Springer Series on Atomic, Optical, and Plasma Physics Volume 63 Series Editors W.E. Baylis Uwe Becker Philip George Burke Robert N. Compton M.R. Flannery Charles J. Joachain Brian R. Judd Kate Kirby Peter Lambropoulos Gerd Leuchs Pierre Meystre For further volumes: http://www.springer.com/series/411 Ingvar Lindgren Relativistic Many-Body Theory A New Field-Theoretical Approach With 97 Figures ABC Ingvar Lindgren Department of Physics University of Gothenburg SE-412 96 Go¨teborg Sweden [email protected] ISSN 1615-5653 ISBN 978-1-4419-8308-4 e-ISBN 978-1-4419-8309-1 DOI 10.1007/978-1-4419-8309-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011921721 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To Eva Preface It is now almost 30 years since the first edition of my book together with John Morrison, Atomic Many-Body Theory [6], appeared, and the second edition ap- peared some years later. It has been out of print for quite some time, but fortunately has recently been made available again through a reprint by Springer Verlag. During the time that has followed, there has been a tremendous development in the treatment of many-body systems, conceptually as well as computationally. Particularly the relativistic treatment has expanded considerably, a treatment that has been extensively reviewed recently by Ian Grant in the book Relativistic Quantum Theory of Atoms and Molecules [2]. Also, the treatment of quantum-electrodynamical (QED) effects in atomic systems has developed considerably in the last few decades, and several review articles have appeared in the field [7, 11, 13] besides the book by Labzowsky et al., Relativistic Effects in Spectra of Atomic Systems [5]. An impressive development has taken place in the field of many-electron systems by means of various coupled-cluster approaches, with applications particularly on molecular systems. The development during the last 50 years has been summarized in the book Recent Progress in Coupled Cluster Methods, edited by C´ˇ arsky, Paldus, and Pittner [14]. The present book is aimed at combining atomic many-body theory with quantum- electrodynamics, which is a long-sought goal in quantum physics. The main prob- lem in this effort has been that the methods for QED calculations, such as the S-matrix formulation, and the methods for many-body perturbation theory (MBPT) have completely different structures. With the development of the new method for QED calculations, the covariant evolution operator formalism by the Gothenburg atomic theory group [7], the situation has changed, and quite new possibilities appeared to formulate a unified theory. The new formalism is based on field theory, and in its full extent the unification process represents a formidable problem, and we can in this book describe only how some steps toward this goal can be taken. This book is largely based upon the previous book on Atomic Many Body Theory [6],anditisassumedthatthe reader has absorbed most of that book, particularly Part II. In addition, the reader is expected to have basic knowledge in quantum field theory that is explained in vii viii Preface books such as Quantum Theory of Many-Particle Systems by Fetter and Walecka [1] (mainly parts I and II), An introduction to Quantum Field Theory by Peskin and Schroeder [12], and Quantum Field Theory by Mandl and Shaw [10]. The material of this book is largely based upon lecture notes and recent publi- cations by the Gothenburg Atomic-Theory Group [3, 4, 7–9], and I want to express my sincere gratitude particularly to my previous coauthor John Morrison and to my present coworkers, Sten Salomonson and Daniel Hedendahl, as well as to the previous collaborators Ann-Marie Pendrill, Jean-Louis Heully, Eva Lindroth, Bj¨orn As´˚ en, Hans Persson, Per Sunnergren, Martin Gustavsson, and H˚akan Warston for valuable collaboration. In addition, I want to thank the late pioneers of the field, Per-Olov L¨owdin, who taught me the foundations of perturbation theory some 40 years ago, and Hugh Kelly, who introduced the diagrammatic representation into atomic physics – two corner stones of the later developments. Furthermore, I have benefitted greatly from communications with many other national and international colleagues and friends (in alphabetic order), Rod Bartlett, Erkki Br¨andas, Gordon Drake, Ephraim Eliav, Stephen Fritzsche, Gerald Gabrielse, Walter Greiner, Paul Indelicato, Karol Jankowski, J¨urgen Kluge, Leonti Labzowsky, Peter Mohr, Debashis Mukherjee, Marcel Nooijen, Joe Paldus, Vladimir Shabaev, Thomas St¨ohlker, Gerhard Soff , Joe Sucher, Peter Surjan, and many others. The outline of the book is the following. The main text is divided into three parts. Part I gives some basic formalism and the basic many-body theory that will serve as a foundation for the following. In Part II, three numerical procedures for calculation of QED effects on bound electronic states are described, the S-matrix formulation, the Green’s-function, and the covariant-evolution-operator methods. A procedure toward combining QED with MBPT is developed in Part III. Part IV contains a number of Appendices, where basic concepts are summarized. Certain sections of the text that can be omitted at first reading are marked with an asterisk (*). References 1. Fetter, A.L., Walecka, J.D.: The Quantum Mechanics of Many-Body Systems. McGraw-Hill, N.Y. (1971) 2. Grant, I.P.: Relativistic Quantum Theory of Atoms and Molecules. Springer, Heidelberg (2007) 3. Hedendahl, D.: Towards a Relativistic Covariant Many-Body Perturbation Theory.Ph.D. thesis, University of Gothenburg, Gothenburg, Sweden (2010) 4. Hedendahl, D., Salomonson, S., Lindgren, I.: . Phys. Rev. p. (to be published) (2011) 5. Labzowski, L.N., Klimchitskaya, G., Dmitriev, Y.: Relativistic Effects in Spactra of Atomic Systems. IOP Publ., Bristol (1993) 6. Lindgren, I., Morrison, J.: Atomic Many-Body Theory. Second edition, Springer-Verlag, Berlin (1986, reprinted 2009) 7. Lindgren, I., Salomonson, S., As´˚ en, B.: The covariant-evolution-operator method in bound- state QED. Physics Reports 389, 161–261 (2004) 8. Lindgren, I., Salomonson, S., Hedendahl, D.: Many-body-QED perturbation theory: Connec- tion to the two-electron Bethe-Salpeter equation. Einstein centennial review paper. Can. J. Phys. 83, 183–218 (2005) Preface ix 9. Lindgren, I., Salomonson, S., Hedendahl, D.: Many-body procedure for energy-dependent perturbation: Merging many-body perturbation theory with QED. Phys.Rev.A73, 062,502 (2006) 10. Mandl, F., Shaw, G.: Quantum Field Theory. John Wiley and Sons, New York (1986) 11. Mohr, P.J., Plunien, G., Soff, G.: QED corrections in heavy atoms. Physics Reports 293, 227–372 (1998) 12. Peskin, M.E., Schroeder, D.V.: An introduction to Quantun Field Theory. Addison-Wesley Publ. Co., Reading, Mass. (1995) 13. Shabaev, V.M.: Two-times Green’s function method in quantum electrodynamics of high-Z few-electron atoms. Physics Reports 356, 119–228 (2002) 14. C´ˇ arsky, P., Paldus, J., Pittner, J. (eds.): Recent Progress in Coupled Cluster Methods: Theory and Applications. Springer (2009) Gothenburg, Sweden Ingvar Lindgren Contents 1 Introduction ................................................................... 1 1.1 Standard Many-Body Perturbation Theory ........................... 1 1.2 Quantum-Electrodynamics............................................. 2 1.3 Bethe–Salpeter Equation ............................................... 3 1.4 Helium Atom: Analytical Approach .................................. 4 1.5 Field-Theoretical Approach to Many-Body Perturbation Theory.... 5 References...................................................................... 7 Part I Basics: Standard Many-Body Perturbation Theory 2 Time-Independent Formalism .............................................. 13 2.1 First Quantization ...................................................... 13 2.1.1 De Broglie’s Relations ....................................... 13 2.1.2 The Schr¨odinger Equation ................................... 14 2.2 Second Quantization ................................................... 16 2.2.1 Schr¨odinger Equation in Second Quantization .............. 16 2.2.2 Particle–Hole Formalism: Normal Order and Contraction.................................................... 18 2.2.3 Wick’s Theorem .............................................. 19 2.3 Time-Independent Many-Body Perturbation Theory ................. 20 2.3.1 Bloch Equation ............................................... 20 2.3.2 Partitioning of the Hamiltonian .............................. 21 2.4 Graphical Representation .............................................. 25 2.4.1 Goldstone Diagrams .......................................... 25