an unaccelerated atom below the accelerated mirror emits is a pure state, not a thermal state (a mixed state); albeit, the probability distribution over Rindler energies is proportional to a thermal factor. There is thus some crude approximate form of the equivalence principle in play here. Hawking showed that a black hole emits thermal radiation. While an observer at infinity sees the black hole as in some sense stationary, a static observer or atom near the horizon is accelerated with constant acceleration. The Hartle–Hawking state of the field near the black hole looks like a thermal state to such a static observer, but looks much more like a vacuum state to a freely-falling observer. We can again look at two cases, the one analyzed by Hawking, in which the atom is accelerated and near the horizon, while the state is the vacuum state as far as the horizon is concerned (although it is a state in thermal equilibrium with a temperature inversely-proportional to the mass for an observer far away). The second case is where the atom is in free fall into the horizon, while the state of the field is the socalled Boulware vacuum (the analog of the Rindler vacuum in the curved spacetime of the Schwarzschild metric of a non-rotating black hole), where a distant observer sees nothing coming out of the black hole. IV. Acceleration Radiation and the Equivalence Principle In this section, we discuss acceleration radiation from atoms which do not accelerate, and show the approximate equivalence between atoms freely-falling into a Schwarzschild black hole and stationary atoms (in Minkowski space) in the presence of an accelerating mirror. See Fig. 9. When Einstein first formulated the equivalence principle he was mainly concerned with the laws of classical physics. Ginzburg and Frolov in their review paper 20 mentioned that: “The question of whether or not the equivalence principle holds for the description of phenomena for which their quantum nature is important is by no means trivial.”

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Fig. 9. The three physical cases which we consider: (A) An atom uniformally accelerated in Minkowski space in the presence of a stationary mirror with the Minkowski vacuum. (B) A stationary atom in Minkowski space in the presence of an accelerating mirror and the Rindler vacuum. (C) An atom in free-fall in the Schwarzschild metric in the Boulware vacuum. In all three sub-figures, we indicate the probability of atomic excitation (atomic frequency ω) in the first case or with an excitation probability at high frequency for the electromagnetic field mode with frequency ν. Cases (B) and (C) are similar because in both cases, the atom is freely-falling, but still emits radiation. Case A is a physically-different case, because the atom has non-zero proper acceleration, and it is the atom that is thermally-excited, giving physically-different results.

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Fig. 10. Atoms in the ground state |b are freely-falling in a Schwarzschild black hole metric, where the state of the field is the Boulware vacuum. The atoms, though inertial and moving along geodesics, emit acceleration radiation. When the atoms are released at random times from infinity, the outgoing field is thermal.

Here we discuss acceleration radiation of an atom freely-falling in the gravitational field of a static BH. The equivalence principle tells us that the atom essentially falls “force-free” into the BH, that is, the atom’s acceleration is equal to zero. How then could it emit something which looks like acceleration radiation? To answer this question we consider modes of the field in the reference frame of the black hole. In the Schwarzschild metric the field modes are stationary, even though they are modified by the gravitational field of the BH. However, in the reference frame of the freely falling atom the field modes are changing with time. The equivalence principle is manifested as a symmetry between emission by a static atom in Minkowski spacetime in the Rindler vacuum (discussed in the previous section), and an atom freely falling in a gravitational field of a BH in the Boulware vacuum. Moreover, there is an analogy between the Rindler horizon and the BH event horizon. Indeed, the time-radius part of Schwarzschild metric interval, ds2 =

r − rg 2 2 r c dt − dr2 , r r − rg

(57)

which could be approximated near the event-horizon r = rg by ds2 

r − rg 2 2 rg c dt − d¯ r2 , rg r − rg

(58)

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and using the coordinate r¯ such that r − rg = er¯/rg to describe space-time events outside the event horizon, the time-radius Schwarzschild interval becomes  (59) ds2  er¯/rg c2 dt2 − dr 2 , which is the interval of the Rindler space metric, Eq. (11). Comparing with the interval of Rindler space, Eq. (11), we obtain an effective acceleration corresponding to a free fall near the event horizon d2 r¯ 1 =α= . 2 ds 2rg

(60)

Next we consider an atom launched radially from the event horizon with an initial radial velocity V0 = cdr/ds (see Fig. 10b). Using the Schwarzschild metric in Eq. (57), the equations of atomic radial motion are

dr ds

2

V02 rg − 1, + c2 r dt V0 = 2 r  . ds c 1 − rg =

(61)

For V0  c we find the following solution r V2 s2 = 1 + 02 − 2 , rg c 4rg

2rg V0 + cs rg ln t= . c 2rg V0 − cs In terms of the coordinate r¯, the atomic trajectory is    4rg2 V02 1 2 −s . r¯ = rg ln 4rg2 c2

(62)

(63)

The trajectory of the atom near the BH event horizon, given by Eqs. (62) and (63), has the same form as the trajectory of the atom fixed in Minkowski spacetime at z0 = 2

V0 rg c

(64)

viewed in the Rindler coordinates (10) when relating the acceleration in the Rindler case to the effective acceleration near the BH, Eq. (60). Since near the event horizon the Schwarzschild metric (57) can be approximated as the Rindler metric (59), the probability of atomic excitation and photon emission for an atom falling into a Schwarzschild black hole is given by the same expressions, (18) and (20), only where α and z0 are replaced with the corresponding values, (60) and (64), respectively.

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V. The “Bogoliubov” Path to Unruh Radiation In this section, we present yet another interpretation of Unruh radiation. It could be understood as a difference of perspective between two observers. For simplicity, we ˆ t) to represent the photons. This field is an operator, consider a real scalar field Φ(z, which can be expanded in different basis sets. Let us consider two observers — a stationary and an accelerating observer (in Minkowski space) — which naturally have two basis sets to describe the modes of the field. The stationary observer has the line element ds2 = c2 dt2 − dz 2 , while the accelerating observer’s line element z 2 ), which is obtained from the stationary observer’s line is ds2 = e−2¯z/ (c2 dt¯2 − d¯ element by transforming to “accelerating” coordinates, Eq. (10). The normal modes φω in each coordinate systems are different, satisfying the wave equation √ 1 √ ∂μ −gg μν ∂ν φω = 0 , (65) −g where g μν is the metric, which could be read-off from the expression for the line element, and g is its determinant. Using Eq. (65), with the metrics corresponding to Minkowski (stationary) and Rindler (11) (accelerating) observers, the normal modes for the stationary and accelerated observer both satisfy [(∂0 )2 − (∂1 )2 ]φ = 0, albeit in different coordinate systems. So in both cases the normal modes are complex exponentials, but in terms of different coordinates. The stationary observer’s modes z , t¯) specified in Rindler coordinates, are φν , evaluated at some spacetime event, (¯   1 1 ¯ ¯ ¯ (66) φν (¯ z , t¯) = √ e−iν/c z(¯z,t)−ct(¯z,t) = √ e−iν/c exp[(ct−¯z)/] ν ν and for the accelerating observer 1 ¯ φν (¯ z , t¯) = √ e−iν/c(¯z−ct) . ν

(67)

So in the right Rindler wedge, the two observers describe the field as  ˆ z , t¯) = Φ(¯ φν (¯ z , t¯)ˆ aν + φ∗ν (¯ z , t¯)ˆ a†ν ν

=



z , t¯)ˆbν¯ + φ¯∗ν¯ (¯ z , t¯)ˆb†ν¯ . φ¯ν¯ (¯

(68)

ν ¯

Using the orthogonality of the modes, φν (z, t), φν¯ (z, t) = δν,¯ν , where the innerproduct is given by Eq. (25), we see that we could obtain a ˆν ’s in terms of the ˆbν¯ ’s,   ˆ t) = a ˆν = φν (z, t), Φ(z, αν ν¯ˆbν¯ + βν ν¯ˆb†ν¯ , (69) 



ν ¯

 = φν , φ¯∗ν¯ . Alternatively, one can obtain the ˆbν¯ ’s 

where αν ν¯ = φν , φ¯ν¯ , and βν ν¯ in terms of the a ˆν ’s,   ˆbν¯ = φ¯ν¯ , Φ ˆ = ˆν − βν∗ν¯ a ˆ†ν , αν∗ ν¯ a n

(70)

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where we have used the properties of the inner-product (25), ∗

∗

f, g = g, f = − g ∗ , f ∗ = − f ∗ , g ∗ .

(71)

Particles in the vacuum We can use Eq. (69) to make calculations, for instance, the number of S particles in the S¯ vacuum is   †    2 ˆν a ˆ n = 0S¯ a ˆν 0S¯ = |βν ν¯ | (72) ν ¯

and using Eq. (70), we find that the number of S¯ particles in the S vacuum is      †  2 ˆ n ¯ = 0S ˆbν¯ˆbν¯ 0S = |βν ν¯ | . (73) ν

An interesting symmetry is that in both cases, the number of particles in the other 2 frame’s vacuum is given by a summation of |βν ν¯ | ; albeit, the two quantities involve summations over different indices. If we use the Unruh–Minkowski modes for the modes φν ,

iν/c u − iλ e−πν/2c lim , (74) φν (u) =

 sinh(πν/c) λ→0+ whose annihilation operator corresponds to a superposition of plane wave √  eiν (ct−z) / ν  annihilation operators α ˆν ,  

  ν dν (iν  )iν/c  iν/c a ˆν = Γ 1 + − (−iν ) (75) α ˆν . c iν  eπν/c αν ν¯ and βν ν¯ are e− 2 ν/c αν ν¯ = !  δν ν¯ , 2 sinh πν/c π

e− 2 ν/c βν ν¯ = !  δν ν¯ , 2 sinh πν/c π

(76)

we find that the number of Rindler photons in the Minkowski vacuum state, and the number of Unruh–Minkowski photons in the Rindler vacuum state are both   1 1 ˆ¯ =  , (77) ˆ n = n 2 exp 2πν/c − 1 which is the Planck factor corresponding to the temperature of TU = a/2πckB . An accelerating observer in Minkowski vacuum Notice that the Minkowski-space mode φ in Eq. (66) is only defined in the right Rindler wedge, see Fig. 5. However, the extension to the rest of Minkowski space (into the left Rindler wedge) is unique if we demand that it correspond to an annihilation operator, and that it not have any creation operator “components” (for all

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values of the frequency parameter ν). To correspond to an annihilation operator, it must have positive-norm, and demanding that its norm be positive for all ν, we find that it is  e−πν/2c φ¯∗L left wedge, 1 ν , R (78) φν =

πν/2c R ¯ 2 sinh(πν/c) e right wedge. φ , ν

There is another family of Minkowski modes, φLν , which is concentrated mostly in the left Rindler wedge,  eπν/2c φ¯Lν , left wedge, 1 L φν =

(79) −πν/2c ∗R ¯ 2 sinh(πν/c) e right wedge. φ , ν

Consider a two-level atom with constant acceleration in the right Rindler wedge, with trajectory given by Eq. (4). In its frame, the atom interacts with the mode φ¯R ν in the right Rindler wedge, Eq. (10), which corresponds to the annihilation operator ˆbR . Thus, the time evolution of the state of the field–atom system is given by the ν ˆ (first-order time-dependent perturbation theory) time-evolution operator U   1 τ −iντ ˆ ˆ U 1+ dτ σ ˆ † eiωτ ˆbR , (80) νe i 0 which means that the atomic excitation process is accompanied by the annihilation of a a right Rindler wedge photon. For the Minkowski observer, however, the mode which the atom interacts with is zero in the left wedge, and he describes the annihilation operator ˆbR ν using Eqs. (78) as  1 πν/2c R −πν/2c †L ˆbR =

a ˆ + e a ˆ . (81) e ν ν ν 2 sinh(πν/c) Thus, since a ˆR ν |0M = 0, the time-evolution operator, operating on the initial Minkowski vacuum state, is  τ 1 1 −iντ ˆ ˆ e−πν/2c U 1+

dτ σ † eiωτ a ˆ†L . (82) ν e i 2 sinh(πν/c) 0 VI. Periodicity Trick for Unruh Temperature Now we will give a “trick” for deriving the Unruh temperature. The trick is to argue that, in the Rindler metric, the time coordinate must be periodic in the imaginary direction and this imaginary periodicity implies that Rindler spacetime has a temperature. The original derivation of the Unruh temperature using periodicity in imaginary time may be found in a paper by one of us, 11,21,22 following a similar derivation of the Hawking temperature. 23 Quantum field theory at finite temperature is periodic in imaginary time, with periodicity t → t + iβ ,

(83)

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where β = 1/kB T . One way to see this is by looking at the thermal average, which possesses the property   ˆ Q(t ˆ  ) = Q(t ˆ  )Q(t ˆ + iβ) . Q(t) (84) ˆ operator and the invariIndeed, using the equation for the time evolution of the Q ance of the trace under cyclic permutation, we obtain   ˆ ˆ −iHt/ ˆ ˆ ) ˆ Q(t ˆ  ) = 1 Tr e−β Hˆ eiHt/ Q(0)e Q(t Q(t) Z ˆ H 1  i Hˆ (t+iβ) ˆ ˆ ˆ  Q(0)e−i  (t+iβ) e−β H Q(t = Tr e  ) Z 1  ˆ ˆ  ˆ = Tr e−β H Q(t )Q(t + iβ) . (85) Z Equation (84) is commonly referred to as the Kubo–Martin–Schwinger (KMS) condition. Since the ordering of the field operators on the two sides are interchanged, the corresponding periodicity along the imaginary time direction is referred to as “periodicity with a twist.” Now let us assume that state of the field is the Minkowski vacuum |0M . That is, in the inertial reference frame the temperature is equal to zero. Then the zero temperature average over this state can be written as ˆ z)Φ(t ˆ  , z  ) |0M , G(t, z; t , z  ) = 0M | Φ(t,

(86)

ˆ z) is the field operator at the spacetime event (z, t). where Φ(t, Since the vacuum is Lorentz-invariant, the two-point function (86) must depend only on the Lorentz-invariant spacetime interval c2 (t − t )2 − (z − z  )2 . If we make a coordinate transformation into the Rindler spacetime using Eq. (10) to express the interval in terms of the Rindler coordinates, the average (86) depends on  

2 ct¯ ct¯ 2  2  2 2 z¯/ z¯ / e sinh sinh −e c (t − t ) − (z − z ) =    

2  ct¯ ct¯ z¯/ z¯ / . (87) − e cosh cosh −e   Hence, because of the periodicity of hyperbolic sine and cosine functions under the addition of the imaginary increment 2πi to their argument, we have 

 c 1 ¯ 2π ¯ i = sinh(ct¯/) cos(2π) , t+ sinh t = sinh   c 

 c 1 ¯ 2π ¯ i = cosh(ct¯/) cos(2π) , (88) cosh t = cosh t+   c and we conclude that in the Rindler spacetime the two-point function G(t¯, z¯; t¯ , z¯ ) obeys the KMS condition, namely

2π i, z¯ . (89) G(t¯, z¯; t¯ , z¯ ) = G t¯ , z¯ ; t¯ + c

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Comparing this with Eq. (84), we see that 2π/c = 2πc/α = /kB TU , which yields the Unruh temperature α . (90) TU = 2πckB In other words, when viewed from a uniformly accelerating frame (i.e., the Rindler frame), the two-point function computed in the Minkowski vacuum appears to satisfy the KMS condition (84). Therefore, one may conclude that with respect to the Rindler observer, the Minkowski vacuum looks like a thermal reservoir of temperature TU . VII. Conclusions We revisit Unruh Radiation and arrive at the effect by different means. Using a quantum-optics route, we treat both the accelerating atom and accelerating mirror cases, which we also treat using the Unruh–Minkowski modes. The case of an atom freely-falling into a black hole is also discussed, and we discuss its relation to Einstein’s Equivalence Principle. Then, we show how the effects could be obtained from Bogoliubov transformations, and finally, we show the relation to the KMS condition, of which Schwinger is among the namesakes. Acknowledgments MOS, JSB, and AAS would like to thank the Robert A. Welch Foundation (Grant No. A-1261), the Office of Naval Research (Award No. N00014-16-1-3054), and the Air Force Office of Scientific Research (FA9550-18-1-0141) for their the support. DNP and WGU are supported by the Natural Sciences and Engineering Council of Canada. MJD is supported in part by the STFC under rolling grant ST/P000762/1. WPS thanks Texas A&M University for a Faculty Fellowship at the Hagler Institute for Advanced Study at Texas A&M University and Texas A&M AgriLife for support of this work. He is also a member of the Institute of Quantum Science and Technology (IQST) which is financed partially by the Ministry of Science, Research and Arts Baden-W¨ urttemberg. References 1. R. Kubo, Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems, J. Phys. Soc. Jap. 12, 570 (1957). 2. P. C. Martin and J. Schwinger, Theory of Many-Particle Systems, Phys. Rev. 115, 6 (1959). 3. S. W. Hawking, Black hole explosions?, Nature (London) 248, 30 (1974); S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975). 4. J. D. Bekenstein, Black holes and entropy, Phys. Rev. D. 7, 2333 (1973). 5. M. O. Scully, S. Fulling, D. Lee, D. Page, W. Schleich and A. A. Svidzinsky, “Quantum optics approach to radiation from atoms falling into a black hole,” Proc. Natl. Acad. Sci. U.S.A. 115, 8131 (2018).

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6. A. A. Svidzinsky, J. S. Ben-Benjamin, S. A. Fulling and D. N. Page, Excitation of an atom by a uniformly accelerated mirror through virtual transitions, Phys. Rev. Lett. 121, 071301 (2018). 7. W. G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14, 870 (1976). 8. M. O. Scully, V. V. Kocharovsky, A. Belyanin, E. Fry and F. Capasso, Enhancing Acceleration Radiation from Ground-State Atoms via Cavity Quantum Electrodynamics, Phys. Rev. Lett. 91, 243004 (2003). 9. A. Belyanin, V. V. Kocharovsky, F. Capasso, E. Fry, M. S. Zubairy and M. O. Scully, Quantum electrodynamics of accelerated atoms in free space and in cavities, Phys. Rev. A. 74, 023807 (2006). 10. W. Rindler, Kruskal space and the Uniformly Accelerated Frame, Am. J. Phys. 34, 1174 (1966). 11. S. M. Christensen and M. J. Duff, Flat space as a gravitational instanton, Nucl. Phys. B 146, 11 (1978). 12. W. G. Unruh and R. M. Wald, What happens when an accelerating observer detects a Rindler particle, Phys. Rev. D 29, 1047 (1984). 13. W. Rindler, Visual horizons in world models, Monthly Notices of the Royal Astronomical Society 116, 662 (1956). 14. G. T. Moore, Quantum theory of electromagnetic field in a variable-length onedimensional cavity, J. Math. Phys. 11, 2679 (1970). 15. A. Einstein and N. Rosen, The Particle Problem in the General Theory of Relativity, Phys. Rev. 48, 73 (1935). 16. O. Levin, Y. Peleg and A. Peres, Quantum detector in an accelerated cavity, J. Phys. A 25, 6471 (1992). 17. S. A. Fulling and J. H. Wilson, The Equivalence Principle at Work in Radiation from Unaccelerated Atoms and Mirrors, Phys. Scr. 94, 1 (2018). 18. A. O. Caldeira and A. J. Leggett, Path Integral Approach to Quantum Brownian Motion, Physica 121A, 587 (1983). 19. S. A. Fulling, Nonuniqueness of canonical field quantization in Riemannian space-time, Phys. Rev. D 7, 2850 (1973). 20. V. L. Ginzburg and V. P. Frolov, Vacuum in a homogeneous gravitational field and excitation of a uniformly accelerated detector, Usp. Fiz. Nauk 153, 633 (1987). 21. J. S. Dowker, Thermal properties of Green’s function in Rindler, de Sitter, and Schwarzschild spaces, Phys. Rev. D 18, 1856 (1978). 22. J. S. Dowker, Quantum field theory on a cone, J. Phys. A: Math. and Gen. 10, 115 (1977). 23. G. W. Gibbons and M. J. Perry, Black Holes in Thermal Equilibrium, Phys. Rev. Lett. 36, 985 (1976).

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Non-thermal fixed points: Universal dynamics far from equilibrium Christian-Marcel Schmied, Aleksandr N. Mikheev and Thomas Gasenzer Kirchhoff-Institut f¨ ur Physik, Ruprecht-Karls-Universit¨ at Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany

In this article we give an overview of the concept of universal dynamics near non-thermal fixed points in isolated quantum many-body systems. We outline a non-perturbative kinetic theory derived within a Schwinger–Keldysh closed-time path-integral approach, as well as a low-energy effective field theory which enable us to predict the universal scaling exponents characterizing the time evolution at the fixed point. We discuss the role of wave-turbulent transport in the context of such fixed points and discuss universal scaling evolution of systems bearing ensembles of (quasi) topological defects. This is rounded off by the recently introduced concept of prescaling as a generic feature of the evolution towards a non-thermal fixed point. Keywords: Far-from-equilibrium quantum dynamics, non-thermal fixed points.

1. Introduction Dynamics of isolated quantum many-body systems quenched far from equilibrium has been an object of intensive study during recent years. Examples range from the post-inflationary early universe, 1,2 via the dynamics of quark–gluon matter created in heavy-ion collisions 3,4 to the evolution of ultracold atomic systems following a sudden quench of, e.g., an interaction parameter. 5,6 Yet, despite great efforts, there are many open questions remaining and little is known about the structure of possible pathways for the evolution of such systems. Various scenarios have been discussed for and observed in ultracold atomic gases, including integrable dynamics, 7–10 prethermalization, 11–15 generalized Gibbs ensembles (GGE), 16–20 critical and prethermal dynamics, 21–24 many-body localization, 25,26 relaxation after quantum quenches, 27,28 wave turbulence, 29–31 decoherence and revivals, 32 universal scaling dynamics and the approach of a non-thermal fixed point, 33–36 as well as prescaling. 37 The rich spectrum of different possible phenomena highlights the capabilities of experiments with ultracold gases, as well as the gain obtained with quantum systems as compared to classical statistical ensembles. To theoretically study such out-of-equilibrium phenomena in quantum manybody systems, a wide range of tools and techniques provided by non-equilibrium quantum field theory is used. A central object in calculating the non-equilibrium time evolution of a many-body system is the so-called Schwinger–Keldysh closed time contour. Evolving quantities such as correlation functions of physical observables in time corresponds to evaluating, e.g., path integrals along such a closed contour. The technique was first introduced by Julian Schwinger in 1961 38 and further developed by Mahanthappa and Bakshi, 39–41 who were focussing on bosonic systems. Around the same time, Konstantinov and Perel developed a diagrammatic

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scheme for evaluating transport quantities in non-equilibrium systems. They used a time contour with forward and backward branches in time together with an imaginary-time path whose length is given by the inverse temperature. 42 The framework of non-equilibrium quantum field theory was then advanced by Kadanoff and Baym in 1962. 43 In their work, they also showed a pathway to kinetic equations. Keldysh proposed the closed-time path technique in 1964 and introduced a convenient choice of variables via the so-called Keldysh rotation. 44 To acknowledge all this work, the closed time contour is referred to as the Schwinger–Bakshi–Mahanthappa– Keldysh, in short Schwinger–Keldysh formulation.a The fermionic case was later considered by Larkin and Ovchinnikov in the context of superconductivity. 49 In summary, based on the work of Schwinger, it became possible to describe a wealth of non-equilibrium phenomena in quantum many-body systems, including non-thermal fixed points which are the subject of the present contribution. Our article is organized as follows. In Sec. 2 we introduce the basic concepts of non-thermal fixed points for which we outline, in Sec. 3, a non-perturbative kinetictheory description based on the Schwinger–Keldysh contour. A complementary approach in the form of a low-energy effective field theory description is presented in Sec. 4. We address the relation between (wave) turbulence and non-thermal fixed points in Sec. 5 and compare the analytical predictions with numerical simulations in Sec. 6. We finally discuss, in Sec. 7, the recently proposed concept of prescaling as a generic feature of the evolution towards a non-thermal fixed point. Our brief overview, which tries to give a short introduction without aiming at a full review, closes with an outlook to future research in the field, see Sec. 8. 2. Non-thermal fixed points The theory of non-thermal fixed points in the real-time evolution of, foremost closed, non-equilibrium systems, is inspired by the concepts of equilibrium and near-equilibrium renormalization-group theory, 50–52 see Fig. 1 for an illustration. The basic concept is motivated by universal critical scaling of correlation functions in equilibrium. When using a renormalization-group approach, a physical system is basically studied through a microscope at different resolutions. Close to a phase transition one observes that the correlations look self-similar, i.e., the same no matter which resolution is used. In this case, shifting the spatial resolution by a multiplicative scale parameter s causes correlations between points with distance x, denoted by C(x; s), to be rescaled according to C(x; s) = sζ f (x/s). Hence, the correlations are solely characterized by a universal exponent ζ and the universal scaling function f . A fixed point of the renormalization-group flow is reached when a change of the scale s does not change C by any means. In that case the scaling a Cf.

Ref. 45, using a different order of the names. See Refs. 46 and 47 for more detailed historical notes. Taking into account the close connection to the Kadanoff–Baym kinetic equations near equilibrium, including an imaginary-time branch, the approach has also been referred to as the Baym–Kadanoff–Schwinger–Mahanthappa–Bakshi–Keldysh formalism. 48

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Fig. 1. Schematics of a non-thermal fixed point 33 based on the ideas of a renormalization group flow. Depending on the initial condition, an out-of-equilibrium system can approach a non-thermal fixed point during the time evolution. In the vicinity of such a fixed point, the system experiences critical slowing down (indicated by the tightly packed red arrows). As a consequence, correlation functions C(k, t) show scaling behavior in space and time according to C(k, t) = tα f (tβ k), with a universal scaling function f . The associated self-similar evolution is, in general, characterized by non-zero universal scaling exponents α and β. Universal scaling close to a non-thermal fixed point is understood to occur as a transient phenomenon on the way to equilibrium (indicated by the trajectory leading away from the fixed point). Figure adapted from Ref. 35.

function takes the form of a pure power law f (x) ∼ x−ζ . In a realistic physical system, the scaling function f is, in general, not a pure power law but retains information of characteristic scales such as a correlation length ξ. Thus, the system may only approximately reach the fixed point. Taking the time t as the scale parameter, the renormalization-group idea can be extended to time evolution of systems (far) away from equilibrium. The corresponding fixed point of the renormalization-group flow is called non-thermal fixed point. In the scaling regime near a non-thermal fixed point, the evolution of, e.g., the time-dependent version of the correlations discussed above is determined by C(x, t) = tα f (t−β x), with two universal exponents α and β which assume, in general, non-zero values. The associated correlation length of the system changes as a power of time, ξ(t) ∼ tβ . Note that the time evolution taking power-law characteristics is equivalent to critical slowing down, here in real time. We remark that, depending on the sign of β, increasing the time t can correspond to either a reduction or an increase of the microscope resolution. The scaling exponents α and β together with the scaling function f allow to determine the universality class associated with the fixed point. 34,53 Hence, the evolution of very different physical systems far from equilibrium can be categorized by means of their possible kinds of scaling behavior. While a full such classification is still lacking, underlying symmetries of the system are expected to be relevant for the observable universal dynamics and thus for the associated universality class. Whether a physical system can approach a non-thermal fixed point and show universal scaling dynamics, and, if so, which particular fixed point is reached, in general depends on the chosen initial condition. A key ingredient for the occurrence of self-similar dynamics is an extreme out-of-equilibrium initial configuration.

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initial distribution after quench

particles removed by strong cooling quench

Fig. 2. Self-similar scaling in time and space close to a non-thermal fixed point. The sketch shows, on a double-logarithmic scale, the time evolution of the single-particle momentum distribution n(k, t) of a Bose gas for two different times t (solid and short-dashed line). Starting from an extreme initial distribution marked by the red long-dashed line, being the result of a strong cooling quench, a bi-directional redistribution of particles in momentum space occurs as indicated by the arrows. Particle transport towards zero momentum as well as energy transport to large momenta are characterized by self-similar scaling evolutions in space and time according to n(k, t) = (t/tref )α n([t/tref ]β k, tref ), with universal scaling exponents α and β, in general, different for both directions. Here, tref is an arbitrary reference time within the temporal scaling regime. The infrared transport (blue arrow) conserves the particle number which is concentrated at small momenta. In contrast, the energy, being concentrated at high momenta, is conserved in the redistribution of short-wavelength fluctuations (green arrow). See main text for details. Figure adapted from Ref. 54.

As an illustration we consider the time evolution of a dilute Bose gas in three spatial dimensions after a strong cooling quench, 55 see Fig. 2 as well as Refs. 34, 54, 56 and 57. An extreme version of such a quench can be achieved, e.g., by first cooling the system adiabatically such that its chemical potential is 0 < −μ  kB T , where the temperature T  Tc is just above the critical temperature Tc separating the normal and the Bose condensed phase of the gas, and then removing all particles with energy higher than ∼ |μ|. This leads to a distribution that drops abruptly above a momentum scale Q (see Fig. 2). If the corresponding energy is on the order of the ground-state energy of the post-quench fully condensed gas, (Q)2 /2m  |μ|  gρ, with g = 4π2 a/m, with scattering length a and atom mass m, then the energy of the entire gas after the quench is concentrated at the scale Q  kξ , with healing√ length momentum scale kξ = 8πaρ. Most importantly, such a strong cooling quench leads to an extreme initial condition for the subsequent dynamics. The post-quench distribution is strongly over-occupied at momenta k < Q, as compared to the final equilibrium distribution. This initial overpopulation of modes with energies ∼ (Q)2 /2m induces inverse particle transport to lower momenta while energy is transported to higher wavenumbers, 34,56,57 as indicated by the arrows in Fig. 2. The rescaling is thus characterized by a bi-directional, in general non-local redistribution of particles and

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energy. Furthermore, in contrast to the case of a weak quench leading to a scaling evolution in which, typically, weak wave turbulence is induced, 55,58 here the inverse transport is characterized by a different, strongly non-thermal power-law form of the scaling function in the infrared (IR) region. While the spatio-temporal scaling provides the “smoking gun” for the approach of a non-thermal fixed point, in all cases examined so far, this steep power-law scaling of the momentum distribution, n(k) ∼ k −ζ , has been observed and reflects the character of the underlying transport, see Fig. 2. The evolution during this period is universal in the sense that it becomes mainly independent of the precise initial conditions set by the cooling quench as well as of the particular values of the physical parameters characterizing the system. In the vicinity of the non-thermal fixed point, the momentum distribution of the Bose gas rescales self-similarly, within a certain range of momenta, according to n(k, t) = (t/tref )α n([t/tref ]β k, tref ), with some reference time tref . The distribution shifts to lower momenta for β > 0, while transport to larger momenta occurs in the case of β < 0. A bi-directional scaling evolution is, in general, characterized by two different sets of scaling exponents. One set describes the inverse particle transport towards low momenta whereas the second set quantifies the transport of energy towards large momenta. Global conservation laws — applying within a certain, extended regime of momenta — strongly constrain the redistribution underlying the self-similar dynamics in the vicinity of the non-thermal fixed point. Hence, they play a crucial role for the possible scaling evolution as they impose scaling relations between the scaling exponents. For example, particle number conservation in the infrared regime of long wavelengths, in d spatial dimensions, requires that α = dβ. The resulting transport in momentum space can emerge from rather different underlying physical configurations and processes. For example, the dynamics can be driven by the conserved redistribution of quasiparticle excitations such as in weak wave turbulence 34,55 but also by the reconfiguration and annihilation of (topological) defects populating the system. 56,59 The latter dynamics is often connected to the concept of superfluid turbulence, see Sec. 5. If defects are subdominant or absent at all, e.g., in multi-component systems, the strongly occupied modes exhibiting scaling near the fixed point 34,55 typically reflect strong phase fluctuations not subject to an incompressibility constraint. They can be described by the lowenergy effective theory discussed in Sec. 4. 60 The associated scaling exponents are generically different for both types of dynamics, with and without defects. 34,59,61 The existence and significance of strongly non-thermal momentum power-laws, requiring a non-perturbative description reminiscent of wave turbulence was proposed by Rothkopf, Berges and collaborators in the context of reheating after early-universe inflation, 33,62 generalized by Scheppach, Berges, and Gasenzer to scenarios of strong matterwave turbulence, 63 and to the case of topological defects by Nowak, Sexty, Erne, Gasenzer et al., 56,61,64–66 see also Refs. 59, 67–72.

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Fig. 3. Schematics of prethermalization 12,13,83 (left) as compared to non-thermal fixed points (right) based on the ideas of a renormalization group flow. The panels depict possible time evolutions in a sub-manifold of the space of many-body states. The arbitrarily chosen axes are given by running ‘coupling’ parameters. These are related, e.g., to mass and coupling parameters in an effective Lagrangian, or to the various Lagrange multipliers of a generalized Gibbs ensemble (GGE). During the time evolution, a system can quickly approach a prethermalized state (red line). Due to conservation laws such a state retains memory of the initial conditions and then slowly drifts to the thermal fixed point (green). If, in contrast, a non-thermal fixed point is approached (right panel), prescaling, associated with partial fixed points can quickly arise, cf. the discussion in Sec. 7. The prescaling is associated with a symmetry being conserved during the flow while other symmetries are already broken. Depending on the particular choice of initial condition, different, non-universal early-time evolutions (blue trajectories) occur before the system enters the universal scaling regime. Along the red line, correlation functions C(k, t) can show approximate scaling behavior in space and time according to C(k, t) = tα f (tβ k), with a universal scaling function f . In case of prethermalization, one expects α = β = 0 ( 0) such that the system gets (almost) stuck when reaching the red line. Left figure taken from Ref. 15.

Universal scaling at a non-thermal fixed point in both space and time was studied by Pi˜ neiro-Orioli, Boguslavski, and Berges for relativistic and non-relativistic O(N )-symmetric models, 34,73 see also Refs. 74 and 75, and discussed in the context of heavy-ion collisions 4,53,76,77 as well as axionic models. 72 At this point we remark that the concept of non-thermal fixed points includes scaling dynamics which exhibits coarsening and phase-ordering kinetics 78 following the creation of defects and non-linear patterns after a quench, e.g., across an ordering phase transition. We emphasize, however, that coarsening and phase-ordering kinetics in most cases are being discussed within an open-system framework, considering the system to be coupled to a heat bath. Moreover, most theoretical treatments of these phenomena do not take non-linear dynamics and transport into account. A common property of the universal evolutions is scaling behavior with evolution time as scaling parameter. The associated scaling is reminiscent of equilibrium criticality at a continuous phase transition. 51,52,79 The system rescales in space with some power of the evolution time, which looks like zooming in or out the field of view of a microscope in real time. To a certain extent, slowed-down dynamics and scaling in the evolution time can be seen as analogues of the universality in equilibrium critical phenomena in non-equilibrium systems. 50,78,80–82

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Prethermalization 13,15,84 is another transient out-of-equilibrium phenomenon which is closely related to non-thermal fixed points and has originally been described using ideas from renormalization-group theory, 11–13,85 see Fig. 3 for an illustration. During the prethermalization stage, a system approaches a (partially) universal intermediate state that is, in general, still out of equilibrium with respect to asymptotically long evolution times. The emerging intermediate state is characterized by conservation laws relevant for the observable considered. Universality of the state allows for a mathematical description in terms of a limited set of parameters and/or functions which solely depend on the corresponding set of symmetries obeyed in the time evolution. Mode occupancies usually become quasi-stationary during prethermalization such that they show a trivial time evolution ∼ t0 . In this case one expects the associated scaling behavior of correlations to be governed by the universal scaling exponents α = β = 0 ( 0). We emphasize, though, that for prethermalization, as opposed to the approach of a non-thermal fixed point, one typically does not have to take into account non-linear transport between different momentum excitations. The characteristics of a universal intermediate configuration during the prethermalization stage are independent of the particular physical realization and of the specific initial condition. In case of a state being only partially universal, the dynamics is dominated by the universal characteristics while non-universal properties remain. Such non-universal properties can depend on the particular initial state of the system. The generalized Gibbs ensemble (GGE) 16–20 is an example for a partially universal state occurring during the prethermalization stage. A GGE contains a limited set of conserved operators which are directly related to intrinsic symmetries of the Hamiltonian describing the system under consideration. However, the values of the Lagrange multipliers associated with the conserved quantities are non-universal as they depend on the initial values of these quantities. The GGE is a direct generalization of thermodynamical ensembles. If only the total energy and the particle number are conserved, it reduces to the grand-canonical ensemble with Lagrange multipliers given by the temperature and the chemical potential. For quantum integrable systems it is believed that the GGE is the final state of relaxation, 6,10,86–89 while it has also been discussed in the context of wave-turbulent scaling evolution. 90 3. Kinetic theory of non-thermal fixed points In the previous section we have introduced the basic ideas of non-thermal fixed points. For studying them theoretically we can make use of analytical as well as numerical tools. In this section we will focus on an analytical approach to describe the scaling behavior at such fixed points. For more technical descriptions at various levels of detail see Refs. 55, 75, 91 and 92. For a general analytical treatment of non-thermal fixed points we need to be able to calculate the time evolution of a quantum many-body system out-of equilibrium.

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Suitable techniques are provided by the framework of non-equilibrium quantum field theory (QFT). Using a path integral formulation, all information about the time-evolving quantum system is contained in the so-called Schwinger–Keldysh nonequilibrium generating functional. 91 Correlation functions, which show universal scaling at a non-thermal fixed point, can be obtained by functional differentiation of the generating functional with respect to corresponding sources. To calculate such observables at some instant in time, the system is evolved along a Schwinger– Keldysh closed time path which reflects the nature of non-equilibrium QFT as an initial value problem. This is in contrast to equilibrium QFT, where only asymptotic input and output states are used. The initial configuration of the out-of equilibrium system is contained in the initial density matrix which enters the generating functional. In the majority of cases it is sufficient to choose the initial density matrix to be Gaussian. Calculating the non-equilibrium generating functional in its most general formulation is highly non-trivial. To study the universal scaling behavior at non-thermal fixed points we focus on the evolution of two-point correlators. From these, e.g., (quasi)particle occupation numbers in momentum space can be derived if the system is, on average, spatially translation invariant. Taking the Schwinger–Keldysh description one derives dynamical equations for unequal-time two-point correlators, called Kadanoff–Baym equations. 43 These equations describe the non-equilibrium dynamics exactly but are as non-trivial to solve as the computation of the non-equilibrium generating functional entering these equations is. One is thus held to reduce the complexity of the problem and to obtain approximate dynamical equations that are capturing the physics relevant at a non-thermal fixed point. It turns out that a kinetic theory approach provides such an approximation, see, e.g., Refs. 75, 91 and 93. Below we will briefly outline, following Ref. 55, how to proceed from the Kadanoff–Baym equations in order to derive a kinetic equation that governs the scaling behavior at a non-thermal fixed point. As a first step, the two-point correlators are decomposed into a symmetric and asymmetric part. For bosons, the symmetric part, which is termed the statistical function F , is given by the anticommutator of the field operators at two points in space and time, whereas the spectral function ρ, defined by the commutator, represents the antisymmetric part. To obtain a kinetic equation it is necessary to introduce a quasiparticle Ansatz for the spectral function. In a next step a gradient expansion of the Kadanoff–Baym equations with respect to the evolution time is performed. As universal scaling is reminiscent of a loss of memory about the initial condition, we can formally put the initial time to minus infinity and thus forget about contributions to the equations coming from the initial state. Taking the gradient expansion to leading order and in an equal-time limit, we obtain a Boltzmann-type kinetic equation for the time evolution of the (quasi)particle occupation number. This equation is termed generalized quantum Boltzmann equation (QBE).

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Using a kinetic-theory description enables us to perform a scaling analysis from which the scaling exponents associated with the non-thermal fixed point can be predicted analytically. To outline this analytical approach we consider an N -component homogeneous Bose gas, with O(N ) × U (1)-symmetric interactions, given by the Gross–Pitaevskii Hamiltonian    ∇2 g Φa + Φ†a Φ†b Φb Φa . (1) H = dd x −Φ†a 2m 2 Here, the time and space dependent fields Φa ≡ Φa (x, t), a = 1, ..., N , satisfy Bose equal-time commutation relations, m is the mass of the bosons and the contact interaction is quantified by a single coupling g = 4π2 a/m, defined in terms of the s-wave scattering length a. Note that we use units where  = 1 and that it is summed over the Bose fields according to Einstein’s sum convention. For simplicity of the notation, field indices are suppressed in the following. Within kinetic  † theory, theobject of interest is the occupation number distribution n(k, t) = Φ (k, t)Φ(k, t) . As already mentioned above, the time evolution of n(k, t) is described in terms of a generalized Quantum Boltzmann equation (QBE) ∂t n(k, t) = I[n](k, t),

(2)

where I[n](k, t) is a scattering integral that takes the form  I[n](k, t) = |Tkpqr |2 δ(k + p − q − r) δ(ωk + ωp − ωq − ωr ) pqr

× [(nk + 1)(np + 1)nq nr − nk np (nq + 1)(nr + 1)].



(3)

Here, Tkpqr is the scattering T-matrix and the short-hand notation p ≡  d d p (2π)−d was introduced. The scattering integral (3) describes the redistribution of the occupations nk of momentum modes k with eigenfrequency ωk due to elastic 2 → 2 collisions. In presence of a Bose condensate, the occupation numbers describe quasiparticle excitations. This modifies the scattering matrix and the mode eigenfrequencies. Here, we consider transport entirely within the range of a fixed scaling of the dispersion ωk ∼ k z , with dynamical scaling exponent z, such that processes leading to a change in particle number are suppressed. We capture collective scattering effects beyond 2 → 2 scattering in the T-matrix (see Subsec. 3.1). Two classical limits of the QBE scattering integral I[n](k, t) exist. The usual Boltzmann integral for classical particles is obtained in the limit of n(k, t)  1. In case of large occupation numbers n(k, t)  1, termed classical-wave limit, the scattering integral reads  |Tkpqr |2 δ(k + p − q − r) δ(ωk + ωp − ωq − ωr ) I[n](k, t) = pqr

× [(nk + np )nq nr − nk np (nq + nr )].

(4)

The QBE reduces to the wave-Boltzmann equation (WBE) which is subject of the following discussion as we are interested in the universal dynamics of a neardegenerate Bose gas obeying n(k, t)  1 within the relevant momentum regime.

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3.1. Properties of the scattering integral and the T-matrix Scaling features of the system at a non-thermal fixed point are directly encoded in the properties of the scattering integral. For a general treatment that governs the cases of presence and absence of a condensate density, we focus on the scaling of the distribution of quasiparticles, in the following denoted by nQ (k), instead of the single-particle momentum distribution n(k). Note that in case of free particles, with dispersion ω(k) = k 2 /2m ∼ k z , i.e. dynamical exponent z = 2, we obtain nQ ≡ n. For Bogoliubov sound with dispersion ω(k) = cs k and thus z = 1, the scaling of nQ differs from the scaling of n due to the k-dependent Bogoliubov mode functions characterizing the transformation between the particle and quasiparticle basis, n(k)  (gρ0 /cs k)nQ (k), for k → 0, in general n(k) ∼ k z−2+η nQ (k), with anomalous exponent η. Here, cs denotes the speed of sound of the Bogoliubov excitations and ρ0 is the condensate density. Using a positive, real scaling factor s, the self-similar evolution of the quasiparticle distribution at a non-thermal fixed point reads  (5) nQ (k, t) = sα/β nQ sk, s−1/β t . We remark that by choosing the scaling parameter s = (t/tref )β we obtain the scaling form stated in the example in Sec. 2. As the scattering integral, in the classical-wave limit, is a homogeneous function of momentum and time, it obeys scaling, provided the scaling of the quasiparticle distribution in (5), according to I[nQ ](k, t) = s−μ I[nQ ](sk, s−1/β t),

(6)

with scaling exponent μ = 2(d + m) − z − 3α/β. Here, m is the scaling dimension of the modulus of the T-matrix |T (k, p, q, r; t)| = s−m |T (sk, sp, sq, sr; s−1/β t)|.

(7)

At a fixed instance in time, the T-matrix can have a purely spatial momentum scaling form. Consider a simple example of a universal quasiparticle distribution at a fixed time t0 , which, at least in a limited regime of momenta, shows power-law scaling, nQ (sk) = s−κ nQ (k),

(8)

with fixed-time momentum scaling exponent κ. The T-matrix is then expected to scale as |T (k, p, q, r; t0 )| = s−mκ |T (sk, sp, sq, sr; t0 )|,

(9)

with mκ being, in general, different from m. Note that Eq. (8) in realistic cases is regularized by an IR cutoff kΛ or, respectively, a UV cutoff kλ to ensure that the scattering integral stays finite in the limit k  kΛ or k  kλ . Generally, the scaling hypothesis for the T-matrix, Eq. (7), does not hold over the whole range of momenta. In fact, scaling, with different exponents, is found within separate limited scaling regions which we discuss in the following.

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= (a)

(b)

99

+ (c)

Fig. 4. Graphical representation of the resummation scheme. (a) The two lowest-order diagrams of the loop expansion of the two-particle irreducible effective action (or Φ functional) which lead to the Quantum–Boltzmann equation and thus to the bare coupling g within the perturbative region. Solid lines represent the propagator G(x, y), black dots the bare vertex ∼ gδ(x − y). (b) Diagram representing the resummation approximation which replaces the diagrams in (a) within the IR regime of momenta and gives rise to the modified scaling of the T -matrix. (c) The wiggly line is the effective coupling function entering the T -matrix, which corresponds to a sum of bubble-chain diagrams. Figure taken from Ref. 55.

Perturbative region: Two-body scattering For the non-condensed, weakly interacting Bose gas away from unitarity the T matrix is well approximated by |Tkpqr |2 = (2π)4 g 2 .

(10)

As the matrix elements are momentum independent we obtain mκ = m = 0. It can be shown that Eq. (10) represents the leading perturbative approximation of the full momentum-dependent many-body coupling function. In presence of a condensate density ρ0 ≤ ρ, sound wave excitations become √ relevant below the healing-length momentum scale kξ = 2gρ0 m. Within leadingorder perturbative approximation, the elastic scattering of these sound waves is described by the T-matrix |Tkpqr |2 = (2π)4

(mcs )4 3g 2 . kpqr 2

(11)

Here, √ the√speed of sound of the quasiparticle excitations cs is given by mcs = kξ / 2 = gρ0 m. For the Bogoliubov sound we obtain the scaling exponents mκ = m = −2. The above perturbative results are in general applicable to the UV range of momenta. However, scaling behavior in the far IR regime, where the momentum occupation numbers grow large, requires an approach beyond the Boltzmann, leading-order perturbative approximation as perturbative contributions to the scattering integral of order higher than g 2 are no longer negligible. Collective scattering: Non-perturbative many-body T -matrix To do so, we use a non-perturbative s-channel loop resummation derived within a quantum-field-theoretic approach based on the two-particle irreducible (2PI) effective action or Φ-functional. The resummation procedure is schematically depicted in Fig. 4. It is equivalent to a large-N approximation at next-to-leading order and enables to calculate an effective momentum-dependent coupling constant geff (k)

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Fig. 5. Left panel: Effective coupling geff (k0 , k)/g in d = 3 dimensions as a function of the spatial momentum k = |k|, on a double-logarithmic scale. The graph shows cuts along k0 = 0.5εk (dark solid lines) and k0 = 1.5εk (transparent solid lines), where εk = |k|2 /2m. Different colors correspond to different IR cutoffs kΛ which are set by the scaling form of the occupation number distribution entering the non-perturbative coupling function. All momenta are measured in units of the ‘healing’-length momentum scale kΞ = (2gρnc m)1/2 of the non-condensed particle density ρnc under n(k). kΞ sets the scale separating the perturbative region at large momenta from the non-perturbative collective-scattering region within which the coupling assumes the form given in Eq. (14). Right panel: Contour plot of the effective coupling function geff (k0 , k)/g as a function of E = 2mk0 and momentum k = |k|. The data is depicted for kΛ = 10−3 kΞ . The two cuts shown in the left panel correspond to the black dashed lines. The quasiparticle distribution nQ (k) ≡ n(k) was chosen to scale with κ = 3.5 such that we are in a regime where the effective coupling assumes the universal scaling form (14). Figures adapted from Ref. 55.

which replaces the bare coupling g. The effective coupling also changes the scaling exponent m of the T-matrix within the IR regime of momenta. In particular, geff (k) becomes suppressed in the IR to below its bare value g. This ultimately leads to an even steeper rise of the (quasi)particle spectrum. For free particles (z = 2) in d = 3 dimensions we obtain 2 (εk − εr , k − r), |Tkpqr | = (2π)4 geff

(12)

where εk − εr and k − r are the energy (εk = |k|2 /2m) and momentum transfer in a scattering process, respectively. The resulting momentum-dependent effective coupling function geff (k0 , k) along two exemplary cuts k0 = 0.5εk and k0 = 1.5εk in frequency-momentum space, for three different IR cutoffs kΛ , is shown in the left panel of Fig. 5. At large momenta, the effective coupling is constant and agrees with the perturbative result, i.e., one finds geff = g. However, below the characteristic momentum √ scale kΞ = 2gρnc m, the effective coupling deviates from the bare coupling g. Within a momentum range of kΛ  k  kΞ ,

(13)

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the effective coupling is found to assume the universal scaling form  2  ε − k02  k , (κ > 3) geff (k0 , k)  2ρnc εk

101

(14)

independent of both, the microscopic interaction constant g, and the particular value of the scaling exponent κ of nQ . Here, ρnc = ρtot − ρ0 denotes the noncondensed particle density. Below the IR cutoff, i.e., for momenta k < kΛ , the effective coupling becomes constant again. The dependence of the scaling form (14) on E = 2mk0 and k is visualized in the right panel of Fig. 5. We remark that the simple universal form of the effective coupling (14) only requires a sufficiently steep power-law scaling of the quasiparticle distribution nQ (k) ∼ k −κ and an IR regularization kΛ ensuring finite particle number. Making use of the scaling properties of the effective coupling, geff (k0 , k) = s−γκ geff (sz k0 , sk) ,

(15)

we obtain γκ = 0 in the perturbative regime and γκ = 2 in the collective-scattering regime for free particles with z = 2. In combination with Eq. (12) we find the corresponding scaling exponent of the T-matrix to be mκ = 2. The same analysis of the effective coupling can be performed for the Bogoliubov dispersion with z = 1. In contrast to free particles the scaling exponent of the T-matrix reads mκ = 0, see Ref. 55 for details. 3.2. Scaling analysis of the kinetic equation We are now in the position to determine the scaling properties of the Bose gas at a non-thermal fixed point. Here, we focus on the case of a bi-directional self-similar evolution as obtained after performing a strong cooling quench, recall the example introduced in Sec. 2. For a detailed discussion of the scaling behavior occurring after weak cooling quenches, where only a few of the high-energy particles in the thermal tail are removed from the system, we also refer to Ref. 55. To quantify the momentum exponent κ leading to a bi-directional scaling evolution we study the scaling of the quasiparticle distribution at a fixed evolution time as stated in Eq. (8). As the density of quasiparticles  dd k nQ (k) (16) ρQ = (2π)d and the energy density

 Q =

dd k ωQ (k)nQ (k) (2π)d

(17)

are physical observables, they must be finite. We assume that the momentum distribution is isotropic, i.e., nQ (k) ≡ nQ (k) and given by a bare power-law scaling nQ ∼ k −κ . The exponent κ then determines whether the IR or the UV regime dominates quasiparticle and energy densities. For a bi-directional self-similar evolution

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the quasiparticle density has to dominate the IR and the energy density the UV, due to their different scaling with k. Hence the scaling exponent κ has to fulfill d ≤ κ ≤ d + z.

(18)

Note that the quasiparticle distribution requires regularizations in the IR and the UV limits in that case as already introduced before in terms of kΛ and kλ , respectively. According to the scaling hypothesis the time evolution of the quasiparticle distribution is captured by Eq. (5), with universal scaling exponents α and β. Global conservation laws strongly constrain the form of the correlations in the system and the ensuing dynamics. Hence, they play a crucial role for the possible scaling phenomena as they imply scaling relations between the exponents α and β. Conservation of the total quasiparticle density, Eq. (16), requires α = dβ.

(19)

Analogously, if the dynamics conserves the energy density, Eq. (17), the relation α = (d + z)β

(20)

must be fulfilled. Obviously, the scaling relations (19) and (20) cannot both be satisfied for nonzero α and β if z = 0. This leaves us with two possibilities: Either α = β = 0 or the scaling hypothesis (5) has to be extended to allow for different rescalings of the IR and the UV parts of the scaling function. In the following we denote IR exponents with α, β and UV exponents with α , β  respectively. Making use of the global conservation laws as well as of the power-law scaling of the quasiparticle distribution, nQ ∼ k −κ , one finds the scaling relations α = dβ,

(21)

β  (d + z − κ) = β(d − κ).

(22)

This implies ββ  ≤ 0, i.e., the IR and UV scales kΛ and kλ rescale in opposite directions. We remark that these relations hold in the limit of a large scaling region of momenta, i.e., for kΛ  kλ . Note that energy conservation only affects the UV shift with exponent β  , Eq. (22), while particle conservation gives the relation Eq. (21) for the exponent β in the IR. With this at hand we are finally able to derive analytical expressions for the scaling exponents based on the kinetic theory approach. Performing the s-channel loop-resummation, the effective coupling geff can be expressed by the retarded oneloop self-energy ΠR , which is defined in terms of the statistical and spectral function encoding the mode occupations and, respectively, the dispersion relation as well as the density of states of the system. The aforementioned anomalous dimension η appears as a scaling dimension of the spectral function. The particle and quasiparticle distributions are obtained by frequency integrations over the statistical function.

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The resulting, most general scaling relations for the (quasi)particle distributions then read  (23) nQ (k, t) = sα/β nQ sk, s−1/β t , nQ (k, t0 ) = sκ nQ (sk, t0 ) ,  n(k, t) = sα/β−η+2−z n sk, s−1/β t , n(k, t0 ) = sκ−η+2−z n (sk, t0 ) = sζ n (sk, t0 ) .

(24) (25) (26)

To show possible differences in the scaling behavior of the particle and quasiparticle distributions we added the relations for the particle distribution which scales as n(k) ∼ k z−2+η nQ (k) relative to the quasiparticle number, see beginning of Subsec. 3.1. Note that the momentum scaling of n(k) is characterized by the scaling exponent ζ according to Eq. (23). Zeroes of the scattering integral in the kinetic equation correspond to fixed points of the time evolution. From a scaling analysis of the QBE one obtains the scaling relation α = 1 − βμ.

(27)

Making use of the scaling of the T-matrix within the different momentum regimes as well as the global conservation laws of the system, one finds the scaling exponents by means of simple power counting to be α = d/z, α = β  (d + z),

β = 1/z,

β  = β(3z − 4 + 2η)(z − 4 − 2η)−1 ,

κ = d + (3z − 4)/2 + η,

ζ = d + z/2.

(28) (29) (30)

On the grounds of numerical simulations in Ref. 94, the IR scaling exponent β = 1/z has been proposed. Note that the exponents stated in Eq. (29) are usually not observed as the UV region is dominated by a near-thermalized tail. During the early-time evolution after a strong cooling quench, an exponent ζ  d + 1 was seen in semi-classical simulations for d = 3 in Refs. 34 and 56, for d = 2 in Ref. 65, and for d = 1 in Ref. 66. Numerically evaluating the kinetic equation in d = 3 dimensions also resulted in κ  4, see Ref. 95. For a single-component Bose gas in d = 3 dimensions, the IR scaling exponents have recently been numerically determined to be α = 1.66(12), β = 0.55(3), in agreement with the analytically predicted values. 34 For the Bose gas, the above stated exponents are expected to be valid in d = 3 dimensions as well as in d = 2. The one-dimensional case is rather different due to kinematic constraints on elastic 2 → 2 scattering from energy and particle-number conservation. We finally remark that the development of non-linear and topological excitations in combination with strong phase coherence is likely to modify the results presented, potentially through an appropriate modification of the scaling exponents z and η.

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4. Low-energy effective field theory In the previous section, collective phenomena that modify the properties of the scattering matrix were taken into account by means of a non-perturbative coupling resummation scheme. Alternatively, one can think of the idea to reformulate the theory in terms of new degrees of freedom in the first place, such that the resulting description becomes more easy to treat in non-perturbative regions. Since the nonperturbative behavior appears at low momentum scales, it is suggestive to use a lowenergy effective field theory (LEEFT) approach. 96,97 This typically implies a choice of suitable degrees of freedom describing the physics occurring below a chosen energy scale. Classical examples of low-energy effective field theories include the Fermi theory of β-decay, 98 the BCS theory of superconductivity 99,100 and the XY-model of superfluidity. 101 Even approaches to quantum gravity can be made using low-energy effective field theories. 102 In the following, we will outline the (Wilsonian) LEEFT approach to the description of non-thermal fixed points in a multicomponent Bose gas. 60 The ideas are based on the treatment of the aforementioned XY-model. The key observation is, that the N -component Gross–Pitaevskii model, see Eq. (1), offers a natural separation of scales. An analysis of the classical equations of motion, obtained within a density-phase representation of the field, Φa = √ (0) ρa exp{θa }, shows that, at low momenta, density fluctuations δρa = ρa − ρa (0) around a mean density ρa are suppressed by a factor of ∼ |k|/kΞ compared to phase fluctuations θa (around a constant background phase). Here, kΞ = [2mρ(0) g]1/2 is " (0) the healing-length momentum scale associated with the total density ρ(0) = a ρa . Hence, density fluctuations can be integrated out to obtain the low-energy effective action Seff of the system. Furthermore, the model provides two types of eigenmodes: N − 1 Goldstone excitations with a free-particle-like dispersion ω1 (k) = · · · = ωN −1 (k) = k2 /2m, which correspond to relative phases between different components, and a single 1/2   related Bogoliubov quasiparticle mode with ωN (k) = k2 /2m k2 /2m + 2gρ(0) to the total phase. This suggests that the physics below the scale kΞ is well-described by the dynamics of phonon-like quasiparticles, although two sorts of quasiparticles are present. The low-energy effective action corresponding to these quasiparticles contains interaction terms with momentum-dependent couplings indicating the fact that the resulting theory is non-local in nature, as is expected for a LEEFT. 60 Moreover, taking the large-N limit, this action becomes diagonal in component space up to O(1/N ) corrections and thus breaks up into N independent replicas. This means that the phases θa of the different components decouple in the limit of large N . Taking the limit N → ∞, the Bogoliubov mode is no longer present suggesting that relative phases are dominating the dynamics of the system. The N → ∞ effective

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action in momentum space is found to be 60  1 −1 θa (k, t)iDab Seff [θ] = (k, t; k , t )θb (k , t )  k,k ,C 2  3  k1 · k2 − θa (k1 , t) θa (k2 , t)∂t θa (k3 , t) δ ki {ki },C 2mN g1/N (k3 ) i=1  4   (k1 · k2 ) (k3 · k4 ) θ (k , t) · · · θ (k , t) δ ki . + a 1 a 4 2 {ki },C 8m N g1/N (k1 − k2 ) i=1

105

(31)

Here, C indicates the integration over a Schwinger–Keldysh contour and Dab is a free inverse propagator −1 (k, t; k , t ) = iDab

  (2π)d δ(k + k ) δab δC (t − t ) −∂t2 − (k2 /2m)2 . N g1/N (k)

(32)

In the above expression, the momentum-depending coupling g1/N (k) = gk2 /2kΞ2 ≡ gG (k)/N was introduced, which remarkably coincides with the universal coupling obtained in the non-perturbative resummation within the 2PI formalism, 55 see Sec. 3. The index G of the coupling refers to the relevant Goldstone excitations in the large-N limit.

4.1. Spatio-temporal scaling To analyze the scaling behavior at a non-thermal fixed point we proceed as in Sec. 3 by evaluating the QBE in Eq. (2). Instead of the quasiparticle distribution nQ we consider the distribution of phase-excitation quasiparticles fa (k, t) = θa (k, t)θa (−k, t) . We again drop the indices in the following to ease the notation. The scattering integral has two contributions arising from 3- and 4-wave interactions in the effective action action (31), I[f ](k, t) = I3 (k, t) + I4 (k, t) .

(33)

The form of the 3- and 4-point scattering integrals can be inferred from the effective action to be  |T3 (k, p, q)|2 δ(k + p − q) δ(ωk + ωp − ωq ) I3 (k, t) ∼ p,q   × (fk + 1)(fp + 1)fq − fk fp (fq + 1) , (34)  I4 (k, t) ∼ |T4 (k, p, q, r)|2 δ(k + p − q − r) δ(ωk + ωp − ωq − ωr ) p,q,r   × (fk + 1)(fp + 1)fq fr − fk fp (fq + 1)(fr + 1) , (35)

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where the corresponding T -matrices are given by gG (k) gG (p) gG (q) , 8 ω(k) ω(p) ω(q) gG (k) · · · gG (r) |T4 (k, p, q, r)|2 = |λ(k, p, q, r)|2 , 2ω(k) · · · 2ω(r) |T3 (k, p, q)|2 = |γ(k, p, q)|2

(36) (37)

with interaction couplings (k · p) ω(q) + perms , m gG (q) (k · p)(q · r) + perms . λ(k, p, q, r) = 2m2 gG (k − p) γ(k, p, q) =

(38) (39)

Here, ‘perms ’ denote permutations of the sets of momentum arguments. The scattering integrals scale, analogously to Eq. (6), with exponents μ3 = d + 4 − 2z + γ − 2α/β,

(40)

μ4 = 2d + 8 − 5z + 2γ − 3α/β,

(41)

where γ = 2(z − 1) is the scaling exponent of the effective coupling geff (k) = s−γ geff (sk). We remark that the subscript of the coupling is chosen as a general notation covering both cases of z = 2 as well as z = 1. Using the scaling relation in Eq. (27) one can, in principle, derive a closed system of equations allowing to determine the scaling exponents α and β. However, since, for different values of the dimensionality d and the momentum scale of interest, one term in the scattering integral can dominate over the other one, it is more reasonable to analyze them independently. To close the system of equations, an additional relation is then required, which can be provided by either quasiparticle number conservation, Eq. (19), or energy conservation, Eq. (20), within the scaling regime. Taking these constraints into account we obtain l=3: l=4:

1 1 , β = , 4 − 2z + γ 4 − 3z + γ 1 1 β= , β = . 8 − 5z + 2γ 8 − 7z + 2γ β=

(42) (43)

In the large-N limit (z = 2, γ = 2), the resulting scaling exponents read β = 1/2,

α = d/2

(44)

for both, 3- and 4-point vertices, and β  = −1/2,

α = −(d + z)/2

(45)

for the 4-point vertex, while, at the same time, for the 3-point vertex, no valid solution exists. We point out that the above exponents are equivalent to the respective exponents derived in the large-N resummed kinetic theory for the fundamental Bose fields, for the case of a dynamical exponent z = 2, and a vanishing anomalous dimension η = 0, cf. Subsec. 3.2.

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One can ask whether both 3- and 4-wave interactions are equally relevant. To answer this question, a comparison of the spatio-temporal scaling properties of the scattering integrals, for a given fixed-point solution f (k, t), is required. Focusing on the conserved IR transport of quasiparticles, for which α = dβ, we obtain −μ3 = d − 2,

(46)

−μ4 = d − 4 + z.

(47)

In the large-N limit, for which z = 2, one finds μ3 = μ4 . Hence, the relative importance of the scattering integrals I3 and I4 should remain throughout the evolution of the system. 4.2. Scaling solution In the following we briefly discuss the purely spatial momentum scaling. The scaling of the QBE at a fixed evolution time t = t0 implies κ = −μκ,l , where μκ,l is the spatial scaling exponent of the corresponding scattering integral, Il (k, t0 ) = s−μκ,l Il (sk, t0 ). Power-counting of the scattering integrals, together with the above stated scaling relation, gives κ3 = −μκ,3 = 4 + d + γ − 2z,

(48)

κ4 = −μκ,4 = 4 + d + γ − 5z/2.

(49)

For a given κl , and assuming the large-N limit (z = 2 and γ = 2), one finds that μκ,3 − μκ,4 = κl − d ≥ 1.

(50)

Hence, the 4-wave scattering integral is expected to dominate at small momenta, k → 0. This implies that, at the non-thermal fixed point, the quasiparticle distribution f (k, t) ∼ k −κ is characterized by the momentum scaling exponent κ = κ4 = d + 1. The result appears to contradict the previous analysis of the spatio-temporal scaling, which, in the large-N limit, showed equal importance of I3 and I4 . We emphasize, however, that the scaling exponents α and β corresponding to the spatio-temporal scaling properties are obtained from relations which are independent of the precise form of f (k, t) but only require the scaling relation f (k, t) = (t/tref )α f ([t/tref ]β k). Hence, the questions which vertex is responsible for the shape of the scaling function and which of the vertices dominates the transport can be answered independently of each other. See Ref. 60 for further discussion. 4.3. The case of a single-component gas, N = 1 While originally being derived for the limiting case of N → ∞, one can also apply the LEEFT to a single-component (N = 1) Bose gas. In this case, the theory describes, in the IR limit, the scattering of modes with linear Bogoliubov dispersion (z = 1 and γ = 0). The scaling exponent β then reads l=3:

β = 1/2,

(51)

l=4:

β = 1/3.

(52)

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In addition, the relation −μ3 = d−2 > −μ4 = d−3 is found. Hence, as the evolution time increases, the scattering integral I3 , for Bogoliubov-like quasiparticles, starts to win against I4 such that the value of the scaling exponent β is predicted to be β = 1/2. We can conclude then that transport of Bogoliubov quasiparticles towards the IR, dominated by 1 → 2 and 2 → 1 interaction processes, is described by the scaling exponents α = d/2 and β = 1/2. Following the same procedure as for z = 2 one can determine the fixed-time momentum scaling exponent κ. The analysis reveals that 4-wave interactions dominate such that κ = κ4 = d + 3/2. However, keeping in mind that I3 becomes more relevant as time increases, one rather expects, in the N = 1 case, the 3-wave interaction to dominate the purely spatial scaling fixed-point equation as well. Under these circumstances, the theory rather predicts the exponent κ = κ3 = d + 2

(z = 1)

(53)

to represent the momentum scaling in the long-time scaling limit. 4.4. Relation to predictions of the non-perturbative kinetic theory As already pointed out, the scaling exponents derived within the LEEFT remarkably coincide with earlier findings from non-perturbatively resummed kinetic theory. 55 In principle, however, there is a priori no reason for them to coincide since they correspond to different degrees of freedom. One needs therefore a translation between the quasiparticle distribution fa (k, t) characterizing the phase-angle excitations and the scaling of the particle number distribution na (k, t) encoded in the fundamental Bose field. Under the assumption that the non-thermal fixed point is Gaussian with respect to quasiparticle excitations in the phase degree of freedom, it can be shown that the scaling properties derived in the large-N limit, defined by the exponents z = 2, α = d/2, β = 1/2, and κ = d + 1, are consistent with the scaling properties derived within the non-perturbative approach, for the case of a dynamical exponent z = 2, and a vanishing anomalous dimension η = 0. While the idea that a non-thermal fixed point is Gaussian may sound odd in the first place, a simple scaling analysis shows that this can indeed be the case. The t Schwinger–Keldysh action S(t) = C,tref dt L(t ) integrated from a reference time tref to the present time t, according to the scaling hypothesis, should have a form S(t) = S(s1/β tref ) = s−dS S(tref ),

(54) β

with canonical scaling dimension dS = [S] and scale parameter s = (t/tref ) . Using the dynamical canonical scaling dimension of the phase-angle field θa at the non-thermal fixed point, [θa ] = −α/2β = −d/2, we obtain the canonical scaling of the quadratic, cubic and quartic parts of the effective action, [S (2) ] = 2z − γ − 1/β

= 0,

(55)

2[S (3) ] = d + 4 + 2z − 2γ − 2/β

= d + 4 − 2z ,

(56)

] = d + 4 − γ − 1/β

= d + 4 − 2z ,

(57)

[S

(4)

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where the 3-vertex was taken twice as it occurs in even multiples in any diagram contributing to the proper self-energy, and inserted, in the respective second equations, γ = 2(z − 1) and β = 1/2. If the non-thermal fixed point is Gaussian in the IR scaling limit, the conditions 2[S (3) ] > [S (2) ] ,

[S (4) ] > [S (2) ]

(58)

need to be fulfilled, which is the case in any dimension d > 0, as d + 4 − 2z > 0 for z < (d + 4)/2. The Gaussianity of the fixed point is also supported by the fact that the integrals Il [f ] scale to zero in the infinite-time limit, Il [f ](k, t) = (t/tref )(μκ,l −μl )β Il [f ](k, tref ). This can be inferred from their scaling exponents μl defined in Eqs. (46)–(49): For z = 2, one obtains μ3 = μ4 = 2 − d, −μκ,3 = d + 2, −μκ,4 = d + 1, such that (μκ,l − μl )β ≤ −3/2, independent of d. We remark that to improve upon the above performed analysis, time-dependent correlation functions need to be analyzed, e.g., within a functional renormalizationgroup approach. 62,103,104 Furthermore, we emphasize that Gaussianity of the nonthermal fixed point here refers to phase quasiparticles only, while in terms of the fundamental fields the fixed point can easily appear to be non-Gaussian. 5. Wave-turbulent transport In the previous sections we have discussed self-similar scaling dynamics at a nonthermal fixed point. Such dynamics is characterized by bi-directional, non-local transport of particles or energy leaving a global quantity such as particle or energy density invariant in time. This is reminiscent of the transport and scaling characterizing wave-turbulent cascades. In these cascades, analogously to fluid turbulence, universal scaling is expected in a certain interval of momenta, termed the inertial range. Within the inertial range of a wave-turbulent cascade, transport occurs locally, from momentum shell to momentum shell, leaving the transported quantity within such a momentum shell constant in time. This transport process can be described by a continuity equation in momentum space. In a dilute Bose gas, quantities other than the kinetic energy can be locally conserved in their transport through momentum space. In contrast to fluid turbulence, this is due to the compressibility of the gas which allows a variety of wave turbulence phenomena to arise. 29,30 Taking into account particle number and energy as alternative possible conserved quantities, the respective continuity equations characterizing the local conservation laws are written as 29 ∂t N (k, t) = −∂k Q(k),

(59)

∂t E(k, t) = −∂k P (k).

(60)

These continuity equations impose relations between the temporal change of a density and the momentum divergence of a current. In particular, Eq. (59) describes the time evolution of the radial particle number N (k) = (2k)d−1 πn(k) driven

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by the radial particle current Q(k) = (2k)d−1 πQk (k). The evolution of the radial energy E(k) = (2k)d−1 πω(k)n(k), Eq. (60), where ω(k) is the dispersion of mode excitations with momentum k, is determined by the radial energy current P (k) = (2k)d−1 πPk (k). Scaling solutions of the continuity equations are generally studied within a waveBoltzmann kinetic approach. Wave-turbulent scaling behavior is obtained if Q(k) and P (k) are independent of momentum k which corresponds to a stationary distribution of particles or energy, respectively. Weak wave turbulence is the mathematically best-controlled case. It is found within the perturbative region, i.e., in the UV regime of momenta, where the mode occupancies are sufficiently small such that the usual wave-Boltzmann equation is valid, meaning that the scattering T-matrix is solely given by the bare coupling g, see also Subsec. 3.1. One finds a direct relation between the continuity equations and the QBE in Eq. (2). 29 The theory of weak wave turbulence therefore rests on the analysis of stationary solutions of the QBE. By means of power counting the UV scaling exponents characterizing the weak wave turbulence are found to be 29,55 UV ζQ = d − 2/3,

ζPUV = d.

(61)

In general, the energy flux corresponds to a direct cascade to larger k, whereas the particle flux constitutes an inverse cascade. The character of the fluxes is entirely determined by the properties of the physical system. We emphasize that also the self-similar evolution at a non-thermal fixed point can be described by transport equations (59), (60), however, in general with a non-local flux such that the quantity being transported does not remain constant within a given momentum shell. Given a positive scaling exponent ζ, momentum occupation numbers n(k) ∼ k −ζ eventually grow large in the deep IR. In that limit, the flux enters the collective scattering region where the effective many-body T-matrix is given by a non-perturbative coupling geff (k) and weak wave turbulence ceases to be valid, see Ref. 55 for details. The concept of wave turbulence, as well as the description of spatio-temporal scaling near a non-thermal fixed point, are based on a wave-Boltzmann kinetic approach and thus on a quasiparticle Ansatz for the excitations in the system. While such an approach has been developed also for the IR regime of strong occupancies, cf. Secs. 3 and 4, it does not account for the effects of (quasi) topological excitations. These excitations are observed in many different systems such as single- and multicomponent dilute Bose gases in different dimensionalities, cf. the corresponding numerical results discussed in Sec. 6. Analytical predictions for the respective scaling exponents and functions derived from first principles are a subject of current research in the field. 6. Topological defects vs. the role of fluctuations While we have focussed, so far, on analytical treatments of non-thermal fixed points, we present, in the following, numerical simulations of dilute Bose gases,

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corroborating analytical predictions as well as showing phenomena beyond analytics. As pointed out in Sec. 2, universal scaling at a non-thermal fixed point can be driven by either (quasi) topological defects populating the system or by strong fluctuations of the phase, if defects are subdominant or absent. In the following we first present results obtained in vortex dominated single-component Bose gases. We will then show universal scaling at a non-thermal fixed point caused by relative-phase fluctuations in an (N = 3)-component Bose gas. As the scaling behavior occurs in a regime of strongly occupied modes, the time evolution can be computed by means of semi-classical simulations. Using the so-called truncated Wigner approximation, 105,106 we follow the evolution, starting from a noisy initial configuration, by evaluating many trajectories according to the classical equations of motion. In the simplest case of a single-component Bose gas, the equation of motion is the Gross–Pitaevskii equation   ∇2 2 + g|Φ(x, t)| Φ(x, t). (62) i∂t Φ(x, t) = − 2m Equation (62) can be mapped to a continuity equation for the density ρ = |Φ| and an Euler-type hydrodynamic equation for the superfluid velocity v = ∇ arg(Φ)/m such that the dynamics of the Bose gas can be interpreted in terms of superfluid flow. For positive g, possible solutions of this equation include topological configurations such as (dark) solitons and vortices. 107,108 Solitons are quasi-topological defects which in general travel with a fixed velocity but are nondispersive, i.e., stationary in shape and stable in d = 1 dimension. Vortices are topologically stable solutions in d > 1 dimensions which form the superfluid analogies of eddy flows in classical fluids. In d = 3 dimensions, point vortices extend to vortex lines or loops around which the fluid rotates. 2

6.1. Defect dominated fixed points To illustrate the scaling behavior of a vortex gas at a non-thermal fixed point, we consider the time evolution of an isolated two-dimensional Bose gas whose initial field configuration is chosen such that the condensate density in position space varies between zero and some maximum value. This can be achieved by macroscopically populating a few of the lowest momentum modes of the system. Vortices are then created within shock waves forming during the non-linear evolution of the coherent matter-wave field. 64 Alternatively, one can populate momentum modes up to a maximum scale kq , with the corresponding phases in each mode chosen randomly (so-called box initial condition) which also leads to the creation of vortices. 109 The approach of a non-thermal fixed point is marked by a self-similarly diluting ensemble of vortices and anti-vortices. From a turbulence point of view, the scaling behavior is characterized by an inverse cascade of particle excitations, possibly accompanied by a direct energy cascade towards the UV. During the time evolution the system runs through different stages, see Ref. 67 for details. On short time scales the dynamics is driven by scattering between the

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Fig. 6. Direct kinetic-energy and inverse particle fluxes, P (k) and Q(k), at an evolution time where a bimodal momentum distribution has emerged, see Ref. 65. Note the logarithmic k-axis. A positive kinetic-energy flux is seen in the UV, a negative particle flux in the IR. Figure taken from Ref. 67.

macroscopically occupied modes. In the next stage of the evolution, one observes strong phase and density gradients forming due to the non-linear evolution. Those phase gradients lead to the formation of vortices and anti-vortices. During the stage of unbinding those vortex–anti-vortex pairs and diluting the defects, the evolution slows down and the correlations evolve self-similarly. A bimodal distribution emerges characterizing the approach of the non-thermal fixed point, cf. Fig. 2. A UV exponent ζPUV  d = 2 indicates the weak-waveturbulence prediction in Eq. (61) to apply. In the IR, a steep power-law exponent IR ζQ  d + 2 = 4 is found, which agrees well with with the prediction for the Porod tail from the theory of phase ordering kinetics. 78 The power law arises from the algebraic fall-off of the superfluid velocity |v| ∼ 1/r with distance r from the core of a single vortex. The Porod law also indicates correlations in the distances between the defects which in the above case are randomly distributed. It appears at momenta larger than the mean inverse distance between vortices and anti-vortices and smaller than the inverse core size. At late times, after the last vortical excitations have disappeared, the entire spectrum becomes thermal, i.e., exhibits the standard Rayleigh–Jeans scaling, n(k) ∼ T /k 2 . Note that, in d = 2 dimensions, the exponent describing weak wave turbulence in the UV is identical to the exponent in the Rayleigh–Jeans regime. Signs of a weak-wave-turbulence exponent of ζPUV  d = 3 have been observed when performing the simulation in d = 3 dimensions. 65 In the vicinity of the non-thermal fixed point, the system picks the exponents IR  d + 2 due to the fluxes underlying the stationary but nonζPUV  d and ζQ equilibrium distributions. Studying the radial particle and energy flux distributions Qk and Pk , see Fig. 6, on time scales within the scaling regime defined by the

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Fig. 7. Depending on the strength Δ of the initial cooling quench, where Δ characterizes the initial high momentum decay k−Δ , a Bose gas in d = 3 dimensions can thermalize in a near-adiabatic manner to a Bose–Einstein condensate. Alternatively, it can first approach a non-thermal fixed point (NTFP). Near the fixed point the evolution is critically slowed down and the spectrum is characterized by a steep IR power law, n(k) ∼ k−5 . Such scaling behavior is found for strong cooling quenches, Δ  3. Furthermore, dynamical scale separation of the compressible (blue points) and incompressible (red points) components of the velocity field of the gas (cf. Ref. 65) occurs. Note that the incompressible component shows the transverse vortical flow. The quantumpressure component is depicted by the grey points. The radial momentum k is given in units of the healing length ξ = [2mgρ]−1/2 , with mean density ρ, and the time t in units of τ = mξ 2 . Note the double-logarithmic scale. Figure adapted from Ref. 67.

bimodal structure of the momentum distribution, one finds that the transport process can be interpreted in terms of an inverse particle transport in the IR and a direct energy transport in the UV. Note that the non-zero energy flux in the UV underpins a weak-wave-turbulence cascade while, by the exponent ζ it is indistinguishable from thermal scaling. At late times, thermalization causes the kinetic-energy flux P to almost vanish. However, Q still reshuffles particles and therefore energy, with the zero mode acting as a sink, keeping the system out of equilibrium close to the non-thermal fixed point. In the numerical simulations discussed above, the system was initialized in a specific configuration that caused the approach of a non-thermal fixed point. In the following, we will address the question of the relevance of the strength with which the system is driven away from thermal equilibrium for the approach of the fixed point. The corresponding numerical simulations were performed in

d = 3 dimensions. 56 We parametrize the initial field in momentum space, Φ(k, 0) = n(k, 0) exp{iϕ(k, 0)}, in terms of a randomly chosen phase ϕ(k, 0) ∈ [0, 2π) and a density n(k, 0) = f (k)νk , with νk ≥ 0. For each momentum k, the νk are drawn from an exponential distribution P (νk ) = exp(−νk ). The resulting occupation number spectrum is flat at low k and falls off according to the function f (k) = fΔ /(k0Δ + k Δ ). The parameter Δ controls the deviation from a thermal decay with Δ = 2. k0 is a momentum cutoff and fΔ the normalization. The generated initial configurations are directly overpopulated momentum distributions.

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200

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0

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Fig. 8. Snapshots of the time evolving hydrodynamic velocity field v = ∇arg(Φ)/m of a 2D Bose gas in a square volume with periodic boundary conditions, starting from a lattice of non-elementary vortices with alternating winding numbers w = ±6, arranged in a checker-board manner (panel (a)). Color encodes the modulus of the velocity |v|. The orientation of the  velocity field  2is indicated by the black flow lines. Panels (b)–(d) show snapshots at times t = 300, 103 , 104 ξh , √ with healing length scale ξh = 1/ 2mgρ. The initial vortices quickly break up into clusters of elementary vortices and anti-vortices with w = ±1 which are marked by the orange and green dots, respectively. A strong vortex clustering is present in the early non-universal stage of the evolution (panel (b)). It leads to strong coherent flows during the later stages shown exemplarily in panels (c) and (d), where the vortices and anti-vortices mutually annihilate in a strongly anomalous manner. Figure taken from Ref. 59.

At sufficiently late evolution times, the occupation number spectra developing from different initial Δ differ strongly, see Fig. 7. For Δ  3, the system approaches a non-thermal fixed point, characterized by a bimodal structure of the spectra, with a power-law behavior n(k) ∼ k −5 in the IR and n(k) ∼ k −2 in the UV. The bimodal structure decays towards a global n(k) ∼ k −2 at very long times (not shown). For Δ  3, the momentum distribution goes over directly to thermal Rayleigh–Jeans scaling n(k) ∼ T /k 2 . The larger the quench strength Δ, the closer the system approaches the non-thermal fixed point, see Ref. 56 for details. The numerical simulations indicate that cutting away sufficiently much population at high momenta initially is necessary if the system is supposed to approach the non-thermal fixed point. Hence, only strong cooling quenches allow for the build-up of a steep population far into the IR and a bimodal scaling evolution, see also the discussion in Subsec. 3.2.

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So far, we have discussed the scaling exponents characterizing the power-law scaling of the momentum distribution, n(k) ∼ k −ζ , at a non-thermal fixed point in a vortex gas. Depending on the initial momentum distribution, the system is either attracted to a non-thermal fixed point or simply relaxes back to thermal equilibrium. In general, more than one attractor can exist for the dynamical evolution of the system. Consequently, different types of universal evolution with different power laws for each attractor are found. Which type of evolution is realized depends on the macroscopic properties of the initial state and on the stability properties of the attractors only. Preparing far-from-equilibrium states by imprinting phase defects, i.e., quantum vortex excitations, into an otherwise strongly phase-coherent two-dimensional Bose condensate, the approach of two different non-thermal fixed points can be triggered. 59 This initial state leads to coherent hydrodynamic propagation of vortices on the background of the otherwise phase-coherent gas, with little sound excitations present, which plays a key role for the observation of scaling. Different kinds of initial states are realized by varying the number of defects, their arrangement, and their winding numbers. A strongly anomalous non-thermal fixed point as well as a standard dissipative fixed point related to coarsening according to the HohenbergHalperin model A 50,78 have been identified in numerical simulations. 59 The anomalous fixed point is approached if the coupling of the defects to the background sound fluctuations is sufficiently suppressed. Starting from a lattice of non-elementary vortices with alternating winding numbers w = ±6 arranged in a checker-board manner, the vortices were found to arrange within clusters of elementary defects with either positive or negative winding, w = ±1, such that the formation of closely bound vortex–anti-vortex dipoles is suppressed, see Fig. 8 for snapshots of the corresponding velocity field. The clustering leads to a steep power law scaling n(k) ∼ k −5.7 , i.e., an exponent ζ  5.7. The subsequent scaling evolution is driven by the mutual annihilation of vortices and anti-vortices that proceeds in a strongly anomalous manner. The defect dilution at the anomalous fixed point is much slower than in the standard dissipative case. The anomalous fixed point is characterized by a universal scaling exponent β = 1/(2 − η)  1/5 that governs the self-similar evolution of the momentum distribution, in the IR regime of momenta, according to n(k, t) = (t/tref )α f ([t/tref ]β k),

(63)

with universal scaling function f and some reference time tref within the temporal scaling regime. Due to particle number conservation within the regime of low momenta and times considered in the numerical simulations the scaling exponents are related according to Eq. (19) such that α = dβ  2/5. The large anomalous exponent η  −3 is related to a large dynamical exponent z = 2 − η  5. The observed strongly slowed scaling can be interpreted as being due to mutual defect annihilation following three-vortex collisions and has recently been found consistent with experimental data. 110–112

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  mean defect distance d (t) ξh

103 Nd (ti ) = 2400 Cd (ti ) : 16 × 16 × 6 Cd (ti ) : 8 × 8 × 6 d ∼ t1/2 d ∼ t1/5 102

101

103

104   time t ξh2

105

106

Fig. 9. Mean defect distance d as function of time, starting from different initial vortex configurations at time ti = 0. The blue triangles depict the evolution from a random distribution of Nd (ti ) = 2400 elementary vortices and anti-vortices. The green squares and red circles correspond to the evolution from an irregular square lattice of 16 × 16 and 8 × 8 non-elementary vortices with winding numbers w = ±6 as in Fig. 8. The growth of the mean defect distance is well described by power laws d (t) ∼ t βd with βd = 1/5 (anomalous fixed point, solid line) and βd = 1/2 (Gaussian fixed point, dashed line), respectively. Depending on the initial condition chosen, the system can also first approach the anomalous fixed point before it shows faster scaling reminiscent of the Gaussian fixed point (see data marked by green squares). Units as in Fig. 8. Note the double-logarithmic scale. Figure taken from Ref. 59.

In contrast to the above scenario, starting from a random spatial distribution of elementary defects, with equal number of vortices and anti-vortices, the system approaches the standard dissipative fixed point characterized by the scaling exponents β  1/2 and η  0. Particle number conservation leads to α = dβ  1. Due to the vanishing anomalous exponent this fixed point was referred to as the (near) Gaussian non-thermal fixed point, while the numerical findings are also compatible with a small but non-zero anomalous exponent. 59,94 The observed scaling is associated with the mutual annihilation of elementary vortices and anti-vortices randomly distributed on the phase-coherent background. The Porod exponent ζ  4 is consistent with a dilute ensemble of randomly distributed vortices in d = 2 dimensions. 61,65,78 The distinctly different scaling behavior, emerging from the two initial vortex configurations chosen, becomes clearly visible in the time evolution of the mean defect distance d , see Fig. 9. As the mean defect distance sets the characteristic IR length scale of the system, its scaling evolution is described by the IR scaling exponent β according to d (t) ∼ tβ . Interestingly, depending on the number of non-elementary vortices in the initial configuration, the system can first approach the anomalous fixed point and subsequently show faster scaling reminiscent of the Gaussian fixed point. A similar behavior has been observed in experiment. 112 A transition between different scalings has also been found in a relativistic φ6 model with attractive quartic interactions. 72

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Fig. 10. (a) Universal scaling of the occupation number n1 (k) ≡ n1 (k, t) near a non-thermal fixed point, according to Eq. (63). The inset shows the evolution starting from a ‘box’ momentum distribution n1 (k, t0 ) = n0 Θ(kq −|k|) (grey line), with n0 = (4πkq3 )−1 ρ(0) and kq = 1.4 kΞ , at five (0) is the constant mean background density and different  evolution times (colored dots). Here, ρ kΞ = 2mgρ(0) marks the momentum scale associated with the corresponding healing length of the system. The collapse of the data onto a universal scaling function, with reference time tref = 31 tΞ , shows the scaling in space and time. Within the time window tref = 200 tΞ ≤ t ≤ 350 tΞ , we extract the scaling exponents α = 1.62±0.37, β = 0.53±0.09. (b) Universal scaling dynamics of the correlator measuring the spatial fluctuations of the relative phases, C12 (k, t) = |(Φ†1 Φ2 )(k, t)|2  for the same system. The collapse of the data onto a universal function, which has a similar shape as for n1 , shows that the scaling behavior at the non-thermal fixed point is driven by relative phase fluctuations. Within the same time window as stated in (a), we extract the scaling exponents α = 1.48 ± 0.18, β = 0.51 ± 0.06. The scaling exponents are in good agreement with the analytical predictions 55,60 of β = 1/2 and α = dβ = 3/2. The evolution time is measured in (0) /2π, the momentum in terms of the inverse healing-length scale Ξ−1 = k . units of t−1 Ξ Ξ = gρ Figure taken from Ref. 37.

6.2. Fluctuation dominated fixed points If topological defects are subdominant, the scaling behavior of a Bose gas at a non-thermal fixed point can be different. To illustrate this, we consider an (N = 3)-component dilute Bose gas in d = 3 dimensions, quenched far out of equilibrium. 37,60 The system is described by an O(3) × U (1) or U (3) symmetric Gross–Pitaevskii model with quartic contact interaction in the total density, see Eq. (1). As for the d = 3 vortex case above, the initial far-from-equilibrium state at time t0 is given by a ‘box’ momentum distribution na (k, t0 ) = n0 Θ(kq − |k|), which is constant up to some cutoff scale kq . The initial phase angles ϕa (k, t0 ) of the Bose √ fields in Fourier space, Φa (k, t0 ) = n0 exp[iϕa (k, t0 )], are chosen randomly on the circle and thus uncorrelated.b Such an initial condition can be realized by means of a strong cooling quench, cf. Sec. 2. After a few collision times, the system shows universal scaling indicating the approach of the non-thermal fixed point.

b Note that the phase angles of the fundamental fields Φ (k, t) in Fourier space are different from a the Fourier transforms of the phase angles in position space, ϕa (k, t) = θa (k, t).

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The time evolution of the occupation number n1 (k) as well as of the correlator measuring the spatial fluctuations of the relative phases, C12 (k, t) = |(Φ†1 Φ2 )(k, t)|2 , is depicted in Fig. 10. For both observables we obtain a collapse of the data onto a universal scaling function. This collapse shows the universal scaling in space and time according to the scaling hypothesis in Eq. (63). Within the time window tref = 200 tΞ ≤ t ≤ 350 tΞ , the extracted scaling exponents are α = 1.62 ± 0.37, β = 0.53 ± 0.09 for the scaling of the occupation number n1 and, respectively, α = 1.48 ± 0.18, β = 0.51 ± 0.06 for C12 . The scaling exponents agree well with the analytical predictions of β = 1/2 and α = dβ = 3/2. 55,60 The scaling collapse of the relative phase correlator C12 clearly shows that the scaling behavior at the non-thermal fixed point is driven by relative phase fluctuations. The spatial scaling exponent ζ  4 characterizing the occupation number distribution n1 (k) ∼ k −ζ also confirms analytical predictions neglecting defects. 55 The physical picture is that the O(3) × U (1) symmetric interactions suppress total density fluctuations while allowing the densities of the separate components to be shuffled around freely provided that the total density stays constant. In this way, also relative phase fluctuations can occur, which reflect the counter-motion of particles and correspond to strongly excited Goldstone modes. 7. Prescaling So far, we have discussed the scaling behavior of dilute Bose gases at a non-thermal fixed point. It remains, though, an unresolved question how precisely quantum many-body systems evolve from a given initial state to such a fixed point. As a typical feature of this evolution towards the fixed point we propose prescaling, 37 see also the illustration in the right panel of Fig. 3. Prescaling, 113 motivated by the concept of partial fixed points, 85 means that certain correlation functions, already at comparatively early times and short distances, scale with the universal exponents predicted for the fixed point. The fixed point itself will only be reached much later in time and, in a finite-size system, may not be reached at all. During the stage of prescaling, weak scaling violations occur for the correlations at larger distances. Such scaling violations only slowly vanish. In fact, it turns out that they affect not only the scaling exponents but in particular also the shape of the associated scaling functions. The existence of prescaling has been proposed on the basis of numerical simulations of an (N = 3)-component dilute Bose gas in d = 3 dimensions, quenched far out of equilibrium. 37 The initial far-from-equilibrium state at time t0 is given by the configuration presented in Subsec. 6.2. Scaling behavior at a non-thermal fixed point is commonly extracted from momentum-space correlators. 34,95 Prescaling, however, is more easily seen in position-space correlations. An intuitive choice, based on the momentum-space (1) treatments, is to study the first-order spatial coherence function ga (r, t) = Φ†a (x+r, t)Φa (x, t) , see Fig. 11(a). For long evolution times it is found to approach

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Fig. 11. (a) Time evolution of the first-order coherence function g1 (r) = g1 (r, t) = Φ†1 (x + r, t)Φ1 (x, t) showing violations of universal scaling at larger distances (five different times, colored dots). The scaling form (64), with inverse coherence length scale kΛ (t) fitted to match the first zero of the sine, is depicted by the yellow dotted lines. The polynomial approximation of the sinc as given in the legend is visualized by the grey dashed lines. At later times, the numerical results agree well with the single-scale function up to the first zero in r. At the latest time shown finite-size effects already become relevant. (b) Corresponding second-order coherence function (2) g12 (r, t) = Φ†1 (x + r, t)Φ2 (x + r, t)Φ†2 (x, t)Φ1 (x, t) measuring the spatial fluctuations of the relative phases between components 1 and 2 (colored dots, same times as in (a)). The inset shows ˜ g (2) (t¯−β r, t the rescaled correlation function t¯−α ˜ = −0.15, and t¯ = t/tref . ref ), with β = 0.6, α 12 The collapse of the data onto a single function, for short distances r, indicates that the scaling (1) violations are considerably weaker than for g1 . Here, tref = 31 tΞ denotes the reference time. −1 The evolution time is measured in units of tΞ = gρ(0) /2π, the distance r is given in terms of the  healing-length scale Ξ = 1/ 2mgρ(0) , with constant mean background density ρ(0) . Figure taken from Ref. 37. (1)

(1)

the exponential × cardinal-sine form

  −kΛ (t) |r| sinc kΛ (t) |r| . ga(1) (r, t) ≈ ρ(0) a e (0)

(64)

While the particle density ρa is uniform, the phase oscillates and fluctuates on a scale given by the inverse coherence length kΛ . At the fixed point, this length scale rescales in time according to kΛ (t) ∼ t−β , with universal scaling exponent β. Note that the form of the first-order coherence function, Eq. (64), differs from a pure exponential obtained analytically within a Gaussian approximation of the relation between the phase-angle and the phase correlators, see Ref. 60 for details. As fluctuations of local density differences, in contrast to the fluctuations of the total density, are not suppressed, Goldstone excitations of the relative phases can become relevant. An observable sensitive to the relative phases θa −θb is the second(2) order coherence function gab (r, t) = Φ†a (x + r, t)Φb (x + r, t)Φ†b (x, t)Φa (x, t) . The time evolution of this correlation function, see Fig. 11(b), reveals weaker scaling (1) violations than obtained for g1 , indicating that the fixed point scaling is driven by relative phase fluctuations. (1) (2) A temporal scaling analysis of the correlation functions g1 (r, t) and g12 (r, t) provides a direct way to extract the scaling exponent β. If, however, the fixed-point scaling is not fully developed, the time evolution of the correlations is not described

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Scaling exponent βi

(a) 0.6

(b) (1)

(2)

g12 (r)

g1 (r)

0.5

0.4 β1 β2

200 tref

300 + Δt [tΞ ]

400

200 tref

β3 β4

300 + Δt [tΞ ]

400

Fig. 12. Prescaling of position-space correlations. (a) Scaling exponents βi characterizing the time evolution of the inverse coherence length scale kΛ,i (t) ∼ t−βi for i = 1, 2, 3, 4. The kΛ,i (t) (1)

are extracted by means of a polynomial fit of the first-order coherence function g1 (r, t), shown in Fig. 11(a), up to order r4 at small distances r. The index i corresponds to the associated order of the polynomial. See main text for details. (b) Scaling exponents βi obtained from an analogous (2) polynomial fit of g12 (r, t), see Fig. 11(b). Prescaling is seen by the scaling exponents βi settling in (1)

to, within errors, constant values for the lower orders of the polynomial fit. While g1 (r) exhibits (2) scaling violations, g12 (r) extracted for times tref +

already shows scaling up to order The scaling exponent βi  0.5, Δt  250 tΞ , is in good agreement with the predicted scaling exponent β = 1/2. The βi (t) are averaged over the time window [tref , tref + Δt] with Δt = 146 tΞ as well as over a set of fits with different fit ranges. Errors are given by the standard deviation of the exponents of the set. Units as in Fig. 11. Figure from Ref. 37. r4 .

by a single scale kΛ (t). During prescaling we expect approximate scaling to emerge on short distances and to subsequently spread towards longer distances. To be independent of the particular form of the scaling function we make use of a general polynomial fit of the form g(r, t)  c0 + c1 kΛ,1 (t) r + c2 [kΛ,2 (t) r]2 + c3 [kΛ,3 (t) r]3 + c4 [kΛ,4 (t) r]4 + O(r5 ), at short distances r, to study the scaling behavior of the different types of correlations. Note that the fit is applied to distances r  5 Ξ avoiding the short-distance thermal peak present in the correlation functions. The scaling exponents βi , associated with the order i of the polynomial fit, are obtained by taking the logarithmic derivative of kΛ,i (t) with respect to the time t and averaging it over a fixed time window Δt. The resulting exponents βi for (1) (2) both, g1 and g12 , for i = 1, 2, 3, 4 are depicted in Fig. 12. The exponents shown are additionally averaged over a set of fits with different fit ranges to account for fluctuations arising from the choice of a particular fit range. Prescaling is quantitatively seen by the scaling exponents settling in to stationary values for the lower orders of the polynomial expansion. However, scaling in the higher orders is not yet fully developed for the times considered in the numerical simulations. This leads to the scaling violations observed in Fig. 11. Comparing

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Figs. 12(a) and (b), we find that different observables can enter the stage of prescaling on different time scales. Hence, establishing the full scaling function and the associated scaling exponents is, to some degree, observable-dependent. The value βi  0.5 found at late evolution times for the scaling of kΛ,i (t) ∼ t−βi , (2) (1) for i = 1, 2, 3, 4, parameterizing g12 , and for i = 1, 2 in the case of g1 , is in good agreement with the analytically predicted value β = 1/2 by means of the LEEFT, see Sec. 4. As the LEEFT approach covers the limiting cases of N = 1 and N → ∞, the numerical results suggest that the universality class does not depend on N . This reflects that the U (N ) symmetry is broken during prescaling while the U (1) symmetries are still intact as long as no condensate is present.

8. Outlook In this article we discussed the concept of non-thermal fixed points on the basis of a dilute Bose gas. We outlined a kinetic theory as well as a low-energy effective field theory approach which allowed for analytical predictions of the scaling behavior at the non-thermal fixed point. While the scaling evolution of the fundamental fields is considered within non-perturbative kinetic theory, the low-energy effective field theory describes the dynamics and scaling of phase excitations in the system based on a perturbative approximation in the non-linearities. We presented a variety of numerical studies corroborating the analytical predictions. By treating the dilute Bose gas within a low-energy effective field theory, it was possible to predict scaling exponents, characterizing the time evolution of the system at a non-thermal fixed point, in the limiting cases of N = 1 and N → ∞. A rigorous approach to analytically study the scaling evolution at intermediate N is missing so far. In addition, the Luttinger-liquid based description neglects defects of all kinds in the system. It is an interesting pathway for the future to derive a low-energy effective theory in presence of defects. Based on the analytical treatments as well as numerical studies it can be concluded that universal dynamics at non-thermal fixed points can emerge in rather different manners depending on the properties of the systems. On the one hand, the dilution of defects can drive the scaling evolution leading to strong wave turbulence in the IR regime of momenta. If defects are subdominant in the system, relative phase excitations can play a crucial role for the observed self-similar evolution. Tuning the initial condition of the system enabled to identify the key features leading to the approach of a non-thermal fixed point. Whether the system shows self-similar universal scaling dynamics or directly relaxes to thermal equilibrium crucially depends on the strength of cooling quenches applied to the system. We furthermore discussed the possibility that a system can be attracted to more than one fixed point. In the case of a dilute Bose gas in two spatial dimensions, the initial vortex configuration played the key role whether the system exhibits strongly anomalous or standard diffusive universal scaling.

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Conducting numerical simulations of a three-component Bose gas in three spatial dimensions, the existence of prescaling as a feature of the evolution towards the nonthermal fixed point has been proposed. As the system prescales on comparatively short times scales it can be studied in present-day experimental systems. In this brief overview we focussed on N -component Bose gases characterized by O(N )-symmetric interactions. Multicomponent spinor Bose gases break the O(N ) symmetry of the models discussed here due to the presence of spin-spin interactions and a possible quadratic Zeeman energy shift arising from external magnetic fields. Spinor Bose gases are extremely well controlled in present-day experiments. Non-equilibrium dynamics following quantum quenches has been investigated in several experimental systems. 114–116 Recently, universal scaling with exponent β  0.5 has been observed experimentally in a spin-1 Bose gas confined in a quasi one-dimensional trapping geometry. 35 Realizing experimental observations of the system for evolution times up to several seconds gives the opportunity to address universal scaling dynamics in such systems. Numerical simulations of a one-dimensional spin-1 Bose gas revealed a scaling exponent of β  0.25. While the purely one-dimensional dynamics of the spin-1 Bose gas is driven by defects populating the system, the experimentally observed scaling is different in nature as defects are absent in the quasi one-dimensional setting. Including interactions that break the O(N ) symmetry of the models into the analytical approaches presented in this article is part of ongoing research.

Acknowledgments The authors thank I. Aliaga Sirvent, J. Berges, K. Boguslavski, R. B¨ ucker, I. Chantesana, S. Diehl, S. Erne, F. Essler, M. Karl, P. Kunkel, S. Lannig, D. Linnemann, A. Mazeliauskas, B. Nowak, M. K. Oberthaler, J. M. Pawlowski, A. Pi˜ neiro Orioli, M. Pr¨ ufer, R. F. Rosa-Medina Pimentel, J. Schmiedmayer, J. Schole, T. Schr¨ oder, H. Strobel, and C. Wetterich for discussions and collaborations on the topics described here, in particular A. Pi˜ neiro Orioli and K. Boguslavski for their careful reading of the final manuscript. This overview article has been written for the proceedings of the Julian Schwinger Centennial Conference and Workshop held in Singapore in February 2018. T.G. thanks the Julian Schwinger Foundation for Physics Research for support and the Institute of Advanced Studies at Nanyang Technological University, Singapore, for its hospitality. Original work summarized here was supported by the Horizon-2020 programme of the EU (AQuS, No. 640800; ERC Adv. Grant EntangleGen, Project-ID 694561), by DFG (GA677/8 and SFB 1225 ISOQUANT) and by Heidelberg University (CQD).

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112. S. Johnstone, A. J. Groszek, P. T. Starkey, C. J. Billington, T. P. Simula and K. Helmerson, Order from chaos: Observation of large-scale flow from turbulence in a two-dimensional superfluid, arXiv:1801.06952v2 (2018). 113. C. Wetterich, Private communication (2018). 114. J. M. Higbie, L. E. Sadler, S. Inouye, A. P. Chikkatur, S. R. Leslie, K. L. Moore, V. Savalli and D. M. Stamper-Kurn, Direct nondestructive imaging of magnetization in a spin-1 Bose-Einstein gas, Phys. Rev. Lett. 95, 050401 (2005). 115. L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore and D. M. Stamper-Kurn, Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose-Einstein condensate, Nature 443, 312 (2006). 116. J. Guzman, G.-B. Jo, A. N. Wenz, K. W. Murch, C. K. Thomas and D. M. StamperKurn, Long-time-scale dynamics of spin textures in a degenerate f = 1 87 Rb spinor Bose gas, Phys. Rev. A 84, 063625 (2011).

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Baryon isospin mass splittings Lai-Him Chan Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA I have proposed that in association with the strong isospin breaking mu = md there must be induced hyperfine isospin breaking effect which has to be taking into account in order to fully explain isospin splittings of hadrons. Recent precise measurements of 11 charm and bottom baryon isospin mass splittings has provided strenuous tests on theoretical models. I review and update my original works with new formulation and methodology using new baryon basis convenient for quark mass interpolation such that all baryon isospin mass splittings can be summarized by simple analytic functions of constituent quark masses, charges and spins. The predicted mass splittings are in excellent agreement with experimental measurements beyond expectation.

1. Introduction Isotopic spin symmetry is the best approximate flavor symmetries. The symmetry is easily identified in the mass spectrum and the mass splittings within the isospin multiplets are at most a few MeV. Such small mass splittings have posted a great challenge both to the experimentalists to measure them accurately and to the theorists to calculate them. Isospin symmetry is obviously broken by the well-understood electromagnetic interaction which contributes a positive Coulomb energy opposite to the observed the mass splitting p − n. The fact that the neutron can be heavier than the proton can only come from other source, namely strong interaction. Quantum Chromodynamics (QCD) is composed by a collection of quarks with different masses conventionally referred to as flavors but otherwise no flavor enters into the dynamics anywhere. The masses are considered to be completely independent parameters in QCD and they in fact should have been used to define flavors rather than the other way around. There is no reason why the up quark mass mu and the down quark mass md should be equal. The inequality can naturally serve as a strong isospin splitting mechanism. However, why the mass difference, md − mu , should be comparably to the order of a few MeV (≈ α md/u ) remains an unsolved mystery. In principle the strong isospin splittings can be directly calculable from QCD given the up and down quark masses as parameters. However, the small magnitudes of the isospin splittings varying from a few tenth to at most a few MeV are beyond the limit of Lattice Gauge Calculation for sometimes to come. If the up and down quark masses serve exactly the same function as the other flavor quark masses and no other external flavor breaking enters into the strong

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interaction, flavor strong breakings must be an analytic functions of quark masses subjected to analytic continuation, extrapolation and interpolation of quark masses. Soon after the discovery of the existence of the charm quark and the charm hadrons in 1977, I have suggested 1 that in addition to strong isospin splittings coming from the up and down quark mass difference there must be also induced isospin splittings mass splittings in association with the mass difference similar to those found in other flavor splittings. Numerically the calculation can be accomplished by interpolating the mass difference from the heavy quark masses all the way from the heavy quark masses difference down to the small isospin quark mass difference without introducing any new parameter. This task can be made much easier if a simple analytic function of quark masses can be found theoretically or phenomenologically to estimate hadron masses to a good approximation. Such is the case for the De R´ ujula–Georgi–Glashow model 3 which introduces a hyperfine interaction symmetry breaking term transforming like the color magnetic moment– magnetic moment interaction. With the scaling of this term as a parameter in additional the quark masses, all baryon masses can be described up to 10% of the corresponding flavor mass splittings. One would expect that the extrapolation to the isospin splitting using the same model would be just as successful, namely deducing the contribution from the induced hyperfine interaction to the isospin splittings with possible uncertainty of 10%, which typically is about order of 0.1 MeV. Our conjecture is that if two models, dynamically or phenomenologically, agree on their predictions on the baryon strong mass splittings depending on the quark masses only, they must also predict baryon isospin mass splittings consistent with each other. My original paper for the hadron isospin splittings calculation 40 years ago 1 and its extension 2 to include the bottom quark 30 years ago may appear to be unnecessarily cumbersome. The nomenclature for particles has even become completely obsolete and definitely needs proper translation: C → Σc , S → Ξc , A → Ξc and χ → Ξcc . However, the predictions of all hadron isospin splittings remain valid without contest. At the time I optimistically expected that prediction of the charm baryon isospin splittings can be tested experimentally within a few years and perhaps even ten years. However, as the result of a large cancellation between the Coulomb energy contribution and md −mu contribution for the charm baryon, most of the isospin splittings of the charm baryons are very small of the order of fraction of a MeV. While such sensitivity is good for discrimination for various theoretical models, it has become extremely challenged for experimental measurements to limit uncertainties to such degree of accuracy. It is not until the recent years meaningful tests of the predictions has become available. I am very happy to take this occasion of Julian Schwinger Centennial Conference to report that all original predicted charm baryon isospin splittings agree extremely well with the latest experimental values beyond expectation. A few bottom baryon isospin mass splittings measurements also suggest that this trend is going to continue for the bottom baryons.

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In this paper I recast the calculation of the matrix elements of the electromagnetic and the strong interaction isospin symmetry breakings terms by using a more physical basis for baryon, namely characterized by quark masses, spins and charges, rather than the flavor and spin quantum numbers. With such basis all isospin symmetry breaking terms are practically diagonalized and the corresponding matrix elements can be virtually read off directly. Finally I also present the evidence why it is reasonable to assume at the constitute quark level that quark has the normal Dirac magnetic moment by calculating the baryon electromagnetic moments using the same assumption. 2. Baryon Basis and Interactions The original model 1 assumes that hadrons are supermultiplets of SU(8) and singlet in color, SU(8) being the the covering group of flavor SU(4) and spin SU(2). The SU(8) symmetry is broken by spin–spin interaction and the quark masses. The contributions to the isospin splittings originate from two independent sources, the electromagnetic contribution and the strong interaction contribution. The electromagnetic contribution of the isospin symmetry breaking to the Interaction Hamiltonian operator is the commonly acceptable approach consisting the spin independent Coulomb interaction transformed like Q ⊗ Q and the spin dependent magnetic interaction transforms like μ  ⊗μ  . Q is the charge operator. μ  is the quark Dirac electromagnetic moment operator, the validity of which can be supported by the calculation of baryon magnetic moments. The source of the strong isospin splittings comes from the intrinsic quark mass difference md = mu and also the hyperfine interaction operator transforming like  s  c , where μ  c is the Dirac color magnetic moment operator m . μ c ⊗ μ COLOR HYPERFINE

M=

MAGNETIC HYPERFINE

MASS   ELECTROSTATIC           si sj si sj μ + mi + k(αsD)4m2

·  + αC Qi Qj − αD4m2u Qi Qj

· , mi mi mi mi i i>j i>j i>j       STRONG HADRON MASS SPLITTINGS



ELECTROMAGNECTIC MASS SPLITTINGS



(1)



ISOSPIN SPLITTINGS

MStrong = μ +



mi + k(αs D)4m2u

i

MEM = αC

 i>j

Qi Qj − (αD)4m2u

 si sj ·

, mi mj i>j  i>j

Qi Qj

si sj ·

, mi mi

(2)

(3)

where mi , Qi , and si are the effective constituent quark mass, charge and spin of the ith quark respectively. k = 23 is the SU(3) color factor. α and αs are the fine-structure constant and the effective QCD coupling constant, respectively. μ, C, and D are constant. As in QCD, the only source of flavor symmetry breaking is from the quark mass differences.

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Given the pattern of the physical quark masses, md − mu  ms − md  mc − ms  mb − mc , three quarks in a baryon can always be arranged such that |m1 − m2 |  |2m3 − m1 − m2 |. In that configuration, the eigenstate of s2 and s12 2 = (s1 + s1 )2 are, for all practical purpose in a very good approximation, eigenstate of the quark mass operator. Therefore the baryons are eigenstates completely characterized by their constituents quark masses, charges, and the total spin s and subspin s12 .

3. Baryon Masses The Strong interaction contribution to the baryon masses can be obtained readily, Strong (m1 , m2 ; m3 ) = μ + (m1 + m2 + m3 ) (4) Bs,s 12  1 2 [2s12 (s12 + 1) − 3] + (αs D)m2u 3 m1 m2

  1 1 3 1 + s(s +1) − s12 (s12 + 1) − + . m3 m1 m2 4

After factoring out the color part, the baryon wave function must be totally symmetric. The totally symmetric wave-function implies the absence of a state with s = 23 and s12 = 0, and a state with m1 = m2 and s12 = 0. By neglecting the isospin violation, Eq. (4) can be used to determine the parameters from the baryon mass spectrum: m = mu = md = 335.7 MeV, ms = 512.5 MeV, mc = 1674 MeV, mb = 5024 MeV, and αs D = 73.3 MeV.

4. Baryon Isospin Splittings All baryon strong flavor mass splittings derived from this set of parameters are within the order of 10% of their experimental values. It is reasonable to expect that with md − mu = 0, Eq. (4) would give the correct strong contribution to the isospin splittings with accuracy with order of 10% of the splittings, which is equivalent to a small fraction of one MeV. The electromagnetic interaction contribution to the baryon masses can be obtained readily, EM (m1 , m2 ; m3 ) = (αC)(Q1 Q2 + Q2 Q3 + Q3 Q1 ) Bs,s 12  Q1 Q2 [2s12 (s12 + 1) − 3] − (αD)m2u m1 m2

  Q2 3 Q3 Q1 + s(s + 1) − s12 (s12 + 1) − + . m3 m1 m2 4

(5)

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The baryon isospin splittings can be completely summarized by the following equations:

2 Q−1 Q (m , m ; m ) − B (m , m ; m ) = m − m − (αC) Q − Bs,s (6) q q d q q u d u s,s12 12 3  

 1 1 3 2 (md − mu ) mu (αs D) Q− − (αD) s(s + 1) − s12 (s12 + 1) − , −2 mq 4 3 mu 2 3

2 Q−1 Q (m , m ; m ) − B (m , m ; m ) = m − m − (αC) Q − (7) Bs,s d q h u q h d u s,s12 12 3 

 2 (md − mu ) 2 mu [2s12 (s12 + 1) − 3] (αs D) − (αD) Q − Qh − − mq 3 mu 3    3 2 (md − mu ) mu (αs D) − (αD)Qh , − s(s + 1) − s12 (s12 + 1) − mh 4 3 mu where q is any quark, h the heaviest of the three quarks in the baryon and Q is the total charge. The only undetermined parameters can be determined by the known mass splittings n − p, Σ0 − Σ+ and Σ− − Σ0 : md − mu = 2.66 MeV, αC = 2.96 MeV, αD = 1.18 MeV.

(8)

Before I engage to compare the predictions of the isospin splittings in this calculation with the experimental measure values accumulated many years until very recently, it is reasonable to ask whether the fitted parameters are physically reasonably meaningful and what can be learned from them. • It is important to note that both C = 1r B = 406 MeV and D = 2π 3 3m2u δ (r) B = 162 MeV are positive as expected in quark model but by no mean guaranteed in the fitting process. They can be measured in principle by the Lattice Gauge Theory. • By comparing the values of αD and αs D, we can deduce the value of αs , 77.3 MeV 1 αs D α= = 0.52 αD 1.18 MeV 137 This value is quite compatible with the αs (Q2 ) extrapolation to the energy scale of a couple GeV region according to the latest PDG summery. 4 • The result that the down quark mass is heavier than the up quark mass as motivated by the neutron proton mass difference apparently works well with all baryon isospin splittings. Even though the mass difference is not associated with any electromagnet effect, md −mu = 2.66 MeV is very close to α mu = 2.44 MeV. αs =

Up to the present time there are 18 baryon isospin splittings measurements available in the PDG, 4 seven light baryons, seven charm baryons and four bottom baryons. In my calculation, I have provided explicit simple analytic formula for the baryon isospin mass splittings depending only on the constituent quark masses, charges and spins with three isospin breaking parameters.

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The contributions from various components to baryon isomultiplet splittings are summarized in Table 1. The quark mass difference and the coulomb energy provide the comparable dominant contributions while strong hyperfine and the electromagnetic contributions give similarly smaller contributions. The sign of the Coulomb contribution is positive for isoscalar quark charge − 13 . The reinforcement of the two large terms makes isospin splittings of the strange baryon and the bottom baryon relatively large, of the order of few MeV. However, the sign is negative for isoscalar quark charge 23 . Such is the case for the charm baryons. The large cancellation yields much smaller isospin splittings for the charm baryons, of order of 0.1 MeV. The delicate balance between various components becomes an extremely useful tool to discriminate various theoretical models. It is in these cases that the small contributions from the strong and the electromagnetic hyperfine interactions become important. The spins and the charges also contribute to the contrast contributions to the hyperfine splittings when their small contributions become important. It is not very surprising that after using three light baryon isospin splittings to determine the three parameters, the other four baryon isospin splittings fall well in line. However, when the seven small charm baryon isospin splittings measurements slowly surfaced one by one through out the years, it is absolutely amazing that they are all in complete agreement with my predictions within experimental and theoretical uncertainties without exception. The strong isospin splittings from the hyperfine interaction induced by the up and down mass difference are obtained directly through mass interpolation without introducing any new isospin breaking parameter and their contributions are listed in Table 1. Their magnitudes are mostly within a few tenth of one MeV. Their uncertainties, as expected by the mass extrapolation are about 10%, can only be at most 0.1 MeV. However, models without such contribution would not come even close to getting the charm baryon isospin splittings correct. − Σ0c , Lichtenberg pointed out 7 that in the case of the isospin splitting Σ++ c there exists a large cancellation between the mu − md and the Coulomb contribution to the isospin splitting and a measurement of this splitting would be the best hope for distinguishing among different models. 7 Four recently measured charm isospin mass splittings and the predictions of various theoretical models are shown − Σ0c = 0.22 ± 0.10 MeV has not only valin Table 2. The small magnitude of Σ++ c idated Lichtenberg’s large cancellation but also has eliminated, by many standard deviations, all but the only one model that includes the strong hyperfine induced isospin splittings. 1,2 The predicted splittings consistent with the measured values are indicated in bold letters. It is of most statistically significant that while only five out of 46 entries from other thirteen models are consistent with the measured values, all four of my predictions are well in agreement with experimental values. My isospin splittings 1,2 are shown in Table 1. With no exception, the predictions are in good agreement with all 16 experimental values with only three isospin breaking parameters determined by the light baryon isospin splittings. In addition to the

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Table 1.

n−p Σ− Σ−

Baryon isospin splittings in MeV.

Mass

Strong

Electric

Mag.

Total

Experiments 4

2.66

−0.78

−0.99

0.40

1.29

1.29

−

Σ+

5.32

0.24

0.99

1.44

7.98

8.08 ± 0.08

−

Σ0

2.66

0.12

1.97

0.12

4.88

4.81 ± 0.04

Σ∗− − Σ∗+

5.32

−1.30

0.99

−0.12

4.90

4.4 ± 0.7

Σ∗−

−

Σ∗0

2.66

−0.65

1.97

−0.65

3.33

3.5 ± 1.2

Ξ−

−

Ξ0

2.66

1.02

1.97

1.04

6.69

6.85 ± 0.21

−

Ξ∗0

2.66

−0.51

1.97

−0.52

3.60

Σ0c − Σ++ c

5.32

−0.47

−4.93

−0.23

−0.32

Σc 0 − Σc +

Ξ∗−

Σ∗0 c

−

Σ∗0 c

3.2 ± 0.6 −0.22 ± 0.00

2.66

−0.23

−0.99

−0.71

0.73

0.9 ± 0.4

Σ∗++ c

5.32

−0.94

−4.93

0.72

0.17

−0.01 ± 0.15

Σ∗+ c

−

2.66

−0.47

−0.99

−0.24

0.97

0 .98 ± 2 .3

Ξ0c − Ξ+ c

2.66

0.77

−0.99

0.78

3.22

3.0 ± 0.24

−

Ξ+ c

2.66

−0.10

−0.99

−0.58

1.00

1 .4 ± 1 .3

−

Ξ∗+ c

2.66

−0.33

−0.99

−0.10

1.24

0 .79 ± 0 .27

++ Ξ+ cc − Ξcc

2.66

0.31

−3.95

−0.63

−1.61

−−

2.66

−0.16

−3.95

0.32

−1.13

−−

5.32

−0.68

0.99

0.51

6.13

4.2 ± 1.1

2.66

−0.34

1.97

−0.34

3.95

−−

5.32

−0.84

0.99

0.35

5.82

3±1

2.66

−0.42

1.97

−0.42

3.80

−−

2.66

0.77

1.97

0.78

6.18

5.9 ± 0.6

2.66

−0.20

1.97

−0.21

4.22

−−

2.66

−0.28

1.97

−0.29

4.07

5.54 ± 1.4

2.66

0.10

1.97

0.11

4.84

−−

2.66

−0.05

1.97

−0.05

4.53

−−

2.66

0.23

−0.99

−0.48

1.43

−−

2.66

−0.03

−0.99

0.21

1.86

−−

2.66

−0.10

−0.99

0.13

1.70

−−

Ξ0 c Ξ∗0 c Ξ∗+ cc

−

Ξ∗++ cc

+ Σ− b − Σb 0 Σ− b − Σb ∗+ Σ∗− b − Σb ∗− Σb − Σ∗0 b 0 Ξ− − Ξ b b 0 Ξ− b − Ξb ∗0 Ξ∗− b − Ξb − Ξbb − Ξ0bb ∗0 Ξ∗− bb − Ξbb

Ξ0cb Ξ0 cb Ξ∗0 cb

− Ξ+ cb − Ξ+ cb − Ξ∗+ cb

135

four recent precise measurements which have provided very strenuous conditions, Table 1 also includes three other measured charm baryon mass splittings and four bottom baryon mass splittings (in italic) with large uncertainties also consistent with my predictions. The four bottom baryon isospin mass splittings measurements with sizable uncertainties listed in PDG. 4 Two, Ξc and Ξ∗c , are consistent with the predictions.

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Table 2.

Isospin mass splittings of the charm baryon isotriplets in MeV. + ∗0 Σ∗+ c −Σc

Ξ0c − Ξ+ c

0 .8

1 .6

−.06

2 .4

−0 .84

2 .51

−6

−4

4

—

−3 .33

−2 .49

−3 .62

−4 .54

+ 0 Σ+ c −Σc

0 Σ+ c −Σc

Lichtenberg 7

3 .5

Itoh 8

6 .5

Lane 9

Reference

MIT

bag 10

Ono 11

6 .10

2 .24

1 .63

1 .77

Wright 12

−1 .4

−2 .0

—

3.1

1 .6

−1 .8

−2 .0

—

Isgur 13 Richard 14

−2

−2

−2

−2

Hwang 15

3 .0

− 0.5

2 .7

2 .1

Sinha 16

1 .5

−0 .3

—

0 .4

Capstick 17

1 .4

−0 .2

−0.1

—

Cutkosky 18

0 .84

−0 .40

1 .00

2.91

Varga 19

1 .2

−0 .36

—

2.83

Chan 1,2

0.32

−0.73

−0.16

3.22

0.22±0.01

−0.9±0.4

0.01±0.15

3.0±0.24

Experiments 4

+ ∗− ∗+ The predictions Σ− b − Σb = 6.13 MeV and Σb − Σb = 5.82 MeV lie outside the uncertainties of the measured values 4.2 ± 1.1 MeV and 3.0 ± 1.0 MeV, though the earlier measurements 6 are 7.5 ± 2.5 MeV and 7.5 ± 3.0 MeV. Hopefully more better measurements in the future may clarify the situation.

5. Doubly Heavy Baryons Very recently baryon with two charm quarks have been observed 20 and it has generated interest of predicting the isospin splittings of baryons with two heavy quarks. In a recent paper, Karliner and Rosner 21 claim to first calculate, + +0.76 Ξ++ MeV, cc − Ξcc = 1.41 ± 0.12 +0.03 Ξ0bb − Ξ− MeV, bb = −4.78 ± 0.06 0 +0.39 MeV Ξ+ cb − Ξcb = −1.69 ± 0.07

using a simple method which is essentially the same as in my papers 1,2 but without any reference to my works. With no surprise the results are not different from + Ξ++ cc − Ξcc = 1.61 MeV,

in my 1977 paper, 1 and Ξ0bb − Ξ− bb = −4.84 MeV

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0 in my 1985 paper. 2 Their Ξ+ bc and Ξbc are mixed states but not proper particles 0 eigenstates. The two proper states should be Ξ+ cb − Ξcb = −1.43 MeV for s12 = 0 + between up/down quark and the charm quark and Ξcb −Ξcb0 = −1.86 MeV for s12 = 1. Unfortunately these states have yet to be observed and their mass differences may not be available in the near future.

6. Baryon Magnetic Moments For the hyperfine interaction, electromagnetic or color, I have assumed quark inside the baryon behaves like a Dirac particle with the standard Dirac dipole moment inverse proportional to its mass. The question is whether the assumption is reasonable and what is if any evidence to support it. Here I use the same approach to calculate the baryon magnetic moments using the magnetic moment operator, 2 + μ  3, μ  =μ 1 + μ Qi where μi = m is the magnetic moment of the ith quark. The electromagnetic i moments and the transition moments of the baryon can then be readily obtained using my new baryon basis,

s12 (s12 +1)− 34 s (μ1 + μ2 + 2μ3 ) + (μ1 + μ2 − 2μ3 ), (9) 2 2(s + 1) ! 2s (μ1 − μ2 ), (10) μ(s,1)→( 12 , 0) (μ1 , μ2 ; μ3 ) = 3 ! 1 (μ1 + μ2 − 2μ3 ). μ( 32 ,1)→( 12 ,1) (μ1 , μ2 ; μ3 ) = − (11) 3 μs,s12 (μ1 , μ2 ; μ3 ) =

As shown in Table 3, with no additional parameter, the calculated values of eight magnetic moments are in good agreement with the measured values to the order of 10%. Such consistency gives further support to the assignment of the quark charge, magnetic moment and mass. The transition magnetic moments for the Table 3.

Baryon magnetic moments in nuclear magneton.

Baryon

Theory

Experiment 4

μp

2.79

2.79

μn

−1.85

−1.91

μΣ+

2.69

2.458 ± 0.010

μΣ −

−1.03

−1.160 ± 0.025

μ(Σ0 →Λ)

−1.61

−1.61 ± 0.08

μΛ

−0.61

−0.613 ± 0.04

μΞ 0

−1.44

−1.25 ± 0.014

μΞ−

−0.51

−0.6507 ± 0.0025

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heavy baryons are given by 1 1 √ μΣ∗0 →Λc = √ μΣ∗0 →Λb = μΣ0c →Λc = μΣ0b →Λb = μΣ0 →Λ 3 c 3 b which gives ΓΣ0c →Λc γ = 89 keV, ΓΣ∗0 = 220 keV, ΓΣ0b →Λb γ = 142 keV and c →Λc γ = 190 keV. ΓΣ∗0 b →Λb γ Unfortunately the heavy baryons are too unstable to have their magnetic moments measured and the branching ratios are too small to test their transition moments. However, the agreement in the light baryon sector gives further evidence in support to the assumption that quark in the baryon behaves like a Dirac particle with the standard Dirac dipole moment inverse proportional to its mass.

7. Conclusion In this report I have presented a complete review on the isospin mass splittings of the baryons, theoretical calculation as well as a comprehensive comparison with the latest experimental results. The basic physical idea and the results from its prediction of the papers 30 to 40 years ago 1,2 have remained valid. However, the methodology and the notations have gradually become obsoleted through the years and the verification from the experimental data has not yet been available then. This is a perfect time, at this occasion of Julian Schwinger Centennial Conference, to put a proper prospective on such an extremely successful effort to understand the important part of the baryon static properties, namely the isospin mass differences and the magnetic dipole moments. The almost perfect agreement of all baryon mass isospin splittings and baryon magnetic moments with the experimental measurements with only three parameters is certainly very much statistically significant that the basic ideas and the method of their implementation is basically correct. There are other predictions not yet tested should be just as accurate and are subjected to be tested by future experiments. With this renewal of my original works 30 to 40 years ago, it is my intention and hope that my presentation would shed some light and eventually lead to the ultimate understanding of the isospin splittings. For the time being and hopefully ultimately this is the explanation of the baryon isospin splittings.

Acknowledgments I am grateful to the sponsors the Julian Schwinger Foundation, the Institute of Advanced Studies, and the Centre for Quantum Technologies for their support. I would like to thank Berge Englert for his tireless assistance to make my visit truly a pleasure. About all I am most thankful to have been a student of Professor Schwinger and to have many helpful discussions with and encouragements from his students through out my career.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

L.-H. Chan, Phys. Rev. D 15, 2478 (1977). L.-H. Chan, Phys. Rev. D 31, 204 (1985). A. De R´ ujula, H. Georgi and S. L. Glashow, Phys. Rev. D 12, 147 (1975). M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018). K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 030001 (2018). K. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 075021 (2010). D. B. Lichtenberg, Phys. Rev. D 16, 231 (1977). C. Itoh, T. Minamikawa, K. Miura and T. Watanabe, Prog. Theor. Phys. 54, 908 (1975). K. Lane and S. Weinberg, Phys. Rev. Lett. 37, 717 (1976). N. Deshpande, D. Dicus, K. Johnson and V. Teplitz, Phys. Rev. D 15, 1885 (1977). S. Ono, Phys. Rev. D 15, 3492 (1977). A. C. D. Wright, Phys. Rev. D 17, 3130 (1978). N. Isgur, Phys. Rev. D 21, 779 (1980) [Erratum: ibid. 23, 817 (1981)]. J. M. Richard and P. Taxil, Z. Phys. C 26, 421 (1984). W.-Y. P. Hwang and D. B. Lichtenberg, Phys. Rev. D 35, 3526 (1987). S. Sinha, Phys. Lett. B 218, 333 (1989). S. Capstick, Phys. Rev. D 36, 2800 (1987). R. E. Cutkosky and P. Geiger, Phys. Rev. D 48, 1315 (1993). K. Varga, M. Genovese, J. M. Richard and B. Silvestre-Brac, Phys. Rev. D 59, 014012 (1999). LHCb collaboration, Phys. Rev. Lett. 119, 112001 (2017). M. Karliner and J. L. Rosner, Phys. Rev. D 97, 094006 (2018). S. Capstick and S. Godfrey, Phys. Rev. D 41, 2856 (1990).

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Multiquark states in the Thomas−Fermi quark model and on the lattice Walter Wilcox∗ and Suman Baral† Department of Physics, Baylor University, Waco, TX 76798-7316, USA ∗ walter [email protected] † suman [email protected], [email protected]

We describe work being done at Baylor University investigating the possibility of new states of mesonic matter containing two or more quark–antiquark pairs. To put things in context, we begin by describing the lattice approach to hadronic physics. We point out there is a need for a quark model which can give an overall view of the quark interaction landscape. A new application of the Thomas–Fermi (TF) statistical quark model is described, similar to a previous application to baryons. The main usefulness of this model will be to detect systematic energy trends in the composition of the various particles. It could be a key to identifying families of bound states, rather than individual cases. Numerical results based upon a set of parameters derived from a phenomenological model of tetraquarks are given. Keywords: Lattice QCD; octaquarks; quark matter; tetraquarks; Thomas–Fermi model.

1. Introduction and Motivation Lattice Chromodynamics (QCD) is the main tool used by particle physicists to investigate the properties of baryons and mesons within the context of the strong interactions. The technology and algorithms of lattice applications are constantly improving. The path integral approach, pioneered by Feynman, is done automatically via Monte Carlo simulation. The quark degrees of freedom, including color, spin and particle/antiparticle, are incorporated into a quark “mass matrix,” which is used to define quark and hadron propagation functions. The lattice scale is set by observed renormalization group behavior. In addition, the lattice “link” variables as depicted in Fig. 1 play the role of the gluon degrees of freedom. Figure 2 shows the heavy quark–antiquark potential for mesons extracted from lattice gluonic combinations. 1 Lattice configurations can be quenched or dynamical. Quenched lattice configurations suppress background quark–antiquark loops in order to limit computer time requirements. Dynamical or nonquenched calculations can accomodate light, strange and charmed quark loops, and are now used in all realistic lattice calculations. Figure 3 gives the present Particle Data Group (PDG) summary of extractions of the strong coupling constant, showing that lattice QCD now results the smallest error bars. 2 Finally, Fig. 4 gives a contemporary depiction of the state of lattice QCD spectrum calculations, showing light quark baryons and mesons as well as states with both hidden and explicit charm and bottom. 3 These results are impressive, and confirm that QCD is the correct theory of the strong interactions.

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ν

μ 
ν

ν 
+

ν 
μ




Fig. 1.

Fig. 2.


μν

μ 



μ

The link variables on a hypercubic lattice.

The heavy quark–antiquark potential for quenched and dynamical configurations.

Tetraquark and pentaquark states are now known to exist from Belle, 4–7 BESIII, 8,9 LHCb, 10–14 and other collaborations. Are there also states of many more quarks? Could there be an analog to nuclear/atomic systems for heavy/light quark systems? As the quark content increases, it becomes computationally expensive and time-intensive to do the lattice calculations. Every state must be investigated separately, which means a great deal of analysis on Wick contractions and specialized computer coding in lattice QCD. In addition as one adds more quarks the states will become larger and the lattice used must also increase in volume. There is a need for quark models which can help lead expensive and spatiallimited lattice QCD calculations in the right direction in the search for high quark states. The Thomas–Fermi (TF) statistical model has been amazingly successful in the explanation of atomic spectra and structure, as well as nuclear applications.

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Maltmann (Wilson loops) PACS-CS (SF scheme) ETM (ghost-gluon vertex) BBGPSV (static potent.)

ALEPH (jets&shapes) OPAL(j&s) JADE(j&s) Dissertori (3j) JADE (3j) DW (T) Abbate (T) Gehrm. (T) Hoang

structure + – e e jets & shapes functions

ABM BBG JR NNPDF MMHT

lattice

HPQCD (Wilson loops) HPQCD (c-c correlators)

τ-decays

Baikov Davier Pich Boito SM review

143

(C)

GFitter CMS (tt cross section)

electroweak precision fits hadron collider

April 2016

Fig. 3.

Recent measurements of the strong coupling constant, αs .

The atomic applications by Schwinger and Englert have brought it to its highest point. Our group has adopted the TF model and applied it to collections of many quarks. 15–17 One would expect that the TF quark model would become increasingly accurate as the number of constituents is increased, as a statistical treatment is more justified. The main usefulness will be to detect systematic trends as the parameters of the model are varied. It could also be key to identifying families of bound states, rather than individual cases. We have now extended the TF quark model to mesonic states in order to investigate the stability of families built from some existing mesons and observed new exotic states, concentrating on heavy-light quark combinations. Although our model is nonrelativistic, we will see that this assumption is actually numerically consistent as quark content is increased.

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Fig. 4.

A compilation of lattice QCD hadron mass calculations.

2. TF Quark Model The Thomas–Fermi model treats particles as a Fermi gas at T = 0. It builds in Fermi statistics, but is not fully quantum mechanical. It does not have a quantum mechanical wave function, but instead a central function related to particle density which is determined by filling states up to the Fermi surface at each physical location. It gives accurate atomic binding energies for large numbers of electrons in atomic systems. It is exactly what is called for in this situation with many quarks. The TF model is better than “bag models,” 18–20 which do not intrinsically include the Coulomb interactions for large numbers of particles. The challenge we face of course is that we are extrapolating from small numbers of particles, where the model is less accurate, to large numbers. However, our goal is to detect systematic trends in particle states, which we believe will be manifest in this many particle theory. Explicit spin interactions can be included in the model, but some limitations are present. In the atomic TF model (and in nuclear applications) the up and down spin 1/2 states are treated as degenerate. One cannot do this in particle physics! In our treatment, the spin quantum number is separated out as another “flavor.” Spin of course is not conserved in particle interactions, only the total angular momentum corresponds to a conserved quantum number. However, this breaks rotational symmetry. It is best to keep the states in a maximal spin “up” or “down” orientation. Then classical and quantum states are “maximally compatible.” Such a treatment has been developed for baryons. 16 In our present limited treatment for mesons, we have not yet included explicit spin interactions, but we can take one level of spin degeneracy into account.

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The types of interactions between the particles can be categorized as: 21 Color−Color Repulsion (CCR) Interactions between quarks with same colors is repulsive with coupling constant 4/3g 2 . The interactions are red–red (rr), green– green (gg) and blue–blue (bb). Color−Color Attraction (CCA) Interactions between quarks with different colors is attractive with coupling constant −2/3g 2 . The interactions are rb, rg, bg, br, gr, and gb. Color−Anticolor Repulsion (CAR) Interactions between quarks and antiquarks with different color/anticolors is repulsive with coupling constant 2/3g 2 . The interactions are r¯b, r¯ g , b¯ g , b¯ r, g¯ r, and g¯b. Color−Anticolor Attraction (CAA) Interactions between quarks and antiquarks with same color/anticolors is attractive with coupling constant −4/3g 2 . The interactions are r¯ r, b¯b and g¯ g. Anticolor−Anticolor Repulsion (AAR) Interactions between antiquarks with same anticolors is repulsive with coupling constant 4/3g 2 . The interactions are r¯r¯, ¯b¯b and g¯g¯. Anticolor−Anticolor Attraction (AAA) Interactions between antiquarks with different anticolors is attractive with coupling constant −2/3g 2 . The interactions are r¯¯b, r¯g¯, ¯b¯ g , ¯b¯ r, g¯r¯, and g¯¯b. Table 1 shows the various color-averaged couplings within mesons with η number of quark–antiquark pairs. We find that, on average, quarks only interact with antiquarks in such systems; i.e., the sum of the products of the couplings and probabilities for CCR, CCA as well as AAR, AAA interactions vanish. If we add the Table 1. The coupling constants and probabilities for certain types of quark and antiquark interactions in mesons. Interaction type

Symbol

Coupling

CCR

Pii

4 2 g 3

CCA

Pij , i = j

− 23 g 2

CAR

P¯ij , i = j

2 2 g 3

CAA

P¯ii

− 43 g 2

AAR

P¯ii

4 2 g 3

AAA

P¯ij , i = j

− 23 g 2

Interaction probability (η − 1) 18(2η − 1) η−1 9(2η − 1) 2(η − 1) 9(2η − 1) (η + 2) 9(2η − 1) (η − 1) 18(2η − 1) (η − 1) 9(2η − 1)

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remaining product of coupling and probabilities from Table 1, we get − 43 g 2 /(2η − 1), very similar to the baryon case. 16 The negative sign indicates that the system is attractive because of the collective residual color coupling alone, even in the absence of volume pressure. This gives rise to a type of matter that is bound, but does not correspond to confined mesonic matter, as discussed in Ref. 15. We are interested in confined matter and will need to add a bag vacuum pressure term to the energy to enforce this. 18 The TF differential equation is constructed and systems with heavy-light quark content are examined. Three types of heavy quark/antiquark (Q ¯ light quark/antiquark (q or q¯) mesonic systems are defined and investigated. or Q) ¯ QQQ ¯ Q, ¯ etc.), Z-meson (QQ¯ ¯ q q, QQ¯ ¯ q q QQ¯ ¯ q q, etc.) and D-meson Charmonium (QQ, ¯ ¯ ¯ (Qq, Qq Qq, etc.) family types are examined. These will be referred to as “Case 1,” “Case 2” and “Case 3,” respectively, in the following. We will examine the charmed quark case below. The phenomenological parameters we need for our model are the strong coupling constant αs , the bag constant B, the charm quark mass, mc , as well as the light quark mass, m1 . Previously, we used baryon phenomenology to obtain these parameters, 17 but we now wish to attempt a more realistic fitting using mesonic states. Since we do not yet include spin interactions in our model, we need to weight spin-split states to “remove” this interaction for our  model fits. Assuming 2 for two spin 1/2 1 · S the interactions are proportional to a spin splitting term, S particles, we weighted masses of 1S states such that 1 3 (ηc (1S)) + (J/Ψ (1S)) = 3069 MeV, 4 4 for charmonium. For the mass of the D meson we weight particles such that 1 3 (D) + (D∗ ) = 1973 MeV, 4 4 where we are also averaging over charge states of D and D∗ . In addition, for the Case 2 mass we spin-weight the J = 1, C = −1 tetraquark states as 22 1 3 (Zc (3900)) + (X(4020)) = 3990MeV. 4 4 This weighting comes from a model where the light quark spin dominates the mass splitting of these two spin 1 states. We solved the differential equations using an iterative implementation of NDSolve in Mathematica. We searched parameter

space such that the model χ2 was minimized using a grid search, obtaining χ2 = 1.05 MeV in the mass evaluations, an almost perfect fit. We obtained αs = 0.217, B 1/4 = 103.5 MeV and charm quark mass mc = 1530 MeV. Our light quark mass, m1 = 306 MeV, we take from our previous TF baryon spectrum fit. 16 Note that our conference paper Ref. 17 contains a numerical error in the calculation of the Case 3 type family binding energies, which is corrected here. A more detailed explanation of our parameter fitting assumptions will be given in Ref. 23. Note we have not yet completed a more comprehensive examination of the physics associated with inclusion of the b-quark sector.

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We will examine TF spatial functions and energies for the three cases defined above. In nuclear physics, one inspects the binding energy per nucleon in order to assess the relative stability of a given nucleus. We will do a similar investigation here. Thus, the important figure of merit in these evaluations is the total energy per quark, for if this increases as one adds more pairs, the family is unstable under decay to lower family members, whereas if it decreases, the family is stable. The actual quark mass dependence does not play a role in these considerations and so will not be included in our energy plots. The degeneracy status of the quarks will play an important role in these considerations. For η pairs of quarks, we have Case 1:

n · g = η,

Case 2: n1 · g1 + n2 · g2 = η, Case 3:

n1 · g1 = η,

n2 · g2 = η.

n, n1 , n2 are the number particles in a given state, and g, g1 , g2 are degeneracies. The index “1” refers to light quarks and “2” refers to heavy quarks. For charmed quarks, g2 = 1, 2 only, whereas for light quarks we may have g1 = 1, 2, 3, 4. 3. Model Results First, let us discuss the behavior of the particle density wave functions. We use the dimensionless parameter x such that r = Rx where

2/3 3πη a . R≡ (8αs /3) 2 αs is the strong coupling constant and a ≡ /(m1 c). The particle density function of charmonium, proportional to (f (x)/x)3/2 , drops smoothly with increase in distance and has a discontinuity at the boundary due to the volume pressure, as seen in Fig. 5(a). The density function of Z-mesons has a long tail for the light quarks, while for the heavy charmed quark the value is large and is concentrated near the origin, as seen in Fig. 5(b). This suggests an atomic-like structure with heavy charm, anticharm quarks at the center while light quarks and antiquarks spread out like electrons. Figure 5(c) is an enlargement of the density function of the light quark wave function for the Z-meson. It drops down abruptly until it reaches the boundary of the heavy quark wave function, then inflects and cuts off. In the case of D-mesons, Fig. 5(d), the density function of light and heavy quarks are relatively closer. We increased the quark content and compared density functions of a family of multi-mesons in all three cases. We observed similar density functions for a given multi-meson family regardless of the quark content. Figure 6 includes 10 possibilities for the physical radius of the various mesons, compared to a generic baryon with three equal mass flavors. The radius is plotted versus quark number and compared with a generic baryon with three degenerate light flavors. We observe that the curve of the radius plot for each case tends to flatten out for larger numbers of quarks. case1nf1xmax refers to quark families

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Fig. 5. TF density functions as a function of dimensionless distance x for the various cases: (a) Charmonium (Case 1) (b) Z mesons (Case 2) (c) Light quark density function for Z mesons (d) D meson (Case 3).

of charmonium with no degeneracy, g = 1. In this case all the charm quarks have the same spin and hence cannot occupy the same state. case1nf2xmax is instead the plot of the charmonium family with g = 2. In this case, spin up and down is assigned to a pair of charm quarks. The physical radius of case1nf1xmax being larger than case1nf2xmax reflects this fact. Note that the dotted lines refer to the inner boundary associated with the charmed quark in Cases 2 and 3. For Case 2 the difference between dotted and continuous lines is the largest. case2nf1x1 and case2nf1x2 refer to the radius plot of the outer and inner boundary of the quark family of Z-mesons with g2 = 1, respectively, while case2nf2x1 and case2nf2x2 refer to the same type of plot with g2 = 2. Similarly for Case 3. The Z- and D-meson family members are found to have equally large outer boundaries. There are three types of energies in this model: kinetic, potential and volume. The kinetic energy per quark depends strongly on the meson family, as seen in Fig. 7. Six separate lines are given (three meson cases and two degeneracies) and compared with the generic baryon. We see that the energy per quark is relatively small and tends to decrease slowly, which seems to provide some justification for this nonrelativistic model. Figure 8 shows the corresponding graph for the potential energy. These energies curve upward and level off as more quark pairs are introduced. The volume energies in Fig. 9 are relatively flat. The case2nf2 and case3nf2 results in these figures deserve some extra comments. These lines are determined by the most degenerate state, and thus the smallest energy per quark,

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−15

x 10

case1nf1xmax case1nf2xmax case2nf1x1 case2nf1x2 case2nf2x1 case2nf2x2 case3nf1x1 case3nf1x2 case3nf2x1 case3nf2x2 baryonnf3

3.5

Physical distance( m )

3 2.5 2 1.5 1 0.5 0

Fig. 6.

0

5

10 15 Number of quarks

20

25

Physical radius of the various mesons in this study versus quark content.

case1nf1 case1nf2 case2nf1 case2nf2 case3nf1 case3nf2 baryonnf3

90

80

Kinetic Energy

70

60

50

40

30

20

Fig. 7.

0

5

10 15 Number of quarks

20

25

Kinetic energy per quark (in MeV) versus quark content.

available for the given quark content. If we denote the total number of quarks and antiquarks as N (= 2η), the N = 8, 16 and 24 quark cases for case2nf2, which all have g2 = 2, are associated with g1 = 2, g1 = 4 and g1 = 3, respectively. Likewise, the N = 4, 8, 12, 16 and 20 quark cases for case3nf2 are associated with g1 = 2, g1 = 4, g1 = 3, g1 = 4 and g1 = 2, respectively.

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0

−10

Potential Energy

−20

−30

case1nf1 case1nf2 case2nf1 case2nf2 case3nf1 case3nf2 baryonnf3

−40

−50

−60

0

Fig. 8.

5

10 15 Number of quarks

20

25

Potential energy per quark (in MeV) versus quark content.

case1nf1 case1nf2 case2nf1 case2nf2 case3nf1 case3nf2 baryonnf3

60

50

Volume Energy

40

30

20

10

0

Fig. 9.

0

5

10 15 Number of quarks

20

25

Volume energy per quark per quark (in MeV) versus quark content.

Figure 10 is our final result. It shows the total energy per quark without the mass term, i.e., the sum of kinetic, potential and volume energy, plotted against the quark content. The generic baryon rises slowly for increasing quark content, implying these are unstable; i.e., a higher quark content state can decay into lower members of the same family. The Case 1 mesons rise quickly and then continue the

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Energywithoutmass

100

80

60

40

case1nf1 case1nf2 case2nf1 case2nf2 case3nf1 case3nf2 baryonnf3

20

0

Fig. 10.

0

5

10 15 Number of quarks

20

25

Total energy per quark without mass terms (in MeV) versus quark content.

rise more slowly; these are also unstable. Similarly, the Case 2 and 3 nondegenerate mesons show increases in energy per quark similar to Case 1. In contrast, for both the Case 2 and 3 degenerate families, there are initial downward, then upward tendencies. In fact, the downward jumps from N = 4 to N = 8 to N = 16 for Case 2 implies octaquark and hexadecaquark states which should be stable against decay into lower family members. In addition, the downward jumps from N = 2 to N = 4 to N = 8 for Case 3 implies stable tetraquark and octaquark states. Our Case 3 tetraquark results can be compared to a lattice calculation. Reference 24 reports two-point function results for quark flavor content c¯c¯ud. Their present calculations are limited to a range of pion masses down to ∼ 150 MeV, and show binding of 22(11) MeV. This would be expected to become more strongly bound at physical quark mass. Our Case 3 results from Fig. 10 show a downward jump in energy per quark from N = 2 to N = 4 of about 10 MeV, corresponding to a total binding energy of about 40 MeV. The comparison is very encouraging. 4. Conclusions and Acknowledgments We have motivated a description of multi quark-pair meson states using the Thomas–Fermi statistical approach. After specifying the explicit interactions and summing on colors, we have formulated system interactions and energies. We have investigated three cases of mesonic states: charmonium family (Case 1), Z-meson family (Case 2) and D-meson family (Case 3). We have not yet included explicit spin interactions in our model, but we can take one level of degeneracy into account in our two-TF function construction. As we have said before, the goal of such a program is to prepare the way for more detailed lattice calculations.

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In this study, we have surveyed the physically relevant parameter space of the TF quark model, looking for relative stability and connections to known phenomenology. We have observed interesting patterns of single quark energies. Similar to our findings for baryons, the energy per quark is slowly rising for the Case 1 mesons, implying family instability. Our Case 2 and 3 findings are the most interesting. In these cases we see an actual decrease in the energy of introduced quark pairs. For Case 2 this would superficially indicate that stable octaquark and hexadecaquark versions of the Z-meson exist, and that stable tetraquark and octaquark versions of the charmed D-meson exist. The preliminary results of Ref. 24 gives evidence from the lattice that Case 3 meson tetraquarks are indeed stable against decay into D D∗ mesons. Our first order of business as we extend the model will be the inclusion of bmeson states. In addition, we also need to evaluate and include explicit quark spin interactions to bring our meson model up to the same level of development as the TF baryon model. These interactions can be determined from the nonrelativistic ground state wave functions of these states and the associated TF function probability. Further extensions of this model would be to examine systems with heavy central charge, similar to atomic systems, or baryonic states such as generalized pentaquark families. We thank the Baylor Quantum Optics Initiative and the University Research Committee of Baylor University for their partial support of this project. We thank the organizers of the Schwinger Centennial Conference for their invitation. We also thank N. Mathur for helpful discussions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

G. Bali et al., Phys. Rev. D 62, 054503 (2000). M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018). A. Kronfeld, Ann. Rev. Nucl. Part. Sci. 62, 265 (2012). S. K. Choi et al. (Belle), Phys. Rev. Lett. 91, 262001 (2003). S. K. Choi et al. (Belle), Phys. Rev. Lett. 100, 142001 (2008). A. Bondar et al. (Belle), Phys. Rev. Lett. 108, 122001 (2012). Z. Q. Liu et al. (Belle), Phys. Rev. Lett. 110, 252002 (2013). M. Ablikim et al. (BESIII), Phys. Rev. Lett. 110, 252001 (2013). M. Ablikim et al. (BESIII), Phys. Rev. Lett. 115, 112003 (2015). R. Aaij et al. (LHCb), Eur. Phys. J. C 72, 1972 (2012). R. Aaij et al. (LHCb), Phys. Rev. Lett. 110, 222001 (2013). R. Aaij et al. (LHCb), Phys. Rev. Lett. 112, 222002 (2014). R. Aaij et al. (LHCb), Phys. Rev. Lett. 115, 072001 (2015). R. Aaij et al. (LHCb), Phys. Rev. Lett. 118, 022003 (2017). W. Wilcox, Nucl. Phys. A 826, 49 (2009). Q. Liu and W. Wilcox, Ann. Phys. 341, 164 (2014); erratum, to be published. S. Baral and W. Wilcox (DPF 2017) arXiv:1710.00418. E. Farhi and R. Jaffe, Phys. Rev. D 30, 2379 (1984). J. Madsen and J. M. Larsen, Phys. Rev. Lett. 90, 121102 (2003). J. Madsen, Phys. Rev. Lett. 100, 151102 (2008).

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21. P. Sikivie and N. Weiss, Phys. Rev. D 18, 3809 (1978). 22. M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018), Sec. 90, Tables 90.2 and 90.3. 23. W. Wilcox and S. Baral, to be published. 24. P. Junnarkar, N. Mathur and M. Padmanath, arXiv:1810.12285v3.

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Are dyons the preons of the knot model? Robert J. Finkelstein Physics & Astronomy, University of California, Los Angeles, 475 Portola Plaza, Los Angeles, California 90095, USA fi[email protected]

We consider the possibility that the preons defined by the SLq(2) extension of the Standard Model may be identified with Schwinger dyons. The SLq(2) extension is here presented as a model that may exist in either a currently observable electric phase or in a magnetic phase that is predicted but currently unobservable.

At this centennial for Julian, I would like to repeat the words that he spoke at the memorials for Tomonaga and George Green. As he said of them and now is true of himself, “Julian lives on in the minds and hearts of the many people whose lives he touched and graced. And he is, in a manner of speaking, alive, well, and living among us.” I would like to pursue one of the seminal thoughts that he has left with us. 1. Introduction In 1948 Dirac 1 attempted to widen Maxwell theory by the introduction of magnetic poles. This idea was further developed in 1969 with Schwinger’s paper 2 entitled “A Magnetic Model of Matter” where it was suggested that the strong nuclear coupling stemmed from the Dirac magnetic field, and it was further proposed that the most elementary particles, which he named “dyons,” carried both electric and magnetic charge. Since any preon is presumably smaller and heavier than the leptons and quarks, any preon picture suggests a very strong binding force — which presents a theoretical challenge that has not yet been met. There were, however, three phenomenological papers, namely: Harari 3 (1979), Shupe 4 (1979), and Raitio 5 (1980), that successfully represented the empirical data on leptons and quarks in terms of a simple preon model. Beginning in 2005, in total ignorance of these phenomenological papers, I began to study the possibility of extending the standard model of elementary particles, in admitting topological degrees of freedom for the field particles by replacing the field operators Ψ(x) by ˜ j  (x)Dj  (q) Ψ(x) → Ψ m,m m,m

(1.1)

j where Dm,m  (q) is an irreducible representation of the knot algebra SLq(2), while j ˜ Ψ (x) satisfies the Lagrangian of the Standard Model after its modification m,m j 6–11 by the form factors generated by the adjoined Dm,m Unexpect (q) factors. edly, this topological model which we shall call the “knot model” agrees with the phenomenological models of Harari, Shupe, and Raitio.

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In this extension of the standard model, the states (j, m, m )q of the SLq(2) algebra are postulated to be restricted by the topological conditions 1 (N, w, r + o) (1.2) 2 where (j, m, m )q labels a state of the quantum knot and (N, w, r) labels the 2d projection of a corresponding oriented classical knot. In this correspondence each quantum state (j, m, m )q is labeled by a classical knot (N, w, r) so that the quantum kinematics is restricted by the spectrum of a classical knot. Since this restriction is on the states of the SLq(2) algebra and not on states of the standard model, it limits only the new degrees of freedom and does not disturb the preexisting symmetries of the Standard Model. In (1.2) N , w, and r are respectively the number of crossings, the writhe, and the rotation of the 2d projection of the corresponding classical knot. Here o is an odd number required by an otherwise unacceptable difference in parity between the two sides of (1.2). We set o = 1 for the simplest knot, the trefoil. Equation (1.2) then describes a correspondence between a state of the quantum knot (j, m, m )q and a 2d-projected classical knot (N, w, r). The dynamical evolution of the field is still described by quantum field theory but the quantum dynamics is kinematically constrained by classical knot topology. If the four elements (a, b, c, d) of the fundamental representation of SLq(2) are assumed to be creation operators for fermionic preons, then the creation operators for the simplest composite preonic structures that are topologically stable, are the j/2 four quantum trefoils: D w r+1 where j = 3, w = ±3 and r = ±2. It then turns out 2 2 that these are the creation operators for the four families of elementary fermions 3/2 3/2 3/2 as follows: charged leptons, D 3 3 ; neutrinos, D− 3 3 ; down quarks, D 3 − 1 ; and up (j, m, m )q =

2 2

2 2

3/2

2

2

quarks, D− 3 − 1 . This preon representation of leptons and quarks by the knot model 2 2 is in essential agreement with the preon models of Harari, Shupe, and Raitio. There are now four independent approaches, including the present approach, based on the same empirical data, that suggest the same preonic model of leptons and quarks. These preons have, therefore, at least a virtual existence and in fact the only real question is whether they have independent degrees of freedom and can be observed, or whether they are lumps of field that concentrate mass and charge with no independent degrees of freedom, and therefore are bound. We next consider a Schwinger dyon model in which the elementary particles, the dyons, carry both electric (e) and magnetic charge (g), in contrast to the models where the particles carry only electric charge. We shall consider a SLq(2) dyon field which can exist in two phases, distinguished by two values of the deformation parameter q as follows: an e-phase where qe =

e g

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and a g-phase where qg =

g . e

In both phases we assume g  e and therefore qg  qe . We also assume that e and g, and therefore q, are in general energy dependent, and we speculate that the dyon field may undergo transitions between the two phases over cosmological times. It is further assumed that the elementary field particles in both phases are preons that carry both e and g charge, and that the creation operators of these dyonic preons are members of the fundamental representation of the SLq(2) algebra. To connect with observation, we study the possibility that there are composite particles in the e phase, which are currently observed as leptons and quarks, while the corresponding particles of the g phase are too massive to be currently produced or observed.

2. The Two Charge Model We first consider a generic field theory where the field quanta have two couplings that may be expressed in the coupling matrix

0 α2 εq = . (2.1) −α1 0 The two couplings α1 and α2 are assumed to be dimensionless and real and may be written as

e g (α1 , α2 ) or (α2 , α1 ) = √ , √ (2.2) c c where e and g refer to a specific two charge model and have dimensions of an electric charge. We assume that e and g may be energy dependent√and normalized at relevant energies. The reference charge is the universal constant c. We shall interpret the two fields presented by (2.2) as describing parity conjugate fields like the electric and magnetic fields. The fundamental assumption that we make on this coupling matrix is that it is invariant under SLq(2) as follows T εq T t = T t ε q T = ε q

(2.3)

where t means transpose and T is a two dimensional representation of SLq(2): ab T = . (2.4) cd By (2.3) and (2.4) the elements of T obey the knot algebra: ab = qba ac = qca

bd = qdb cd = qdc

ad − qbc = 1 da − q1 cb = 1

bc = cb q1 ≡ q −1

(A)

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where q=

α1 α2

(2.5)

so that the two couplings normalize the algebra through their ratio. If also det εq = 1

(2.6)

α1 α2 = 1.

(2.7)

one has

If the two dimensionless couplings (α1 , α2 ) are expressed in terms of e and g, where e and g are the electroweak and “gluon”-like charges, or electric and magnetic charges, then eg = c.

(2.8)

Then (2.8) implies that qe is the fine structure constant: e (2.9) qe = g 1 e2 ∼ . (2.10) and qe = c 137 If g represents magnetic charge, then (2.8) is like the Dirac requirement according to which the magnetic charge is very much stronger than the electric charge. 1 If the magnetic pole is very much heavier as well, it may be observable only in deep probes of space, i.e. at early and not at current cosmological temperatures, or at currently achievable accelerator energies. Since the knot form factors associated with the two phases are highly dependent on the deformation parameter q, the energy dependence of e and g will be quite different in the e and g phases. We shall assume that magnetic poles do exist and shall study the possible extension of knot symmetry to magnetic charges. The (2j + 1)-dimensional representation of SLq(2), constructed on the Weyl monomial basis, may be expressed as follows  j Dmm Ajmm (q|na , nb , nc , nd )ana bnb cnc dnd . (2.11)  = na ,nb ,nc ,nd

Here a, b, c, d satisfy the knot algebra (A) and na , nb , nc , nd are summed over all positive integers and zero that satisfy the following equations: 9,10 na + nb + nc + nd = 2j,

(2.12)

na + nb − nc − nd = 2m,

(2.13)



(2.14)

na − nb + nc − nd = 2m .

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Here 9,10



Ajmm (q|na nb nc nd )

n+ 1 ! n− 1 ! = n+ 1 ! n− 1 !

 12

n+ 1 ! n− 1 ! na 1 ! nb 1 ! nc 1 ! nd 1 !

159

(2.15)

where n± = j ± m, n± = j ± m , and n q = 1 + q + · · · + q n−1 , with n 1 = n q1 where q1 = q −1 . The two dimensional representation, T , already introduced, now reappears as the j = 12 fundamental representation of SLq(2), 1 ab 2 Dmm (2.16)  = cd = T. In a physical model with the εq coupling we interpret (a, b, c, d) in (2.11) as creation j operators for (a, b, c, d) particles, which we have termed preons. Then Dmm  (a, b, c, d)  as given by (2.11) is the creation operator for the quantum state (j, m, m )q containing (na , nb , nc , nd ) preons.

j 9 3. Noether Charges Carried by Dmm  Knots

The knot algebra (A) is invariant under Ua (1) × Ub (1) :

a = eiϕa a, d = e−iϕa d,

b = eiϕb b,

(3.1)

c = e−iϕb c.

The transformation, Ua (1) × Ub (1), on the (a, b, c, d) of SLq(2) induces on the j 10 Dmm  of SLq(2) the corresponding transformation j j     Dmm  (a, b, c, d) → Dmm (a , b , c , d ) 

j = ei(ϕa +ϕb )m ei(ϕa −ϕb )m Dmm  (a, b, c, d)

(3.2)

j = Um (1) × Um (1)Dmm  (a, b, c, d) j and on the field operators as modified by the Dmm 

˜ j  → Um (1) × Um (1)Ψ ˜j  Ψ mm mm

(3.3) ˜ j  (x) Ψ mm

where the modified field operators have been expressed in (1.1) as j Dmm  (q|a, b, c, d). For physical consistency any knotted field action that is allowed must be invariant under (3.3) since (3.3) is induced by Ua × Ub transformations that leave the defining algebra (A) unchanged. There are then Noether charges associated with Um and Um that may be described as writhe and rotation charges, Qw and Qr , since m = w2 and m = 12 (r + o) for quantum knots.

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For quantum trefoils we have set o = 1, and we now define their Noether charges: w 2 1  Qr ≡ −kr m ≡ −kr (r + 1) 2

Qw ≡ −kw m ≡ −kw

(3.4) (3.5)

where kw and kr are undetermined charges. N/2 The generic model based on D w r+1 has been worked out in some detail as a 2

2

SLq(2) extension of the standard lepton-quark model at the electroweak level. 9–11 It is successful when formulated as a preon theory at the electroweak level. Being a new model, however, it presents some unanswered questions and in particular it does not predict whether the preons are bound or are in fact observable. Since the hypothetical preons are much smaller and heavier than the leptons and quarks, a very strong binding force is required to permit one to regard the leptons and quarks to be composed of three observable preons. The binding force could be gravitational and it could also be dyonic as suggested by Schwinger, or it could be both. To study the dyonic model one assumes that the preons are dyons. The question that we examine here is whether there is a formulation and interpretation of the SLq(2) topological algebra such that the knot extension of the standard model can be reinterpreted and reparametrized at high energies to realistically also describe a dyonic Lagrangian of observable dyons. To approach this question we begin to summarize the SLq(2) extension of the standard model by first restricting the states described by the field operators, ˜ j  (x)Dj  (q), to states obeying the postulated relations (1.2) Ψ mm mm (j, m, m )q =

1 (N, w, r + o) 2

(1.2)

and also obeying the empirically based relations 9–11 6(t, −t3 , −t0 ) = (N, w, r + 1)

(3.6)

(j, m, m )q = 3(t, −t3 , −t0 ) .

(3.7)

or by (1.2)

Here t and t3 refer to isotopic spin and t0 refers to hypercharge. Equation (3.7) holds for j = 32 and t = 12 as shown in Table 1. Table 1 describes an empirical correspondence between the simplest fermions (t = 12 ) and the simplest knots (N = 3), which are the classical trefoils, and so reveals an unexpected relation between the simplest fermions and the simplest knots. By Table 1: (N, w, r + 1) = 6(t, −t3 , −t0 ). By postulates (1.2) and (3.8): (j, m, m )q = 3(t, −t3 , −t0 ).

(3.8) (3.9)

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Table 1.

Empirical support for (N, w, r + 1) = 6(t, −t3 , −t0 ).

Elementary particles

t

t3

t0 Classical trefoil N

w

N/2

r r + 1 Dw 2

leptons

quarks

161

− 12 − 12

(e, μ, τ )L

1 2

(νe , νμ , ντ )L

1 2

1 2

− 12

(d, s, b)L

1 2

− 12

1 6

3

3 −2

−1

(u, c, t)L

1 2

1 2

1 6

3 −3 −2

−1

3

3

3 −3

2 2

r+1 2

3/2

3

D3

3

3/2 D 33 −2 2

3 2 2

3/2 −1 2 2

D3 D

3/2 −1 −3 2 2

Note: The symbols ( )L designate the left chiral states. The topological labels (N, w, r) on the right provide a way to label the left chiral states. The last column describes the states of the quantum trefoil.

The trefoil has three crossings, two values of the writhe, and after its 2d projection two values of the topological rotation. Only for the particular row-to-row correspondences shown in Table 1 do (3.6) and (3.7) hold, i.e., each of the four families of fermions labeled by (t3 , t0 ) is uniquely correlated with a specific (w, r) 3/2 classical trefoil, and therefore with a specific state D w r+1 of the quantum trefoil. 2

2

The t3 doublets of the standard model now become the writhe doublets (w = ±3) of the knot model. With this same correspondence the leptons and quarks form a knot rotation doublet (r = ±2). We now repeat earlier work that refers explicitly to the e-phase. It is repeated here since this development is compatible with the generic two charge model and it therefore also provides a possible description of the g-phase, as well as the e-phase. Retaining the row to row correspondence established in Table 1, it is then possible to compare in Table 2 the electroweak charges, Qe , of the most elementary fermions with the total Noether charges, Qw + Qr , of the simplest quantum knots, which are the quantum trefoils. There are four charges Qe to fix the two constants kw and kr . One sees that Qw + Qr = Qe is satisfied for charged leptons, neutrinos and for both up and down quarks with a single value of k: e (3.10) k = kr = kw = 3 and also that t3 and t0 then measure the writhe charge and rotation charges respectively: Qw = et3 ,

(3.11)

Qr = et0 .

(3.12)

Then Qw + Qr = Qe provides an alternative meaning of Qe = e(t3 + t0 ) of the standard model.

(3.13)

t3

Quantum Trefoil Model t0

Qe

(N, w, r)

N/2

Dw 2

3/2

(e, μ, τ )L

1 2

− 12

− 12

−e

(3, 3, 2)

D3

(νe , νμ , ντ )L

1 2

1 2

− 12

0

(3, −3, 2)

D

(d, s, b)L (u, c, t)L

1 2 1 2

− 12

1 6

− 13 e

(3, 3, −2)

2 e 3

(3, −3, −2)

Qe = e(t3 + t0 )

(j, m, mq ) =

1 2

1 6

Qw

r+1 2

−kw

3 2 2

3/2 −1 2 2

−kw

3/2 −1 −3 2 2

1 (N, w, r 2

3 2

 −kw − 32

3/2 3 −3 2 2

D3 D

Qr

+ 1)

3



2

3

−kw − 2

−kr −kr

Qw + Qr

3 2

3 2



1



1

−kr − 2 −kr − 2

Qw = −kw w Qr = −kr r+1 2 2

− 32 (kr + kw ) 3 (k 2 w

− kr )

1 (k 2 r

− 3kw )

1 (k 2 r

+ 3kw )

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Standard Model (f1 , f2 , f3 )

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Table 2.

by (3.8) and (3.9)

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In SLq(2) measure Qe = Qw + Qr is: e Qe = − (m + m ), 3 where (m, m ) =

1 (w, r + 1). 2

163

(3.14)

(3.15)

Then e Qe = − (w + r + 1) 6

(3.16)

for the quantum trefoils, that represent the elementary fermions. Then the electroweak charge is a measure of the writhe + rotation of the trefoil. The total electroweak charge in this way resembles the total angular momentum as a sum of two parts where the knot rotation corresponds to the orbital angular momentum and where the localized contribution of the writhe to the charge corresponds to the localized contribution of the spin of a particle to the angular momentum, i.e. the writhe and rotation correspond to the spin and orbital angular momentum, respectively. In (3.16) o = 1 contributes a “ground state charge” resembling the ground state energy of the quantum oscillator. We may now try to extend (3.15) beyond the trefoil, where o = 1, to an o that depends on the knot. Then e Qe = − (w + r + o). (3.17) 6 The total SLq(2) charge sums the signed clockwise and counterclockwise turns that any knotted energy–momentum current makes both at the crossings and in one circuit of the 2d-projected knot. In this way, the “handedness” or chirality of the “knot particle” determines its electroweak charge, so that chirality reduces electroweak charge to a geometrical concept similar to the way that curvature of space–time geometrizes mass and energy. This topological measure of electroweak charge, which is suggested by the leptons and quarks, goes to a deeper level than the also exact standard electroweak isotopic-spin measure that was originally suggested by the approximate equality of masses in the neutron–proton system. As here defined, quantum knots carry the charge expressed as both t3 + t0 and m + m . The conventional (t3 , t0 ) measure of charge may be based on SU(2) × U(1) while the (m, m ) measure of charge is based on SLq(2). These two different measures are related at the j = 32 level by Eq. (3.7): (j, m, m )q = 3(t, −t3 , −t0 ), where leptons and quarks are both t = 12 isospin and j = 32 knot particles, and preons are j = 12 knot particles.

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4. The Fundamental Representation We next extend this analysis beyond j = 32 , describing leptons and quarks, and in particular to the fundamental representation j = 12 . This extension to other states of j is here√intended to include as well a specialization of the generic model (α1 , α2 ) to (e, g)/ c where e is the electroweak coupling and g is a hypothetical “magnetoweak” coupling. We continue with the description of the electroweak phase. As far as we now know, a magnetoweak phase may be constructed along the same lines with k = e/3 replaced by k = g/3, but then the missing experimental support of the g-phase, containing g-leptons and g-quarks, must be regarded as a currently unverified prediction of this formulation of the dyon model. We continue with the extension of (3.14) from j = 32 , representing leptons and quarks, to j = 12 , representing preons,

1/2

Dmm (q) =

1 2

− 12

1 2

a

b

− 12

c

d

Qe = (−e/3)(m + m )

(B)

where the Noether charge in SLq(2) measure is Qe = (−e/3)(m + m ). Hence there is one charged preon, a, with charge − 3e or − g3 and its charge antiparticle, d, and there is one neutral preon, b, with its antiparticle, c. We define the particle–antiparticle relation with respect to either electric or magnetic charge. If j = 12 , then N = 1 by the postulate (1.2) relating to the corresponding classical knot 1 (4.1) (j, m, m )q = (N, w, r + o). 2 The corresponding a, b, c, d classical labels of the preons cannot therefore be described as knots since they have only a single crossing. The preon labels can, however, be described as 2d-projections of twisted loops with N = 1, w = ±1 and r = 0, where the two loops forming the twist have rotations that cancel. Having tentatively interpreted the fundamental representation in terms of preons, labeled by twisted loops, we next consider the general representation labeled by the knot (N, w, r)r coordinates. j Interpretation of all Dmm  (q|a, b, c, d) j Dmm  =



Ajmm (q|na , nb , nc , nd )ana bnb cnc dnd .

(2.11)

na ,nb ,nc ,nd j Every Dmm  , as given in (2.11) being a polynomial in a, b, c, d, can be interpreted as a creation operator for a superposition of states, where each state has na , nb , nc , nd preons.

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3/2

165

3/2

The creation operators for the charged leptons, D 3 3 ; neutrinos, D− 3 3 ; down 2 2

3/2

2 2

3/2

quarks, D 3 − 1 ; and up quarks, D− 3 − 1 are described by (2.11), where the general 2

2

2

2

j polynomial representation of Dmm  reduces to the following monomial representations of the quantum trefoils: Quantum Trefoils

3/2

3/2

3/2

3/2

D 3 3 ∼ a3

D− 3 3 ∼ c3

D 3 − 1 ∼ ab2

D− 3 − 1 ∼ cd2

charged leptons

neutrinos

down quarks

up quarks

2 2

2 2

2

2

2

(C)

2

Here the (j, m, m )q indices are empirically determined in Tables 1 and 2. Then (B) implies that charged leptons and neutrinos are composed of three a-preons and three c-preons, respectively, while the down quarks are composed of one a- and two b-preons, and the up quarks are composed of one c- and two d-preons, in agreement with the Harari, Shupe and Raitio models, and with the experimental evidence on which their models are constructed. Note that the number of preons equals the number of crossings ((j = N2 = 32 ) in (B)). There are only four families of “elementary fermions” differing by the two possibilities for the writhe and the two possibilities for the rotation of the projected quantum trefoil. Each of the “elementary fermions” has three states of excitation, ¯ 3/2  D3/2  , that appear in the Higgs mass term determined by the eigenstates of D mm mm as modified by the knot form factor. 11 The masses in the “electric” and “magnetic” phase are very different since qg  qe . The discussion up to this point identifies the Noether charge with the electroweak charge appearing in the empirical Tables 1 and 2. Since the corresponding empirical support for a physical g-phase has not yet been seen, we are speculating that the g-phase has not yet been observed because it lies at a higher energy that is so far unobservable, and that the currently observable universe is an e-state of the dyon field. A higher mass of the g-phase is consistent with its higher deformation parameter, q, and should be observable in deep probes of space.

The Knotted Electroweak Vectors We continue with the extension of j = 3/2, representing leptons and quarks, to j = 1/2, representing preons, and to j = 3, representing knotted electroweak vectors. To achieve the required Ua (1) × Ub (1) invariance of the knotted Lagrangian (and to retain the associated conservation of t3 and t0 , or equivalently of the writhe and rotation charge), it is necessary to impose topological and empirical restrictions on the knotted vector bosons by which the knotted fermions interact. For these electroweak vector fields we assume the t = 1 of the Standard Model and therefore

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j = 3 and N = 6, in accord with both (1.2) and (3.7), 1 (N, w, r + o), 2 (j, m, m )q = 3(t, −t3 , −t0 )

(j, m, m )q =

(1.2) (3.7)

that hold for the elementary fermion fields and that we now assume for the knotted vector fields as shown in Table 3. Table 3.

Knotted electroweak vectors (j = 3). 3t D−3t 3 −3t0

Q

t

t3

t0

W+

e

1

1

0

3 D−3,0 ∼ c3 d 3

W−

−e

1

−1

0

3 ∼ a3 b 3 D3,0

W3

0

1

0

0

3 ∼ f (bc) D0,0 3

The charged Wμ+ and Wμ− are described by six preon monomials. The neutral vector Wμ3 is the superposition of four states of six preons given by 3 = A(0, 3)b3 c3 + A(1, 2)ab2 c2 d + A(2, 1)a2 bcd2 + A(3, 0)a3 d3 D00

(4.2)

j according to the general representation of Dmm  in (2.11), which is reducible by the algebra (A) to a function of the neutral operator bc.

5. Presentation of the Knot Model in the Preon Representation 10 The elements (a, b, c, d) are assumed to be creation operators for both e and g preons that carry both e and g charges. The creation operator for a general state of the j knot model is given by the knot representation of Dmm  as a function of (a, b, c, d) and (na , nb , nc , nd ). This representation (2.11) implies the constraints (2.12), (2.13), (2.14) on the exponents in the following way:  j Ajmm (q|na , nb , nc , nd )ana bnb cnc dnd (2.11) Dmm  = na ,nb ,nc ,nd

where (na , nb , nc , nd ) are summed over all positive integers and zero that satisfy the following Eqs. (2.12)–(2.14), and where (a, b, c, d, ) satisfy the knot algebra (A) na + nb + nc + nd = 2j,

(2.12)

na + nb − nc − nd = 2m,

(2.13)



(2.14)

na − nb + nc − nd = 2m .

The two relations defining the quantum kinematics and giving physical meaning to j Dmm  , are again the postulated (1.2): (j, m, m )q =

1 (N, w, r + o) 2

field (flux loop) description

(1.2)

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and the semiempirical (3.7) that holds at (j, t) = (3/2, 1/2): (j, m, m )q = 3(t, −t3 , −t0 )L

particle description.

(3.9)

The preceding relations (1.2) and (3.9) imply two complementary physical interpretations of the SLq(2) relations (2.12)–(2.14). By (1.2) and (2.12)–(2.14) one has a semiclassical field description (N, w, r˜) of the quantum state (j, m, m )q as follows. The (N, w, r) are topological coordinates that may characterize either a binding field or a flux loop. In (2.14), where r˜ ≡ r + o and o is the parity index, r˜ may be termed “the quantum rotation,” and o the “zero-point rotation.” By (3.7) one has a particle description (t, t3 , t0 ) of the same quantum state (j, m, m )q , t = 16 (na + nb + nc + nd ), t3 = − 16 (na + nb − nc − nd ), t0 =

− 16 (na

(5.1)

− nb + nc − nd ).

In (5.1), (t, t3 , t0 ) are to be read as SLq(2) preon indices agreeing with standard SU(2) × U(1) notation only at j = 32 . In general, however, t3 measures writhe charge, t0 measures rotation charge and t measures the total preon population or the total number of crossings of the flux-loop.

6. Interpretation of the Complementary Equations Continued We now present an alternative particle interpretation of the flux loop equations (2.1): N = na + nb + nc + nd ,

(6.1a)

w = na + nb − nc − nd ,

(6.1b)

r˜ = na − nb + nc − nd .

(6.1c)

Equation (6.1a) states that the number of crossings, N , equals the total number of preons, N  , as given by the right side of this equation. Since we assume that the preons are fermions, the knot describes a fermion or a boson depending on whether the number of crossings is odd or even. Viewed as a knot, a fermion becomes a boson when the number of crossings is changed by attaching or removing a geometric curl

. This picture is consistent with the view of a curl as an opened preon

loop, which is in turn viewed as a twisted loop . Each counterclockwise or clockwise classical curl corresponds to a preon creation operator or antipreon creation operator respectively. Equations (6.1b) and (6.1c) may also be read as particle equations as follows. Since a and d are creation operators for antiparticles with opposite charge and

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hypercharge, while b and c are neutral antiparticles with opposite values of the hypercharge, we may introduce the charged (νa ) and neutral (νb ) preon numbers, similar to baryon and lepton numbers ν a ≡ na − nd ,

(6.2)

ν b ≡ nb − nc .

(6.3)

Then (6.1b) and (6.1c) may be rewritten in terms of preon numbers as νa + νb = w (= −6t3 ),

(6.4)

νa − νb = r˜ (= −6t0 ).

(6.5)

By (6.4) and (6.5) the conservation of the preon numbers (νa , νb ) and also of the charge and hypercharge (t3 , t0 ) is equivalent to the conservation of the writhe and rotation, which are topologically conserved at the 2d-classical level. In this respect, these quantum conservation laws for preon numbers correspond to the classical conservation laws for writhe and rotation. 7. Graphical Representation of Corresponding Classical Structures The representation (2.11) of the four classical trefoils as composed of three overlapping preon loops is shown in Fig. 1. In interpreting Fig. 1, note that the two lobes of all the preon loops make opposite contributions to the rotation, r, so that the total rotation of each preon loop vanishes. When the three a-preons and c-preons are combined to form charged leptons and neutrinos, respectively, each of the three labeled circuits is counterclockwise and contributes +1 to the rotation while the single unlabeled and shared (overlapping) circuit is clockwise and contributes −1 to the rotation so that the total r for both charged leptons and neutrinos is +2. For quarks the three labeled loops contribute −1 and the shared loop +1 so that r = −2. In each case the three preons that form a lepton trefoil contribute their three negative rotation charges. The geometric and charge profile of the lepton trefoil is thus similar to the geometric and charge profile of a triatomic molecule composed of neutral atoms since the electronic valence charges of the atoms, which cancel the nuclear charges of the atoms, are shared among the atoms forming the molecule just as the negative rotation charges which cancel the positive rotation charges of the preons are shared among the preons forming the trefoils. There is a similar correspondence between quarks and antimolecules. 8. Model of Preonic Trefoil with Binding Field Since one may interpret the elements (a, b, c, d) of the SLq(2) algebra as creation j operators for either preonic particles or flux loops, the Dmp may be interpreted as a creation operator for a composite particle composed of either preonic particles (N  , νa , νb ) or flux loops (N, w, r˜) where νa and νb are charged and neutral preon

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(w, r, o) 3/2

Charged Leptons, D 3 3 ∼ a

(w, r, o) 1/2

3

a-preons, D 1 1

2 2

ε

ijk 

2 2

aj _

o ai / _

o ak ?



3/2



2 2

cj 

o 

ci / _

ck

_

 ci

_

(−1, 0, 1)

(−3, 2, 1) 3/2

d-quarks, D 3 − 1 ∼ ai b2 2

 

ai

2

bk o ?

?

_

2

2



di

_

1/2

d-preons, D− 1 − 1

2

O /

(1, 0, −1)

(3, −2, 1)

3/2

ijk

? bi

u-quarks, D− 3 − 1 ∼ ci d2 2

2



bi

/

1/2

b-preons, D 1 − 1

2

εijk

ci

(1, 0, 1) c-preons, D− 1 1

2 2

ε

ai

1/2

Neutrinos, D− 3 3 ∼ c3

ε

?

(3, 2, 1)

ijk

169

?

/ dk o

_

? di

(−3, −2, 1)

(−1, 0, −1)

Fig. 1. (Color online) Graphical representation of corresponding classical structures: Preonic j structure of elementary fermions. Q = − 6e (w + r + o), and (j, m, m )q = 12 (N, w, r + o). Dmm  = N

D w2 2

r+o 2

. The clockwise and counterclockwise arrows are given opposite weights (∓1) respectively.

The (rotation/writhe) charge is measured by the sum of the weighted (black/red) arrows.

numbers and N  is the total number of preons. These two complementary views of the same particle may be reconciled as describing N -preon systems bound by a knotted field having N -crossings with the preons at the crossings as illustrated for N = 3 in Fig. 2. In the limit where the three outside lobes become small or infinitesimal compared to the central circuit, the resultant structure will resemble a three-particle system tied together by a string.

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(w, r, o)

(w, r, o) Neutrinos,

εijk

3/2 D− 3 3 2 2

∼c

. .. 

Charged Leptons, D 3 3 ∼ a



_

_

εijk o ck (−3, 2, 1)

3/2

d-quarks, D 3 − 1 ∼ ab2 2

εijk / ai

.. . ?

ai /

aj

_

_

2

εijk

o ak ?

(3, 2, 1)

bk o

ci (3, −2, 1)

_

2

.. . O

/



?

o

3/2





. .. 

u-quarks, D− 3 − 1 ∼ cd2

2

bj

3

2 2

cj



ci /

3/2

3



dj

?

/ dk o

(−3, −2, 1)

Fig. 2. Leptons and quarks pictured as three preons bound by a trefoil field. The preons conjectured to be present at the crossings are suggested by the blue dots at the crossings of the lepton-quark diagrams, or at the crossings of any diagram with more crossings. These diagrams may also be described as superpositions of three twisted loops.

9. Alternate Interpretation In the model suggested by Fig. 1 the parameters describing the preons and the parameters describing the flux loops may be understood as codetermined by the eigenvalues of a common Hamiltonian. On the other hand, in an alternative interpretation of complementarity, the hypothetical preons conjectured to be present in Fig. 2 carry no independent degrees of freedom and may simply describe concentrations of energy and momentum at the crossings of the flux tube. In this interpretation of complementarity, (t, t3 , t0 ) and (N, w, r˜) are just two ways of describing the same quantum trefoil of field. In this picture the preons are bound, i.e. they do not appear as free particles. This view of the elementary particles as nonsingular lumps of field or as solitons has also been described as a unitary field theory. 10 The physical models suggested by Fig. 1 may be further studied in the context of gravitational and “preon binding” with the aid of generalized preon Lagrangians similar to that given in Ref. 11. The Hamiltonians of these three body systems can be parametrized by degrees of freedom characterizing both the preons (j = 12 ) and the binding field (j = 1). The masses of the leptons and quarks (j = 32 ) can be inferred from the eigenvalues of this Hamiltonian in terms of the parameters describing these three body systems. There is currently no experimental guidance at these conjectured energies. These three body systems are, however, familiar in different contexts, namely

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H3 composed of one proton and two neutrons: P N 2 , P composed of one down and two up quarks: DU 2 , N composed of one up and two down quarks: U D2 , which are similar to the preon models U composed of one c and two d preons: cd2 D composed of one a and two b preons: ab2 where U and D are up and down quarks. In order to treat the preonic three body problems in a way similar to the other three body problems one needs to include in the Lagrangian all the terms expressing the degrees of freedom generated by the knot insertions. The Higgs mass term will also acquire left and right factors coming from the adjoined knot factors. All knot ¯ j  ...Dj  factors. insertions will be products of the D mm mm 10. Lower Representations We have so far considered the states j = 3, 32 , 12 representing electroweak vectors, leptons and quarks, and preons, respectively. We finally consider the states j = 1 and j = 0. In the adjoint representation j = 1, the particles are the vector bosons by which the j = 12 preons interact and there are two crossings of the associated classical knot. These vectors describe the interaction of the preons, including the formation of the binding field, and are different from the j = 3 vectors by which the leptons and quarks interact. If j = 0, the indices of the quantum knot are (j, m, m )q = (0, 0, 0)

(10.1)

and by the rule (1.2) for interpreting the knot indices on the left chiral fields 1 (N, w, r˜) = (j, m, m )q = (0, 0, 0). (10.2) 2 Then the j = 0 quantum states correspond to classical loops with no crossings (N = 0) just as preon states correspond to classical twisted loops with one crossing. Since N = 0, the j = 0 states have no preonic sources of charge and therefore no electroweak interaction. It is possible that these j = 0 hypothetical quantum states are realized as electroweak non-interacting loops of field flux with w = 0, r˜ = r + o = 0, and r = ±1, o = ∓1 i.e. with the topological rotation r = ±1. The two states (r, o) = (+1, −1) and (−1, +1) are to be understood as quantum mechanically coupled. If, as we are assuming, the leptons and quarks with j = 32 correspond to 2d projections of knots with three crossings, and if the heavier preons with j = 12 correspond to 2d projections of twisted loops with one crossing, then if the j = 0 states correspond to 2d projections of simple loops, one might ask if these particles

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with no electroweak interactions, which are smaller and heavier than the preons, are among the candidates for “dark matter.” If these j = 0 particles predated the j = 12 preons, one may refer to them as “yons” as suggested by the term “ylem” for primordial matter. 11. Speculations about an Earlier Universe and Dark Matter If the Dyons are Preons One may speculate about an earlier universe before leptons and quarks had appeared, when there was no charge, and when energy and momentum existed only in the SLq(2) j = 0 neutral state as simple loop currents of gravitational energy-momentum. Then the gravitational attraction would bring some pairs of opposing loops close enough to permit the transition from two j = 0 opposing simple loops into two opposing j = 12 twisted loops. A possible geometric scenario for the transformation of two simple loops of current (yons) with opposite rotations into two j = 12 twisted loops of current (preons) is shown in Fig. 3. To implement this scenario one would expect to go beyond the earlier considerations of this paper. Without attempting to do this, one notes according to Fig. 3 that the fusion of two yons may result in a doublet of preons as twisted loops, which might also qualify as Higgs doublets. In the scenario suggested by Fig. 3 the opposing states are quantum mechanically entangled  c and may undergo gravitational exchange scattering. The a doublet of Fig. 3 is similar to the Higgs doublet which is independently required to be a SLq(2) singlet (j = 0) and a SU(2) charge doublet (t = 12 ) by the mass term of the Lagrangian described in Refs. 8 and 9. Since the Higgs mass contributes to the inertial mass, one should expect a fundamental connection with the gravitational field at this point. If at an early cosmological time, only a fraction of the initial gas of the quantum loops, the yons, had been converted to preons and these in turn had led to a still smaller number of leptons and quarks, then most of the mass and energy of the universe would at the present time still reside in the dark loops while charge and o r=1 r˜ = 0

/ +

r = −1 r˜ = 0

 /

Two j = 0 neutral loops with opposite topological rotation

gravitational attraction

 _ ? a preon r=0 wa = +1 Qa = − 3e Fig. 3.

+

r=1 r˜ = 0

O

r = −1 r˜ = 0



interaction causing the crossing or redirection of neutral current flux shown below

 ? c preon r=0 wc = −1 Qc = 0 _

Creation of preons as twisted loops.

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current and visible mass would be confined to structures composed of leptons and quarks. In making experimental tests for particles of dark matter one would expect the SLq(2) j = 0 dark loops to be greater in mass than the dark neutrino trefoils where j = 32 . 12. Summary Comments on the Magnetoweak Phase To express the correspondence between electroweak and magnetoweak sectors that suggests magnetic monopoles, we have assumed that the magnetoweak charges have the same topological origin as the electroweak charges so that they are also describable as quantum trefoils. There are then magnetoweak as well as electroweak charged leptons, quarks and preons. If all masses, both electroweak and magnetoweak, are fixed by the corresponding Higgs terms, then all masses are proportional to the ¯ qj (m, m )Dqj (m, m ) as follows: 11 eigenvalues of D   ¯ qj (m, m )Dqj (m, m )|n = f (q, β, n) n|D (12.1) where |n are the eigenstates and n labels the three states of excitation in a family of either the observed electroweak leptons and quarks or the predicted magnetoweak particles. Here β is the value of b on the ground state |0 . Since (12.1) is a polynomial in q and β and of degree determined by n, the three lepton and quark masses in a family may be parametrized by q, β and n. The masses of the electroweak and corresponding magnetoweak charged preons, leptons and quarks, could then be vastly different if the fine structure constant is close to its present value; since qe and qg would be very different 2 2 e qe = qg c

2 1 ≈ . (12.2) 137 The masses would also be highly dependent on n. Since all form factors generated by the Dqj (m, m ) also depend on q in a major way, the dynamics of the magnetoweak phase will accordingly differ seriously from the dynamics of the electroweak phase. The corresponding e-form-factors and masses have so far been only roughly discussed and there has not been an adequate parametrization of the dynamics of the electroweak phase in terms of the available experimental data. The corresponding experimental data is not available for the magnetoweak phase. In both cases one would be exploring the SLq(2) extension of models with either gravitational or dyonic binding. At currently accessible accelerator energies, one may speculate that only the e-phase of the dyon field is observable, but the g-phase of this field is not, because the masses of the g-particles puts them out of reach of modern accelerators. On the other hand, it is possible that the cosmological temperatures of the early universe may be high enough to reveal a few g-particles, including magnetic monopoles.

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Acknowledgments I thank E. Abers, C. Cadavid, and J. Smit for help in the preparation of this talk. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

P. A. M. Dirac, Phys. Rev. 74, 817 (1948). J. Schwinger, Science 165, 757 (1969). H. Harari, Phys. Lett. B 86, 83 (1979). M. Shupe, Phys. Lett. B 86, 87 (1979). R. Raitio, Physica Scripta 22, 199 (1980). R. J. Finkelstein, Int. J. Mod. Phys. A, 20 (6487) (2005), R. J. Finkelstein, Int. J. Mod. Phys. A, 22 (4467) (2007), R. J. Finkelstein, Int. J. Mod. Phys. A, 24 (2307) (2009). R. J. Finkelstein, Int. J. Mod. Phys. A 29, 1450092 (2014). R. J. Finkelstein, Int. J. Mod. Phys. A 30, (2015). R. J. Finkelstein, Phys. Rev D 89, 125020 (2014). R. J. Finkelstein, Phys. Rev 75, 1079 (1949).

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The Schwinger−DeWitt proper time algorithm: A history Steven M. Christensen Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC 27514, USA [email protected]

Publications by Julian Schwinger in 1951 and Bryce DeWitt in 1963 led to the Schwinger– DeWitt proper time algorithm. Among other things, this algorithm resulted in the development of regularization techniques for dealing with divergences in quantum field theory in curved spacetimes. Studies of the algorithm and its extensions have brought about many important developments in physics, mathematics, and computer software. In this presentation, I will discuss a few of these and how they came about.

I am very grateful to the organizers of the Julian Schwinger Centennial Conference for inviting me to attend. My goal here is to discuss how two publications, one by Schwinger in 1951 and the other by Bryce DeWitt in 1963, led to many developments in physics, mathematics, and computer software. Schwinger’s paper, 1 On Gauge Invariance and Vacuum Polarization, currently has nearly 4,500 references on the inSPIRE database 2 and continues to grow almost daily. If you follow the tree of papers that reference that paper you find thousands of other publications that can trace back to it. One of the most significant of those publications is DeWitt’s series of lectures entitled Dynamical Theory of Groups and Fields 3 given at the Les Houches summer school session in 1963, Relativity, Groups, and Topology. 4 As a graduate student of DeWitt’s, one of my first jobs when we moved from The University of North Carolina at Chapel Hill to the University of Texas at Austin in January of 1972, was to organize his preprint and reprint collection. He told me that when I did this, I should look for the above paper by Schwinger with the strong suggestion, “Read Schwinger.” I did find the paper, and as with much of the Dynamical Theory of Groups and Fields, I understood almost none of it at the time. However, the paper did introduce me to key concepts like gauge invariance, Green’s functions, divergences, renormalization, and the proper time parameter — all of which slowly led me to some understanding of what the problems in quantum field theory were. Section 17 of DeWitt’s book became a constant companion in my graduate and postdoctoral years. It defined the equations, like those in the following image, that present the proper time algorithm for a very general propagator.

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Equation (17.54) in the image provides a curved space ansatz based on flat spacetime methods. The σ(x, x ) is the geodetic interval function, the covariant form of 12 (x − x )2 . The object Δ(x, x ) is the VanVleck–Morette determinant. This

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leads to Eqs. (17.56)–(17.58), the so-called recursion relations. Substituting these relations into the expression of the Green’s function, we can isolate the divergent parts of the function that appear in the Hadamard function G(1) (x, x ).

We see in Eq. (17.62) that the G(1) (x, x ) has terms that are quadratically and logarithmically divergent when x → x . In order to find the values of the an coefficients in the x → x coincidence limits, we first have to find the limits for covariant derivatives of σ and Δ. DeWitt’s initial calculation of the coincidence limits for the first four covariant derivatives is given in the next image.

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This calculation is only good for getting to the a1 coefficient in the series expansion. To get to a2 or higher, we need six or more derivatives. Further, we would then have to find the coincidence limits of Δ and derivatives of the lower a0 and a1 terms. These calculations are so tedious that hand calculations seemed almost impossible. As I will mention later, the calculations must be done perfectly; otherwise serious consequences can occur. After 1963 not much was done with this technique for several years. DeWitt did modify the algorithm to replace the mass term in the expansion with a term involving the Riemann tensor. This was done, but buried in his 1967 paper Quantum Theory of Gravity III. Applications of the Covariant Theory, 5 in Section 7.

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The script R term replaces the mass term in the algorithm in Eq. (7.24). Modifying and extending the algorithm in other ways was done by Parker and Toms, DeWitt, Kennedy and Fulling, and many others in the following years. The event that brought about the reuse of the Schwinger–DeWitt technique was the 1972 Les Houches summer school on Black Holes. In 1968, Leonard Parker published his paper Particle Creation in Expanding Universes 6 bringing on new interest in quantum field theory in curved spacetimes and the need for understanding the divergences in the theory. The notions of regularization and renormalization were of interest again. Jacob Bekenstein showed the relationships between the second law of thermodynamics and black holes in the paper Black Holes and Entropy, 7 published in 1973. At the summer school, Hawking, Bardeen, Carter, and others expanded on the relation of entropy and the horizon of a black hole. This led Hawking to write his famous 1974 paper, Black Hole Explosions. 8 At this point, there was no obvious link to these results and Schwinger–DeWitt, but that would soon change.

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Once we thought black hole radiation was possible, we had to study the socalled backreaction problem, that is, exactly how does spacetime change as the hole evaporates? Mathematically, we want to solve the equations   Gμν = Tμν , where the vacuum expectation value of the matter stress tensor, in some vacuum state we choose, can be written as       Tμν = Tμν div + Tμν ren . We want to find the divergent terms in the expectation value and eventually create a way to remove them, thus leaving a renormalized stress tensor to set equal to the Einstein tensor. This problem was discussed at length in DeWitt’s Physics Reports paper, Quantum Field Theory in Curved Spacetime, 9 published in 1975. There, he indicated how one may use the Schwinger–DeWitt procedure. In the summer of 1974, he suggested that I compute in detail the structure of the stress tensor divergences for my dissertation topic. As an aside, it was during this time that I met Schwinger when he came to Austin to give a series of lectures. He, Bryce, and I had lunch at the UT faculty club, and I spent that hour in awe of the conversation I was a part of. Schwinger’s lectures, like Bryce’s, were precise and advanced, requiring rapt attention and note taking. I can still picture those experiences with great joy. The basic stress tensor equations I used started with the action for a scalar field with a coupling to the Riemann scalar and a mass term. The special case where ξ = 16 and m = 0 is the conformal scalar field example, which plays an important role later.

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In this, we used the Hadamard expansion and any coincidence limits needed. The method came to be known as point-splitting regularization. I wrote a dissertation 10 and later two papers 11,12 that give the details of the divergences for spin 0, 12 , and 1 fields in general spacetimes. The next two images show the quartic, quadratic, logarithmic, and finite terms generated by the point-splitting technique. We see that the results are very complex. The actual calculations took me six months of eightyhour weeks to complete. One tiny error would have invalidated the results. Once in 1975, when I expressed to DeWitt the difficulty of finding consistent answers, he just responded sternly, “It’s a calculation, JUST DO IT!” Another side note. When I submitted the papers to the Physical Review, the referees approved them quickly, but the editor did not. The editor did not like calling the technique point-splitting and insisted on calling it point-separation. Everyone used the splitting name, but that argument did not go anywhere. He said “a point cannot be split.” But I wondered how it could be separated without first splitting it? I lost the argument much to my embarrassment when the papers came out and some of my colleagues chuckled or were just mystified. The ultimate results published in the second paper were

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To use these results in a specific spacetime, we must find a way to compute the vacuum expectation value via something like a mode-sum and then find out what the above divergences and finite terms look like in that spacetime. In all of this, there was one item that DeWitt objected to strongly:

This shows that even when the field is conformally invariant, implying that the trace of the stress tensor is zero, the finite part introduced by the regularization procedure is not trace free. This is the now famous Weyl or trace anomaly with the form indicated above. DeWitt told me that this result in my dissertation had to be wrong, but that I should work out why later. When I left Texas to be a postdoc at King’s College, London, I shared the office with Michael Duff. I found out that he and Derek Capper had also found the trace

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anomaly via dimensional regularization a year earlier. Duff describes this event in his wonderful summary paper, Twenty Years of the Weyl Anomaly. 13 We did eventually convince DeWitt that the anomaly was real. Duff and I determined that each of our results matched and resulted from the a2 coefficient in the Schwinger– DeWitt expansion. While at King’s College, Steve Fulling also shared the office with Duff and me. Fulling and I decided to see what the consequences of the anomaly were in a two- and four-dimensional Schwarzschild spacetime. Following on work done by Fulling, Davies, Unruh, Wald and others, we tried to solve the conservation equations T μν;ν = 0 in that spacetime. It turned out that, in two dimensions, we were able to show that the existence of Hawking radiation was exactly equivalent to the nonzero trace anomaly. Our paper Trace Anomalies and the Hawking Effect 14 showed the following result in the four-dimensional case:

← −

Through analysis in different vacuum states—Hawking–Hartle, Boulware, and Unruh—we get some idea of the nature of the function.

The full backreaction problem has yet to be solved to this day though it is important in understanding the nature of the horizon and singularity of a black hole. Starting in the late 1970s, Duff and I began to study what the an coefficients told us about issues in the various supergravity theories popular at the time. In several papers, 15,16 we found that the a2 coefficients for fields of spin 0, 12 , 1, 3 2 , and 2 could be used to derive index formulas relating the zero modes of the fields to the topological invariants: the Euler number and the Pontryagin number. In this we rederived the Atiyah–Singer index formula for a Dirac operator, the

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Hirzebruch signature formula, and the Gauss–Bonnet theorem. In that effort we received valuable input from such luminaries as Weinberg, Bott, and Erd¨ os. As I found in years before, it took us months to carefully compute a table of the structures of the a2 coefficients for spins up to 2 (we called the a2 coefficients b4 for a variety of consistency reasons related to work in mathematics). Third side note. In our paper, 16 we obtained answers very different from some other well-known researchers in the field, and we perhaps too blatantly pointed out their errors. These errors led to some notions about the early universe that were significantly wrong. This convinced us that doing the enormous calculations had to be done even more carefully than we had believed possible. At this point, the tediousness of the calculations reinvigorated my thoughts of using computers to do the work.

To give an idea of just how big the an coefficients calculations can get, the next image shows some computer output hanging from a tree in my front yard. This is just one intermediate step in the process. The printout consists of 32 single-spaced pages, two columns per sheet. This is one equation. The calculation requires that each term be compared with the others and combined if possible. Doing this by hand was just not going to be accurate. We often had to do the work five times independently to find errors, and even then the same error might still be made.

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Back in 1974, I had convinced DeWitt that computers might be useful in such calculations, at least as a check on hand calculations. Already in 1968 I had learned one of the first symbolic manipulation systems called FORMAC, developed at IBM. Unfortunately, FORMAC was not appropriate in our case as it had no pattern matching system to combine the terms in our massive results. Bryce knew of some other work being done and put me in touch with Martinus Veltman in Michigan. Veltman had developed a system called Schoonship that could be run on the Texas mainframe. He very kindly sent us a copy. It turned out that it also would not do what we needed, so we gave up on computerizing anything then. While at the University of North Carolina at Chapel Hill, I came back to this computerization issue in the early 1980s when desktop workstations of significant power came into use. I again contacted Veltman and he said he knew of a fellow at Caltech who was writing a system that might have pattern matching functionality. This was Stephen Wolfram, then a young grad student at Caltech. I contacted Wolfram and told him what I wanted to do. He said he was working on a system called SMP whose main functions included pattern matching as a key element. He

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told me I would need a UNIX-based computer system with a C compiler and as much memory as I could get. This led me to a brand new company, at that time with just nine employees, called Sun Microsystems. I visited the company in Silicon Valley and found that I needed roughly $30,000 to buy one of their Sun-1 systems. In the following year, I was able to apply to the National Science Foundation to purchase one of those workstations. It did not take long before I was able to get SMP running and learned what it could and could not do. Yet another side note. My contacts with Sun and Wolfram Research led to a 35-year relationship with both companies, including consulting on scientific and open-source software. At that time, SMP could not do what I needed, so I started writing my own C language code to specifically do just the Schwinger–DeWitt calculations. Coincidentally, Wolfram and I both ended up at the University of Illinois at UrbanaChampaign in the late 1980s. Wolfram had started to create what now is called Mathematica, and I got to be one of the first users outside his development team. With this alpha test version of his software, the pattern matching functions were much more powerful and complete than those in SMP. The C code I had by that time spent two years trying to write was replaced in two weeks by much better code written in Mathematica. I was able to start Schwinger–DeWitt calculations very quickly and to write my own functions to simplify the large equations generated. I soon found out that my colleague at the University of Wisconsin-Milwaukee, Leonard Parker, was also working to create tensor analysis functionality for Mathematica. We decided to join forces and in 1988 created MathTensor. Over the next few years, we put it out for beta testing and eventually had over 200 new functions and objects to do very sophisticated tensor manipulations. In 1994, we formed a company and over the years sold more than 2,000 copies to physicists, engineers, and many others researchers we had no idea were using tensors. A book 17 that acted as a manual for the software was also created. So, what did all of this 20-year effort lead to in the Schwinger–DeWitt calculations? I found that in a day with MathTensor, I could write code that would do coincidence limits of all the objects needed to compute up to as ; the run time for this calculation is only 12 seconds! The first page of the code to do the σ derivative coincidence limits is shown below. This was amazing compared to months of work back in grad school. Furthermore, we showed that it was possible with something like Mathematica and MathTensor to extend such calculations to a3 and a4 . This is an ongoing effort.

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There is another element of these calculations that needs to be understood. As we compute to higher and higher covariant derivatives of σ, Δ, a0 , a1 , a2 , and above, we generate complex products of Riemann tensors, Ricci tensors, Riemann scalars, and their derivatives. We need to apply all possible symmetries and identities of these objects to reduce the number of terms to some basic set. It turns out that people had already been looking at this problem for a while. For example, in

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1975, Peter Gilkey published his paper The Spectral Geometry of a Riemannian Manifold, 18 in which he calculated the terms in the a3 coefficient. In 1992, Fulling, King, Wybourne, and Cummins in their paper Normal forms for tensor polynomials: I. The Riemann tensor 19 discussed ways to find a preferred basis for converting an arbitrary Riemann tensor polynomial into what they call its “normal form.” Using the representation theory of symmetric and other groups, they could list all the products of Riemann tensors. Some products of three Riemann tensors are shown in the next image.

Earlier, in 1980, I had looked at the issue of finding such a basis in a brute force way for curved manifolds with torsion. I did not find a basis at that time, but did list the possible structures that might appear and some of the relations between them. This was another clue to the need for computerization of such things. Here are two pages from my paper Second- and fourth-order invariants on curved manifolds with torsion. 20 There has recently been new interest in such manifolds with regard to Schwinger–DeWitt coefficients. Calculations to computerize these are underway.

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Another important development with regard to using a computer to solve the Riemann tensor basis problem for the Schwinger–DeWitt algorithm was development of the Schur software system by Brian Wybourne. Wybourne created Schur as an interactive package for calculating properties of representations of Lie groups and symmetric functions. I worked with Brian to compile his code for various systems and, for a number of years before his death, we sold the software to support its development. Later, the code was released as open-source. 21

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In the years since the 1970s, many papers and books have been published, outlining the details of the Schwinger–DeWitt method (also called the heat kernel method). I will list just a few of these. There are many more and even more still being published. Heat Kernel Method and its Applications by Avramidi. 22 Heat Kernel and Quantum Gravity by Avramidi. 23 The Generalized Schwinger-DeWitt Technique in Gauge Theories and Quantum Gravity by Barvinsky and Vilkovisky. 24 Transport Equation Approach to Calculations of Hadamard Green Functions and Non-coincident DeWitt Coefficients by Ottewill and Wardell. 25 (Wardell has also developed a Mathematica-based system for doing these calculations.) Stress Tensor for a Scalar Field in a Spatially Varying Background Potential: Divergences, Renormalization, Anomalies, and Casimir Forces by Milton, Fulling, Parashar, Kalauni, and Murphy. 26 We see that since 1951 and 1963, the work of Schwinger and DeWitt has led to exciting developments in physics, mathematics, and computer software. These developments have not only contributed to major theoretical advances, but have also resulted in many papers, books, and uses in applied research. In my case, they led to my creation of three companies that have sold thousands of software packages, paying back in taxes any grant money that may have contributed to my work over the years. I suspect neither Schwinger nor DeWitt would have imagined any of this. Acknowledgments I wish to thank Berge Englert and the conference organizers, especially Yee Jack Ng, my colleague at UNC-Chapel Hill for inviting me to the very interesting event in Singapore. Thanks to Steve Fulling, Ivan Avramidi, and Andrei Barvinsky for their guiding thoughts on the history presented here. I also greatly appreciate the careful reading of the manuscript by Victoria Baehren. References 1. J. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951). 2. http://inspirehep.net/record/113. 3. B. DeWitt, Dynamical Theory of Groups and Fields (Gordon and Breach, 1965). 4. C. DeWitt-Morette and B. DeWitt, Relativity, Groups and Topology: Lectures Delivered at Les Houches During the 1963 Session of the Summer School of Theoretical Physics, University of Grenoble (Gordon and Breach Science Publishers, 1964). 5. B. S. DeWitt, Quantum theory of gravity. III. Applications of the covariant theory, Phys. Rev. 162, 1239 (1967). 6. L. Parker, Particle creation in expanding universes, Phys. Rev. Lett. 21, 562 (1968). 7. J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973). 8. S. W. Hawking, Black hole explosions, Nature 248, 30 (1974).

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9. B. S. DeWitt, Quantum field theory in curved spacetime, Phys. Rept. 19, 295 (1975). 10. S. M. Christensen, Covariant coordinate space methods for calculations in the quantum theory of gravity, PhD thesis, The University of Texas at Austin (1975). 11. S. M. Christensen, Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method, Phys. Rev. D 14, 2490 (1976). 12. S. M. Christensen, Regularization, renormalization, and covariant geodesic point separation, Phys. Rev. D 17, 946 (1978). 13. M. J. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11, 1387 (1994). 14. S. M. Christensen and S. A. Fulling, Trace anomalies and the Hawking effect, Phys. Rev. D 15, 2088 (1977). 15. S. M. Christensen and M. J. Duff, New gravitational index theorems and supertheorems, Nucl. Phys. B 154, 301 (1979). 16. S. M. Christensen and M. J. Duff, Quantizing gravity with a cosmological constant, Nucl. Phys. B 170, 480 (1980). 17. L. Parker and S. M. Christensen, Math Tensor: A System for Doing Tensor Analysis by Computer (Addison-Wesley, 1994). 18. P. B. Gilkey, The spectral geometry of a Riemannian manifold, J. Differential Geom. 10, 601 (1975). 19. S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins, Normal forms for tensor polynomials. I. The Riemann tensor, Class. Quantum Grav. 9, 1151 (1999). 20. S. M. Christensen, Second- and fourth-order invariants on curved manifolds with torsion, Journal of Physics A: Mathematical and General 13, 3001 (1980). 21. https://sourceforge.net/projects/schur/. 22. I. G. Avramidi, Heat Kernel Method and its Applications (Birkh¨ auser, Cham, 2015). 23. I. Avramidi, Heat Kernel and Quantum Gravity (Springer, 2000). 24. A. O. Barvinsky and G. A. Vilkovisky, The generalized Schwinger-Dewitt technique in gauge theories and quantum gravity, Phys. Rept. 119, 1 (1985). 25. A. C. Ottewill and B. Wardell, Transport equation approach to calculations of hadamard green functions and non-coincident dewitt coefficients, Phys. Rev. D 84, 104039 (2011). 26. K. A. Milton, S. A. Fulling, P. Parashar, P. Kalauni and T. Murphy, Stress tensor for a scalar field in a spatially varying background potential: Divergences, “renormalization”, anomalies, and Casimir forces, Phys. Rev. D 93, 085017 (2016).

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From Julian to Jupiter: Unanticipated outcomes∗ Margaret Kivelson Earth, Planetary and Space Sciences, UCLA, Los Angeles, CA 90095-1567, USA Climate and Atmospheres and Space Physics, University of Michigan, Ann Arbor, MI 48109-2143, USA [email protected]

I never expected that my PhD work on quantum electrodynamics, completed under the tutelage of that superstar of Physics, Julian Schwinger, would serve as a foundation for a life in research that focused on problems that rarely include an . I shall talk about my path from QED to the exploration of magnetized plasmas in the solar system and mention some areas of interest to me, including theoretical studies of the interactions of particles with magnetohydrodynamic waves, analysis of magnetic reconnection in the Earth’s magnetosphere, studies of the structure and dynamics of planetary magnetospheres, and inferences regarding properties of the moons of Jupiter and Saturn from their interactions with the flowing plasma in which they are embedded. This celebration of the 100th anniversary of Julian’s birth gives me an opportunity to celebrate his influence on diverse studies by describing some of my contributions to plasma- and planetary-science.

I often wonder how, as a newly minted graduate of Radcliffe, I summoned up the nerve to ask a distinguished and seemingly aloof professor to be my advisor. Yet I did. Julian welcomed me to join the dozen students studying under his direction, the only woman among more than 70 doctoral students whom he mentored over his career. It is important to state that Julian never seemed to treat me differently from the men in our cohort, so I ploughed through the graduate program with confidence shattered only occasionally by my own recognition of the truly extraordinary abilities of others in my cohort. Over a few years, I mastered many tools of physics in the classroom, in the graduate student office where Julian’s students learned from one another, and in the infrequent hours of one-on-one interaction with our advisor. Julian turned out not to be quite so aloof as I had thought. My discussions with him were remarkably fruitful. He zeroed in on my problems and quickly identified what I should try next, leaving me with enough new ideas to keep me busy for months. That was fortunate because I moved to California before completing my dissertation. Despite the distance, which was far more of an obstacle before jet planes handled civilian travel, I was able to finish my dissertation on Bremsstrahlung of Ultrarelativistic Electrons with little additional guidance. But, although I learned from Julian about the inevitability of field theory and its power in accounting for fundamental properties of quantized systems, my road as a physicist post-PhD went in a different directions that I found both challenging, and rewarding.

∗ Transcript

of the video lecture recorded for the Schwinger Centennial.

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Julian’s name opened opportunities on the job market that might not otherwise have developed. At the RAND Corporation, the problems I was asked to solve were remote from quantum electrodynamics (QED), but Julian’s lessons were relevant beyond the limitations of subject matter. Julian’s students learned to think in both mathematical and physical terms, always being aware of what can be measured (as we are reminded in the title of his Quantum Mechanics text† ). At RAND, I worked on problems such as: properties of electrostatic waves in plasmas, papers in which I used concepts of QED to sum over all orders in the coupling constant, and equation of state of hydrogen at megabar pressures, of which the only examples were H-bombs and the deep interior of Jupiter. Unfortunately, I never ran into a problem for which I could introduce the “well known Gegenbauer polynomials” that Julian had introduced with a flourish in one lecture that I well remember. My interests diverged even farther from QED when, a decade after arriving in California, I accepted a research appointment at UCLA to work on problems related to radio-frequency emissions from Jupiter. Those studies introduced me to Jupiter’s magnetic field and its interaction with the Galilean moons. I started learning about the properties of space plasmas and soon accepted an invitation to join a group studying Earth’s magnetosphere. Using data from a spacecraft in Earth’s magnetosphere and working with a stimulating group of collaborators, I found evidence of magnetic reconnection, then a suspect concept, between the planetary magnetic field and the field of the solar wind. Soon I was analyzing data and developing theory to understand the properties of magnetohydrodynamic waves in Earth’s magnetosphere. I had no idea at the time that within the next decade I would find myself distracted from studies at Earth after becoming the magnetometer Principal Investigator on a NASA mission to Jupiter, ultimately named Galileo. Some of the problems related to Jupiter (and later Saturn) that have attracted my attention and which I investigated with collaborators and students are the following: • The dynamics of Jupiter’s magnetosphere. A magnetosphere is a region of space around a magnetized body within which the planetary magnetic field is largely confined by a flowing plasma. Magnetospheres of magnetized planets such as Earth and Jupiter form within the solar wind, a tenuous plasma that flows radially from the sun at speeds that exceed the fastest wave speeds of the large scale system. Jupiter’s magnetosphere differs from Earth’s because of the importance of rotation-dominated stresses, and I have examined the effects of rotation on plasma-field interactions. The spatial scales of the magnetospheres of Earth and Jupiter are imposed partly by the strength of the planetary magnetic field (an order of magnitude larger at Jupiter) and partly by the dynamic pressure of the solar wind (an order † Julian Schwinger, Quantum Mechanics, Symbolism of Atomic Measurements, edited by BertholdGeorg Englert (Springer Verlag, New York, 2001).

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of magnitude smaller at Jupiter); another factor of two in spatial extent arises because of Jupiter’s rapid rotation, which stretches the field radially away from the rotation axis. At Earth, gravity balances centrifugal force at 6.6 Earth radii, not far from the distance of ∼ 10 Earth radii to which the solar wind confines the magnetosphere along the Sun–Earth line. At Jupiter, the balance occurs at 2 planetary radii in a magnetosphere that extends of order 100 planetary radii towards the Sun. Rotational effects are significant everywhere outside of a 2RJ cylinder aligned with Jupiter’s spin axis. Plasma is pushed down rotating flux tubes towards the magnetic equator, forming a comparatively high density structure called the plasma sheet. In the outer regions of the magnetosphere, rotational stresses can cause bubbles of plasma to break off and be lost to space. Rotational stresses break the symmetry of phase space distributions, which can lead to heating of the plasma; I have tried to understand how and where such heating occurs. • Ganymede’s magnetic field and magnetosphere. Data from the Galileo magnetometer proved rich in information about the moons of Jupiter. It was already known to be covered with volcanoes, whose existence was understood in terms of the massive tidal stresses that accounted for heating of magma below the surface. However, the other moons were thought to be inert, having cooled over eons since their formation. That picture was overturned when we discovered a planetary magnetic field at Ganymede, the largest moon. It is still not totally clear how the iron core of Ganymede has retained enough heat to provide the fluid layer required for a planetary dynamo but the evidence is incontrovertible. Ganymede forms an unusual magnetosphere through its interaction with slowly flowing plasma trapped in Jupiter’s magnetic field, and studies of this system have been helpful in understanding the role of reconnection in a different plasma regime. • A global scale, subsurface ocean at Europa. The surface of Europa is low density matter referred to as ice and dominated by water. The layer is probably about 100 km thick and the ice at the surface is solid. How does a magnetometer discover that there is an ocean under the ice? By detecting the signature of a magnetic field induced by the time-variation of Jupiter’s field at Europa’s orbit, a response that requires a conductor far better than solid ice. A slightly salty, global scale, liquid water ocean accounts well for the observations. The initial inference that the signal was that of an induced field was confirmed late in the Galileo mission when we found that the magnetic perturbations at a different phase of the time-varying driving signal had reversed polarity. The link between water and life is so great that our discovery has generated enthusiasm for searching for life on Europa, and so Europa is now the target of the Europa Clipper mission, to be launched in 2022. That mission will look for places near the surface

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that seem conducive to life, places to be targets of later missions. It will also collect data that will reveal the structure and extent of the ocean that we know must be there. • Planetary period oscillations at Saturn. Post Galileo, I was lucky to join the Cassini magnetometer team to study yet another planetary magnetosphere. There I have focused on the mysterious signals that pulse at a period close to the period of planetary Saturn’s rotation period, but varying on time scales of year. The period varies slightly from year to year, so not too rapidly to be imposed directly by planetary rotation. The only plausible way to drive time-varying magnetospheric periodicities at close to the planetary rotation period is through structures fixed in the rotating upper atmosphere that couple to the ionosphere through collisions and to the magnetosphere through field-aligned currents. I have been exploring this model and have been able to demonstrate that it can account for all of the pulsating properties of the system. The missing step is to understand how the structures of the upper atmosphere form. I shall conclude these remarks by mentioning briefly my interactions with Julian in the years beyond those of graduate study. After the Schwingers moved to Los Angeles, long after my graduate studies ended, my husband, Daniel, and I became not only colleagues of Julian’s (at UCLA) but close friends of both Julian and Clarice. We shared an enthusiasm for food and wine (I recall with pleasure a few that included a bit too much of the wine) whether at dinners in our homes or at the great restaurants of Los Angeles and elsewhere. We (not Clarice) greatly enjoyed skiing and tennis independent of expertise, and took delight in lively conversation on topics from collecting art to politics. What a privilege to have been given a chance to work and to play with one of the great scientists of the ages!

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Margaret Kivelson (2017)

The RAND Corporation in Santa Monica

Julian Schwinger (1960s)

UCLA campus

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Jupiter’s inner magnetosphere and moon interactions; image courtesy of Jet Propulsion Laboratory, Pasadena, CA.

Magnetic reconnection at Earth’s magnetopause.

Jupiter’s magnetosphere is far larger than Earth’s, with field stretched radially by centrifugal effects; image from Kivelson and Bagenal.

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Ganymede’s magnetic field.

Artist’s rendition of the Europa Clipper passing near Jupiter’s moon, Europa.

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A global scale, subsurface ocean and magnetosphere at Europa.

Planetary period oscillations at Saturn are modeled in a magnetohydrodynamic simulation that pulses periodically.

Julian admired the work of the Pueblo potter, Maria Martinez, and treasured one of her pots (not this one).

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Julian Schwinger — Recollections from many decades Stanley Deser Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA Physics Department, Brandeis University, Waltham, MA 02454, USA [email protected]

I present some reminiscences, both personal and scientific, over a lifetime of admiration and friendship with one of the Grandmasters of our subject.

Dear students, friends, and admirers of Julian Schwinger, or all three. We are here to celebrate and commemorate a century since Julian’s birth. He only lived three quarters of that period, unfortunately dying far too young at 76, but left us a great legacy. Being in this Conference’s history section, I will try to discuss the life and work as I saw it, minus technicalities. I knew Julian for three-fifths of his life, a reasonable fraction. It began when I arrived as a graduate student in the fall of 1949. I didn’t know much physics, nor did I know who Julian was, but I was soon educated on the latter. In fact, I sat in on three of his quantum mechanics courses, all different. Like everybody else who wanted to do theory, I was convinced that Julian should be my mentor. He was willing to accept just about anybody, but he was chary with his time, as all his students know. Since he has saved my life so many times, I feel I should begin by giving some examples. The system at Harvard in my day, if you wanted to do theory, required you to take a qualifying exam, usually in something called Math and Mechanics, which covered various sins. One was supposed to bone up on that during one’s second academic year; a jury of one’s would-be advisor plus two other people was then convened. The day duly came and Julian arrived, flanked by Abe Klein and Bob Karplus, two up-and-coming assistant professors whose careers depended critically on sufficiently impressing Julian so they could get good positions elsewhere — one didn’t get promoted from within. They had discovered something in their latest calculations, some particularly uninteresting but technical stuff called dilogarithms, which are now, I suppose, taught in kindergarten but in those days unknown to anyone — certainly to me. They proceeded to show Julian how brilliant and clever they were, at my expense, so that after the first few words, I was totally excluded from everything, and after an hour and a half of this, they turned to me and pityingly asked me a question like what two plus two was, at which point I couldn’t even have answered one plus one. And so this terrible ordeal ended, I walked out, and two minutes later Julian came and said, “you realize you failed your qualifying

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exam,” and I said “yes,” and there was a little pause and he went on, “don’t worry about it.” I think this miracle (and miracle it was — no one else failed M&M) may have been due to my performance on an advanced electrodynamics course I had just taken with him. Then I started on my thesis. I think I probably saw Julian for a total — just on the upper limit — of about ten hours during those two years. One day, in the spring of my fourth year, I asked Julian, when would I could possibly think of finishing up. When he replied “right now if you want” — this was shortly before the strict Harvard deadline for submitting a thesis, I was not going to let this opportunity slide; somehow it all got done and typed on a Bible paper, only available in one place in the world, and bound in one particular way, and all the rest of it. Although the thesis was mediocre, I was handed my PhD by James Bryant Conant, in his last year of a long tenure as president of the university. Rescue number two was a bit more indirect. In those days, Julian would simply phone Oppenheimer at the Institute for Advanced Study, tell him who his latest graduates were and Oppenheimer would take them, no applications or recommendations. Unfortunately, the year before mine, Julian’s choice at that point was a very strange guy, we’ll not name names, who was found in his first year at the Institute climbing the wall of some estate in Princeton, something frowned upon at such a rich community. The whole thing was handled very well, all airbrushed out. He disappeared, and I’m told became a successful psychoanalyst, but that could be apocryphal. In any case, Oppenheimer was taking no chances, so he told Julian that his two picks, Roger Newton and I, had better show up and pass a psychiatric exam. In those days Oppenheimer still had his clearance, so two FBI agents were guarding his files; I walked past with trepidation, but all Oppenheimer did was ask me what my thesis was about, the title of which I told him. He immediately told me (a) what was in the thesis and (b) why it was wrong. He was way off the mark, at least on point (a); he had no idea whatsoever, but that was in his style. At least I didn’t have any obvious tics. I was vetted also by the younger permanent people at the Institute, and Roger also passed with flying colors. We were installed at the Institute where I had my two years, and not so much contact with Julian. However he saved me because when I arrived at the Institute not too sure what to do, I was immediately pounced on by Murph Goldberger and Walther Thirring who were both visiting there. They said, “you must know all of Julian’s tricks, so let’s get moving and apply them to the following project.” Of course I didn’t know Julian’s tricks, but it in fact provided my first successful extra-PhD experience and did use some of them after all. I should mention — going back a bit — that before you start on a thesis, you’re given a little test problem by your advisor. Julian gave me the little test problem, of which I had no idea whatsoever at all what to do, the reason was that this little problem was the beginning of his celebrated National Academy of Sciences series that to this very day is a standard tool. So when he

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showed me what he had done, I realized why I hadn’t a clue as to what to do. Well, that too he accepted. So I learned from that that one should do onto others and give would-be graduate students a certain amount of leeway, perhaps not as much as he gave me, but still. Then came my second postdoc stage. After the two years at the Institute, I went off for two years to the Niels Bohr Institute in Denmark, which was a difficult period for me. I only wrote one paper, which was furthermore wrong, although wrong in an interesting way. In any case, in those days especially, I hadn’t realized that, once you go into exile, you no longer exist in the United States, because you’re not in any loop. Fortunately, Julian came by that summer, visiting Denmark with Clarice, and he again saved my life by offering me one year as his assistant as an Instructor at Harvard, while I found my footing back home. That was truly critical, because being married having a baby, it was clear that I needed some sort of a job. He then also recommended me for my first faculty position at Brandeis. So, this was the support I got from Julian: his faith in me was truly beyond any requirement. His greatest confidence in me occurred much later. I was an invited visitor to UCLA, where Julian had moved, and used to stay in his house. Once I came it was during one of those oil embargos when you couldn’t get any gas for your car, especially in California. Julian lived in Bel Air, which had, and has still, for all I know, one and only one gas station, at some chi-chi little shopping center. It was going to open at 7 AM until the gas ran out by 7:30, and Julian was of course in a terrible quandary because 7 AM is too late for staying up and far too early for getting up. I was still on Eastern time, so 7 AM suited me fine, but would he entrust his precious sports car? He agonized all evening and then finally handed me the keys, gave me a three-hour lecture on how to drive, and I’m sure had a very restless night. I arrived at 7 AM, surrounded by all the neighborhood Bentleys and Rolls-Royces, chauffeurs waiting in line, but I did manage to snag sufficient gas for the next period and avoided having any dents in the car, which Julian inspected carefully. Our relations became more even with time. In particular, after the birth of supergravity in ’76, Julian asked me to come for a weekend tutorial for him and his entourage at UCLA. So there, on a Saturday at some ungodly early hour like 10 AM, we started on a full Soviet-style two day session; Julian would say, “I don’t understand what this is,” and I replied “come on, Julian, you invented it all,” and reminded him of the Rarita–Schwinger equation, which he did indeed vaguely remember — they had actually a fairly ugly form for it — but they had found it. I suggested that in fact Julian should have discovered supergravity, as he had all the tools. He was of course one of the few people in the United States, back when he was student in the ’30s, who even knew general relativity. Just like he knew quantum mechanics. You all know the famous story of how when he was flunking out of City College, Lloyd Motz brought him to Rabi to try to get him a transfer to Columbia; he must have been in the teens. He was put on a bench while they

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argued about some quantum mechanics problem unsuccessfully, and then he spoke up, adding the one word that explained it all. Motz and Rabi finally realized he was in the room, and the rest is history. He was also an extremely cultured person, and though very shy, he was really quite up on a number of extra-physical things that one might not have expected. I realized that also during the year that I was his assistant in ’57–58, because I was then in the position that we graduate students used to envy. When we were students, with his assistants, instead of seeing us he would walk off to the restaurant of his choice, there was only one at Harvard Square those days that was even semiedible, and leave us in the lurch. So this time it was my turn to leave the students in the lurch, and we would talk about all sorts of things, physics, and non; a great educational opportunity. The range of Julian’s discoveries and inventions, formal as well as directly physical, is, as we know, enormous. He was of course a great master of Green functions and everything related to gauge fields. He was one of the predecessors of the weak interaction theories that were then soon developed — Shelly Glashow was his student. Julian was extremely active not only during the great triumph of QED, quantum electrodynamics, but after that stayed very much in touch with developments. However, and this is my own theory, the Moses complex, that great men who are handed the truth from up above, are fated never to set foot in the promised land, and Julian is certainly an example. He felt at a certain point, especially in his last times at Harvard, that he was more or less sidetracked from what was then the main line, and that’s when he made his motto “If you can’t join ’em, beat ’em,” the origin of source theory, which engaged him for quite a long time, and which of course was a very interesting way to look at quantum field theory, although not really as productive as he might have hoped. He also had his engagements with cold fusion, and I think it was all part of a reaction — the Moses reaction, not being able quite to go to the next stage, such as it is, our standard model “promised land.” But he provided an enormous amount of impetus through his students. I remind you that Julian had seventy-some PhDs to his credit. That is an amazing number, four of whom — if you count that way — were Nobel Prize winners, not bad. Roy Glauber, Shelly Glashow, Ben Mottelson and Walter Kohn, although he got it in Chemistry, was very much a Schwinger product. In fact, my first quantum mechanics course was taught by Walter, it was one of the many Schwinger QM variants. So, his influence both with the early PhDs and of course later on with the cohort at UCLA, many of whom will be speaking and reminiscing here, should be very much counted as part of his contribution to our subject. But really — when you think about it, there’s almost nothing we do in theory that doesn’t somewhere bear Schwinger’s imprint. In fact, in the old days, there was a great form that you could fill out in order to publish a paper: it started with “According to Schwinger . . . ,” then you would put in the equation of your

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choice, and then went on and said “. . . now using the Green function appropriate to this problem we discover that . . . ,” and the paper was guaranteed to get into the Physical Review. That was not so far from true. But of course, Julian was much more than that. He was involved in early postwar nuclear physics; I took a full year’s nuclear physics course from him. This was really dirty nuclear physics and its phenomenology, effective range theory, scattering and bound states, again from an effective theory point of view, it was a very powerful tool. It was not quarks, but it was a way to understand, classify, and normalize low-energy nuclear physics, as it was then practiced, a forefront field. He had an enormous effect on classical electrodynamics. We all know that during the war his work on waveguides and propagation was not only very useful to the war effort, but really began a whole field of investigations in that time — and then of course, QED as I mentioned. I still remember his lecturing, his derivation, first of course of α/2π, as he did it in class, and then of the Lamb shift, the really great achievements of QED. He did it in real time on the blackboard, sans notes. I have since taught the Lamb shift and I couldn’t do any better than Julian had done 20 years earlier. Sometimes, of course, like all great men he faltered. There was an early famous incident. Every Wednesday afternoon the joint theoretical seminar used to oscillate between Harvard and MIT. Julian started one, claiming to have completely solved the closed form of QED, which was clearly going a little too far. Finally, at some point Francis Low pointed out that in order to do this Julian had assumed that a four-point function was simply the product of two two-point functions, and of course if you assume that, you’re talking about a free theory. So that particular attempt did not work. I mention this to say that Julian, like all great men, was of course fallible, but fallible in a very trivial way, never on the real essentials. I think the only person who was infallible was Enrico Fermi; he was called the pope because of that. But in Julian’s frontier explorations, you had to take the risk of being wrong in order to get anywhere. He also was interested in general relativity later, so when we — ADM: Arnowitt, Misner, and I — especially Arnowitt and I at the end of my year at Harvard began our work on the theory, Julian got interested and wrote a couple of papers. He also tried to do something about supersymmetry which didn’t quite work out. There is actually very little in modern theory that he was not at least cognizant of if not actively pursuing, so that the brain never really stopped working. For example, he set Wally Gilbert, George Sudarshan and me on dispersion work during that Harvard year that is still useful. As I said, later on he became a little bit isolated from the mainstream community, but that’s not that he was unaware of what they were doing, he knew what was going on, and although he never made it into the standard model, he certainly laid its groundwork in many directions. The work that he did in electrodynamics evolved into the heroic calculations that his graduate students — Charlie Sommerfield, for

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example — utilized to do the two-loop magnetic anomalous moment corrections. That Julian stayed away from Feynman diagrams is a trivial difference which has wrongly been blown up. Whether you have a Green function or a line makes no difference; at the end of the day, you are doing the same integrals. I give these — among many other — examples to indicate the sheer scope of Julian’s inventions. In the early days, the Rarita–Schwinger equation, which was the first serious attempt at going beyond spin 1 — was completely virgin ground at that point, and of course the significance of going beyond spin 1 and its problems did not emerge until very much later, again very much before its time as often was with Julian and his work. During the war, he, like all the other physicists, concentrated on applications — all were either in Los Alamos or the Rad Lab, where Julian chose to go. It was not until after the war and he was lured to Harvard that he was able to really go full speed again in fundamental physics. The aura that Julian had around him at Harvard was really unparalleled and well-earned. It was a sad day for Harvard when he was lured to UCLA, although they did quite well with Steven Weinberg as a replacement, I should immediately add. UCLA was greatly enlivened by the presence of Julian and his group, and although I suspect that eternal sunshine and tennis had something to do with his move, he was really quite happy at UCLA. He died of a cancer with a guaranteed lethal outcome, but I think he enjoyed his life to the very end. Clarice was his ideal companion. He has left the memories we all know for his successors. He has made us all his ex-students. Even people who were not yet alive when he died benefited from the foundations that he laid to our field. I can think of no better exemplar in every respect. It was, for me, a great honor to be his student, as I’m sure will be echoed by everyone in this room. Thank you. Aknowledgments This work was supported by grants NSF PHY-1266107 and DOE#desc0011632. It is a transcription of an invited video contribution to the Schwinger 100 Symposium, Singapore 2018.

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Schwingerians∗ Charles Sommerfield Department of Physics, Yale University, New Haven, CT 06520-8120, USA

Schwinger influenced many physicists during his productive lifetime. I recall my interactions with him and others in this group as I look back over my career. I include both the professional and the personal.

I want to thank the organizers for inviting me to speak here, even though at a distance in time and space. I would very much like to have been here in person but, as the old song says, “I don’t get around much anymore.” I begin by referring to the last time that I heard Julian give a physics talk. The occasion was Ken Johnson’s 65th birthday and the topic was sonoluminesence. He started off by asking the audience to indulge him for his decision to read his remarks. Now I ask the same myself. I will talk about my interactions with Julian over the years and I expect that there will be much overlap with what other speakers have to say. Also, I cannot vouch for the accuracy of some of these reminiscences. Much time has passed. I came to Harvard from Brooklyn College (as had Abe Klein and Stanley Deser before me) in 1953. I originally did not want to apply to Harvard for I found its catalog to be a bit unimaginative. However, my good Jewish mother told me “Charles, do me a favor. Apply to Harvard. It’s a good school.” During my first year as a graduate student Julian was away. I took a traditional course in quantum mechanics from Willis Lamb (who was visiting from Columbia) and Wendell Furry (then being hounded by the McCarthy Committee). When Schwinger returned he lectured on advanced theoretical nuclear physics, which featured such topics as effective range theory. His lectures were spell-binding. The problem was that when I reviewed them later, there were things that seemed plucked out of thin air. I just couldn’t reproduce some of his arguments. I came to the realization that Julian would have made a very good used-car salesman. He would convince you that black was white, using such phrases as “it is very natural that” or “we are led to.” Julian was always late for his noon class. He told me later that he had always intended to get there on time but somehow had never quite made it. Since the graduate dining hall closed at 1:30 this tardiness was often a difficulty for some of us. ∗ Transcript

of the video lecture recorded for the Schwinger Centennial.

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He would invariably end his lectures standing by the door and immediately disappear. This made it difficult to ask questions. In fact, there was an apocryphal story to the effect that the last student who had asked a question in one of Julian’s classes, had met an untimely end during the following summer break. Nevertheless, undaunted, I raised my hand one day just as he got to the exit and announced that I had three questions. I don’t remember what they were, but he stopped, listened to them, answered them politely, and continued out the door. It appears that I am still around to tell the story. When the fall semester started in 1955, 11 students showed up in his office seeking a thesis adviser. I won’t try to name them all here. Julian assigned us three problems, one of which involved the anomalous magnetic moment of the electron. There was much collaboration among us supplicants in developing the solutions. He had us return one at a time to discuss what we had submitted. At my meeting with him, he suggested that I continue the calculation of the anomalous moment to the next (fourth) order. Two other assignments that I still remember are those to Marshall Baker to continue the development of the determinantal method (known outside Harvard as the N/D method) and to Shelly Glashow to “think about vector mesons.” Things ended up with Julian agreeing to serve as thesis adviser of all 11 of us. I turn now to my thesis. The magnetic moment could be determined from either the tensor part of the electromagnetic interaction vertex or from the dependence of the self-energy of an electron on an applied magnetic field. The first method had been used by Bob Karplus and Norman Kroll at the Institute for Advanced Study in 1949 and the result was in full agreement with the experiments at the time. Schwinger wanted me to use the other method, while respecting gauge invariance at every step. Many years later Roy Glauber told me that the faculty was not entirely happy that a graduate student had been given such a problem. The self-energy (or mass operator) in an applied electromagnetic field had been calculated in a gauge-invariant manner, in a thesis by Schwingerian Roger Newton and applied to electron–photon scattering. I looked up Newton’s thesis. It described the most complicated calculation I had ever seen. Just stating the result had required a five-page equation. And if this wasn’t enough these were followed by about 13 more pages of definitions of the multitude of symbols used in the first five. I started to feel very sick. After some serious thought, however, I realized that since I would have to work only to first order in the magnetic field, things would simplify considerably, to say the least. I must mention that I never took a formal course on quantum field-theory (or as Julian referred to it, the theory of quantized fields) at Harvard simply because at the time there wasn’t one. I learned the subject from a series of lectures at the Naval Research Laboratory in Washington from Schwingerian Richard Arnowitt. This necessitated my driving completely across Washington from the old National Bureau

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of Standards campus in the Northwest, where I was working for the summer, to the extreme south of the city. Schwingerian Danny Kleitman attended the lectures as well. They were essentially an expanded version of Julian’s two very concise papers on coupled fields in the Proceedings of the National Academy of Sciences. Making an appointment to see Julian for help was such a complicated process that by the time the appointed time rolled around I often had answered any questions I had myself. Was this his method of motivating self-reliance? After working away for a while, however, there was a question that I couldn’t answer myself. Did merely repeating a calculation demonstrate sufficient originality to justify a Ph.D.? In addition, I didn’t think I was learning very much physics from this enterprise. Julian did not respond directly to this but told me the following: Kroll was known to be very careful in his work and so the calculation was probably correct. On the other hand, Karplus could sometimes be a bit sloppy and the result could very likely be wrong. Hence, he put the odds of correctness at 50-50 and that if I got a different answer he would give me two degrees, never alluding to what would happen otherwise. I later found out that while I had been plugging away there had been a new measurement which disagreed with the old one. As I proceeded along, I kept finding algebraic errors. When I extrapolated my changing results to the day of the Harvard commencement, they converged right to the Karplus and Kroll value. Fortunately, things stopped changing at about the end of February 1957. In the meantime, Schwingerian Paul Martin had gone to the Niels Bohr Institute in Copenhagen and had spoken to Andr´e Petermann, a postdoc with the Swedish theoretician Gunnar K¨allen. Martin told Petermann about my work but the result he gave him was an earlier one that was no longer valid. Well, this nearly drove Petermann out of his mind, for Petermann had placed rigorous bounds on the theoretical value and neither my preliminary result nor that of Karplus and Kroll fell within those bounds. In the end, however, after both of our calculations were completely finished they were in agreement with each other but not with Karplus and Kroll. We agreed to cite each other’s work when published. However, Schwinger and K¨ allen had had a somewhat acrimonious discussion (at least on K¨ allen’s part) at a recent Rochester Conference (now called the International Conference on High-Energy Physics) and K¨allen had forbidden Petermann to mention my work. Petermann’s apology to me was profuse. In retrospect, I can’t remember how all of these interactions took place in the ancient age before e-mail. Incidentally, Karplus and Kroll later found their error in the simple addition of fractions. I spent the summer of 1957 at a place near Baltimore called RIAS, the Research Institute for Advanced Studies set up by the Martin Company, which made airplanes. There I renewed my connection to Bill Rarita who had been one of my professors at Brooklyn College. He regaled me with stories about his time working

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with Julian at Berkeley. Modestly he said that his major contribution to their joint research was making sure that Julian got to his obligations on time. He also told me about Julian’s obsessions with fancy cars. I must mention that also at RIAS at that time was Lou Witten, the father of you-know-who, who was then in elementary school. In the fall I drove across the country to Berkeley to start my postdoc. Berkeley represented quite a change in atmosphere compared to Cambridge. After all this was California in its heyday. Everything was much more informal. There were a few Schwingerians around: Bob Karplus (with whom I was to collaborate), Chuck Zemach (a student of Roy Glauber), and Marshall Baker, with whom I had shared an office in B-24 for two years and was now at Stanford, a mere 25 miles away. One other thing about the reach of Julian’s influence. At some point Zemach gave a colloquium in which he displayed many of Schwinger’s mannerisms: for example pacing back and forth with his head down, the use of the phrase “how shall I say it,” and so on. This was a period when dispersion relations and bootstraps were in their ascendance. I really liked my two years at Berkeley and might have been tempted to stay in California when out of the blue I got a telegram from Julian urging me to return to Cambridge as his postdoc. I interpreted this as my promised second degree. The position offered had been occupied by Ken Johnson while I was still a student and by George Sudarshan during the two years I was away. As far as I could tell the main obligation of the job was to have lunch with Julian and other faculty members on Wednesdays, usually at Chez Dreyfus. There were times when everyone else gave up waiting as Julian spoke to students and I was the only one left. I remember one occasion emerging from Joyce Chen’s as the sun was setting. Kurt Gottfried suggested that I look at the Thirring model in two dimensions using Julian’s methods. Ken Johnson (who had gone down river to MIT) was involved in similar work and we collaborated for a while. However, I wanted to use a strictly canonical approach while Ken was more adventurous. His method led directly to the discovery of various anomalies while mine resulted in a discussion about the implicit dependence of currents on external fields. I also generalized the model to include a massive vector meson. I don’t know how much Julian relied on this work when he introduced what is now known as the Schwinger model in which the four-point Thirring coupling is removed and the vector meson is made massless. I once got a look at Julian’s notebooks when he asked me to give a couple of lectures for him while he was away. As I remember they were rather concise. At one time he asked my help in marking some papers. I remember the names of two of the undergraduate students, who did quite well: Steve Adler and Fred Goldhaber. Although he was by no means a social butterfly, ironically Schwinger became the anchor of some of the social life in Cambridge. I think of Ruth and Burt Malenka

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and Ann and Paul Martin as the chief organizers of this aspect of the Schwingerians’ activities. In 1961 I started a faculty appointment at Yale University, Harvard’s arch rival. But before I got married in 1969 I would spend many weekends in Cambridge and as few as possible in New Haven. I was an almost constant guest at Danny and Sharon Kleitman’s house. And occasionally I would visit Julian and Clarice in their apartment. I also remember visiting them some years later when I was in Los Angeles. To this day I regret not having made the drive from New Haven on the night Schwinger won the Nobel prize. This reminds me of the traditional invited lectures of Nobel laureates that Schwinger and Feynman gave at the next meeting of the American Physical Society. It seemed as though they had traded mind sets. Julian started drawing pictures, which I think were associated with source theory, while Feynman concentrated on the abstract definition of current operators. In this talk I have used the term Schwingerian for Julian’s students, postdocs and other close associates. There were others who did not fit into these categories. In particular I want to mention Herb Fried and Bruno Zumino. And of course all of the people attending this meeting are welcome to consider themselves members of the fold. I end by quoting from a note of condolence I wrote to Clarice. It described how on one occasion at the Chez Dreyfus lunch, as the waitress took orders going clockwise around the table, Sidney Coleman, sitting to the right of Julian ordered luncheon steak (medium-rare I think) with carrots and mashed potatoes. This was Julian’s invariable lunch order. Coleman continued: “and the gentleman on my left will have the same.” In turn, Julian then said “the gentleman on his left will not have the same” and proceeded to order something else. Julian was his own man and could not be stuffed into a pigeon hole.

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Drell−Yan mechanism and its implications Tung-Mow Yan Cornell University, Ithaca, NY 14853, USA

After a brief account of my Harvard years 1963–1968, we review the history of theory and experiments on the lepton pair production in high energy proton–proton scatterings. We begin with the original ideas of Drell and me based on parton model. We then discuss the QCD improved version. Experimental implications are considered and we conclude that the process has become an important tool for discovery of new particles associated with physics beyond the standard model.

I would like to tell you a story which started 48 years ago and it still continues today. It is the story about a research project Sid Drell and I collaborated on the lepton pair production in high energy pp collisions. 1 1. My Harvard Years 1963–1968 I grew up in Taiwan. After I finished my college there, I went to Harvard for my PhD. The year was 1963, and I stayed there for five years. During those five years, I learned lots of physics. Most of it I learned from Julian Schwinger. I took Quantum Mechanics (251 a & b) from him. I was lucky. Because he did not give that course in the next four years. It was my first real contact with quantum mechanics. As you all know, his course was so unique compared with all the quantum mechanics given elsewhere. In the next four years, he gave Quantum Field Theory (QFT) (253 a & b). Though the title was the same, its content was different every year. It covered relativistic QFT, many body problems, particle phenomenology, even source theory. I took lots of notes for these courses. I have consulted these notes from time to time for my research and teaching. When I finished my PhD in 1968, I was more or less ready for the real world. I owed my physics training and learning to Julian’s teaching. I went to SLAC for my postdoc. At that time the experimentalists were working on Deep Inelastic Electron Nucleon Scattering. 2 Because the elastic electron–nucleon scattering cross-sections drops off rapidly with increasing momentum transfer, people were not sure what to expect from the inelastic scatterings. Let me summarize the status of strong interactions at the time. 2. Particle Physics in 1960s There was no acceptable QFT for strong interactions. The main ideas around that time were:

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(a) SU(3) flavor group proposed by Gell-Mann and Ne’eman for particle classification. 3,4 (b) Quarks: these are the simplest objects corresponding to three reps of SU(3). If they exist, then Gell-Mann and Zweig 5,6 showed that meson = q q¯ bound states

(1)

baryons = qqq bound states

(2)

But quarks have fractional electric charges and violate spin-statistic connection. Color 7 was invented to restore the connection, and all hadrons were proposed to be color singlets. This leads to a separate SU(3) color group. Eventually, this is part of QCD. (c) Current algebra. 8 The vector and axial vector currents which appear in the electroweak interactions satisfy SU(3)×SU(3) commutation relations. Gell-Mann 8 proposed that these commutation relations be exact even if the underlying symmetry was broken. Many low energy predictions follow from current algebra and soft pion theorem (see Ref. 9). Steve Adler 10 who is here at the conference derived in 1966 from Current Algebra sum rules for neutrino–nucleon scattering that show that cross-sections at large momentum transfer are comparable to those for a point like target. Bjorken 11 used isospin rotation to obtain an inequality for e–N scattering that its cross section is also comparable to a point-like target. His result could be tested by the MIT-SLAC experiment 2 going on at the time. Steve’s result is so unexpected, I believe it was the starting point of the important theoretical development of Bjorken scaling, 12 Operator Product Expansion (OPE), 13 QCD, 14 asymptotic freedom, 15 etc. It was the first hint that hadrons are made of point-like constituents.

3. Deep Inelastic Electron Nucleon Scattering When I arrived at SLAC in 1968, SLAC’s two-mile electron linear accelerator had already started operation with a major project to study the inelastic electron– nucleon scattering: 2 e− + N → e− + anything which is a generalization of the elastic scattering in that now the final state can be anything consistent with conversation laws. The kinematics is depicted by Fig. 1 and its differential cross-section in the rest frame of the nucleon is given by   θ θ 8πα2  2 d2 σ 2 2 2 2 = (ε ) (q , v) cos (q , v) sin W + 2W , (3) 2 1 dε d cos θ (Q2 )2 2 2

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Fig. 1.

where W1 and W2 are called structure functions which contain all the strong interaction properties of the nucleon. And q 2 = −Q2 < 0

(4)

is the square of momentum transfer and P ·q (5) M is the electron energy loss in the rest frame of the nucleon. To understand Adler’s sum rules, Bjorken proposed that the structure functions satisfy a scaling property known as Bjorken scaling 12 at large momentum transfers: v=

lim M W1 (q 2 , v) = F1 (x),

(6)

lim vW2 (q 2 , v) = F2 (x),

(7)

Q2 →∞

Q2 →∞

where Q2 , 0 < x < 1. (8) 2M v Bjorken scaling prediction was promptly confirmed by the MIT-SLAC collaboration experiment. 2 x=

4. Parton Model (1968) Feynman 16 interpreted the Bjorken scaling as the point-like nature of the nucleon’s constituents when they were incoherently scattered by the incident electron. Feynman named the point-like constituents partons. This is the parton model. In this Q2 model, the variable x = 2M v is the fractional longitudinal momentum of the struck parton, and the structure functions F1 and F2 are related to the longitudinal momentum distribution functions of the partons. Feynman left open the possibility that the partons need not be the quarks. However, theorists quickly identified the partons with the quarks. A nucleon consists

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of three “valence” quarks which carry the nucleon’s quantum numbers and a “sea” of quark–antiquark pairs. This identification led to many predictions for electron and neutrino (and antineutrino) scatterings from a nucleon (see Ref. 17). 5. Lepton Pair Production In late 1960s, J. H. Christenson, G. H. Hicks, L. Lederman, P. J. Limon, B. G. Bope, and E. Zavatini 18 studied the reaction at BNL: p + U → μ+ μ− + X

(9)

for proton energies 22–29 GeV and the muon pair mass 1–6.7 GeV. Drell and I studied this process 1 to see if we could apply parton model in this case. We picked an infinite momentum frame (e.g. CM frame) to exploit time dilation, and showed that time duration of the external probe τprobe is much shorter than the lifetimes of relevant intermediate states τint.states , i.e., τprobe  τint.states

(10)

then the constituents could be treated as free during the interaction. Thus, the cross-section in the impulse approximation is the product of the probability to find the particular parton configuration and the cross-section for the free partons. Consider the lepton pair production from two initial hadrons: P1 + P2 → + − + X.

(11)

In the parton model for deep inelastic electron scattering, the point-like constituents of the proton are scattered incoherently by the incident electron. Similarly, we expect that the lepton pair in the above process is produced by the annihilation of one parton from one hadron and an antiparton from the second hadron. The pair production by the parton–antiparton annihilation satisfies the criteria of impulse approximation (see Fig. 2).

Fig. 2.

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It is easily shown that the fractional longitudinal momenta of the annihilating partons satisfy τ = x1 x2 =

Q2 , s

(12)

where Q2 and s are, respectively, the pair mass squared and the square of the C.M. energy of the initial energy of the initial hadrons. The rapidity of the pair is given by y=

1 x1 ln . 2 x2

(13)

The predictions stated in our original paper 1 are: (1) The magnitude and shape of the cross-section are determined by the parton and antiparton distributions measured in deep inelastic lepton scatterings; 4πα2 1  dσ = x1 f p(x1 )x2 f p¯(x2 ), (14) dQ2 dy 3Q4 Nc p

(2)

(3) (4)

(5)

dσ where a color factor Nc is included in anticipating QCD; the cross section Q4 dQ 2 depends only on the scaling variable τ = Q2 /s; If a photon, pion, kaon, or antiproton is used as the projectile, its structure functions can be measured by lepton pair production.a This is the only way we know of to study the parton structure of a particle unavailable as a target for lepton scatterings; The transverse momentum of the pair should be small (∼ 300–500 MeV); In the rest frame of the lepton pair, the angular distribution is 1 + cos2 θ with respect to the hadron collision axis, typical of the spin 1/2 pair production from a transversely polarized virtual photon; The same model can easily be modified to account for W boson productions.

The lepton pair production is the first example of a class of hard processes involving two initial hadrons. These processes are not dominated by short distances or light cone. So the standard analysis using operator product expansion is not applicable. But the parton model works.

6. Asymptotic Freedom, QCD, and the Parton Model Only non-Abelian gauge theories 14 possesses asymptotic freedom. In particular, SU(3) color gauge theory has been accepted as the theory of strong interactions. a For

a review of the status of pion structure functions measured by pion induced lepton pair production, see Ref. 19.

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The asymptotically free gauge theory QCD 14,15 makes the parton model almost correct, namely for deep inelastic processes we have QCD = parton model + small corrections.

(15)

In the modern language, the impulse approximation is replaced by the more precise concept of factorization which separate the long distance and short distance physics and the condition for impulse approximation now becomes Q2  Λ2QCD ,

(16)

where Λ2QCD is a typical momentum scale in QCD. Asymptotic freedom leads to logarithmic corrections to the structure functions fi = fi (x, ln Q2 ).

(17)

Factorization for the lepton pair production works in QCD, but in a more complicated manner and it has taken the hard work of many people and many years to establish.b The main complication comes from new feature of initial and final state interactions between the hadrons. The result is fairly simple to state  1  1 dσ AB = dξA dξB fa/A (ξA , Q2 )fb/B (ξB , Q2 )Hab , (18) dQ2 dy XA XB a,b

where the sum over a and b are over parton species. The parton distribution functions are the same as those in deep inelastic lepton scatterings with the understanding that Q2 is its absolute value. The function Hab is the parton level hard scattering cross-section computable in perturbative QCD and is often written as Hab =

dˆ σ . 2 dQ dy

(19)

7. The Process as a Tool for New Discovery It is now generally accepted that a Drell–Yan process refers to a high energy hadron– hadron collision in which there is a sub-hard process involving one constituent from each of the two incident hadrons. New physics always manifests itself in production of new particle(s), and the ordinary particles do not carry the new quantum number of the new physics. To discover new physics in a hadron–hadron collider therefore requires annihilation of the ordinary particles to create these new particles. Thus, the Drell–Yan mechanism is an ideal tool for the new discoveries. Let us mention three important discoveries in the recent past which had employed this process to help: b For a most recent review on the subject of factorization and related topics see Ref. 20 and the many references cited therein.

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• It was used to design the experiments at CERN that discovered the W and Z bosons. 21 • The process was also crucial in the discovery of the top quark at Fermilab. 22 • The discovery of the Higgs Boson at CERN in 2012 23 was perhaps the most dramatic example of the utility of the process. The Higgs Boson is the last particle that appears in the Standard Model to have been found. 8. Conclusions Since the first experiment at BNL and the naive model proposed to understand it, both experiments and theory have come a long way. It is interesting to note that our original crude fit did not remotely resemble the data. We went ahead to publish our paper because of the model’s simplicity and our belief that future experiments would be able to definitely confirm or demolish the model. It is gratifying to see that the successor of the naive model, the QCD improved version, has been confirmed by the experiments carried out in the last 40 years. Lepton pair production process has been an important and active theoretical arena to understand various theoretical issues such as infrared divergences, collinear divergences leading to the factorization theorem in QCD for hard processes involving two initial hadrons. The process has been so well understood theoretically that it has become a powerful tool for discovering new physics. We can expect to find new applications of this process in the future. References 1. S. D. Drell and T.-M. Yan, Phys. Rev. Lett. 25, 316 (1970); S. D. Drell and T.-M. Yan, Ann. Phys. (N.Y.) 66, 578 (1971). 2. W. Panofsky, in Proc. 14th Int. Conf. on High Energy Physics (Vienna), eds. J. Prentki and J. Steinberger (CERN, Geneva, 1968). 3. M. Gell-Mann, California Institute of Technology Synchrotron Laboratory Report No. CTSL-20, (1961), unpublished. 4. Y. Ne’eman, Nucl. Phys. 26, 222 (1961). 5. M. Gell-Mann, Phys. Lett. 8, 214 (1964). 6. G. Zweig, CERN preprints 401, 412 (1964), unpublished. 7. O. W. Greenberg, Phys. Rev. Lett. 13, 598 (1964); M. Y. Han and Y. Nambu, Phys. Rev. B 139, 1006 (1965). 8. M. Gell-Mann, Physics 1, 63 (1964). 9. S. L. Adler and R. F. Dashen, Current Algebras and Applications to Particle Physics (W. A. Benjamin, Inc, 1968). 10. S. Adler, Phys. Rev. 143, 1144 (1966). 11. J. D. Bjorken, Phys. Rev. Lett. 16, 408 (1966). 12. J. D. Bjorken, Phys. Rev. 179, 1547 (1969). 13. K. G. Wilson, Phys. Rev. 179, 1499 (1969). 14. QCD is the SU(3) color non-Abelian gauge theory of C. N. Yang and R. Mills, Phys. Rev. 96, 191 (1954). 15. D. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973); H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).

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16. R. P. Feynman, Phys. Rev. Lett. 23, 1415 (1969); J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975 (1969); these authors had worked on similar ideas independently of Feynman. 17. F. Close, An Introduction to Quarks and Partons (Academic Press, 1979). 18. J. H. Christenson, G. H. Hicks, L. Lederman, P. J. Limon, B. G. Pope and Zavatini, Phys. Rev. Lett. 25, 1523 (1970), and Phys. Rev. D 8, 2016 (1973). 19. W. C. Chang and D. Dutta, Int. J. Mod. Phys. E 22, 1330020 (2013), arXiv:1306.3971. 20. The rapporteur’s talk by M. Neubert at LHCP 2014, Columbia University, New York, 2–7 June 2014: https://indico.cern.ch/event/279518/other-view?view=standard 21. UA1 collaboration (G. Arnison et al.), Phys. Lett. B 122, 103 (1983); UA2 Collaboration (G. Banner et al.), Phys. Lett. B 122, 476 (1983). 22. CDF Collaboration (F. Abe et al.), Phys. Rev. Lett. 74, 2626 (1995); D0 Collaboration (S. Abachi et al.), Phys. Rev. Lett. 74, 2632 (1995). 23. CMS Collaboration, Phys. Lett. B 716, 30 (2012); ATLAS Collaboration, Phys. Lett. B 716, 1 (2012).

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My years with Julian Schwinger: From source theory through sonoluminescence Kimball A. Milton Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA www.nhn.ou.edu/∼milton [email protected]

I recall my interactions with Julian Schwinger, first as a graduate student at Harvard, and then as a postdoc at UCLA, in the period 1968–81, and subsequently. Some aspects of his legacy to physics are discussed. Keywords: Quantum field theory, source theory, quantum mechanics, magnetic charge, Casimir effect

1. The Birth of Source Theory When I came to Harvard in 1967, Julian Schwinger had already heeded the advice he gave in his Nobel Lecture to find an alternative to quantum field theory. 1 He was increasingly concerned that conventional field theory, which he had so very largely developed, was becoming physically remote. Renormalization was supposed to connect the fundamental fields with the physical particles observed in the laboratory. His attempts to include gravity and dual electrodynamics in the framework which so gloriously accommodated quantum electrodynamics sparked his frustration, as did the general feeling then that quantum field theory could not describe strong interactions. Surely, Schwinger had mused, there had to be a more direct way to confront the phenomena of nature. So within a year, he did come up with a more phenomenological approach, which he dubbed source theory. Source theory papers started appearing in 1966, 2 first applied to electrodynamics, 3 and then to chiral symmetry. 4 The emphasis was on effective Lagrangians, and the avoidance of infinities. In some sense, source theory blended dispersion relations and field theory. That is, typical calculations of processes involving virtual particles involved constructing a situation where real particles were exchanged between effective sources, and the resulting amplitude was “space-time extrapolated” to the general situation, with the amplitude being written as a spectral form. At least in simple cases, the process was straightforward and quite effective. I learned the theory first from the detailed notes of Wu-yang Tsai, taken the first year I was at Harvard, when I foolishly did not sit in on Julian’s lectures. By the end of that year, I had approached the great man and asked if I could work with him. To my surprise, he was quite encouraging, so I worked hard the next three years to justify his faith in me.

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2. What Was It Like to Work with Julian? Typically, at Harvard, he had 12 students, at various stages of development. He was available only on Wednesdays, after lunch, first come, first served (advance booking required). I recall staying up most of the night on Tuesdays working feverishly, then getting up at the ungodly hour of 8:30 to post my name near the top of the list kept by his secretary, who would arrive at 9:00. If you were near enough to the top, and Julian returned from lunch early enough, you would receive admittance. Once you entered Julian’s office in the afternoon, there was no time pressure, although you didn’t want to appear to be too stupid. If the phone rang, it was invariably ignored. Time with the master was unmetered! All his students had distinct problems, none of which coincided with what Julian was working on himself, but after a few minutes of explanation, he would come up with a valuable suggestion. It might not work, but it would take a week or two to follow the ideas through, and 30 minutes or so of consultation every week or two was more than sufficient to keep progress on track. Other than these weekly or biweekly meetings, we saw Julian in his classes. All his students, and some faculty, sat in on his courses on quantum mechanics and field theory. (Questions were not encouraged!) The lectures were like musical performances, and he held the audience in rapt attention. Every time he taught a course, the content was completely new, so it was advantageous to attend every reincarnation. Much original research appeared in his lectures, and sometime never anywhere else! (An example was the first appearance of the Bethe–Salpeter equation.) My oral exam, in 1969, on a derivation of the Lamb shift via “sidewise dispersion relations,” devolved into a dispute between Paul Martin and Schwinger on the fundamentals of source theory, but I emerged unscathed, being able to speak for no more than five minutes. I was rewarded with a copy of his recent Brandeis lectures. 5 Even easier was my defense of my thesis, two years later, largely on trace anomalies it turned out. It consisted of an excellent lunch at a French restaurant in Westwood. (I successfully answered Julian’s single question, on my birthplace.) 3. The Schwinger’s Move to California As I was working on stress tensors (which I still am), Clarice and Julian took a sabbatical to Japan, a wonderful experience for them. There he completed his first volume of Particles, Sources, and Fields. 6 When they returned in Fall 1970, Julian announced his plan to move to UCLA, which was met with great consternation by his gang of students. However, he invited his three senior students, me, Lester DeRaad, Jr., and Wu-yang Tsai, to accompany him as his assistants (postdocs). The move was accomplished in February 1971. (They were greeted by the San Fernando earthquake!)

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3.1. Why did Julian leave Harvard? Of course, he was unhappy with the reception of source theory at Harvard, by his own former students, Paul Martin and Shelly Glashow, and others — but this was not the chief reason for relocating. Julian had become obsessed with exercise, after the death in 1958 of Wolfgang Pauli to pancreatic cancer, so he took up skiing and tennis, and California offered more opportunities. He could, and did, even have a pool. For years, his assistant at the MIT Radiation Laboratory during the war, David Saxon, who had become chair of the physics department at UCLA (eventually he would become president of the whole University of California system), had been urging Julian to move to UCLA. He finally accepted, much to the regret of Bostonbred Clarice. Saxon and Schwinger both assumed many students would flock to him, as they had when he joined Harvard in 1946. But this was not to be. 3.2. The sourcerer’s apprentices I arrived at UCLA soon after the Schwingers did, while Lester and Wu-yang arrived in the summer of 1971. We formed a close research group at UCLA for several years. Wu-yang stayed till 1976, when he left for Coral Gables, while Lester left in 1978, although he stayed in Southern California; I left the following year for Ohio State University in Columbus, when my wife was offered a job there in the Dance Department. As grad students, we never had any social contact with the great man. That changed with our change in status: We had lunch with him (usually including Bob Finkelstein and visitors) once a week, and occasionally were invited to the Schwinger’s home in Bel Air, with its magnificent view of LA. Julian was always a gracious host. He was not status conscious, and had a remarkable ability to listen to others. An example was the occasion when Julian brought my wife’s nephew, a sullen teenager, into the conversation by an insightful remark about Thelonious Monk. 4. Interactions with Feynman Although Schwinger and Richard Feynman were both living in the Los Angeles area for more than 20 years, they rarely socialized. A story I learned from Berge Englert is that once the Feynmans were invited to the Schwinger’s home, and were having a good time, until another couple arrived, which spoiled the evening for the Feynmans. When they did meet at conferences they were always very cordial, and had great mutual respect, which dated back to the encounters at Shelter Island and the Poconos, during their parallel development of quantum electrodynamics. 5. Particles, Sources, and Fields “If you can’t join ’em, beat ’em” was the motto to his three-volume exposition of source theory, 6–8 which contains a wealth of information about fields with arbitrary

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spin, and included many detailed calculations in QED. The third volume even includes a very complete calculation of the fourth order electron anomalous magnetic moment, which was first correctly calculated by his student Charlie Sommerfield. 9,10 As mentioned, the first volume was written during his sabbatical in Japan, while the second and third (unfinished) volumes were completed at UCLA. He abandoned this book project just as he was about to embark on strong interactions, being diverted by his attempt to understand deep inelastic scattering. Although we three apprentices diligently proofread the later volumes, Julian forgot to acknowledge our help in the published books!

6. Collaborations with his “Assistants” With the dramatic discovery in 1974 of what is now called the J/ψ particle, Julian was quick to come up with an explanation, involving a “hidden sector,” maybe “dyons.” 11,12 (The latter was based on his Science article, “Magnetic Model of Matter,” written in 1969. 13 ) We joined in with papers on the decay of ψ(3.7) to ψ(3.1). 14 His work on “renormalization group without renormalization or a group” 15,16 led to parallel papers with me, 17 eventually finding applications in QCD. 18

6.1. Magnetic charge Julian’s last papers before the “source theory revolution” were on electric and magnetic charge renormalization. 19–21 He revisited the subject periodically thereafter, with variations on the eg = nc/2 quantization condition. 22,23 (He bemoaned the lack of experimental evidence for magnetic monopoles, “If only the Price had been right.” 24 ) This led to the joint “Dyon–dyon scattering” paper, with rainbows and glories. 25 The entirely separate but joint interest of Luis Alvarez 26 and Julian Schwinger in the quest for magnetic charge would eventually lead to the OU search for magnetic monopoles, which set the best limit on their masses for a decade, 27 until LHC data became available. 28

6.2. Deep inelastic scattering Julian’s last major effort in “particle physics” was his effort to describe the dramatic scaling phenomena discovered in deep inelastic scattering 29,30 without reference to Feynman’s partons or Gell-Mann’s quarks of which he had disparaged the naming. 13 His approach involved double spectral forms, related closely to the Deser–Gilbert– Sudarshan representation. 31–35 We followed on with some explicit related calculations, 36,37 but eventually showed that although the double spectral representation was generally valid, it required anomalous spectral regions, which invalidated some of the expected behaviors. 38

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6.3. QED in background fields Julian also revisited the electrodynamics of particles in strong magnetic fields, harking back to the famous “Gauge Invariance and Vacuum Polarization” paper, 39 his most cited paper, and what I regard as the first source theory paper, even though it was written in 1951. This work recalled his days at the Rad Lab during World War II, where he worked out the theory of synchrotron radiation, 40,41 independently of the Russians. 42 The sequel to Schwinger’s 1949 paper appeared in 1973, 43 and collaborative papers with Tsai and Erber appeared subsequently. 44,45 Independent papers based on Schwinger’s powerful formalism continued to emerge. 46,47 6.4. The triangle anomaly Sometimes the interactions were bottom-up. What is called the axial-vector anomaly was discovered by Julian in his famous 1951 paper. 39 We postdocs showed in 1972 that there were radiative corrections to neutral pion decay into two photons, in apparent (but not actual?) disagreement with the Adler–Bardeen theorem. 48,49 Julian later confirmed our result, a correction by a factor of 1 + α/π, but instead of seeking accommodation with received wisdom, wrote a joint confrontational paper. Before the paper was submitted, Julian gave a talk on the subject at MIT, which was badly received. The remains of that unpublished paper exist in the third volume of Particles, Sources, and Fields. 8 6.5. Supersymmetry With the emergence of supersymmetry, and especially its local version, supergravity, Julian regretted he had not thought of it first, since he had expounded the multispinor basis of particles with integer and half-integer spin in Particles, Sources, and Fields. 6 A command performance by Stan Deser in 1977 led to his own reconstruction of the theory, 50 but with negligible impact (according to the current Inspire HEP database, this paper received only 12 citations). The same fate befell the follow-up paper by Bob Finkelstein, Luis Urrutia, and me in which we reconstructed supergravity following Schwinger’s lead. 51 6.6. Casimir effect Invariably, Julian put his current research into his lectures, which of course we always attended. He became intrigued with how the Casimir effect could be understood without the zero point energy which seemed not to be present in the source theory approach, inspired, I believe, by conversations with Seth Putterman. This lecture got quickly written up 52 and then, with Lester and myself we rederived the Lifshitz theory, 53 which included the infamous “Schwinger prescription,” concerning how to treat the thermal corrections, an amazingly ongoing controversy. We went on to rederive the surprising result, first found by Tim Boyer, 54 that the

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Casimir self-stress on a perfectly conducting spherical shell is repulsive. 55 I’ve been strongly bound to the Casimir effect ever since. 56 7. Books We have already mentioned his monumental three-volume Particles, Sources, and Fields. But his continual lectures led to several other volumes. 7.1. Classical electrodynamics For the first time since the War, Julian taught graduate electrodynamics at UCLA, and we assistants sat in. We started turning the lecture notes into a book, and ended up with a contract with W.H. Freeman. Julian then began to pay attention, decided it didn’t sound enough like himself, and began a nonconvergent series of revisions. He worked on it for the better part of a decade, abandoning the rewriting when he got to radiation theory. After his death, we turned it into the book which exists today, 57 with the publisher cycling from Addison-Wesley to Perseus and now Taylor&Francis. 7.2. Understanding space and time Julian and astronomer George Abell developed a BBC course with the Open University (1976–79). The bulk of the course dealt with cosmology; Julian’s part was to explain relativity to a general audience. A “Robie the Robot” graced the Schwinger’s living room thereafter. It aired at a “good time” on BBC2, but rather obscurely on KCET in Los Angeles, because its release coincided with that of Carl Sagan’s Cosmos. Unlike Sagan, Julian was not a natural performer. “Not God’s gift to presenting” was how he was described by his producer Ian Rosenbloom. However, a Scientific American Library volume Einstein’s Legacy 58 did memorialize this effort, which is full of interesting historical anecdotes and insights. 7.3. Quantum mechanics For the first time Schwinger began to teach quantum mechanics to undergraduates at UCLA. (Of course, at Harvard, half his audience for his graduate quantum mechanics courses consisted of bright Harvard undergrads.) He taught many subjects in very original ways: the framework was the “measurement algebra” that he had developed in the early 1950s based on an analysis of Stern–Gerlach experiments on polarized atoms. Some notes based on these ideas were published rather obscurely (for physicists) at the end of that decade in the Proceedings of the National Academy of Sciences; 59–63 most famous were the Les Houches lectures of 1955, which appeared in print only in 1970 (together with reprints of some of the PNAS papers). 64 But the definitive record of his lectures only was published under the editorship of Berge Englert. 65

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More impact, of course, followed from Schwinger’s Quantum Action Principle, which originated at the same time. This was immediately applied to his reformulation (his third) of quantum electrodynamics. 66,67 We cannot trace that profound redevelopment here, but note that a rather accessible volume describing his action principle based largely on his lectures over the years has now appeared. 68

7.4. Electromagnetic radiation It will be recalled that much of the impetus for Schwinger’s rapid scaling of the peak of quantum electrodynamics derived from his wartime work on radar theory at the Radiation Laboratory. (Ironically, it was thought then that “radiation” was a word that would not suggest classified work was in progress.) Of course, toward the end of the war, he gave brilliant lectures on his researches to the other lab workers, and many of these were transcribed by David Saxon, and mimeographed. There was supposed to be a published volume of these lectures, but that never materialized. Many years later, after the successful publication of Classical Electrodynamics, 57 Alex Chao of SLAC and Chris Caron of Springer persuaded me to turn the archival materials into a book. I did so to the best of my ability, and supplemented it with some bits of the Schwinger archive at UCLA which had not found their way into the previous textbook. The resulting volume contains also reprints of a number of papers of Schwinger on waveguides, synchrotrons and synchrotron radiation, and diffraction. 69

8. Family and Diversions 8.1. My marriage to Margarita Ba˜ nos Alfredo Ba˜ nos, Jr., was a colleague of Julian’s at the Rad Lab during the War. Alfredo moved to UCLA thereafter, eventually divorcing his first wife and marrying Alice (a great scandal at the time), and they had a child Margarita. She grew up and became a dancer. Naturally, Clarice thought to introduce Margarita, who in 1975 had just returned from six years with the Royal Ballet School in London, to one of Julian’s apprentices. Although I was second choice (Lester DeRaad was first, but, unbeknownst to Clarice, he was already taken), it did work out, and this year we celebrated our 40th anniversary with a most memorable trip to Bali after the Singapore Centennial.

8.2. V. Sattui Winery The Schwingers became significant investors in this winery when it was relaunched in 1976. In those days, the winery and the BBC efforts were a frequent topic of our lunch conversations, which ranged widely. On Julian’s death, in 1994, the winery introduced a special Cabernet Sauvignon to commemorate their famous partner.

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8.3. 60th birthday celebration I helped organize, along with Bob Finkelstein and Margy Kivelson, his Fest in 1978. Julian was rather unhappy about the affair, for he thought of it as a retirement celebration. But he later apologized publicly to me and Margy when he received the Monie Ferst award at Georgia Tech in 1986. Dick Feynman gave a wonderful talk at the banquet for Schwinger’s Fest. He recounted his encounters with Julian at Los Alamos and Pocono. “Although we’d come from the ends of the earth with different ideas, we’d climbed the same mountain from different sides.” His remarks were not included in the 60th birthday volume 70 but were in the one assembled for Julian’s 70th, 71 the Fest then being a somewhat more low-keyed affair. 9. The Later Years 9.1. Thomas–Fermi and Humpty-Dumpty I left UCLA, as noted above, in 1979, when I went to Ohio State as a Visiting Associate Professor. But I didn’t resign my semi-permanent position at UCLA until 1981, when I accepted a regular faculty position at another OSU, this time in Oklahoma. Berge Englert became my replacement as Julian’s assistant in November of that year. He immediately joined Schwinger in renewed explorations of the Thomas–Fermi model of atoms, which Julian had started to analyze in his undergraduate quantum mechanics course. 72,73 Lester helped at first, 74 but then Berge became the chief calculator, and an impressive series of papers followed. 75–81 Englert left UCLA for a position in Munich in 1985, but their collaboration continued. Marlan Scully, who was visiting the University of Munich in 1987 involved them in a series of papers questioning whether one can reunite beams of atoms which have been separated by a Stern–Gerlach measurement, with an unsurprising negative answer; 82–84 Humpty-Dumpty cannot be put back together again. 9.2. Cold fusion and sonoluminescence In 1989 began one of the most remarkable examples of “pathological science” 85 with the announcement by Pons and Fleischman, noted chemists, of the discovery of cold fusion. 86 Of course, fusion occurs at some very low rate at ambient temperature due to quantum tunneling, but they claimed to see significant energy released. The history of this sad affair is given in the book by Huizenga. 87 It bears on our story because Schwinger, nearly alone among physicists, took the report seriously and believed he could explain it. The rejection of his paper by Physical Review Letters led him to resign his fellowship in the American Physical Society and to demand that the “source theory” index category be deleted, as he would no longer use it! (The journal complied, even though the PACS scheme was beginning to be widely used by other journals, and the source theory category was of course used by others.) Englert then helped him get the paper published in Zeitschrift f¨ ur Naturforschung; 88 in spite of negative reviews, the second paper was published in

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Zeitschrift f¨ ur Physik D, 89 but accompanied by an editorial note disavowing any responsibility for the validity of the conclusions by the journal! Third and fourth papers were rejected and never published, although Schwinger wrote three short notes to the PNAS. 90–92 Englert reports that eventually Schwinger began to doubt whether his theory was entirely correct, and doubts about the experimental evidence rapidly accumulated. But until his death, he thought there was something right about the phenomena and his explanation of it. So Julian turned his attention to another seemingly impossible phenomenon, that of sonoluminescence. Again he learned about this from his good friend Seth Putterman. In “single-bubble sonoluminescence,” a tiny bubble of air injected into water and submitted to suitable ultrasonic acoustic vibrations undergoes rapid collapse and expansion, which can persist for months. Near the point of minimum radius (a factor of 10 smaller than the maximum) a flash of visible light is emitted, carrying a total energy of about 10 MeV. For a review, see Ref. 93. The dynamics of the bubble seem rather well understood, but the mechanism for the light emission remains poorly understood to this day. Julian immediately thought: “dynamical Casimir effect”; the rapidly changing boundary conditions might convert virtual photons into the real ones seen in the observations. So he proceeded to write a series of papers, the first two being followups on his first Casimir effect paper in 1975. 94,95 The balance of the work appears in a series of short notes in PNAS, his favorite journal where he could avoid scrutiny by cynical referees. 96–102 Unfortunately, he had forgotten, or perhaps never realized, that before I had left UCLA I had considered the Casimir effect for a dielectric sphere; 103 such was the basis of his estimates, which he never made very precise. In fact, at our last encounter, at the annual Christmas party at the Ba˜ nos’ home in Westwood, to which Clarice and Julian invariably came (Julian’s job being to hide the three wise men in the Christmas tree), Julian suggested we work together to put the theory of sonoluminescence on a firm footing. But that was not to be. Two months later, he was diagnosed, like Pauli, with pancreatic cancer, and he died in July 1994. I did go back and work out the theory; unfortunately, the conclusion was that the energy balance was too small by a factor of a million to explain the copious production of photons seen in sonoluminescence. 104–106 10. Schwinger’s Legacy Julian Schwinger spent nearly as many years at UCLA as at Harvard; the former from 1946–70, the latter 1971–94. The contrast seems dramatic: he published 150 papers in his Harvard years, many of which were groundbreaking and founded new fields. (Besides the obvious field theory developments, for example, think of the Keldysh–Schwinger formalism. 107 ) In contrast, the 80 published during the UCLA period seemed more reactive, for instance, suggesting alternatives to the Weinberg–Salam–Glashow theory 108 or an alternative approach to the renormalization group 15,16 or to supersymmetry, 50 or even wrong-headed as in the cold-fusion

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papers. 88 One could blame much of this on his heroic attempt to reformulate field theory free from infinities, his source theory. Had he not been so confrontational in his demand that students totally divorce themselves from the conventional approach, but recognized that he was developing an effective action approach which offers numerous computational advantages, reception to his ideas would have been much more favorable and he would have attracted more students. (He did invent the concept of effective action, after all.) One could argue that the fact he had only some five students at UCLA as compared to (depending on how one counts) nearly 70 at Harvard led to his increased isolation. But maybe it was not Schwinger who changed, but the world around him. In 1965 conventional field theory, which Julian had largely developed, seemed to have reached a dead end, and current algebra and dispersion relations seemed the way forward to understand hadronic physics. Julian tried to find a third way, but just as he was getting launched, the electroweak theory (for which he had laid most of the groundwork 109 ) was shown to be consistent and experimentally verified. Source theory might be efficient, but it was not necessary, and so was ignored, since only by contact with the master could you become initiated. It is unfortunate that Julian gave up his reconstruction of field theory just at the point where he was turning to strong interactions, which was what had impelled him to start this development. He could have contributed, like Feynman, to the elucidation of the non-Abelian theory of quantum chromodynamics, which is still more of a framework than a precisely calculable theory like QED. Instead he turned to his “dispersive” approach to deep inelastic scattering, which led to limited insight. His most cited papers from the UCLA years, with over 200 citations each according to INSPIRE, are those on the Casimir effect. 53,55 His work on this subject lives on, even though we practitioners embrace the notion that the effect reflects the change of zero-point energy or field fluctuations which Julian rejected. This work has technological applications, and undoubtedly has something to say about the accelerating universe we live in. For much more information about the life and work of Julian Schwinger, please see the biography Ref. 110, updated in Ref. 111. This presentation grew out of Ref. 112. Acknowledgments I thank Berge Englert and all the other organizers of the Julian Schwinger Centennial for putting together such a wonderful memorial, and we are all indebted to the Julian Schwinger Foundation for financial support of the Conference. I am grateful to Berge for his encouragement of my writing of this essay. References 1. J. Schwinger, “Relativistic Quantum Field Theory,” in Nobel Lectures—Physics, 1963–1970 (Elsevier, Amsterdam, 1972). [Reprinted in Physics Today, June 1966.]

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

19. 20. 21. 22. 23. 24. 25. 26.

27.

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J. Schwinger, Particles and sources, Phys. Rev. 152, 1219 (1966). J. Schwinger, Sources and electrodynamics, Phys. Rev. 158, 1391 (1967). J. Schwinger, Chiral dynamics, Phys. Lett. 24B, 473 (1967). J. Schwinger, Particles and Sources, (Gordon and Breach, New York, 1968). J. Schwinger, Particles, Sources, and Fields. Vol. 1, (Addison-Wesley, Reading, Mass., 1970). J. Schwinger, Particles, Sources, and Fields. Vol. II, (Addison-Wesley, Reading, MA, 1973). J. Schwinger, Particles, Sources, and Fields. Vol. III. (Addison-Wesley, Advanced Book Classics, 1989) ISBN-13: 978-0738200552. C. M. Sommerfield, Magnetic dipole moment of the electron, Phys. Rev. 107 328 (1957). A. Petermann, Fourth order magnetic moment of the electron, Helv. Phys. Acta 30, 407 (1957). J. Schwinger, Interpretation of a narrow resonance in e+ -e− annihilation, Phys. Rev. Lett. 34, 37 (1975). J. Schwinger, Speculations concerning the ψ particles and dyons, Science 188, 1300 (1975). J. Schwinger, A magnetic model of matter, Science 165, 757 (1969). J. Schwinger, K. A. Milton, W.-y. Tsai and L. L. DeRaad, Jr., Resonance interpretation of the decay of ψ  (3.7) into ψ(3.1), Phys. Rev. D 12, 2617 (1975). J. Schwinger, Photon propagation function: Spectral analysis of its asymptotic form, Proc. Nat. Acad. Sci. USA 71, 3024 (1974). J. Schwinger, Photon propagation function: A comparison of asymptotic functions, Proc. Nat. Acad. Sci. USA 71, 5047 (1974). K. A. Milton, Spectral forms for the photon propagation function and the Gell-MannLow function, Phys. Rev. D 10, 4247 (1974). K. A. Milton and I. L. Solovtsov, Analytic perturbation theory in QCD and Schwinger’s connection between the beta function and the spectral density, Phys. Rev. D 55, 5295 (1997). [hep-ph/9611438]. J. Schwinger, Electric- and magnetic-charge renormalization. I, Phys. Rev. 151, 1048 (1966). J. Schwinger, Electric- and magnetic-charge renormalization. II, Phys. Rev. 151, 1055 (1966). J. Schwinger, Magnetic charge and quantum field theory, Phys. Rev. 144, 1087 (1966). J. Schwinger, Sources and magnetic charge, Phys. Rev. 173, 1536 (1968). J. Schwinger, Magnetic charge and the charge quantization condition, Phys. Rev. D 12, 3105 (1975). P. B. Price, E. K. Shirk, W. Z. Osborne, L. S. Pinsky. Evidence for the detection of a moving magnetic monopole, Phys. Rev. Lett. 35 487 (1975). J. Schwinger, K. A. Milton, W.-y. Tsai, L. L. DeRaad, Jr. and D. C. Clark, Nonrelativistic dyon-dyon scattering, Ann. Phys. (N.Y.) 101, 451 (1976). R. R. Ross, P. H. Eberhard, L. W. Alvarez and R. D. Watt, Search for magnetic monopoles in lunar material using an electromagnetic detector, Phys. Rev. D 8, 698 (1973). G. R. Kalbfleisch, W. Luo, K. A. Milton, E. H. Smith and M. G. Strauss, Limits on production of magnetic monopoles utilizing samples from the D0 and CDF detectors at the Tevatron, Phys. Rev. D 69, 052002 (2004) [hep-ex/0306045].

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28. B. Acharya et al. [MoEDAL Collaboration], Search for magnetic monopoles with the MoEDAL forward trapping detector in 13 TeV proton-proton collisions at the LHC, Phys. Rev. Lett. 118, 061801 (2017) [arXiv:1611.06817 [hep-ex]]. 29. E. D. Bloom et al., High-energy inelastic e-p scattering at 6◦ and 10◦ Phys. Rev. Lett. 23, 930 (1969). Bibcode:1969PhRvL..23..930B. 30. M. Breidenbach et al., Observed behavior of highly inelastic electron-proton scattering. Phys. Rev. Lett. 23, 935 (1969). Bibcode:1969PhRvL..23..935B. 31. J. Schwinger, Source theory viewpoints in deep inelastic scattering, Proc. Nat. Acad. Sci. 72, 1 (1975) [Acta Phys. Austriaca Suppl. 14, 471 (1975)]. 32. J. Schwinger, Deep inelastic scattering of leptons. Part 1., Proc. Nat. Acad. Sci. USA 73, 3351 (1976). 33. J. Schwinger, Deep inelastic scattering of charged leptons, Proc. Nat. Acad. Sci. USA 73, 3816 (1976). 34. J. Schwinger, Deep inelastic sum rules in source theory, Nucl. Phys. B 123, 223 (1977). 35. J. Schwinger, Deep inelastic neutrino scattering and pion-nucleon cross-sections, Phys. Lett. 67B, 89 (1977). 36. W.-y. Tsai, L. L. DeRaad, Jr. and K. A. Milton, Verification of virtual Comptonscattering sum rules in quantum electrodynamics, Phys. Rev. D 11, 3537 (1975) Erratum: [Phys. Rev. D 13, 1144 (1976)]. 37. L. L. DeRaad, Jr., K. A. Milton and W.-y. Tsai, Deep inelastic neutrino scattering: A double spectral form viewpoint, Phys. Rev. D 12, 3747 (1975) Erratum: [Phys. Rev. D 13, 3166 (1976)]. 38. R. J. Ivanetich, W.-y. Tsai, L. L. DeRaad, Jr., K. A. Milton and L. F. Urrutia, Anomalous spectral regions in source theory, in Themes in Contemporary Physics (Julian Schwinger’s Festschrift), eds. S. Deser, H. Feshbach, R. J. Finkelstein, K. A. Johnson and P. C. Martin, North-Holland, Amsterdam, 1979, p. 233 [Physica 96A, 233 (1979)]. 39. J. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951). 40. J. Schwinger, On radiation by electrons in a betatron, 1945 unpublished. Transcribed by M. A. Furman, in A Quantum Legacy: Seminal Papers of Julian Schwinger, ed. K. A. Milton (World Scientific, Singapore, 2000), pp. 307–331. 41. J. Schwinger, On the classical radiation of accelerated electrons, Phys. Rev. 75, 1912 (1949). 42. D. Iwanenko and I. Pomeranchuk, On the maximal energy attainable in a betatron, Phys. Rev. 65, 343 (1944). 43. J. Schwinger, Classical radiation of accelerated electrons. II. A quantum viewpoint, Phys. Rev. D 7, 1696 (1973). 44. J. Schwinger, W.-y. Tsai and T. Erber, Classical and quantum theory of synergic synchrotron-Cherenkov radiation, Ann. Phys. (N.Y.) 96, 303 (1976) [Ann. Phys. (N.Y.) 281, 1019 (2000)]. 45. J. Schwinger and W.-y. Tsai, New approach to quantum corrections in synchrotron radiation, Ann. Phys. (N.Y.) 110, 63 (1978). 46. W.-y. Tsai, New approach to quantum corrections in synchrotron radiation. 2., Phys. Rev. D 18, 3863 (1978). 47. K. A. Milton, L. L. DeRaad, Jr. and W.-y. Tsai, Electron pair production by virtual synchrotron radiation, Phys. Rev. D 23, 1032 (1981). 48. L. L. DeRaad, Jr., K. A. Milton and W.-y. Tsai, Second order radiative corrections to the triangle anomaly. I, Phys. Rev. D 6, 1766 (1972).

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49. K. A. Milton, W.-y. Tsai and L. L. DeRaad, Jr., Second order radiative corrections to the triangle anomaly. II, Phys. Rev. D 6, 3491 (1972). 50. J. Schwinger, Multispinor basis of Fermi-Bose transformations, Ann. Phys. (N.Y.) 119, 192 (1979). 51. K. A. Milton, L. F. Urrutia and R. J. Finkelstein, Constructive approach to supergravity, Gen. Rel. Grav. 12, 67 (1980). 52. J. Schwinger, Casimir effect in source theory, Lett. Math. Phys. 1, 43 (1975) 53. J. Schwinger, L. L. DeRaad, Jr. and K. A. Milton, Casimir effect in dielectrics, Ann. Phys. (N.Y.) 115, 1 (1978). 54. T. H. Boyer, Quantum electromagnetic zero point energy of a conducting spherical shell and the Casimir model for a charged particle, Phys. Rev. 174, 1764 (1968). 55. K. A. Milton, L. L. DeRaad, Jr. and J. Schwinger, Casimir selfstress on a perfectly conducting spherical shell, Ann. Phys. (N.Y.) 115, 388 (1978). 56. K. A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy, (World Scientific, Singapore, 2001) 301 pp. 57. J. Schwinger, L. L. DeRaad, Jr., K. A. Milton and W.-y. Tsai, Classical Electrodynamics (Perseus/Westview, New York, 1998) ISBN-13: 978-0738200569. 58. J. Schwinger, Einstein’s Legacy (Scientific American Books, New York, 1986). 59. J. Schwinger, The algebra of microscopic measurements, Proc. Natl. Acad. Sci. USA 45, 1542 (1959). 60. J. Schwinger, The geometry of quantum states, Proc. Natl. Acad. Sci. USA 46, 257 (1960). 61. J. Schwinger, Unitary operator bases, Proc. Natl. Acad. Sci. USA 46, 570 (1960). 62. J. Schwinger, Unitary transformations and the action principle, Proc. Natl. Acad. Sci. USA 46, 883 (1960). 63. J. Schwinger, The special canonical group, Proc. Natl. Acad. Sci. USA 46, 1401 (1960). 64. J. Schwinger, Quantum Kinematics and Dynamics (Benjamin, New York, 1970). 65. J. Schwinger, Quantum Mechanics: Symbolism of Atomic Measurements, ed. B.G. Englert (Springer, Berlin, 2001). 66. J. Schwinger, The theory of quantized fields. 1., Phys. Rev. 82, 914 (1951). 67. J. Schwinger, The theory of quantized fields. 2., Phys. Rev. 91, 713 (1953). 68. K. A. Milton, Schwinger’s Quantum Action Principle (Springer Briefs in Physics, Cham, 2015). 69. K. A. Milton and J. Schwinger, Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators (Springer, Berlin, 2006). 70. S. Deser, H. Feshbach, R. J. Finkelstein, K. A. Johnson and P. C. Martin, eds., Themes in Contemporary Physics, (Julian Schwinger’s 60th birthday Festschrift) (North-Holland, Amsterdam, 1979). [Physica 96A (1979)]. 71. S. Deser and R. Finkelstein, Themes in Contemporary Physics II (Julian Schwinger’s 70th birthday Festschrift) (World Scientific, Singapore, 1989). 72. J. Schwinger, Thomas-Fermi model: The leading correction, Phys. Rev. A 22, 2353 (1980). 73. J. Schwinger, Thomas-Fermi model: The second correction, Phys. Rev. A 24, 2353 (1981). 74. L. DeRaad, Jr. and J. Schwinger, New Thomas-Fermi theory: A test, Phys. Rev. A 25, 2399 (1982). 75. B.-G. Englert and J. Schwinger, Thomas-Fermi revisited: The outer regions of the atom, Phys. Rev. A 26, 2332 (1982).

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76. B. G. Englert and J. Schwinger, The statistical atom: Handling the strongly bound electrons, Phys. Rev. A 29, 2331 (1984). 77. B. G. Englert and J. Schwinger, The statistical atom: Some quantum improvements, Phys. Rev. A 29, 2339 (1984). 78. B. G. Englert and J. Schwinger, The new statistical atom: A numerical study, Phys. Rev. A 29, 2353 (1984). 79. B. G. Englert and J. Schwinger, Semiclassical atom, Phys. Rev. A 32, 26 (1985). 80. B. G. Englert and J. Schwinger, Linear degeneracy in the semiclassical atom, Phys. Rev. A 32, 36 (1985). 81. B. G. Englert and J. Schwinger, Atomic binding energy oscillations, Phys. Rev. A 32, 47 (1985). 82. M. O. Scully, B.-G. Englert and J. Schwinger, Spin coherence and Humpty-Dumpty. I. Simplified treatment, Found. Phys. 18, 1045 (1988). 83. M. O. Scully, B.-G. Englert and J. Schwinger, Spin coherence and Humpty-Dumpty. II. General theory, Z. Phys. D 10, 135 (1988). 84. M. O. Scully, B.-G. Englert and J. Schwinger, Spin coherence and Humpty-Dumpty. III. The effects of observation, Phys. Rev. A 40, 1775 (1989). 85. I. Langmuir, lecture given in 1953, published in Physics Today, October 1989, p. 36. 86. M. Fleischmann, S. Pons and M. Hawkins, Electrochemically induced nuclear fusion of deuterium, J. Electanal. Chem. 261, 301 (1989) 87. J. H. Huizenga, Cold Fusion: The Scientific Fiasco of the Century (Oxford University Press, 1993). 88. J. Schwinger, Cold fusion: A hypothesis, Z. Nat. Forsh. A 45, 756 (1990). 89. J. Schwinger, Nuclear energy in an atomic lattice, Z. Phys. D 15, 221 (1990). 90. J. Schwinger, Phonon representations, Proc. Natl. Acad. Sci. USA 87, 6983 (1990). 91. J. Schwinger, Phonon dynamics, Proc. Natl. Acad. Sci. USA 87, 8370 (1990). 92. J. Schwinger, Phonon Green’s functions, Proc. Natl. Acad. Sci. USA 88, 6537 (1991). 93. M. P. Brenner, S. Hilgenfeldt and D. Lohse, Single-bubble sonoluminescence, Rev. Mod. Phys. 74, 425 (2002). 94. J. Schwinger, Casimir effect in source theory II, Lett. Math. Phys. 24, 59 (1992). 95. J. Schwinger, Casimir effect in source theory III, Lett. Math. Phys. 24, 227 (1992). 96. J. Schwinger, Casimir energy for dielectrics, Proc. Natl. Acad. Sci. USA 89, 4091 (1992). 97. J. Schwinger, Casimir energy for dielectrics: Spherical geometry, Proc. Natl. Acad. Sci. USA 89, 11118 (1992). 98. J. Schwinger, Casimir light: A glimpse, Proc. Natl. Acad. Sci. USA 90, 958 (1993). 99. J. Schwinger, Casimir light: The source, Proc. Natl. Acad. Sci. USA 90, 2105 (1993). 100. J. Schwinger, Casimir light: Photon pairs, Proc. Natl. Acad. Sci. USA 90, 4505 (1993). 101. J. Schwinger, Casimir light: Pieces of the action, Proc. Natl. Acad. Sci. USA 90, 7285 (1993). 102. J. Schwinger, Casimir light: field pressure, Proc. Natl. Acad. Sci. USA 91, 6473 (1994). 103. K. A. Milton, Semiclassical electron models: Casimir selfstress in dielectric and conducting balls, Ann. Phys. (N.Y.) 127, 49 (1980). 104. K. A. Milton and Y. J. Ng, Casimir energy for a spherical cavity in a dielectric: Applications to sonoluminescence, Phys. Rev. E 55, 4207 (1997) [hep-th/9607186]. 105. K. A. Milton and Y. J. Ng, Observability of the bulk Casimir effect: Can the dynamical Casimir effect be relevant to sonoluminescence?, Phys. Rev. E 57, 5504 (1998) [hep-th/9707122].

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106. I. H. Brevik, V. N. Marachevsky and K. A. Milton, Identity of the van der Waals force and the Casimir effect and the irrelevance of these phenomena to sonoluminescence, Phys. Rev. Lett. 82, 3948 (1999) [hep-th/9810062]. 107. J. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2, 407 (1961). 108. J. Schwinger, How massive is the W particle?, Phys. Rev. D 7, 908 (1973). 109. J. Schwinger, A theory of the fundamental interactions, Ann. Phys. (N.Y.) 2, 407 (1957). 110. J. Mehra and K. A. Milton, Climbing the Mountain: the Scientific Biography of Julian Schwinger, (Oxford University Press, 2000). 111. K. A. Milton, In appreciation Julian Schwinger: From nuclear physics and quantum electrodynamics to source theory and beyond, Physics in Perspective 9, 70-114 (2007). 112. K. A. Milton, Reminiscences of Julian Schwinger, arXiv:1709.00711, to appear in Berge Englert’s new edition of Julian Schwinger’s Quantum Mechanics: Symbolism of Atomic Measurements.

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The statistical atom∗ Julian Schwinger and Berthold-Georg Englert† Department of Physics, University of California, Los Angeles, CA 90024

This review is the updated and enlarged version of a talk delivered by J. S. on the occasion of the 1982 meeting of Nobel laureates at Lindau, and of talks given by B.-G. E. at several West German universities and Max Planck institutes in 1984.

1. Introduction The title of this review indicates the two main themes of the subject. The generality of “The Statistical Atom” emphasizes the ambition of dealing, not with a specific chemical element, but with the Periodic Table as a whole. And the word statistical points to the method applied in these investigations. Statistics has two different meanings here. First, many-electron systems obey Fermi–Dirac statistics, of course. Second, and more to the point, some properties of atoms can be studied by looking first at situations involving large numbers of electrons. Let us supply some evidence for the practicality of such a statistical approach. The most primitive theoretical model neglects the inter-electronic interaction, thus treating the electrons as independently bound by the nucleus. But even if fermions do not interact they are aware of each other through the Pauli principle. Therefore, such noninteracting electrons (NIE) will fill the successive Bohr shells of the Coulomb potential with two electrons in each occupied orbital state. Since the degeneracy of the shell with principal quantum number n is 2n2 -fold, the total number, N , of electrons in ns filled shells is

3

ns  2 1 1 1 2 2n = − N= ns + ns + . (1) 3 2 6 2 n=1 The total binding energy for a nucleus of charge Z is even simpler, −E =

ns  n=1

2n2

Z2 = Z 2 ns , 2n2

(2)

which uses the single particle binding energy Z 2 /(2n2 ). [Here and in the sequel we adopt atomic units, which measure energies in multiples of twice the Rydberg ∗ This

review was written in 1985 as an invited article for Physikalische Bl¨ atter, but the editor did not like what we submitted and did not put the paper into print. A few years later, it was to be the very first paper in a new physical chemistry journal, which never came into existence. After decades in limbo, the review is now a fitting contribution to these proceedings. † Now at Centre for Quantum Technologies and at Department of Physics, National University of Singapore; and at MajuLab, Singapore.

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constant, me4 /2 = 27.21 eV, and distances in multiples of the Bohr radius, a0 = A.] If we now invert Eq. (1) for large N , we get the asymptotic 2 /(me2 ) = 0.5292 ˚ energy formula

1/3

−1/3 3 1 3 1 −E ∼ N N = ns = − + + ··· . (3) Z2 2 2 12 2 This is certainly a good approximation if the number of electrons is large enough. But suppose we apply it for a small number, say N = 2, where the exact value of ns is one? Well, this asymptotic formula produces ns ∼ = 1.0000297

for

N =2

(4)

which is in error by only 0.003%! For N = 10, the deviation from ns = 2 is substantially smaller: just 0.0001%. We take this practically perfect agreement as strong encouragement for trying a similar large-N approximation for realistic atoms, where the electrons do interact. But before going into those details, we point out that the primitive NIE-atoms can supply realistic qualitative answers. For example, the total binding energy of neutral systems (N = Z) can be written as −E 1 2 2Z

= 2.289Z 1/3 − 1 + 0.1456Z −1/3 + · · · ,

(5)

a structure that we shall meet again for real atoms, but with somewhat different numerical factors. Then consider atomic size. The dimension of a Bohr shell is specified by the square of the principal quantum number divided by the charge of the nucleus. For neutral atoms this says that n2s ∼ Z −1/3 , (6) Z which again is qualitatively correct for the main body of electrons. Another kind of support for the statistical, i.e., large-N treatment of atoms comes from a look at properties of real atoms. A good example is the total binding energy, as presented in Fig. 1. Each individual circle in this figure represents the result of a Hartree–Fock (HF) calculation,1 which energies agree reasonably well with experimental values where they are available. Yet there is no understanding supplied by these individual calculations, at integer values of Z, for the fact that these binding energies behave so remarkably regularly as a function of the atomic number Z. Why then should one expect the large-N approximation to be useful? Simply because such a regular dependence on Z (or N ) cries out for a formula like Eq. (5), and the statistical approach is likely to produce it. atomic size ∼

2. General Approach The basic simplification provided by the large number of electrons in the statistical atom is the possibility of introducing an average potential in which the electrons can be considered to be moving independently. That effective potential, V , describes

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8 .............................................................................................................................................................................................................................................................................................................................................................................................. 7 6

−E 5

1 2 2Z

4 3

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

2 ``` `

```

``

``` ```

````

````

```` `````

``````

``````

```` ```````` ```````

`````````

` ` ` ` ` `` ``

`

`` ` `` `` ` `` `` `` ` ` ` `` `

`` ` `` `` ` `` ``

` `` ``

1 ` 0 0

25

Fig. 1.

50

Z

75

100

125

HF binding energies for Z = 1, . . . , 120.

both the interaction with the nucleus and the electron–electron interactions. We thus start with the one-particle Hamiltonian (atomic units again) 1 2 p +V , (7) 2 and use it to write down the total one-particle energy and the total number of electrons when all states with energy less than −ζ are occupied, H=

E1p = tr{H η(−H − ζ)} , N = tr{η(−H − ζ)} .

(8)

The combination H + ζ, that appears in the argument of Heaviside’s step function η, invites rewriting E1p as E1p = tr{(H + ζ)η(−H − ζ)} − ζN ≡ E1 − ζN .

(9)

This sum of single particle energies counts every electron pair interaction twice. In order to obtain the real energy, we therefore have to subtract the electron–electron interaction energy once. If we disregard the exchange interaction for a start, this is just the electrostatic energy of the electronic cloud. It is most advantageously evaluated in terms of the integrated square of the electric field. Thus, the total energy is given by 

 2  Z 1 ∇ V + − ζN , (10) E = E1 − (dr ) 8π r where −Z/r is the potential of the nucleus, which has to be subtracted from V because only the field produced by the electrons is asked for.

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The advantage of the particular combination of energies in Eq. (10) is its stationary property under variations of V and ζ. First notice that the response of E1 to infinitesimal changes of V exhibits the electron density n,  (11) δV E1 = (dr ) n δV . Then we indeed find a vanishing first order change of E, i.e., δV E = 0, in consequence of Poisson’s equation

1 2 Z . (12) n=− ∇ V + 4π r Likewise, the variation of ζ produces no first order change, because  ∂E1 = (dr ) n , N= ∂ζ

(13)

where the last equality is a simple consequence of the fact that V and ζ appear in E1 only as the sum V + ζ. Equation (12) is a differential equation for the potential V , while Eq. (13) is an algebraic equation that determines ζ. We thus have just enough information to find both V and ζ for given Z and N . Obviously, the essential difficulty in this general approach is the evaluation of the trace in Eq. (9) for an arbitrary potential V , and then the subsequent calculation of the density n, needed in Poisson’s equation (12). Hartree’s way of solving this problem is to write down the one-particle Schr¨ odinger equation for H of (7); look for eigenvalues and eigenfunctions; then square the wave functions to finally produce the density, which in turn leads to a new potential to be used for the next iteration. This method imitates the exact treatment of hydrogen. But the idea of the average potential is best justified at the other end of the Periodic Table, where there are many electrons. So a different way of evaluating the trace of Eq. (9) is called for. It was invented more than half a century ago by Thomas and, independently, by Fermi.2 3. The Thomas–Fermi Model The Thomas–Fermi (TF) approximation is based on the following idea. Although it is true that the potential V changes substantially from deep within the atom to far outside, it should not vary significantly over the range important for a single electron, provided there are many electrons. If that is so, it will be a reasonable approximation for E1 to sum the classical single-particle energies 12 p 2 + V (r ) over those cells in phase space that are occupied. The counting is left to quantum mechanics; two electrons per phase space volume of (2π)3 [= (2π)3 in atomic units]. Thus, we write



 (dr )(dp) 1 2 1 2 p p E1 = 2 + V + ζ η − − V − ζ . (14) (2π)3 2 2

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The step function η cuts off the momentum integral at the (r -dependent) maximal momentum,

(15) P = −2(V + ζ) , so that

 E1 =



 1 1 5 (dr ) − = (dr ) − P [−2(V + ζ)]5/2 . 15π 2 15π 2

(16)

The density is found by differentiating the integrand with respect to V , in accordance with (11), thus producing, after insertion into Poisson’s equation (12),

1 2 Z 1 3/2 [−2(V + ζ)] =− ∇ V + . (17) n= 3π 2 4π r This is the TF equation for V . It has some simple but fundamental implications. Far inside the atom, the potential is that of the nucleus, −Z/r, large and negative. Moving outwards the potential becomes less and less negative, finally equaling −ζ, after which the argument of the square root in (17) changes its sign. So there, at a certain distance r0 , the TF atom has, in general, a sharp edge, beyond which the density is zero. The picture is too simple to describe the exponential decrease of the density in the outer regions of the atom. At the edge, the potential is just the Coulomb potential of the net charge Z − N , implying ζ=

Z −N = −V (r0 ) , r0

(18)

thus ζ=0

for

Z=N.

(19)

The differential equation (17) requires an additional condition at the otherwise undetermined distance r0 . It is supplied by the continuity of the electric field, being the field of net charge Z − N , or Z −N dV (r0 ) = ; dr r02

=0

for

Z=N.

(20)

The boundary conditions (18) and (20) are supplemented by the obvious one at r = 0, Z for r → 0 , (21) r and together they specify a unique solution of (17) for every N ≤ Z. Equation (19) tells us that neutral TF systems are filled to the brim with electrons. There are no negative ions in this approximation. Let us take a closer look at the TF potential for neutral systems. It is useful to measure V as a multiple of the potential of the nucleus by introducing the TF function F (x), V →−

Z V = − F (x) , r

(22)

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F (x) ... .......... ....... ... ... ... ... ... . ....... .. ... ... ... ... .. ....... ... ... ... ... ... . ....... ... 3 ... ... ... ............................... ... ... . ........................................................................................................................................................................................................................................................................................................................................................

0.5

0

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

1 .....

1 − Bx

... ... ... ... ... ... ... ... ... .... ..... ..... ...... ......... ............ 144/x ................ ......................... ................................................. ..............................................................

0

2

6

Fig. 2.

10

x

The Thomas–Fermi function F (x).

the argument of which is related to the physical distance r by 2/3 1 3π Z 1/3 r with a = x= . a 2 4

(23)

The constant a is chosen such that the differential equation for F , [F (x)]3/2 d2 F (x) = , (24) dx2 x1/2 also called the TF equation, is free of numerical factors. For neutral atoms, ζ = 0, the boundary conditions (18) and (20) can be satisfied only at infinity, F (∞) = 0 ,

F  (∞) = 0 ,

(25)

and (21) translates into F (0) = 1 .

(26)

Please notice that both the differential equation (24) and the boundary conditions (25, 26) do not refer to Z. The TF function is a universal function for all Z. The potential V itself does, of course, depend on Z; first through the factor Z/r, but then also because of the Z dependence of the TF variable x of Eq. (23). The factor Z 1/3 there implies the same shrinking of larger atoms that we have already observed for NIE, see Eq. (6). Figure 2 shows a plot of F (x), which is well known numerically. For our purposes the initial slope B, F (x) ∼ = 1 − Bx

for

x1

(27)

is important; its numerical value is (approximately) B = 1.588 .

(28)

The physical significance of B is apparent when (27) is inserted into (22), producing B Z V (r) ∼ = − + Z 4/3 r a

for

r∼ = 0,

(29)

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8 .............................................................................................................................................................................................................................................................................................................................................................................................. 7

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

TF

6

−E 5

1 2 2Z

4 3

2 ``` `

```

``

``` ```

````

````

```` `````

``````

``````

`````` ````````` ` ` ` ` ` ` ` HF ``````

` ` ` ` ` `` ``

`

`` ` `` `` ` `` `` `` ` ` ` `` `

`` ` `` `` ` `` ``

` `` ``

1 ` 0 0

Fig. 3.

25

50

Z

75

100

125

Comparison of HF binding energies and the TF prediction, Eq. (32).

inasmuch as the additive constant is the interaction energy of an electron, near the nucleus, with the main body of electrons. We can use it to immediately write down the change in energy caused by an infinitesimal change of the nuclear charge Z to Z + δZ. It is the analogous electrostatic energy of that additional nuclear charge,3 where a minus sign is needed to connect with the known energy, which is that of an electron, δE = −

B 4/3 Z δZ . a

(30)

3 B 7/3 Z 7a

(31)

The consequence E=−

is the TF formula for the total binding energy of atoms. Comparison of the numerical factor in −E 1 2 2Z

= 1.537Z 1/3

(32)

with the one of the leading term for NIE [Eq. (5)] shows that the electron-electron interaction reduces the atomic binding energy by roughly one third. A look at Fig. 3 shows that Eq. (32) does reproduce the general trend of the atomic binding energies. Although the need for refinements is clear, it is remarkable how well TF works despite the crudeness of the approximation that it represents. In Fig. 3 the continuous statistical curve is closer to the integer-Z HF circles at small Z values than at large ones. This is deceptive, however, since it is the fractional difference that counts. This relative deviation decreases with increasing Z. For

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0.4 .........................................................................................................................................................................................................................................................................................................................

.. ... ... .. ... ..... .... .. ... .... ... ... 0.3 ... ... ... ... . ... . . −4/3 ... . . Z D(r) ... ... ... ..... ... ... ... 0.2 .. ... .... ... ... ... .... .. .... ... .... ..... .... ...... 0.1 .... ...... ....... .... ......... ........... .... ∼ r .............. ... .................... ................................. ... ..................∼ ...........r........................ ... 0 ... . ... ... ... ... .. ... ... ... ... .... ... .. ... ... . . ... . ....... ..... ... ... ... .. ... ..... ... .. ... ... ... ... ... ... ... ... ... . . ... .. ....... .... ... ... ... .. ... ... .... ... ... ... ... ... ... ... ... ... ... . . ... ....... .. .... ... ... ... .. ... 1/2 ... .... ... ... ... .... ... −4 .. ... ... ... ... ......................................................................................................................................................................................................................................................................................................................

0

Fig. 4.

2

4

Z 1/3 r

6

8

10

Radial density D = 4πr2 n, as predicted by TF.

Z = 10, 20, 30, 60, 90, and 120 its amount is 29, 24, 21, 17, 15, and 13 percent, respectively. There are also obvious deficiencies of the simple TF model, graphically demonstrated by a plot of the TF density in Fig. 4. At small distances the radial density grows like the square root of r, not proportional to the square of r as required by a finite density at r = 0, whereas the drop off at large distances goes like r−4 , which is so slow that one never really gets outside the TF atom. We have noted earlier that neutral TF atoms have their edge, r0 , at infinity. 4. Validity of TF Before we can improve TF we must find out where it fails. Recall that the derivation started with the assumption that V is slowly varying. What does this mean? The quantum standard of length associated with an individual electron is its de Broglie wavelength, λ. The potential does not change significantly over this range, if |λ∇V |  |V | .

(33)

Substantial changes in V occur on a scale set by the distance r, so that criterion (33) requires that λ  r.

(34)

On the other hand, λ is the inverse momentum (we ignore factors of 2 or π for this kind of reasoning), which in turn is given by the square root of the potential, see Eq. (15). In short, we have, as criterion for the validity of TF, the relation

(35) r |V |  1 .

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0.4 ................................................................................................................................................................................................................................................................................................................

... ... ... .. ... ... .... . ... .. ... ... ... . ... ... ... .. ... .. ... .. ... .. ... .. ... .. ... . −4/3 . . D(r) .. Z ... ... ... ... ... .... .... .. .... ... ..... .... ..... ...... .... ∼ Z ....... ........ .... .......... ... ............ ................ .. ....................... ... ....................................... ∼Z .................. .... . 0 ... . ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... .. .. ... ... ... ... ... ... ... ... ... ... ... .... ... .. ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... .............. ... −2/3 ... ... ... ... .. ... .... ... ... ... .... ... .. ... 1/3 .. ... ... ............................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................................

0

Fig. 5.

x

10

Regions of failure of TF, illustrated by the radial density as a function of x.

Upon introducing TF variables, this reads

Z 1/3 xF (x)  1 .

(36)

First, we learn here, that for a given x, TF is reliable only if Z is large enough. Second, there is information about the regions where TF cannot be trusted for given Z. At short distances, F (x) practically equals unity, and the left-hand side of (36) is of the order of unity, when x ∼ Z −2/3 , or r ∼ 1/Z. Consequently, there is an inner region of strong binding where TF fails. Then, at large distances, where F (x) ∼ 1/x3 , the inequality (36) is not obeyed, once x is of the order Z 1/3 , or r ∼ 1. Now we find the outer region of weak binding to be also described inadequately by TF. The entire situation is thus as shown in Fig. 5. The two shaded areas are badly treated by TF, whereas the intermediate region of the atom is dealt with rather accurately. And, the bigger Z, the less important the shaded regions are. We conclude that (in some sense) TF becomes exact for Z → ∞. Nice, but in the real world Z isn’t that large, the more so Z 1/3 , which obviously is the relevant parameter. It ranges merely from one to roughly five over the whole Periodic Table. Clearly, modifications aimed at improving TF are called for.

5. Improving TF. Strongly Bound Electrons Despite their relatively small number, the electrons close to the nucleus have such a large binding energy that the leading correction to the TF energy formula (32) stems from a better description of these strongly bound electrons. This can be achieved rather simply, with a very rewarding outcome. Here is how it goes.

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

E ...............................................................................................................................................................................................................................................

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

r

−ζ ..... ..... ..... ..... ..... ..... ..... ..... .......................................................

..... ..... V (r) ... . . .. .... ... .. .... (ns + 1)th shell .. − 12 Z 2 /(ns + 1)2 ................. .. −ζs ..... ..... ......... .. . − 12 Z 2 /n2s ............. . ns th shell ... .. ........ ........ . .........

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

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

Fig. 6.

Concerning corrections for strongly bound electrons; see text.

In the region we are talking about now, the vicinity of the nucleus, the potential is well approximated by (29); it is the Coulomb potential plus a (small) constant. In other words, the strongly bound electrons feel practically only the force of the nucleus, while the interaction with other electrons is negligible. Consequently, we shall treat the neighborhood of the nucleus as occupied by NIE, filling a certain number, ns , of Bohr shells. Formally, the one-particle energy spectrum is divided at a binding energy ζs which separates the strongly bound electrons from the rest, see Fig. 6. This ζs is, of course, not a uniquely defined physical quantity. But it is not arbitrary. First, ζs denotes a binding energy that is large on the TF scale, but small on the Coulomb scale, because we do not want to correct for just the 1s shell, but for all relevant Bohr shells. Symbolically this means Z 4/3  ζs  Z 2 .

(37)

Second, ζs is related to the number of shells, ns , that are treated specially. So ζs has to be sandwiched by the binding energies of the ns th and the (ns + 1)th shell. This can be expressed by ns < √

Z < ns + 1 , 2ζs

(38)

which exhibits the combination of Z and ζs that will be relevant in a moment. The union of (37) and (38) shows that the total number of specially treated strongly bound electrons, Ns [∼ n3s , cf. Eq. (1)], is a small fraction of all electrons, Ns  Z .

(39)

To obtain the change in the binding energy generated by the improved description of the innermost we have to do two things: remove the wrong TF value,  √electrons,  which is Z 2 Z/ 2ζs , and then add the correct Bohr energy of Eq. (2). Thus the

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

a

6

−E 5

1 2 2Z

4 3

2 ``` `

```

``

``` ```

````

````

```` `````

``````

``````

```` ```````` ```````

`````````

` ` ` ` ` `` ``

`

`` ` `` `` ` `` `` `` ` ` ` `` `

`` ` `` `` ` `` ``

` `` ``

b

1 ` 0 0

25

50

Z

75

100

125

Fig. 7. Comparison between binding energies as predicted by HF (circles); TF (curve a); TF with corrections for strongly bound electrons (curve b).

correction to the binding energy is Z Δ(−E) = −Z 2 √ + Z 2 ns . 2ζs

(40)

This looks ambiguous, inasmuch as it contains ns and ζs , both being quantities that for ζs is given in (38), lack a unique value. However, for fixed ns , the possible range  so that averaging over this range assigns the value −Z 2 ns + 21 to the first term on the right of (40). This implies 1 Δ(−E) = − Z 2 . 2 What we have found is the next term in the energy formula; it now reads −E 1 2 2Z

= 1.537Z 1/3 − 1 .

(41)

(42)

Since the strongly bound electrons involved in this correction are basically noninteracting, it could have been anticipated that the additional term is identical with the respective one in the formula for NIE, Eq. (5). Again we compare with the HF energies. Figure 7 also gives a plot of the previous TF curve to emphasize the significant improvement. 6. Improving TF. Quantum Corrections and Exchange The corrections we discuss next come from the main body of the electrons. First we remark that Eq. (14) is in error to the extent that the quantum effects introduced in (33) are significant. So we now consider corrections associated with the finiteness

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of ∇V . Inasmuch as this is a vectorial quantity, the leading energy correction is of second order4 in the parameter of (33), |λ∇V |/|V |, and therefore produces an energy change ∼ Z 5/3 . For details the reader is referred to Ref. 5, from which we quote the leading quantum correction to the energy,   2 1 (43) (dr ) −2(V + ζ) . ΔEqu = − 18π 3 The derivatives of the potential that occur initially have been removed both by partial integrations and by utilizing the TF equation for V , Eq. (17). For neutral atoms, the corresponding supplement to (42) is now obtained by inserting the TF potential (and ζ = 0, of course) into (43), which leads to   Z 5/3 ∞ 2 (44) −0.2699Z 5/3 . dx F (x)2 = ΔEqu = − 2 16a 0 11 [The integral has the numerical value 0.6154.] Again, as for the leading Z 7/3 term, this Z 5/3 contribution is roughly two thirds of the Z 5/3 term for NIE in Eq. (3). There is a second effect that also produces a Z 5/3 correction to the energy — the exchange interaction of the electrons. In contrast with the electrostatic interaction energy of each electron with the other electrons, constituting Z electrons at a distance ∼ Z 1/3 , which is of order Z/Z −1/3 = Z 4/3 , exchange is limited to electrons 1/2 ∼ Z −2/3 ; thus the with overlapping wave functions at a distance ∼ λ ∼ 1/|V | −2/3 2/3 = Z . The explicit result exchange energy of each electron is of the order 1/Z of the calculation is5, 6   2 1 ΔEex = − 3 (dr ) −2(V + ζ) 4π  9 9 −0.2699Z 5/3 . (45) = ΔEqu = 2 11 It supplements (44) and yields the final statistical energy formula −Estat 1/3 − 1 + 0.5398Z −1/3 . 1 2 = 1.537Z 2Z

(46)

A plot of the successive levels of approximation is given in Fig. 8. The marvelous agreement of (46) with HF — the curve goes right through the circles — is a great triumph of the statistical method. One now understands why these atomic binding energies are so regular. They are a property of the ensemble of electrons, no individuality is (yet!) recognizable. 7. History The three terms of (46) are associated with certain names, and we welcome the opportunity to give a brief historical account. The subject started with Thomas’s paper of November 1926.2 He could have, but did not derive the leading term of the binding energy formula. The first to write down Eq. (32), in July 1927, was Milne7 who — being an astrophysicist — recognized the similarity of the TF equation (24)

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

a aa aa a a a a a a a a aa ca a a a a a a a a a b a a aa a a a aa a a a

6

−E 5

1 2 2Z

4 3

2 a a

a

a

a

1 a 0 0

25

50

Z

75

100

125

Fig. 8. Comparison between binding energies as predicted by HF (circles; Z = 1, 2, 3, 6, 9, . . . , 120 shown only); TF (curve a); TF with corrections for strongly bound electrons (curve b) ; the statistical binding energy formula (46) (curve c).

with Emden’s equation for spheres of polytropic perfect gases, held together by gravitation. Milne’s numerical factor was about twenty percent too small, which accidentally improved the agreement with the then available experimental data. Fermi’s first paper on the statistical theory of atoms was published in December 1927.2 It contains a remarkably good numerical solution for F (x) [he calls it ϕ]; e.g., the initial slope B is given as 1.58. Fermi also noticed the connection between the total binding energy and this constant, so that he can claim fatherhood of Eq. (31). His numerical factor is, of course, much better than Milne’s — only half a percent short of the modern value. We are told that Fermi was unaware of Thomas’s work until late in 1928, “when it was pointed out to him by one (now unidentified) of the foreign theoreticians visiting Rome.”8 There are two probable candidates for this anonymous person: Bohr and Kramers, whose encouragement is acknowledged by Thomas in his paper.2 The credit for the first highly accurate calculation of F (x) belongs to Baker.9 His work was published in 1930, long before the age of high-speed computers, and contains a value for B which is exact to 0.03%. We honor Baker by assigning his initial to this number. Now to the next term in (46), the correction for strongly bound electrons. While it has, of course, always been recognized how badly the innermost electrons are represented by TF, it would take the surprisingly long time of 25 years until Scott came up with the energy correction of Eq. (41), in 1952.10 However, his derivation — he calls it a “boundary effect” and treats it accordingly — has not been widely accepted. The general feeling concerning Scott’s correction was that “it seems difficult

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to give a completely clearcut demonstration of the case.”11 This was delivered — in the spirit of the treatment reported above — by one of us in 1980,12 another 28 years later. Recently, we showed an elegant derivation of Scott’s term by making use of the scaling properties of TF with corrections for strongly bound electrons.13 Scott, in the very same paper,10 was also the first to give a Z 5/3 term in the energy formula. However, being unaware of the quantum corrections, he considered merely the exchange contribution of Eq. (45), thus accounting for nine elevenths of the last term in (46). Again it took many years before, in 1981, the quantum correction, Eqs. (43, 44), was evaluated by one of us.5 From then on, the statistical energy formula was known.14 Of course, there has been important work on extensions of TF by other authors. The exchange interaction was first considered by Dirac, as early as 1930.15 [He was possibly reacting to a remark by Fermi at the end of a talk presented at a 1928 conference in Leipzig,16 which Dirac also attended.] But Dirac did not deal with exchange energy, just with the implied modification of the TF equation. An expression for this energy, equivalent to (43), was first given by Jensen in 1934,17 who also on this occasion corrected for an inadvertance of Dirac, whose exchange effect was too large by a factor of 2. However, there is no doubt that it was Scott who for the first time evaluated the exchange energy perturbatively, arriving at (45). Maybe both Dirac and Jensen were just thinking that one should not talk about the second correction before the first one is known . . . The first attempt at including the nonlocality of quantum mechanics was performed by v. Weizs¨ acker in 1935.18 He derived a correction to the kinetic energy which has the serious drawback that it cannot be evaluated in perturbation theory — the outcome would be infinite. From our investigations of quantum corrections5, 6 it has become clear that a consistent treatment requires a simultaneous, correct handling of the strongly bound electrons. Why didn’t Scott do exactly that? There are two reasons. First, Scott’s “boundary effect” theory of the vicinity of the nucleus cannot be directly implemented into the energy functional. And second, the language used by v. Weizs¨acker, Scott, and others is based on the electron density as the fundamental quantity, whereas these problems are most conveniently discussed by giving the potential the fundamental role. 8. Shell Effects Although we were justifiably pleased with the striking agreement of the statistical curve and the HF crosses in Fig. 8, the story is not yet finished. Let us now look at this plot as though through a microscope. Figure 9 shows the relative deviation between Eq. (46) and HF. One sees that from Z = 22 on, the 0.2% level is reached; after Z = 56 the accuracy is better than 0.1%. There is an obvious, steady increase in agreement as Z becomes larger, but this is accompanied by interesting fluctuations. It turns out that Fig. 9 is not the appropriate way of looking at these oscillations. Instead of the relative difference between HF and Eq. (46) we now present the absolute deviation, divided by Z 4/3 , which is the anticipated order of

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(EHF /Estat − 1) [%]

0.6

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

`

0.4

```

0 −0.2

`

`

`

`

`

−0.4 0

Fig. 9.

````

`

0.2

251

`

`

`` ` ````

` ``

``

` ` `` ` `` ` `` `` ` ` ` ` ` ` `

```

```

` ` ` `` `` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` `` `` ` `` ` `` `` ` `` `` ` `` `` ` `` `` ` `` `` ` `` ` `

``

25

50

75

Z

100

125

Relative deviation, in %, between HF binding energies and the statistical formula (46).

the next term in the binding energy formula, and plotted not as a function of Z itself, but against Z 1/3 which is the significant measure of the number of electrons. This is Fig. 10. We are confronted with an unexpectedly regular oscillatory curve that is not only well defined for large Z but even reaches all the way down to hydrogen, Z = 1. Can the statistical approach be employed at all to explain such a behavior? Well, although oscillatory, the Z-dependence is still smooth and certainly not irregular. However, before embarking on a calculation, we first have to gain a qualitative physical understanding of the underlying phenomena. Everybody’s immediate reaction to Fig. 10 is that the oscillations are the filling of atomic shells. If this were true, surely the atoms with closed shells, the inert gases — He (Z = 2), Ne (10), Ar (18), Kr (36), Xe (54), Rn (86), and another one with Z = 118 for which the chemists have not yet invented a name — would sit on ............................................................................................................................................................................................................................................................................................................................................. ... ... ... ... . ..... ....... .. ... ... ... ... .... .. .. ... . ... ....... ...... .... .... .. .. ... ... . . ... ....... ...... .... ... ... .... ... .. . . ... ....... ...... ... ... .... ... ... ... . ... ... ...... ...... ..... ... ... .... ... .. . . ... ....... ..... . ...........................................................................................................................................................................................................................................................................................................................................

↓

−(EHF − Estat ) Z 4/3

0.06

∗`

`

↓

`

↓

`

` ` ` ` `∗`

↓

↓

````

`````

↓

↓

`

````` ∗``` ` ` ` ` ` ` ` ` `` ` ` ` ` ` ` ` ∗`` ` ` `` ` ` ` ` ` 0.02 ` ∗` ` ` ` ` ` ` ` `` ` 0 . . . . . . . . . . . . . . . .` . . . . . . . . . .` `. . .` . . . . . . . . . .``. . .```. . . . . . . . . . . . `` `` ```` `` ` `` 0.04

−0.02 −0.04

`

` `` ``` ``

`

∗

∗``

`

1

2

3

Z 1/3

4

5

Fig. 10. Absolute deviation between HF binding energies and the statistical formula (46). Stars mark the location of inert gas atoms, and the arrows point to them.

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prominent places of the curve? Figure 10 shows these locations, and on first sight the inert gas atoms do not seem to follow any pattern related to the oscillatory curve. They are, however, also not randomly distributed over the oscillatory curve, but show a clear tendency to be close to its maxima. We infer, therefore, that there is a connection between the energy oscillation and the existence of closed atomic shells. These two phenomena are manifestations of one underlying physical effect. To answer the question what effect that is, let us recall the origin of atomic shells. The reason for their being is the existence of quantum numbers in a spherically symmetric potential: angular quantum number , and radial quantum number nr . But  and nr alone would not account for shells; we also need the fact of energetic degeneracy. States with differing quantum numbers may have almost the same binding energy. This is, of course, familiar for the Coulomb potential where the energies depend only on the principal quantum number n = nr +  + 1, leading to the 2n2 fold degeneracy that we have made use of in Eqs. (1) and (2). Thus, in atoms containing NIE, the maximal radial quantum number and the maximal angular quantum number are equal. Not so in the real world, where the ratio of the two is roughly 2 : 1 [uranium, e.g., possesses 7s electrons (nr = 6) and 5f electrons ( = 3)]. The degeneracy of the weakly bound outermost electrons is certainly not of Coulombic type. We can learn more about it from another look at the Periodic Table, this time at the last row. There the 7s, 6d, and 5f electrons are filled in, but not in a given order, instead they compete with each other — a sure sign of degeneracy. In an -nr diagram, Fig. 11, these three states do not lie on a straight line; degenerate states are connected by bent curves which are the steeper, the larger  is. Deep inside the atom, we expect Coulombic degeneracy for the strongly bound electrons. In this situation, states with the same binding energy do lie on a straight line in the -nr diagram. In Fig. 11 this is illustrated by the 2s and the 2p state. It is clear that a theoretical description of the oscillations in Fig. 10 must be based on a detailed energetic treatment of those few electrons with least binding energy. This view is supported by the relative size of the effect we are looking for, which is of order 1/Z as compared to the leading Z 7/3 term, or, like one electron compared to the totality of Z electrons. After these preparatory remarks, it is no surprise that we now attempt to evaluate the effective (i.e., including ζ) single particle energy E1 of Eq. (9) by performing the sum over the quantum numbers, E1 =



2(2 + 1)(Enr , + ζ) η(−Enr , − ζ) .

(47)

nr ,

Here we exhibit the spin and angular momentum multiplicity, and have the step function select those states with Enr , < −ζ. We shall relate the individual energies to the potential V and the quantum numbers, not through the eigenvalue problem associated with Schr¨odinger’s equation, but by means of the semiclassical

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7

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

nr .......

6s 6 ........s...... 5 4 3 2

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

s6d

2s 1 s 0

2p

s

0 Fig. 11.

s5f

1

2

3



4

Energetic degeneracy in a large atom.

quantization rule, 1 1 nr + = 2 π

 2 1/2   + 12 dr 2 Enr , − V − , 2r2



(48)

which is usually derived by the WKB method. Equation (48) is known to produce the exact energy eigenvalues for a few simple potentials, notably the Coulomb and the oscillator potential. The exactness for Coulombic potentials is significant, since it assures correct treatment of the strongly bound electrons. For other “smooth” potentials, (48) gives very good approximate values for Enr , . Certainly, for our purposes, Eq. (48) is good enough. But even with the simplifications provided by employing (48), the double sum of (47) is not easily evaluated. The main reason for that is the implicit definition of Enr , in (48) where it is the radial quantum number that is expressed as a function of  and Enr , . Both quantum numbers are accompanied by an added 12 in Eq. (48); we shall therefore simplify matters by introducing new variables according to ν ≡ nr + so that now 1 ν= π

1 , 2 

λ≡+

1 , 2

Eν,λ ≡ Enr , ,

1/2 dr  2 2r (Eν,λ − V ) − λ2 . r

(49)

(50)

In both equations, (48) and (50), the domain of integration is the classically allowed region where the argument of the square root is positive. Before proceeding with our investigations of Eq. (47) let us make sure that the TF potential can be expected to be useful here. Its insertion into (50) produces the

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(a)

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

.........

(b)

1.6 ..................

0

6

1 00 .01 0 0.

5 nr

−

Z E/

4 3

3 4/

=

2

1 0.

−

1

3 4/

Z E/

=

0

1

10

0

0.2 0.4 0.6 0.8 λ/Z 1/3

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

... 7 ............... c

E=

.... ... . 1.4 ..... ...... ..... ... ..... .... . . 1.2 ..... ......... ...... ..... ... .. ..... .... ... .... ..... .... ..... ... .. 1.0 .... .... .... .... ... .. . . 1/3 .... ... ... . . . . . ν/Z .... .... ... 0.8 ......... .... ... ... ..... .... ... .. .... ....... ..... .... ...... .... . .... ...... .... 0.6 .... .... ....... ..... . .... ...... . . ..... .... .... ...... .... ..... . ... .... 0.4 ..... .... ............... ..... .... . ..... .... ............... ..... .... ......... ..... 0.2 ......... . . . .... ............ ..... ..... .. . . ..... . ...................................................................................................... 0

1

soccupied cunoccupied

s7s s s

c s

6p

s

s

s

s

s

s

s

s

s 4f s

c

0

1

2

4

c

s

5d

c

s



3

Fig. 12. (a) Energetic degeneracy in the TF potential. (b) TF prediction for occupied states in Ra (Z = 88).

following ratio of maximal values for ν and λ: ν(E = 0, λ = 0) 1.659 Z 1/3 = = 1.79 , λ(E = 0, ν = 0) 0.928 Z 1/3

(51)

which is in reasonable agreement with the corresponding number for a large atom, e.g., uranium, 6+ νmax = λmax 3+

1 2 1 2

= 1.86 .

(52)

In Fig. 12(a) we see ν/Z 1/3 as a function of λ/Z 1/3 for several E, demonstrating the different character of degeneracy for small and large binding energies; a plot that has a striking similarity with Fig. 11. This becomes even more convincing when we ask for the specific states available in a large atom, say radium, Z = 88. Figure 12(b) shows perfect agreement between the TF prediction and experimental observations; the E = 0 curve of Fig. 12(a) separates the occupied states from the unoccupied ones, selecting exactly those that are spectroscopically known to be available. This is another way of presenting the results of Fermi’s application of the statistical theory of atoms to the systematics of the Periodic Table, the great historical triumph of TF, published in his second paper on the subject in January 1928.19 With this reassurance, the discussion of Eq. (47) is now continued. A more useful way of writing this sum over quantum numbers employs ν and λ of (49) and replaces the summation, e.g., over , by an equivalent integration over λ, with the

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

ν

0

E

=

−

ζ

Fig. 13.

E=

0

.... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ..... ... ..... ........... ... . ............................... ... ............................................ ... ..................................................... ... ... ................................................................................. ... .................................................................................. ......................................................................... ... . . . . ... .......................................................................................................................... ... ..................................................................................................................................................................... ... ............................................................................................................... ... .......................................................................................................................................... .......................................................................................................................................... ..... . . . .. . .. .. .. . .. . .. . . . .......................................................................................................................................................................................................................................................... ..... ............................................................................................................................................................................................................................................ ... .............................................................................................................................................................................. ... ......................................................................................................... .. .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. . λ

Domain of integration in Eq. (54) is shaded area.

aid of the Poisson sum formula, ∞ 

 2(2 + 1)A+ 12 = 4

=0

=4

∞

dλ λ 0 ∞ 

∞ 

  δ λ −  − 12 Aλ

=−∞



∞

(−1)k

dλ λ ei2πkλ Aλ .

This, with the analogous procedure for nr and ν, turns (47) into  λE  ν(E=−ζ,λ) ∞  E1 = 4 (−1)k+j dλ λ ei2πkλ dν ei2πjν (Eν,λ + ζ) . k,j=−∞

0

(53)

0

k=−∞

(54)

0

We got rid of the step function by making the limits of integration explicit; the domain of integration is the area between the ν, λ axes and the curve belonging to E = −ζ in Fig. 13. So far we have done nothing but rewrite Eq. (47). Now we shall illustrate this structure by picking out the j = 0 terms. In other words, we concentrate on λ oscillations only, disregarding ν oscillations, which corresponds to a simplified picture in which only angular momentum is quantized, not the radial motion. We call this TF, short for -quantized Thomas–Fermi. The absence of the exponential in the ν integral for j = 0 enables us to change variables from ν to E (for fixed λ) which is desirable in view of the implicit definition of Eν,λ in Eq. (50). Here is how it goes: dν (E + ζ) = d[ν (E + ζ)] − ν dE ;

(55)

the total differential has zero value at both limits of the integral, and ν possesses a

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simple algebraic dependence on E, allowing further integration,     1 dr  2 ∂ 3/2 − 2r (E − V ) − λ2 . dν (E + ζ) → dE ∂E 3π r3 The TF result for E1 is then   ∞ 3/2 4  dr  2 2r (−ζ − V ) − λ2 (−1)k dλ λ ei2πkλ . E1T F = − 3 3π r

(56)

(57)

k=−∞

This equation was, in some sense, known to Hellmann in 1936,20 but certainly not in this form. His formula used the original sum over  [easily obtained by reversing (53)], and was expressed in terms of densities, one for each value of . This was a clumsy way of writing it, which unfortunately kept both Hellmann and Gomb´as, who devoted a chapter in his classical textbook21 to the matter, from realizing the important fact that the k = 0 contribution to this sum is just the TF expression (16), once the λ integration is carried out.22 With that essential piece of information, we split E1TF into E1TF and the rest,   ∞ 3/2  8  dr TF TF k−1 E1 −E1 = (−1) cos(2πkλ) , (58) dλ λ 2r2 (−ζ −V )−λ2 3π r3 k=1

and, since this right-hand side must be a small correction to TF, we are justified in using the TF potential for its evaluation. The leading TF oscillation can be easily identified. To that end, we first replace λ by its maximum value times the cosine of an angle θ, 1/2  cos θ ; (59) λ = 2r2 (−ζ − V ) this leads to E1TF

−

E1TF

 ∞ ∞ 5/2 dr  2 8  k−1 2r (−ζ − V ) = (−1) 3 3π r 0 k=1  π/2 × dθ cos θ sin4 θ cos(z cos θ)

(60)

0

with

 1/2 z = 2πk 2r2 (−ζ − V ) .

(61)

Then the asymptotic form of the θ-integral in Eq. (60) for large z [compare Eq. (35)] is employed in identifying the leading oscillatory term,  ∞ 5/4   1  (−1)k−1 ∞ dr  2 TF 2r (−ζ − V ) cos z − 14 π . (62) Eosc = − 3 3 5/2 π r k 0 k=1

The last step is a stationary phase evaluation of the remaining r-integral for V = VTF and ζ = 0. The explicit form of the resulting leading TF oscillation is ∞  (−1)k TF 4/3 Eosc = −0.4805Z sin(2πkλ0 ) (63) (πk)3 k=1

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0.06 0.04

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

`

`

`

`

`

`

0.02

−Eosc Z 4/3

`

` ``` ` ``

`

`

`

`

`

`

`

````` ```

``` ``` ``` `

`

`

`

`

`

`

`

``

``````` ` `` ``` ```` ```` `` `` ` `````````

` `

`

` ` ` ` ` ` ` ` ` 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .`. . . . `. . . . . . . . . . . . . . . .` . . . . `.` . . . . . . . . . . . . . . ` ` ` ` ` ` `` ``` ` ``` `` ` `

−0.02

−0.04 ` 1

2

3

4 Z

Fig. 14.

257

5

1/3

Comparison of HF oscillations (circles) with leading TF oscillations (solid line).

wherein λ0 is the maximal value of of λ [cf. (51)],  1/2 = 0.928Z 1/3 . λ0 = max 2r2 (−VTF ) r

(64)

The high power of k in the denominator in the sum of (63) assures us that a smooth function is represented by this Fourier series; indeed, it is a repeated piece TF of a cubic polynomial. Let us compare Eosc with the HF oscillations of Fig. 10, which is plotted in Fig. 14. It looks very encouraging because a number of details are quite right: first, the overall amplitude factor Z 4/3 ; second, the periodicity Z 1/3 → Z 1/3 + 1.08; third, the phase. TF of (63) is not the entire oscillation. It accounts for roughly Obviously, Eosc half the amplitude, but does not show any sign of the intriguing structure that evolves at the maxima with increasing Z 1/3 . On the other hand, Eq. (63) was obtained by picking out only the leading contribution to the j = 0 term of the double sum in Eq. (54). Naturally, a better result should be obtained by evaluating the whole sum. This is somewhat involved, however, and we have described details elsewhere.23 We restrict the present discussion to general aspects. The analysis shows that one has to pay attention to two major things. One is that the leading contribution of asymptotic amplitude ∼ Z 4/3 does not suffice, the next-to-leading oscillation (∼ Z 3/3 for large Z) is also needed; this is reminiscent of the smooth part of the energy formula, where the leading TF term also did not give satisfactory results. The other one is the fact that extrapolation from the large-Z domain to the small-Z 1/3 region in question has to be done with extreme care; this is contrary to the situation of the smooth curve, where extrapolating was easy. Here then is the final semiclassical oscillation: Fig. 15.

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0.08 .........................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.. ... ... ... + .. ... + ... + ... + + ... ... +++ ... + + .... . . . . . ... .... + ... ... ... ... + ... ... ... + ... ... .. ......... .... . . ....... ... ... .... + ... . . ... ... .. ... . ... .. + .. ........ .. . . . . . . . ... ... . ... ........ ... .... ... .. ... .... .. .. ..... .... .. ..... .. .. .. ... . .... . . ... . .. .. . . ... .. . . . . ... . . . . .. ... .. .. . .. . ......... . . . . . . . . . . .... . . . . . . . . ... .. . .. .. ......... .. .. .. + .. .. ... .. .. ... .. .... ... ... ........ .. .. ... .. .. .. .. .. ... .. ... ... ... .. ... .. .. .. .. .. .. .. ... ..... .. ........ ... . .. .. . . .... ... . . . . . . . . . .. .. .. ... .. .. . . ..... .. .......... . . . .. .. . + ... . . . . . . . . . . .... ... .. .. .. . .. ... . .. + . .. . .. ... . .. . . . . . . . .. ... ... . . .. .. .. .. . . . .. . . . ... . . . . . . . . . . .. .. .. .. .. . .. .. + ... .... . ...... . . . . . ... . . . . ...... . .. .. .. . .. ... . . . . . ... .. . . .. .. .. ... .. .. .. .. . . .. . ... . ... ... . .. .. .. .. .. .. .. ... ... .. .. .. .. .. .. .. ... ... . . . . . . .. .. . .. .. .. . . ... ... . . . . .. .. .. . . . .. . . . . .... . . . . . . . .. .. .. . . ....... . . . . . ...... .... . . . . .. .. ... . . .. . .. . ... . . ... ... . ... ... .. ... . ... .. .. .. . . . ... . . . . .. .. .. . . . .. .. .. ... ... .. .... ... .. .. .. .. .. ... .. .. . .. ... . . . .. .. .. ... .. .. .. . .... ... . . . .. .. .. ... .. .. .. . + . . .. ... . . . ..... . .. .. . ... .. .. .. . . . .. ... ... ... .. .. ... .. ... .. ... . ... ... . ... .. .. .. .. .. .. . ... . ....... . ... .. .. .. .. . ... .. .. .. .. .... ...... .. .. . ... .. . . ... ..... ... .. .. . . ... .. .. .. ....... .... ..... . ................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0.06

`

`

`

`

0.04

`

`

`

0.02

−Eosc Z 4/3

` ``` ` ``

`

`

`

`

`

`

`

```

``````

` ` ``` ` ` `

`

`

`

`

`

`

`

`

`

```````` ` `` ``` ```` ``` `` `` ` ````````` ` `` ` `

`

` ` ` ` . . . . . . . . . . . . . . . . . . . .`. . . . .` . . . . . . . . . . . . . . `.` . . . . .` ` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ` 0 ` ` ` ` ` ` ` `` ``` ` ``` `` ` `

−0.02

−0.04 `

−0.06 1

2

3

4 Z

5

1/3

Fig. 15. Comparison of HF oscillations (circles) with the semiclassical ones (solid line) and with experimental data (crosses, corrected for relativistic effects).

No doubt, our calculation provides a clear understanding of these binding energy oscillations. Both the HF curve and the semiclassical one have the same periodicity, the same amplitude, the same phase, and the same general shape of plain minima, and maxima with an evolving structure. True, the HF oscillations show less of this structure, but we never expected perfect agreement. [One should not be misled by the obviously missing constant ∼ Z 4/3 that shifts the semiclassical curve down. The calculation concentrated on oscillatory terms, systematically discarding all smooth contributions.] While this comparison of two independent theoretical predictions is entertaining, more to the point is to see what experiment tells. Total binding energies of atoms have been measured (by stepwise ionization) for the first 20 members of the Periodic Table only.24 After correcting for relativistic effects and subtracting the smooth background of Eq. (46) these experimental values are given by the crosses in Fig. 15. They demonstrate that the oscillations we have been considering are really present in nature, both period and amplitude being of the theoretically predicted size. Neither HF nor the semiclassical curve represent an exact quantitative description of experiment, while the qualitative agreement is about the same. 9. Concluding Remarks We have been focusing on nonrelativistic binding energies throughout this review. Naturally, there has been work on relativistic corrections to the statistical binding

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energy formula (46), although the results are not yet quite satisfactory.12, 26 This field remains to be tilled. Other aspects of the theory concern various properties of atoms, such as densities, diamagnetic susceptibilities, electric polarizabilities, etc. Here one encounters different problems. For the described energy considerations it was sufficient to treat all modifications of TF in first order perturbation theory (even the oscillations). For the investigation of a given atomic system with specified Z and N , this approach is not practicable. Instead, the corrections for strongly bound electrons, the quantum improvements, and exchange have to be incorporated consistently into the energy functional, thus leading to a new differential equation for the potential [with only a remote resemblance to the TF equation (17)]. This is an entirely different story; we told it recently.6, 13, 25 Then there are generalizations of the statistical theory to other objects than atoms: molecules, solids, nuclei, even neutron stars, etc. We do no more than mention them. Acknowledgment One of us (B.-G. E.) gratefully acknowledges the generous support by the Alexander von Humboldt-Stiftung, which granted a Feodor Lynen fellowship. References 1. The use of HF results [J. P. Desclaux, At. Data Nucl. Data Tbls 12, 311 (1973)] is dictated both by the subsequent nonrelativistic discussion, and the lack of experimental binding energies beyond Z = 20. The factor 12 Z 2 is taken out of the energy for a convenient restriction in the numerical range. 2. L. H. Thomas, Proc. Cambridge Philos. Soc. 23, 542 (1926); E. Fermi, Rend. Lincei 6, 602 (1927). Their reasonings differ from the one presented here. 3. The accompanying infinitesimal increase of the number of electrons does not change the energy because of the null value of ζ, Eq. (19), as follows from this implication of Eq. (10): ∂E/∂N = −ζ. 4. For consistency, it is also necessary to include a contribution from the second derivative of the potential; it gives a similar result. 5. J. Schwinger, Phys. Rev. A 24, 2353 (1981). 6. B.-G. Englert and J. Schwinger, Phys. Rev. A 29, 2339 (1984). 7. E. A. Milne, Proc. Cambridge Philos. Soc. 23, 794 (1927). 8. F. Rasetti, in E. Fermi, Collected Papers, E. Amaldi et al. eds., (Univ. of Chicago Press, 1962), p. 277. 9. E. B. Baker, Phys. Rev. 36, 630 (1930). 10. J. M. C. Scott, Philos. Mag. 43, 859 (1952). 11. N. H. March, Adv. Phys. 6, 1 (1957). 12. J. Schwinger, Phys. Rev. A 22, 1827 (1980). 13. B.-G. Englert and J. Schwinger, Phys. Rev. A 29, 2331 (1984). 14. Strictly speaking, Eq. (46) can already be found in G. I. Plindov and I. K. Dmitrieva, Phys. Lett. 64A, 348 (1978); but their derivation of the Z −1/3 term is hardly satisfactory, because the model used contained arbitrary cut-offs of the density for small

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15. 16. 17. 18.

19.

20. 21. 22.

23. 24. 25. 26.

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and large distances. As they say, such an atom “reminds one of an apricot without a stone.” Also, these authors did not obtain the Scott term correctly. P. A. M. Dirac, Proc. Cambridge Philos. Soc. 26, 376 (1930). Ref. 8, pp. 291–304. H. Jensen, Z. Phys. 89, 713 (1934). C. F. von Weizs¨ acker, Z. Phys. 96, 431 (1935). A deficiency of Weizs¨ acker’s approach caused his result to be too large by a factor of nine which, unfortunately, has induced people to consider that numerical factor as an adjustable parameter. E. Fermi, Rend. Lincei 7, 342 (1928). Fermi apparently used too crude an approximation for the integral needed to calculate the maximal value of ν. With his number, the ratio of Eq. (51) would be 1.46, but no harm done. H. Hellmann, Acta Physicochimica URSS 4, 225 (1936). P. Gomb´ as, Die statistische Theorie des Atoms und ihre Anwendungen (Springer, Vienna, 1949). N. H. March and J. S. Plaskett, Proc. Roy. Soc. London A 235, 419 (1956), also extracted E1TF out of Eq. (47), but they did not have a systematic way of dealing with the remainder. B.-G. Englert and J. Schwinger, Phys. Rev. A 32, 26 (1985); ibid. 32, 36 (1985); ibid. 32, 47 (1985). C. E. Moore, Nat. Stand. Ref. Data Ser., Nat. Bur. Stand., Vol. 34 (1970). B.-G. Englert and J. Schwinger, Phys. Rev. A 29, 2353 (1984). I. K. Dmitrieva and G. I. Plindov, J. Phys. (Paris) 43, 1599 (1982).

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Julian Schwinger and the semiclassical atom Berthold-Georg Englert Centre for Quantum Technologies and Department of Physics, National University of Singapore MajuLab, Singapore [email protected]

In the early 1980s, Schwinger made seminal contributions to the semiclassical theory of atoms. There had, of course, been earlier attempts at improving upon the Thomas– Fermi model of the 1920s. Yet, a consistent derivation of the leading and next-to-leading corrections to the formula for the total binding energy of neutral atoms, −

E 1 = 0.768745Z 7/3 − Z 2 + 0.269900Z 5/3 + · · · , e2 /a0 2

had not been accomplished before Schwinger got interested in the matter; here, Z is the atomic number and e2 /a0 is the Rydberg unit of energy. The corresponding improvements upon the Thomas–Fermi density were next on his agenda with, perhaps, less satisfactory results. Schwinger’s work not only triggered extensive investigations by mathematicians, who eventually convinced themselves that Schwinger got it right, but also laid the ground, in passing, for later refinements — some of them very recent.

1. Introduction Julian Schwinger’s groundbreaking work on quantum electrodynamics first and then, more generally, on quantum field theory and the physics of elementary particles is, of course, at the center of his legacy. In addition, he made seminal contributions to many other topics, among them the semiclassical theory of atoms. This work of his is summarized in an essay of 1985 for non-expert readers that did not get published in his lifetime. 1 I shall recall here some aspects of our collaboration from mid 1981 to early 1985 and also mention later work that was triggered by Schwinger but did not involve him. 2. Schwinger’s Papers of 1980 and 1981 While teaching a course on quantum mechanics in the late 1970s, Schwinger was intrigued by the Thomas–Fermi (TF) model and the systematic deviation of the TF approximation for the total binding energy of an atom from the Hartree–Fock (HF) values (curve a and circles in Fig. 1). This was crying out for an understanding. Schwinger responded to the challenge with his derivation (in 1980 2 ) of the leading correction that had been conjectured by John Scott in 1952, 3 now known as the Scott correction. Schwinger did not know Scott’s paper before a referee pointed it out. Although the inclusion of the Scott correction improved the binding energies substantially (curve b in Fig. 1), there remained a systematic error. In a paper of

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8 ............................................................................................................................................................................................................................................................................................................................

6

−E 1 2 2 2 Z e /a0

4

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

2 a a a

a

a

a

a

a

a

a aa

0 0

aa

a aa

◦: a: b: c:

25

aa

a

aa aaa a a  aa

aaa

a

c ↓a a a a a a a a aa

b

Hartree–Fock energies TF TF + 1st correction TF + 1st & 2nd correction

50

75

100

125

Z Fig. 1. Total binding energy of neutral atoms, as a function of the atomic number Z, in Rydberg units (e2 /a0 ) and scaled by 12 Z 2 . The circles represent the Hartree–Fock values, and curves a, b, and c show, respectively, the Thomas–Fermi approximation by itself, and with the leading and the next-to-leading corrections included. See also Fig. 8 in Ref. 1.

1981, 4 Schwinger took care of this as well, again giving a clear-cut demonstration of what had been conjectured before; 5 this takes us to curve c in Fig. 1 that goes right through the circles of the HF values. The consecutive approximations are the three terms in Eq. (46) of Ref. 1, which are recalled in the Abstract. 3. Mathematicians in action Schwinger’s derivation of the two leading corrections to the TF energy of atoms in Refs. 2 and 4 convinces any theoretical physicist, but the standards of mathematical physics are different. Earlier, in 1977, Elliot Lieb and Barry Simon had given a proof that the TF approximation is asymptotically exact as Z → ∞; see Ref. 6 for the precise statement. Now, two other duos of mathematicians dealt with Scott’s leading correction and Schwinger’s next-to-leading correction. First, Heinz Siedentop and Rudi Weikard in a series of papers 7–11 published 1986–1989 showed that Scott’s correction is indeed the leading correction when Z is large enough. Then, Charles Fefferman and Luis Seco devoted another series of papers 12–20 to the second correction, published 1989–1995, eventually confirming that Schwinger’s result is correct. These three important pieces of mathematical research illustrate how, in the history of this subject matter, the link between theoretical physics and mathematics has been a one-way road: The physicists provide conjectures and the mathematicians convert them into theorems. There hasn’t been any benefit for the physicists

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in return, beyond being assured that they got it right (which, by their own standards, they knew already). The work by Lieb and Simon did not yield a suggestion on how to improve on the TF approximation, nor did the Siedentop–Weikard proof indicate how to go beyond the Scott correction, and an analogous remark applies to the accomplishments by Fefferman and Seco. While it is true that the task chosen by each of the three duos was limited to proving what was conjectured, it would have been also nice to get an idea about where to look for the next improvement. 4. Summer of 1981: A Homework Assignment Schwinger finished the work on the corrections to the TF energy by the end of 1980 (Ref. 4 was submitted in December 1980), and then began studying the corresponding modifications of the TF density. Lester DeRaad had already provided numerical solutions to a modification of the TF differential equation, 4,21 d2 φ φ3/2 = +φ dy 2 y 1/2

(1)

with φ(0) =

48π Z = 1.461Z (22)3/2

and

φ(∞) = 0 ,

(2)

where φ(y) is the auxiliary function in terms of which the quantities of interest can be computed; see Ref. 21 for details. As expected, this improved on the TF approximation for the electron density at large distances — the values obtained for diagmagnetic susceptibilities, essentially the squared distance from the nucleus weighted by the density, were much better than the TF values — but, so the paper summarizes, the “outcome of this test [. . . ] hardly warrants proclaiming a successful conclusion to the search for an extrapolation of the TF model into the outer regions of the atom.” Accordingly, this search continued and led Schwinger to considering 3 # φ 1 d2 φ + =y (3) dy 2 y 3 with φ(0) = 1.461Z

and

φ(y0 ) = 0 ,

1.461(Z − N ) = −y0

dφ(y0 ) , dy0

(4)

which he gave to me as a homework assignment in the summer of 1981 (see Fig. 2). At this time, he spent part of a sabbatical leave in T¨ ubingen, hosted by my Ph.D. supervisor Walter Dittrich. I had recently submitted my thesis and was waiting for the defense, and so I was available when Schwinger needed someone with programming skills. It was easy to get the numbers, such as a table of y0 values for different Z and N , and the numbers passed the various tests that Schwinger could subject them to.

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Fig. 2. The homework assignment: Schwinger’s modification of the TF differential equation in his hand writing. The white spot is a drop of correction fluid.

He then gave me a copy of the draft paper he was working on, and I finally understood the origin of Eq. (3) and the physical meaning of φ(y). While my feedback on the draft and other input did have a bearing on the published version, 22 it was hardly enough to earn co-authorship. Schwinger, however, thought differently and my objection to being first author were gently and firmly pushed aside: “There is much to be said in favor of the alphabet.” The electron densities from the formalism based on Eq. (3) do improve on the TF approximation. They give reasonably good values for the diamagnetic susceptibilities, better than the HF numbers for larger atoms,∗ and this was good motivation for taking the next step. 5. 1984 and 1985: Two Trios of Papers By the time Ref. 22 was finalized and submitted, I had been recruited by Schwinger as a postdoc, and we were already working on the follow-up project that yielded a trio of papers in 1984. 23–25 Equations (1) and (3) were modifications of the TF equation that incorporated Paul Dirac’s approximate exchange energy 26 and Carl Friedrich von Weizs¨acker’s inhomogeneity correction, 27 both responsible for the second correction, but not Scott’s leading correction. While this is acceptable when the focus is on the outer reaches of the atom, it is somewhat inconsistent. Scott’s correction addresses the failure of the TF approximation in the vicinity of the nucleus where the singularity of the Coulomb potential results in large changes of the potential energy over short distances. Therefore, it was necessary to treat ∗ In

those days, HF numbers were systematically too large by about 10%.

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the strongly bound electrons differently. 23 While Dirac’s exchange energy could be used without any essential modification, Weizs¨acker’s inhomogeneity correction required a refinement that accounts for higher-order terms, and Schwinger designed an ingenious method for that, where a certain averaging† over simpler Weizs¨ackertype expressions is performed. 24 An extensive numerical study confirmed that the resulting modification of the TF model was worth the effort. 25 It is, however, not truly satisfactory because of an unresolved problem with the handling of the strongly bound electrons. That involves a separation on the energy scale, and the electrons with a binding energy less than the threshold value are treated in the TF fashion (with modifications). The electrons with larger binding energy are regarded as dominated by the Coulomb potential of the nucleus, with corrections accounted for by perturbation theory. As a consequence of the rather different treatments, there is an unphysical dependence on the choice of the threshold value. While this can be systematically removed in the resulting correction to the energy (this is an important detail in Schwinger’s derivation of Scott’s correction 2 ), we did not manage to get rid of it in the density.‡ Meanwhile, we were wondering about the difference between the HF energies in Fig. 1 and those of curve c. As Figs. 9 and 10 in Ref. 1 show, this difference is a notso-regular oscillatory contribution, which suggests that the degeneracy associated with energy shells — Niels Bohr’s shells 31 modified by the repulsive forces between electrons — is important here. This suggestion was not misleading and eventually provided an understanding of these energy oscillations, reported in another trio of papers, published in 1985. 32–34 Here, a crucial observation is that the TF approximation can be obtained by first expressing the energy as a sum over WKB energies, and then replacing the sum by an integral. 32 This integral is a zeroth-order approximation of the sum, and the hierarchy of higher-order terms contains oscillatory terms that can be extracted systematically. The leading oscillatory term contributes 34 ∞ 2 

(−1)k sin(2πkλ0 ) a0 (πk)3 k=1

1 e2 2 = 0.3206 λ0 − λ0 Z 4/3 4 a0

−0.4805 Z

4/3 e

(5)

to the energy, where λ0 = 0.928Z 1/3 and λ0 denotes the difference between λ0 and its nearest integer; see also Eq. (63) in Ref. 1. After the TF energy (∝ Z 7/3 ), Scott’s leading correction (∝ Z 6/3 ), and Schwinger’s second correction (∝ Z 5/3 ), this contribution (∝ Z 4/3 ) is the next in line. As of today (2019), there is no † Strictly speaking, it is not an average because the terms are weighted by the Airy function Ai(x), which is assuredly positive for x > 0 but not for x < 0. See Ref. 28 for a recent benchmarking exercise. ‡ The situation is remarkably different in momentum space where the Scott-corrected density does not suffer from this problem. 29,30

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Fig. 3. Schwinger’s appreciation, in 1981, of functionals with both the density and the potential as independent variables.

mathematical proof of its correctness, and so we have to be content with the evidence presented in Ref. 34. Consult Ref. 35 for a detailed account of all these developments. 6. Functionals of Both the Density and the Effective Potential The argument presented in Ref. 22 proceeds from a functional for the energy that has the single-particle density, the electrostatic potential, and the chemical potential as independent variables. Schwinger wrote this down without giving a derivation but justifying it by its consequences, and I must have just accepted it. It is indeed consistent with a remark by Schwinger in a letter of 1981 that he wrote in response to the detailed feedback I gave on the draft paper mentioned above; see Fig. 3. I am pretty sure that I did not appreciate the flexibility of this approach at the time. In hindsight, I regard the functional in Ref. 22 as very clever and just right for the purpose at hand, but not as a useful starting point for more general developments. We did not use anything quite like this again and, instead, switched to a formalism based on energy functionals of the particle density, the effective potential, and the chemical potential as independent variables. The effective potential is the sum of the external potential (here: the Coulomb potential of the nucleus) and an interaction potential obtained from the response of the interaction energy (here: direct and exchange electrostatic energy) to variations of the density. This was a natural thing to do, it was intuitive, and no justification seemed necessary — it was clearly an obvious matter for Schwinger, an example of his phenomenal intuition about physics and its mathematical language. As for myself, I took some time to comprehend fully what was so obvious to him. Eventually, at the time of writing Ref. 36, with a precursor in Ref. 35, I saw how this is systematically connected to the standard density-functional theory as formalized by Pierre Hohenberg and Walter Kohn 37 — and this turned the densitypotential functional into a central tool for all of my subsequent work on related topics. The connection is actually quite simple: just subject the kinetic energy term in the Hohenberg–Kohn density functional to a Legendre transformation.

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The concept of density-potential functionals is a contribution of lasting value, perhaps Schwinger’s most important contribution to the field. These functionals are more flexible than density-only functionals, and they facilitate systematic approximations that go beyond formal gradient expansions, for which the Weizs¨acker term of 1935 27 is the prototype. In particular, there is a density-potential functional for the Scott-corrected TF model but no density-only functional. As another example, I mention the somewhat amusing case of gradient corrections to the TF model in two dimensions. Papers in the early 1990s established — or so it seems — that there are no such corrections, all terms in a formal power series of the density gradient have vanishing coefficients; see, for example, Refs. 38 and 39. On the other hand, the TF approximation to the kinetic energy is not exact, whether in one, two, or three dimensions, and so one has the puzzling situation of a non-exact approximation with all systematic correction terms vanishing. The puzzle disappears as soon as one recognizes that there are nonzero gradient corrections in the density-potential functional, and they can be evaluated perturbatively. 40 There is also an analog of the density-potential functionals in momentum space, where one gets an energy functional with the momentum-space density, the effective kinetic energy, and the chemical potential as independent variables. The hierarchy is repeated: We have the momentum-space version of the TF model 36 (which has succumbed to the power of mathematics 41 ), of the Scott-corrected TF model, 30 and of a model with exchange energy and gradient corrections included as well. 42 Finally, there is a recent, novel attempt at approximating the potential part of the density-potential functional. There is a connection with the single-particle propagator that was already exploited in Ref. 24 and led then to the Airy-averaged expressions mentioned above, with an improvement over a related method by Eugene Wigner 43 and John Kirkwood. 44 Now, one can alternatively approximate the propagator by a factorization into terms that refer only to the kinetic energy or only to the potential energy — a technique introduced and developed by Hale Freeman Trotter 45 and Masuo Suzuki 46 — and this yields very good approximations without a gradient expansion. 47 The applications thereof are work in progress. 7. Summary Schwinger’s papers of 1980 and 1981 on the leading and the second correction to the TF energy are well known and have triggered extensive mathematical studies. His insight that one should treat the density and the effective potential as independent variables on equal footing is just as important and crucial for ongoing research. Acknowledgments Julian Schwinger, always kind and generous, taught and guided me from 1981 till his premature death in 1994. His lessons, both formal and informal, were invaluable. I was a greenhorn when he recruited me as his postdoc at UCLA. When I moved to the University of Munich forty months later, I was a physicist.

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References 1. J. Schwinger and B.-G. Englert, The Statistical Atom, these Proceedings, pp. 237–260. 2. J. Schwinger, Thomas–Fermi model: The leading correction, Phys. Rev. A 22, 1827 (1980). 3. J. M. C. Scott, The Binding Energy of the Thomas–Fermi Atom, Philos. Mag. 43, 859 (1952). 4. J. Schwinger, Thomas–Fermi model: The second correction, Phys. Rev. A 24, 2353 (1981). 5. G. I. Plindov and I. K. Dmitrieva, Notes on the nonrelativistic binding energy for neutral atoms, Phys. Lett. 64A, 348 (1978). 6. E. H. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules and solids, Adv. Math. 23 22 (1977). 7. H. Siedentop and R. Weikard, On the leading energy correction for the statistical model of the atom: Non-interacting case, Abh. Braunschweig Wiss. Ges. 38, 145 (1986). 8. H. Siedentop and R. Weikard, Upper bound on the ground state energy of atoms that proves Scott’s conjecture, Phys. Lett. 120A, 341 (1987). 9. H. Siedentop and R. Weikard, On the leading energy correction for the statistical model of the atom: interacting case, Comm. Math. Phys. 112, 471 (1987). 10. H. Siedentop and R. Weikard, On the leading correction of the statistical atom: lower bound, Europhys. Lett. 6, 189 (1988). 11. H. Siedentop and R. Weikard, On the leading correction of the Thomas-Fermi model: lower bound, Invent. Math. 97, 159 (1989). 12. C. L. Fefferman and L. A. Seco, An upper bound for the number of electrons in a large ion, Proc. Natl. Acad. Sci. USA 86, 3464 (1989). 13. C. L. Fefferman and L. A. Seco, The Ground-State Energy of a Large Atom, Bull. AMS 23, 525 (1990). 14. C. L. Fefferman and L. A. Seco, Asymptotic Neutrality of Large Ions, Comm. Math. Phys. 128, 109 (1990). 15. C. L. Fefferman and L. A. Seco, Eigenvalues and Eigenfunctions of Ordinary Differential Operators, Adv. Math. 95, 145 (1992). 16. C. L. Fefferman and L. A. Seco, Aperiodicity of the Hamiltonian Flow in the ThomasFermi Potential, Revista Math. Iberoamericano 9, 409 (1993). 17. C. L. Fefferman and L. A. Seco, On the Dirac and Schwinger Corrections to the Ground-State Energy, Adv. Math. 107, 1 (1994). 18. C. L. Fefferman and L. A. Seco, The Eigenvalue Sum for a One-Dimensional Potential, Adv. Math. 108, 263 (1994). 19. C. L. Fefferman and L. A. Seco, The Density in a One-Dimensional Potential, Adv. Math. 107, 187 (1994). 20. C. L. Fefferman and L. A. Seco, The Density in a Three-Dimensional Radial Potential, Adv. Math. 111, 88 (1995). 21. L. L. DeRaad, Jr. and J. Schwinger, New Thomas–Fermi theory: A test, Phys. Rev. A 25, 2399 (1982). 22. B.-G. Englert and J. Schwinger, Thomas–Fermi revisited: The outer regions of the atom, Phys. Rev. A 26, 2322 (1982). 23. B.-G. Englert and J. Schwinger, Statistical atom: Handling the strongly bound electrons, Phys. Rev. A 29, 2331 (1984). 24. B.-G. Englert and J. Schwinger, Statistical atom: Some quantum improvements, Phys. Rev. A 29, 2339 (1984).

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25. B.-G. Englert and J. Schwinger, New statistical atom: A numerical study, Phys. Rev. A 29, 2353 (1984). 26. P. A. M. Dirac, Note on exchange phenomena in the Thomas atom, Proc. Cambridge Philos. Soc. 26, 376 (1930). 27. C. F. von Weizs¨ acker, Zur Theorie der Kernmassen, Z. Phys. 96, 431 (1935). 28. M.-I. Trappe, Y. L. Len, H. K. Ng and B.-G. Englert, Airy-averaged gradient corrections for two-dimensional Fermi gases, Ann. Phys. (NY) 385, 136 (2017). 29. K. Buchwald and B.-G. Englert, Thomas–Fermi–Scott model: Momentum-space density, Phys. Rev. A 40, 2738 (1989). 30. M. Cinal and B.-G. Englert, Thomas–Fermi–Scott model in momentum space, Phys. Rev. A 45, 135 (1992). 31. N. Bohr, On the Constitution of Atoms and Molecules, Philos. Mag. 26, 1 (1913). 32. B.-G. Englert and J. Schwinger, Semiclassical atom, Phys. Rev. A 32, 26 (1985). 33. B.-G. Englert and J. Schwinger, Linear degeneracy in the semiclassical atom, Phys. Rev. A 32, 36 (1985). 34. B.-G. Englert and J. Schwinger, Atomic-binding-energy oscillations, Phys. Rev. A 32, 47 (1985). 35. B.-G. Englert, Semiclassical Theory of Atoms, Lecture Notes in Physics 300 (Springer, 1988). 36. B.-G. Englert, Energy functionals and the Thomas–Fermi model in momentum space, Phys. Rev. A 45, 127 (1992). 37. P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964). 38. A. Holas, P. M. Kozlowski and N. H. March, Kinetic energy density and Pauli potential: dimensionality dependence, gradient expansions and non-locality, J. Phys. A: Math. Gen. 24, 4249 (1991). 39. J. Shao, Exact kinetic energy functional of noninteracting fermions, Mod. Phys. Lett. B 7, 1193 (1993). 40. M.-I. Trappe, Y. L. Len, H. K. Ng, C. A. M¨ uller and B.-G. Englert, Leading gradient correction to the kinetic energy for two-dimensional fermion gases, Phys. Rev. A 93, 042510 (2016). 41. H. Siedentop and V. von Conta, Statistical Theory of the Atom in Momentum Space, Markov Processes and Related Fields 21, 433 (2015). 42. M. Cinal and B.-G. Englert, Energy functionals in momentum space: Exchange energy, quantum corrections, and the Kohn–Sham scheme, Phys. Rev. A 48, 1893 (1993). 43. E. Wigner, On the Quantum Correction For Thermodynamic Equilibrium, Phys. Rev. 40, 749 (1932). 44. J. G. Kirkwood, Quantum Statistics of Almost Classical Assemblies, Phys. Rev. 44, 31 (1933). 45. H. F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc. 10, 545 (1959). 46. M. Suzuki, Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems, Commun. Math. Phys. 51, 183 (1976). 47. T. T. Chau, J. H. Hue, M.-I. Trappe and B.-G. Englert, Systematic corrections to the Thomas–Fermi approximation without a gradient expansion, New J. Phys. 20, 073003 (2018).

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Fond memories of Julian and Clarice, especially involving Moshe Flato and Noriko Sakurai Daniel Sternheimer Department of Mathematics, Rikkyo University, Tokyo 171-8501, Japan Institut de Math´ ematiques de Bourgogne, F-21078 Dijon, France [email protected]

We start with some general remarks related to Julian Schwinger’s works in, and attitude to, physics, in particular in relation with Moshe Flato and me. Then we recall some notable events with Moshe (70s to 90s) and (since the 80s) with Noriko Sakurai, and also with Clarice Schwinger and me this millennium. A few photos illustrated the talk, most of which are reproduced at the end of this contribution.

The first part of this short and quite personal presentation deals with some surprisingly intertwined scientific events, going from Julian’s early years in Science to our contributions in the last decades of the past century, and perspectives for further development. The second part centers around the Schwinger house in Bel Air, which many knew of but not so many visited to a significant extent. 1. From Isidor Rabi to Deformation Quantization and Beyond 1.1. Isidor Rabi, Julian Schwinger and Moshe Flato 1.1.1. Rabi and Schwinger As is known, Isidor Rabi (1898–1988) was Julian’s thesis adviser. Rabi was a giant in modern physics. Among many important achievements one can note the Nuclear Magnetic Resonance (NMR), for which he was awarded the Nobel Prize in 1944, and the creation of the Brookhaven National Laboratory (BNL) in 1947 (in the former US Army Camp Upton), which in turn (also thanks to him) inspired the creation of CERN in 1952. Though (or maybe because) being considered as an awful lecturer, which required many to head for the library to try and work out what he had taught, Rabi had a feeling for discovering brilliant young (male) students, a number of whom were eventually Nobel Laureates. He joined the faculty of Columbia in 1929, the first Jew to get such an appointment, and was promoted to full professor in 1937. Later, in 1964, he became there the first University Professor (which in particular meant he was not required to teach courses), becoming Emeritus in 1967 but remaining active until his death in 1988. In 1934 (when he was 16) Julian Schwinger completed his first research work (unpublished, “on the interaction of several electrons”). His first nine papers were published in 1935–1937 in the Physical Review, at that time the “nec plus ultra”

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in physics publications. Among them, published in 1937, are the sixth, titled “Depolarization by Neutron-Proton Scattering” with Rabi, and the eighth, titled “The Scattering of Neutrons by Ortho- and Parahydrogen” (as well as the fifth by the same title, in the preceding volume of Phys. Rev.) with a physicist who became famous. (Some might say infamous because he is considered as the “father” of the H-bomb, though his contribution in the subject was corrected in an essential way by our friend Stanislaw Ulam, who did not insist in sharing the dubious honor.) The eighth paper is the first included in the selection of Julian’s “lesser” papers, edited by Moshe Flato with Chris Fronsdal and Kim Milton. These are reproduced, with incisive comments by Julian, in Ref. 1. Julian’s comment on the first paper in the selection reads “because I, not my distinguished colleague, wrote it.” On the following paper in the selection, “The Neutron-Proton Scattering CrossSection,” published in Phys. Rev. in 1939, his comment is: “An experimental paper. The result has remained valid over the years.” Also in 1939, at 21, Julian received his PhD with Rabi. The rest is history, related in many places. Among later papers selected by Julian, one should mention his 1957 paper3 in Annals of Physics, “A theory of the fundamental interactions” (Ref. [82] in his publication list), described by the reviewer in Mathematical Reviews as “a ‘connected series of speculations’ about elementary particles and their interactions.” Julian’s comments, very much to the point, read: “A speculative paper that was remarkably on target: VA weak interaction theory, two neutrinos, charged intermediate vector meson, dynamical unification of weak and electromagnetic interactions, scale invariance, chiral transformations, mass generation through vacuum expectation value of scalar field. Concerning the idea of unifying weak and electromagnetic interactions, Rabi once reported to me: ‘They hate it.’ ” That was about 10 years before the independent and important works by Sheldon Glashow (who in 1959 got his PhD at Harvard with Schwinger and in 1961 extended the Schwinger scheme for that unification), Abdus Salam (in 1968, after earlier works) and (in 1967, after doubting it before) Steven Weinberg. Their schemes for unifying these symmetries are the reason for which they were awarded together the Nobel Prize in 1979. In that context there is an interesting anecdote, which we have learned from a first hand witness and is not well known. At the time of the Nobel ceremony in Stockholm in December 1979, Salam and Weinberg were interviewed on the Swedish radio. At some point Weinberg quoted a sentence from the Bible. Salam remarked that there is a similar one in the Qu’ran, to which Weinberg answered: “There also we published it before!” In 1959 Rabi was appointed a member of the Board of Governors of the Weizmann Institute of Science, and during a trimester he gave a series of lectures at the Hebrew University of Jerusalem, invited by Giulio Racah. One of the students in the class was Moshe Flato, possibly the best student Racah ever had (and he had quite a few excellent ones). Moshe not only was able to follow these lectures but also kept asking pertinent questions, which Rabi often could not answer.

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1.1.2. From symmetries to deformation quantization In 1965 Moshe and I submitted to the Physical Review Letters (PRL) a contribution criticizing that of Lochlainn O’Raifeartaigh, published there the same year. In the latter paper was “proved” that the so-called “internal” (unitary) and external (Poincar´e) symmetries of elementary particles can be combined only by direct product. In our rebuttal Moshe insisted that we write that the proof (of O’Raifearthaigh, who by the way became a good friend after we met) was “lacking mathematical rigor,” a qualification which incidentally (especially at that time) many physicists might consider as a compliment. The formulation was provocative. Moshe made it deliberately because he felt that we were going against something that, for a variety of reasons, most in the main stream wanted to be true. Remark. The “theorem” of O’Raifeartaigh was formulated at the Lie algebra level, where the proof is not correct because it implicitly assumes that there is a common domain of analytic vectors for all the generators of an algebra containing both symmetries. In fact, as it was formulated, the result is even wrong, as we showed later with counterexamples. The result was proved shortly afterward by Res Jost and, independently, by Irving Segal, but only in the more limited context of unitary representations of Lie groups. In those days “elementary particle spectroscopy” was performed mimicking what had been done in atomic and molecular spectroscopy, where one uses a unitary group of symmetries of the (known) forces. As a student of Racah, Moshe mastered these techniques. The latter approach was extended somehow to nuclear physics, then to particle physics. That is how, to distinguish between neutrons and protons, Heisenberg introduced in 1932 “isospin,” with SU (2) symmetry. When “strange” particles were discovered in the 50s, it became natural to try and use as “internal” symmetry a rank-2 compact Lie group. In early 1961 Fronsdal and Ben Lee, with Behrends and Dreitlein, all present then at UPenn, studied all of these.6 At the same time Salam asked his PhD student Ne’eman to study only SU (3), in what was then coined “the Eight-Fold way” by Gell’Mann because its eightdimensional adjoint representation could be associated with mesons of spin 0 and 1, and baryons of spin 12 . Since spin is a property associated with the “external” Poincar´e group, it was simpler to assume that the two are related by direct product. Hence the interest in the “O’Raifeartaigh theorem.” For this and much more see e.g. Section 2 in Ref. 11 and references therein.

The Editors of PRL objected to our formulation. In line with the famous quote by Einstein (“The important thing is not to stop questioning, curiosity has its reason for existing.”) Moshe insisted on keeping it “as is.” The matter went up to the President of the American Physical Society, who at that time was Felix Bloch, who consulted his close friend Isidor Rabi. [In short, Rabi discovered NMR, which is at the base of MRI, due to Bloch.] Rabi naturally asked who is insisting that much, and when he learned that it was Moshe, he said: “If he insists he must have good reasons for it. Do as he wishes.” The Editors of PRL followed his advice. Remark. We met PRL Editor Sam Goudsmit only a year later at BNL, during our first visit to the US, but he knew well our host there, my cousin Rudolph Sternheimer who

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spent almost all of his career at BNL and is known in particular for having discovered and studied extensively the “Sternheimer effect,” a quadrupole effect in atomic physics (NMR is a dipole effect) that was the first effect not to get a Nobel prize because it is very small and was often neglected. Incidentally, though many physicists do not understand the problem we raised, our objections must have been considered serious by some because in 1970 Sydney Coleman found it necessary to write (with Mandula) a different proof of the fact, in the S-matrix context. We are among the few to notice that there also was an unstated hypothesis hidden in the notations. But a few years later their proof was instrumental in stressing the importance of the newly (re)discovered notion of supersymmetry, which otherwise could then have been considered as heretic because Pauli seems to forbid having both bosons and fermions together in one entity.

The following year (1966), at a conference in Gif-sur-Yvette on “the extension of the Poincar´e group to the internal symmetries of elementary particles” which Moshe (then 29) naturally co-organized, Christian Fronsdal told Moshe: “You wrote that impolite paper.” This was the beginning of a long friendship, which lasts to this day and is at the origin of important scientific works. Fronsdal (now 87 and not retired) invited us many times to UCLA, where we naturally met Julian. As we mention below, Julian developed a very close relationship with our friend Robert (Bob) Finkelstein, who incidentally is now still active at 102 (he contributed a scientific paper to the Schwinger Centennial in Singapore) and had been Fronsdal’s adviser. That is how, by transitivity as we say in mathematics, started our friendship with Julian. In the mid 70s, when we were writing what is now considered as the foundational papers on deformation quantization,5 we naturally discussed these with Julian during our visits to UCLA. [I also met him in 1975 at a conference in Austin TX, where I lectured on a work with Flato and Lichnerowicz, on deformations of Poisson brackets, which a year later evolved into deformation quantization. That was after his arrival at UCLA (in 1972) and well before 1982 when his more conventional former Harvard colleague Steven Weinberg moved to Austin, after being awarded the Nobel Prize in 1979.] Julian liked our somewhat nonconventional approach, so much that he sent to Annals of Physics the two main papers, then (1977) UCLA preprints. That is how, indirectly thanks to Rabi, were published our (by far) most cited papers, published in 1978 (they have well over a thousand citations and counting), something quite rare in mathematical physics. Remarkably, as we discovered later (perhaps because of Julian’s modesty), the discrete spectrum (n + 2 ) of the harmonic oscillator H = 12 (p2 + q 2 ) can be read off a formula for what turns out to be the Moyal trace (integral over phase space) of what we call the “star exponential” for the harmonic oscillator, the tool with which we obtained that spectrum in an autonomous manner (without operators). The latter “trace” formula had been obtained already in 1960 by Julian (Ref. [98] in Ref. 1 with the interesting comment: “Quantum Phase Space”).

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Already in 1949–1951, Dirac had written:7 “Two points of view may be mathematically equivalent and you may think for that reason if you understand one of them you need not bother about the other and can neglect it. But it may be that one point of view may suggest a future development which another point does not suggest, and although in their present state the two points of view are equivalent they may lead to different possibilities for the future. Therefore, I think that we cannot afford to neglect any possible point of view for looking at Quantum Mechanics and in particular its relation to Classical Mechanics. Any point of view which gives us any interesting feature and any novel idea should be closely examined to see whether they suggest any modification or any way of developing the theory along new lines.”

What Dirac had in mind was probably what soon became known as the “Dirac constraints formalism,” for quantizing systems in the presence of constraints, which he developed then. That was later systematized in a geometric context (in modern terminology, coupled second class constraints reduce flat phase space to a symplectic submanifold, and first class constraints reduce further to what we called e.g. in Ref. 5 a Poisson submanifold). What we did with deformation quantization was to take even more seriously Dirac’s approach and quantize in an autonomous way (without Hilbert space operators) systems on general symplectic and Poisson manifolds. 1.2. Julian and Moshe at UCLA and in France 1.2.1. UCLA In the 70s and 80s, the “Theoretical Elementary Particles” (TEP) group was a very interesting place for us to visit, which we did for periods varying from a quarter to a few weeks. Our intensive collaboration with Fronsdal was of course the main reason for these visits, but it gave us the opportunity to meet Julian in a personal environment. For a variety of reasons Julian had limited local interactions with his colleagues at UCLA, where he arrived in 1972 (a couple of years after Jun Sakurai). The most notable and probably the closest exception is our common friend Bob Finkelstein. Indeed, though he was friendly with people he met, Julian was “clinically shy,” as my late wife Noriko Sakurai told me at some point. Noriko was very close to Julian and Clarice. Surprisingly for many, Moshe was also shy. That was noticed by our friend Yvette Chassagne, with whom Moshe worked closely when she was Chairperson of Union des Assurances de Paris (where, at her initiative, Moshe established a Scientific Council). She added that Moshe hid his shyness under the exuberant personality of a true “tsabar” (name given to Jews born in Israel who are like the fruit of the cactus, spiny outside and sweet inside). That may be one of the reasons why there was immediate chemistry between Moshe and Julian. It is known that Julian had a mixed relationship with his Harvard colleagues because he always pursued independent research, different from mainstream fashion. His shyness may have caused him to resent that. But keeping an open mind and

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asking questions was exactly Moshe’s approach to Science. (Moshe did not suffer too much from it: early in his career he found support from mentors who appreciated his talent and attitude, and eventually he was granted “scientific refuge” in the more open French mathematical school.) However, as Julian told us then, his move from Harvard was mainly because he felt that he needed more sun and physical exercise (in particular tennis), and was becoming stressed because of the attention required by the many brilliant students he attracted at Harvard. Another reason was the insistence of his friend David Saxon (who had been Dean of Physical Sciences). Julian was disappointed when, soon after his arrival, Saxon left LA to high level positions in the UC system at Berkeley (and later at MIT, coming back to UCLA as Emeritus only in the 90s). Avoiding the Faculty Club, where most of his colleagues were having lunch, Julian developed the habit to have a quiet lunch in a nice restaurant of Westwood (close to, but not on, the Campus), usually on Mondays with Bob Finkelstein and (maybe) a couple of others. Moshe attended these lunches, which were an occasion to exchange personal views on the development of theoretical physics and other topics of common interest. 1.2.2. Julian the wine grower, in France in 1989 Quite naturally Moshe invited Julian to visit him in France, both in Paris where he lived most of the week (when in France) and in Dijon where he had created (in 1968) his mathematical physics “laboratory” within the Department of Mathematics. Julian came in 1989, shortly after turning 71. Moshe had invited him as visiting professor in Dijon, as he did with many colleagues, who (like him) stayed most of the time in Paris and came with him to Dijon (usually Thursday and Friday, the days he was teaching). The guests were giving a seminar talk late in the first day, followed by a very nice dinner in town. Julian’s stay in Dijon was longer, in part because the university wanted him to give a special lecture, but also because he was interested in visiting Burgundy for oenological reasons. Indeed Julian had inherited some money from his father. He decided to invest it all in 12% of the Vittorio Sattui winery in Napa valley, the largest share except for that of Darryl (a.k.a. Dario) Sattui, the grandson of the founder, who was then trying to restart the winery. Julian’s investment was crucial for that. [Clarice was at first not supportive of the move, which turned out to be a very wise investment.] Every year there was a stockholder’s meeting. Noriko participated with Clarice in some of these (and I in a couple of them), staying in the house of Darryl. It was a superb experience. It turned out that, for stupid bureaucratic reasons, Julian could not be appointed as visiting professor. For that the University would have needed to provide him French health insurance for the duration of his stay, which was totally impossible for someone past 70. Of course Moshe managed to get Julian’s coming covered from other sources.

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During Julian’s stay Moshe arranged that he becomes “Chevalier du Tastevin” in a ceremony at the famous Chˆ ateau du Clos de Vougeot. Julian was also invited to a special dinner in Dijon, for wine growers, in which every guest was bringing a bottle of his own vineyard. Though “his wine” was more of the Bordeaux style, Julian brought a bottle of “his” Zinfandel, a popular variety originally from Europe where it is mostly known as Primitivo. After tasting some of the wines of the guests, Julian told them that they are welcome to keep his bottle but not to open it, because it was no match for the superb Burgundy wines brought by the guests. The visit occurred at the time when Julian, well in line with his non-conventional attitude to science, became interested in the claims about “cold fusion.” In his talk in Dijon he developed elaborate calculations on the subject, of a kind that only his virtuosity permitted, showing that a phenomenon of cold fusion was theoretically possible. His talk was attended by the scientific journalist of a leading French newspaper, who had come for the occasion. Moshe had to comment that what Julian’s calculations show is that there is a theoretical possibility for such a phenomenon, albeit probably with a small cross section, which does not mean that what Pons and Fleischmann pretended to have observed was of that kind. [It is now commonly admitted that what was observed was, if anything, a chemical reaction, not a nuclear reaction.] 1.2.3. QED and around Schwinger, K¨ all´en, and Flato As reported in Ref. 10, there were warm but critical relations between Julian Schwinger and Gunnar K¨ all´en, another leading figure in quantum electrodynamics (QED) of that time, who died in October 1968 at 42 in the crash of the plane he was piloting. Young K¨ all´en’s work had placed him in Julian Schwinger’s “Hall of Fame of QED.” Moshe and I had met Gunnar at the above-mentioned 1966 conference in Gifsur-Yvette. There was immediate chemistry between Gunnar and Moshe, neither of whom were mincing their words. At some point during the conference dinner (Gunnar was seated between Moshe and I) Moshe asked Gunnar, who was very influential in the Nobel Committee (especially for particle physics) what are the chances of Murray Gell-Mann for the Nobel Prize. The question was natural since there was then strong support in the community for that. [Indeed, while in the 50s Gell-Mann, working on the so-called “strange particles,” was not really mainstream, that changed completely in the 60s and he became a kind of “guru,” however not for an unconventional person like Moshe, see Section 1.1.2 above.] Gunnar’s answer was “over my dead body.” One should never say such a sentence. [Gell-Mann, now 88, got the Nobel Prize in 1969, which did not prevent Moshe from asking a number of (im)pertinent questions to the laureate during his traditional scientific lecture at the Royal Institute of Technology (KTH) in Stockholm, on the conformal symmetry, which we were familiar with since 1965 thanks to Roger Penrose; Gell-Mann’s only answers were “good question,” but that is another story.]

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Shortly after their meeting in Gif, Moshe and Gunnar started preparing a French-Swedish meeting, to be held in December 1968 at KTH, Stockholm, where a friend of us (Gilbert Karpman) was the French scientific attach´e. It became naturally the first Gunnar K¨ all´en colloquium. A second was held in Paris in June 1970. Then it was extended to include Poland in December 1972 and March 1974. That indirectly gave rise to what are now known as the International Congresses of Mathematical Physics in Moscow in December 1972 (the week following our Warsaw meeting), and lead to the creation of the International Association of Mathematical Physics (following our suggestion at the 1974 Warsaw meeting). The logo M∩Φ was introduced in 1972 in Moscow by Nikolay Bogolyubov, whom we knew well since the Moscow 1966 ICM (International Congress of Mathematicians), after which he invited us to Dubna. So, very indirectly (through Moshe’s impact) Julian’s QED contributed to these developments. As is well known the absolutely fundamental work for which Schwinger was awarded the Nobel Prize in 1965 is relativistic quantum field theory of electrodynamics. He got it (essentially for his 1948–49 papers) together with Shin’ichiro Tomonaga (whose papers in Japan, in 1946, based on a 1942 paper during WWII, were unknown in the US), both of whom had a more analytical approach, and Richard Feynman (whose 1949 papers followed a more intuitive approach with his “diagrams”). The equivalence of the two approaches was shown in 1949 by Freeman Dyson, who missed the prize in great part because it is traditionally attributed to at most three scientists. Somehow Dyson was not included in later prizes, though there would have been at least one occasion and it is not yet too late since at 94 he is still alive and active. Moshe had an immense admiration for the powerful approach of Julian. He used to compare Julian to Jesus whose somewhat abstract teachings were popularized, with the success we know, by St. Paul (Moshe compared Feynman to the latter). On the front page of his “tour de force” 1997 monograph,9 which established rigorously classical electrodynamics (and much more), Moshe stressed his view by writing (in capital letters): “THIS MONOGRAPH IS DEDICATED TO THE MEMORY OF THE CHIEF CREATOR OF QUANTUM ELECTRODYNAMICS, A GIANT IN CONTEMPORARY PHYSICS, A GREAT HUMAN BEING AND UNFORGETTABLE FRIEND — JULIAN SCHWINGER.” 2. Fond Memories from Bel Air Few know that when Julian got the Nobel Prize in 1965, the finances of the Nobel Foundation were at a minimum. As he told us, his share (1/3) permitted him barely to buy the Volvo with which he traveled from coast to coast in the US. But in 1972 (the year Julian moved to UCLA) Baron Stig Ramel was appointed executive director of the Nobel Foundation, a responsibility he was to carry for 20 years, and the financial situation came back to what it had been at the beginning, if not better. Ramel told us that in 1991, at the festive dinner celebrating 90 years

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of Nobel Prizes, to which all living laureates were invited; most came (including Salam, who was already quite sick, but not including Julian . . . ). Moshe (who had been asked to nominate for the Nobel Prize from 1971 to his untimely death in 1998) and I were invited. Moshe sympathized with his neighbor, a charming lady who introduced herself as Gell-Mann’s fianc´ee (that is also another story). My neighbor was a much older charming lady, Frances Townes who inter alia mentioned that she had noticed a “phase transition” with fellow Nobel Laureates, at some point their wives become much younger, adding that she (at that time, 1991) was happily married for 50 years with her husband Charles, who got the 1964 Prize for quantum electronics (they remained so until his death in 2005). The same is true for Julian and his wife Clarice. 2.1. Noriko Sakurai at Bel Air One of the colleagues at UCLA with whom Julian had friendly relations was a brilliant and somewhat unconventional physicist, Jun (J.J.) Sakurai, who was much younger (born in 1933) and had been a student at Harvard. Their relations soon extended to their wives, Clarice and Noriko. J.J. traveled a lot, often with Noriko (when that was possible because of their sons). Noriko used to say jokingly: “Marry a physicist and see the world.” [Eventually, after being widowed for a long time and traveling with me around the world, she “fell from the frying pan into the fire” and married me, a mathematician, in 2008.] Yet Moshe and I did not have much interaction with J.J. because when we visited UCLA he was often away, and then we were using his office. In 1982, while visiting CERN, J.J. died suddenly from aneurysm rupture. That caused a huge emotion in the community. In his memory Noriko established in 1984, also with family and friends, the J.J. Sakurai Prize for Theoretical Particle Physics with the American Physical Society, A number of the laureates eventually got the Nobel Prize. But for many “physics wives” Noriko’s status changed soon from being the wife of a promising physicist (who in addition was giving superb parties at her house in Cashmere Street in Westwood) to almost nobody. Some behaved differently, in particular Norma Finkelstein and especially Clarice. A few years later, Noriko chose to sell the house in Westwood (at the bottom of the market) and move her residence to Tokyo, in a house she built on the land where (before WWII) stood the house in which J.J. was born. But she needed to come back to LA for a month or two every Spring. Very generously, in spite of being generally not very open, Julian and Clarice offered her to stay then in their Bel Air house. That house had two parts separated by living quarters, one with the Master bedroom and another with two rooms, one of which was then occupied by Clarice’s mother. Noriko was offered hospitality in the other room. So their already close relations became even closer. In fact, for many years, Noriko’s US address was the Bel Air house (10727 Stradella Court).

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In the Spring of 1994, Julian was happy to tell Noriko before her return to Tokyo: “All these years I have been trying to lose weight, now it worked!” Nobody could have then imagined that the “silent killer” was already well at work. Shortly afterward, when the first symptoms appeared, Julian was diagnosed with an advanced stage of pancreatic cancer, having little more than two months to live. Clarice continued to invite Noriko to her house, with a twist. In the meantime her mother had died and Clarice was diagnosed with Parkinson. So she moved to the other part of the house with a helper and, when in LA, Noriko was given her bed in the master bedroom. In 1997 Clarice left to visit her family on the East Coast and Noriko, not keen in staying alone in the house, arranged for her a superb trip to Peru (in particular Machu Picchu). When both were back and Noriko was about to fly back to Tokyo, Clarice told her: “Why don’t you stay a week more, Moshe and Daniel are coming.” She did, and that’s how we met again at a wonderful party in the Finkelstein’s house in Santa Monica, and developed close relations. Shortly afterward we visited Noriko in Tokyo during one of our trips around the world. And suddenly there was another shock in the scientific community: Moshe, barely 61, died of brain hemorrhage in Paris in November 1998. 2.2. Clarice, Noriko and me At the exceptional first conference in Moshe’s memory, in Dijon in September 1999, alongside with many VIPs, Noriko read Clarice’s short speech (p. 10 in Ref. 8): “Mosh´e was a man to be treasured, and we did. We admired his brilliance, respected his integrity, enjoyed his wit, were touched by his compassion and loved the joy he brought. I trust it will be understood that ‘we’ means Julian and me. I know the meeting in Dijon will be moving. While I hadn’t expected to be able to join, I am sorry that I can’t.” Noriko added a short sentence of her own: “I mourn the unexpected and untimely loss of an irreplaceable friend. Mosh´e was a man amongst men.” Shortly afterward, during a trip around the world, I stopped in Tokyo. Noriko had made us the friendly offer to stay in a guest room at her Tokyo house (her two sons officially left it when they married). Not keen in staying alone in the hotel near the Kabuki-za where Moshe and I (both Kabuki fans) were often staying, I accepted. We didn’t realize that the whole neighborhood would see, in the middle of the day, a “Gaijin” arrive at the house with suitcases. Normally Japanese travel in groups for 1 or 2 weeks, but Noriko was traveling alone for 1 or 2 months, which is unusual. So when I appeared everything became clear. Of course it was all wrong but her reputation of dignified widow evaporated in an instant. Our relations became more intimate some time afterward, and during my/our travels I joined her in the Bel Air house, sleeping in Julian’s bed in the master bedroom. Unfortunately Julian’s exceptional talent for physics did not perspire through the bed, but I got something from the room. Well in evidence (in that orderly room) there was an A4 post (coming from Julian’s office at UCLA) which read:

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“This Place Was Built From CHAOS Into A SHINING EXAMPLE OF DISORDER.” That fit even better the “hoarder” I am, as many may have seen, so I made a couple of photocopies which I keep both in Paris and in Tokyo (see photo).

Copy from a sign that had been in Schwinger’s office at UCLA.

Noriko and I kept visiting for several years the Bel Air house, at Clarice’s insistence, so we became a kind of “family.” In that context it is impossible not to mention Margaret (Margy) Kivelson, now 90 and still active in science, who had been Julian’s (only female) student at Harvard (she got her PhD in 1957) and is since 1967 at UCLA (in Earth and Space Sciences). During many years, the latter part of which I witnessed first hand (see e.g. the two photos below, taken in 2007 at Margy’s house in Pacific Palisades), and especially when some personal attention was needed, she cared for Clarice until the end, being around when (and only when) her most efficient intervention was needed.

Noriko and Clarice looking fondly at Noriko’s grandson Johan, September 2007, Bel Air house next to the pool.

Clarice, Margy and Noriko with Harold and Suzy Ticho and Daniel, May 2007 in Margy’s house in Pacific Palisades.

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Though Internet was then not as widespread as it is now, staying in the house was not most convenient because I needed to go to UCLA to use it. (The house was somewhat isolated, which is nice, but the phone line was not good enough for connecting.) Noriko would drive me to UCLA and pick me up from there. Still I have the fondest of memories from our stays in the house, and of the home feeling we got from Clarice. One of them is from the superb lunch we had in April 2004 at the Bel Air hotel with David and Shirley Saxon (see photo below).

Clarice with David and Shirley Saxon, Bel Air hotel, April 2004.

Clarice and Margy being happy, May 2007 in Margy’s house in Pacific Palisades.

Clarice with Bob Finkelstein, Noriko Sakurai and Daniel Sternheimer, Westwood, April 2001 (Palace Seafood Chinese, Wilshire in Brentwood).

Julian Schwinger with Freeman Dyson in 1993.

At some point I visited UC Berkeley, shortly before the annual stockholders’ meeting of the Sattui winery. Noriko and Clarice (then on a wheelchair) had planned

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to meet me at Oakland airport, where I would come with a rented car. When they arrived at LAX it turned out that Clarice did not have with her a photo ID, which had become required for air travel inside the US. Somehow Noriko managed to get Clarice through with her wheelchair. Nowadays that would probably not be possible, though with Clarice one cannot be sure. The stay at the Sattui estate was superb, and the way back went fine also. Eventually Clarice’s physical condition got worse and she needed four helpers (day and night, weekdays and weekends), but her mind was still sharp to the end. Though Clarice would have wanted us to keep coming to the house when in LA, Noriko decided that we should stay in Westwood, which we did for some years, including when Noriko (diagnosed somewhat early but not early enough, at a yearly check-up, also with pancreatic cancer) was treated at UCLA. Still we managed to meet Clarice often, even going out with her to movies and restaurants. Clarice was a special person. Occasionally she would have to go to hospital for treatment. About a year before she died, Clarice was in so poor condition that the hospital sent her back home, to die there. The next day her condition improved dramatically. Her time had not yet come. She outlived also Noriko (by over a year), and died at 93 in January 2011. References 1. M. Flato, C. Fronsdal and K. A. Milton (eds.), Selected Papers (1937–1976) of Julian Schwinger, Mathematical Physics and Applied Mathematics Vol. 4 (D. Reidel, 1979). 2. J. Schwinger and E. Teller, The scattering of neutrons by ortho- and parahydrogen, Phys. Rev. 52, 286 (1937). 3. J. Schwinger, A theory of the fundamental interactions, Ann. Physics 2, 407–434 (1957). 4. J. Schwinger, The special canonical group, Proc. Nat. Acad. Sci. U.S.A. 46, 1401–1415 (1960). 5. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization I. Deformations of symplectic structures, Ann. Physics 111(1), 61–110 (1978); II. Physical applications, ibid. 111(1), 111–151 (1978). 6. R. E. Behrends, J. Dreitlein, C. Fronsdal and B. W. Lee, Simple groups and strong interaction symmetries, Rev. Mod. Phys. 34, 1–40 (1962). 7. P. Dirac, The relation of Classical to Quantum Mechanics, 2nd Can. Math. Congress, Vancouver 1949 (U. Toronto Press, 1951), pp. 10–31. 8. G. Dito and D. Sternheimer (eds.), Mosh´e Flato, the man and the scientist, in Conf´erence Mosh´e Flato 1999, Vol. I (Dijon), pp. 3–54, Math. Phys. Stud. No. 21, (Kluwer Acad. Publ., Dordrecht, 2000). 9. M. Flato, J. C. H. Simon and E. Taflin, Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations, Memoirs Amer. Math. Soc. 127(606) (1997). 10. C. Jarlskog (ed.), Portrait of Gunnar K¨ all´en, A Physics Shooting Star and Poet of Early Quantum Field Theory (Springer, 2014). 11. D. Sternheimer, ‘The important thing is not to stop questioning’, including the symmetries on which is based the standard model, in Geometric Methods in Physics, 7–37, Trends Math. (Birkh¨ auser Basel, 2014).

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Speeches by V. F. Weisskopf, J. H. Van Vleck, I. I. Rabi, M. Hamermesh, B. T. Feld, R. P. Feynman, and D. Saxon, given in honor of Julian Schwinger at his 60th birthday Berthold-Georg Englerta and Kimball A. Miltonb a Centre for Quantum Technologies and Department of Physics, National University of Singapore MajuLab, Singapore [email protected] b Homer

L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA [email protected]

In February 1978 Julian Schwinger’s 60th birthday was celebrated with a SchwingerFest at UCLA. This chapter consists of transcripts of historical talks given there.

The UCLA Physics Department organized a two-day celebration of Julian Schwinger’s 60th birthday in February 1978, to which many of his former students, collaborators, and friends came. Scientific sessions were held during the days of the Symposium, and after the banquet there were a number of extended remarks given by some of the distinguished participants. The mostly scientific talks were published in a special issue of Physica, which also appeared as a stand-alone volume. 1 The purely historical talks were deposited in the AIP archives. The present chapter aims to make the latter talks more accessible. They are retyped from transcriptions made at the time by one of the authors (KAM) from no longer existent audiotapes; we present them verbatim to preserve the freshness and vitality of the presentations. Some extracts of these talks have appeared in Schwinger’s biography. 2 The talks by Weisskopf, Rabi, and Hamermesh were given during the plenary sessions, while Feld’s, Feynman’s, and Saxon’s talks (the latter with some interjection by Rabi) were delivered after dinner. Van Vleck was unable to come, having suffered a mild heart attack en route, but a telegraphic message was delivered by telephone to KAM, and it was included in Weisskopf’s remarks. Feynman’s lecture was previously published in the volume celebrating Schwinger’s 70th birthday; 3 we include it here for completeness. There are many wonderful stories recounted here, and it is very interesting to see how the stories are modified when told by different participants in the history. The story of Victor LaMer is a case in point. Since none of these great men are now with us, preserving their memories is important for the future development of our science, which is as much about personalities as it is about technical developments.

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We will not make any attempt here to offer corrections. We can remark on a few clarifications: (i) Since evidently titles of talks were excluded both from the program (which has since disappeared) and the transcripts, we are uncertain which talk was entitled “Schwinger and the Boom in Theoretical Physics,” referred to by Rabi. (ii) Herman Feshbach’s talk referred to by Hamermesh appeared in Ref. 1, p. 17. (iii) Weisskopf, Rabi, and Saxon quoted different numbers for the quantity of Julian’s Ph.D. students. The final official count is 73, but that includes five students receiving their degrees after the SchwingerFest, so Saxon (and therefore Schwinger) was essentially correct. Of course, as noted by several, this number is largely meaningless, since Julian Schwinger was the greatest teacher many of us have ever had, and many justly regard him as their master. References 1. S. Deser, H. Feshbach, R. J. Finkelstein, K. A. Johnson and P. C. Martin, eds., Themes in Contemporary Physics: Essays in honour of Julian Schwinger’s 60th birthday (North-Holland, Amsterdam, 1979) [reprinted from Physica, vol. 69A (1979)]. 2. J. Mehra and K. A. Milton, Climbing the Mountain: The Scientific Biography of Julian Schwinger (Oxford University Press, Oxford, 2001). 3. S. Deser and R. J. Finkelstein, Themes in Contemporary Physics II: Essays in honour of Julian Schwinger’s 70th birthday (World Scientific, Singapore, 1989).

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Victor Frederick Weisskopf (and John Hasbrouck Van Vleck) Thank you for your very kind words. It is a very special pleasure for me to be the first chairman in this, in the celebration of the sixtieth birthday—a young boy— a man whom I love, admire and just like. Before becoming again sentimental, I’d like to say a few, so to speak, administrative remarks. You know the program unfortunately had to be changed for this afternoon because our colleague Van Vleck has had a mild heart attack. I am told it is nothing serious; he’s not in danger, but it was not possible for him to come here and to address you, and I would like right away for us to read a statement of Van Vleck, which he sent by telephone, which I have here. He says: “Very disappointed to miss the ceremonial session and frustrated to be hospitalized so near to goal line.” He had to be in San Diego at that time. “Abigail and I send our congratulations, not only to Julian but also to Clarice.” (Applause for Mrs. Schwinger, who was present, taking pictures.) “Rabi can claim that Julian is a product of New York culture,” so he says here, “but we claim Clarice as a proper Bostonian. Columbia is to be felicitated for its audacity and liberality in giving Schwinger a travelling fellowship to Wisconsin in 1937 so that he could get a good education right after his doctorate. This was the golden year in theoretical physics in Madison with Schwinger, Wigner, and Breit all on the campus at the same time. I need not elaborate on his achievement while at Harvard except to mention that the Karplus and Schwinger paper on line breadth is a classic which has been a guideline for much of my subsequent research in this field. I congratulate Schwinger not merely on his past research, but projected into the future. A few years ago I commented in the Harvard alumni magazine on how three former members of its faculty, Kendall, Oldenburg, and Webster, were still publishing at the age of 83. With Schwinger’s sustained productivity in research displayed by his starting at 17, he should do at least as well and a simple calculation shows that he will still be publishing in 2001.” Now, as to the administrative matters, Herman Feshbach was so kind to agree to speak this afternoon, though this takes away one evening for preparation. Take this into account (laughter). Before I now give the word to the first speaker—Rabi, naturally, who else, could be, should be, the first speaker at a Schwingerfest—I would like to take this opportunity as the chairman to say just a very few sentimental words. I knew Schwinger, perhaps, before the War, but really I got acquainted with him personally only when I also came to Cambridge after the end of the Second World

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War in 1946. We came to Harvard and that was, of course, an extremely exciting time. Remember, not only because we all came back from war work, but that was the time when the new quantum electrodynamics, the renormalization theory, Lamb shift, magnetic moment, etc., when all this was born in these fantastic conferences that were held—organized by Oppenheimer—in Shelter Island and different places every year in which every conference meant a big step forward and these big steps were, to a great extent, done by Julian himself. But what I would like to say, a few words, I feel I have to, is the wonderful times that we had together. I think it was roughly on the average once a month that we had lunch together at some strange places. Some of them still exist. The food was not always very good, but the conversation was good, and somehow I, for me, to get regularly in touch with him was a very great thing, and I’m really unhappy that he has moved to Los Angeles, although flying today, or yesterday, from Boston to Los Angeles—Boston where there is one meter of snow—I sort of understand at least part of the reason, but still I’m very unhappy about it. Let us say a little more, because there is something which is so valuable in having Julian around in the community of physicists and that is what I would like to call, for lack of a better word, his style. I find, and I’m sure that some of you will agree with me, that within physics, within science in general, contemporary science, there is not enough variation of style. There is a sort of fashion style and that, of course, unfortunately is amplified by the fact that we are forced to publish on two pages or three pages in the terrible magazine called Physical Review Letters, instead of having room to place leisure, so to speak, to develop style, and therefore there’s either no style or a common style, and somehow Julian has kept and developed a style different from us, fortunately. I’m not making a value judgment, although I could, because it is not so much that it is a better style, but that it is a different style, which is so necessary in physics. But all I want to say is, that we should be grateful to Julian that he has produced and kept and developed an individual style, and even more so strive also to develop more individual styles so that physics becomes more colorful. And if anybody has contributed to the color of physics in this sense, it is Julian, and we should be grateful to him for that. I don’t want to say more because there will be other people who can express all this much better and I therefore hasten to give the floor to Rabi whom I don’t need to introduce. Isidor Isaac Rabi I prepared this lecture very carefully. I arrived yesterday. Spent all morning. But I left it in the hotel, so you’ll be spared all that. I once read a book review by a former professor of mine at Cornell. In reviewing this book he said the author was at his best when he quoted, so I will not be at my best but if you come to my hotel room . . . I don’t know just where to begin since Van Vleck sent this wonderful message, but I’ll get to it. This is an extraordinary occasion and somehow 1918 must have

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been an extraordinary year. I didn’t investigate it fully, but the year 1918, separated only by about three months, we had our birthday child here, Julian, and a little younger, in the direction I’m pointing, is another one—this was February, the other one born in May—Feynman, and not only were they born at this same point which is pretty good measurement, pretty good aiming, from an experimental physicist it’s something to hit so close, but they work in the same subject, in essentially the same way. So in a certain sense there’s still over here in the audience, we’ll drop some on Dick. I’m surprised we have so few people—maybe they knew the size of the room selected, but there are a few standees. There are a few seats back there. Because I read last night that Julian has had over 100 Ph.D’s, and this is over a fairly long period of time. If wives had been invited, I suppose, and children, I suppose we’d have had the need for a very large room. Now, when I was asked to speak, somehow I didn’t know quite what to say and I couldn’t think of a topic. Then I was called up by Mr. Milton and finally was pressed to give a topic, so I gave it and it was “Physics When Schwinger was a Boy.” Now boy in this sense is more the western usage. I’ve seen cases where you had postdocs, married, wondering how to support family, a little gray around the edges, and yet they were referred to by professors as “our kids.” So it’s only in this exaggerated sense that I meant Julian was a boy. I’m going to trace a little bit of the career of our hero. Coincidence and fortune plays an extraordinary part in our lives. Now Julian Schwinger was born in 1918. In a certain sense a most unfortunate time to be born because physics was finished at that point. You had the great movement of Einstein, of Bohr, of Sommerfeld, Einstein again, and the classical quantum theory was pretty well set by 1918, including when Einstein came again. With 1917, the famous 1917 paper, and the year before, general relativity. So you had relativity, quantum theory, highly developed. A theory of light, there it was, 1918. Fortunately, the war intervened around that time. No, the war came to an end. It came a little earlier and the war came to an end and then there was several years of frustration. Well, it didn’t bother Julian very much because he was busy growing up in a beautiful part of New York City, and in 1925 came the beginning of the second breakthrough. The second coming of quantum, namely the beginning of the establishment of quantum mechanics. It didn’t bother him at all, because at that time he was seven. This was all happening. But I was also thinking, if he had been born in time, let’s say, to participate in the great events of 1905 when Einstein was establishing quantum, or later, in 1913, when Bohr gave the theory of the atom, and so on. But all that passed him by. And then came 1925 and all the great men of that period—Heisenberg, Dirac, Pauli, Schr¨ odinger, de Broglie and many, you might call minor characters, but they were not minor by any absolute standards. By then, Julian was going to elementary school, so things went on this way. I can’t give many details of his actual boyhood except that his biography is very brief. You cannot figure it out. I didn’t know when he started high school,

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but I was told that in his family, he was not the bright one. The one who got all the recognition and all the prizes was his older brother and Julian was the dumb one—like those super unclear , but in this case it was rather reversed. But he did go to what I consider the best high school in the whole United States. The Townsend Harris High School, which is preparatory to City College and does it in only three years. And I’m told that after getting through Townsend Harris, after graduating from Townsend Harris, college is a breeze. What Julian did, graduate from that and he went to City College and that’s where I begin to make contact with him and I imagine the period, since he was a very bright lad, must have been around 1934. Is that right, Julian? There’s very little in his autobiography. I looked it up in those Nobel Prize lectures and, whereas others have a long tale of what they did and how they did it, Julian only describes what his aims were, how he became interested in physics at a very early age. Anyway, he arrived there at City College where he did very badly. I looked at his transcript and he was flunking English. I said to Julian, “Here you are flunking English. You speak very well. What happened?” “I have no time to do those themes.” Because at the time he was working on what was the real problem to him at that time and not solved until a decade or so later—quantum electrodynamics. Anyway, he came to my attention through Lloyd Motz, who was then teaching at City College, and talked about this young student there, who was having great troubles there, flunking, very bad. I said, “Bring him up,” and he did bring him up and there was this child he brought in and meanwhile—no, I was talking to Motz about the paper of Einstein on quantum development and he said, “Somebody’s waiting outside.” “Bring him in.” He brought this kid in—I should say boy but boy is somehow a bad word nowadays. “Sit down,” and then we started talking about the paper because this was my way of reading a paper—taking some student and explaining it, so to speak. At one point there was a dispute and up pipes this voice and solved the whole problem through the sufficiency theorem. I’m not using the right word—what you need of orthogonal functions to give you a complete set. I was impressed. So then, I said, “What about transferring to Columbia?” He was willing to agree, so I got his record and so on, and brought them to admissions. “I’d like to give this young fellow a scholarship.” He looked at him, said, “With this record we wouldn’t admit him.” I said something very tactless—I said, “Suppose he were a football player.” I suspect it was the wrong thing to say, but I was never very tactful. Still, the problem remained. Just then Hans Bethe happened to come to New York and Julian showed me something on which he was working. It was quantum electrodynamics. I showed it to Hans and asked, “What do you think?” He liked it. I said, “Well write me a letter.” No man has great honor in his own country, so with this letter, Julian was admitted and I must say he was a reformed man. The change of location of about a mile, from 120th Street to 130th Street and he made Phi Beta Kappa. I never asked him closely about what happened at that point, but he did make Phi Beta Kappa and so this is the boy part.

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Of course, what was happening in physics was beginning to sink in. It had the revolution, the second revolution, the third, whatever they call it, starting in 1925, and that revolution was completed more or less by 1929. Quantum mechanics was understood. We had the Dirac equation, transformation theories, the theory of the electron, Pauli–Heisenberg quantum electrodynamics, Fermi’s version and there were others. Dirac had just about shot his bolt except for the magnetic pole and the whole thing Julian missed. We come to the next place where the difficulties just became very great. Quantum electrodynamics—beautiful. But what’s beautiful if it doesn’t converge? How many infinities can you live with and still call it beautiful? So, those difficulties were there and somehow or other Julian found out about these things. I don’t know how. I think he found out something about it in—when he entered City College, and he first heard then I think of the existence of quantum mechanics. Not the old quantum theory, because I’m saying that he got his education from the Encyclopedia Britannica. I can’t give you the volumes and numbers, but that was his mathematical education. When he came to City College he had heard of Heisenberg. Julian read the book, I forget what’s its name. And also Dirac was mentioned. So by the time he came to Columbia as a sophomore or a junior, he was very conversant with Dirac’s papers, which I think is really extraordinary. It’s not something you would think would appeal right off to a reader of the Encyclopedia Britannica. But he was doing very well with it. I know I was impressed. And as I say, he was graduated from Columbia. His undergraduate degree and then his graduate work. He was a great help to me because he was in my course in quantum mechanics and whenever I had to go away, I’d ask Julian, who was an undergraduate, to take the class. I can assure you it was a great improvement. He’s a much better teacher than I ever was. One of the people in that same class was Bob Marshak and it was a bad time for being in the same class with Julian, no matter how clever you were. Anyway, he entered graduate work and after a short time—and this is where Van Vleck’s story comes in—I thought that he had about had everything at Columbia that we could offer. By “we,” as theoretical physics is concerned, is me. So I got him this fellowship to go to Wisconsin, with the general idea that there were Breit and Wigner, and they could carry on. It was a disastrous idea in one respect because, before then, Julian was a regular guy. Present in the daytime. So I’d ask Julian, I’d see him from time to time, “How are you doing?” “Oh, fine, fine.” “Getting anything out of Breit and Wigner?” “Oh yes, they’re very good, very good.” I asked them. They said, “We never see him.” And this is my own theory, I’ve never checked it with Julian, that—there’s one thing about Julian you all know, I think he’s an even more quiet man than Dirac. He is not a fighter in any way. And I imagine his ideas and Wigner’s and Breit’s or their personalities didn’t agree. I don’t fault him for this, but he’s such a gentle soul, he avoided the battle by working at night. He got this habit of working nights—it’s pure theory, it has nothing to do with the truth.

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When he came back to Columbia after one year, a changed man. He was taking a course with Uhlenbeck who was a visiting professor there. Some problems of statistical mechanics, a graduate students course, but then came time for the final examination, these oral exams. I saw George and he said, “What should I do about Schwinger?” And, “What do you mean?” “He’s taking the course and he hasn’t even shown up to make arrangements for the examination.” “Well, that’s bad.” So I got hold of Julian and I said, “Now look Julian, that’s not polite,” and, not a man to give offense, so he went to Uhlenbeck and arranged for an examination. He tried to arrange it with him for ten o’clock at night. Well, George is no weak character. Ten o’clock in the morning, and at this sacrifice Julian appeared. Well George told me later, what astonished him was, not only did he answer his questions, he’d done a little research in the field. But what astonished him most, since he hadn’t appeared in class, he did these things in his, George’s notation—by osmosis which unclear anyway. We are now talking of the period just before the war where things were going great guns in a way—now I’m talking purely of experimental physics—things were going great guns and Oppenheimer, the Oppenheimer school was the place to go. So Julian did get a National Research Council Fellowship to go there. I thought he should go to Pauli’s, but he thought Oppenheimer was a more interesting physicist and he went there. Spoke to Oppenheimer about it. He said yes. It was very good for him to have some contact with Snyder, who was one of Oppenheimer’s students or something like that, and Julian went and I spoke to Oppenheimer later and he was terribly disappointed. He came to the point of writing a letter to the National Research Council suggesting that Julian go somewhere else because it took a man like Oppenheimer quite a bit to get used to Julian. Pauli once referred to Oppenheimer’s students as being Zunicker. Somebody who knows enough German knows what this means—people who nod their heads, and Julian wasn’t that way— that and his hours. However, he thought better of it and he soon learned to not only respect but to love him and that is how it went, and then came the war. Meanwhile, except for Fermi, and the explanation of the beta-ray spectrum, and so on, nothing terribly much happened of a fundamental nature in physics although there was the experimental discovery of the mesotron. Well, the war came and Julian got sucked into it. At first he started at Chicago in the metallurgical project and was persuaded to come to Cambridge for the radiation lab. I had nothing to do with it very much except that I approved of it and, well, it seems that he came—he just came. And weeks and months later, the garage kept on calling the university, what to do with Julian’s car? They went to his room and they discovered numbers of uncashed checks. He was a real scholar. Now, from that rather rarefied region of meson physics and of course quantum electrodynamics, came the very practical basic problems of the radiation laboratory, the war laboratory in Cambridge. And in this period, it was interesting to see, if you were leaving approximately five o’clock in the afternoon, leaving your lab, there was

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a single, solitary figure coming up against the current—in other words, you could see that his life work was sort of perceptive in this way—against the current. He was coming to work. A story, which although it sounds apocryphal is true. They’d be working all day in this theoretical group headed by Uhlenbeck and didn’t finish by quitting time and there was this integral on the blackboard, so when they came in the next morning everything was rubbed out except the solution to that integral. This was the gremlin working nights. Julian lectured, twice a week, during that whole period on work in progress, which generally was of the form of translation of wave guide theory into the lumped constant theory and as soon as Julian made a breakthrough in one of these things, these other bright boys around the place, like Dicke, and so on, just invented all sorts of stuff. It all went into weapons of some kind but they started with this breakthrough of the translation of wave guide theory. The radiation lab, at the end, published 27 volumes of the radiation lab series. Most of you who are old enough have seen those and studied from them. There was all the lore of that period. There was supposed to be a volume by Julian but the editor didn’t know how to handle Julian and didn’t get this volume, so Julian’s very fundamental work there is missing. I think some of it was published later by Marcuvitz. And this, more or less, takes us up to that period. Now a few more general remarks that will apply to many members of the audience. James Franck once told me—it was quite well known at that time—that there are three stages in a man’s life: werden, sein, bedeuten—the coming, the being, and the signifying. And well, theoretical physicists make a distinct translation—he is born, he gets his doctorate degree, and then on an impetus works very hard, and by the age of 30, he has the Nobel Prize. Not in hand, not at all, but the work is there, he sort of knows it, but he isn’t sure. Then there’s a long stretch of time—to be or not to be, as Shakespeare said. Sometimes it takes—it took Einstein about 15 years, and Van Vleck, I’m not good at mental arithmetic any more, but it took much longer than that, it must be something like 40 years—but by the age of 30, give or take a few, for theoretical physicists you’ve done it. And the rest is consolidation, power, things of that sort. And then comes a later period which gradually merges into the—I don’t know quite what to call it, a stuffed shirt would be too strong a term—but it’s something else, it’s certainly not the same thing. So this is an interesting celebration. I was 60 once. It’s to some extent a difficult period in one’s life. There’re not so many here who are there, but quite a few who will soon be there. So I will in general make a few remarks about suggestions if you’re expecting then to live a long life, and how to do it. Number one, act your age. You’re crazy if you try to compete with the young fellows. This is especially true for experimental people. You don’t have the quickness, the stamina. You have something else. But what is called productivity in the normal sense, you won’t find. You go through the careers of the great men of our era, Einstein, Bohr, Heisenberg,

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Pauli, Dirac, Schr¨odinger, de Broglie, after the great surge, you don’t do it again. Of course, you’d expect them to do more. The daughter of one of my colleagues watched him blow a smoke ring, and she said, “This is an O, Daddy, now blow a T.” He’s not going to do it. But it is terribly important to this mission which Julian has fulfilled, and I don’t know if he meant it on purpose, to sort of scatter his seed: 80 Ph.D’s under his direction. I think that adds up easily to the combination of Bohr, Einstein, Dirac, Pauli, Heisenberg, Schr¨odinger, de Broglie—and that’s quite a challenge to the future. So I’m eagerly awaiting to hear the results. Some of them are right here before me. Some of the new ideas make me uncomfortable. And that’s as it should be. It’s in the direct tradition. I know how uncomfortable Enrico Fermi was with the later developments of quantum mechanics, and we all know how Einstein fought this gallant fight over a great period of his life. He wasn’t at all lacking in ability, but what was happening in a sense, has happened. Now we’re—I’ve jumped ahead of myself because before the philosophy part I intended to put in, it’s gone by—I’ve left out the greatest period of all. The post-war period, the immediate post-war period. When—sorry Van isn’t here, I’d have to rub his nose in it, this sort of thing—Columbia came to the rescue of New York, Julian and Dick—were the experiments of Lamb and Retherford and Nelson and Nafe, which Julian immediately interpreted as an extra moment of the electron and within a very short time, just about the time when both of these gentlemen were 30, we had the completion of the quantum electrodynamics as a finished business and at the same time, the discovery of the meson—they came one after the other. One of the new, challenging fields, and ushered in the title of the next talk, “Schwinger and the Boom in Theoretical Physics,” the big boom did come right after that. So I’m not going to make any peroration about Julian, except to say that to me, my life with him was extremely unclear and a great inspiration. I’ve learned much of what I know from either fellow graduate students or my own graduate students and one of my regrets, as I’m retired, is I don’t get any more graduate students to teach me, but it was a good time while it lasted. Again, final statement: Don’t overdo it. Morton Hamermesh These are just to keep me company. It’s a great pleasure to be here and I thought on this occasion I might start by telling you the title of this talk which the organization committee refused, apparently, to print in the program. I think it was called “When We were One and Twenty.” At least it was when I was one and twenty, he was younger. I remember at City College, this was around 1934, I was a mathematics student. I regarded physicists as an absurd bunch of people. They always fiddled around making all sorts of strange approximations and they were hardly what I call pure. There was one person whom I knew very slightly and that was Julian who apparently had been a student at Townsend Harris where he had as his teacher

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Irving Lowen, who later came to NYU and, in fact, taught a number of us. I know he taught me Dynamics and Relativity. And then Julian arrived at City College and I knew him slightly until we became students in a class together and I don’t know if he remembers this, but maybe he’ll be sorry when I tell about it. This was in 1934, ’35. We were in a class called Modern Geometry. It was taught by an old dodderer named Frederick B., I think his name was Reynolds. That’s it: Frederick B. Reynolds. Thank you. Frederick B. Reynolds. He was head of the Math Department. He really knew absolutely nothing. It was amazing. But he taught this course on Modern Geometry. It was a course in projective geometry from a miserable book by a man named Graustein from Princeton, and Julian was in the class, but it was very strange because he obviously never could get to class, at least not very often, and he didn’t own the book. That was clear. And every once in a while he’d grab me before class and ask me to show him my copy of this book and he would skim through it fast and see what was going on. And this fellow Reynolds, although he was a dodderer, was a very mean character. He used to send people up to the board to do a problem, and he was always sending Julian to the board to do problems because he knew he’d never seen the course, and Julian would get up at the board. And of course projective geometry is a very strange subject, the problems are trivial if you think about them pictorially, but Julian never would do them this way. He would insist on doing them algebraically and so he’d get up at the board at the beginning of the hour and he’d work through the whole hour and he’d finish the thing and by that time the course was over and anyway, Reynolds didn’t understand the proof, and that would end it for the day. And that was my introduction to Julian. The only other thing I remember from the time was that apparently Julian used to help various of the young instructors at City College who were working on their Ph.D. theses at Columbia. The usual thing was, you taught at City College and you worked at Columbia and he was sort of the unofficial adviser to various people and this well-known habit of his of working late at night and not appearing until then, I’m sure started then. All sorts of people bugged him about problems they wanted solved and so this, I think, was his means of keeping away from them. Then a little after that I remember that Julian disappeared and went to Columbia. Through the good offices of Rabi and some others, he went there, and he no longer had to make classes. Apparently that was the idea. I went on and finished at City College and started at NYU and after about a year I decided I was getting sick and tired of taking courses. I wanted to do some research, except that my Professor, Otto Halpern, just couldn’t be bothered with his students until they’d really ripened and he would leave us all alone: “Go do something, don’t bother me, I can’t be bothered.” And at that time Hy Goldsmith, whom I had known at City College, and Julian were at Columbia and I guess I talked to them and somehow or other—it was a very mysterious process. Columbia was an amazing place in those days. Really.

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A golden age for physics. They didn’t care whether you had a job there, whether you were a student there. You simply walked in and worked. No one paid me, you understand. I didn’t need them to pay me. I had an assistantship at NYU, but all you had to do was come in and you could work. I knew lots of people who worked at Columbia who had absolutely nothing to do with the place. You just knew somebody and so you came to the party. And I remember very well that the first piece of research that I worked on was something having to do with a paper that Herman mentioned today. It was a paper on the widths of nuclear energy levels by Manley, Goldsmith, and Schwinger, and I was looking for something to do and I came there and Julian said, “Well there is this thing that should be worked out and I haven’t done it because it takes a little more work. Here’s what you do.” It was some calculation about albedo of neutrons and I didn’t know what he was talking about. I mean, I really was miserably educated. And then a process began which recurred very often after that. The calculation involved a great deal of information about Bessel functions and in general, special functions, and Julian said, “I’ll show you how to do the calculation,” and he started off, “now this is the differential equation satisfied by Bessel functions and then here’s how you get the solution.” And then he went on this way for about, I think, it must have been about five days. He put in an enormous number of hours, by which time he had reproduced at least three-quarters of Watson’s Bessel Functions, but this apparently was a technique he liked to use. It was very helpful to me because I began to see how you did physics. At least mathematical physics. And I remember the great pride I had when my name appeared in a footnote in a paper in the Physical Review. I mean, it was my maiden effort. And this was the beginning of a whole series of things, where we had a very interesting method of working. It was along about this time that I started to help Julian and Goldsmith and Cohen, Bill Cohen, in some further experiments in nuclear physics, including, of course, the neutron-proton interaction that Herman mentioned and—life was very strange. I would work up at NYU or City College, come to Columbia around three o’clock, start doing calculations. Julian would appear some time between four and six and we would have a meal, which was my dinner and his breakfast, and then we would begin the evening’s work, which was a strange combination of theoretical and experimental work. We were experimenters, if you can call us that. That is, we were capable of putting foils in front of a radon beryllium source and measuring transmissions through them and activations, like grabbing the foils, running down the hall of Pupin—it was on the top floor—running like crazy, putting the foil on a counter, and taking a reading. And then we would run back, put them up again, and start doing theoretical work. And we would work rather strange hours. It seemed to me that we would work usually to something like midnight or 1 a.m., and then go out and have a bite to eat. This would mean two or three hours during which I would get educated on some new subject.

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I learned group theory from Julian, and I must admit I forgot it all immediately, but as I recall, I had all of Wigner’s book given to me, plus a lot more at the time and this was a regular process we went through and I think this must have gone on for a year or so, and we started doing calculations on ortho-paradeuterium and on orthoparahydrogen, scattering of neutrons, and this involved just an unbelievable amount of computation. And we would work on this nights and there was this wonderful theoretical seminar at that time in New York. There was a joint theoretical seminar of Columbia, NYU, and anybody else in the City who could come to these things, and it was a sort of a battlefield. My professor, my official professor, Halpern, would take on anybody and he was a rather testy fellow and loved getting into arguments. He would just take the greatest pleasure in taking on Gene Feenberg, or Fermi, when Fermi came, or anybody else. It just didn’t matter. There were just violent fights. I can recall giving a seminar there where, it seems to me, I prepared for this seminar for a month—a little bit with Julian’s help—and then started to talk and I said something and Fermi objected and Halpern came to my defense, and as I recall, I never got another word in edgewise. Never spoke another word through that hour. Well, the work went on for a while and we got all these computations done, except that this was a period, as I recall it was around 1938, beginning of ’39, and I think Julian was getting ready to go off to Berkeley, and the paper was done and we were going to write it up and I looked upon this as my magnum opus. You know, I was going to be doing a thesis with Halpern, but who cared about that. This was really great stuff. Then we started to write the paper. The only trouble is that at this time Julian was already very much interested in the tensor forces, and I remember very well helping him with some calculation involving the coupled differential equations that you get. And I was a great reader of the literature and I was always telling about interesting problems and unfortunately one day I mentioned the absorption of sound in gases and that started him off on an enormous amount of work which I don’t think he ever published, as far as I can tell. But he did all sorts of calculations on this and there I was, trying to get him to write a paper and he’s a rather finicky writer—maybe he isn’t so finicky any more—but I can recall that there were only a few weeks left before he was to leave and there was the paper and we were still in the first paragraph and every night we would start, we would write six or seven lines, and we wouldn’t get it done, and here I could see the time slipping, and I would go home and I would cuss hell out of him—to myself. And at one point I contemplated murdering him, but I didn’t. He went off to Berkeley, paper not done, and then in 1940, suddenly there comes a telegram, “Please send all the calculations,” and I packed up a pile of stuff about this high, shipped it off, heard nothing till suddenly some letters appeared in the Physical Review. There were some experiments by Alvarez and Pitzer, and a short note by Julian with the calculations, and I just gave up on him, did a thesis quick, got it done and then, in 1941, I suddenly discovered I had a job through the character.

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He actually had managed to find a job for me at Stanford. It seems to me he was at Berkeley and I believe it was he who talked to Felix Bloch. No? OK, then I’m under delusions. I’ve given him credit for my job. But the next time I saw Julian was in Cambridge. I came to the Harvard Radio Research Lab in ’43 and Julian arrived there about the same time, at the radiation lab, and we saw each other and he said to me, well, you know, we really ought to write that paper. That’s a great idea. It turned out, of course, he really had a point. He had found a very neat trick for reducing all this unbelievable amount of calculation that we had to do to what then amounted to four days of work and so we did it all over again very, very quickly and the paper was finished in about two weeks, I think, of writing. He had improved his style by then and it was published, I think, in ’46 and then another one in ’47. Well, essentially what I’m trying to say is that I think I should claim that I’m Julian’s first student. I believe I learned more from him than I learned from anybody else. In fact, I think he’s the only one from whom I ever learned anything. I find it very hard to learn from other people, but in his case I would say he showed he was a great one-on-one teacher as well as fine lecturer. It’s really a great pleasure to be here. There are lots of other people in addition to Julian, whom it really gives you great pleasure to see. There’s a certain special pleasure in seeing people whom you’ve known for a long time, with whom you’ve worked, with whom you’ve had miserable troubles and, on the other hand, gotten great pleasure. For one thing, it reminds you that you were once young and I hope there will be lots more reunions of the same kind. Thank you. Bernard Taub Feld Well, I must say that I feel like the aging vaudeville comedian who comes on after a couple of acts that are impossible to follow. What with Rabi’s reminiscences about the good old days of physics in New York and then Morty’s reminiscences about doing physics in the mid to late 30’s at Columbia with Julian around, all I can add to this picture is to note that I was around more or less at that time and I was going to talk—and I guess I still have to talk—briefly this evening about some aspects of life growing up as a physicist or trying to become a physicist in New York in that period. But I must say that my view is a somewhat different one from, certainly from Rabi’s and to a large extent from Morty’s. You might call it a worm’s eye view of growing up in New York in that period. I was an undergraduate at CCNY a few years after Julian had already become a legend and by the time I got to Columbia, Julian was just taking off for Berkeley, so I followed enough afterwards to have absorbed a great deal of the legend which Julian very rapidly became in New York, although fortunately, I did get to know Julian somewhat in that period and I guess I will talk a little bit about that. You’ve heard a number of stories about actually people who worked in New York in that period but I guess there was something very special about the period of, let’s say, the 1930’s. Not only in New York but in the United States, as far as physics was

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concerned. It was really a period when the United States, when physics in the United States developed at a fantastically rapid rate. I mean, before, let’s say in the early ’20’s, there were a few isolated instances of good physics being done in the United States. There was of course Michelson, and Millikan, and Rowland, but you could count them on the fingers of one or two hands. They were great experimentalists, but it was really not in the mainstream of modern physics. In a certain sense, the United States was dragged into the mainstream of modern physics in the late ’20’s as a consequence of the efforts of a few people. These were young physicists who obtained their Ph.D.’s in the United States and decided that what was really going on of interest was all going on in Europe—the development of the new quantum mechanics. And they somehow or other got themselves fellowships and went off to Munich or to Zurich or to Gottingen and they worked with the greats who were then developing—or to Copenhagen—who were then developing quantum mechanics and came back determined to introduce the modern physics into the United States. We’re fortunate to have one of them with us tonight. Rabi was certainly one of these pioneers. In fact, Hamburg, too. Yes. I had it written down here but I wasn’t reading. I would say that modern physics in the United States owes a tremendous debt to two people. Two of these pioneers—Rabi was one, Robert Oppenheimer was the other—who came back determined that they were going to start schools of modern physics in the United States. Oppie went off to the West Coast and started a school of theoretical physics at Berkeley and Pasadena, and Rabi settled in New York and turned New York in the period of the ’30’s, or was certainly one of the people who was primarily instrumental in turning New York in that period, into what one might refer to as a kind of experimental Copenhagen. I don’t know whether that’s the best description but it certainly was a center where young people could come and could do physics. Now there were a number of things that happened in New York in the ’30’s which, of course, contributed—at least three. Two of those were in a sense historical vicissitudes and in fact, really, they represented very tragic events in world history and by a kind of what you might call a principle of the ill wind, the effect in New York, or in the Eastern part of the United States and to some extent all over the United States, was to turn, was to build up modern physics in the United States. One of those was the Depression, the Great Depression of the ’30’s and the other was the horror of the development of Nazism in Germany. Consequence of the Depression was that there were in New York at that time, in that period, a fairly large number of young, aspiring, bright, extremely bright physicists, or aspiring physicists, who needed jobs and were willing to work long hours and low pay and there were lots of jobs available in the—or at least a fair number of jobs available at that time—teaching physics in the City College system in New York. In CCNY, in Brooklyn College and Queens College and Hunter College, and they were able to get jobs. If I recall correctly a normal teaching job in New York at that time was something like teaching 20 or 25 hours a week and the pay was something

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like $1500. At that time that was regarded as pretty good—$1600, was it?—and a number of people in this room did that. Now, the other thing about New York was that there was Columbia University and Rabi was there and the head of the department, George Pegram, was a very farsighted man and with, I suspect, a certain amount of prodding from Rabi, Columbia took advantage of the presence of all those people and as Morty just pointed out, any aspiring, bright, young physicist could do research at Columbia. There was Rabi’s lab, where there was a cyclotron, and all these people gathered around Columbia to do research and gathered around them all the bright people they could find. So this was a special period and the final aspect was that there was in New York City at that time this reservoir of young, aspiring, mainly children of immigrant parents, upward mobile, with a kind of tradition of learning and respect for learning and hard work, who were very anxious to take advantage of this situation and who were the fodder out of which the physicists were turned, so it was this combination of things. Now, to be absolutely fair, there was more to physics in the United States than just New York and vicinity and the West Coast. There was a great hinterland, which most New Yorkers thought of as a sort of prairie, that started out on the other side of the Palisades and then ended up in the Rockies, which was mainly useful for producing corn and steak and hay fever in the summer, but there were some things going on there. In Chicago, for example, there was the Compton effect and there was the school of work in cosmic ray physics, which was quite important, and then there was Michigan which, at least in New York at that time was regarded as a sort of a glorified summer colony in physics where the great Europeans would come in the summer to lecture and if you were lucky you could get to go and listen to them, and they had Goudsmit and Uhlenbeck, and Fermi came to lecture, and so on. That was a great place. But after that there was Berkeley and Pasadena and New York and surroundings—and surroundings meant, oh, Princeton and then Cornell and the other aspect, of course, was that all the great European physicists who fled from Nazi Germany got funneled through New York and some of them stayed and some of them stayed long enough to give some courses in New York. Teller and Bethe, who didn’t stay in New York, nevertheless, as I recall, came back in this period and taught quantum mechanics to graduate students at Columbia and in any event— of course Fermi and Szilard, when they came they just stayed, at Columbia, but everybody passed through so that for a student at that time it was really a great paradise. Well, that’s more or less the atmosphere into which Julian flashed in the early ’30’s and where I came along a few years later. And as I say, by that time there were a certain number of legends surrounding Julian. It’s amusing to recall some of these legends because they are legends which refer to some of the same things that Rabi was telling us about and you got a first-hand account this afternoon of some of these stories, but when I tried to recall them in the last few weeks, trying to think of what was going on in New York and my recollections of the stories about

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Julian, the stories were, somehow or other, just slightly different. It’s interesting to compare the legends which didn’t take long to grow up—as I say I came along just a year or two after Julian. For instance, there was a legend at CCNY that Julian was first discovered by Irving Lowen. Irving Lowen was then teaching at Townsend Harris, and the story is that Irving came across this kid sitting in the library reading the Physical Review and he looked over his shoulder and there was this kid reading Dirac and so Irv thought, well, here’s another one of these smart-aleck kids that, you know, we get them every once in a while, so he quizzed him about what he was reading and Julian allegedly was not only capable of telling him what he was reading but also told him what needed to be done to complete what Dirac hadn’t completed in this particular paper. The other legend—or rather the story you got from Rabi—was how Julian shifted from CCNY to Columbia. Now the story that went the rounds at CCNY at the time when I just got there was a little different. It had to do with a chairman of the department, a man called Corcoran, who was—well, he was not the greatest physicist who had ever been around. A rather crotchety old Irishman, and allegedly when Lloyd Motz had taken Julian under his wing, he took Julian to Corcoran and said look here’s this kid and he really knows more physics, too much physics, to have to take these elementary physics courses. Why can’t we let him take some of the advanced courses and Corcoran is alleged to have said, “Over my dead body. As long as I’m chairman of this department, no smart-ass kid is going to be allowed to skip taking my course in elementary particle physics.” In elementary physics, excuse me. So Motz was supposed to have taken Julian to Rabi and Rabi straightened it all out by getting a scholarship to Columbia. That was the story. Now we heard the real story from Rabi but it’s interesting to compare some of these stories. There was another story which parallels the one Rabi told, a story about Julian’s taking a course with Uhlenbeck. That wasn’t the way the story went when I got into Columbia. The story had to do with a chemistry professor and I’m not sure I remember his name—Kimball, I think, LaMer. This was a chemistry course that Julian was supposed to have had to take. And now, the story was, it is probably completely apocryphal—that Rabi got a telephone call one day from Professor LaMer who said, “This kid, this prot´eg´e of yours, this Schwinger, I’m going to have to fail him.” And Rabi said, “Why?” “Well, he’s taking my course and he hasn’t appeared at a single lecture all term.” And Rabi was supposed to say, “Well, why don’t you fail him?” To which LaMer was supposed to have answered, “The trouble is, he just took the final and he got 100 on it.” And now, Rabi is supposed to have answered, “Well, look, are you a man or a mouse? If you want to fail him, fail him.” Besides, the story is supposed to go on, Rabi said, “How can you pass up this opportunity to go down in history as the man who failed Julian Schwinger!” Well, now, these were the apocryphal stories that were going the rounds when I arrived on the scene. I got to CCNY in 1935 and actually when I came to CCNY

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as a freshman, I thought I was going to be majoring in history. This came about for a number of reasons, one of which was that I had a great memory for dates, and history in those days, at least in the high school courses that I took, was the memorization of a long series, a semi-infinite series of dates and since I was great at memorizing it, I thought I was obviously a born historian. So I thought I was going to be a history major but I also had an idea that maybe science would be fun and so I had taken a course, I’d taken physics—this was at Boys High School, which was a rather good high school, but the man who taught physics was not really much of a physics teacher. I remember two things. One, his great pedagogical tour de force was he would sometimes if it was a nice, cold, crisp winter day, he would go shuffling across the room and then open a bunsen burner and light it by a static electric discharge from the tip of his finger and he thought that this was really, you know, the way you taught electricity to a bunch of high school students. This was great physics. The other thing is I got an “A” in the course, but the reason I got an “A” in the course was that the final exam contained a lot of simple questions but the one question that everybody else failed on, which I got like that, was the question of “Name three forms of Ohm’s Law.” I had no difficulty in naming the three forms of Ohm’s Law. Some of you may not know what the three forms of Ohm’s Law are, but I still remember them very well. They are: V = RI, I = V /R, R = V /I. That was passed for physics, so physics to me was not a very interesting subject. Then I came to CCNY and then for some reason or other—I’m not sure whether it was required or whether I just thought it might be fun to try it again anyway. Anyhow, I took elementary physics and then I had real luck because my first instructor in elementary physics was Morty Hamermesh. He was just a graduate student then and he was teaching elementary physics at CCNY and all I can say is that after that term I was no longer a history major and after that sort of I felt that my unclear . And the second thing was that I met Hy Goldsmith and this was right in the middle of the period that Morty was talking about. In my sophomore year I was taking a course with Hy, and Hy was a dilettante and he was a very interesting guy because he was very lazy. But on the other hand he had two things—one he had an encyclopedic knowledge of what was going on and he had really good taste. He knew what was important and what wasn’t important. But at that period, I guess, it was just a period I think, probably when Morty and the other people working with him at Columbia had gotten to be pretty tired, gotten pretty tired of running up and down the hall with the foils and so I was recruited, as a sophomore then, to do the running. I guess Marty doesn’t remember but I spent six months at Columbia doing the sprinting. I was a pretty good sprinter. I didn’t know anything else but they were studying resonances in rhodium, and rhodium—I’ve forgotten what the mean life is now, but it’s really very short. You had to take those foils and sprint that 50 yards from the irradiation to the Geiger counter and I was the fastest sprinter they could find. I was a real good sprinter then, so I made out real well. As a result of that I not only got to hang around at Columbia at night but

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even when they went up to see Julian to consult on the theory or when something had gone wrong with the experiment and they got bored and just went up to talk to Julian, I was allowed to go up with them, and so I got to listen. (From the audience: “Did they pay you by the trip or the minute?”) Listen, the payment was—I was allowed to go along and that was great stuff, you know. A sophomore in college doing that. That was really being a student. Well, that was my introduction and as I say, I sort of got to know Julian because I was allowed to sit in on these sessions when they worked out the theory and in that period, I think, Julian was really on the way to becoming an experimentalist. You never saw him running down that hall, but I just looked up the Physical Review the other day and in the period from ’37 to ’39, Julian published six papers and three of those were experimental and I suspect—I don’t know, I’ve never asked Julian—I have a suspicion that one of the reasons that he ran off to Berkeley—there probably were other reasons in ’39—but one of them was to get away from these mad experimentalists who didn’t give him any peace, who would come wandering up at all hours of the night to disturb his work and who were trying to turn him into an experimentalist—God forbid. But in any case, by the time I got to Columbia, Julian had left, had just left and I didn’t meet Julian again until 1943, when I was in Chicago and Julian arrived at the metallurgical laboratory to spend something like six months, I guess it was—two months. Anyhow, it was in the summer, a few months in the summer of ’43. I’d been there a year and I was pretty sophisticated by then. In fact, I was doing experiments. That was the period in Chicago after the chain reaction had been proved, and what the people in Chicago were doing was designing the Hanford Reactor. Now that was not easy because nuclear engineering didn’t exist. They were really discovering and inventing nuclear engineering. This was Wigner and Szilard and Fermi and Wheeler and a crowd of young people who were working with them. They were all physicists but in a pinch a physicist, a theoretical physicist make pretty good engineers. In fact, they were all first-class engineers. No question unclear and Julian turned out—I didn’t realize that he’d been doing engineering already at the radiation lab, but, Julian, as you heard this afternoon, is also a pretty good engineer and he demonstrated it that summer. I guess there were a number of problems that remained to be solved, mainly to help in the design of Hanford, and these were in transport theory and some of them were more or less difficult and Julian was working on some of them and the things I was doing were not really directly connected with the Hanford thing and the Hanford thing was the most, the design of Hanford was the most important. Anyhow, there was the need of some kind of a match between Julian and the metallurgical laboratory, because Julian didn’t come in until some time in the evening and most people were already gone, so the contact wasn’t well established, so I sort of elected myself in that period to be the matchmaker. I had known Julian from Columbia, slightly, but at least well enough that I figured that I might be able to help in

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making the match between Julian and the project. It was really a very interesting process because what would happen was that I would go around in the afternoon— not every afternoon, but occasionally—go around to my friends in Wigner’s group and sort of try to smell out what were the problems with which they were having trouble. Things that were giving them some difficulty. And then some time in the late evening, maybe 10 o’clock or 11 o’clock or something, I would wander off to Julian’s office and wander in and Julian would be sitting there at his desk, typically, this was a very hot summer in Chicago, and Julian was a very fastidious dresser in those days. He never took off his coat. In Chicago I never saw Julian without his jacket. He’d be sitting there with his white shirt and tie, tie never even loosened, jacket on, with a pad and a paper, he’d be scribbling furiously, working on some problem on the pad, with his handkerchief, supersaturated handkerchief in the left hand, mopping the sweat off his brow as he worked, and I used to wander in and sit down, and wait and at some point Julian would pause to catch his breath and I would kind of interrupt him and try to get his attention away from whatever he was doing and I usually succeeded and not only because I was a pretty persistent guy, but because Julian is a nice guy and if you sort of bother him, he’ll pay some attention to you, and after a while I would get him interested in the particular problem I had in mind. I’d start talking about it and Julian would get interested and then he would go to work on it. He’d get up to the blackboard and I would start taking notes. As Julian worked on the problem, I would be taking notes and sometimes, you know, that could be pretty hectic. I don’t know any of you who saw Julian work in those days. Julian is ambidextrous and he can, he has a blackboard technique that uses two hands, and frequently, when he really got carried away, he would be solving two equations, one with each hand, and trying to take notes could be a hectic job. Well, at some point, either we would finish the problem or the dawn would start to break in the eastern horizon, and we would decide it was time to quit and then often, we would go to have breakfast together. We would get into Julian’s sleek black Cadillac and go to the nearest all-night eatery, where we would both have breakfast of, I think, it was a steak. So we would have our steak breakfast and then off we would go to our respective beds. Now this wasn’t all that either I or Julian did that summer, but I suspect that many of the knotty problems which the Hanford engineers weren’t able to cope with, were solved in that summer as a result of that particular technique. Well, after a while Julian went off to the radiation lab, to get back to the radiation lab, and I went off to Los Alamos and we didn’t meet again till we both arrived in Cambridge after the war, but that history is one that everybody is familiar with, so I guess that this is a good point to stop my worm’s eye view of the good old days when physics was really bursting out in New York. I tried to think of a title for this talk and the only title I could think of that really fits it, is “Some Reminiscences from a Well Spent Youth,” I think.

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Richard Phillips Feynman (D. Saxon: And now it seems to me very important at this stage, in the interest of accuracy and versimilitude that we have a New York physicist who actually speaks like one: Dick.) Not only that, but I thought this was Schwinger’s birthday party, not Columbia University’s. In 1935 I wanted to go to Columbia University and I applied there. It was during the Depression and it was hard to get money, and they said yes, but you have to take our examination and it will cost you $15 and if you don’t pass, we keep the $15. I did not pass, they kept the $15, and I’ve been cheated by Columbia ever since. Instead of getting a good education, I had to go to MIT while the expert was there. Therefore, it was not until I went to Los Alamos that I got a chance to meet Schwinger. He had already a great reputation because he had done so much work as the others have described and I was very anxious to see what this man was like. I’d always thought he was older than I was because he had done so much more, at the time I hadn’t done anything. And he came and he gave us lectures. I believe they were on nuclear physics. I’m not sure exactly the subject, but it was a scene that you probably have all seen once: the beauty of one of his lectures. He comes in, with his head a little bit to one side. He comes in like a bull into a ring and puts his notebook down and then begins—and the beautiful, organized way of putting one idea after the other, everything very clear from the beginning to the end. You can imagine for a lecturer like me, what a sensation it was to see such a thing. I was supposed to be a good lecturer according to some people, but this was really a masterpiece. Each one of the lectures was a great discourse while what I did was a talk on something. So I was very impressed and the times I got then to talk to him, I learned more, and then we went off and met many times in the different conferences, which always was a pleasure, but the greatest conference was the one at Pocono, I think it was, where we tried to describe—each of us had worked out quantum electrodynamics and we were going to describe it to the tigers. He described his in the morning, first, and then he gave one of these lectures, which are intimidating. They’re so perfect that you don’t want to ask any questions because it might interrupt the train. But the people in the audience like Bohr, and Dirac, Teller, and so forth, were not to be intimidated, so after a bit there were some questions. A slight disorganization, a mumbling, confusion—it was difficult. We didn’t understand everything, you know. But after a while he got a good thing. He would say, “Perhaps it will become clearer if I proceed.” So he continued this, continued it. Then I was supposed to go on in the afternoon, and Bethe, Hans Bethe said to me, “Look Feynman,” he says, “notice every time Schwinger tries to explain a physical idea how it gets all wrapped up in it, so you can’t explain anything to these guys, you better make it mathematical.” Well, Schwinger could make it mathematical. I only thought I could make it mathematical, so when my time came I started out by trying to make an equation that would do everything. They wanted to know where that equation came from. So then I’d start to describe

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physical ideas and things went very badly. Dirac would say things like—first of all I must explain our methods were entirely different, as you all know. I didn’t understand about those creation and annihilation operators. I didn’t know how they worked, that he was using, and I had some magic way, from his point of view, and Dirac would always interrupt me saying, when I was explaining about how I was going to work out positrons and so on, “Is it unitary?” And again, I’d say, again, well, I remembered his trick and I said, “Perhaps it will become clear as we proceed.” But Dirac was not put off, and like the Raven kept saying, “Is it unitary?” And not being quite sure what unitary meant, I said, “Is what unitary?” So he said, “The matrix that carries you from the past to the future,” and then I was saved because I said I had no matrix to do that because some of my positrons are already in the future. Like coming in from the wrong end. And then I tried to go on and I explained how you could disregard the exclusion principle in intermediate states and then Professor Teller said, “You mean that Helium can have three electrons in the S state for a little while?” I said, “Yes.” That was chaos, and then, all the time I was pushed back, away from the mathematics into my so-called physical ideas until I was driven to the point of describing quantum mechanics as an amplitude for every path, for every trajectory that a particle can take there’s an amplitude, and Professor Bohr got up and explained to me that already in 1920 they realized that the concept of a path in quantum mechanics—that you could specify the position as a function of time—that was not a legitimate idea and I gave up at that point. But the thing that I really, why I told this story, is that right after that, we got together in the hallway and although we’d come from the ends of the earth with different ideas, we’d climbed the same mountain from different sides and we could check each other’s equations. We compared our results because we had worked out problems and we looked at the answers and kind of half described how the terms came. He would say, “Well I got a creation and then another annihilation of the same photon and then the potential goes . . . ” “Oh, I think that might be what,” I’d draw a picture that looks like this. He didn’t understand my pictures and I didn’t understand his operators, but the terms corresponded and by looking at the equations we could tell, and so I knew, in spite of being refused admission by the rest, by conversations with Schwinger, that we had both come to the same mountain and that it was a real thing and everything was all right, so then after that we had many conversations together, at Rochester Conference, and so on, and he’s become, and I’ve always had a great pleasure talking to him and with his gentleness and sense of humor, and so on and so on, I’m very happy to be here tonight to wish him a happy birthday and also to wish that we can continue our collaboration as we have been doing before because I’m kind of working on quantum chromodynamics. My work to produce high energy results and I’m getting a bit confused and I need the kind of elegance and style that Professor Weisskopf mentioned, and I’d like to talk to you about it some time, so after this stuff’s over, I’m going to come back again and tell you. Thank you very much. Congratulations, Julian.

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Closing remarks by David Saxon and Isidor Isaac Rabi D. Saxon Now I would like to finish with a few rather brief remarks. Perhaps I’ll begin by adding to Professor Rabi’s rules for a long and satisfying life an additional rule which I think is equally important to the one you enunciated, Professor Rabi . . . I. I. Rabi Before you finish I’d like equal time on that Rabi story. Julian was an undergraduate at Columbia, as I explained, and he suddenly leaped from a student of low standing, precarious standing, at City College, to getting a Phi Beta Kappa at Columbia. It’s not that our standards were lower than the City College’s, although I assure you, their standards were very high. But something happened in this transplant. Anyway, Columbia at that time had anticipated the matrix mechanics and students were classified in a matrix with two indices. One was ordinary credit and the other was maturity credit. And you had a certain weight of ordinary credits and a certain weight of maturity credits. One Sunday morning I was called up by the dean, Dean Hawkes, and he said, “What shall I do about Schwinger?” I said, “What’s the problem?” He said, “He has enough credits to graduate but he hasn’t enough maturity credits.” It seemed to me absurd. How can you talk about things that way? So I said, “Well, you have your rules. I don’t know what you can do about it.” I wasn’t going to make a great plea. See how the thing’d work. Well, he was a real man, and on Sunday, he was a religious person, he said, “I’ll be damned if I won’t let Schwinger graduate because he doesn’t have enough maturity credits.” Of course this gave me great faith. And then LaMer came along. He was, for a chemist, awfully good. A great part of his life work was testing the Debye–H¨ uckel theory rather brilliantly. But he was this rigid, reactionary type, as described. He came along, he had a mean way about him. He said, “You have this Schwinger? He just didn’t pass my final exam.” I said, “He didn’t? I’ll look into it.” So I spoke to a number of people who’d taken the same course. And they had been greatly assisted in that subject by Julian. So I said, “I’ll fix that guy. We’ll see what character he has.” “Now, Vicky, what sort of guy are you anyway, what are your principles? What’re you going to do about this?” Well, he did as said flunk Julian, and I think it’s quite a badge of distinction for him, and I for one am not sorry at this point, they have this black mark on Julian’s rather elevated record. But he did get Phi Beta Kappa as an undergraduate, something I never managed to do. D. Saxon Thank you, Professor Rabi. Earlier today you were commenting on some rules for a long and satisfying life, and I want to add one to it which I expect most people here will agree about, and that is you only ought to talk about things you really know something about. Not all of us observe it, of course. I intend to take my own

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advice very seriously and therefore I’m not going to talk about physics but instead about universities, which is my present field of specialization. (From the audience: That means you know anything about it?) The hard way. The hard way. One of the most difficult problems that we who have responsibility for university administration have to contend with, is an astonishingly forceful tide of anti-intellectualism, which takes many forms. I’m not going to try to run through all of the manifestations of that phenomenon, but there’s a particularly troublesome one which is the repeated contention that in our great universities, research somehow comes at the expense of teaching. That scholarship only happens if people neglect teaching. Now I believe that’s manifestly false, but it’s been extraordinarily difficult to persuade people of that fact and therefore I was especially struck by the fact that earlier today both Professor Weisskopf and Professor Rabi called attention to that dimension of Julian’s contribution. Professor Weisskopf mentioned that there is something like 100 Ph.D. students of Julian’s, Professor Rabi, 80. The most striking thing about that is the difference between 100 and 80 is 20, which is more than most people have under any circumstances. Now, it so happens that probably never before in my life have I had a chance to get one up on Professor Weisskopf. Never before in my life have I had a chance to get one up on Professor Rabi. And to do it to both at the same time is really remarkably improbable, but I performed an experiment. I’ll tell you what I did. I asked Julian and I know the answer. The answer is 67. But that happens to be an aside. I couldn’t resist it. The first remark to be made is that Julian, in addition to all the work he did in physics, and incidentally, the unpublished work equals in volume, or perhaps exceeds in volume, the published work. I know that for I’ve seen a very substantial part of the unpublished work. Let me also mention that if you want to talk about Julian’s students, you need also to talk about his collaborators because his collaborators have almost always also been his students. Morty referred to that. Morty was a collaborator of Julian’s, not a formal student of his. I was a collaborator of Julian’s, not his student in the formal sense. The number of people I know—Rarita, Gerjuoy, Marcuvitz, Frank Carlson, Harold Levine, John Miles, Albert Hines, and I suspect, to some degree, Edward Teller with a 19 year old Julian and Isador Rabi with a 17 year old Julian, collaborators of him were also in some way his students. But in addition to that, as has been pointed out, the auditors in his courses, from other institutions often—Black, Jackson, and so on—were also his students. I think the message here is a very clear one to me: that great physicists, great scholars, are also great teachers. I expect almost all of you in this room would agree that that was the case. Why is it that we are so little believed when we try to make that point? Why? Why does that statement fall on deaf ears when all of us live it and know it. Perhaps, may I suggest, because we have not been willing enough to inform others about our subject. Not merely for their good, but also for our own. I’m sure each of you

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will have some thoughts on that problem but I expect each of you is as far from a solution to it as I am. I am desperately in need of a solution. That’s a problem that’s ever present with me and it is worth serious thought, let me assure you. Well, let me conclude by remarking that we’re celebrating a great occasion this weekend. We’re celebrating it at UCLA which gives me very special pleasure. Vicky, I have to say that because even though you’re unhappy that Julian left Cambridge, I am convinced beyond any argument that I am largely responsible for the fact that he’s at UCLA. I believe it to be true and therefore it gives me very special pleasure. Celebrations such as this, I think, are occasions which convey very much mixed emotions. On the one hand, the joy of recognizing a life of extraordinary accomplishment, the joy of being again with old friends, but also there’s the bittersweet taste of inexorable passing of the years, not only for Julian but for us all. Thank you very much.

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