E-Book Overview
This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. In addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantum mechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann) variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature. This book is written in a concise style with careful attention to precise mathematics formulation of methods and results. Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory. Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.
E-Book Content
Quantum Mechanics for Mathematicians Leon A.Takhtajan
Graduate Studies in Mathematics Volume 95
American Mathematical Society Providence, Rhode Island
Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 81-01.
For additional information and updates on this book, visit
www.ams.org/bookpages/gsm-95
Library of Congress Cataloging-in-Publication Data Takhtadzhian, L. A. (Leon Armenovich) Quantum mechanics for mathematicians / Leon A. Takhtajan.
p. cm. - (Graduate studies in mathematics ; v. 95) Includes bibliographical references and index. ISBN 978-0-8218-4830-8 (alk. paper) 1. Quantum theory. 2. Mathematical physics. I. Title. QC174.12.T343
2008
530.12-dc22
2008013072
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissionclams . org. © 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.
The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/
1098 7G 5432 1
131211100908
To my teacher Ludwig Dmitrievich Faddeev with admiration and gratitude.
Contents
Preface
xiii
Part 1. Foundations Chapter 1. Classical Mechanics §1. Lagrangian Mechanics 1.1. G