Quantum Mechanics of Non-Hamiltonian and Dissipative Systems MONOGRAPH SERIES ON NONLINEAR SCIENCE AND COMPLEXITY SERIES EDITORS Albert C.J. Luo Southern Illinois University, Edwardsville, USA George Zaslavsky New York University, New York, USA ADVISORY BOARD Valentin Afraimovich, San Luis Potosi University, San Luis Potosi, Mexico Maurice Courbage, Université Paris 7, Paris, France Ben-Jacob Eshel, School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel Bernold Fiedler, Freie Universität Berlin, Berlin, Germany James A. Glazier, Indiana University, Bloomington, USA Nail Ibragimov, IHN, Blekinge Institute of Technology, Karlskrona, Sweden Anatoly Neishtadt, Space Research Institute Russian Academy of Sciences, Moscow, Russia Leonid Shilnikov, Research Institute for Applied Mathematics & Cybernetics, Nizhny Novgorod, Russia Michael Shlesinger, Office of Naval Research, Arlington, USA Dietrich Stauffer, University of Cologne, Köln, Germany Jian-Qiao Sun, University of Delaware, Newark, USA Dimitry Treschev, Moscow State University, Moscow, Russia Vladimir V. Uchaikin, Ulyanovsk State University, Ulyanovsk, Russia Angelo Vulpiani, University La Sapienza, Roma, Italy Pei Yu, The University of Western Ontario, London, Ontario N6A 5B7, Canada Quantum Mechanics of Non-Hamiltonian and Dissipative Systems VASILY E. TARASOV Division of Theoretical High Energy Physics Skobeltsyn Institute of Nuclear Physics Moscow State University Moscow, Russia AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2008 Copyright © 2008 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. 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Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-53091-2 ISSN: 1574-6917 For information on all Elsevier publications visit our website at books.elsevier.com Printed and bound in The Netherlands 06 07 08 09 10 10 9 8 7 6 5 4 3 2 1 Preface This book is the expanded version of lectures on quantum mechanics, which au- thor read for students of the graduate level and which have been published in Russian. The main attention is given to the consecutive and consistent descrip- tion of foundations of modern quantum mechanics. Difference of the suggested book from others is consistent use of the functional analysis and operator alge- bras. To read the text, preliminary knowledge of these sections of mathematics is not required. All the necessary information, which is beyond usual courses of the mathematical analysis and linear algebra, is included. To describe the theory, we use the fact that quantum and classical mechanics are connected not only by limiting transition, but also realized by identical mathe- matical structures. A common basis to formulate the theory is an assumption that classical and quantum mechanics are different representations of the same total- ity of mathematical structures, i.e., the so-called Dirac correspondence principle. For construction of quantum theory, we consider mathematical concepts that are the general for Hamiltonian and non-Hamiltonian systems. Quantum dynamics is described by the one-parameter semi-groups and the differential equations on operator spaces and algebras. The Lie–Jordan algebraic structure, Liouville space and superoperators are used. It allows not only to consistently formulate the evo- lution of quantum systems, but also to consider the dynamics of a wide class of quantum systems, such as the open, non-Hamiltonian, dissipative, and nonlinear systems. Hamiltonian systems in pure states are considered as special cases of quantum dynamical systems. The closed, isolated and Hamiltonian systems are idealizations that are not ob- servable and therefore do not exist in the real world. As a rule, any system is always embedded in some environment and therefore it is never really closed or isolated. Frequently, the relevant environment is in principle unobservable or is unknown. This would render the theory of non-Hamiltonian and dissipative quan- tum systems to a fundamental generalization of quantum mechanics. The quantum theory of Hamiltonian systems, unitary evolution, and pure states should be con- sidered as special cases of the generalized approach. Usually the quantum mechanics is considered as generalization of classical me- chanics. In this book the quantum mechanics is formulated as a generalization of modern nonlinear dynamics of dissipative and non-Hamiltonian systems. The quantization of equations of motion for dissipative and non-Hamiltonian classical systems is formulated in this book. This quantization procedure allows one to de- rive quantum analogs of equations with regular and strange attractors. The regular v vi Preface attractors are considered as stationary states of non-Hamiltonian and dissipative quantum systems. In the book, the quantum analogs of the classical systems with strange attractors, such as Lorenz and Rössler systems, are suggested. In the text, the main attention is devoted to non-Hamiltonian and dissipative systems that have the wide possibility to demonstrate the complexity, chaos and self-organi- zation. The text is self-contained and can be used without introductory courses in quan- tum mechanics and modern mathematics. All the necessary information, which is beyond undergraduate courses of the mathematics, is presented in the book. Therefore this book can be used in the courses for graduate students. In the book the modern structure of the quantum theory and new fundamental results of last years are described. Some of these results are not considered in monographs and text books. Therefore the book is supposed to be useful for physicists and mathe- maticians who are interested in the modern quantum theory, nonlinear dynamics, quantization and chaos. The book consists of two interconnected parts. The first part is devoted to the quantum kinematics that defines the properties of quantum observable, states and expectation values. In the second part, we consider the quantum dynamics that describes the time evolution of the observables and states. Quantum mechanics has its mathematical language. It consists of the operator algebras, functional analysis, theory of one-parameter semi-groups and operator differential equations. Although we can have some sort of understanding of quan- tum mechanics without knowing its mathematical language, the precise and deep meaning of the physical notions cannot be obtained without using operator alge- bras, functional analysis, etc. Many theorems of operator algebra and functional analysis, etc. are easy to understand and use, although their proofs may be quite technical and time-consuming to present. Therefore we explain the meaning and significance of the theorems and ask reader to use them without proof. The author is greatly indebted to Professor George M. Zaslavsky for his invalu- able suggestions and comments. Thanks are expressed also to Edward E. Boos, Vyacheslav A. Ilin, Victor I. Savrin, Igor V. Volovich, colleagues of THEP di- vision, and my family for their help and invaluable support during the work on the book. Finally, the author wishes to express his appreciation to Elsevier for the publication of this book. Vasily E. Tarasov Moscow September 2007 Contents Preface v A Very Few Preliminaries 1 PART I. QUANTUM KINEMATICS 9 Chapter 1. Quantum Kinematics of Bounded Observables 11 1.1. Observables and states 11 1.2. Pre-Hilbert and Hilbert spaces 11 1.3. Separable Hilbert space 16 1.4. Definition and examples of operators 18 1.5. Quantum kinematical postulates 20 1.6. Dual Hilbert space 21 1.7. Dirac’s notations 23 1.8. Matrix representation of operator 24 Chapter 2. Quantum Kinematics of Unbounded Observables 27 2.1. Deficiencies of Hilbert spaces 27 2.2. Spaces of test functions 28 2.3. Spaces of generalized functions 31 2.4. Rigged Hilbert space 33 2.5. Linear operators on a rigged Hilbert space 36 2.6. Coordinate representation 40 2.7. X-representation 43 Chapter 3. Mathematical Structures in Quantum Kinematics 47 3.1. Mathematical structures 47 3.2. Order structures 49 3.3. Topological structures 50 3.4. Algebraic structures 52 vii viii Contents 3.5. Examples of algebraic structures 57 3.6. Mathematical structures in kinematics 66 Chapter 4. Spaces of Quantum Observables 69 4.1. Space of bounded operators 69 4.2. Space of finite-rank operators 71 4.3. Space of compact operators 72 4.4. Space of trace-class operators 74 4.5. Space of Hilbert–Schmidt operators 78 4.6. Properties of operators from K1(H) and K2(H) 79 4.7. Set of density operators 81 4.8. Operator Hilbert space and Liouville space 83 4.9. Correlation functions 87 4.10. Basis for Liouville space 88 4.11. Rigged Liouville space 91 Chapter 5. Algebras of Quantum Observables 95 5.1. Linear algebra 95 5.2. Associative algebra 96 5.3. Lie algebra 96 5.4. Jordan algebra 98 5.5. Involutive, normed and Banach algebras 100 5.6.