Table Of ContentSpringer 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).
