Springer Monographs in Mathematics Springer Science+Business Media, LLC N.P. Landsman Mathematical Topics Between Classical and Quantum Mechanics With 15 Illustrations , Springer N.P. Landsman Korteweg-de Vries Institute for Mathematics University of Amsterdam Plantage Muidergracht 24 Amsterdam 1018 TV The Netherlands Mathematics Subject Classification (1991): 8ISIO, 8IPXX, 58FXX, 8IRXX, 81TXX Library of Congress Cataloging-in-Publication Data Landsman, N.P. (Nicolaas P.) Mathematical topics between classical and quantum mechanics / N.P. Landsman. p. cm. Includes bibliographical references and index. ISBN 978-1-4612-7242-7 ISBN 978-1-4612-1680-3 (eBook) DOI 10.1007/978-1-4612-1680-3 1. Quantum theory-Mathematics. 2. Quantum field theory Mathematics. 3. Hilbert space. 4. Geometry, Differential. 5. Mathematical physics. 1. TitIe. QCI74.I7.M35L36 1998 530.12---dc21 98-18391 Printed on acid-free paper. © 1998 Springer Science+Business Media New York Origina1ly published by Springer-Verlag New York, Inc.in 1998 Softcover reprint ofthe hardcover 1s t edition 1998 AII 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), 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 of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Camera-ready copy prepared from the author' s JJ.TEiX files. 987 6 5 4 3 2 1 lSBN 978-1-4612-7242-7 I realize that the disappearance ofa culture does not signify the disappearance of human value, but simply of certain means of expressing this value, yet the fact remains that I have no sympathy for the current of European civilization and do not understand its goals, if it has any. So I am really writing for friends who are scattered throughout the corners of the globe. Our civilization is characterized by the word "progress". Progress is its form rather than making progress one of its features. Typically it constructs. It is oc cupied with building an ever more complicated structure. And even clarity is only sought as a means to this end, not as an end in itself For me on the contrary clarity, perspicuity are valuable in themselves. I am not interested in constructing a building, so much as in having a perspicuous view of the foundations of typical buildings. Ludwig Wittgenstein Preface Subject Matter The original title of this book was Tractatus Classico-Quantummechanicus, but it was pointed out to the author that this was rather grandiloquent. In any case, the book discusses certain topics in the interface between classical and quantum mechanics. Mathematically, one looks for similarities between Poisson algebras and symplectic geometry on the classical side, and operator algebras and Hilbert spaces on the quantum side. Physically, one tries to understand how a given quan tum system is related to its alleged classical counterpart (the classical limit), and vice versa (quantization). This monograph draws on two traditions: The algebraic formulation of quan tum mechanics and quantum field theory, and the geometric theory of classical mechanics. Since the former includes the geometry of state spaces, and even at the operator-algebraic level more and more submerges itself into noncommutative geometry, while the latter is formally part of the theory of Poisson algebras, one should take the words "algebraic" and "geometric" with a grain of salt! There are three central themes. The first is the relation between constructions involving observables on one side, and pure states on the other. Thus the reader will find a unified treatment of certain aspects of the theory of Poisson algebras, oper ator algebras, and their state spaces, which is based on this relationship. Roughly speaking, observables relate to each other by an algebraic structure, whereas pure states are tied together by transition probabilities (in both cases topology plays an additional role). The discussion of quantization shows both sides of the coin. One side involves a mapping of functions on the classical phase space into some operator algebra; at the other side one has coherent states, which define a map from the phase space itself into a projective Hilbert space. The duality between these sides is neatly exhibited in what is sometimes called Berezin quantization. viii Preface The second theme is the analogy between the C* -algebra of a Lie groupoid and the Poisson algebra of the corresponding Lie algebroid. For example, the role played by groups and fiber bundles in classical and quantum mechanics may be understood on the basis of this analogy. Thirdly, we describe the parallel between symplectic reduction in classical me chanics (with Marsden-Weinstein reduction as an important special case) and Rieffel induction (a tool for constructing representations of operator algebras) in quantum mechanics. This provides an interesting example of the mathematical similarities alluded to above, and in addition leads to a powerful strategy for the quantization of constrained systems in physics. Various examples illustrate the abstract theory: The reader will find particles moving on a curved space in an external gauge field, magnetic monopoles, low dimensional gauge theories, topological quantum effects, massless particles, and 8-vacua. On the other hand, the reader will not find path integrals, geometric quantization, the WKB-approximation, microlocal analysis, quantum chaos, or quantum groups. The connection between these topics and those treated in this book largely remains to be understood. Prerequisites, Level, and Organization of the Book This book should be accessible to mathematicians with a good undergraduate education and some prior knowledge of classical and quantum mechanics, and to theoretical physicists who have not completely abstained from functional analysis. It is assumed that the reader has at least seen the description of classical mechanics in terms of symplectic geometry, and knows the standard Hilbert space description of a quantum-mechanical particle moving in R3. The reader should be familiar with the basics of the theory of manifolds, Lie groups, Banach spaces, and Hilbert spaces, say at the level of a first course. The necessary concepts in operator algebras, Riemannian and symplectic geometry, and fiber bundles are developed from scratch, but some previous exposure to these subjects would do no harm. It is suggested that the reader start by going through the informal Introductory Overview as a whole. The main text is of a technical nature. The various chapters are logically related to each other, but can be read almost independently. To study a given chapter it is usually sufficient to be familiar with the preceding chapters merely at the level of the Introductory Overview. Some technical details will, of course, depend on previous material in a deeper way. One should by all means go through the list of conventions and notation below. In the interest of clarity and continuity, no credits or references to the literature are given in the main text. These may be found in the Notes, which in addition contain comments and elaborations on the main text. If no reference for a particular result is given, it is either standard or new (we leave this decision to the reader). Conventions and Notation ix The author would be happy if glaring omissions in the notes or references were pointed out to him. In the Index, entries refer only to the location where an entry is defined and/or occurs for the first time. Conventions and Notation Unless explicitly indicated otherwise, or obvious from the context, our conventions are as follows. General • The (Roman) chapter number is used only in cross-referencing between dif ferent chapters. In such references, numbers in brackets refer to equations and those without refer to paragraphs (e.g., 1.2.3) or to sections (such as 1.2). • The symbol • means "end of proof". The symbol 0 stands for "end of incomplete proof". • The equation A := B means that A is by definition equal to B. • The abbreviation "iff" means "if and only if". Li • An index that occurs twice is summed over, i.e., ajaj := ajaj. • Projections between spaces are denoted by T; in case of possible confusion we write TE->Q for the pertinent projection from E to Q. • The symbol f means "restricted to". • The symbol Ix stands for the function on X that is identically one. • We put 0 E JR+ but 0 f/. N. Functional Analysis • Vector spaces are over C, and functions are C-valued. Vector spaces over JR are denoted by VIR etc.; spaces of real-valued functions are written, for example, COO(P, JR). The only exception to this rule is formed by Lie algebras 9, which are always real except when the complexification 9c is explicitly indicated (this occurs only in 111.1.10, III.l.l1, and IV.3.6). • The space Co(X), where X is a locally compact Hausdorff space, consists of all continuous functions on X that vanish at infinity; the space of all compactly supported continuous functions on X is denoted by Cc(X), and the bounded continuous functions form Cb(X). These are usually seen as normed spaces under the sup-norm 11/1100 := sup I/(x)l. XEX • When X has the discrete topology (relative to which all functions are continu ous), we often write l(X),lc(X),lOO(X),lo(X) for C(X), Cc(X), L OO(X), and Co(X). x Preface • The topological dual of a topological vector space V is denoted by V*; hence the double dual is V**. The action of () E V* on v E V is denoted by ()(v). Multilinear forms a are similarly denoted by a(vI' ... , vn). + • When confusion might arise otherwise, we write X + Y for X Y in VI EB V2, + where X E VI and Y E V2 (for example, in V EB V the expression X Y would be ambiguous, denoting either X + Y +0, where X + Y E V ~ V EBO c V EB V, + or X+Y, orO+X Y). Hilbert Spaces • Inner products (, ) in a Hilbert space 1-l are linear in the second entry and antilinear in the first. • If K is a closed subspace of a Hilbert space 1-l, then [K] denotes the orthogonal projection onto K. If \II E 1-l, we write [\II] for [C\II]. • The symbol S1-l denotes the space of all unit vectors in 1-l. The projective space of 1-l is called 1P7t; hence IPCN = ClPN- I• • The symbols ~(1-l), ~o(1-l), ~ I (1-l), ~lh(1-l) stand for the collections of all bounded, compact, trace-class, Hilbert-Schmidt operators on 1-l. The unit operator in ~(1-l) is called lL We write VJtN(C) for ~(CN). • When A and B are operators on 1-l, the symbol [A, B) stands for the commutator AB - BA. We also use {A, B}1i := i[A, B]/Ii. • In the context of the previous item, or more generally when A and B are elements + of a Jordan algebra or a C* -algebra, A 0 B denotes ~ (A B B A). In all other situations, 0 has its usual meaning of composition; i.e., when f and g are suitable functions, one has f 0 g(x) := f(g(x». • We say that two Hilbert spaces are naturally isomorphic if they are related by a unitary isomorphism whose construction is independent of a choice of basis. • The Hilbert space L2(JRn) is defined with respect to Lebesgue measure. Our convention for the inner product is the one mainly used in the physics literature. Its motivation, however, is mathematical. Firstly, each \II E 1-l defines a linear functional on 1-l by \11(<1» := (\II, <1», without the need to change the order. Secondly, the convention is the same as for "inner products" taking values in a C*-algebra, which for good reasons are always taken to be linear in the second entry; see IV.2. C* -Algebras • The set of self-adjoint elements in a C* -algebra sa is called salR. Its state space is S(sa), and its pure state space is P(sa). • The unitization of a C*-algebra sa is called san. • States on a C* -algebra are denoted by w; pure states are sometimes also called p, a, or 1/f. The state space of sa is called S(sa); the pure state space is denoted by P(sa).