Methods of Noncommutative Analysis: Theory and Applications (De Gruyter Studies in Mathematics)

de Grunter Studies in Mathematics 22 Editors: Heinz Bauer • Jerry L. Kazdan • Eduard Zehnder de Gruyter Studies in Ma...

Author: Vladimir E. Nazaikinskii | V. E. Shatalov | Boris Yu Sternin

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de Grunter Studies in Mathematics 22 Editors: Heinz Bauer • Jerry L. Kazdan • Eduard Zehnder

de Gruyter Studies in Mathematics 1 Riemannian Geometry, 2nd rev. ed., Wilhelm P A. Klingenberg 2 Semimartingales, Michel Métivier 3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup 4 Spaces of Measures, Corneliu Constantinescu 5 Knots, Gerhard Burde and Heiner Zieschang 6 Ergodic Theorems, Ulrich Kren gel 7 Mathematical Theory of Statistics, Helmut Strasser 8 Transformation Groups, Tammo tom Dieck 9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii 10 Analyticity in Infinite Dimensional Spaces, Michel Hervé 11 Elementary Geometry in Hyperbolic Space, Werner Fenchel 12 Transcendental Numbers, Andrei B. Shidlovskii 13 Ordinary Differential Equations, Herbert Amann 14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and Francis Hirsch 15 Nevanlinna Theory and Complex Differential Equations, Ilpo Laine 16 Rational Iteration, Norbert Steinmetz 17 Korovkin-type Approximation Theory and its Applications, Francesco Altomare and Michele Camp iti 18 Quantum Invariants of Knots and 3-Manifolds, Vladimir G. Turaev 19 Dirichlet Forms and Symmetric Markov Processes, Masatoshi Fukushima, Yoichi Oshima, Masayoshi Takeda 20 Harmonic Analysis of Probability Measures on Hypergroups, Walter R. Bloom and Herbert Heyer 21 Potential Theory on Infinite-Dimensional Abelian Groups, Alexander Bendikov

Vladimir E. Nazaikinskii • Victor E. Shatalov • Boris Yu. Sternin

Methods of Noncommutative Analysis Theory and Applications

w

6

Walter de Gruyter Berlin • New York 1996

Authors.

Vladimir E. Nazaikinskii Moscow State Institute of Electronics and Mathematics Technical University 3/12 B. Vuzovskii per Moscow 109028, Russia

Victor E. Shatalov, Boris Yu. Sternin Department of Computational Mathematics and Cybernetics Moscow State University Vorob'evy Gory Moscow 119899, Russia

Series Editors

Heinz Bauer Mathematisches Institut der Universitdt BismarckstraBe 1 1/2 D-91054 Erlangen, FRG

Jerry L. Kazdan Department of Mathematics University of Pennsylvania

Eduard Zehnder

209 South 33rd Street

ETH-Zentrum/Mathematik RdmistraBe 101 CH-8092 Zürich

Philadelphia, PA 19104-6395, USA

Switzerland

1991 Mathematics Subject Classification. 35-02; 22 Exx, 44-XX, 47-XX, 81R50 Keywords Functional analysis, partial differential equations, Fourier integral operators, asymptotic theory, representation theory, Yang-Baxter equations, quantum groups

0 Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability Library of Congress Cataloging-in-Publication Data

Methods of noncommutative analysis : theory and applications / Vladimir E. Nazaikinskii, Viktor E. 'S. hatalov, Boris Yu. Stemin p. cm. — (De Gruyter studies in mathematics ; 22) Includes bibliographical references and index. 1. Geometry, Differential 2. Noncommutative algebras 3 Mathematical physics. I. Shatalov, V. E. (Viktor Evgen'evich) II Stemin, B. Yu. III. Title. IV. Series. QC20.7.G44N39 1996 95-39641 515'.72—dc20 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication Data Nazajkinskij, Vladimir E.: Methods of noncommutative analysis : theory and applications / Vladimir E. Nazaikinskii ; Victor E. Shatalov , Boris Yu. StemM. Berlin ; New York : de Gruyter, 1996 (De Gruyter studies in mathematics ; 22) ISBN 3-11-014632-0 NE: atalov, Viktor E.:; Stemin, Boris J.:; GT

-

C) Copyright 1995 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typeset using the authors'TeX files: I. Zimmermann, Freiburg — Printing. Gerike GmbH, Berlin. Binding: Dieter Mikolai, Berlin. — Cover design: Rudolf Hübler, Berlin.

Preface

Noncommutative analysis, that is, the calculus of noncommuting operators, is one of the main tools in contemporary mathematics. Indeed, the theory of differential and pseudodifferential operators, various problems of algebra, functional analysis, and theoretical physics deal with functions of noncommuting operators. This was clearly understood by such outstanding scientists as H. Weyl, I. Schur, R. Feynman and many others. It is therefore not surprising that the development of mathematics required creating noncommutative analysis clear and convenient in applications. R. Feynman was clearly a pioneer in the field. As early as 1951, he noticed in his paper "An operator calculus having applications in quantum electrodynamics", that noncommutativity of operators can be accounted for by introducing numbers or indices showing the order of action of the operators. This apparently simple remark served ,. as a starting point for the noncommutative operational calculus created by V. Maslov in the early 70's. Maslov's ideas have been developed in numerous papers, and very deep and important results have been obtained. In particular, they show that noncommutative operator calculus is intimately related not only to traditional mathematical and physical theories but also to rapidly developing new ones. Thus, noncommutative analysis occurs to be useful in such mathematical fields as the theory of geometric and asymptotic quantization, representation theory, the theory of quantum groups, and so on. In this book the reader will find a lot of examples from functional analysis, algebra, representation theory, and the theory of differential equations where non commutative analysis is involved. Unfortunately, up to now there does not exist a sufficiently simple exposition of noncommutative analysis which might serve as an introduction to the subject for scientists who are just beginning to get acquainted with this area and that is why this book has been written. It is primarily addressed to those who are not specialists in noncommutative analysis. At the same time, even the experienced mathematician can find in this book many new and interesting topics. Noncommutative analysis gives a new outlook both on quite traditional and modern mathematical topics such as representation theory, operator theory, the theory of (pseudo)differential operators, Yang—Baxter equations and others.

vi Acknowledgements. We express our kind gratitude to Professor Victor P. Maslov, whose great influence inspired our work and whose advice was of much use for us. This book has been written under the support of the Chair of Nonlinear Dynamic Systems and Control Processes, Moscow State University, and of Max-PlanckArbeitsgruppe "Partielle Differentialgleichungen und Komplexe Analysis", Institut fiir Mathematik, Universitdt Potsdam. We are thankful to the heads of these Departments — Professor Stanislav V. Emel' yanov and Professor Bert-Wolfgang Schulze. We are also grateful to Dr. Manfred Karbe, whose advice was truly invaluable. Finally, we are cordially thankful to Mrs. Helena R. Shashurina, who prepared the manuscript for the publishers. Moscow — Potsdam, 1994

The Authors

Contents

Preface I

v

1 Elementary Notions of Noncommutative Analysis Some Situations where Functions of Noncommuting Operators Arise 1 1 Nonautonomous Linear Differential Equations of First Order. 1.1 T-Exponentials 1 1.2 Operators of Quantum Mechanics. Creation and Annihilation 4 Operators 7 Differential and Integral Operators 1.3 Problems of Perturbation Theory 10 1.4 14 1.5 Multiplication Law in Lie Groups 16 1.6 Eigenfunctions and Eigenvalues of the Quantum Oscillator 20 T-Exponentials, Trotter Formulas, and Path Integrals 1.7 2 Functions of Noncommuting Operators: the Construction and Main 23 Properties Motivations 23 2.1 The Definition and the Uniqueness Theorem 26 2.2 33 Basic Properties 2.3 Tempered Symbols and Generators of Tempered Groups . 42 2.4 The Influence of the Symbol Classes on the Properties of Gen2.5 erators 45 48 2.6 Weyl Quantization 51 Noncommutative Differential Calculus 3 52 The Derivation Formula 3.1 54 The Daletskii—Krein Formula 3.2 55 Higher-Order Expansions 3.3 60 Permutation of Feynman Indices 3.4 66 The Composite Function Formula 3.5 70 The Campbell—Hausdorff Theorem and Dynkin's Formula 4 70 Statement of the Problem 4.1 71 The Commutation Operation 4.2 74 4.3 A Closed Formula for in (e B e A ) 77 A Closed Formula for the Logarithm of a T-Exponential . . 4.4 Summary: Rules of "Operator Arithmetic" and Some Standard Tech5 84 niques

viii

CONTENTS

5.1 5.2 5.3

Notation Rules Standard Techniques

85 86 87

II Method of Ordered Representation 94 1 Ordered Representation: Definition and Main Property 94 1.1 Wick Normal Form 94 Ordered Representation and Theorem on Products 1.2 97 Reduction to Normal Form 1.3 99 Some Examples 2 104 2.1 Functions of the Operators x and —iha/ax 105 2.2 Perturbed Heisenberg Relations 107 Examples of Nonlinear Commutation Relations 2.3 108 2.4 Lie Commutation Relations 110 Graded Lie Algebras 2.5 115 Evaluation of the Ordered Representation Operators 117 3 Equations for the Ordered Representation Operators 3.1 117 How to Obtain the Solution 121 3.2 3.3 Semilinear Commutation Relations 125 4 The Jacobi Condition and Poincaré—Birkhoff—Witt Theorem 132 4.1 Ordered Representation of Relation Systems and the Jacobi Condition 133 4.2 The Poincaré—Birkhoff—Witt Theorem 138 Verification of the Jacobi Condition: Two Examples 4.3 142 5 The Ordered Representations, Jacobi Condition, and the Yang—Baxter Equation 144 6 Representations of Lie Groups and Functions of Their Generators . . 156 6.1 Conditions on the Representation 156 Hilbert Scales 6.2 158 Symbol Spaces 6.3 161 6.4 Symbol Classes: More Suitable for Asymptotic Problems . 167 III Noncommutative Analysis and Differential Equations 171 1 Preliminaries 171 1.1 Heaviside's Operator Method for Differential Equations with Constant Coefficients 174 1.2 Nonstandard Characteristics and Asymptotic Expansions . . 179 182 Asymptotic Expansions: Smoothness vs Parameter 1.3 1.4 Asymptotic Expansions with Respect to an Ordered Tuple of 185 Operators 186 1.5 Reduction to Pseudodifferential Equations 1.6 Commutation of an h -1 -Pseudodifferential Operator with an Exponential 189

CONTENTS

ix

1.7

2

3

4

5 6

7

Summary: the General Scheme 191 Difference and Difference-Differential Equations 193 2.1 Difference Approximations as Pseudodifferential Equations 194 2.2 Difference Approximations as Functions of x and 8x± 196 2.3 Another Approach to Difference Approximations 198 Propagation of Electromagnetic Waves in Plasma 200 3.1 Statement of the Problem 200 3.2 The Construction of the Asymptotic Expansion 202 3.3 Analysis of the Asymptotic Solution 205 Equations with Symbols Growing at Infinity 208 4.1 Statement of the Problem and its Operator Interpretation . 208 4.2 Asymptotic Solution of the Symbolic Equation 210 4.3 Equations with Fractional Powers of x in the Coefficients . 212 Geostrophic Wind Equations 216 Degenerate Equations 224 6.1 Statement of the Problem 224 6.2 Localization of the Right-Hand Side 225 6.3 Solving the Equation with Localized Right-Hand Side 229 The Asymptotic Solution in the General Case 6.4 232 Microlocal Asymptotic Solutions for an Operator with Double Characteristics 233

IV Functional-Analytic Background of Noncommutative Analysis Topics on Convergence 1 What Is Actually Needed'? 1.1 Polynormed Spaces and Algebras 1.2 1.3 Tensor Products 2 Symbol Spaces and Generators 2.1 Definitions 2.2 S" Is a Proper Symbol Space S"-Generators 2.3 3 Functions of Operators in Scales of Spaces Banach Scales 3.1 S"-Generators in Banach Scales 3.2 3.3 Functions of Feynman-Ordered Selfadj oint Operators

242 242 242 249 257 260 260 263 268 270 270 273 278

Appendix A. Representation of Lie Algebras and Lie Groups 287 Lie Algebras and Their Representations 287 1 . Lie Algebras, Bases, Structure Constants, Subalgebras 1.1 287 Examples of Lie Algebras 1.2 288 Homomorphisms, Ideals, Quotient Algebras 289 1.3 Representations 1.4 290 The Associated Representation ad. The Center of a Lie Algebra 291 1.5

x

CONTENTS

1.6 1.7

2

3 4

The Ado Theorem 291 Nilpotent Lie Algebras 292 Lie Groups and Their Representations 292 2.1 Lie Groups, Subgroups, the Gleason—Montgomery—Zippin Theorem 292 2.2 Examples of Lie Groups 293 2.3 Local Lie Groups 294 2.4 Homomorphisms of Lie Groups, Normal Subgroups, Quotient Groups 294 Left and Right Translations. The Haar Measure 295 Left and Right Regular Representations 3.1 295 Representations of Lie Groups 3.2 296 297 The Relationship between Lie Groups and Lie Algebras 4.1 The Lie Algebra of a Lie Group 297 Examples 298 4.2 The Exponential Mapping, One-Parameter Subgroups, Coor4.3 dinates of I and II Genera 299 Evaluating the Commutator with the Help of the Mapping exp 301 4.4 Derived Homomorphisms 302 4.5 Derived Representation 4.6 303 The Lie Group Corresponding to a Lie Algebra 4.7 305 The Krein—Shikhvatov Theorem 4.8 307

Appendix B. Pseudodifferential Operators Elementary Introduction 1 2 Symbol Spaces and Generators Pseudodifferential Operators 3

311

Glossary

327

Bibliographical Remarks

351

Bibliography

357

Index

371

311 317 321

Chapter I

Elementary Notions of Noncommutative Analysis

1 Some Situations where Functions of Noncommuting Operators Arise In this section we consider a few examples of problems from different areas of mathematics and physics whose study requires the usage of functions of noncommuting operators. In fact, each of these problems can be studied by its own inherent methods, but if considered as a whole they suggest that there should be a universal apparatus that would permit us to consider them all from a common point of view. Indeed, such an apparatus is already developed. It is called noncommutative analysis; the aim of his section is to illustrate its main features by simple examples and, in particular, to introduce the so-called Feynman indices, which play the main role in the machinery of noncommutative analysis. It should be clearly noted that the examples given in the following were not chosen at random. In the course of our exposition we will return to them repeatedly and show how the ideas and methods of the theory developed work in simple situations.

1.1 Nonautonomous Linear Differential Equations of First Order. T-Exponentials The equation X = A(t)x

can easily be integrated in quadratures if x function:

E

R1 and A(t) is a given continuous

t

x(t) = exp (

fo

A(t) dt) x(0).

No more complicated is the case in which x A (e) commute with each other:

E

Rn but the n x n matrices A (t) and

[A(t), A(e)] tf A(t)A(t I ) — A(t I)A(t) = 0,

for any t, t'.

(I.1)

(I.2)

I. Elementary Notions of Noncommutative Analysis

2

The solution is expressed by the same formula, where exp stands for the matrix exponential, defined as the sum of the convergent series exp(B) =

a° Bn E . n!

n=0

Indeed, under condition (I.2) we have ( t+At

f 0

exp

_

A(t) dt)

t

t+At

— exp (f A(t) dt) 0

exp (

f t

A(t) dt)

(since the arguments of the exponentials commute with each other, one can expand the exponentials into Taylor series and repeat verbatim the proof of the identity ea eb = ea+b valid for the case in which a and are numbers). Since

b

( t+At

f 0

exp

t+At

A(t) dt --= 1

+f t

A(t) dt +

it is easy to differentiate x (t) with respect to t and prove that x (t) satisfies the original equation.

However, the simple formula fails if we do not require that the commutator (1.2) is equal to zero. In general, we can only write down the solution in the form of the following limit: x(t) = lim exp(ANAtN) exp(AN-1 AtN_i) .. . exp(Ai Ati) x(0) 1 N—>.00

max Ati —>.0 i

where

0 = to < ti < • • • < tn = t ,

Ati = ti = ti — ti—i

and Ai = A (0i ) for some Oi E [ti_i , tit In physics literature the limit on the right-hand side of the latter equality is called a T -exponential (see, e.g., [SS]) and is sometimes t

denoted by T-exp (J A (r) dr x(0). Generally speaking, 0

t

t

T- exp ( / A(r) dr x(0) 0 exp 0

f A(r) dr 0

since the identity exp(A) exp(B) = exp(A ±

B),

which could be used to represent the expression under the limit sign as the exponential of an integral sum, is not valid if [A, B] 0 0, that is, if A does not commute with B. However, Feynman invented a fairly simple notation enabling us to sidestep this

1. Some Situations where Functions of Noncommuting Operators Arise

3

difficulty and, in a sense, pay no attention to the fact that the matrices A(t) do not commute with each other for different values of t. Namely, we equip matrices with indices prescribing their arrangement in products: the greater the index, the further to the left stands the corresponding matrix. In writing, these indices will be placed over the letters denoting matrices; for example, 12

AB = BA, 1

12

2,

22

12

(A -I- Br = A +2AB ± B = A2 ± B 2 + 2B A, 2 „ 1 (A + B)' = A" + 3BA2 ± 3B 2 A ± B 3 , etc. In this notation, we see that 1 1 2

exp(A ± B) =

E

(A

2

+ B)n n!

1 2 = exp(A) exp(B) = exp(B) exp(A).

Note that the order of the factors exp(Ai Ail) in the product coincides with the order of the points Oi on the real axis. Hence we can rewrite x(t) as X (t) =

el ON lim exp(Ai Ati + • • • + AN AtN) X (0).

N—).00 max Ati—>.0 i

Formally, the argument of the exponential is the integral sum of some integral; let us write down this integral explicitly:

t

e x(t) = exp (f A(9) dO) x(0). 0

Note that equation (I.1) is a special case of the last equation. Indeed, if {A (t)} is a family of commuting matrices then their arrangement in products is irrelevant, and the index 19 over A(0) may as well be omitted. As a result, we obtain formula (I.1). At first it may seem that the expression on the right-hand side of the last equation is no more than yet another notation for the T-exponential. But this is a false impression: its meaning can be defined t

directly as the result of substituting the family M (t )} of matrices ordered by the parameter

argument y(t) into the functional

t for the

t F [y] = exp

(f y(r) dr) . 0

Thus we have introduced indices over operators (in the example considered these operators are matrices, which, in fact, does not matter). They will be referred to as Feynman indices, in honor of their inventor.


4

Let us sum up their simplest (and evident) properties. (i) Indices over operators indicate their arrangement in products; namely, an operator with lower index always stands to the right of an operator with greater index. Thus, it is the order relation between the indices that counts, not the values of the indices themselves (which may be changed without affecting the operator expression). (ii) The mutual order of indices over commuting operators plays no role at all. In particular, if all operators involved in an expression commute with each other, then the indices can be omitted.

1.2 Operators of Quantum Mechanics. Creation and Annihilation Operators In quantum mechanics, the state of a system of particles is described by the wave function T, which is a normalized element of a Hilbert state space 7-1; observables correspond to Hermitian operators in 7-I. The quantity A = (T, AT), where ( • , • ) is the inner product in 7-1, is called the expectation value of the observable A in the state T, and (AA) 2 = (T, (A — A) 2 T) is called the variance of A in the state T. One says that A has a definite value a in a state T if A T = a T, that is to say, T is an eigenvector of A with eigenvalue a. It is easy to see that in this case A = a and (AA) 2 = O. The converse is also true: if (AA) 2 = 0, then

I I A tli — A ti/ 11 2 = ((A — A) tli, (A — A) 40 = ( W, (A — A) 2 '1 ) = 0 (recall that A is a Hermitian operator), so that A T = AT. Heisenberg 's Uncertainty Principle claims that the variances of two observables A and B corresponding to canonically conjugate variables in classical mechanics (such as the position q i and the dual momentum pi) satisfy the uncertainty relation (AA) (AB) > h, where h is Planck's constant. In particular, the observables A and B cannot be measured "simultaneously", that is to say, they cannot have definite values in some state T simultaneously. Indeed, the Uncertainty Principle implies that if AA = 0, then AB = oo. Thus, A and B have no common eigenvectors. On the other hand, commuting Hermitian operators always have a complete system of common (generalized) eigenvectors. We conclude that A and B do not commute with each other, and the algebra of observables is not commutative.


5

In the Schro5dinger representation the states tif are functions of the physical coordinates x = (x 1 , . . . , x 3N), where N is the number of particles in the system, kii = W(x) e '1-1 = L2(R3N ) (we assume that the particles have no internal degrees of freedom). The observables X i associated with the coordinates x i are merely the operators of multiplication by x i , X i = x i , whereas the observables Pi associated with the momenta pi are differentiation operators, pi = — ih a lax i . Moreover, if some classical variable F has the form of a function F = F(x, p), then the associated quantum observable is given by

fr = F(X, P)t-f F (x, —ih

a

axi

) •

(1.3)

In fact, the quantization procedure F iis ambiguous and the last formula is only valid modulo lower-order terms with respect to h. We point out that this formula itself is ambiguous: since the operators x and — ihalaxi do not commute with each other, one should fix their ordering, e.g., by equipping them with Feynman indices. However, the said ambiguity can be "hidden" in some concrete problems. For example, if H is a Hamiltonian of the form N

2 Pi

H(x, p) = E - ± V (x),

2m

then, due to its additive structure, the choice of Feynman indices of

x and

— ih

a

.

axi

is unimportant, and the energy operator N h2 if = H(x, p) = — —Ai + V(x) i=1 2m

E

(where Ai is the Laplacian w.r.t. the coordinates of the i th particle) is defined uniquely 1 . Relativistic quantum mechanics deals with systems with a variable number of particles. Here the so-called occupation number representation is convenient. Let us consider it in the simplest model version. Suppose that there is only one type of particle in the system considered, and each particle can occupy one of n distinct basis states2 . Let us represent the state space 7-t as the Hilbert sum of one-dimensional subspaces -

7-1 =

-

.ii,...,ino

1 The picture would be much more complicated should we consider the quantization in "generalized"

coordinates and momenta. 2 1n realistic systems the number of possible states, as a rule, is infinite.


6

where 7-1h ,..., in is the space of states such that exactly jk particles occupy the kth basis state, k = 1, .. . , n. The numbers ji , . . . , in are referred to as occupation numbers. The space No = H0,...,0 is called the vacuum subspace and corresponds to the state with no particles at all. The structure of the state space is known to depend on the spin of the particles. Namely, there are two possibilities: if the spin is integral, then the particles obey Bose—Einstein statistics, that is, each state can be occupied by an arbitrary number of particles; if the spin is half-integral, then we have Fermi—Dirac statistics, that is, at most one particle can occupy each given state. In the first case, the sum is taken over all nonnegative ji, . . . , in , whereas in the second case the sum is finite and extends over ji G {0, 1 } . According to the type of statistics, the particles are referred to as bosons or fermions. To each of the n basis states there corresponds a creation operator aiE and an annihilation operator ak— , the adjoint of aiE . These operators "create" and "destroy" particles in the kth basis state, that is,

4-c N11—lk....in C Nil—A -FL-in , ak— N11-1k—jn C N .i1-1k -1 —in (it is assumed that H11 ....in = {0} provided that at least one of the indices jk is negative or, in the case of fermions, greater than 1). The operators aP and ak— satisfy the following commutation relations: [at, ail+ = [ai, aï]± = 0, [at, aï 1J+ = SO I.

Here

8k1

is the Kronecker delta, I the identity operator, and [A, B]= AB ± BA

the (anti)commutator of operators A and B (the upper sign is taken for fermions and the lower for bosons). The Wick normal form of an operator A acting on the space H is its representation in the form A = E C ai , ... , cen Ph-4n (at) al . . . (a;ban (aT)fil .. . (a , /3)

(an- )fin ,

that is, the representation in which all creation operators in each monomial stand to the left of all annihilation operators. The Wick normal form is very convenient, e.g. for evaluating vacuum expectations (the expectations in the vacuum state To E No, I I To I l = 1). Indeed, we have

( 'I's , Alpo)

=

E cc,i,...,cinp.,...,finc(at)ai ... (a n-F ) n To, (al ... ( a,-,- )Pn To)

=

(01, P) co...(),


7

since all terms with (a, fi) 0 (0, 0) vanish. Clearly, any polynomial (series) of creation and annihilation operators can be reduced to the Wick normal form by permuting all creation operators to the left with the help of the commutation relations. From the viewpoint of noncommutative analysis, the Wick normal form of an operator A is none other that its representation as 2

2

1

1

21 +

def

= f (a a– )

(the Feynman indices were assigned taking into account that the creation operators commute with each other, as do the annihilation operators). The function f (z, w) = f (z i , . . . ,z, w 1 , . . . , wn)

=E

7

al

7 an „f3 1 " • 4-n uv 1 • "

wn n

(c1,13)

is called the Wick symbol of the operator A [11]. It is easy to check that the Wick symbol is unique. In the case of fermions the creation operators (and the annihilation operators) no longer commute with each other, so they all should get different Feynman indices: 2n

1

n+1 n

The problem of calculating the Wick normal form of an operator acting on 7-1 will be considered in Subsection 1.1 of Chapter II.

1.3 Differential and Integral Operators The theory of linear partial differential equations deals with differential operators of the form a

P=

E a(x) (—axa )

9

iceinz

where X E EV, a = (a1 , ... , an ) is a multi-index, and m is the order of P. Obviously, the operator P can be interpreted as a function of the operators xi and alaxj, j = i „...,n: 1)

2

P = f (x, where

f (x, ) =

a

ax

E ace (x) e . lal - In

8


For technical reasons (since —ia/ax is a self-adjoint operator in L2(EV)), the operator P is usually represented in a somewhat different form, namely

1

. a

(2

P = g x, —t —) , ax where

g(x, p) =

E aa (x)(ip)'. icii'm

2

Thus, a differential operator is a function of the Feynman-ordered operators x and 1 —id/8x with symbol a polynomial in p. A solution (or almost-solution) of a differential equation

Pu=f can often be represented in the form of some integral operator applied to f. For example, if P is an elliptic operator, then the solution modulo smooth functions can be represented via a pseudodifferential (singular integral) operator,

u(x) = f K(x, x — y) f(y)dy (see, for example, [7911). If P is a hyperbolic operator, then the solution can be represented via a Fourier integral operator (see [81], [137], and others). Let us show, formally for now, that the pseudodifferential operator can be represented as a function 1 2 of the Feynman-ordered operators x and —ia/ax. To this end, we make the change of variables y = x ± r in the integral and obtain

u(x) = f K(x, — r) f(x ± r)dr = f K(x, — r)etalax f (x) dr,

where et a /ax is the operator of translation by r, which can be defined as the solution U of the operator Cauchy problem

au au ar = ax'

Ui r=0 = id .

Set

f

H(x, p) = K(x,—r)e uP dr.


9

Then 1

u (x) =

dr f (x) = H 1, —i

2n-

— 8

I f (x),

where = (2n- )n K (x , — r) is the Fourier transform of H (x , p) with respect to the variable p. The properties of the kernel K(x, x — y) imply that H(x, p) is a homogeneous function of p smooth for p O. We also have 1

H

(2

x,

a

f 11(x, Oe i“x— Y ) f (y)dy

f (x) =

Dx

Fourier integral operators admit a similar representation. Let us illustrate this fact on a simple example. Suppose that the Fourier integral operator has the form

43.f = f e"))a(x, p) f (13) dP,

(1.4)

where (p) is the Fourier transformation of the function f(x) (such operators give, in particular, an almost-solution of the Cauchy problem

I

2

: i auf = MX, —I" aX \14 )'

ult=0 = f (x),

1 2 for small values of t; here H(x, — ialax) is a first-order pseudodifferential operator). The operator (1.4) can be represented in the form f = f K(x, x — y) f (y)dy, where

K 1 , z) = Fp,z {e i(1)(x'P) a(x, p)}, (I) (x , p) = S(x, p) — xp, and F- z is the inverse Fourier transform. Now computations similar to those above show that

i)f = ei cp (x2 ,—ialax) a x2 , __ i

(

(

f.


10

1.4 Problems of Perturbation Theory Let A and B be two noncommuting operators. We will consider a function f (A ± 8B) of the operator A ± 8B, where e is a small parameter, and try to expand it in powers of B. This is a typical problem of perturbation theory; the operator A is considered as the operator of the nonperturbed problem and EB as a small perturbation. For example, consider the operator Cauchy problem du

dt

= (A + e B) u, ult=0 = E

(1.5)

(here E is the identity operator). Suppose that the solution of the unperturbed problem (s = 0) is known. Then the "corrections" for small s can be found by expanding the function f (A ± EB), where f (x) = etx , in powers of s.

If the operators A and B commute, the solution of this problem is well known. It is given by the Taylor formula 82

f (A ± 8B) = f (A) + 813 J.' (A) + — B 2 f" (A) ± • • • . 2 However, this equation fails if [A, B] 0 0. Indeed, consider the simplest example in which f (x) = x2 . We have f (A + 8B) = A2 ± 8(AB ± BA) ± whereas

82

f (A) ± e B f i (A) ± — B 2 f " (A) = A2 ± 28B A + 2 and these two expressions differ by 8 [A , B]. Hence there is the problem of giving a counterpart of the Taylor expansion for noncommuting A and B. Computing the derivative d f (A + 813) de would be a first step in that direction (this provides the desired expansion with the accuracy 0 (8 2)); let us compute this derivative for the particular case in which f (x) = (X — x) -1 (that is, we consider the perturbation theory for the resolvent RA (A)). The following resolvent formula is valid (see, e.g., [1971) (X — C) -1 — (X — Art = (X — C) -1 (C — A)(X — A) -1 (this can easily be checked by multiplying both sides of the last equation by (X — C) on the left and by (X — A) on the right). By inserting C = A + 8B in this formula, we obtain f (A ± 8B) - f (A) = (X - A - 813) -1 8B (X. - A)-1.


11

Hence, it is clear that 2

d f (A ± eB)][— ds

= (X

B

A) l B (X — A) -1 =

6=0

3

1.

(X

Obviously, the same "naive" method permits us to calculate higher-order derivatives with respect to 8; however, we refrain from doing so, since our computation uses the particular form of the function f heavily and is by no means universal. The general formulas of perturbation theory for f (A ± EB) will be given in Section 3 and now we consider yet another example. Let us compute the expansion in powers of 8 of the exponential e A+8B (recall that this expansion is the perturbation theory series for the solution of the Cauchy problem (I.5) at t = 1). For simplicity, we assume that A and B are bounded operators on a normed space. Thus, we intend to find the coefficients Pi, P2, . . . of the expansion e A-1-03 = e A ± s pi ± 82p2 +....

(I.6)

We will use the extraction formula for T-exponentials [54]:

t t t T-exp [f (A(r) ± B(r)) dri= T-exp (f A(r) dr) T- exp (i C(r) dr) 0 0 0

?

where C(r) is the operator family given by the formula

—1 t t C(t) = [T- exp (f A (r) dr)1 B(t) T - exp (f A (r) dr) ,

t that is, C(t) is the conjugate of B(t) by the operator T- exp ( f A(t) dr).

0

Proof The proof is quite simple. For brevity, rewrite the desired formula as UA-FB

(t) = UA (t)Uc (t).

Both sides are equal to the identity operator if t = O. We can differentiate both sides with respect to t, using the definition of a T-exponential (see Subsection 1.1). We obtain

a

—

(uA +B(0) = (A(t) ± B(t)Wit+B(t)

8t

and

a — at (LIA(ouc (0) = mouA(ouc(t)+uA(t)c(ouc(t) = (mt)+B(ow Awuc(t),


12

since

C(t)

=

UA(t) -1 B(t)UA(t).

This implies immediately that Uit+B(t) = UA(t)Uc(t) for all t.

0

We choose A(t) and B(t) to be constant families of operators, A(t) B(t) ..- EB, and obtain the formula

a.-

A and

1 eA-FeB = eA T - exp

(C(t) dt ,

E

o where

C (t) = e —rA Be ta .

As in Subsection 1.1, we rewrite the T-exponential using Feynman indices and expand it into the Taylor series in powers of E:

1 T - exp

(e f C(r) dr) —

exp

(e 1 Cl(r) dr)

0 1

0 r

= 1 ± s f C(v)

62

(Pr + — 2

0

2

1

(f

Cl(r) dr)

±

•

o

The coefficient of E on the right-hand side is equal to

1

1

f0 C(r) dr = 0

e —r A B e" dr

(since in the coefficient of E all C(r) enter additively and are not multiplied by one another, their Feynman indices play no role and can be omitted). Let us calculate the coefficient of 62 • We have (1

2

1

1

11

0 r f C (r) dr = f C(r) dr f C (0) de = f f C (r)C (0)

0

0

r

o

0

00

The order of factors in the integrand is governed by the sign of the difference r — 0: we have r 0 C (r)C (0) = C(r) C(0) for r > O, and

(0) = C (0) C (r) C (r)C


13

for r < 0, that is, the factor with the smaller argument always stands to the right of the other factor. We divide the integration square [0, 1] x [0, 1] into two triangles r > 0 and r < B and note that both of them give the same contribution to the integral. We retain only one of these triangles. Then the factor 1/2 cancels out, and we obtain the following expression for the coefficient of s2 : r f / C(v)C(0) 1( 0 0

= 1 f e– "Be ("")A Be eA dO 0

dr.

0

Finally, we find that

e A+cB = eA

, -r

1 8 f e (1--r)A B e r Ad.(

0 1

±

62 f 0

( t

f e (1—r)AB e(r-6)A Be 9.4 —a

dr ± • •

0

Thus, we have calculated the coefficients Pi and P2 of the expansion (1.6) for e ll-FEB . Using Feynman indices again, we can write 1 3 1 2 --r)A-FrAdr, Pi = B f e(1 o I ( r 3 1 5 24 f (1–t)A-F(r--0)A-FOA de d r. e = BB 12 f 0

0

It is easy to predict the formulas for P3, P4, etc., but we leave that to the reader. We remark that the choice of two above examples is not accidental. Actually, the exponent et At plays an important role in the definition of functions of noncommuting operators (see Subsection 2.4 below) for tempered symbols. Similarly, as is well known, the resolvent RA (A) of the operator A can be used for the definition of f (A) for symbols f (A) analytic in a neighborhood of the spectrum of the operator A. These examples are also of use when considering Weyl quantization (see Subsection 2.6 of this Chapter).


14

1.5 Multiplication Law in Lie Groups The extraction formula can be read from right to left:

t T - exp (f A (t) dt)T

t t - exp (f /3 (r) dr) T - exp (f C(t) dt) , o

O

o

where r

r

--1

C(t) = A(t) ± T exp (f A(r) dr) B(t) [T exp (f A(r) dr)1 -

-

0

0

(we have changed the notation slightly so as to show some respect for the alphabetic ordering). In this form it has the following meaning: the product of two T-exponentials is again a T-exponential. Furthermore, if [A(t), B (0] = A(t)B(t) — B(t')A(t) = 0 for all t and t', then the exponent C(t) has the form C(t) = A(t) ± B(t) (indeed, we

t can permute B(t) and T exp f A(t) dr , which follows easily by considering the -

0

passage to the limit (see Subsection 1.1) or the corresponding Cauchy problem). A and B(t) B. Now let A (t) and B(t) be constant families of operators, A(t) Then for t = 1 we obtain

1 eA eB = T exp ( f C(t) dt), -

0 where

C(t) = A + e tA Be —tA .

Thus, the product of two "common" exponentials of operators is represented as a T exponential. There arises the natural question whether this product can be represented as a "common" exponential, that is to say, whether one can find an operator C such that 21 e A+B C=e . (I.7) -

This is an important problem in the theory of Lie groups. It was partially solved already in the XIX century (see [19 1 ). Namely, it was shown that the element C can be expressed as the series 1 C=A±B±—[A,B]+•••

2


15

convergent for small l'All and IIBII, where the dots stand for the sum of commutators of the operators A and B of order > 2 (the order of a commutator is the number of left (or right) commutation brackets it contains). Let A and B be elements of some matrix Lie algebra; then eA and e B are elements of the corresponding local Lie group realized by matrices, and the above relations mean that the multiplication law in a Lie group is uniquely determined by the commutation operation in its Lie algebra. This is just the statement of the famous Campbell–Hausdorff theorem. An explicit expression for all terms of the series was found about half a century later by E. B. Dynkin [43], who showed that

(_ om-2

co

C

.E

M

m=1

k1+li>1,ki,li>0

where [Si

def

,

.

.

.

,

[Aki , B/1 , . . . , Ak m

z

ki !ii ! . . .

,

km !Im !

Bi n

,

]

'

1

SN ] = — [Si, [S2 . . .[SN–i, Sri] . . .l. N

In Section 4 we obtain in an elementary way, in the framework of noncommutative analysis, a closed formula expressing the operator C via A, B and the commutation operation. Let us use the considered problem to make a comment on the notation used. We 21

seek an element C such that "C = In D, where D = e A+B " , It would be desirable to convert the phrase in quotes into a formula. Unfortunately, we cannot write 21

C = ln(eA ±B ), since this means the following: "take the function f (x, y) = ln(ex+Y) and substitute 2

1

A, y 1--> B into it". Taking into account that ln(ex+Y) = x + y, we would obtain C = A + B. In order to solve the arising problem we extend our notation by introducing the so-called autonomous brackets [[11. We write X 1-+

2 1 C = ln([e A+B ]l); the meaning of this notation is as follows: "calculate the operator in the brackets [[ ][ and forget about Feynman indices occurring inside [H]; instead, use the resultant operator in the subsequent computations as an indivisible entity". The introduction of autonomous brackets is motivated by the simple observation that operator expressions involve, in addition to traditional arithmetic operations and function evaluation, only one new operation, namely, substitution of operators (equipped with Feynman indices) instead of numerical arguments. And we do not have any convenient means to denote recursive application of this operation. Recall that the order of computation in arithmetic expressions is governed by brackets. We just introduce here a new type of brackets, responsible for substitution of operators.


16

1.6 Eigenfunctions and Eigenvalues of the Quantum Oscillator Let us now use a well-known problem of quantum mechanics to show how the above notation works. Let H be the energy operator of the one-dimensional quantum oscillator, 2 = 1

(D

ax

2x21

where h is the Planck constant, co > 0 the frequency of the oscillator, and x e R. Consider the eigenvalue problem for I1 in the space L2 (R 1 ):

!IT (x) = E I (x) ,

tlf

E L2(R 1 ).

One should find the values of E for which this equation has nontrivial (nonzero) solutions W. It is easy to check that H can be represented in the form

= hco (a+ a— ± 1/2), where

a

a+ = (2hw)-112 (-ih— ±i cox) ax

are the creation and annihilation operators (it is not mere chance that this term coincides with the one already introduced in Subsection 1.24 this is related to the decomposition of a free electromagnetic field into oscillators, frequently used in quantum theory). Thus, our problem is reduced to the eigenvalue problem for the operator 21

a +a = a+ a: 21

a+ a- ‘11 (x) = Xtlf (x)

We seek the solution in the form

1 W(x) = G (a + , a- )v(x), 2

1

(I.8)

where v(x) is an arbitrary function and G( , y) is an unknown symbol to be defined. 21

2

1

It suffices to require that the product of the operators a+ a— and G (a+, a— ) be equal 2

1

to A.G (a+, a). Using the introduced notation, we can rewrite the cited product in any of the following forms: 21

2

1

43

2

1

2

1

a + a ] [[G(a+ , a- )11 = a+ aG (a+ , a- ) = a+ a- [G (a+ , a- )]] 321

2

= a+ a- [[G(a+ ,

1

= BA,

1. Some Situations where Functions of Noncommuting Operators Arise 2

17

1

where B = a+ a- and A = G (a+, a- ). (The index over the left autonomous bracket in the last case is the Feynman index to be assigned to the operator obtained by computing the expression inside the autonomous brackets). We obtain the equation 2

1

2

1

a+ a- [[G (a+ , a- )]] = A.G (a+ , a- ) 2

(I.9)

1

for the operator G (a+ , a- ) . Our plan is to reduce this equation to an equation with respect to the symbol G , y). 2 1 To this end, we should represent the left-hand side as a function of a+ and a- . Should a+ and a- be arbitrary operators, such a representation would be impossible. However, a+ and a- satisfy the commutation relation clef

+

_

[a+ , a] = a a - a -+ a = -1, which enables us to compute the symbol of the product on the left-hand side of (I.9). 2 1 Compute first the product W_ = al[G(a+, a- )]1. Here we apply a trick standard in "operator arithmetic". Clearly, we have 3

2

1

W_ = a- G(a+, cr), 2

3

and the problem is to permute a+ to the last place so as to identify the arguments a1 and a- . Note that if G(y,, y) were a linear function, this problem could be solved immediately by applying the commutation relation, which can be rewritten in the form 3

21

- a- ) = 1. If G( , y) is a polynomial in , one could apply induction on the order of G(, y). In fact, we can avoid this cumbersome procedure and obtain the result for general symbols as follows: 3

W_ =

4

1

3

2

1

4

1

a- G (a+ , a- ) ± a- (G(a+ , a- ) - G(a+ , a - )) 3

4

1

3 2

4

3G 2 4

1

= a- G(a+, a- ) ± a- (a+ - a+)— (a+ , a+ , a- ), 4 where

SG

G(, Y) - G( 71, Y) - 77


18

is the difference derivative of G(, y) by 4. The first term is already in the desired form, and we transform the second one by changing the Feynman indices:

4 sH 2 4 1 a- (a+ - a- )— (a+ , a+ , a- ) 3

4 sH 1 5 0 = a- (a+ - a- )— (a+ , a+ , a- ) 3

32

32

33 2

4

sH 1 5 0

= [[a- (a+ - a+)]]— (a+ , a+, a- ) 3 (we can insert autonomous brackets since none of the operators outside them have a Feynman number in the interval [2, 4]). The commutation relation says that the operator in the autonomous brackets is equal to 1, and the last expression takes the form BG 1 5 0 BG 2 2 1 3G 2 1 = — (a+ , a+ , a- ) =

c3

c3

cS

(the difference derivative becomes the usual derivative on the diagonal = 77). Finally, we obtain 22

W_ = H (a+ , a- ), where

aG

H(, y) = yG(, y) + --(, y).

n

We can now compute the product 2 W+ =

2

1

1

a+ a- [[G(a+ , al )]] = a+ W_ = al[11 (a+ , a- A.

In fact, this is trivial: 2 W+ =

1

3

2

1

2

2

1

2

1

al[H(a+ , a- )]] = a+ H (a+ , a- ) = a+ H (a+ , a- ) = 11±(a+ , a- ),

where

y) = 4 11 (4, Y). (The change of Feynman indices is valid, since their order over noncommuting operators was preserved.) Let us substitute the result of our computation into the left-hand side of (1.9) and equate the symbols of operators on both sides of the resulting equation. For G(, y) we obtain the equation

a

(y + —) n G(4,

Y) = ) .G(4,

y).

The remaining part of the solution is, in fact, purely technical. We have obtained an ordinary differential equation, whose general solution has the form

G(, y) = e-- Y e c(y),


19

2

where c(y) is an arbitrary function. We substitute a+ instead of and a- instead of y into G(4, y) and substitute the resulting operator in (1.8): 2 1 2 1 klf(x) = e -a+ (a+)1 c(a+)v(x). Extracting (a+)À , we obtain (x) = (a+) )` o (x),

where

21

q)(x) =

1

Ca+a-c(a- )v(x).

Although v(x) is an arbitrary function the same is not true of ço(x). In fact, we 21 1 intend to show that the operator exp (-a+a- )c(a- ) is one-dimensiona1 3 . Indeed, apply a- to ça (x): 1

21

q), (x) =

[P-a+ c(a- )Jlv(x).

We compute the operator on the right-hand side in the same way as W_ and obtain 1

21

2

1 = f (a+ , a+),

where

a

f(, y) = y e+4.Y c(y) — e+sY c(y) Henceforth,

O.

a

(x) = (2wh) -112 (h— wx) g9(x) = 0, ax

so that ço(x) = const . e - x 2 /th .

This function decays together with all its derivatives more rapidly then any polynomial, so for each integer nonnegative the function tlf (x) lies in L2 (R2 ). Thus the function •

(a+)ne —x 2 colih =

(.

n W(x)=

COX

012. 1

—

ax

n e—x2w/ih

is an eigenfunction of a+ a and hence of ñr and the corresponding eigenvalue is ,

En = coh(n ± 1/2). This is a classical result in quantum mechanics.

Remark 1.1 How to prove that there are no other eigenvalues? The simplest way is to prove that the system of functions obtained is complete in L2 (R2 ); however, one can also prove directly that the function (a+)x yo(x) does not lie in L2 (R 1 ) for noninteger 3 That is, its range is one-dimensional.

L Elementary Notions of Noncommutative Analysis

20

1.7 T-Exponentials, Trotter Formulas, and Path Integrals Now that we are acquainted with autonomous brackets, let us return to the T-exponentials introduced in Subsection 1.1. Consider the SchrOdinger equation

1 avi a 2 —i— H x, —i— , t * = 0, x at + 0x

ERn ,

t

E Ri ,

with time-dependent energy operator (we assume a system of units in which Planck's constant h equals 1). Its solutions can be expressed via the T-exponential, 1

J

f [r[H (x2 , —i—d , r) Adr) *0(x), , ax

(x , t) = exP ( —i

0

where *0 (x) is the initial value. The integrand in the exponent contains autonomous brackets. It would be interesting to remove them, so let us try it (as we shall see, the result is quite intriguing). To this end, consider first the T-exponential t

F(t) = exp f

r

12

[[f (A,

B, r)]lch ,

o where A and B are bounded operators (e.g., matrices). By definition,

lim

F(t)=

N

1

2

12

exPaLf(A, B, -EN)116aN)... exPaif (A, B, ralAti),

At=max Ati —03

i

where 0 = to < ti < • • • < have 12

exp(Lf (A, B,

tN = t, Ati = ti — ti_i , and ri c [ti -i, tit However, we 12

ti)][Ati) = 1 ± f (A, B, 1

ti)Ati + 0 (At 2 )

2

= exp(f (A, B, ri)Ati) ± 0 (At 2). The remainder terms 0 (At2 ) result in 0 (At), which vanishes as At —> 0 in the overall product. Hence the autonomous brackets can be removed, t

exp f [[ f (A, B , r)]]ch r 1 2

0

,

2N-12N

=

lim

N—). co At=max Ati—)-0

i

12

exp(f( A , B , rN)AtN) — exp(f (A, B , ri)A 11)•


21

It is natural to denote the limit on the right-hand side of the last equation by

I

t

r r+0

exp ( f f (A, B , v)dr) o that is, we have

r

exp

r

t

12

f 1[ f (A, B, r)Adt = exp (f f (A,r r+0 B,

where the "+0" over B stands there to indicate that at each time y the operator B acts after the operator A. Hence, the autonomous brackets can safely (and almost without trace) be removed in T-exponentials. The 0 (At2 ) argument is no longer usable with functions of unbounded operators. However, under appropriate functional-analytic conditions (not to be discussed here) the conclusion remains the same. Consider the simplest case in which

f (x, y, Then

r

t)

=x ± y.

12

exp (I' [[ f (A, B , -c)]]dt = ell A+1311!

,

0

and, by setting ti = tIN, i = 0, .. . , N, we obtain the Trotter product formula [180] e liti+mr _ lim e(tIN) B e(t IN) A . ..e (tIN)B e (r1 N) A . — N—)-00

N

pairs of factors

We see that the rule of removing autonomous brackets in T exponentials is merely a generalization of the Trotter formula. Let us return to the T-exponential solution of the Schriidinger equation. According to our reasoning, we get -

r

t

0 c, t) = exp ( —i f H

r+0

x

0

a

dt)Vfo(x)

ax

or 2k-1

N

Ox,

t) =

lim

N—>co At=max Ati—>0 {

i

11 exp (—i H ( 2xk k=1

, —i

— a , rk) Atk) ik o (x)} ax


22

The operators included in the product can be expressed by the formula 2k-1 )

2xk , - i -1 , tic Atk u(x)

exp ( —iH

ax

1 f e ip(x—y) e —(x,p,rk)Arku /ky,d) .1 = — yup, (27r)fl R2

(this is one of the usual representations of a Fourier integral operator, see Subsection 1.3). Substituting this expression into the preceding equation, we obtain N -1

ilf(x,t) =

f

lim

N- *cc At=max A ti —>.°R2Nn

E

exp { i (xk+i ( pk — xk) k=0

i

—H(xk+i, pk,rk)Atk

NI—r -1

*o(xo) I I

dxkdpk

where xk and Pk are n-vectors, the expression pk(xk+i — xk) is the inner product of Pk by (xk+i — xk), and xN = X. Let us interpret (xk, Pk) as the point at time t = tk of some trajectory (q(t), p(t)) in the phase space. Then, as Atk —›- 0, we have xk+i — xk = 4(ric)Atic +

and the expression in the exponent can be considered as a Riemann sum for the action integral t

fo [p(r)4(r) — H(q(r), p(r),r)]dr. Hence, the limit on the right-hand side of the obtained equation is denoted by t Vi(x, t) = ff [dq][ 3" 1 ] exp 27r

/if [p(r)dger) o

dr

H(q(r), p(r),r)]dr ifro(xo)dx0;

the inner integral is known as the Feynman path integral; it is taken over all phase trajectories (P(r), q(r))te[0,ti such that q(0) = xo and q(t) = x.

Thus, at the formal level, the path integral expression for the solution of the Schr6dinger equation can be obtained directly from the T-exponential expression by removing the autonomous brackets.

2. Functions of Noncommuting Operators: the Construction and Main Properties

23

2 Functions of Nonconunuting Operators: the Construction and Main Properties In the preceding section we considered several examples of functions of noncommuting operators and described convenient notation (Feynman indices and autonomous brackets) permitting one to define the arrangement of operators in operator expressions. However, we were acting rather intuitively, having no definition for functions of noncommuting operators. The meaning of operator expressions is clear as long as the functions involved are reasonably simple (e.g., polynomials or exponents). However, the examples considered in Section 1 suggest that, in view of applications, noncommutative analysis cannot be limited to elementary functions. One is forced to deal with expressions of the form f (Ai, . , An ), where A 1 , . . . , An are given operators and the symbol f (x i , . . . , x n ) is a function of quite general form. Hence, let us consider what meaning can be assigned to such expressions and study their basic properties.

2.1 Motivations We intend to learn how to solve the following problem. Suppose that some operators A 1 , . . . , An are given (that is, elements of an operator algebra A, noncommutative in general) and .Fn is some class of functions of n arguments .x 1 , , xn . Let f E Fn• 1

Then, what is f (Ai , • • • , An)? Let us first study the trivial case in which all the considered symbols are polyno, xn ] is the algebra of polynomials in n variables. Here mials, that is, Fn = C[x i , , x n ) is the polynomial the definition is evident: if f (x i , =

ccq...an

xal

• • ' X nan

al + —Fan A. Hence .Ti should be an algebra (with respect to pointwise multiplication of functions) and we arrive at the following question: Is the homomorphism ILA uniquely determined by the condition /IA (x) = A, i.e., by the value it takes on the function f (x) = x? (with polynomial symbols, the answer is evident). It turns out (see Subsection 2.2) that, under quite general conditions, the answer is "yes" if we consider only continuous homomorphisms. From now on we tacitly assume that the notion of convergence is defined in the symbol spaces Fn and the operator algebra A and all mappings in question are continuous2 . We say that an element A E A is a generator if some homomorphism p„A satisfying the above condition is fixed. 'Throughout the exposition all algebras considered contain the identity element, and homomorphisms are assumed to take 1 into 1. 2 The detailed and rigorous analysis of all questions pertaining to admissible symbol and operator classes, convergence, estimates in function spaces, etc., is postponed until Chapter IV.


25

b) Uniqueness for n > 1. If the operators Ai, j = 1, . .. , n, are generators, that is, the homomorphisms j = 1, . . . , n, are defined and fixed, then conditions 1° and 3° define the action of the mapping ti, i n uniquely on functions f (x i , . . . , x n ) representable as finite linear

combinations N

f (x i , . . • , x n ) = E fi (xi) • • • fsn(Xn), fsk E -F1

(IA 0)

s=1

(such linear combinations comprise the so-called tensor product .F1 0 • • - 0 Fi = 1. ■ ---.. v.-d

On

1

n copies

of n copies of the space Fi). In contrast to the case of polynomial symbols, we cannot hope that the linear combinations (I.10) fill the entire space .Fn , i.e., that Fn = .F1® n . Instead, let us require that each symbol g E Fn can be approximated by such linear combinations. Then the mapping bc 1 n , which is uniquely defined on F1® ", Ai,...,A, uniquely extends to the entire space Yn by continuity. Prior to formal definitions, let us consider two simple examples.

Example 1.1 (Entire functions of bounded operators) Let' A be the algebra of bounded operators in a Hilbert space 'H. For the space .Fn of n-ary symbols we take the space 0(C") of entire analytic functions on C" equipped with the topology of uniform convergence on compact subsets in C". Let A E A be an arbitrary operator. If 00

f (x) =E fkxk E .Fi k=0

00

is an arbitrary function, then the series

E

fk A k clearly converges in the operator norm

k=0

and defines a continuous operator in 'H, which will be denoted by f (A). The mapping ILA : f i-

f (A)

is obviously continuous. Furthermore, f (A) can be obtained as the limit N

f (A) = iim E fkii k k=0

N—>o

of polynomials in A, which implies that ti,A is an algebra homomorphism uniquely defined by the condition tt,i(x) = A. Thus, each element of A is a generator.


26

The definition of functions of several operators is now evident. By passing to the limit we easily show that for each function co

f (x i , ...,xn ) = E fkxciei, ...,xnan E .Fn 1(1 1= 0

and any A 1 , . . . , A, E A we have 1 f

(Ai,

•••,

n An) =

00

la 1=0

and the series on the right-hand side converges in the operator norm.

Example 1.2 (Continuous functions of bounded self-adjoint operators) Let .F„ = C (WI ) be the algebra of bounded continuous functions on Rn with uniform convergence on compact subsets of Rn , and let A be the same algebra as in Example IA. If A E A is a self-adjoint operator, then for each function f (x) E li the operator co

f (A) dg f f (X)d E A( A)

is well-defined (see [42]); here Ex (A) is the spectral function of the operator A, and the Stieltjes integral on the right-hand side is in fact taken over a finite interval. The mapping ttA : f 1- f

(A)

is continuous and coincides with the "natural" one on polynomial symbols; since by the Weierstrass theorem a continuous function on a compactum can be approximated by polynomials, it is clear that IA is an algebra homomorphism uniquely determined by the condition ,u,A (x) = A. Hence A is a generator. A function of several Feynman-ordered operators can be defined as follows: 1 f

(Ai,

••.,

co Do n An) = f • ' • f -

f

(' ... , A. n ) d EA,„(A n ) . . . d Ex i (Ai) ,

00 -00

(the integral on the right-hand side is an iterated Stieltjes integral).

2.2 The Definition and the Uniqueness Theorem Let us now make the construction of the preceding subsection into a precise definition. Let A be some algebra, whose elements will be referred to as operators (in applications of noncommutative analysis, as a rule, A is an algebra of operators in some


27

linear space; however, within the framework of the general theory it is convenient to pay no attention to this fact and assume that A is merely an associative algebra with 1). Furthermore, let a space 3:1 of unary symbols be given. The elements of .FE are functions of a real (or complex) variable x. 3 We assume that .Fi is an algebra with respect to pointwise multiplication of functions and contains the function f (x) x (and, consequently, all polynomials). Definition 1.1 An operator A E A is said to be a generator if there exists a continuous homomorphism

AA : Fi —> A such that istA (x) = A. If A is a generator, then we set f (A) = /LA (f )

for each f

E 3:1.

The property of being a generator obviously depends on the choice of the symbol class F1, so we mention .Fi explicitly, by saying that A is an Fi -generator, whenever ambiguity is possible.

Since the uniqueness of /IA has not yet been proved (see Theorem 1.1 below in this subsection), we temporarily assume where necessary that for all the operators involved the corresponding pt-homomorphism is chosen and fixed. Let Y n be the space of n-ary symbols f (x i , . . . , xn ). According to our strategy, the elements of the tensor product l'i®n should be dense in .Fn . We assume that Fn is obtained from .Fi®n by completion, that is to say, by adding the limits of Cauchy sequences of elements (I.10); we denote this by = -F6 n ' -Fn = -F.1( • • • 6-Fi copies

n

(the hat over 0 denotes completion). Instead, we could impose a weaker condition that the embedding ,F1 '6n c— .Fn be continuous and dense. However, we would have to require additionally that the mapping (1.11) be continuous as a mapping from Fn to A, which holds automatically in our case. This would lead to slight modifications in the subsequent statements.

Let A 1 , .. . , An be generators. The mapping n

: Ti n —> A,

fi (xi) f2(x2) • • • fn(xn) "- fn(xn) • • • f2(x2) fi (xi)

extends by continuity to the entire space F. 3 We do not assume that x ranges over the entire space IR or C.

(I.11) (I.12)


28

Definition 1.2 We set n

1

f (ili,

• •• ,

An) = A

n 1 111,...,An

(f);

1 n a tuple A = (Ai , . .. , An ) of generators equipped with Feynman indices will be referred to as a Feynman tuple. in

il

Functions of a Feynman tuple (A1, . . . , An ) with arbitrary pairwise distinct Feynman indices ji , . . . , jn are defined in a similar way: the factors on the right-hand side of (I.12) should be arranged so that their Feynman indices form a descending sequence.

Remark 1.2 In order to avoid clumsy notation, we assume that all the operators A 1 , . . . , An are Fi -generators with the same symbol class Fi. In principle, one may well consider operator expressions involving generators with respect to various symbol classes. In that case .Fn would be the tensor product of various spaces of unary symbols rather that .T16 n , each space associated with the corresponding operator argument. In the sequel we often make use of a particular case of this situation in which some of the operators occur only linearly or polynomially in the expression considered. If so, the corresponding symbol classes may be chosen to consist of polynomials, which eliminates the necessity of checking any additional conditions (with polynomial symbol, each operator is a generator). Remark 1.3 The presented construction of functions of several operators suggests the following general way of proving this or that identity, assertion, etc. in noncommutative analysis: first, they should be established for decomposable symbols of the form f = fi 0 • • • 0 In and then extended by linearity to .F? n and by continuity to the entire -Fn, cf. the proof of Propositions I.1 and 1.2 given below. Our definitions do not allow coinciding Feynman indices over arguments of an operator expression (later on, when it will be shown that the ordering of Feynman indices over commuting operators does not affect the value of an operator expression, commuting operators will be allowed to bear the same Feynman indices). However, il

the formula f (A, A) is not erroneous. It means the following: one should restrict the function f (x , y) to the diagonal x = y by setting g(x) = f (x, x) and substitute the I operator A into g(x) instead of x. Thus, il

f (A, A) = g(A). The same interpretation is used if f has additional operator arguments, e.g. 112 def

12

f (A, A, B) = g(A, B),


29

where g(x, z) = f (x , x, z). It turns out that in this situation the Feynman indices over an operator A can be moved apart, i.e., set different.

Proposition I.1 (Moving indices apart) i)The operator of restriction to the diagonal [tfl(x) = f (x, x). is a continuous operator from .F2 to Fi. ii) Let A E A be a generator For each symbol f (x, y) E .F2 we have 12

21

1

1

f (A, A) = f (A, A) = f (A, A) (note that the right-hand side is nothing other than [tf](A).) Proof Let f (x, y) E .F2 have the form f = g 0 h, that is, f (x, y) = g(x) h(y), so

that t(g 0 h) = gh.

Since .Fi is an algebra and multiplication in Fi is continuous, the operator t is continuous from .F1(i3:)' F1 = F2 to Fi. Moreover, since ptA is a homomorphism, we have 12

(g 0 h)(A, A) = h(A) g (A) = hg (A) = c[g,® h](A), 2 1 and similarly for (g 0 h)(A, A). Thus the identity in ii) holds on F1 0 .F1 and, by 0 continuity, on the entire F2. The proposition is proved.

Evidently, the statement of Proposition 1.1 remains valid also in the presence of additional operator arguments. The Feynman indices over A can be moved apart but only in such a way that the ordering relation between them and Feynman indices of other operators remains unchanged. More precisely, sm s,n k 1 si l i si f (A, A, Bi, . • ., Bm) = f (A, A, Bi, • • • , Bm)

provided that j, k, and ! all lie in some interval [a, b] that contains none of the numbers m . Since our definitions make it evident that the value of an operator expression S depends on the ordering relation between Feynman indices rather than on the indices themselves, we arrive at the following conclusion: Feynman indices in an operator expression can be changed arbitrarily without affecting its value, provided that the ordering relation between Feynman indices over different operators is preserved. For example, 123

213

115

f (A , A, B) = f (A, A, B) = f (A, A, B),


30 but

123 423 f (A, A, B) 0 f (A, A, B)

in general, since the order of Feynman indices of the first and the third argument has

been changed. The next proposition is stated in its general form (i.e., with an arbitrary number of operator arguments). Proposition 1.2 (Extraction of a linear factor) Let f (x i ,

. . . , x n ) E Fn be a symbol

of the form ,x,) = g(xi , ... , x k )h(x k+i , ... ,x,), il

in

and let A = (A1, . . . , A n ) be a Feynman tuple satisfying the following condition: the Feynman indices j1 , . . . , jn lie in some interval (a, b) that contains none of the indices Then k+1 s h in il ik f ( 11 1, • • • , An) = ffg (Ai, . . . , Ak)lh( A k+1,

in

where s E (a, b) can be chosen arbitrarily.

Remark 1.4 Recall that the autonomous brackets [ill have the following meaning: one should evaluate the expression they enclose and use the result as a single new operator in the subsequent evaluation, no longer caring about the Feynman indices that occurred within the brackets. The index s over the left bracket is just the Feynman il lic index to be assigned to this new operator. The operator g (Ai, . . . , Ak) occurs linearly in this identity, so, according to Remark 1.2, there is no need to impose any additional requirements on this operator. Proof Assume that f (x i , . .. , x n ) is a decomposable symbol, f (x1, • • • , x n ) = ft (xi) • • • fn(xn).

Then g (x i , . . . , x k ) = fi(xi) • • • fic(xk)

and h(x k+i , . . . , x n ) = fk+i(xk-Fi) • • • fn(xn)• il

in

The factors fi (A 1 ), . . . , fkin (Ak) in the product fi (A 1 ) • • • fn (An) stand next to each other due to the condition imposed on the Feynman indices. Hence, we can group these factors together, thus obtaining the factor il ik ilc il fl(A1) • • • fk(Ak) = 8 1 (A1, • • • , Ak)


31

in the considered product. Hence, the proposition is proved for decomposable symbols. By linearity and continuity, it extends to arbitrary symbols g E Yk, h E Yn-k (cf. Remark 1.3). The proof is now complete. 0 Let us consider two examples. We have 15

23

15

223

2

15

g(A, B) h(C , D) = [[h(C , Dj] g(A, B) = K g(A, B), 23

where K = h(C , D). Similarly, we have 2

4

1

5

4

32

1

5

(A — A) sin(B + B) = [[A — Al sin(B + B) = 0, since

2

4

A — A =0. However, 2

4

1

3

(A — A) sin(B + B) 57 0 3

2

4

in general, since the operator B acts between A and A. Proposition 1.1 and 1.2 enable us to prove a theorem stating the uniqueness of the homomorphism ti,A for any generator A.

Theorem 1.1 Suppose that the symbol class .F1 satisfies the following condition: the difference derivative

8

8f 6x

{ (f (x) — f (Y))1(x — y), x 0 x=y nx),

)1 ,

is a continuous operator form .F1 to .7.2. Then for any generator A e A the homomorphism 1.1,A described in Definition 1.1 is uniquely defined. Proof Let A be a generator in A, and let ti,1 and p,2 be two (possibly, different) choices of the homomorphism ,uA. It suffices to prove that pi = g2. For notational convenience, we set C = A, ILA = gi, and p,c = p,2; in other words, we denote the operator argument differently, according to which of the two homomorphisms is used:

f (A) = ,(LA (f),

f(C) = ',Lc (f )•

All that we need is to establish that f (A) = f (C) for every f E .F1. By substituting 2 1 A for x and C for y in the identity 8f f (x) — f(y) = (x — y) — (x , y), sx


32 we obtain

2 1 2 8f 1 2 f (A) — f (C) = f (A) — f (C) = (A — C)—(A, C). Sx By the conditions of the theorem, 8f/8x(x, y) e F2, and consequently, 1

(z — w)Sf/tSx(x, y) E F4. Hence we can apply Proposition 1.1 so as to move the Feynman indices over A apart, and the same thing can be done for C:

1 2 sf 0 3 f (A) — f (C) = (A — C)-(A, C). Sx 1 2 Next, we use Proposition 1.2 to enclose the factor (A — C) in autonomous brackets: 1 2 sf 0 3 2 sf 0 3 21 (A — C)— (A, C) = [A — Cl— (A, C), 8x Sx

and the last expression is equal to zero since

1 2 A —C=A—C=0 (the reader should not be too suspicious about this conclusion, since the homomorphisms IA and pc coincide on the symbol 'p(x) = x by definition). Hence we have 0 shown that f (A) = f (C). The theorem is proved. In what follows we assume that the condition stated in Theorem 1.1 is fulfilled. Remark 1.5 This proof is so simple that it seems to be a mere trick. However, it is perfectly rigorous. Its simplicity is just due to the wizardry of our notation, which displays itself in quite a few places thereafter. Remark 1.6 According to Proposition 1.1, the operator t of restriction to the diagonal acts continuously from ..T2 to F1. Since t 8/8x = Diaz (on the diagonal the difference derivative coincides with the usual one), we see that ,#), is a continuous operator from -F1 to li and consequently all symbols in .F1 (and hence, in each .F,i ) are infinitely differentiable. We could choose another way and postulate that the operator d „ — : Pi —> Pi dx is continuous. Then the formula 1 Sx

0

dx

— t)y)d -c


33

would imply that the condition of Theorem 1.1 is satisfied. However, in that case we would have to require that the domain where symbols are defined is arcwise connected and the mapping (t, f (x)) 1--> f (-c - x ± (1 — t)y) from II x .F1 to .F2 is continuous.

2.3 Basic Properties Let A be an algebra of operators, and let

Fi = . F ,

Y2= .T .C.T.,...,-Fn

= Y6 n , . . .

be algebras of symbols. Given a tuple A = (A 1 , . . . , An ) of generators in A, we intend to study the mapping n : Fn —> A. ILA = Al Ai,...,A,,

In doing so it is convenient to use the regular representationsof the algebra A on itself. Let A E A. The mappings LA: A

—>- A

B I--> LA(B) = AB and

: A —>- A B I--> RA(B) = B A

RA

defined as left and right multiplication by A, respectively, are continuous linear operators on A. The set of continuous linear operators on A will be denoted by L(A), so we can write LA E L (A) and RA E £ (A) . Consider the mapping

L : A —> G(A) A I--> LA. It is a continuous linear mapping called the left regular representation of the algebra A. Similarly, the continuous linear mapping

R : A —>- L(A) A 1--> RA is called the right regular representationof A. Note that L is a homomorphism of algebras LAB = LA LB,


34

whereas R is an antihomomorphism, that is, it inverses the order of factors RAB = RB RA.

Since L(A) is an algebra of continuous linear operators on a linear space, we may well consider the notion of a generator in L'. (A). It turns out that the supply of generators in .C(A) is at least as rich as that in A, as shown by the following theorem: Theorem 1.2 The following conditions are equivalent: i) ii) iii)

A is a generator in A. LA is a generator in L(A). RA is a generator in L(A).

Proof i) .#;> ii). Let A be a generator in A and 1.2,A : F —> A be the corresponding

homomorphism. We define the mapping AL A : .F —> L(A)

by setting ALA

(f ) = LAA(f)•

Clearly, this mapping is a continuous homomorphism as the composition of two continuous homomorphisms, namely of L and ptA. We also have ALA(X) = LA(x) = LA,

and so LA matches the requirements of Definition I.1 . Conversely, let A E A be an operator such that LA is a generator; in other words, for each symbol f (x) E .F the operator f (I, A) = A L A (f) is well-defined. To construct the corresponding mapping ILA, note that for an arbitrary operator B E A we have [LA, RB] dg. LARB — RBLA=

0,

that is, the operators LA and RB commute. Indeed, for any C LA(RBC) = LA(CB) = A(CB) = (AC)B =

E A

we have

RB (AC) = RBLAC,

by the associativity of A. We can now apply an argument similar to that used in the proof of Theorem I.1 to obtain f (L A )R B = RB f (L A ).

(1.13)

Indeed, we have 3

1

2

3

1

2

8f 3

f(LA)RB - RB f ( 1, A) = (f ( 1, A) — f(LA))RB = (LA- LA)RB—(LA, 6x

1 LA).


35

We can move apart the Feynman indices over LA in this expression (Proposition 1.1) and then extract a linear factor (Proposition 1.2) thus obtaining f(LARB —

13 1 2 6f 3 1 LA) = 0, RB f (LA) = FLA — L A)R.131 — (L A,

Sx

since the operator in the autonomous brackets is equal to zero. Since B is arbitrary, equation (1.13) means exactly that f (L A) is the left regular representation of some element of A, which will be denoted by f (A). Specifically, we have f ( 1, A)(x) = f (1 , A)(1 • x) = f (LA) (Rx 1) = Rx f(LA)(1) = ( f(LA)(1)) X,

and consequently, f (A) = f(LA)(1).

The mapping f

p,A (f) = f (A) is clearly continuous; moreover,

f (A) g (A) = f A(1) g(L A)(1) = f (L A)g(LA)(

1 ) = (f

A(1) = f g(A),

so that it is a homomorphism. iii). This can be proved similarly; the only point worth mentioning is that R i) is an antihomomorphism, and at first glance it may seem that the mapping AR A = R

fails to be a homomorphism. However, the image of ,uA is a commutative subalgebra of A, and the restriction of R to it is a homomorphism. The proof is complete. Now there is an evident formula: 1

f (1, A i , • • • ,

An ) =

L1

n

f

for any generators A 1 , . , An E A and symbol f c Tn . The proof goes by linearity and continuity: if f (x) = (xi) . . . fn (xn ) is a decomposable symbol, we have 1

f ( L A1 • • •

LAO

= f n (LA )

fl(L Al) = fn(An) • • • Lifl(A1)

Lfn(An)....f1( 11 1) =

L1

and this equation extends to the entire .Fn by continuity. Similarly, 1 f(R A i , • • • , R A n ) = R n f

n


36

(the operators A 1 , . . . , An on the right come in reversed Feynman ordering, since R is an antihomomorphism). Let us consider a slightly more general situation in which a function contains both L and R arguments. Let f (x , y) E Y2 be an arbitrary binary symbol. What is 2

1

f(LA,RB)? Note that LA and RB commute, and so we can choose their Feynman indices arbitrarily; let us choose them in such a way that the index of LA is greater 1

3

than that of RB; thus, we consider f (L A, R B). This operator is an element of L(A), and to describe it means to describe its action on an arbitrary element C E A. If f is decomposable, f (x , y) = fi (x) f2(y), then we have 3

312

1

f (LA, RB)(c) = fi(LA)f2(RB)c = fi(A)C f2(B) = f (A, B)C .

By continuity, this identity extends to arbitrary symbols f result in a general form.

E F2.

We now can state the

Theorem 1.3 Let A i , ... , A s ,

B i , ... , B,n E A

be generators, and let j1 , . . . , js , k l , . . . , km be their (pairwise distinct) Feynman indices. Then il

is

f (Li Ai, • • • , L A s

,

km k1 RBi, • • • , R B m )(C)

ko—km .is 1c0 — k1 C f (L A i , • • • , L A s , B1 , • • • , Bm ), 1

=

il

where /co and l are chosen in such a way that ko — k1 < 1 < ji for any i and l E {i, . . . , M}.

E

{1, . .. , s}

The proof is modelled on the above argument, and we omit the trivial details. Let us now state and prove the main properties of Feynman operator calculus. We fix some class Y of unary symbols and some generator A E A.

Theorem 1.4 The Feynman operator calculus has the following naturality properties. 10. Let ço : A —> B be a continuous homomorphism of algebras. Then w(A) E B is a generator, and f (ço( A)) = W(f (A)) for any f E Y. 0 2 . Let B E A be an operator such that AB = BA. Then

f (A)B = B f (A) for any f E Y. 3°. Suppose that

AB = BC,

(I.14)


37

where B,C E A and C is a generator Then

f (A)B = B f (C) for any f E Y. 4°. Assume that A is an algebra of operators in a linear space V, B an algebra of operators in a linear space W, and

x : W —› V a continuous linear operator such that

Ax=xC, where C

E

(1.15)

13 is a generator Then we have f (A) x = x f (C)

50 .

for any f E F. As in 4°, assume that A is an algebra of operators in a linear space V. Let be an eigenvector of A with eigenvalue A,

E

V

,U =A. Suppose also that f(A) is defined for any f E F. Then "

f (A) = f (xg for any f E Y.

Remark 1.7 There are no Feynman indices in the statement of this theorem. However, the proof employs Feynman ordering heavily. See Theorem 1.5 for generalization to multivariate symbols.

Proof 1°. Let q) : A —›- B be a continuous homomorphism of algebras. Set f (ço( A)) tf ki. (A)), that is, Py(A) = PIAA.

Evidently, pw (A) is a continuous homomorphism (as a composition of such mappings). Hence, q) (A) is a generator in B, and we obtain the desired result (recall that g y (A) is unique by Theorem I.1). 2°. We proceed as in the proof of Theorem 1.2. We can write 2 3 1 1 6f 3 1 23 f (A) B — B f (A) = B ( f (A) — f (A)) = B (A — A)-' (A, A) . 8x


38

Next, by moving indices apart and extracting a linear factor, we obtain 22 3 1 8f 5 0 f (A)B — Bf (A) = [13 (A — A)]1 — (A, A) = 0 8x

since the expression in autonomous brackets is zero: 23

1

B (A — A) = AB — BA = O. 1 1 3°. In order to prove this, we repeat the above computation with A replaced by C: 2

3

1

f (A)B — Bf (C) =

B(f (A) — f (C) 2 3 1 8f 3 1 = B (A — C)— (A, C) 3x 22 3 1 8f 5 0 = I[B (A — C)]]— (A, C) = O. 8x

Of course, 2° is a particular case of 3°. 4°. The condition of this item can be expressed by the commutative diagram A

V

C

W

V

w

We now make some preparations in order to reduce the proof of this item to that of item 3 0 Consider the direct sum E = V ED W of the linear spaces V and W. Continuous linear operators in E can be viewed as 2 x 2 matrices (

GH E F )

with continuous operator entries

E : V —> V, H : W —>, W, F : W —> V, G : V —> W.


39

Relation (I.15) can be rewritten in the form (A 0 c 0 ) (0 00 x ) = ( 0 x 0 )(A 0O c).

Let A be some algebra of continuous linear operators in E containing the subalgebra 00 x0 ) (e.g., the minimal algebra generated by these A ED B and the operator ( objects). Then the last equation has the form (I.14) in the algebra A. Furthermore, ( A0 is a generator in A. Indeed, one can set 0 C

=

f (( j°

( f (A) 0

0 ) f (C)

By applying item 3°, we obtain

f (( or

( f (A) 0

x 1(1) l' ) ) ( °() X° ) = ( °1 3 f (O c) )

0 0 )

0

) f ( ( 11

( 0 0(0x x0 ) ( f (A) 0

C O)

)'

0 ) f (C)

We perform the necessary multiplications and find that f (A) x = x f (C), as desired. 5°. Strange as it may seem, but this is a particular case of 4°. Specifically, let H c V be the one-dimensional subspace generated by the eigenvector , and let x : C —)- V be the linear mapping such that x (a) = c4 for all a E C. It is evident that x (C) = H. The commutativity of the diagram C

d C

X

X

v

1

A

v

expresses the fact that H = x(C) is an eigenspace of A with eigenvalue A. An 0 application of item 4° yields the desired result. The theorem is proved. We can now state the counterpart of Theorem 1.4 for multivariate symbols. In the following we assume that the class Y of unary symbols is fixed, Y„ = Y @n, and A 1 , . . . , An E A are arbitrary generators.


40

Theorem 1.5 1°. Let g) : A —> B be a continuous homomorphism of algebras. Then we have 1 n n 1 f Will), . . - , kiln)) = g)(f Oh • • • , An))

for any symbol f E •Tn• 2°. Let B E Abe an operator commuting with all Aj,

[Ai, B]F-_-- AJB — BAi =O. 1

n

Then B commutes with functions of A1, ... , An as well, n 1 n 1 n 1 lf (Ai, • • • , An), B] = [Lf Oh ••• , An4B — B f ERAi, . .. , An)II = 0.

3°. Let AiB = BCi, j =1,...,n, where B

E

A, Ci

E

A, j = 1, . . . , n, and each Ci is a generator. Then 1

n n 1 f (A1, . . . , A n )B = Bf (C1, . . . , C n )

for any symbol f E Fn . 4°. Let A 1 , . . . , An E A be continuous linear operators in a linear space V, B 1 , . . . , Bn E B be continuous linear operators in a linear space W, and let

x :W > V —

be a continuous linear operator such that Ai x =x Bi, j =1,...,n (in this case one says that x is an intertwining operator for the tuples (A 1 , . . . , An ) and Assume that each Bj is a generator in B. Then 1

n

1

n

f (Ai, . . . , An) o x = x f (Bi, . .. , BO.

In other words, if x intertwines operator tuples, it also intertwines any function of these tuples. 5°. Let A 1 , . .. , An E A be linear operators in a linear space V, and let E V be a common eigenvector of A 1 , . . . , A n with eigenvalues A = (X 1 , ... , Xn ): Aft = XA, j = 1, ...,n.

2. Functions of Noncommuting Operators: the Construction and Main Properties 1

Then for each f

E .Fn

41

n

the vector is an eigenvector of f (Ai, .. . , A n ) with eigenvalue

f 0`.1 , • • • , Xn) ,

1

n • • ., An) = f f (Ai,

(xi,

• • • ,

Xng•

Proof The proof of this theorem can be carried out in the usual way. Namely, we first consider decomposable symbols; for these symbols the statement of each item follows directly from the corresponding item of Theorem 1.4. Next we extend the result to 0 arbitrary symbols by continuity. Let us make some remarks on Theorems 1.3 and 1.4. These theorems express naturality properties of the Feynman operator calculus: item 1° says that it behaves naturally under homomorphisms of algebras; item 4° says the same about its behaviour under homomorphisms of linear spaces in which the operators act; item 5° states that one may as well restrict consideration to common invariant subspaces where necessary, and finally, items 2° and 3° express quite natural behavior with respect to the commutation operation, in the spirit of von Neumann's definition of functions of operators via

bicommutants. It should be pointed out, however, that item 5° is less important as soon as functions of several operators are considered because, typically, a tuple of noncommuting operators has no common eigenvectors at all, and in any case, a system of such vectors cannot be complete (this is quite understandable since operators commute on each common eigenspace and hence on the sum of common eigenspaces.) It is clear that the assertions of these theorems are far from being independent. In fact, we can draw the following diagram showing the dependence between some of our assertions: The continuity of the mapping 8 ••Y —> .F&F 87

-

2° = 2° 1: tP=uniqueness 0 4 of p,A 30

lo .

We have already mentioned and used some of these implications in our proof. However, let us discuss those implications which remain not elucidated as yet. The implication 4 0 = 3° can be proved in a standard way: we consider the algebra A ED A and rewrite the commutation relation (I.14) in the form

( A 1:)

0\(0

B\ (0 B\(AO 0 c ).

C)0

0)

0 0)

It remains to follow the lines of proof of item 4 0 . The implication 4° = 3 0 can be proved as follows. We consider A, B, and C as linear operators in A acting by multiplication on the left, i.e., we pass to the left regular representation. We have LALB = LBLc,

and item 4° applies. In conclusion let us state the following commutation theorem.


42

Theorem 1.6 Let A, C e A be arbitrary generators; let B, D e A; and suppose that AB = BC ± D. Then for any symbol f

c .F we have 2

3f 3 1

f (A)B = Bf (C) ± D—(A, C). Sx Proof This can be proved by the following computation: 3 1 sf 3 1 2 1 23 f (A)B — Bf (C) = B(f (A) — f (C)) = B (A — C)—(A, C) Sx 23 1 6f 4 0 = B (A — C)— (A, C) (moving indices apart) Sx 22 3 1 3f 4 0 = 1[B (A — C) — (A, C) (extraction of a linear factor) Sx 2

3f

3 1

= D—(A, A), Sx 0

as desired.

2.4 Tempered Symbols and Generators of Tempered Groups Though the discussion of functional-analytic questions has been postponed until Chapter IV, we consider here a particular realization of the construction presented in Subsection 2.2. This realization deals with symbols of tempered growth and the related class of generators and is particularly needed in applications to differential equations. (a) The Symbol Classes. We introduce the space .Fn = Sec (Rn) of n-ary symbols. By definition, the space Soc(Rn) consists of all functions f (y i , . . . , yn ), (y 1 , . . . , yn ) = y e Rn, satisfying the estimate ° y ) " ( — a

f(y)

Ca (1

+ ly Dr,

for any multi-index a = (al , . . . , an ), with some constant r independent of a (but depending on f). A (generalized) sequence fp E S °° (Rn ) is said to be convergent to an f E S °° (Rn ) if there exists an r such that

ysup E Rn

a a (.5T7 ) {

(

4960 — 0

2.

Functions of Noncommuting Operators: the Construction and Main Properties

43

for any multi-index a. The algebra S°° (1W') is complete and the algebraic operations are continuous with respect to this convergence. Moreover, one has scx)(Rn) _ soo(kl) c)... •(:Lbsoo(R1) ,

and the difference derivative is a continuous operator ,iip 1 3 Ts; : S-- (R- ) _>. S °° (R).

These properties (whose proof we cannot carry out as yet since we do not have the material of Chapter IV at our disposal) show that the symbol spaces S°°(Rn) satisfy all the requirements imposed in Subsection 2.2. (b) The Operator Classes. We consider operators acting on a Hilbert scale. A Hilbert scale is a sequence of densely embedded Hilbert spaces • • • C H_2 C H-1 C

Ho

C

Hi

C H2 C - • •

indexed by Z or R. To be definite, we assume that the indexing set is discrete and deal with Z-indexed Hilbert scales. It is assumed that the set H-00

= iEz n Hi

Hi is convergent to is dense in each H. A (generalized) sequence hp E He° = H U jEz h E I1,, if there exists a k such that for /3 > /30 all hp E Hk and hp —> h in Hk. A linear operator A on Hoo is said to be bounded (of order r) in the scale {Hs } if there exists an integer r such that for any s one has -

AHs C Hs+r and the operator AIHs : Hs —> Hs±r

is continuous. The minimal possible r is called the order of A and is denoted by ord A. If the set of possible r does not have a lower bound, we say that ord A = —oo. A bounded operator A in the scale {Hs } is said to be the generator of a semigroup of tempered growth (or simply a tempered generator) if the Cauchy problem du —i— = Au, dt

ul t=o = Uo,

has a unique solution for any uo E Hoo , and there exists an 1 such that for any s E Z the inclusion uo E Hs implies that u(t) E Hs ±/ for all t, is differentiable in Hs+i, and

11u

(

t) 1 1 s +1

co + iti)Niluoi is,


44

where 11h Ilk is the norm of h in Hk and the constants C and N depend on s but are independent of uo E H. Thus, a tempered semigroup in a Hilbert scale grows at most polynomially as t ---> oc. We denote u(t) = exp(iAt)uo.

(c) The Functional Calculus. Let A be the algebra of bounded operators in the scale {Hs }, and let A tempered generator in {Hs } . Let us define the mapping

E

A be a

AA : S 00 (R 1 ) -> A

by setting co

(i + AYn f exp(iAt)(t)dt, PAU') = V-2ri T 00

where the integral is in the sense of strong convergence, f (y)

g (y) .

(y + Om ' 1 f 00 CitY g(y)dy

k (t) .

.,/-27t- i

-00

is the Fourier transform of g(y), and the number m is chosen large enough to ensure that (t) is continuous. It is easy to prove, using our definitions and the properties of the Fourier transform, that any possible choice of m gives the same result and that the resulting operator is bounded in {Hs }. Moreover, /1A takes y into A and is an algebra homomorphism; let us give a formal calculation proving the latter statement: if 00

PA(h) = (i + Arl f exp(iAt)ic(t)dt, ri ,NF-Tf -00

where kW is the Fourier transform of k(y) = (i ± yrnlh(y), then

1-tit (f *A (h) =

(i + Ar±m1 10 i exp(iA(t ± r))g(t)i-c(r) dr 2, 7ri - 00 -00

= (i ± A)m±"" i

N/2-7r-

oo

f exP(in A)

- 00

i° -g (11 - t)k(t) dr -co

dn.


45

By the properties of the Fourier transform of the convolution, the expression in braces in the integrand on the right-hand side of this equation is just the Fourier transform of the product g (y) k(y), and so we obtain ILA (f)i-tA (h) = I-tA(f h),

or f (A)h(A) = f h(A),

as desired.

1 n We can now define f (Ai , . .. , An ) for a function f E S°0 (R) and a Feynman tuple

1 n (Ai, ... , A n ) of tempered generators in a usual way (see Remark 1.3). Obviously, we

get 1

f (Al,

n

( 1 \ n/2 f

... , An) =

27r. i )

_ f (ti, • • • ,tn )exP(iAntn)

...

exp(iAiti)dti ... dtn,

where ( i \n/2 f

tn ) =

27r )

f (y 1 , .. . , yn ) exp(—itiyi — • • • — itnYn)dYi • • • dYn

is the Fourier transform of f and the integral is understood in the sense of strong convergence. We have omitted here several important details, which will be clarified in Chapter IV.

2.5 The Influence of the Symbol Classes on the Properties of Generators The choice of the class .F of unary symbols in fact determines the possible properties of generators in a rather restrictive manner; this is evident in itself from general considerations, and this was confirmed by the example considered in the preceding subsection. Here we present some more simple examples clarifying the subject.

Example 1.3 We begin with a simple remark that the structure of .F is closely related to the possible spectrum of a generator. For instance, if the symbol f (x) = (x — a ) 1

belongs to F, then necessarily a V a (A) for any generator A (here a (A) is the spectrum of A). Indeed, we have (A — a) f (A) = f (A)(A — a) = 1, and so A — a is right and left invertible. Going back to Section 2.4, we see that (x — a) E S 00 (R 1 ) for any nonreal a. Therefore, the spectrum of each generator lies


46

completely on the real axis, so that the polynomial estimates for the growth of exp(i At) become less surprising (in fact, the position of the spectrum of A on the real axis is not sufficient for the polynomial growth of exp(itA); some estimates of the resolvent of A are also needed; however, these estimates can also be derived from the fact that ILA :

f I-- f (A)

is defined and continuous on P0 (R 1 ). Example 1.4 This example is somewhat less trivial. Suppose that the symbol class .F3 contains the function f (y1, Y2, Y3) = e i(n—Ylbq Suppose also that operators A and B satisfy [A, B] =

—

i

(for example, A = it and B = x). Then at least one of the operators A and B is not an T-generator. Indeed, we have, by Theorem 1.10, —

132

123

f (A, B, A)

(5 2 f

4

1 3 5 2 6

= f (A, B, A) + [A, B] 6y2sy3 (A, B, B, A, A) 1 32 3 af 1 3 2 4 4 — 42 (A, B, A, A), = f (A, B, A) — i -

3

since the commutator [A, B] commutes with B. Furthermore, , 07 1, Y2, Y3, Y4) =

'3 Y3 0)/2

( - i (Y3 — Y1)e i(Y3 Y1)Y2 (Y1, Y2, Y3, Y4).

oy3

We use the Leibniz rule for difference derivative

3 3f 3g — ( f (x)g(x)) = —(xi, x2)exi) ± f (x2) - (xi, -x2) 3x 3x and obtain

a

3

SY3 ay2

(yi,

Y2, Y37 Y4) = e1(Y4—Y1»2 ± t: (Y3 - Y1 »(y17 Y27 Y3, y4),

where

e i(Y3 — Y1)Y2 _ ei(Y4 — Y1)Y2 W(Y1, Y2, Y3, Y4) =

Y2 — Y4

By substituting the last expression into the previously obtained expression for 1 23 f (A, B, A), we find that 312

e

213

413

i(A—A)B = e i(A—A)B + ei(A—A)B ±

2

1

1324

(A — A)(p (A , A, A, A).


47

However, 213

e

223

i(A—A)B = e i(A—A)B = 1

by Proposition 1.1, 413

312

e i(A—A)B = ei(A—A)B and

2 (A

1

—

1324

12

1

043 5

1

A)(A, B, A, A) = [[A — AMA, B, A, B) = O.

Thus we obtain 312

312

e i(A—A)B = 1 ± ei(11—A)B ,

which is a contradiction. Thus, A and B cannot be generators simultaneously. Example 1.5 Let r be the Riemann surface of the function .Vi = ,s/x ± iy. It is a double covering of C \In We will consider r as a manifold with local coordinates (x, Y). Consider the operators

a ax

a

and B = —i — ay

,

on the Hilbert space L2 (r) (the inner product on L2(r) is defined by the integral over the Lebesgue measure dxdy). These operators are essentially self-adjoint on the dense subspace C8,°(r) c L2 (T) of smooth compactly supported functions and commute on

C(F), a2 f

82 f

ay aX

aX ay

for any f c c-,T(r). Consequently, they generate unitary groups on L2 (T). Thus, [A, B] = 0, and, according to the permutation formula, we should apparently have 12

21

g(A, B) = g(A, B) for any symbol g E V° (R2 ). However, this is not the case. Indeed, consider the symbol eitzi e itz2 ezi, z2) = (here t and r are parameters). The corresponding operators 12

a

a

21

g(A, B) = e r 75" et T, and g(A, B) = e t k e r 4 are simply the products of the corresponding unitary groups, which are the translations by t and r along the x and y-axis, respectively. But the translations along the two -

48


axes do not commute on I': starting from the point (x, y) = (1, 1) and performing the translations by —2 along both axes, we move to different sheets of the Riemann surface depending on the order of the translations performed. Thus A and B cannot be generators simultaneously in any Hilbert scale including L2 (r). The reason for this phenomenon is that the product of the groups e itA e ir13 is not strongly differentiable on q(r) and Cr (f) is not invariant under these semigroups: as soon as in the course of a translation the support of a function collides with the branching point (0, 0), the function loses its differentiability.

2.6 Weyl Quantization We have given the definition of functions of Feynman tuples of noncommuting operators. The approach was to equip the operators with Feynman indices and arrange them in products according to the ordering of these indices. In the literature one can find another approach to the construction of the functional calculus of several noncommuting operators, known as the Weyl functional calculus or Weyl quantization. We neither study nor use the Weyl functional calculus in this book but only say a few words about it in this subsection. In contrast to Feynman quantization, Weyl quantization is symmetric; this means that neither of the operators acts first or last. In a sense, they all act "simultaneously". Let A 1 , . . . , An be a tuple of operators and f a polynomial of the form f (z i , • • • ,

Zn) = 0Y1Z1+ a2Z2+ • • • ± anZn),

where a1 , . . . , an are numbers and (y) is a univariate polynomial. Then the Weylquantized function of A 1 , . . . , An with symbol f is defined by

fw(A. 1

, • • • ,

AN)

= g9 (11a1A1+ • • • ± anAn11)

(the autonomous brackets are not obligatory here but we let them stand, for clarity). The subscript W is an acronym for "Weyl". Now if f (z p . . . , z„) is an arbitrary polynomial, it can be represented in the form f (z i , • • • , Zn) =

E

W(alzi ± • • • ±

anzn),

w ,ce i , • • •, an where q) are univariate polynomials, a1 , . .. , an are complex numbers, and the sum is finite. We set fw(A i , .. • , An) =

E (P,ai ,

this is well-defined.

49 (Tai Al + • • • ± anAni), an


49

The main property of Weyl quantization is its affi ne covariance. Let M be an n x n matrix and let f (z) = g (m z);

then f (A) = g(M A), where MA is the tuple of operators defined by n

(MA) j = Emi kAk• k=1

To define Weyl quantization for general symbol classes, it is necessary to make some additional assumptions, namely: 1°. Any linear combination ai Ai ± • • • ± an A n is an ..F-generator, where .F is the class of univariate symbols. 2°. The set of functions f (ai z 1 + • • • + anzn) for various f c .T and a1 , . . . , an is dense in the class Tn of n-ary symbols. (The admissible values of the scalars a 1 , • .. , an can be taken either complex or real, depending on the specific case, but simultaneously in 1° and 2° Under these assumptions, the affine covariance property in conjunction with the requirement that f (A) is defined as usual for unary symbols determines the Weyl quantization uniquely. Let us consider two examples.

Example 1.6 (Weyl functions of self-adjoint operators) Let A i , .. . , An be selfadjoint operators on a Hilbert space H. We assume that there exists a dense linear subset D c H such that for any a l , . . . , an e R the linear combination a A = ai A 1 + • • • + an A, is essentially self-adjoint on D (that is, a A is closable and its closure ia a self-adjoint operator on H. Then the exponential U (a) = exp(aA) is well-defined for any a E Rn and is strongly continuous with respect to a (see [179]). We define fw (A i , . . . , AO for f (z) c C7 (Rn) by the formula

1 ) n/2 f f( (a)U(a)da, fw (A i , . . . , A n ) = (27 ri Rn

where f (a) is the Fourier transform of f (Z) and the integral is convergent in the strong sense, or else by the formula

1 ) n/2

fw(A i , ... , An ) = 27i (-

f f(z) 17(z)dz, Rn

where the operator-valued distribution C/(z) is the Fourier transform (in the sense of strong convergence) of U (a) (see [2] and [3]).


50

If D is invariant under A 1 , . , An , then the above definition can be extended to symbols of tempered growth (i.e., symbols growing polynomially as lz I oo with all derivatives). Weyl quantization has a minor advantage from the quantum-mechanical point of view. Namely, if A 1 , . .. , An are self-adjoint, then Weyl quantization takes real-valued functions (Hamiltonian functions in quantum mechanics) into self-adjoint operators. This is not at all essential in Cartesian coordinates since the Hamiltonian function usually has the form 2

H(x, p) = — ± V (x), 2m the position and momenta operators are not mixed and the Feynman and Weyl quantizations give the same result. However, in curvilinear generalized coordinates the Hamiltonian function becomes H(q, p)=

aii(q) pi pi +V(q),

and the choice of ordering is essential. It is Weyl quantization that takes H(q, p) into a self-adjoint operator.

Example 1.7 (Analytic Weyl functions of bounded operators) Let A 1 , , An be , an ) E Cn with bounded operators in a Banach space B such that for any (a l , ± • • • ± lani 2 < 1 the spectrum cr(ai Ai -I- • • • ± an A n ) lies inside the unit ball U in Cn centered at the origin. Let f (z), zee'', be a holomorphic function in a neighbourhood of the unit ball. Then the Cauchy—Fantappiè formula

f cz)

(n - .1)! f (2z j 110=1}

(-1)v -117, 171. )

A • • • A c4v _i A dev±i A • • • A v=1

c4n A

n_

(1 - E

v=1

U, and, consequently, the resolvent is valid for z E U. For ç E 0U we have a (e A) (1 — A) - 1 is well-defined. We use the Cauchy—Fantappiè formula to define the Weyl-quantized function fw (A i , , A n ), (n 1)! f (2.7r (k 1=11 —

, An )

-

) --n

A•••A

cgv-1

A fgv+1 A • • • A

den A 4.

n u =1

We considered here the simplest case in which the domain is the unit ball. Of course, more complicated spectrum structures can be considered as well by using the Cauchy—Fantappiè formula for more complicated domains; we leave this to the reader as an exercise.

3. Noncommutative Differential Calculus

51

3 Noncommutative Differential Calculus The conventional differential calculus deals with the local behaviour of functions; given a function f (x) and a point xo, the main question is as follows: what can be said about f (xo --1-- Ax) as Ax ----> 0? The ultimate answer is that, provided f (x) is a "well-behaved" function, one has f (x0+ Ax)= f (x0)+ f t (x0) Ax ±

( Ax )2 +

f

2

+ f (N) (x0) (Ax)N-1-0(AxN+1), N!

i.e., f (xo + Ax) can be expanded into a Taylor series in powers of Ax. (Of course all of differential calculus is not reduced to this; but this is the essence.) We will pose a similar question for functions of noncommuting operators, and the machinery developed to answer it will be called noncommutative differential calculus. We will see that the matter is rich in subtleties and try to explain them as clearly as possible. Thus, we should consider f (A --I-- AA), where AA is "small", AA —> 0; but what does this requirement mean? We follow the usual practice of perturbation theory and take AA = EB, where E is a small numerical parameter. This permits us to keep track of infinitesimals of different orders easily, since the orders are indicated by the powers of s. In Subsection 1.4 we have already considered two examples of computation for f (A + E B), namely, 1 f (x) = X—x and f (x) = exp(x). Here we deal with the general setting. In the conventional differential calculus, the first-order term f" (x0) Ax of the Taylor expansion is of extreme importance; it bears the special name of the first differential of f, and teaching differential calculus usually begins with its intensive study. By analogy with that case, we are specifically interested in the term of order E in the expression f (A + 613) = f (A) + eCi + 8 2 C2 ± • • • .

It would be wise to ask why such an expansion exists at all; however, as is the case for the conventional calculus, the computational method at the same time provides justification for the expansion. The coefficient of E can be obtained as d Cl =-78 [f (A + 03 )]is.0 ,

(I.16)

i.e., by differentiating with respect to E followed by setting E = O. However, we will find an expression for a more general "derivative", of which this is a particular case.


52

3.1 The Derivation Formula Let A be an operator algebra. A derivation of A is an arbitrary linear mapping

D : A ----> A

(I.17)

satisfying the Leibniz rule D(uv) = (Du)v ± u(Dv), u, v E A. Let A E A be a generator and f E .F a symbol. We claim that (5f 1 3 2 D[f (A)] = [DA] =-(A, A) Bx

(I.18)

for an arbitrary derivation (I.17). The proof will be divided into several stages. First, we consider a special class of derivations.

Definition 1.3 Derivations of the form D(A) = DB (A) ,=- BA — AB, where B is an arbitrary element of A, are called inner derivations of A. It is easy to show that the commutator indeed defines a derivation; the corresponding computation is rather simple and is left to the reader. Proposition 1.3 Formula (I.18) is valid for inner derivations. Proof We need to compute the commutator

[B , f (A)] = B f (A) — f (A)B . So far, there are no Feynman indices in this expression; let us introduce some. Note that they can be chosen independently for either summand on the right; we may prefer 2

1

21

B f (A) — f (A)B ,

52

107

or B f (A) — f (A) B ,

or whichever other choice provided that the indices are consistent with the order of factors. We would like to factor out B, and so we write 2

1

32

2

1

3

[B, f (A)] = B f (A) — f (A) B = B ( f (A) — f (A)). 1

3

Of course, we cannot simply put f (A) — f (A) = 0, because the Feynman index of B lies just between those of the first and the second A.


53

The subsequent computations are quite simple: we obtain, by multiplying and 1

3

dividing by A — A, 1

3

21 3 8f 1 3 f (A) — f (A) [B , f (A)] = B (A — A) = B (A — A) — (A, A) 1 3 Sx A — A 21

1

3

3

(the division by A — A seems to be very dangerous, but in fact is not so; actually, we used the identity 3f f (x) - f (y) = (x - y) - (x, y), 8x which is valid everywhere including the diagonal x = y, where Sf

f (x, x) = f' (x). Sx We now move apart the Feynman indices over the A's, thus obtaining 2 1 2 1 3 sf 1 3 3 8f 0 4 22 1 3 8f 1 3 B (A — A) — (A, A) = B (A — A)—(A, A) = I[B (A — Al— (A , A) Sx 8x Sx (it is easy to see that the introduction of autonomous brackets is valid). It remains to notice that the operator in brackets is equal to

21

3

B(A — A) = BA — AB = [B, A],' and, consequently, 8f 1 3 [B, f (A)] = [B, A]—(A, A). Sx 2

D

The proposition is proved. Theorem 1.7 Formula (I.18) is valid for an arbitrary derivation of A. In order to prove Theorem 1.7, we need the following lemma.

Lemma I.1 The left regular L representation takes each derivation D of A into the inner derivation LDL -1 of L(A). Proof Let D be a derivation of A and A E A. For any C E A

L D(A)(C) = D(A)C = D(AC) — AD(C) by the Leibniz rule; we can thus write LD(A)(C) = D(LAC) — LAD(C) = [D, i.e.,

L D (A) = [D, LA], that is, we have the commutative diagram

La A

](C ),


54

D

A

A

I

Li

L [D,•]

L(A)

which shows that any derivation D is transformed by L into the operator [D, • ] of 0 commutation with D in .C(A). The lemma is proved. Proof of Theorem 1.7. Let D be an arbitrary derivation of A. By Proposition 1.3, 3f 1 3 2 [D, f (LA)] = [D, LA]— (LA, LA), Sx

By Theorem 1.2,

f (1, A) = L f (A) , and so, by Lemma 1.1, we obtain 3f 1 3 2 L D(f(A)) = I, D(A) - (LA, LA) 3x

=L

2

xf 1 3

D (A) (A , A)

(we have applied Theorem 1.2 one more time). Since L is a faithful representation, we obtain 2 sf 1 3 D(f (A)) = D(A)—(A, A), 3x o as desired. The proof is complete.

3.2 The Daletskii—Krein Formula Let us now return to our original problem of computing the coefficient Ci defined in (1.16). To this end, consider the algebra Aft } whose elements are (infinitely differentiable) families of elements of A depending on a numerical parameter t. Clearly, the mapping d A Al r } --> A{t} dt taking each family A(t) E A{t} into its t-derivative is a continuous derivation of the algebra Af t). By Theorem 1.7 we obtain :

3 sf 1 2 d — f (A(t)) = A t (t)— (A(t), A(t)). dt Sx


55

This is the famous Daletskii—Krein formula, obtained by these authors in [29] for the case of self-adjoint unbounded operators on a Hilbert space (their technique involved spectral families). From this we easily derive a formula for the coefficient (I.16). Namely, take A (s) = A + EB, then A1 (E) --. B, and we obtain 2 6f 1 3 Ci = B-8---- (A, A), x

2 sf 1 3 f (A + EB) = f (A) + E B — (A, A) + 0(82 ) . bc

Remark 1.8 If [B, A] = 0, one can take the same Feynman index for both A's on the right-hand side of the last equation, thus obtaining the usual Taylor expression

af

f (A + EB) = f (A) ± E B — (A) +

Ox

3.3 Higher-Order Expansions What we discussed in the preceding subsection was, in fact, the first-order infinitesimal calculus in a noncommutative setting. We have evaluated the differential of f (A), that is, the linear part of the increment f (C) — f (A) w.r.t. the difference B = C — A. It is given by the Daletskii—ICrein formula, which can be rewritten as 8f 1 3 = B — (A, A) B) =- — f (A + 03)] Sx de e=0 def [ d

2

and employs at least the notation of noncommutative analysis, although there was no indication of it being useful on the left-hand side. However, we should like to develop infinitesimal calculus in its full extent, which assumes deriving expansions of arbitrarily high order for the increment cited, with explicitly writing out the remainders. Hence, let us consider the difference f (C)— f (A) more comprehensively. Assume that C — A = EB, where 8, as above, will be treated as a small parameter. Then it becomes much easier to keep track of the orders of various terms in our formulas and to explain convincingly the order of the accuracy of our expansions. The simplest and clearest method to obtain the expansion of f (C) — f (A) = f (A + E B) — f (A)


56

in powers of E is to apply the Daletskii—Krein formula successively. Namely, we can write down the usual MacLaurin expansion (A + s B) — f (A) 2.-. '.

cx) CkE k E f k ' k=1

!

of E k is the kth E-derivative of f (A ± s B) at way, the Daletskii—Krein formula holds for E 0 0 as well:

where the coefficient

Ck

E=

O. By the

3 2 8f 1 d f (A ± sB) = B— (1[A ± en IA + EMI). cle Sx

Thus we can differentiate it with respect to E once more. There will clearly be two terms arising from the differentiation in the first and in the second argument of 8f18x; and the formula itself applies to each of these terms. Hence we obtain 2 4 82 f 1

d2

3

5

f (A ± s B) = 2B B—(1[A ± s B]], 1[A ± EMI, TA + s B]]) de Sx 2

(the factor 2 is due to the presence of two terms, which are equal to each other since 8f18x(x, y) is symmetric, 8f 8f — (x, Y) = — (Y, x), Sx Sx and so it makes no difference with respect to which argument the subsequent derivatives are taken). At this stage, it is easy to predict the general result (and to prove it, which we leave to the reader): dk —

de

24

2k sk f 1

f (A ± EB) = k!B B .. . B

2k+1

3

+ s B[1, 1[A ± EMI, ... , [[A ± s B]]).

Sx k

Thus the factor 1/ k! in the MacLaurin expansion cancels out, and we obtain the Newton formula e° , 2 4 k=1

2k8kf 1 3

Sxk

2k+1

••'

A)

or, forgetting about E, f (C) — f (A) e.'.

Sk f 1 3 °° 2 4 2k 2k+1 E ic _ All[C — A]] . . . [[C — All j (A A . . . A ) . 8xk .--.„--.—., L '

k=1

k copies

k+1 arguments

However, these formulas are still of little practical importance, chiefly because of the mysterious sign " La " in the middle. What does it mean exactly? This is not so easy to explain without going into functional analytic peculiarities, but there is an alternative, stating that we will always be on the safe side if we have an explicit formula for the remainder. Fortunately, such a formula is at hand. -


57

Theorem 1.8 (Newton's formula with remainder) Let A and C be generators. Then for any symbol f (x) and for any positive integer N we have N-12

f (c)

- f (A) =

sk

2k

2k+1 A, ... , A ) + RN, E[c - Al_ Ic - il]](A, E 8.3c k k=1 13

f

where the remainder RN is given by the formula 2

8N f 1

2N

2N+1

3

A ).

RN = I[C — Al ... [ C — All (xN

Proof We proceed by induction on N. Clearly, 3

1

3

1

f (A)

3

1

f (C) — f (A) = f (C) — f (A) = (C — A)

.

C—1 We can move the indices apart and then isolate the factor C — A, 1

1

3

(C — A)

3

f (C) — f (A) 1 3 C—A

0

=

1

(C

4

3 f (C) — f (A) =[ [2 ., A) 0 4 C—A

4

0

Ai, f (C) — f (A) ii

0

4

.

C—A

3

1

Since C — A = C — A, it follows that 2 3f 1 3 f (C) — f (A) = IC — A]]--= (C , A), Sx

(1.19)

which is none other than Newton's formula for N = 1. Let us carry out the induction step N = 1 = N = 2. To this end, we apply the last identity to the difference 4 2 [C —

(5f 1 3 2 5f 1 3 Al —8x(C , A) — I[C — A]]— (A, A). Sx

We obtain

3f

2

TC

—

13

2

8f

13

0

2

32 f —1 1 3

All— (C, A) = TC — Al—(A, A) + [C — ATV — MI T— x2 (A , A, A). Sx Sx

The second term, up to an inessential change in indices, is just the remainder R2. Proceeding in a similar way, we accomplish the proof. D 2

4 This means that we apply the identity (I.19) to

1 1 operators C and A instead of z.

3

the function I[C — A]3f/8x(z, A) substituting

58


The remainder is small in the following sense. If C — A = EB, then E N factors out of RN. Also, the distinguishing feature of Newton's formula is that all its terms, except for the remainder, are "multilinear forms in 2

2k

1[C — Al, ... , [[ C — A]] with various Feynman indices". In the commuting case ([A, C] = 0) Newton's formula is reduced to the conventional Taylor formula with operator arguments. However, the commuting case is a distinguished one; it can be viewed as the "maximally degenerate case" of noncommutativity. Therefore, it is not surprising that there are a large variety of formulas generalizing the Taylor formula to the noncommutative case. Let us get acquainted with one of these formulas. Theorem 1.9 (Taylor's formula with remainder) Under the hypotheses of Theorem 1.8 one has

E _k!1 f (k) (A)[(C1 — A ) ]] + QN, N

f (C) — f (A) =

2,

k=1

where the remainder QN is given by the formula 21

QN =

3N f 1

2

[RC — A) N 1

3

3

8xN

Proof By the usual Taylor formula for functions of numerical arguments,

f(y) - f (x)

N x---% 1 f(-„

= 2 IT;

1 (x) (y - x) k + QN(x, Y),

k=1 •

where QN(X , y) is easily proved to be equal to

0f QN(x,

y) =

2 1 Let us insert y ---> C, x ----> A into the formula for f (y) — f (x). We obtain

N

f(C) - f (A) =

1

Ek=110•

2

1

2

21

f (k) (A)(C — A) k + QN (A,

q.

This yields, by moving indices apart and isolating appropriate factors, the desired formula. The proof is complete. D Let us now make a comparative analysis of the Newton and Taylor formulas. Both formulas are quite easy to obtain; beyond this, there are more differences than similarities.


59

First of all, strange as it may seem, the Taylor formula does not generally provide an expansion in powers of E if C = A --1- EB. We claim that, unless A and B satisfy some algebraic relations, the remainder 21

2 7„ 8N fl1 3

3

Q N = (C - A)" II---j— (C A . . . A) ' 8x N ' '

is of order E regardless of N. To understand this well, let us consider a few of the first terms of the expansion. The point is that although 1

2

C—A=C—A=eB=0(e),

this does not mean that 1 2 „, (C - A)" = 0(E N ).

Indeed, say, 1 2 (C - A) 2 0 (C - A) 2 .

Instead, we have 2 1 .2 _ 2AC, (c, _ A)2 = c2 ± A (c, ..._ A)2 = c2 ± A.2 - AC — CA,

and these two operators differ by CA — AC = [C, A]. However, C = A ± 8B and hence [C, A] = E[B, A], because A commutes with itself. Eventually, we obtain

1 2, (C - Ar = 62 172 ± E[B, A], and this is not 0 (5 2 ) unless [B, A] = O. 1

2,

Next, let us compute (C — A)'. By similar computation, we are led to the relation 1 2 „ (C - A)' = e`[A, BIB — e 2 [A, (B — A) 2] + E[A, [A, B]]. 2, 1 We see that (C — A)" = 0(E) in general. However, if [A, [A, B]] = 0, then 1 2 (C - A) 2 ) = 0(6 2 ). Similarly, if for some mo

[A, [A, ... [A, B] .. .] = 0, ,---„,--, m times


60

then the order in E of summands in the Taylor formula increases with N, though less rapidly than in the Newton formula. This enables us to compute the derivatives (--dd7 ) k f (A + E B) 1 in terms of the derivatives of f (x). Thus, if E=0

[A, [A, B]] = 0,

then we have 1 = f'(A) B — — f"(A)[A, B], 2 6=0 1 , 2 d2 = f"(A)B - — — f m (A)[A, BIB — f (A + EB) 3 3 d82 =o 1 ±- f"(A)[A, B]2 . 4

( Te d f (A + E B))

We see that the derivative of order k of f (A + E B) w.r.t. higher-order derivatives of f taken at A.

E

B]

can be expressed via

Remark 1.9 The Daletskii—Krein formula can also be considered as the usual formula for the differential. Namely, a symbol f can be viewed as a (partial) mapping from A to A that assigns f (A) to each generator A. We may wish to consider the differential of this mapping. According to the general rule, the differential of a mapping q) : A ----->- A is a linear mapping ql,,, : A —> A, i.e., an element of L(A). Having this in mind, we can represent the Daletskii—Krein formula as follows: 3f f(A) = — (L A RA). Sx '

This formula is understood in the sense that df (A, B) = MA)(B) = MA)(B)

=

2 8f 1 3 Sf LA, RA)B = B — (A, A). Sx Sx

—(

We have considered various expansions for f (A + AA). These expansions are closely related to the conventional Taylor's formula in the analysis of functions of the numerical argument. However, there are also several topics specific to noncommutative differential calculus and having no counterpart in the usual calculus. We mean the index permutation formulas and the composite function formulas considered in the following two subsections.

3.4 Permutation of Feynman Indices Given a binary symbol f (x, y) and two generators A and B, one can define f (A, B) 12 2 1 (A, B). can take f (A, B) or f in two different ways using Feynman's approach: one How different will the results be? The answer is given by the following theorem.


61

Theorem 1.10 (Index permutation formula) 12

21

3

82 f

15

24

f (A, B) = f (A, B) + [B, A] —3. . (A, A, B, B) 8y 21 3 B2 f 2 4 1 5 = f (A, B) + [B, A] - (A, A, B, B). 8x8y 12 2 1 consider the difference Proof Let us [[ f (A, B)]] — [[ f (A, B)J]. We can change the indices in this difference arbitrarily provided that (i) the order of operators in each term remains unchanged; (ii) noncommuting operators are assigned different indices. This being done, we can omit the autonomous brackets. In particular, we can write 12

21

12

32

[[f (A, B)]] — [[f (A, B)]] = f (A, B) — f (A, B).

Let us now recall that once all operators in an expression are equipped with indices, we can transform the expression according to the rules of commutative algebra. Hence we have 12

32

1

f (A, B) — f (A, B) = (A — A)

12 32 3 f (A, B) — f (A, B))

1 132 3 f = (A — A)— f (A, A, B).

1 3 (Sx A—A Strange as it may seem, this computation is perfectly rigorous; in fact, it means nothing other than that we take the formula Sf f (x, y) — f (z, y) = (x — z) f (x , y) - f (z, y) = (x — z)— f (x, z; y) x—z

Sx

123

and substitute the operators A, B, A for the variables x, y, z. In the following, we always use the shortened form for computations like this. The relations obtained give 1 2 21 1 3 Bf 1 32 f (A, B) — f (A, B) = (A — A)—(A, A, B). Sx Note that we do not write autonomous brackets on the left-hand side of the last equation, although they must formally stand there; from now on we widely make such abuse of notation provided that this cannot lead to misunderstanding. Changing the indices again according to (i) and (ii), we obtain 12 2 1 f (A, B) — f (A, B)

1 Sf 1 3 2 3 8f 1 3 2 = A(A, A, B) — A — (A, A; B) 8x Sx 3 Sf 1 5 4 3 8f 1 5 2 = A" (A, A, B) — A — (A, A; B) 8x 6x 8f 1 5 2 3 8f 1 5 4 = A[' (A, A; B) — —(A, A; B)]. Sx Sx


62

We transform the right-hand side of this relation in a similar way, by introducing yet another difference derivative, and obtain 1 2 21 34 282f 1524 f (A, B) — f (A, B) = A(B — B)—(A, A; B, B), SxSy or, by (i), 1 2 2,5 82 f 1 5 2 4 2 1 3 3,5 f (A, B) — f (A, B) = A(B — B)— (A, A; B, B). SxSy 3 3,5

We can now extract the linear factor A( B 1 2

2 1

2,5 —

B):

3 3,5

62 f

2,5

1 5 2 4

f (A, B) — f (A, B) = [[A(B — B)]]—(A, A; B, B). 8xSy

This is just the first variant of the index permutation formula since 3 3,5

2,5

A(B — B)= BA — AB =[B, A]. The proof of the second variant of the formula is left to the reader. The theorem is proved.

0

Remark 1.10 Let us point out (though this is trivial) that expressions of the form il

i2

in

il

(A — A) f (B 1, . • • , Bn) i generally do not vanish if [Bi, A] 0 0 for at least one 1 such that ii 3. Proof We apply the first version of the equation in Theorem 1.11 to the second term on its own right-hand side, thus obtaining f (ffA + BA)

1 2 3 62 f 5 61 21 4 f (A + B) + [A, B]6y2(A ±B,A+B,A+ B)

=

3 1 21 45 65 8 84f9 -F[A, B]- (I[A + B]], A+B,A+B,A+B,A+ B).

44

In view of Theorem 1.11, we can omit the autonomous brackets in the third term of the obtained expression and use any admissible indices over the operators, with the resultant remainder being of degree > 3. Thus, we have

62 f 5 61 21 1 2 3 4 f a[A + B]])'.-`-' f (A + B) + [A, B], (A + B, A + B, A + B) 44

2,,

1

31

31

31

3

+IRA, Bfiler f (A +B,A+B,A+B,A+ B), where -2-'-= can be translated as "is equal modulo terms of degree > 3". 5

In the second term on the right-hand side of the last equation we permute the operator A consecu4

3

tively with B and [A, B]: 3,

5

61

21

4

3

32 f 4

61

21

5

[A, B ] ' f (A + B, A + B, A + B) 2-1= [A, B]6y2(A +B,A+B,A+ B) 21 51 3 4 f4 91 7 98 3 6 -1- [A, B][A, B]- (A+ B, A + B, A + B, A + B, A + B)

6 y4


68 4

62 f 3

61

21

5

= [A, B]— (5),2 (A +B,A+B,A+ B) 71 21 6 4 83 f 3 75 +[A , [A, B]]— (A + B, A + B, A + B, A + B) 43

51 21 7 84 f 4 98 91 3 6 A + B). (A + B, A±B,A+ B, A + B, +[A, B][A, 13]44

In the second and third terms of the last expression we can use any admissible order of operators modulo terms of order > 3. Hence, 1

2

4

21

61

5

2f 3 f ([[A + Bil) = f (A + B) + [A, B]6y2(A +B,A+B,A+ B) 33 f 1

2

31

31

3

+[A, [A, B]]— (A +B,A+B,A+ B) Sy 3 64 f 1 31 31 2 31 31 3 + B, A + B, A + B, A + B). (A+ B, A + 2 HA + M211 75-y4 2

3

4

In the second term on the right-hand side of this equation we permute B with A and [A, B]. Similar calculations yield 1

f (IA + Bp

2

4

82f 1

31

3

31

f (A + B) + [A, B]— (5y2 (A + B, A + B, A + B) 83f 1

2 -

31

31

31

3

FHA, [A, B]]+ [[A, B], B]]]—(A + B, A-FB,A+B,A+ B) By3

2, 64f 1

31

31

31

31

3

+3[[[A + Br]1— sy 4 (A + B, A-F B, A -I- B, A + B, A -I- B). The last formula proves the corollary, since Sk f/Bx k (x, ... , x) =

Let us

r (k) 1( x)

Ei

now proceed to the general case.

Theorem 1.12 The following composite function formula holds: 1 2

12

f ([g(A, B )]I ) = f (g(A, B)) 28 78 5 sg 3 4 6 sg 2 7 8 62 f 1 12 ±[A, B]—(A, B, B)—(A, A; B)-(---5 y2 (D(A, B)]], g(A, B), g(A, B)). e5 xi Sx 2, Proof We have 12

12

12

112

f (l[g(A, B) ]1) — f(g(A, B)) = fa[g(A, B)]]) — f (g(A, B)) 112

23 8f 1 12

23

= a[g(A, B )]] —g(A, B))—([[g(A, B)]], g(A, B)) Sy 36 212 45 sf 1 1 2 = a[g(A, B)]] —g(A, B))—([[g(A, B )]I, g(A, B)). Sy

3. Noncommutative Differential Calculus 3

69

212

By permuting the operators A and [[g(A, B)]] in the last expression, we obtain 26 12 12 312 4 5 sf 1 12 g(A, B)) f([[g(A, B)]1) — f(g(A, B)) = (1[g(A, B)]] — g(A, B))— ([[g(A, By 3 12 82 f 1 12 25 45 k g 2 4 5 -I- [A , [[g(A, ([[g(A, g(A, B), g(A, B))- (A, A, B). 3x 1 By 4

Here the first term on the right-hand side vanishes, which can easily be proved by index manipulations (see Proposition IA) 3g 1

3

12

24

[A, [[g(A, B)]]] = [A, B]—(A; B, B). 3x2

The theorem is proved.

Let us now apply the obtained formula to pseudodifferential operators. 2

1

.

l ax ) be an h —l -pseudodifferential operator, and let a Example 1.10 Let P x ( —tha , function f(z) of a single variable z be given. We will rewrite the operator 2

1

f (IP x, ( —iha lax )1) in the form of an h -1 -pseudodifferential operator modulo 0 (h2) . Due to Theorem 1.12 we have 1\

1

a

2

5

a

ih Tx ,x]

sP

(4 6

x,x, —th y;

a

s2 f 1

x — ([1 P (x, —ih—)]], P ax Sz 2

a

2

) (x, —th y;

f([13 (x, —th—)]]) P = (f ax

)

SP (8

x,

2\

(8

a

Dx

—ih Tc

)

7

,P

ax

).

Since [—iha/ax, x]= —ih, we can omit the index 5. Consequently, the indices 4 and 6 can be set equal to each other. Taking into account that

SP

— sx (x,x, p) =

aP(x, p) ax

we get 1 a 2 a (2 f (Up ( x, —ih— ]]) = (f o P) x, —ih— ax :x) 2 2 a ) SP (5 a) P (3 a —ih— x, ih— — — x, —ih— —ih— ax ax ax Sp ax' DP


70

4

32 f 1 (2

x—a[P Sz 2

419

(5

x,—ih— 1, P x,—ih ax ax

)

(5

a

, P x,—ih— ). ax

Computing up to terms of order h 2 , we can change arbitrarily the indices over operators in the second term on the right-hand side of the last formula. Hence, 1

a fa[13 x, ( —ih y; 2

2 _. â) h ax

) JD= (f o P) x, t

1 P ap — (f" o P)] i,—ih—a +0(h2 ).

ax ap

L

—[

ax

Similarly, one can calculate the subsequent terms of this expansion.

4 The Campbell—Hausdorff Theorem and Dynkin's Formula As a sample application of the techniques of noncommutative analysis, let us consider a famous old problem in the theory of Lie algebras and Lie groups. This problem was already mentioned in Subsection 1.5, but here we start from the very beginning and give a more detailed exposition. Those who are not familiar with Lie algebras and Lie groups at all may wish to consult the Appendix, where all the necessary information is provided; however, for better readability we reproduce here some of the definitions.

4.1 Statement of the Problem Let L be a (finite-dimensional) Lie algebra. This means that L is a (finite-dimensional) linear space equipped with a bilinear operation [., •], referred to as Lie bracket and satisfying the following conditions: i) [a, b] = —[b, a] (antisymmetry); ii) [a, [b, c]] = Ra, b], c] + [b,[a, c]] (Jacobi identity). For our aims it suffices to assume that L is realized by square matrices of size mxm, with the usual matrix multiplication l . This assumption is, in fact, unnecessary, and we are only doing so in order to simplify the exposition by avoiding the consideration of unbounded operators. 1 By

Ado's theorem this is the general case.

4. The Campbell—Hausdorff Theorem and Dynlcin's Formula

71

Next, let G be a Lie group, i.e., a manifold equipped with smooth group operations. Again, we assume that, at least locally2 , G is a represented as a matrix group. Then the tangent space Te G to G at the point e is naturally identified with a subspace of the space of matrices, and it can be shown to possess the structure of a Lie algebra. There arises a natural question: given a Lie algebra, can one reconstruct the multiplication law in the corresponding Lie group? Let G be a Lie group and L the corresponding Lie algebra. There is a mapping

exp : L —> G

taking each element X E L into the element exp(X) E G defined as follows: let { g (t)} be the one-parameter subgroup of G defined by the condition AO) = X. Then we set exp(X) = g(1).

Thus we obtain a coordinate system in the vicinity of the neutral element e E G; this coordinate system is referred to as the exponential coordinate system. Let us seek the multiplication law in the exponential coordinate system. If G is commutative, then the multiplication law has the form A • B = A + B. If G is not commutative, there appear correction terms on the right-hand side of the last equation, namely, 1 2

A • B=A+B± — [A, B]-1- • • • . The first correction is equal to 1[A, B] and this is completely determined by the commutation law in L. The Campbell—Hausdorff theorem [19] asserts that the same is true of all subsequent terms of this expansion, which implies that the multiplication law in G can be reconstructed given the commutation law in L. E. B. Dynkin [43] found an explicit exposition for all terms of the series. In this section, following Mosolova [139], we find a closed formula expressing ln(eA e B ) in the form of an integral rather than a series. Then we use the conventional Fourier expansion so as to obtain the expression for ln(e A e B ) via the commutators. First of all, let us consider functions of commutation operators in more detail; this proves to be useful not only in the particular problem considered, but also in the general framework of noncommutative analysis.

4.2 The Commutation Operation We have already seen how important a role commutators play in noncommutative analysis. It is clear that this operation is worth studying. In this subsection we obtain a simple expression for functions of the operator of commutation. 2 That

is, in a neighbourhood of the neutral element.

72


Let A be an algebra and B e A some element. Denote by adB : A —> A the linear operator that takes each element A E A into its commutator with the element B: adB :

A —>- A A 1—> adB (A) = [B, A].

Clearly, ad B (A) : BA — AB = L B(A) — RB (A), that is, adB = LB — RB, where L B and RB are the operators of left and right regular representations introduced in Subsection 2.3. The powers of adB give rise to multiple commutators with B: (adB) k (A) = [B . . .[ B , A] . . 1. k times

Let us find as expression for such commutators using Feynman indices. Note that 3

12

[B, A] = BA — AB = (B — B)A. Similarly, (adB)k (A) = (LB — RB) k (A) = (13 — 113)k /21' , according to the description of functions of the operators LB and RB given in Subsection 2.3. Moreover, for any symbol f (x) we have 3

12

f (adB)(A) = f (B — B)A.

(1.20)

Although this follows directly from the formulas given in Theorem 1.3 let us present the proof for two particular cases in which the functions of operators are defined via the Fourier transform or the Cauchy integral formula.

A. The Cauchy integral formula. Since the mapping f i-± f (A) is continuous, it suffices to consider the case f (x) = (A — x) -1 . Let f (x) = (A — x) -1 , where A 0 a (B)— cr (B), that is, A cannot be represented in the form A = Al —A2, where Xi E a (B),i = 1, 2. Then the function (A — x + y) -1 is analytic in the neighborhood of

4. The Campbell-Hausdorff Theorem and Dynkin's Formula

73 31

2

the product a (B) xcr (B) c C2 , and hence the operator (X - B ± B) A is well-defined. We have

(X - adB)

1

(

3

X—

-

A

1

3

1

2

12

A

B + B)[[

= (X

1 1

3

A-B±B 1 = (X B ± B) 3

A

B + B)

3

B±B 2

4 0

= (X

2)

X-B+B

2

A 3

1

=A

(we used extraction of a linear factor and moving indices apart). Hence the operator X - adB is the left inverse of the operator 1

A-*

2

3

1

A.

Similarly, it can be proved that the operator X - adB is the right inverse. Consequently, 1

Rx(adB)(A) =

2

A. 3 1 X-B ±B D

B. The Fourier transform formula. Let 3

i 2

U(t)A = eit(B-13) A.

We intend to show that U(t)= eitadB.

Indeed, let us differentiate U(t)A by t. We obtain 3 1 , 31 2 4 0 „1 n1 , 2 d —U(t)A = i(B - B)e 1t(13-B) A =i(B - B)ei t 'D - D ) A dt 4

0 2 „!

nl, 2

= (B - 13)[[e`t 'D - D' Al 2 3 1 = i(B - B)U(t)A = i adB(U(t)A).

Moreover, for t = 0 we have U(0)A = A, that is, U(0) = id = eitad B 1

t=0

•

We see that U(t) and eit adB satisfy the same Cauchy problem and hence are equal,


74

which implies the desired result for the case in which functions of operators are defined via the Fourier transform. Let us now prove a technical lemma, which will be used in the next section.

Lemma 1.2 Suppose that A, B,C

E

A are elements such that

e c = eB e A . Then

e adc

d ad = e a B eA .

12

4 03

Proof We have 3

1 2

e adB eadA (D) = eadB (eA—A D) = e B—B e A—A D =

3 4 2

1o

e A+B D e —A—B

But 34

e A+B = e C ,

10

e —A—B = e —C .

Thus,

e adB eadA (D) = ec De —t-, = as desired.

4.3 A Closed Formula for ln (eB eA ) For simplicity, assume that A and B are elements of a Banach algebra A with identity; this means that A is a complete normed linear space, and the multiplication defined on A has an identity element 1 E A and satisfies the inequality

11xY I I

11x11 • 111'11

for any X and Y E A. Throughout the following exposition we only use analytic symbols and define functions of operators via Taylor series or Cauchy integrals. Our intention is to find an explicit expression for the element C = ln(e B e A ). Let us consider the family of elements D(t) = in(etB eA ) with parameter t. Clearly, we have D(0) = A and D(1) = C. Moreover,

e by the definition of the logarithm.

t) = etB e A

4. The Campbell—Hausdorff Theorem and Dynkin's Formula

75

We differentiate the last equation by t and multiply by e —D(t) on the right. This gives the following result:

B =

±,130)) e —D(t) . ( dt -

The latter equation can be transformed with the help of the Daletskii—Krein formula (see Subsection 3.2), which asserts that

d D(t) —e =

1

3

2

e D(t) — e D(t)

D' (t) 3

dt

1

.

D(t) — D(t)

Thus, 3

B=

1

2 e D(t) — e D(t) D' (t) 3 1

o

e — D (t ) .

D(t) — D(t)

There are no operators with Feynman indices between 0 and 1 on the right- hand o

side. Therefore, the "moving indices apart" rule applies to the operators D(t) and I D(t), and their indices can be set equal to each other: 3 1 e D(O—D(t) _ 1

2

B = D' (t)

3

1

.

D(t) — D(t)

According to the results of Subsection 4.2, 3

12

f (D — D)E = f (adD)(E),

where adD is the operator of commutation with D. By taking D = D(t), E = V (t), and f (x) =

ex

—

1

X

we obtain B = f (adD(0)(13' (t)).

We apply the operator f (adD(0) -1 to both sides of this equation and obtain

1

(B)

f (adD(0)

However, by Lemma 1.2, adD -(t-) = ln(e t adB e adA ),

76


and so

1

( ID = oe tad BeadA)(B), `' A)) ead f (1n(et adB

Di (t) =

where

1 ln z = . f (ln z) z-1 Finally, we use the Newton-Leibniz formula and obtain q)(z) =

1

t

C = A + f Di (t) dt = A + f go (e t adB eadA) d t (B) . 0

0

We see that C is expressed as a linear combination of A, B, and their commutators. Indeed, we can expand (p (z) into a Taylor series in powers of z - 1: ln z

°°

v(z) = z - 1 = E( ok k=0

thus,

(Z - 1)k

•

k+1'

00, (et ad B ead A _ ok q)(e tad B e ad A) =E(-1) K • k+1 k=0

Let us represent the exponentials in the form of Taylor series. We obtain

E ti (adB) 1 (adAr 00

k+1

k=0

=

Nc°-■ (-1) k

nm!

(-Frn>0

q)(et adB ead A) =

k

t11±-±ik

Lak+i tiE i +; In >t., n

(adB) 11 (adA) m l ... (adB) 11 (adA) mk ii!mi!•••/k!mk!

k=0

We replace ça (e t ad B eadA ) by the last expression and perform te-by-term rm integration: \--N (adB) 11 (adAr l ... (ad B) Ik (ad Ark (B) C=Ad-B +E (-1)k k+ 1 , 1-1 n (11 + • • • + lk + 1)11!mi! .. .1k!ink! k=1

t i±mi>v

00(

=

Ad-B-1-E k=1

k

1 -

)

E

k + 1 li+mi>0,1;>0,mi>0

[B 11 A m l . . . B ik A m k

ldmi! . . .1k!mk!

13] '

where [1311 A m l . .. Blk A m k B. ] = (11 + • • • + lk + 1) -1 B[B . . .[B [A . . . [A . . .[B . . .[B [A . . . [A, B] • • •]• s........„,, ,,.„-, •-...„-,,..„—• 1k

mk


77

This expansion is slightly different from that obtained by Dynkin. In fact, Dynkin's expansion is more "symmetric" and can easily be obtained by our method if we consider the family D(t) . e tB etA

as was originally done by Mosolova [139]. The only reason for considering our family is that this leads to even simpler calculations. Thus, we have proved that ln(e B e A ) can be expressed as a linear combination of operators A, B and their commutators. There arises a natural question: where is the obtained formula valid? We do not go into a thorough investigation of this question here; let us only mention that p(z) is an analytic function in the disk I z — 11 < 1; e therefore, go (e t adA h ) is well-defined at least so long as \

I le t adBead A _

1 11 < 1 .

The latter inequality is valid at least if II adB II and II ad A I I are small enough (one can easily write down the bounds for I I adB II and I I adA II). Next, if this condition is violated, we can proceed by analytic continuation. The function go (P (t)), where P(t) = etadBeadA can be defined via the Cauchy integral 1 i g9(X) dA. q)( 13 (t))= 2 1.i ir ),,— P(t) ' C

where C is a contour surrounding the spectrum of P (t) and such that go (X) is holomorphic in the closed domain bounded by C. The function q)(z) has the unique ramification point z = 0; it is easy to see that a sufficient condition for go (P (t)) can be defined is that the spectrum of w (P (t)) does not surround the point Z = 0 and does not contain this point. If this is true, we can find a simply connected domain Dt such that Cf ( P ( t ) ) C Dt and 0 (I D. We set C = apt . Note that one should define w (P (0) for each t E [0, 1] in such a way that w(P (t)) depends continuously on t. This imposes an additional condition on the behaviour of the domain Dt as t moves along the interval [0, 1]. For instance, if both A and B are positive definite self-adjoint operators, then the spectrum of P (t) never intersects the negative part of the real axis and everything is all right. We refer the reader to M. V. Mosolova [139] for a detailed discussion of conditions under which the analytic 0 continuation is possible.

4.4 A Closed Formula for the Logarithm of a T-Exponential The reasoning in the preceding subsection can easily be modified so as to solve the following problem.


78

Problem. Let a T-exponential 1

U = exp (fA(8)d6) be given, where A(0) is a family of elements 3 of an operator algebra A. Find an element C E A such that U = ec . This problem is an obvious continuous generalization of the discrete problem of finding log(e B eA ). Again, the method of solving this problem is to use noncommutative differential calculus. To this end, we introduce the family of operators

t fo U(t) = exp ( A(0)de , t E [0, 1], 0

and seek for an operator C(t) such that

U(t) = e t) . Clearly, U(1) = U, and D(1) = C is the desired operator. Let us differentiate both sides of the last equation with respect to t. By the definition of T-exponentials, we have

d — U (t) = A(t)U(t) = A(t)e D (t) • dt

the differentiation on the right-hand side gives 2 ±eD ( ‘it) = D'

dt

3

(t)

1

e D(t) — e D(t)

3

1

D(t) — D(t)

by the Daletskii—Krein formula. Combining the last two equations, we obtain 2

A(t) = D' (t) or

3

3

1

e D(r)—D(t)

1

3 1 D(t) — D(t)

1

D(t) — D(t) 2 (t) = 3 A(t) = f (ad D(t))(A( 1")), 1 e D(t)—D(t) 1 3 Our argument is a bit formal in this subsection since we do not dwell upon the conditions that would guarantee the existence of the T-exponential, which is a rather complicated topic in its own right.

4. The Campbell—Hausdorff Theorem and Dynlcin's Formula

where f (z) =

79

z ez — 1

.

We will make use of the following lemma. Lemma 1.3 The formula r

e f admodO

eadD(t) = exp

0 is valid. Proof. We have 3

12

e admo (B) = e D(t)—Do) B = e mt) Be—D (t ) = U(t)BU(t) 1 for any B E A (cf. Section 4); in view of the definition of U (t), d .„1 —euuD(t)(B) = A(t) o U (t)BU (t) -1 — U (t)BU (t) -1 A = adA(t) (e ad mo (B)). dt On the other hand, r 0 d — exp f admod0 dt 0

(r (B) = adA(t) exp

,

f adAe (ode

(B)

0

by the definition of the T-exponential. We see that both sides of the equation in question satisfy the same first-order equation dX — = adA( t) X dt

and the same initial condition X(0) = id (identity operator). Thus, they coincide for all t. The lemma is proved. 0

It follows from Lemma 1.3 that t

e

admo = log (ffexp (f admod0)1) . 0

Hence we obtain r

e D' (t) = q) (ex p (f ad(ode)l) (A(t)),

0


80 where

q)(z) = f (log z) =

log z

z-l

=

E (1k —+ z)1 k .

ic=0

Since D(0) = 0, we obtain r

r

D(t) = f ço o

ffexp

e f ad(e)d6) ]] A(r)dr 0

and, in particular

t

1

0

C = D(1) = f q) (i[exp (f adA(0)(19)11) A(t)dt . 0

0

This is a closed expression for 1

0

C = log ([[exp (f A( 0)d0)1) 0 via the operators A (t) and their commutators. Proceeding by analogy with the preceding subsection, we can obtain an expansion of C whose Nth term contains Nth-order commutators for any N. To this end, we expand q) in a Taylor series, thus obtaining t 0 (1 — ffexp (f adA(9) de) 1) 0

1

c=

f0 kEc° =°

k

k+1

A(t)dt.

Next, we use the Taylor series expansion of the T-exponential:

t ) 0 exp (f adAmde 0

2

t

e 1 = 1 + f ad A(e)dt9 +

( 0f t 0

0

t

= 1 ± f adme) 0

adAmd0 + . • • t ei

ch9 + f f ad moo adA(02) d02de1 + • • • 00

(in the Nth-order term we divide the integration domain into N! equal simplices on each of which the factors admoi ) appear in a fixed order; it is easy to see that all these simplices give the same contribution, and the factor + I cancels out). On substituting this expansion of the T-exponential into the preceding 'formula we obtain

4. The Campbell-Hausdorff Theorem and Dynkin's Formula

c

81

cc) =EE k ±1 (

s

--

o

k

s=0 k=0

6. i- 1

. . . f admoi ) adA(92) ... ado o 0 x

dOil .. . dOi

• • •

f f ad moo adA(92) . .. adoik ) Oh, . . . d61 A(t)dt. 0

0

The sth term of the outer sum is a finite sum of integrals; this sum involves all sth-order commutators involved in the expansion of C. Let us write out a few first terms of the expansion. We have 1

t

1

1 C = f A(t)dt - - f f adA(0)(A(t))d9dt 2

o

—

f f admoo adA(02) (A (t))&92dOidt

w

th

1

t

1

oo

t

000 t

+1 f f

admoo adA(g2 )(A(t))d92dO1dt ± • • • f 000

3 f f or, in terms of commutators,

1

i

t

C = f A(t)dt - f f [A(0), A(t)]dOdt o oo 1 1 t ei -- f f f [A(6i), [A(02), A(01]d62d9idt 2

1 +3

000

i t t

f f f [A(01), [A(92), A(t)11d02d91dt ± • • • , 000

where the dots stand for a sum of integrals of commutators whose order is greater than or equal to three. Let us consider a simple illustrative example.


82

Example 1.11 Consider the first-order differential equation

au at

au —

ax

± b(t)xu

a(t)—

with the initial condition

u(x, t)Ir=o = uo(x). The solution of this Cauchy problem can easily be found explicitly by straightforward computation. We consider the equation of characteristics .i = a(t), x(0) = xo;

its solution is

r x (xo, t) = xo ± f a(r)dr.

I Set U (xo, t) = u(x(xo, t), t);

then for U we obtain the equation r

dU

dt = b(t) xo ± f a(r)dr u, o

whence

t U(xo, t) = uo(x0)exp (f

e b(6) (x0 ± f a(r)dr) d6 )

o

o

Making the inverse substitution t xo = x — f a(r)dr, 0

we obtain r

r

r

u(x , t) = uo (x — f a(r)dr) exp o

(f

b(6) (x — f a(r)dr) dO)

o

e

Now, let us compute the same solution by means of the T-exponential. Let u(t) denote the solution u(x , t) considered as the function of t with values in functions of x. We have It

u(t) = exp (frA (z)dr) u(0), o


where

83

a

A(r) = a(r)— +b(v)x.

ax

Let us compute the T-exponential for t = to (to is arbitrary) using the formula we have just obtained. We shall see that this leads to the same result. We have

a

a

( 0)— + b(9)x, [A(0), A(t)] =[a ax

± b(t)d= ( a 0)b(t) —

ax

and the commutators of second and higher orders are all zero. Therefore, only two terms are retained in the expansion of the logarithm of the T-exponential. Namely, we have to

log (exp

(f

( to

to ax

0

7

2

o

t

to

+.i1- f

1

—a + (f b(t)dt)x —

f a(t)dt)

0

2

o

o =

t

to

to

1 r A( t)d -r)) = f A(t)dt — — f f [A(0), A(t)]adt

oo to (

f

t

b (t) f a (0)&9 dt

o

o

a

def

a(t) f b(r)d -c dt = C1— ± C2x + A. ax 0

The value of the solution u(x,t) at t = to, say, coincides with the solution at t =1 of the Cauchy problem

{ay _ ,-,1 av at — -- D--i vit=o

= uØ (x )

with coefficients independent of t. We have

( 1 v(x, 1) =

f C2(x —(1 — t)Ci)dt ± X

uo(x — CO exp

o to

to

to

1 f t°a(t)dt) exp { x f b(t)dt — -2- f a(t)dt f b(1)dr

= uo (x

o

o

o

to

t

to

to

1 1 ±2 f a(t) f b(r)dr dt — -2- f a(t) f b(z)dr dt

o

o

o


84

to to to = uo ( — if a(t)dt) exp I f b(t) ( — f a(r)dr) dtl , t o o which coincides with the value obtained by the straightforward method. The method shown in this example is quite useful if for each t the operator A (t) is a linear combination of generators of a nilpotent finite-dimensional Lie algebra,

m

A(t) =

E a., (0 Aj ,

where Ai are the generators. In this case the computation of the T-exponential t 0 exp ( f A(0)d9 is reduced to the computation of exp( E oti (t)Ai (t)), where the o coefficients cei can be obtained by appropriate integrations from the above expansion. Even if the Lie algebra in question is not nilpotent, the problem is greatly simto I plified since we reduce the computation of exp ( f A(0)d0) (operators in an infinite° / dimensional space) to the computation of 1

f

t

(i[exp

o

o (f ad(e)c/81) )

o

which in fact is carried out in the m-dimensional space of the Lie algebra.

5 Summary: Rules of "Operator Arithmetic" and Some Standard Techniques In the preceding sections we have considered a number of notions and statements pertaining to noncommutative analysis. We hope that it is now clear to the reader why noncommutative analysis proves so useful in applications. Here we summarize the main notions and notation introduced and also give a few common techniques, or even "tricks", that are usually employed in noncommutative analysis. So far we have been studying the "operator arithmetic", which comprises the rules used in treating operator expressions and the corresponding notation.

5. Summary: Rules of "Operator Arithmetic" and Some Standard Techniques

85

5.1 Notation The subject of noncommutative analysis is functions of operators, more precisely, numerical functions of numerical arguments substituted by linear operators (or elements of some associative algebra). The definition of such a substitution encounters certain difficulties. First, even for functions of a single argument it is not clear a priori what the substitution means if the function is not a polynomial; second, if there is more than one argument, then an additional ambiguity arises, caused by the fact that a function of commuting numerical arguments cannot in principle carry any information about the arrangement of the arguments. The first difficulty is overcome by the traditional method following from the familiar functional calculus of a single operator. The requirement that the mapping symbol * operator be a continuous homomorphism taking x into A uniquely determines the operator f (A) for each symbol f (x) for an appropriate class of symbol spaces (see Theorem I.1). The second difficulty can be removed by introducing new notation having no counterpart in classical analysis. This new notation includes Feynman indices and autonomous brackets. —

Feynman Indices. In order to determine the order of action for operators occurring in an operator expression, we write an index (a real number) over each of the operators. The factors in all products are arranged so that these indices decrease from left to right (that is, the leftmost operator has the highest Feynman index, and the rightmost, the lowest). For example, 1320

AABC = ABAC. If two operators commute, their indices are allowed to coincide; this does not lead to any ambiguity.

Autonomous Brackets. These are the special brackets [111 used to determine the order of computations in an operator expression. The expression within the brackets is computed first and then is used in the subsequent computations as a single operator (equipped where necessary with a Feynman index, which is written in this case over the left autonomous bracket). In particular, the indices outside and inside the brackets do not affect the action of each other. For example, 112

2

1012

AAB = BA, AffAB]] = ABA; 1

2 ,

(A ± 1

= A2 ± B2 ± 2B A, 2 ,

Br =

(A ± B)2 = A2 ± B2 ± A B B A .

86

I. Elementary Notions of Noncommutative Analysis Nested autonomous brackets are allowed. For example, the expression

1 431 2 AA[[(A + 1[A + B1)21 should be computed as follows. First we compute the inner bracket

C = 1[A + B]]= A ± B, and then the outer one: 1

2,

D = [[(A ± C)]] = A2 + C2 + 2CA

= A2 + (A + B) 2 + 2(A + B)A = 4A 2 ± B2 +3BA ± AB. Finally, the entire expression is computed: 143

AAD = A(4A2 + B2 + 3BA + AB)A = 4A4 + AB 2 A + 3ABA 2 + A2 BA.

5.2

Rules

It follows from the definition of Feynman ordering that it is the mutual order of the Feynman indices, not the indices themselves, that counts. Hence the indices can be changed without affecting the value of an operator expression. Let us now state a rule describing admissible changes of indices. A pair of arguments A, B of an operator expression is said to be critical if the following conditions are satisfied: (a) A and B do not commute, [A, B] 0 O. J

(b) The operator expression cannot be represented in the form E1 ± E2, where A / occurs only in El and B in E2. J

(c) The expression does not contain autonomous brackets such that A is inside the / brackets and B outside the brackets, or vice versa.

The Rule of Changing Feynman Indices. Feynman indices in an expression can be changed arbitrarily provided that for any criticalpair the ordering relation between the Feynman indices of the operators in the pair remains valid.

Examples. 1)

2)

1

2

2

1

12

A+B=A+B (the pair A, B is not critical since condition (b) is violated). 123 112 1 1 f (A, A, B) = f (A, A, B) = g(A, B), where g(x, y) = f (x, x, y). Thus, the changing indices rule includes changing the number of arguments of a symbol by restriction to the diagonal ("moving indices apart", Proposition I.1).

5. Summary: Rules of "Operator Arithmetic" and Some Standard Techniques 5,

321

3)

5

324

87

31

BEA ± BY]] = BEA ± B) 2 1 (the pair B, A is not critical since B and A are separated by an autonomous bracket).

The Rule of Deleting Autonomous Brackets. Autonomous brackets can be removed if for any operator argument within the brackets its Feynman index is in the same relation with the Feynman indices of operators outside the brackets as the index of the brackets themselves. Naturally, this rule also tells us whether we can introduce the autonomous brackets.

Examples. 4

232

1)

4

32

15

1[A(B — B)}B f 3x(B , B) = A(B — B)8 f /8x(B, B). 2

31

2)

15

1

EA ± B) 21I(A

2, —

1

2,1

2,

0 (A ± BY (A — BY

in general, since the Feynman index

1

2

of the A inside the brackets, which have the index 3, is less than that of B outside the brackets. Surprisingly, this is almost all the new notation and rules to be introduced in order to permit us to deal easily with operator expressions.

5.3 Standard Techniques When someone applies differential and integral calculus to solve a particular problem, he or she generally uses a lot of standard techniques such as integration by parts, L'Hôpital's rule, etc., being however scarcely aware of their presence, because they are used often enough to become fairly commonplace. Similarly, noncommutative analysis includes quite a few techniques of its own, which should always be at hand in noncommutative computations. Some of them were already presented in this chapter; and the others will be given in the sequel. Here we recall briefly what we have already seen above. The methods can be divided into a few groups. First, there are some simple transformations following directly from definitions and from the rules of changing Feynman indices and inserting (deleting) autonomous brackets, given in the preceding subsection. It is probably convenient to present them as a number of axioms, as was done in [129 ]. ( pi) (Homogeneity axiom). For any symbol f and number a we have

i[a f (All

nk

ni

A

= (YU (Al

nk

A011.

, nk and m i , . . . , mk be Feynman indices (p,2) (Index changing axiom). Let n 1 , such that ni < ni if and only if mi < mi. Then nk

f

Oh

• • • , Ak) = f

ml

mk

(Ai, • • • , AO.


88

If ni = ni and Ai = Aj = A, then ni

fl

ni

ni

ni

nk

where g(xi, . .. , xi, — • , xi, • • • , xk) = f (xi, • • • , xidix .i =xi • ni

(p,3) (Correspondence axiom). If f (x i , .. • , xk ) = 1, then f (Ai, ni

tity operator); if f (x i , .. . , xk ) =

nk ... , Ak) =

1 (iden-

nk

then f (iii, • • • , Ak) = Aj.

(p,4) (The sum axiom). ni

mi

nk

mi

ni

mi

mi

nk

Ak)± g(B 1, — , BO = U(Ai, ••• , Ak)11+ ffg(B 1, • • • , B 01 (p,5) (Product axiom). If mi < n1 for any i and j, then ni

nk

mi

mi

ni

mi

nk

mi

f (Ai, • . . , Ak)g(M 1, . . . , Bi) = Ef (Ai, ... , Ak)Hg(B 1, • • . , B1)11. ni

nk

(A6) (Zero axiom). If f (Ai, ... , Ak) = 0 and the indices Pi , • • • , PI ,

ri , • - • , rm

satisfy pi < ni and ri > ni for any i and j, then rm Pi P1 ri f (Ai, • . . , Ak)g(B 1, • • • , Bi, Ci, . . . , C m ) = 0 ni

nk

for any generators B 1 , ... , B 1 and C1 , ... , Cm . (T1) (Continuity axiom I). The mapping ni

f (x i , . . . , x k ) i—

nk

f (Ai, . . . , Ak)

is continuous. ni nk (T2) (Continuity axiom II). If fc,(Ai , ... , Ak) —> 0 and the indices Pi, • • • , pi, ri, . .. , rm satisfy the conditions of axiom 04), then ni

nk Pl fa(Al, — , Alag(B1, • • • ,

pi Ti

T,,,

Br, Ci, ••• , C m ) —> 0

for any generators B1 , . . . , B1 and C1 , ... , Cm . Second, there is a group of techniques based on the naturality properties of the mapping symbol 1---> operator, as stated in Theorems 1.4 and 1.5. Actually, the main naturality property is that expressed by item 1 0 in the theorems cited, namely, that if ço:A—>8


89

is a homomorphism of operator algebras, then go takes generators into generators and preserves symbols when applied to functions of generators (i.e., go ( f (A)) is the function of B = g)(A) with symbol f, and a similar statement is true of multivariate symbols). Let us recall some of these techniques. (a) Passage to the left (or right) regular representation. The mapping

A 1-->- LA, where LA (B) = AB , B E A

is an algebra homomorphism (a representation of A in the linear space A) and we have f (LA) = Lf(A), 1

n

f (LA i , • • • ,LA n )= L

1 n f(A1,...,An)

(this was proved in Subsection 2.3 independently of Theorems 1.4 and 1.5). Since the representation is faithful (i.e., its kernel is zero), if suffices to reason in terms of the left regular representation operators, which is often much simpler. For example, in proving the derivation formula 23 3f 1

D[f (A)] = DA— (A, A) Sx

the passage to LA permits us to assume that D is an inner derivation (i.e., given by the commutator with some element), since LDA = [D, LAI

(b) The conjugation transformation. If U is an invertible element of an operator algebra A, then the mapping A i-->- UAU -1 !Ad(A) is an automorphism of A. Consequently, for any generator A a generator as well, and we have 1

n

1

E

A the operator UA U -1 is n

UU (Ai, • • • , An)}IU -1 = f (U AiU -1 , — ,U A n U -1 ) 1

n

for any symbol f (y i , ... , yn ) and any Feynman tuple (Al, . .. , An ) of generators. Further examples of using the naturality property can be found in Chapter II.


90

Along with LA, RA, and Ads, it is often helpful to use two other classes of transformations on A; namely, we speak of adA = LA — RA

(Commutation operator) and allA = LA -I- RA

(Anticommutation operator). Suppose that A is a generator; then adA is a generator provided that f (x —y) E Y2 for any f E .F, and the same is true of anA if f (x+y) E .F2 for any f E F.

These operators satisfy numerous remarkable formulas. Here are some of them: e adA _ Ad L (e A ); [adA , adB] = [anA, anB1 = ad[A, /3];

[adA, anB] = an[a,b]; [LA, LB] = [LA, ac1B] = [LA, ani3 ] = L[a,b]•

Third, there is a very useful trick of considering matrix operators. It was widely used in the proofs of theorems in Section 2. Without loss of generality it can be assumed that all operators considered act in linear spaces (this can always be achieved by passing to the left regular representation). Let A be an algebra of operators on V, B an algebra of operators on W, g an (A, B)bimodule of operators from W into V, and D a (B, A)-bimodule of operators from V into W (we do not exclude the case in which V = W and A = B = g = D). We may consider the algebra U of matrix operators ( A D

C B)' A

E

A,

B E B, C E

g,

and D E D,

acting on V ED W. Computations in such an algebra are sometimes much simpler than in the original algebra. This is chiefly due to the following property.

Lemma 1.4 Let A E A and B E B be generators. Then for any C E g the operator (

A

0

c\

B

. i

s a generator in U, and

f ( () A6 CB

)

(f (A) 0

for any symbol f E F.

2 A f3 1 ) C t, (A , B)

f (B)


91

Proof The above formula clearly defines a continuous mapping .T. —> U. Next, this A C since Sy/Sy 0 B the mapping is a homomorphism. We have mapping takes 1 1—> 1 and y 1—>

f

B )) g (( l ' e

(( l '

= ( f (A)

0

CI (A, 1) Y f (B)

1. It remains to check that

e B )) 2 (

3 1 )

g (A) C (A,B) Y g(B) 0

. f((A)g (A) f (B)g(B)

0

The diagonal entries have the desired form; the superdiagonal entry can be transformed as follows: 2 31 8g 3 1 2 8f 3 1 (A, B)]g(B) = CF(A,B), f (A)I[CT (A, B )]] + 1[C— Sy y 2

where 3f 8g F (x , y) = f (x)— (x , y) ± g(y)— (x , y) Sy Sy 8( f g) f (x)g(x) — f (x)g(y) ± g(y) f (x) — g(y) f (y) = (x, y). = x—y Sy

o

The lemma is proved. Let us consider two examples. (a) Suppose that the identity ac = cb ± d

is valid for some elements a, b, c, d E A where a and b are generators. In matrix form, we can write out this identity as ( (a 0 db )

01

0c ) = ( 01

c ) ( a d) , 0 b 0

or

AC = CA, where A=( a d ) 0 b)

( 1 c \ and c = ( 00)•


92

By Theorem 1.4, 2°, we obtain f (A)C = C f (A)

for any symbol f, or, according to Lemma 1.4, 31

2

(f (a) 0

1

( f (a)

d () a,b) ( 1 c ) Y 0 0 ) f (b)

Y

0

(a, b) )

.

f (b)

Multiplying the matrices on both sides of the last equation, we get

2f 3 1 f (a)c = cf (b) + d — (a, b), Sy

that is, we have obtained yet another proof of Theorem 1.6. (b) Our second example pertains to Lie algebras and employs n x n rather than 2 x 2 matrices. Operators A 1 , .. . , An E A are said to form a representation of a Lie algebra if n [Ai, Ad = Ai ,

Exiik /=1

where A.1jkare numbers, called the structural constants of the Lie algebra. These commutation relations can be rewritten in a very simple form if we introduce the matrices

0

( A1 A3

A= 0

An

J

0

Ai

'

Ai d O

A

...

Ai ) '

and Ai = 114i Ilic,/=1. With this notation, the Lie commutation relations take the form A Ai = (Ai + Ai)A ,

which permits one to obtain various permutation formulas easily; for example, we have, by Theorem 1.4, 3 0 Af (Ai) = f (A i + Ai) A .

Finally, let us note that the techniques of noncommutative analysis widely use the difference derivative in conjunction with moving indices apart; this can be seen throughout this chapter.


93

Of course, there are also numerous formulas such as those derived in Section 3. However, it would be redundant to list all these formulas here, and we refrain from doing so, with the exception of the following ones: d

3

811

2

— f (A(t)) = A' (t) (A(t), A(t)) dt Sy (Daletskii-Krein formula); N-12

f (C) = f (A) ±

2k+1

E i[ c - it]] ... [IC - Al.,3-yk

k=1 2N

2

8k 1 1

2k

8N 1 1

3

2N+1

+I[C - Al ... [[ C - All„ (C, A, . .. , A ) Sy"

(Newton formula);

f (C) = f (A) ±

k=1 f (k) (A)

1

2,

21

2

3

82 1

1 5 2 4

I

3

3

E , A!, iRc _ A)"]] ± FC - Arl— syN (C, A, . .. , A) N-1

(Taylor formula); 12

21

f (A, B) = f (A, B) ± [B, A]

I (A, 8 Y1 6Y2

A; B, B)

(Index permutation formula);

f (11A ± BP

12

f ([[g(A, B)]I)

1

2

3

82 1 4

2

12

= f (g(A, B)) 5 82 2 7 8 8,s, 3 4 6 , 1-(A, A, B)- ,; (A, B, B) = [A, B] t 28 78 62 11 1 2 ag(A, BAI, g(A, B), g(A, B)) X

6y

(Composite function formulas); 3

12

f (ad B)(A) = f (B - B)A

(functions of adB).

1

2

= f (A ± B) ± [A, B] Ty2 (IM ± B]],[[A ± B]], A ± B) 1 2 21 4 3 1 8 2 15 = f (A + B) ± [A, 13. ] 8y 2--(1[A ± B1], A ± B , A ± B),

Chapter II

Method of Ordered Representation

1 Ordered Representation: Definition and Main Property 1.1 Wick Normal Form We begin with an example. In Subsection 1.2 we have posed the problem of calculating the Wick normal form for operators acting on a Hilbert state space, and in fact the answer was given for a rather particular case in Subsection 1.6. Let us recall the matter and clarify the subject. For simplicity, suppose that we deal with the case in which there is only one basic state. So, we are given a Hilbert space li and two operators, a+ and a— , on 7-1 satisfying the commutation relation [a a +,, a— ] m a+ a— — a— a+ =

where / is the identity operator on 7-1 , the operators a+ and a— will be referred to as the creation and the annihilation operator, respectively. For example, we can take 7-1 = L2(R 1 ) and

a a ± = (21m0-1/2 Lih_ 1 ax

where h and co are positive parameters, as was done in Subsection 1.6 in considering the eigenvalue problem for the quantum-mechanical oscillator. The Wick normal form of an operator A acting on 7-1 is a representation of A in the 2 1 (a+, form of a function of the Feynman tuple a— ), namely, 2 1 A = f (a+ , a— )

where

f (z, w) = E cafi zawfi ci,fi_>0

1. Ordered Representation: Definition and Main Property

95

is a function of two arguments, which we call the Wick symbol of A (we avoid discussing 2

1

exact requirements on the symbols and assume that f (a+, a- ) is defined as 1

2

f (a+ , a- ) = E ccep(aice(a—)fi, ci,i50

where the series is assumed to converge, say in the weak sense, on some dense subset of 7-0. Given an operator A on 71, how can we reduce it to the Wick normal form? If A is arbitrary, this question is rather difficult. Therefore, we will not consider this question in full generality, but will rather consider a particular class of operators A, namely, the operators which can be represented as functions of several appropriately numbered occurrences of a+ and a- . First of all, consider the problem of reducing to the Wick normal form the product A = A1 A2, where A1 and A2 are already in normal form, 2

1

2

Ai = fi (a+ , a- ),

A2 =

1

f2(a +, a- ).

In a particular case (A2 is arbitrary and A1 = a+ a- ) this problem has already been solved in Subsection 1.6. However, the general case is not much more complicated. Indeed, suppose for the moment that fi (z, w) = z or fi (Z, tV) = w. For the former, we have 22

3

2

1

2

2

1

alf2(a+ , d ) l = a+ f2(a + , a- ) = a4 12(a+ , a- )

1 g (a+ , a- ), 2

where g(z, w) = zf2(z, w). For the latter, the calculation is a little longer, since we have to move the operator 3

a+ to the first position. In doing so, we use the permutation formula: 2

1

3

2

1

1

2

1

3

3 f2 42

1

a- f[h(a+ , a- )]] = a- f2(a + , a- ) = a- f2(a + , a- ) ± a- , al— (a+' a+ a- ) ' ' Sz However,

[a- , al = I , and so we can continue this chain of equations by identifying the first and the third arguments in 8f2/8z: 2

1

1

2

1

af2 2

1

2

I

a- fff2(a + , a- )]] = a- f2(a + , a- ) ± — (a+ , a- ) = h(a+ , a - ), az

96

II. Method of Ordered Representation

where

af2 az

h(z, w) = wh(z, w) — —(z, w). Denote by 1+ and 1 — the operators taking 12 into g and h, respectively, i.e.,

a

1+ = z, 1— = w — —•

az

Thus, we have

1 alf (a+, a— )] = (1+ f)(a + , a— ), 2 2 1 1 al[f (a+, a— )]] = (1— f)(a+ , a— ). 2

2

1

The operators 1+ and 1 — characterized by this property are called the left ordered 2 1 a representation operators for the Feynman tuple ( +, a). As soon as these operators are evaluated, the problem of reducing the product of operators to the Wick normal form becomes trivial. Indeed, let

h (z, w) = E baffle Wfi . a,6

Then 2

I

1

2

2

II fi (a+ , cl — )11 Th(a + , a— A]

1

= E ba,fi (a ± r(a — )P EU2 (a+ , d)ll. a, 0

We can split off the operators a— and a+ in the products (a+)"(a— )/3 one by one. As a result, each of them will be replaced by the corresponding operator l+ or 1 — acting on 12, and we obtain 2 1 l[h(a + , a — )1

2

1

2

((I + r (1— ) 13 h)(a+ , a— )

U2 (a+ , a — )11 = «,13

=

2 1 2 1 [fl (1+ , r) f2](a + , a— )

(we do not discuss the convergence of the series). Thus, the symbol of the product 2 itfl (a+ ,

1

1

2

1

cr)Hh(a + , ct — )11

can be calculated according to the following recipe:

(11. 1)


97

1 Take the left ordered representation operators 1+and 1— and substitute them into the symbol fi. Then apply the obtained operator to the symbol 12. We see that the introduction of left ordered representation operators in this example enables us to avoid direct calculations with the operators a+ and a— and with functions of these operators. Instead, we consider symbols and operators acting on symbols. In fact, this is the basic idea of the method of ordered representations. Let us now discuss this idea and then proceed to general definitions. 2

1.2 Ordered Representation and Theorem on Products Let

1

n

A = (Ai, ... , An ) be a Feynman tuple in some operator algebra A. We are interested in the problem of reducing an operator A E A to a "normal form". What is a normal form? By this we mean a representation 1 n B = f (Al, .. • , An),

i.e., an operator is in normal form if it is represented as a function of the Feynman tuple A. Let us point out that we make no attempts to reduce arbitrary operators to normal form. Our task is far less general. We start from some operators already in normal form, perform some algebraic operations and try to express the result in normal form. Clearly, the main difficulty is to represent, in the form desired, the product of two operators and the superposition f (ig (A)1). Once this is done, we will actually pass from analysis in the algebra A to analysis in symbol spaces. Let us now give the precise definition. We assume that a class Y . of unary symbols is fixed and .Fn = Y.6 n . 1 n Let A = (Ai, . . . , An ) be a Feynman tuple in an algebra A and suppose that each Ai is an T-generator in A. 1

n

Definition 11.1 A Feynman tuple 1 = (ii, . .. , l n ) of operators li : .Fn —>- Tn ,

j = 1, . . . , n, 1

is called the left ordered representation of the tuple A = two conditions hold: i) for any f E Fn and j = 1, . . . , n we have 1

n

1

n

i, .. . , A n ) if the following

n

Ai Iff (Ai, ... , An)I1 = (li f)(211, ... , An);


98

if a symbol f E Tn does not depend on y, • •

ii)

yn , f = f (Y 1 , • • • , yi ), then (II.2)

f = Yi f

Condition ii) will be referred to as the regularity condition. In view of i), this condition is quite natural; indeed, 1 1+1 j j 1 j AiLf (Ai, • • • , Ai)1 = A if (Ai, ••• , Ai) = Ai f (Ai, • • • , Ai), so that the symbol of this product can be chosen equal to yi f (y i , . , y 1 ), and condition ii) simply says that the operators of left regular representation are consistent with this natural choice l . We intend to establish a theorem generalizing (11. 1) to the "abstract" situation of Definition II.1 . Since we have no reason to assume that it is possible to insert the operators Li into arbitrary symbols f E Tn , we need to be extremely careful in our statement. Let :fr C F be a continuously embedded subalgebra (we do not exclude the case = .7). Assume that the operators Li are .fr-generators in G(Tn ). Such a subalgebra can always be found; in the worst case, it still contains all polynomials. Clearly, = is a continuously embedded subspace in the symbol space F.

Theorem II.1 Under the above assumptions any product 11 lIf (Ai , • • • , 21011 l[g(Ai, • • • , :10 ]1, where f E

frn

and g E Fn , can be reduced to normal form. Namely,

1 1 n 1 1 n n if (A19 • • • An)11 141 (A19 • • • An)1 = [f( 1 19 • • • 9 in)g] (A19 •••

An).

Proof In the following we will widely use the more convenient notation f (A) for

f (Ai,

. , A n ); with this notation, the preceding equation reads f (A)11 [g (A)}1 = [ f (l)g](A),

or even f (A) o g(A) = f (l) g(A) (the small circle on the left-hand side serves as a substitute for autonomic brackets, which look somewhat clumsy in combination with the abridged notation). Condition i) can be rewritten as the commutative diagram is essential to impose this requirement since the uniqueness of the symbol of a given operator is not assumed anywhere. l It


11,

Tn li

99

A

1

1 L Ai II

Tn

A,

where A = AA is the linear mapping sending each symbol f (x i , . . . , xn ) into the coneI n sponding operator f (Ai, . .. , A n ) and L Ai is the operator of left regular representation, L Ai B = AB for any B E A. The operators /i and L A are 5.-generators in £(.Fn ) and L(A), respectively, and the continuous mapping p, intertwines the tuples 1 = (ii, . . . , 1n ) and LA = (L A ! , • • • , L A), as expressed by the diagram. By Theorem 1.4, we have ,

f (L A) itt = kt f (1)

for any symbol f E At . Let us apply both sides of the last equation to an arbitrary symbol g E Tn . We obtain

f (L A) (g (A)) = (f (1)g) (A), whence the statement of the theorem follows immediately, since

f (L A) (g (A)) = f (A)g (A) (see Subsection 2.3). The proof is complete.

0

1.3 Reduction to Normal Form We now consider the case f" = F. In this case the statement of Theorem II. 1 can be given the following interpretation. We introduce a bilinear operation * on .Fn by setting f * g = f (1) g . This operation will be referred to as the "twisted product". It makes Tn into an algebra (possibly nonassociative; the associativity conditions will be discussed later in this chapter). Then Theorem 11.1 states that the mapping

kt : (Tn,*) —> A (where A is equipped with the usual operator multiplication) is an algebra homomorphism. Next, assuming that .'. = F, we can give a somewhat more elegant statement of condition ii) in Definition II.1 .

II Method of Ordered Representation

100

Lemma 11.1 If all li are T-generators in Tn, then condition ii) in Definition 11.1 is equivalent to the following condition: ii') for any g E .F,, one has 1 ell,

• • • ,

n ln) 1 = g(Yi? • • • , Yn) ,

where the operator on the left is applied to the symbol identically equal to 1. Proof ii') = ii). Let f (y i , ... , y) E Tn ; set

g(Yi, • • • , Yi) = Yi

f (Yi, • • • ,

Yi).

By condition ii'), we have =

Xi f(Y1, • • • ,y1)

j+1 I j j 1 j ljf(li, • ..,li)1= li f(li,...,li)1 1

= 1j(f (li, . . . ,

j

l) 1) = lj f(Yi, • • • , y 1 ),

the last equality also due to ii'). ii) = ii'). It suffices to prove ii') for decomposable symbols of the form g(Yi, - • • , Yn) = gl(Y1) • • • gn(Yn)•

Condition ii) implies that the subspace Yi C Tn consisting of symbols independent of yi+1 , . . . , yn is an invariant subspace of the operator /i , and the restriction /i I . _ . coincides with multiplication by yi ; that is, there is a commutative diagram j

Ti Xi

1

Ti where the horizontal arrows are the embeddings. Furthermore, /i is an Y-generator in Tn by assumption and yi is an Y-generator in Ti (for any f E Y the corresponding function of yi is merely the operator of multiplication by f (yi)). Hence the embedding .T1 —> Tn intertwines the T-generators yi and li, and by Theorem 1.4 we have f(0)I F = f (b)

for any f E F.


101

We now apply this identity successively for j = 1, 2, . . . , n, and obtain 1 g(l

, 7n ) 1

,

=

(1 0

gn ln) • . • g2(12) (

= gn(ln) • • • g2(12)gl(Y1) = gn(ln) • • • g3(i3)g2(Y2)gi (Yi) • • •

= g(Yi • • • YO•

This equation extends by continuity to the entire symbol space Tn . The lemma is proved. We have considered the problem of reduction to normal form for the product C1 C2 , is of two operators in normal form. Let us now consider the general case. Let j 1 , and j 1 , . . . , is be two sequences of indices such that ji jr whenever [Ail , Ai r ] 0 0. , . , y s ) be a given symbol. We consider the operator Let fi

C=

Is • • • , Ai s )

and try to reduce it to normal form. 1

Theorem 11.2 Suppose that the left regular representation '1 = (li,

, l n ) of the

1

, An ) exists and that each l is an T-generator in Fn. Feynman tuple A = (Ai, , ys ) e Ps . Then the operator C can be reduced to a normal Assume that ço(y i , form. Namely, Js { , Ai s ) = ço(l

fi

, i 5 )1 } (A).

Proof By analogy with the proof of Theorem 11.1, we see that the mapping .Fn —>

A

A, 1

f Oh

f fi

•••

An) Is

fi

fs

intertwines the Feynman tuples (LA ii . . . , L A is ) and ( / , . . . , 1 is ). Therefore, we obtain Js q)(A zi ,

,

is , LA) (1)

ii = ÇO(L It

Is

, • • . LAi s )

= Ii

= Juitçql

(1)

Is • • • , 1 is ) (1) =

1 f(A i , .

,

An),

11. Method of Ordered Representation

102 where J!

is

f (yi, . . • , y,) = çql ii, -.. , l is )

(1). I=1


Remark II.! We cannot guarantee that the symbol f (yi, . . . , yn ) lies in .f-n - since be invariant under functions of L. However, the paradox there is no reason for F is that we can still compute products of the form f (A) g(A) with such a symbol f; we il

is

need only bear in mind that f (A) can be represented as q)(Ai i , . .. , Ai s ), and so h is f (A) o g(A) = Iço ( 1 i i , ... , 1 is ) g} (A)

Let us now consider how to perform reduction to normal form for composite functions. Let f (z) be a function of a single variable z, let g E .Fn be an n-ary symbol, and

1

n

suppose that the tuple A = (A1, .. . , An ) of T-generators has a left ordered represen-

tation

1

n 1 = (li, ... , I n ) consisting of F-generators.

Theorem 11.3 Let f E .t, - where .--. is some symbol class, and suppose that the operators g(A) and g(1) are .f.- . -generators in A and ,C(Fn ), respectively. Then f (Eg (A)]1) = (f (ffg (MD 1) (A).

In other words, f (Eg(A)1) can be reduced to normal form, f (Eg(A)1) = q)(A), where the symbol q)(y) E Fn is given by the action of the operator f a[g(1)1) on the constant function 1 e F. Proof This is quite standard stuff. Indeed, the mapping p,A clearly intertwines g(1) and g(L A), that is, the diagram

AA

-Fn

A

1 AA

Fn

g(LA)

A

commutes. Next, g(1) and g(LA) are P-generators in £(Fn ) and L(A), respectively, so that we have, by Theorem 1.4, AA

f alg (MD = f (1Ig(L A)1) /1'A


103

and consequently,

f ([[g(A)]]) =

Lf(ff g (A )]) U)

= f (L g (A) ) (1) = f (L g(A)),u,A(1)

= f (ffg(L A )JI) P, A (1) = Ail f (ffg(1)1) (1), whence the assertion of the theorem follows immediately. The proof is complete.

0

We have defined the left ordered representation of a Feynman tuple 1

n

(Ai, . • • , An)•

For symmetry reasons, it is clear that one can define the right ordered representation in a similar way. —1 —n Definition 11.2 A Feynman tuple r = (r 1, ... , r n ) of operators

ri : *Fri —> .Fn , j = 1, . . . , n 1

n

is called the right ordered representationof the tuple A = (A1, . . . , An ) if the following two conditions hold: i) for any f E .Fn and j = 1, ... ,n we have 1 n 1 n Ef 01, • • • , AniliA j = (rj f ) (Ai, . . . , An);

ii)

if a symbol f = f(y i ,..., yn ) G .F does not depend on y l , . . . , yi _ 1 , then

ri f = y 1 f. Remark 11.2 Note that the Feynman indices of the tuple r have the minus sign. This is because the natural ordering of these operators is the opposite of that of A 1 , . . . , An . Let us state, without proof (which follows the lines of that for the left ordered representation) the main properties of the right ordered representation. Theorem 11.4 Let the operators r1 , . .. , rn of the right ordered representation be i'-. - c F. Then generatorsfor some continuously embedded subalgebra i. have (i) for any f E Fn and g E F 1 ... , AniliffeAl, . .. , A n )]] = [g(ri, ... , rn)f] Oi, ... , An n) or, in the shorthand notation,

f (A) g (A) = [g(r) f] (A);

II Method of Ordered Representation

104

(ii) for any f E A. and any sequences of indices i v . • . , is and j1 , .. . , js such that jk 0 ji whenever [Ai k , Ai l ] 0 O we have fi f (Ail , • • • ,

.is -.is —.11 Ai s ) = [f (r 1 , • • • , ris ) 1 1 (A);

(iii) If :F = .F, then condition ii) of Definition 11.2 is equivalent to the condition

ii') f (r) 1 = f (Y)

for any

f

E

Tn;

(iv) let r1 , .. . , rn be T-generators in L(Tn ), and suppose that g E .Fn and f E F, where rf." is some symbol class, and moreover, g(LA) and g(r) are :F-generators in A and Tn, respectively. Then

f ([[g(A)1) = (f a[g(r)]J)1) (A). We remark that the existence of the right ordered representation is equivalent to the existence of the left ordered representation. More precisely, the following statement is valid. 1

n

Theorem 11.5 Suppose that the tuple A = (A1, . . . , A n ) has a left ordered represen1

n

tation 1 = (li, .. . , I n ) such that all the operators li are T-generators. Then there —1 n exists a right regular representationr = (ri , . . . , rn ) of this tuple. It is given by —

1

n

(rj f)(y)= f (li, • • • ,ln)(yd• Proof The proof goes by straightforward verification. Indeed we have

(ri f)(A) = (f (1)yi)(A) = f (A) Ai (the last equality is due to Theorem II. 1 ), and we see that condition i) of Definition 11.2 is satisfied. Next, if f does not depend on y 1 , . . . , yi _ i then we have j

n

j

n

ri f = f(li,...,1n )(yi)= f(li,...,1 n )(1i1) j n = (Yjf)( 11, • • • ,ln)( 1 ) = Yjf(Yj, • • • , Yn) ,

so that condition ii) of Definition 11.2 is also satisfied.

0

2 Some Examples Note that a given Feynman tuple of operators does not necessarily have a left ordered representation. Really, why should it exist at all? We have seen that the principal

2. Some Examples

105

usefulness of the ordered representation is to move the operators in products (or even more complicated aggregates such as composite functions) to the places rigidly prescribed by the Feynman indices. All in all, such an operation requires (perhaps implicit) permutations of the operators involved, and so we may suggest that the existence of a sufficiently rich set of commutation relations is a necessary condition for the existence of the left ordered representation. This guess is supported by the observation that in the example considered in Section 1 (creation-annihilation operators) it was the commutation relation [a+,a— ] = I that enabled us to construct the left ordered representation. Here we consider some extremely simple, but at the same time useful examples of calculating the ordered representation operators. In the next section we distinguish some classes of Feynman tuples (more precisely, of commutation relations) and present general methods for computing the regular representation.

2.1 Functions of the Operators x and

—

iitO/Ox

In Section 1 we have mentioned that differential, pseudodifferential, and Fourier integral operators can be considered as functions of the position and momenta operators x l , .. . , x n and —ialaxi,...,—ialax n . Since the main problems of the theory of differential equations involve computation of products of such operators, it would be useful to calculate the ordered representation for the tuples consisting of these operators. In order to simplify the notation, we make all the calculations for n = 1. Since the operators xi and —i a/axi commute with xk and —i a/axk whenever j 0 k, the result for n > 1 can easily be obtained from the result for n = 1 by attaching the indices. First of all, consider the Feynman tuple 1

. a X, —I—) . ax

(2

Clearly, we have the commutation relation [

a ax

—i—,x1= —i

(we drop the identity operator I in our notation); this relation differs from that for the creation-annihilation operators a+ and a— only by the factor i. Hence it is not surprising that the computations and the result are quite similar to what we have for a+ and a— . Thus, we will just formulate the result, leaving the computation to the reader.

1

Namely, the left ordered representation of the tuple (2x,—ia/ax

. a ix = q, Lialax = P —1 • aq

has the form

IL Method of Ordered Representation

106

A similar computation yields an expression for the right ordered representation operators: rx = q -

a

ap

, r_id/ax = P.

One can also consider the reverse-ordered tuple

For this tuple the ordered representation operators have the form

2

.

a

lx

= q + i—, Lia/ax = P; ap

Tx

= q, r-ia/ax = 13 ± i aq .

a

1 2 (x,-ia/ax) be pseudodifferential operators. By The-

1

(x, - lax)Q and Let P ta

1 1 2 2 ialax)]] is again a pseudodiffer(x, [[Q (x,orem 11.6, the product [[P-ialax)]]

ential operator, 1 1 2 a a )1= H (x, -i--6-.x- ), [[P (i, -i s )]][[Q (i, -i— ax ax

1

where

1 2

a

H(p,q)= P q, p - i— aq (

Q(q, p).

The last formula immediately implies the classical formula [109], [84] for the symbol of the product of pseudodifferential operators. In order to obtain this formula,

1 we expand the operator P q, p - ia/aq into a Taylor series in powers of -ialaq. 2

This poses no difficulties since p and -i a/aq commute, and we obtain 00

H (p, q) r=d

E k=0

(_ok ak p

k!

apk

ak n

(q, p) --..aqk

(a a) . ‘ 7 7

r

Of course, this expansion needs justification by estimating the decay as 'pi -)- oo of the remainder terms, but we omit these routine computations.

2. Some Examples

107

2.2 Perturbed Heisenberg Relations The operators Ao = — ia lax and Bo = x satisfy the commutation relation Bo Ao — A0B0 = i. Consider the "perturbed" relation BA — aAB = i,

where a is a constant, a = 1 + E, E -> 0 valid for some "perturbed" operators A, 13; we do not need to use below the concrete form of these operators. Let us compute the 21

left ordered representation for the tuple (A, B). We have 21

3

21

21

2

A[[f (A, B)]] = A f (A, B) = A f (A, B), so that /A = x (here x and y are the arguments of f, f = f(x, y)). Furthermore, by analogy with Subsection 2.1 we obtain 21

21

3

B[[f (A, B)]]

= B f (A, B) 41 4 sf 2 41 3 32 = B f(aA, B) + B(A – aA)— (A, aA, B). Sx

Moving the indices apart and introducing the autonomous brackets, we obtain 21

1

21

32

4

332

4

Sf 2

41

B[[f (A, BA = B f (a A, B) + [[B (A – a A))]]—(A, a A, B). Sx But

B(A – aA) = BA – aAB = i, so that we obtain 21

1

21

8f2

21

Bfff (A, B)]]= B f(aA, B) -I- 1—(A, aA, B). Sx In contrast to the preceding example, the arguments of the difference derivative do not coincide, so that it does not reduce to the usual derivative. Let Ic, denote the dilatation operator /0,q)(x, y) = w(ax, y). We obtain the left ordered representation operators in the form lA

= x, 1B = yic, +

i

(1 – a)x

(1 – JOE).

lil. Method of Ordered Representation

108

Let us demonstrate how this ordered representation can be applied. Assume that 2 1 we wish to invert an operator of the form f (A, B), i.e., to solve the equation 21

21

Ef (A, B)]] 11)((A, B)11 = 1, where x (x, y) is an unknown symbol. By Theorem 11.1, we obtain the following equation for the symbol x: 2 1

x, ffy/a +

f

(

i (1 — IA) X (x , y) = 1. (1 — a)x

If f (x, y) is a polynomial with respect to y, then this is a difference equation and can be solved explicitly; it may seem more convenient to reduce this equation to a difference equation on a uniform grid, which can be accomplished by the change of variables x = e0 . Under this change of variables, /a is carried into the operator ecia me of translation by a = in a, and the equation takes the form f (e2° , [lye" + lie— ea (1 — eaa/a0 )]]) x (e° , y) = 1. This is a difference equation on a uniform grid with mesh width a.

2.3 Examples of Nonlinear Commutation Relations Let A, B, and C be operators satisfying the relations [A, B] = 1, [A, C] = a(C)A ± I3(C)B ± y (C), [B, C] = E(C)A -I- B(C)B + a (C), where a (x),

13 (.0 , y (x), 6(x),

and a (x) are some symbols. Let us find the left 1 23 ordered representation for the Feynman tuple (A, B, C). The arguments of symbols 1 2 3 will be denoted by (x,y,z),x1--> A, y 1--> B, and z 1-->- C. The last two relations satisfied by A, B, and C can be written in the matrix form 6 (C) ,

) C = S( C) (

AB )

+

(y(C) ) cr (C)

where S(z) is the matrix symbol S(z)

=

(z + a(z) I3 (z) ) E(z) z -I- (5(z)

2. Some Examples

109

Let F(x, y, z) be an arbitrary symbol. By Theorem 1.6,

( A B

irF(

4 =( A

A, ;31,

11,

B 3

12

3

4 (

= F (A, B, S(C))

A)

3F 1 2 3 3 ( y (C) — (A, B, C, S(C))

In the second term of the last expression the ordering of operators is just the required 3 one, whereas in the first term A still needs to be moved to the first position. This, by analogy with the previous example, can be performed by applying the operator x+alay to F. Finally, we obtain /A

( /13

)

x

=

'S(z)

1 1 — iS(z) 2 2

z—

S (Z)

2 Y (Z) /2 kz)

where /s(z) is the operator of substituting the matrix S(z) instead of z into a function,

/s(z)F(Z)

1 2 2 a = exp f (S(z) — z)— az

def

F(z) = F(S(z)). ,

since the operator C acts last, it is clear that lc = z. Our next example is concerned with the phenomenon that the ordered representation may or may not exist depending on the chosen ordering in a Feynman tuple. Let A and B be operators satisfying the permutation relation

AB = B r' A, where n > 1 is an integer. Let us first construct the left ordered representation for the 12

tuple (A, B). We have 2 12 B )J ] = B f (A, B) BI[ f(A, 12

and 12

A [[ f (A, BA

3

12

3

14

= A f (A, B) = A f (A, Jr) =

4 f 12 4 332 I[A(B — B a )]] —(A, B, B n ) Sy

1 12 A f (A, Ba),

and so

2

/A

=

1

1B = y,


110 12

if we choose the ordering (A, B). On the other hand, one can easily check that the 21

ordered representation does not exist at all for the ordering (A, B). Indeed, consider the quotient A of the free algebra with generators A and B modulo the ideal generated by AB-13n A. Each element of A can be uniquely represented in the form E aki Bk A1 , where aki are complex numbers and the sum is finite. The existence of the representation follows from the existence of the left ordered representation operators /A and /B (see above). The uniqueness is also clear: since

1A1B

= 173 1A ,

it follows that the algebra generated by /A and /B is a homomorphic (in fact, isomorphic) image of A; now the coefficients aki can be determined by applying the operator E akilll AI to the function 1:

[Eakak,A](1)=Eakiykx/. Now consider the element BA E A. It has no representation of the form

Ec ki AkB/, for if such a representation could be found, we would have

BA

= Eck/AkBi=EckiBnkiAk,

which is impossible since the system nk/ = k has no integer solutions. This implies that the left ordered representation simply does not exist for the or2 1 dering (A, B). Let us consider two more general examples concerned with Lie algebras.

2.4 Lie Commutation Relations 1

m

Generators of a Lie algebra, special case. Let A = (A1, . . . , Am ) be a Feynman tuple satisfying the following conditions: i) for any j,k E 11, .. . , m} there exists an index s = s (j, k) E { 1, . . . , ml such that [Ai, Ad = —icsAs, where cs is some constant;

2. Some Examples ii)

111

there exists a positive integer N such that any commutator of length > N composed of A 1 , .. . , An is equal to zero, [A 1 , [A 2 , [. . . [AiN_ i , AN] . . .1]] = 0 for any ii , ./2, • • • , i N E {1, • .. , N },

Condition i) means that the operators A 1 , . . . , Am define a representation of a Lie algebra of a rather special form (with a general Lie algebra, the right-hand sides would contain linear combinations of all A1, 1 E {1, . . . , n}; see the next example). The factor —i is introduced so as to ensure that cs are real in case A 1 , .. . , As are selfadjoint operators. Condition ii) is simply the requirement that this Lie algebra is nilpotent; N is referred to as the nilpotency rank. This class of commutation relations was considered in [129]. Note that any nilpotent Lie algebra can be described as a homomorphic image of a nilpotent algebra of the type described above. Let us describe the construction of the left ordered representation for this case. The main (and the only) idea is the same as in Subsection 2.1. Assume that we need to compute the operator /Ai , i.e., in fact, the product m m+1 1 Ai f (A) = Ai f (Ai, ... , Am ) for an arbitrary symbol f = f (x i , .. . , xn ). To this end, the product should be transformed in such a way that the Feynman indices of A . +1 , . . . , Am become greater than m + 1. Using the permutation formula (see Corollary 1.1) a

b

b

a

4b c

a

Aço(B) = Aço(B) ± [A, B]—(B, B), Sx where c > a > b and ço may have additional operator arguments, we obtain, by successive commutation, Ai f (A) = (xi f )(A) ±

m 1+1

6f 1

1

E [4,,A,] —,,...,A, 0 Sx/

i=j+i

m 1+2 A /, ... , Am ).

We now use the expression for the commutator and obtain m

Ai f (A) = (xi f)(A) — i

1+1

3f1

1 1+2

8xI (Ai, . .. , A1, E cs„,„ A s u,0-

m

A1, ... , Am ).

i=j+1

The first summand already has the desired form; as for the other summands, let us move the operator A s om in each of them to its place, thus obtaining remainder terms with second-order difference derivatives and second-order commutators, then we proceed to the remainders, etc. By virtue of nilpotency, the process will terminate after the Nth step. Clearly, at this stage all difference derivatives disappear and transform into the usual ones. Thus we have proved the following theorem.


112

Theorem 11.6 ([129]) Under conditions i) and ii) the Feynman tuple 1 n (Ai, .. • , An)

possesses the left ordered representation li , .. . , ln , and each of the operators li is a differential operator of order < N — 1 with linear homogeneous coefficients.

In fact, it still remains to prove that 1 m f(li, ... ,l m )1 = f (x i , ... , xm )

for any symbol f (x). By Lemma II.!, it suffices to prove that lAi g(x) = xig(x) whenever g(x) is independent of x i +1 , .. • , x,n . However, this follows by construction.

Generators of a Lie algebra, general case. Let us now consider the case in which the operators A 1 , . . . , An are generators of a representation of an arbitrary Lie algebra. This means that n

[Ai, A] =

E

CliciAk,

i , j =1, ...,n,

k=1

where ci.c. if are some (complex) numbers called the structural constants of the Lie algebra. Let us calculate the left ordered representation (ii, . . . , ln ) for the tuple of operators n 1 (A1, . .. , A n ). In order to avoid nonalgebraic difficulties, we consider only polynomial

symbols f (x), x = (x 1 , .. . , xn ). Remark 11.3 As shown below, for polynomial symbols we can accomplish the construction by purely algebraic means. For more general symbols this is not the case; as is shown in the Appendix, in the latter situation it is more convenient to pass to Fourier transforms, which takes the left ordered representation operators into right-invariant vector fields on the corresponding Lie group. Remark 11.4 The constants ci.c. i j are structural constants of a Lie algebra, and therefore they must satisfy the antisymmetry condition and the Jacobi identity. However, our construction does not use these properties. These conditions are discussed in detail in Section 4 of this chapter. Denote by A the associative subalgebra generated by A 1 , . . . , An and by G c A the linear span of A 1 , . . . , A. In the algebra A[[ti , . .. , tn I] of formal power series in t = (t1 , . . . , tn ) with coefficients in A consider the element U(t) = etn A n . . . eti Al .

2. Some Examples

113

Clearly, for any polynomial p(x), x = (x l , . . . , xn ), we hay& A 'i n) = 1P

u (01

at

t=0

(it suffices to check this identity for monomials, which is trivial). Let us find operators Li such that

U(t) = Ai U(t), j = 1,

, n.

To this end, we compute the derivative (a/atq )U(t) using the relationship e St D

_ eadB (D) e St .

Differentiating U (t) and applying the last equation successively, we obtain

a U(t) = a tq

e tn A n . . . e tq+1A q+1 A g e t qAq . . . et i A i r _tnadA n

= te

For each

j

E

{1,

etq+

ad Aq+1

(A q )]U(t).

, n1 denote by ci the matrix with the elements (ci)ki = c11 .

Evidently, the operator adAi acts from L. into L, and if A =

E As As, 1

(11.3)

A., A s , then adAi (A) =

where A s = ci A.s •

Note that the last assertion is valid regardless of whether or not the operators

, A n ) are linearly independent. Therefore, the element in square brackets on the right-hand side of (II.3) belongs to L, and moreover, we have

a U(t) = EA qi(t) Ai u (t) , at where2 Aqj (t) =

1

[e tnCn . . . etq+1Cq+1 ] iq ,

8 kj

(11.4)

if q < n, if q = n.

In particular, Aqj (0) = E, and we see that the matrix A(t) = (Aqj (t)) is invertible in the algebra of formal power series. Multiplying (II.4) by A -1 (t) on the left, we obtain

a

A; 0 U(t) = EAT.: (t) _u (t), q atq 1 Recall that the substitution t = 0 is a well-defined operation in the algebra of formal power series;

this operation assigns to each formal series its constant term. 2 Here [ stands for the (j, q)th entry of the matrix inside the square brackets.

H. Method of Ordered Representation

114 i.e., we can set

where L., ( t, x)

= E A7: (t)x q . q

Now, for any polynomial p(x) we have 1 n AjffP(Ai, ... , An)ii = Aj

1p (-0-18 -. )u(t)1 t=o

(II.5)

= jp (-:i,) AiU(t)1 t.0 =

a

Ep (-.i.)IIIIL; ( 2t,. )r 11u()I t=0

To compute the composition of the two autonomic brackets in the latter expression we use the right ordered representation

rt =t+

a ax

, ra =x Vi

1 2 for the tuple (r,a/at) computed above in Subsection 2.1. With the help of this representation the above mentioned composition can be written in the form

a —

li p

at (

11 2 a )11[[Li(t,— A= q (t, at

a

2

where the function q(t, x) equals

1

q(t, x) = Li

a

(

2 „

x t + —, ax

p(x)•

Substituting this equality in the right-hand side of formula (II.5) we represent the leftI n hand side of this formula in the form h(Ai, ... , An), where the function h(x) is given by 1

2 _1 (

h(x) = q(0, x) =Ex A . q

g ig

a — p(x) ax

'

2. Some Examples

115

(we used here the above expression of the functions Li (t, x) via the matrix A -1 ). Thus, we have obtained the left ordered representation operators in the form 2 1 ( = EXqA7 - ij —) , i = 1 „ — , n. fq aX q It is easy to check that these operators are well-defined on polynomials and satisfy condition ii) of Definition 11.1.

2.5 Graded Lie Algebras We again consider the creation-annihilation operators, as in Subsection 1.1. This time we assume that there are different types of particles in the system, namely, + bosons and fermions. Let b.1. be the creation-annihilation operators for bosons, and + f•.1 be the corresponding operators for fermions. Then the following relations must be valid:

[b7: ,bk+] = [b7,

= [b,

0,

+ [bj , bk ] = —

[ fi , f k + ]

=- f i f k

+ f k f i = Si k I.

These relations are different from the usual Lie relations in that one uses anticommutators instead of commutators for fermion creation-annihilation operators. Thus, the model symmetric with respect to bosons and fermions leads to the consideration of a new mathematical notion, namely, of graded Lie algebras that involve both commutators and anticommutators (see [26]). Formally, the underlying linear space of a graded Lie algebra L is the direct sum L = Lo ED Li. The space L is equipped with a bilinear operation, [, .], that respects the Z 2 gradation, that is,

[Lo Lo] C Lo, [Lo, ,

Li] C Li,

[L1, L1] C Lo,

and if a, b, c are graded elements of L (that is, they each belong either to Lo or to Li), then the graded Jacobi identity holds:

(_1)1a1 I cta , b] , cij --1-. i— 1) 1b1 la i[[b, c], a] ± (-1)1c libi[[c, a], b] = O. Furthermore,

[a, b] = —(-1)iallbi[b, a] (here lai is the gradation of a: we write la I = j if a c Li).

116


A representation of a graded Lie algebra L is a linear mapping of L into an associative algebra such that [a, b] is carried into ab — (_ olallbl b a; in particular, the bracket of two elements of L1 is realized as the anticommutator and the brackets of all other combinations of graded elements are taken into commutators. Let us consider a simple example in which we construct the ordered representation for some operators forming a representation of a graded Lie algebra. Let A and B be operators satisfying the relation AB ± BA =1. This can be regarded as a representation of a graded Lie algebra with A, B E L1, and 1 E Lo. Note that this graded algebra is likely to be infinite-dimensional since the above relation does not form a closed system of graded Lie relations by itself: we have not supplied any data for the anticommutators [A, A] = 2A2 and [B, B] = 2B2 . Formally, this can be considered as a special case (for a = —1) of the perturbed Heisenberg relations considered in Subsection 2.2. 12

Let us consider functions of the Feynman tuple (A, B). Then for any symbol f (x , y) we have 21 3 21 2 2 1 BU (B , A)1 =B f (B , A) =B f (B , A), which implies that /B = x. Proceeding as in the cited example, we obtain 21

3

4

3

21

Afff (B , A)]] = A f (B , A) =A f (— 3 2

± A (B ±

1

B , A)

21 41 4 f(B, A) - f (— B , A) B) 2 4

B±B

3f 1 50 = f(- B, A) A ±ff A (B ± B)1—(B , — B, A). Sx 2 1 1

32

23 2

4

4

Since A (B ± B) = 1, it follows that 21

21

AU(B, A)1 = /Af(B, A),

where lAf(x,Y) = Yf ( — x, y) ±

f (x,

)7) -

f (-x,

)7)

2x

•

12

Thus, the left ordered representation of the Feynman tuple (A, B) is /B = x,

1 — Ix lA = Y Ix ± 2x '

where ./.), is the spatial inversion operator Ix f(x, y)

= f(—x,

y).

3. Evaluation of the Ordered Representation Operators

117

3 Evaluation of the Ordered Representation Operators In Section 2 we have considered several examples of computation for the operators of left and right ordered representation. Here we present a general method to compute these operators for some "distinguished" classes of commutation relations.

3.1 Equations for the Ordered Representation Operators We consider a special class of commutation relations of the following form. Let k < n be a positive integer. Consider the relations

Mi, Ai]

=

0,

j = k ± 1, ... ,n, i = 1, ... ,n,

(II.6)

n

AA i = E§91,-J (Ai ,Ak,,,...,An)Ar,

i, j = 1, . . . , k,

(II.7)

r=1

where Ki (Yi , Yk+l' • • • , yn ) are some symbols; since, according to (II.6), the operators Ai, Ak+i , ... , An are pairwise commuting, their Feynman indices are inessential and therefore omitted. The relations (II.6) and (II.7) can be rewritten in the matrix form AA i = q)j(Ai, Ak+i , ... , A n )A, j = 1, . .. ,n,

(II.8)

where A is the vector operator A = f (A 1 , .. . , An ), and çoi (Ai , Ak+i , ... , An ) is the n x n matrix operator with entries (g0j(Aj, Ak+i, ... ,

I

W il:i (Aj,

Ak+i , ... , An )

if j < k and i _ < k, otherwise.

A »Si r

Example 11.1 Let k = n and

au

0 yi ± Ai ,

0

crnj

(II.9)

)

where Ai are constant matrices. Then, if all aii = +1, we obtain Lie commutation relations; if crii = ±1, we obtain graded Lie algebras, if aii are close to 1, we obtain "perturbations" of Lie algebras. One can also consider the more general case (11.1 0) where the matrix Mi is not diagonal, or even the case in which q)i (yi) is a nonlinear function (such relations will be referred to as strongly nonlinear).

118


First of all, note that it follows from (II.6) that Lk+1 = Yk+1, • • • , L n = yn ; without loss of generality it can be assumed that the operators Li, j = k +1, ,n, commute with all the other left ordered representation operators. Thus we can assume that k = n, which is done throughout the sequel. Along with (II.7) one can also consider the "dual" relations AkAj =

E As s-2ski (Ak). s=1

They transform into (II.7) and vice versa if we pass from the algebra

to the opposite algebra Av, whose multiplication is given by the formula

N'PB = BA. It follows that the left ordered representation of A corresponds to the right ordered representation of A° P , and the same is true for the right ordered representation of A and left ordered representation of A° P . Let us first consider the commutation relations (II.7). Introduce the auxiliary operators Lii, i =1, ,n, j =0, . . . , n satisfying the relations 1 (Lii f)(111,

n j+1 1 1 f (A 1, An) = A

j j+2

, Ai , A j+1,

n+1 , A n)

for any symbol f. In particular,

Li n = LA i and Li() = RA; are the operators of left and right ordered representation respectively. From the moving indices apart rule it follows that Lii = yi, j =1,

,n.

We seek the operators Lii in the form 1

a . = Lij (y, --) ay y

2

Our aim is to derive equations for the symbols Lii (x, p). Introduce the vector

operatorl Gi = t (L 1i , 1 Here t stands for the transposed matrix.

, L ni ), j =0,

,n,


119

with symbol Gi (x, p) = t (Lii (x, p),..., L ni (x, p)).

By Theorem 1.4 we conclude that 1

j-I-1

j+2

j

n+1

(II.12)

Ai f (Ai, ... , A j, A j +i, ... , A n) j

j-1

1

j+2

j +1

rz+1

A j-1, ço j (A), A j +i, ... , A n )

= Ai f (Ai, ... ,

(the operators A1+1 , ... , An on the right-hand side of the latter equation are tensored by the identity matrix). Using the substitution operator (see Subsection 2.3 above) one can write f (Y 1, • • • , (Yi), • • • , Yn) =

e (99(Y))—Yi)a/aYi f (yi , • . • , yn).

Thus it follows that the operators corresponding to the symbols Lij f (yi, • • • , Yn)

and

n

E Lr,

j—i f e(49.1'01.1

') —

Y.1*)a/aY1'l

f

tr '

r=1

( 1)

J k.7 1 f • • • f

Yn )

coincide. We require that the symbols of these operators coincide and use the right 2

1

regular representation of the tuple (y, 8/0y) constructed in the preceding section. We obtain 2

1 )

pi Fi xi +alapi p) = e Gj(x, Li_1(x, p),

where Fj (Y) = ÇO.i (Y)

—

y.

Using the relation Lii = yi, we can write out the following system of equations to define Lo(x, P):

I E

2

1 )

2

p i Fi (x i +aialpii )

pi Fi (xi +alapi

e

... e

140(X? P)

= x1.

(II.14)

k

jk

If this system is solvable, its solution gives the right ordered representation operators; then we can also obtain the left ordered representation operators. Of course, it remains to prove that the solution obtained indeed gives the ordered representation operators. This requires some additional assumptions, and we will return to this question later; the corresponding theorem will be stated only for relations (II.11), since


120

the statements are essentially the same. Let us now consider relations (II.11) in more detail. We define matrix functions Ak (y) by the equations

(II.15)

(Ak (Y ))sj = S-2Isci (Y) — Y 3si, where 34 is the Kronecker delta, and introduce the matrix operator 1 a 2 Uk = eXP ( Ak (Y k) ---, (gXk

The operator of Yk

Uk

•

acts on scalar functions of yk, and the result is n x n matrix functions

7

[ukf(yk)]si = f(Yk + Ak(Yk))s;

(II.16)

7 . = f(ok(yk))si,

where Ok (A) is the matrix with elements

(Qk (Yk))si = asici (Yk), and f (Ok(Yk))si stands for the (s, Dth entry of f (Ok (Yk)). Define operators D51, s, j =1, ... ,n, acting on functions of n variables y l , . • • ,yn by the formula D51

=

[Un X • • • X Uj+11sj,

j

< n,

Dsn = Ssn .

Theorem 11.7 Suppose that the operators il , . . . , ln of the left ordered representation for system (11. 11) exist and are uniquely defined. Suppose also that all the operators 1.1i are invertible. Then the operators i l , .. . , ln satisfy the system of equations

E is Ds; = yi .

(II.17)

s

Proof Let yo(y i , . . . , yn ) be a given symbol. Relations (II.11) can be rewritten in the

form AkA = AS2k(Ak), k = 1, ... , n,

where as above A = 4 (A 1 , . . . , An ). Thus, using successively relations (II.11) and (II.16) we obtain 1

k-1

ÇO(A17 • • • 7

k+1

A k-1, A k, . . . ,

n+1 k A n)A

1 k-1 k k+2 n+1 k+1 = W(Al, • • • , A k-1,S-2k(Ak), A k-I-1, • • • , A n ) A 1 k-1 k n+1 k+1 k+2 = (UkV)(Ak, • • • , A k-1, Ak, • • • , A k+17 • • • 7 A n ) A

1

n„ def „ \ „ = (1 Un • • • UkO(Ak, • • • , An) = lt(k)Wlktil

7 • • • 7

An ),

...


121

where l

=

(lie ... , 1n ),

1 (k) = (1 (k)1 , • . • , 1 (k)n)•

Since all operators U1 are invertible and / is uniquely determined, it follows that /(k) is also uniquely determined by the property 1

k-1

ÇqA1, • • • ,

k+1

A k-1, A

1

n+1 k k, • • • ,

n

A n) A = (1 (k)W)(A k , • • • , A,),

that is, /(k)i = iik. From this it follows that

E is [un . . . ui+ds., = ii; = yj, 1 < j < n — 1. s

Taking into account that ln = yn , we find that the operators ls satisfy system (II.17). 0 The theorem is proved.

3.2 How to Obtain the Solution Let us now solve the equations for the ordered representation operators. Equations (II.14) and (II.17) admit explicit solution in a variety of cases. We shall now analyze these cases and then state a general theorem concerning the solvability of system (II.14). (a) Let the operators A 1 , . . . , An form a Lie algebra, i.e., satisfy the relations [A i ,

A;

] = E4Ak,

(II.18)

where X.kij are structural constants. These relations can be rewritten in the form (II.8) by setting çoi(b)

= xi + Ai,

Fj (Y i) = q).i (Y j) — yi = Ai, (Ai)ik = x,

and we obtain from (II. 13) £1(x, p) =---- ePi Ai ,Ci_1(x, p). Let us compute the left ordered representation

L i (x, I))

4. We have

= e— Pii-1Ai+1 . e — Pj+2Aj+2 ...e—PnAnLn(x,

Thus we can write out the system of equations eke p— n An L n (x , p))i

= Xi

P).

122


and obtain the following familiar expression for L n (cf. Subsection 2.4): Ln = A -1 where

\. . (A(p)) ii . (e— P)+1 A.ii-i .. e —pn An nj •

Note that A(0) = and consequently, the obtained operator is well-defined as a series in powers of 0/8 y . Simple though cumbersome calculations show that the components of Ln satisfy (11.1 8) if and only if the structural constants satisfy the Jacobi identity

E (A4;4 + xlici Ali + A.i.p4i ) = 0 k

for all tuples (j, i, 1, s) (cf. Section 4). (b) Let us now analyze a much less trivial case in which one manages to solve system (II.17) with nonlinear functions S2ski .

Definition 11.3 The system of commutation relations (II.11) is said to be solvable if (a) all matrices Ak (y) in (II.15) are lower-triangular, that is, AL 0 for s < j; (b) the function ç 5 (y) have inverses (12 5 ) 1 (y) for all k and s. Theorem 11.8 If system (II.11) is solvable, then system (II.17) can be solved for ls, s = 1, ...,n (the explicit form of the solution is given below in the proof of the theorem). Proof First of all, let us transform (II.17) to a more convenient form. Let us seek /s in the form n

is :---

E

ykmks,

k=1

where Mk s are now unknown operators. Inserting this into (II.17), we obtain

E

YkMksDsj =

y»

j

= 1, . . . , ri,

k,s

so that it suffices to solve the system of equations Mks D1= Ski j, s

k =1, ... , n.

(II.19)


123

Let us introduce the operators R1 by setting (Rj f)(Y 1, • • • , y n ) =

f (Y i , • • • ,

3 y 1 , (E2ij+ i, j) -1 (Y j+1), • • • , ( 2n j) -1 (Yn),

j = 1, . . . ,n — 1, Rn = 1. Now set

Mss = Rs ,

s = 1, . . . ,n,

M1 =O, s < i, s—i Msj = — Rs Dsi mi.; ,

E

s > j.

1=1

These equations permit us to determine the operators Msi for all s and j. We claim that they satisfy (II.19). To prove this, note that if A is a lower-triangular matrix, then so is f (A), and f(A)1j = f (Ajj) for all j = 1, . . . , n. We conclude that Ds1 = 0 for s < j and DE o R1 = 1, j = 1, . . . , n. Inserting Ms./ into (II.19), we obtain, by the above, n

E

Mks Dsj

E

=

Mks Dsj

s j

s=1

k-1 k-1

= MkkDkj

EE —E {E —

RkDkIMIs Dsj

s=j 1=s k-1

= MkkDkj

1

RkDkl

'wisps;

s=,

/=1

1

If j = k, then only the term MkkDkk = Rj Djj = 1 occurs on the right-hand side of the last equation. Let us show by induction over k j that —

k

E

/wisps.; = So •

s=i There is nothing to prove for k < j. If k > j, then, by the induction hypothesis, 1

k-1 MkkDkj

—

E

RkDkl

1E

/wisps,

s=,

I

k-1 = MkkDkj

—

E

RkDk1 31j

1=j

= MkkDkj — RkDkj = RkDkj — RkDkj =


O. 0

H. Method of Ordered Representation

124

Let us now prove that system (II.14) is solvable in the class of formal power series provided that the functions çoi (yi) have the form (II.10). We seek the solution of (II.14) in the form 00

=

Lso(x, P)

iœi=clof equations for the functions Cs , OE : Then we obtain the following infinite system n

oo

E E cs ,a (x) la 1=0

Vn !

(vn

x • • • x xnvn —41n

•

_ an )!

...

Vi_f_i !

. —ai+1 x vi-i-i (vi ± i — ai + i)! s=1 .1 + 1 aai

[ aai ---Te7 (Wl (X1))Vi • •

(II 20)

aX "1 1 Goi(xi))1 is

a X J. j

= x 1 . .. xjvj+1 ... x nnv , VI

(vi = 0,1, ...; i = 1, ... ,n).

Indeed, let us make the substitution ..x -± y, p -±

— aay

yivl . . . ynvn. Then in (II.14) and apply both sides of this equation to the monomial we will have yr .. . y 7 +1 . . . ynvn on the right-hand side of this equation and on the left-hand side we can use the fact that (p, x + 0/8p) is the right regular representation of the tuple

1 2 (a lay, )7). We obtain 1

ERN

2 ( y,

k

a

—) ay

= E L k 0 (Y2 ' —

a° y)

E vi+i

oo

n

= E E csa(y) s=1 la1=0

X

3(79 f.

E

ay.j

J

1

2

ky •1vi I • •

yriin )

ik

1 k

1

2

leFi (Y-dala Yi ...e F' (Y 1)a laY1 YI+1 ' • •

ynn i

[(

vn ! (vn

—

Wi (Yi» vi — (Sol (Y1)) vi Lk

vi±i!

an)!

• • • (Vi + I —

Vi4-1 — Crj+1

cei±i)! Yi+1

. . ...nvn —an •

Y

aai

(Wi

(Y./ )) vi • • • ai. (49 1 (Y1))1 "Y1

j,s

and we arrive directly at (II.20). Let q)i(yi) have the form (II.10). Then the summation in (II.20) ranges over a < y, and (II.20) can be rewritten in the form n vi ! . . . I), ! E[mJv.i cs,v (X) is s=1

_mil

3. Evaluation of the Ordered Representation Operators vi

= x 1 ..

vi+1

.x i

125 n

. . . Xnv n +

E E pi,,„ v(x)ci,,, (X), ct m,

142


Thus we have proved (11.34) for 1 > m (and, by antisymmetry, for m < 1). If m = 1, then both sides of (11.34) are zero, so that everything is all right in this case,

too. It remains to note that the passage from (11.33) to (11.34) was an equivalence. The lemma is proved.

Remark 11.6 The proof of Lemma 11.4 could be substantially shorter if we used some standard facts from the theory of Lie algebras. However, we have preferred to give a direct proof since it shows with full clarity that there is a direct algebraic relationship between the Jacobi condition for the operators / 1 , .. . , /n of the left regular representation and the Jacobi condition for the structural constants of the Lie algebra. It is because of this relationship that the term "generalized Jacobi condition" has been introduced.

4.3 Verification of the Jacobi Condition: Two Examples In the preceding subsection we have checked the generalized Jacobi condition for commutation relations. Here we accomplish such a verification for the examples given in Subsections 2.2, 2.3, and 2.5. The perturbed Heisenberg relations. The left ordered representation constructed in Subsection 2.2 for the Feynman tuple (A, B) satisfying the relations BA — aAB = i has the form lA = x, 1B = yla ±

i

(1 — a)x

(1 — la),

where /OE is the dilatation operator, Ia f (x, y) = f (ax, y). Let us check that 1B1A — alAlB = i•

The variable y can be considered as a parameter. Therefore, we can assume without loss of generality that our symbols depend only on x. Let f (x) be an arbitrary symbol. We have 1131A f = 1B(xf)

—

(y Joe ±

i

(1 —

= ayxf (ax) +

a)x(1-- Ia )) (xf (x)) ix ( f (x) — a f (ax))

(1 — a)x

'

or, on cancelling out the factor x, . f (x) — a f (ax)

1B1A f (x) = ayxf (ax) + t

1— a

Furthermore, 1A1B f (x) = x 1 B(.f (x)) = xYf (ax) ± t

. f (x) — f (ax) 1— a

'

4. The Jacobi Condition and Poincaré—Birkhoff—Witt Theorem

143

We obtain, by combining the last two inequalities, f (x) — af (ax) 1— a Pax)) a (xyf(ax) ± i f(x) = if(x), 1— a

(1 B l A — alAlB) f (x) = axyf (ax) + i

—

that is, the operators /A and /B satisfy the same perturbed Heisenberg relation as the operators A and B themselves. Thus we may conclude that the twisted product 1 (f * g)(x , Y) = f (l A , 1 B )(g (x , Y)) clef

2

defines the structure of an associative algebra on the symbol space .F2, or, in other words, functions of A and B form a hypergroup. Of course, this argument applies to the case a = —1 as well, so that for the simple example of a graded Lie algebra considered in Subsection 2.5 we can also claim that the functions of A and B form a hypergroup. However, the situation is quite different as soon as a closed system of graded Lie relations defining a finite-dimensional graded Lie algebra is considered. For example, consider the relations [A , B] + -• AB ± BA = 1, [A, A ] 2A2 = 0, [B, B ] 4_ -.-a 2B 2 = 0.

The last two relations are not needed in calculating the ordered representation operators; nor do /A and /B satisfy li = / /23 = 0, and we see that the generalized Jacobi condition, together with the Poincaré—Birkhoff—Witt property is lost. The remedy may be to consider symbols as functions of commuting and anticommuting arguments (see, e.g., [12] and [110]). The nonlinear commutation relations of Subsection 2.3. Let us check that the operators /A , /B, and lc, constructed there satisfy the relations UA ,1B ] = 1, UA, lc] = a(1C )L A + 13(101B + Y (lc), UB, lc] = 13* (101A + 3(/c)//3 ± a(lc). We have ( ii3 A

x + 19, ) , 1-15(z) ( y(z) )] — ) / e v),(x y z) = PS(z) ( -t- z--_,S(z) a (Z) y a ) (p(x, y, z) -r-I ZS(Z)I5(z) ( = Sqc)Is(z) ( x+ -5-Y Y

Y (zz ) Cr

)

/cço(x,

y, z)

(.x ' Y' z).


144

By adding and subtracting S(z) in the numerator, we obtain lA 1

( tB)

lc = S(lc) S(lc)( 1A ) + ( "C) 1B (TUC). ).

It remains to check that UA , //31 = 1. Exercise 11.1 Check that

UA , / B] = 1 if and only if tr(S(z) — z) --. O. 123

Thus, the last condition guarantees that functions of (A, B, C) form a hypergroup.

5 The Ordered Representations, Jacobi Condition, and the Yang Baxter Equation In this section we clarify the relationship between the generalized Jacobi condition for the ordered representation operators introduced in the preceding section and the Yang—Baxter equation playing an important role, e.g. in the theory of quantum groups. In Section 4 we studied the classical Poincaré—Birkhoff—Witt theorem. Letus recall its statement. The theorem claims that if L is a Lie algebra with basis A 1 , . . . , An and U the enveloping algebra of L, than the ordered monomials E (Z+ U {O }) '

form a basis in U. This result can be viewed as follows. The algebra U is generated by generators A 1 , .. . , An and Lie relations between these generators. The PoincaréBirkhoff—Witt theorem provides a linear basis in U constructed from A 1 , . . . , A,. Here we discuss the same problem for general algebras defined by generators and relations. As we have learned in the above exposition, the main problem which is solved by introducing the ordered representation operators is as follows. Suppose that we are given an algebra A, generated by the operators A 1 , . . . , An and the relations il

(co (Ail , . .. ,

is

40 = 0}„EQ

(11.35)

(for simplicity, we assume that all symbols co (y i , .. . , ys ) are polynomials; the number s of arguments of this polynomial is allowed to depend on co). We are interested

1

n

in whether any element of A can be represented in the form P (Ai , . . . , An ) for an appropriately chosen symbol P(y i , . . . , yn ). It was shown that this is true if and only if there exist operators / 1 , . . . , ln in the space of n-ary symbols such that 1

n

A1 11 f (A 1, . . . , A)]1 =

1 n (ii f)(Al, .. . , An)

(11.36)

5. The Ordered Representations, Jacobi Condition, and the Yang—Baxter Equation

145

for any j = 1, ... ,n and any symbol f (y i , . . . , yn ). With polynomial (or power series) symbols it is not hard to show that provided that the operators / 1 , ... , in exist at all, they can be chosen so that 1

n

(11.37)

f(11, • • • ,ln)(1)= f(Yi, • • • ,y n )

for any symbol f (y i , • • • , yn ). Using the operators / 1 , . . . , ln we represent any element il ik l'ir = Vi(Ai i , ... , Ai k ) E

A_

as an ordered polynomial, 1 n ik 1fr (A ji, • • • , A jk) = f(Ai, .. • , An), il

where il

ik

f(Yi, • • • , Y n) = lif (1 j 1 , • • • , 1 jk )(1),

i.e., we can bring the operator gf into normal form. From this viewpoint, formula (11.37) means that the normal form of an operator already in normal form is just the operator itself. We can arrange things in a slightly different way. Let Fn be the space of n-ary symbols. We have the linear mapping it

=

, :F, —> A i ,...,A,

A

it 1

1

n

(II.40)

f (iii, • • • , An),

f F-*

whose image consists of operators in normal form. Since any operator lif E A can be brought into normal form (see (11.38) — (11.39)), we see that A is an epimorphism. Moreover, by (11.38) — (11.39) we have I

n if RA1, • • • , AnAi

where the symbol

h(y i , . .. , y n )

1

n

1

n

lig(Ai, ... , An)II = h(Ai, ... , An),

of the product can be expressed by the formula 1

n

1

n

h(y i , ... , yn ) = II f(11, ... , 1,)111Ig (11, ... ,111)1(1). Since, according to (11.37), we have

1

(II.41)

n

g(li, . . . ,ln)(1) = g(Yi , • • • , Y n),

(11.42)

IL Method of Ordered Representation

146 (11.42) reduces to

1 h(y i ,

Denote

, Y n ) = f(i1, .

, n)(g(y)).

n f * g = f (li, • • • , in)(g(Y))•

This is a bilinear operation on Fn and the element 1 E F is the two-sided identity with respect to the operation * (indeed, 1 is the right identity by (11.37) and the left identity since 1 (11.45) 1* g =1(11, ... ,1n)(g)= g. The mapping (II.40) is an algebra homomorphism, and it is epimorphic. There are two important closely related questions. (1) Is (Fn , *) an associative algebra (or, in Maslov's terminology [130], is (Fn , *) a hypergroup)? (2) Is the mapping (II.40) a monomorphism? The implication (11.37) 44. (11.36) is easy, since if (II.40) is a monomorphism, then it is an isomorphism taking (Tn ,*) into an associative algebra. So we can concentrate on condition (11.37). It turns out (see Theorem 11.10) that the mapping (II.40) is an isomorphism if and only if the left ordered representation operators / 1 , . . . , /n themselves satisfy relations (11.35), i.e., in

)

=0

(11.46)

for all co E Q. This condition is called the generalized Jacobi condition. Hence, we have the equivalence Generalized Jacobi condition = 4a is an isomorphism.

(11.47)

Let us study the right-hand side of (11.47) in more detail. With polynomial symbols, it can be restated as follows: The ordered monomials

(11.48) form a (linear) basis in the algebra A. This property is known as the Poincaré—Birkhoff—Witt (PBW) property and the basis (11.48) as the PBW basis. One can even define a finer property as follows. Let AO ) c A be the subspace spanned by the unordered monomials of order < j. Clearly,

.0) C AM C • • • C A (i ) C fl (i +1) C • • •

(11.49)

is an increasing filtration on A. Definition 11.7 One says that the PBW property is satisfied in A up to order s if for k form a basis in the any k < s the unordered monomials (11.48) with k1 + • • • + k space


147

The major advantage of the new definition is that the PBW property (= the PBW property up to order oo in terms of this definition) can be verified step-by-step, via, say, induction on k. However, this is not so easy, and in practice it is much simpler to invert the emphasis and to find some (necessary) conditions on the constants involved in the commutation relations (11.35) than to prove these conditions to be sufficient for the PBW property for any given order. Let us consider some examples.

Example 11.2 Enveloping algebra of a Lie algebra. Let A be the algebra generated by A 1 , . . . , A n and the relations

[ A1 , Ad t AJ Ak — AkA; . Eciik Ai.

(II.50)

/ Let us assume that A satisfies the PBW property and infer some consequences on . This is quite easy. First of all we have the constants cljk. [Ai, Ak] ± [Ak, Ai] = 0,

(II.51)

E(clik + clki ) A1 = 0. /

(11.52)

that is, according to (II.50),

Assuming that the PBW property is valid for A up to order 1, that is, the A1, 1 = 1, . .. , n, are linearly independent, we readily obtain that / . ,... . ik / = —L„I./

(11.53)

Furthermore, consider the Jacobi identity [[Ai, Ak], Ad + c.p. = 0,

(11.54)

where c.p. stands for the two summands obtained from [[A1, Ak], Al] by cyclic permutations. Using (II.50) twice, we obtain

Ecsikcsr or + c.p. = 0,

(11.55)

s

where c.p. denotes the terms obtained by cyclic permutations of the indices j, k, and 1. Again using the linear independence of the Asi , we obtain n s ,,r E (csjk crs l + cslj sk cr ± Ckit'sl i\ =u

.

(11.56)

s

The identities (11.53) and (11.56) are known, respectively, as the antisymmetry condition and the Jacobi condition for the structural constants of a Lie algebra. Let us


148

emphasize that they both follow from the assumed PBW property up to order 1 (!) in A (the algebra A is called the enveloping algebra of the Lie algebra determined by the structural constants c s. j k ). Let us also point out that it is not obvious at this stage how to infer the PBW property of A (at least for order 1) from these identities. Nevertheless, the PBW property holds in this situation to arbitrary order (the Poincaré—Birkhoff—Witt theorem). The standard "combinatorial" proof of this theorem can be found in any advanced textbook on Lie algebras. In this book we have given another proof. First, the left ordered representation of the relations (II.50) was constructed (see Section 4), and then it was proved that the identities (11.53) and (11.56) imply that the generalized Jacobi condition holds for the operators of the left ordered representation.

Example 11.3 Consider the algebra A defined by generators A, B, and C and the relations BA =aAB +(IC,

(11.57)

BC = yCB ± vB, CA= PAC ± ti,A,

where a, y, )5, a, y, and pc are some prescribed constants. Let us assume the PBW property and infer some equations involving these constants. First of all, it is easy to see that ap y 0 0. Indeed, assume the opposite, say, a = 0. Then BA crC = 0, (11.58) —

that is, BA and C are linearly dependent and the PBW property up to order 2 fails. Next, consider the monomial ABC and transform it into CBA using the relations (11.57) in two different ways. Method 1. Interchange first B and C. We have

(11.59)

ABC = A(yCB ± vB)= vAB ± yACB.

In the last term we interchange A with C and with B, thus obtaining 1 vAB ± yACB = vAB + y CAB (– – IIAB)

P

= (v

P

y ApY ) AB ± —CBA– Ya C2 . Pa Pa —

(II.60)

Method 2. Let us first interchange B and A in the product ABC. Then we have, by analogy with the above, Y ABC = —CBA -I- .111' BA– –a C2 . af3 a af3

(II.61)


149

Comparing (II.60) and (1 1.61) we see that

#

AB +

2

(

C——C = v a

af3

—

gy P —

AB — YOE C2 . Pa

(11.62)

Assuming the PBW property up to order 2 we see that one should have o- ( v — p,)

afi

o= -.; (1

fi )

and

v — ,u, = vf3 — ,u,y . In other words, either

a = 0 and v(1 — /3) = ,u(1 — y) Or

f3 = y and v = ,u.

(11.65)

Note that we have only proved that conditions (II.60) or (11.65) are necessary for the PBW property to be satisfied and not vice versa. To prove the converse statement, we again calculate the left ordered representation operators. This can easily be done by analogy with numerous other examples that can be found in the preceding sections. 123

Choose the Feynman ordering C, B, A. We omit the calculations and only present the final result. It reads lA = Ys, 1B =

— ayv IY3 Ivi-7 2lY' l +a ,uy317 ,8, I , cryi I)( 3"8 1)712' Y + y2 ( la Y3

IC = ylIf73 .1y12/1/

(11.66)

_ L ip iyliy,1 ± itty3 //y33,1 , y Y3 2

where the notation /zŒ, /z1145 , and g"6')' stands for the following combinations of dilatations and difference derivatives:

(11: f)(z) = f (az),

(I:43 f)(z) = — 81x (az, Pz), 82 f W AY f)(z) = — (az, fiz, yz). 8x 2

(11.67)

It is a routine but useful exercise to check that the operators (11.66) satisfy the original commutation relations (11.57) if and only if one of conditions (11.64) and (11.65) is satisfied.

150


Example 11.4 (Graded algebras) Let G be a finite abelian group, and let

A = ED

(11.68)

Ag

gEG

be a G-graded algebra with 1. Recall that this means that the underlying linear space A is the direct sum of linear spaces A g , g E G, and that

Ag Ah C

(11.69)

Ag-Fh

for any g, h E G. In particular, it is assumed that 1 E A0, where 0 E G is the neutral element of G. Assume that A is generated (as an algebra) by a finite number of elements A1, .. ., A. These elements are homogeneous, that is, Ai

E Agi , i =

1, . . . ,n,

and that the following permutation relations are valid:

A i A; — wij A; Ai

= E vicj Ak

( 11 .70)

k

where coii and Xii`i are some prescribed constants (for homogeneity reasons, the sum on the right-hand side of ( 11 .70) extends only over k such that

(II.71)

gk = gi + gi;

all the other Xiii. are zero). We shall assume that the coii depend not on i and j themselves but rather on gi and gi: coii = co (gi , gi). (11.72) The numbers coii and Xkii will be referred to as the scale and structural constants of the algebra A, respectively. Assume that the PBW property is valid in A with respect to the tuple A 1 , . . . , An , and deduce some equations involving the scale and structural constants. Let (11.73) A" c A(1) c • • • C A (s) c A(8± 1) c • • • be the filtration in A induced by the degree of unordered monomials in A 1 , .. . , An . Clearly, one has (11.74) A(i) = e AV gEG -

'

where

(11.75)

Aog ) = A(i ) n Ag.

Consider the associated graded algebra

B = °EIS A (i) I A (i —1) --. i=0

06 B (i)

i =0

(11.76)


151

(here we set A(-1) = {0}). Then, clearly, each BO is G-graded, that is, B(1) = e3, B(i) geG

g '

(11.77)

and, moreover, B has the PBW property with respect to the tuple B1 , . . . , Bn , where each Bi is the image of the corresponding Ai in A(') / Am . The operators Bi satisfy the "homogeneous" permutation relations that can be obtained from (II.70) by omitting the linear term, lost in the course of factorization: Bi Bi = co(gi, gi)BiBi.

(11.78)

Consider the product BiBi Bk and transform it into BkB1Bi using the relations (11.78). We have Bi Bj Bk =

co (gi , gi)Bi Bi Bk

= W(gi , gi)co(gi, gk)Bi Bk Bi

= co (gi , gi)w(gi , gk)ci.)(gi, gk)BkBi Bi and the same factor results if we first interchange 131 and Bk. Hence, the PBW property imposes no restrictions on the constants co(gi, g1) alone, except for (z)(gi, gi ) = co(gi , gi ) -1

(11.79)

In particular, if all the constants A.lici are zero, then B is isomorphic to A and it is easy to see that the PBW property holds in A up to any order. Let us now proceed to the general case of nonzero constants A. We assume that the scale constants co (g , h) satisfy the following condition: co(g, h ± k) = co (g , h)c o (g , k).

(II.80)

If we define a bracket in A by setting [A g , Ah] = A g Ah — co (g , h)AhA g

for A g E Ag and Ah

E Ah,

(II.81)

then the Jacobi identity

[A g , [Ah, Ak]i = [[A g , Ah], Ad + co(g, h)[Ah,[Ag, Ak]i

(11.82)

holds for any homogeneous elements A g E A g , Ah E Ah, and Ak E Ak. Applying the identity (11.82) to Ai , Ai , and Ak and taking into account that the third-order terms in this identity necessarily cancel out, we obtain

E xiii (Ai Ai

— co (gi , gi)Ai Ai) = E

x(Aiiik — co (gi, goAkm

i

i

+co (gi , gi)

E ),/ik (Aj Ai 1

—

co (gi , gi)Ai Ai),

(11.83)


152 or

(x i k x/i7 _ xii; x1,nk _ co(gi, gi)X4 k X7i )A ni = O. E i,m

(11.84)

Assuming the PBW property of order 1, we obtain the following equation tying up the scale and structure constants of the graded algebra: — x li i x 'ink — co ( g i , gi ) x li k ) ; ) = 0

(11.85)

i

for all i, j,k,m =1, ...,n. It can be shown by constructing the ordered representation and computing the commutators that conditions (11.79), (II.80) and (11.85) guarantee that the PBW property holds in the algebra A.

Example 11.5 (The Faddeev—Zamolodchikov algebra) In the quantum inverse scattering problem method the key role is played by the Faddeev—Zamolodchikov algebra, defined as follows. PP;) be a tensor whose indices range from 1 to n. Consider the algebra Let R = (R cia A generated by the operators 4 where a, 13 = 1, . . . , n and the relations

R au/ PP' ApY 13 A Y', R YY/ AP' AP = PP , a/ a'

(11.86)

where a, a', y, y' = 1, . .. , n. Such an algebra is called a Faddeev—Zamolodchikov algebra. It can be proved that the PBW property up to degree 3 holds in a FaddeevZamolodchikov algebra A provided that the tensor R satisfies a certain equation known as the Yang—Baxter equation. This equation can be written out as follows. The tensor R can be viewed as an element of the space

R E Mat(n, C) 0 Mat(n, C), R=

E as 0 vs,

(11.87) (11.88)

where as and vs are (n x n) matrices. Define the elements R 12 , R 13 , R23 E Mat(n, C) 0 Mat(n, C) 0 Mat(n, C) by setting .-.12

K

'Eas 0 Vs Q!,

R 13 =

E 0-s 0 I ® vs,

R 23 =E/Oas0vs, where / is the identity matrix.

(11.89)


153

The Yang—Baxter equation reads

R 12 R 13 R23 =

R23R13 p12 4‘

(II.90)

Lengthy but routine computations show that (II.90) guarantees the PBW property in the Faddeev—Zamolodchikov algebra up to degree 3. Unfortunately, the ordered representation operators cannot be written out in such a general setting, so that it remains unknown whether the PBW property holds up to arbitrary degree. However, in some particular cases the ordered representation can be computed explicitly. From now on we use the term "Yang—Baxter equation" for any equations following from the PBW property for the constants in the commutation relations.

Example 11.6 (The Sklyanin algebra) Consider the algebra A with generators Ao, A1,

A2,

and A3 and the relations [A0, Ai ] = i (AoAi + Ai A0), [Ak, Al l

=

(AkAl + AiAk),

(II.91)

valid for any cyclic permutation (i, j, k) of the indices 1, 2, 3. The constants g-id occurring in equation (II.91) satisfy the Yang—Baxter equations provided that —

Jkl

=

3

t7k

,

where all Jk are constants. Sklyanin [168] showed that the PBW property holds in A. However, for general „Z. the ordered representation is not known. Let us consider the particular case in which

= ‘72 = 1 and 33 = 1 + co, where for simplicity we assume co > 0 and co 0 1. The relations (II.91) take the form [A0, Al] = i (A2A3 + A3A2),

[A0, A2] = —iw(A1A3 + A3A1), [Ao, A3] = 0, Ak] = i(A0A1 + A1A0),

(11.92)

where (j, k,l) is an arbitrary cyclic permutation of the indices (1, 2, 3). We choose in A another set of generators given by the formulas a± = Ai ± A2, b± = ±A0 4fo5A3.

(11.93)

154


These generators are linear combinations of the original generators. We choose the 1

3

2

4

Feynman ordering (a_, a+, b_, b+) and construct the regular representation for this ordering. For the new generators the relations (11.92) take the form

[b+, a+] = ±A/FD(b+a+ ab), [b_ , a+] = /5(b _a + a+b_), [b+, b_] = 0, [at , a _] = o (b2+ —

(11.94)

In a slightly different form, these relations can be rewritten as

b+a+

= ota+b+,

b_a+

=

b+a_

= (71 a_b+,

b_a_

=

[b+, b_]

= 0,

[a+a_] = ce

where we denoted a=

1

a+b_, (11.95) 1 (b 2_ b2

+ AFL)

(11.96)

Let us proceed to computing the left ordered representation. First, note that [b_, b+] = 0, and these operators act last. Hence, no commutations are necessary in computing the composition 1

2

3

4

b±i[f (a_, a+, b_, b+)]], and consequently, we obtain lb+ = y4,

1b_ =

)3.

Furthermore, we have

1 2 3 4 5 1 2 3 4 a+fff (a_, a+ , b_, b+)]] = a+ f(a_, a+, b_, b+) 1 4 ) 3 (1 2 2 a+ f a_, a+, ab_, a

(11.99)

by virtue of (11.95) and the permutation formulas from Chapter I. We arrive at the conclusion that (II.100) la+ = y2Iya3 IyY a , where

f)(z) = f (a z) , that is, Ift is the dilatation with respect to the variable z.

(II.101)


155

Similar but somewhat more complicated computations yield the following formula for the representation operator / a_ . We have

la_

=-

1

,2

[yi + (1 + a )2

2 (PI y2 Y2

y

2 2 Y4

-1) + a —(I y2lia2— 1) Y2

a lia Y4 Y3

(II.102)

The constructed operators are readily verified to satisfy relations (11.95). For example, let us check the commutator [la, la]. We have 1a + 1a_ —

=

r 2 a 2 2 1/a2 a iY3 i L.)) (i 2 — 1 ) + a y4 (IY2 Y4 3 Y2 ,2

—

(

„Y ' )

a

Y2

— 1)1/ a / 1/a (1 ±a 2 ) Y4 y3

,

2 l ice` 2 -7 4 — (IY2 a — 1) + a - (1)72

1) y2(1 + a ) -2 .

Y2

Performing elementary transformations we find that

c e2 _ 1) ± a) -2 1a2yi (Iya22 1) ± y,24. (iyy y2 + y32 — — y32a2 1ra2

a— 1 a+ 1

2 kY4

2\ Y3

)=

2 T 1 /ce 2 y4 1 y2 +

a-1 (72 a

± k'b+

2 21 y4 72 \

)•

The other relations in (11.95) can be checked in a similar way. Now let us proceed to the general scheme. Suppose that we are given an algebra A determined by generators A 1 , . .. , An and relations (11.35) involving certain parameters X,. We suppose that there are sufficiently many such relations so that any element lr E A can be put into normal form 1 =

f

Oh

•••,

An).

How can we study the Poincaré—Birkhoff—Witt property in the algebra A? The usual technique is as follows. First, one considers triple products (cf. Examples 11.2 and 11.3) and uses permutations performed in various orders to infer the conditions on the parameters for which the PBW property holds up to degree < 1, 2 or 3. This procedure is considerably simpler than that of computing the ordered representation, and it permits one to eliminate from consideration all the parameter values for which the PBW property fails in low orders. The equations obtained at this stage are the Yang—Baxter equations. Then one assumes that the parameters take only the allowed values and computes the ordered representation (if possible). Finally, the operators of the ordered representation are substituted into the original commutation relations and this makes it clear whether the PBW property holds in the algebra A.


156

Hence, the situation is as follows. The Yang—Baxter equation is in general only a necessary condition for the PBW property to hold up to a certain low order. The ordered representation, though often hard to obtain, always gives a precise answer (that is, a necessary and sufficient condition for the PBW property to be valid).

6 Representations of Lie Groups and Functions of Their Generators As is clear from the preceding considerations, noncommutative analysis provides the following method of dealing with commutation relations: Given a set 0 of commutation relations, find some reasonable class of representations and an appropriate symbol class such that the operators 1 f

with admissible symbols

f

(A) =

f

n

(Ai,

•••,

An)

are well defined and the composition formula

1

n 1 n [Lf Oh • • • , AnlEg(Al, • • • , An)Il =

1

Lf (Li,

1 n n • • • , Ln) gii (Al, • • • , An)

is valid; Furthermore, try to produce a reasonably explicit expression for the left regular representation operators L 1 , . .. , L n ; Then it becomes possible to use the above calculus for investigating such problems as the Cauchy problem and the inversion problem for the operator f (A). Here we present a partial realization of this programme for Lie commutation relations, that is, the commutation relations which define finite-dimensional Lie algebras. Let us now proceed to precise formulations.

6.1 Conditions on the Representation 1

n

We intend to study functions of the operator tuple A = (A1, . .. , An ) whose components satisfy Lie commutation relations. What does this mean exactly? This means that the operators A1, .. . , An act in some linear space E, and that n

[A j , Adu = —i

E

Clik

Ai

U

/=1

for any u E E, where clik are the structural constants of some Lie algebra g in some basis ai, .. . , an E Ç. Using notions and notations of representation theory, one may

6. Representations of Lie Groups and Functions of Their Generators

157

write

A1 = —i r (ai), j = 1, . . . , n, where r:

g ---> End(E)

is a representation of g in the algebra of endomorphisms of E (see Section 1). The reasons for introducing the factor (—i) will soon become clear. Thus, we are able to develop the calculus for polynomial symbols, or in terms of formal series. This might be interesting in itself, but it does not cover even the simplest applications to the theory of differential equations (see Chapter III below). Thus we have to impose some further conditions which make it possible to consider wider symbol classes. First, we require that g be a real Lie algebra (that is, g is an n-dimensional vector space over R; in particular, Clik E R), and that g be a nilpotent Lie algebra, that is, any commutator

[A11, ... , [Ai s _i , Ais ] . . .] = 0 once s > N, where N < oc is the nilpotency rank of Ç. Violation of these requirements would lead to considering analytical symbols with certain restrictions on the supports of their Fourier transforms; these topics are beyond the framework of this book. Since the definition of f (A) for nonpolynomial symbols uses some notions of convergence (see Chapter IV below), our model should necessarily involve conditions of a functional-analytic nature. We introduce the simplest version of conditions of this sort:

—

E is a dense linear subset in some Hilbert space H;

—

The following Nelson condition is satisfied: the operators A1, j = 1, .. . , n and n

D= 1 + EA. ' j=1 are essentially self-adjoint on E. These conditions are very important. Indeed, Nelson's theorem (see Nelson [149]) guarantees that under these conditions there exists a unitary representation

T : G --->- Aut(H) such that t is a derived representation of T; more precisely, r = LIE•

The representation T plays a crucial role in what follows.

Ii. Method of Ordered Representation

158

6.2 Hilbert Scales We have stated the conditions we impose on the operators A. Our next task is to de1 n .. . , A n ) act. It is not convenient scribe the function space, in which the operators f (Ai, to construct the calculus in the space H itself, since unbounded symbols (which are of practical interest) give unbounded operators. To avoid this difficulty, we use special Hilbert scales associated with H, whose construction uses the operators A1, ... , An (see the propositions given below). We postpone using the nilpotency requirement as far as possible; for now we only use the much weaker requirement that G is unimodular, that is, its Haar measure dg is biinvariant (see Appendix).

Proposition II.! Let A be a strongly positive essentially self-adjoint operator, defined on a dense linear subset E of a Hilbert space H, invariant with respect to A (that is, AE C E). Consider the following family of norms on E: ,L 3Lk u \l/2 K ) Ilullk = (u,

r-17 E g.., U E

E

(here A is the closure of A; in the last formula one may take A instead of A for k > 0). We denote by H I` the completion of E with respect to the norm Il • Ilk. The following statements are valid: (1) {H k }kEz is a Hilbert scale, 110 = H. (2) For k > 0, Hk may be naturally identified with the domain of the operator A k/2. (3) A 112 :ti-k ---> Hk-1 is an isomorphism for k > 0 and extends to such isomorphism for k < O. (4) The bilinear form (u, 0 may be uniquely extended by continuity to the pairing Hk x fl –k ---> C, which induces an isomorphism (H –k ) Ld (H k )*. Proof Since A. is strongly positive and self-adjoint, all powers (A)' are well defined (see Dunford, Schwartz [42]). Set Wk = DC-e 12) fork > 0, Ilullwk = ilullk, Wk = (147–k )* for k < O, with the identification W0 = (W°)* obtained via the scalar product (., .)H in the space H. Clearly, properties (1) — (4) are satisfied for the spaces W k , and we must only prove that Hk = W k for k > 0 (the case k < 0 then follows automatically). It suffices to show that E is dense in W k in the topology of Wk . Assume the opposite is true. Then there exists a nonzero vector E Wk such that

(, u)k = 0 for all u

E

E.

We have

(, u)k = (, A k u) = ( 3,k , u) = 0 for any u E E. Since E is dense in H, ,&11 = 0, and thus 4' = 0, because Ker Ak is ril trivial. The proposition is proved.


159

Proposition 11.2 Let G be a Lie group with Lie algebra g, and let r:

g --> End(E)

be a representation of g in a dense linear subset E of a Hilbert space H, satisfying Nelson's conditions. Let g, c g be an ideal, and let ai, . . . , ak be a (linear) basis of gi , A; , —ir (ai), j = 1, .. . , k. Denote by

T : G -- Aut(H) the unitary representation of G in H, associated to r by Nelson's theorem, and set k

Au=

E iOu J +u, u

E E.

j=1

The following statements are valid: (1) A is a strongly positive essentially self-adjoint operator on E; the corresponding Hilbert scale does not depend on the choice of the basis (ai, . . . , ak) up to a norm equivalence. (2) The representation T extends to a bounded strongly continuous representation in the space Hs for any s; there is an estimate

ilT (g) ull s __ C s (ii Ad g a Ilulls, where Cs is a constant independent of u, and (Ii Ad glpgi is the norm of the restriction of the operator Ad g on the invariant subspace gi. Proof 1) Since g1 is a subalgebra, it generates the corresponding subgroup G1 C G. The operators A1, ... , Ak are generators of the unitary representation T 1G 1 . By

Nelson's theorem, the operator A is essentially self-adjoint. Positivity of A is clear. The fact that the corresponding Hilbert scale does not depend on the choice of the basis up to norm equivalence is a consequence of the following lemma.

Lemma 11.5 Let {H k ) be the Hilbert scale described above. For any integer s the norm equivalence k

illills+1 fd Mulls +

E iiiipils

holds for u E E.

We postpone the proof of this lemma until the end of the proof of Proposition 11.2. 2) It suffices to prove the estimate for positive s and then use conjugation. We proceed by induction on s. The estimate is valid for s = 0, since T is a unitary representation, and therefore, II 7' (g)li = 1 for all g E G. To perform the induction step s = s + 1, we use Lemma 11.5.

160


By this lemma, it suffices to estimate Ilk T (g)u Il s . We have

Ai T (g) = T (g) (T (g) -1 Ai T (g)) = — i T (g)r (Ad g(ai)), thus IlAiT(Oulls = II T(g)r(Ad g(ai))/411,

5_ I T (g)11s - II Ad gligi . Ilulls+i• The latter inequality together with Lemma 11.5 immediately yields the desired estimate for the new value of s. The proposition is proved. 0

Proof of Lemma 11.5. Case 1. s > 0. For s = 0 we have

n Illiq = CO ± Ai ± • • • + AD u, 0 = 11 14 1I 2 + E HA;

uH2,

j=1

and the statement of the lemma holds. For s > 0 we use the identity

n [A1,

n

A] = Dili, Ai] = —i E ci ( A, A k + Akm. .

k=1

k,I=1

Using the latter relation, we obtain

n 114+1 = ((1 ± E A . ) A s u, u)=Hull s2 j=1

n +E(AjA s u, A1

u)

j=1

n = Illills2 + E HA; uHs2 +

(P2s(11 1, ... ,

An ) u, u)

j= 1

n

cik ((A i iik + A k A ;) Ai As-iu,

+ E

u) ,

j ,k,I=1

where P2s (Ai, ... An) is an unordered polynomial of degree < 2s. The last term equals zero due to antisymmetry, so that

n 114+1 =11 11 11, + E HA; /411. + (P25(Ai, ... , An) u, u ) . ; =1

By induction on s we prove that k IlUilk+1 ' illillk + E mph j=1

for all k E {0, ... , s — 1}


161

l(P2s (Ai, ... , A n ) u, u)I < C Hue. First of all, the first relation implies the second, since (P2s (Ai, . . . , A n )u, u) is a sum of terms of the form (A1 1 ... Aisi u, A11 . . . Ai n u), where si , s2 < s. Thus, it suffices to verify the first relation. The statement is valid for s = 1 (see above). The induction step goes as follows. We have

n

Hu e+,

calue + E HAE4e), j =1

n

2 —CI I U 11 s. II U lis+1 ?-. E IlitiU Il s22 j=1

Moreover, one has

ilue + 1

Hue.

Multiplying this by (c + 1) and adding the result to the penultimate inequality, we obtain

n

(C + 2) Ilue-Fi

iiiiii, + E liApe, j=1

as desired. Case 2. s 0 if we use the relations [A» A --t i _ —A—[A1, A ] 6,--1; instead of the polynomial h25 (Ai, ... , A n ) we get a polynomial in Ai, ... , A n and 0 A -1 of weight < 2s, where the weight of Ai is 1, and the weight of A -1 is —2. Thus, we can associate a scale {Hs} to any ideal gi above). Of course, g1 = g would be the simplest choice.

c g (see Proposition 11.2

6.3 Symbol Spaces Let us now define admissible symbols. For this purpose, we introduce a special function space on the Lie group G. Definition 11.8 Let G be a Lie group with Lie algebra g, gi c g an ideal. We denote by 1)± (G, go the space of distributions q) on G representable both in the form §9 =

E

lalm,ak+1,...,an=0

laraa


162

and in the form

=

E

ra(rta),

icrimotrk+,,•••,cen=0

where 1 = O p ...,ln ) is a tuple of right-invariant vector fields on G, forming a basis in g, with / 1 , ... , /k being a basis in Ç 1 , m = mco, ,uct are measures on g summable with the weight II Ad gllsoi for any s > 0, r = r1 , . . . , rn is a basis in the space of left-invariant vector fields on g, with r1, ... , ri being a basis in Ç1; /2„ satisfy the same conditions as If q) E 'P+ (G,

go, then the operator T (q)) = f q)(g)T (g)dg G

may be defined as follows: for q' (g)

E

=

1' ii3OE (g)

l'Imotrk+1 , --an=°

we set T ( 0 tf

E

lalm,ak+i

( -1 ) 1cri f pl, a (g) [la T

an =0

(g)]dg .

(II.103)

G

Proposition 11.3 The integral (II.103) strongly converges in the scale {Hs} and defines a continuous operator

T (q)) : W _>. Hs — m (S0 ) for any s E Z. Proof We have la T (g) u = (i Ar T (g) u = T (g) (i Ad g (A))" u.

Thus, ilia T (g) ulls—mm

c IIT (g)ll s—m(ço)---s —no) II Ad gil a Hulls < ci II Ade s iluiis•

Since the measure t.ta is integrable with the weight II Ad glisoi , the integral converges in Hs— m (v) when applied to u and defines a bounded operator. 0 The proposition is proved. The following two assertions describe the product for operators of the form T (0.

6. Representations of Lie Groups and Functions of Their Generators Proposition 11.4 The space P+ (G, lification ('Pi * (P2)

163

go is an algebra with respect to the group mol-

=f

Wi(h)402(h -1 g) dh.

G

Proposition 11.5 Let wi, q)2 E P+ (G, ai). Then T(1) 7' (q)2) = T (Sol * 'p2). Proof of Proposition 11.4. Denote

4. (g) = il Ad gligi ; since Ad(gi g2) = Ad gi o Ad g2, we have (g h) First of all, note that it suffices to prove the statement for the case where both q)i and ÇO2 are measures on G. Indeed, if q;11 = la Pt, q)2 = rfi ii, then (Pi * go2 = rx rfi (A *T). Next, let us verify that 'pi * q;02 is a well defined measure dn G. Let f G Co(G) be a test function. We have = f çoi(h) 402 (h -1 g) f (g) dh dg

GxG = f q) 1(h) q)2, (k ) f (hk ) dh dk, GxG

since the Haar measure is invariant. Since f (hk) is bounded, and fl , go2 are summable on G, the latter integral converges. Let us now prove that 'pi * 'p is absolutely integrable with the weight (g)s for any s. We have

< IWi * q)21, r > =

f IWi (h) W2(k)I e (hk) dh dk GxG

- oo of solutions to the Helmholtz equation is reduced to the problem of finding asymptotics expansions with respect to smoothness for the wave equation'. The reverse transition (from smoothness asymptotics to parameter asymptotics) is not as simple as the direct one. First, asymptotic expansions with respect to smoothness are usually considered in all variables rather then a single variable. Second, the transition to the Fourier transform in t in the wave equation is simplified by the fact that the coefficients of this equation do not depend on this variable. If we tried to pass to the Fourier transform in the variable x, we would not obtain a differential equation for it unless the coefficients of the equation are polynomials. To deal with each of these difficulties separately, we first consider the equation aiOE(x, (t

)

— x , t) — 0 03 x ) a u

(III.9)

and construct asymptotic solutions with respect to smoothness in t alone. According to what was said above, we perform the Fourier transform t k in equation (III.9) and obtain the equation

E +Ice15..m

a)

(x, k) =0.

a• -- (ik)i Ja ak

We seek its solutions in the form

î (x, k) = eikS(x)

M (x )

k -1 vi (x )

.) = e ikS(x) çqx, k).

The asymptotic solution of the original equation has the form u(x, t) =

t j eiks(x),, ( ,. it) ). /

7

t is the inverse Fourier transform. where Note that the latter formula can be rewritten as

a u(x,o= eigx( — ialat) w (c,—i— at )s(t).

(III.10)

It is somewhat difficult to substitute this expression into the equation directly, since the coefficients depend on t and thus do not commute with the operator (—i that in spite of the evident relationship described here both theories were developed independently for a long time. The reason probably resides in the difficulties which we discuss below. 1 Note

184

III. Noncommutative Analysis and Differential Equations

involved in the asymptotic expansion (in fact, —i8/0t is the "large parameter" of our expansion). This indicates that noncommuting operators may be useful here. Clearly, the fact that t does not commute with —ia/at directly affects the form of the corresponding Hamiltonian function. In conclusion, let us try to find the general form of asymptotic expansions with respect to smoothness of solutions to differential equations. We use formula (111.1 0) as a starting point. Clearly, one should use derivatives with respect to all variables. To simplify the notation, we consider independent variables (x 1 , .. . , xn ) (i.e., we do not distinguish t explicitly). Analyzing formula (111.1 0), we arrive at the conclusion that the general form of the asymptotic expansion with respect to smoothness is 2 2

1) , a

isx,-1 7, u(x)= e (

where

I

ço (i,—i -1) f (x),

ax

f (x) is an arbitrary function of x, S(x, p) is first-order homogeneous in

p=

(p 1 , ... , pn ), and w(x, p) is given by a formal series

60 (x, P) = goo(x, p)+ q)_1(x, p) ± • - • . of homogeneous functions (the subscript denotes the degree of homogeneity). We see that i) the operators (—i

a

axl '•••'

.

a

1 axn

play the role of "large parameters" in the expansions with respect to smoothness; ii) the terminology of noncommutative analysis is convenient when dealing with expansions of the form (111.1 0). In more general cases, the operators defining the asymptotic expansion can be chosen in some other way. For example, if we deal with simultaneous asymptotic expansions with respect to smoothness and growth at infinity, we will usually introduce the set of operators

( — i a/axi, . . . , — i a/axn , x l , . . . , xn ) , which itself is not commutative. In the next subsection we give a sketch of the general theory of asymptotic expansions with respect to a tuple of (noncommuting) operators. 2 We

ignore focal (caustic) points for the sake of simplicity.

1. Preliminaries

185

1.4 Asymptotic Expansions with Respect to an Ordered Tuple of

Operators As was explained in the preceding section, in order to construct the general theory of asymptotic expansions, it is necessary, first of all, to choose and fix a tuple of operators A = (A 1 , . . . , A n ), which will serve as large parameters for asymptotic expansions in question. Clearly, this tuple cannot be chosen completely arbitrarily (but rather should satisfy some conditions we do not discuss here; see [133]). We assume that our operators are self-adjoint operators in some Hilbert space l H. The choice of the tuple A determines the type of asymptotic expansion and it is now appropriate to give a precise definition of this basic notion. For this purpose, we introduce a scale of spaces Hs (A) associated with the tuple A in the following way. Let D C H be a linear subset of the common domain of all possible products of powers of operators A 1 , . .. , A N . The set D is assumed to be dense in H. We consider the following family of norms on D:

ii u ii s2 --= 11 (1 + Ai + • • • + A 2N Y/ 2 u The space Hs (A) is defined as the completion of D with respect to the norm11 . 11 s .

Definition 111.2 A function ii is called an asymptotic approximation of order s of a function u if u — ii E Hs (A). Thus, if H = L2(R 1 ), N = n, and

a axi

(

a axn

then we can take D = C(7° (Rn ) , and the scale Hs (—i0/8x) is none other than the usual Sobolev scale. Asymptotic expansions obtained in this way are those with respect to smoothness. If we choose another tuple of operators, say,

a 9 • • • 9

-

i

a - 9

X1

9 • • • 9

X

n) 9

then the norms in the spaces Hs (—i8/0x, x) are defined by the formulas

11u(x)11s = 11( 1 + 1 x 1 2 _ 6.) si2u(x)1 dx. An advantage of these norms is that they are Fourier-invariant. Along with smoothness, these norms allow for the decay at infinity and prove to be useful in studies dealing with pseudodifferential equations (see Shubin [163]). 1 For

example, one can often take H

= L2.


186

The second stage in the construction of asymptotic expansions with respect to the tuple (A 1 , . . . , A N ) is to represent the differential operator of the original problem as a function of the (generally noncommuting) tuple (A 1 , . . . , A N ), that is, to represent the original equation in the form 1 N H(Ai, ... , AN) ti =

(111.11)

!

Of course, such a representation is generally ambiguous and sometimes impossible. In constructing asymptotic expansions with respect to smoothness, along with the operators (—j 8/0x) defining the scale Hs, one needs to introduce the operators x 1 , .. . , xn not involved in the definition of the asymptotic expansion. Even at this stage one can obtain different representations of the original equation by choosing different ordering of operators. For simplicity, we do not dwell upon the fact that not all of the operators A 1 , . . . , A N can be involved in the definition of the scale Hs (A). The first two stages described are not algorithmized as yet. The choice of the tuple A and the representation of the original equation must be done in a special way for each equation and is a matter of art rather than craft. However, we try to show in the following that as soon as the tuple A is chosen and the representation obtained, the remaining stages are automatically determined by the calculus of noncommuting operators.

1.5 Reduction to Pseudodifferential Equations With the asymptotic expansion constructed in Section 2 as an example, let us try to find asymptotic solutions, in the sense of Definition 111.2, of equation (III. 1 1 ) in the form 1 N 14 = F(A1, • • • , AN)

f

A sufficient condition that u be a solution of equation (III. 1 1) is that the operator

1

N

F(Ai, ... , AN) be a solution of the operator equation 1

N 1 N [MAL • • • , AN)]I t[RA1, ... , ANA = 1. 1

N

This is an equation for the symbol F(z i , ... , z n ) of the operator F(Ai, ... , AN). Here we consider the case, studied in Chapter II, in which the following condition is satisfied:

1

N

The product of any two functions of the ordered tuple A = (A1, . . . , AN) can be 1 N represented as a function of the same tuple, i.e., the functions of (A1, . .. , AN) form an operator algebra. As was shown in Chapter II, this is the case if the tuple A = (A 1 , .. . , A N ) has a left ordered representation on the space of symbols. Let 1 = (I 1 , .. . , I N ) be this

1. Preliminaries

187

representation. Then 1 [H(Ai, •••

,

N 1 N 1 N AN) ]1[F(Al, • • • , ANA = W(Al, • • • , AN),

where 1 N Ço(Yi, • • • , Y N) = H( 1 1, • • • , 1 N)(F).

Thus we obtain the equation 1 N H(11, ... , 1 N)

(F) = 1

for the symbol F(y i , . .. , y N ). Of course, this equation has little practical value unless 1

N

we can calculate the left ordered representation operator H (11, . . . , 1 N) explicitly; we have discussed these problems in Chapter II, and we shall also find numerous examples in the following sections. Here we only recall that "triangular" commutation relations lead to differential operators / 1 , . . • , /N of the left ordered representation; if the symbol H is a polynomial in the variables whose associated operators are nontrivial, then 1

N

1 N) is a differential operator as well. So, what conclusion can we derive from our considerations? The problem of solving equation (III.11) has been reduced successively to the operator equation and then to the equation for the symbol. At first glance it seems that this "symbolic" problem is no simpler than the original one. This impression would be true if we considered exact solutions of equation (III.11) rather that asymptotic ones. But in order to obtain asymptotic solutions we should solve the operator equation modulo operators of sufficiently low negative order in the scale 1-1S (A). Accordingly, the "symbolic" equation should be solved up to functions of z 1 , . . . , Zn that decay at infinity sufficiently 1 N rapidly (this clearly requires "good" estimates for the operators F (Ai, .. . , AN) in the scale Hs (A) and concerns the choice of appropriate symbol classes). Thus, the operational calculus permits us to reduce the problem of constructing asymptotic solutions to equation (111.11) with respect to an arbitrary tuple of operators to asymptotic expansion of a solution to the symbolic equation with respect to growth at infinity. On passing in the latter equation to the Fourier transform, we reduce an "arbitrary" asymptotic problem to a problem of constructing asymptotic expansions with respect to smoothness. We can now return (at a new level of understanding) to the problem posed in Subsection 1.2: What is the "correct" characteristic equation (or, what is the same, the Hamiltonian) for a given differential equation if we consider asymptotic solutions with respect to a tuple A = (A 1 , . . . , A N )? With the above information in mind, we should perform the following steps in order to construct the Hamiltonian. 1)

Compute the operators of the left ordered representation.


188 2)

1 Compute the operator H (11, . . . , 1 N) and pass to the Fourier transform by , z N ) (or only the part of these variables that corresponds to the type (z 1 , of asymptotic expansion consid ered).

The "standard" Hamiltonian function of the operator obtained is the Hamiltonian function corresponding to the original equation for the given type of asymptotic expansions. Let us consider the construction of the characteristic equation for the Helmholtz equation. To represent this equation in the form (III. 1 1), we introduce the following set of operators: =

a

a

—, ay

A2 =

ax

A3 = k, B1= x,

B2 = y.

The Helmholtz equation now becomes 12

12

12

2 2 [A1 +A2 — A 3 n 2 (Bi, B2)] u =

(the assignment of Feynman indices uses the fact that some of these operators commute with each other). Let us write out the left ordered representation operators, using the following correspondence between the operators and the arguments of symbols: Zi 4-->- A1, Z2 ÷-> A2, Z3 4---> A3, wi 0.

With this choice of B, the perturbed equation can be rewritten in the form

h2 Olin — 1 + IN+ 1 — 2Vfn } V ((n — 1)a)*n—i — EVi n _i = 0, 2(1 — é') 2 where *n = i,k(na).

Hence, we have arrived at a difference approximation of the Schriidinger equation on the uniform grid fx = na}; the example is of course trivial, but in less evident situations the idea may be helpful.


200

3 Propagation of Electromagnetic Waves in Plasma Let us consider the construction of high-frequency asymptotic expansions for a wave generated in plasma by a point source. This problem is a classical one (see [120]) and is characterized by an interesting physical phenomenon. Aside from the usual rays predicted by geometric optics, there occur so-called transient rays; in the illuminated region their contribution is of an order less than that of geometric optics rays, but they still give the leading term of the asymptotic expansion in the umbral region. Here we show that the appearance of transient rays is caused by turning on the point source at the initial moment of time (t = 0). Concurrently, we compute the socalled diffraction coeffi cient, which is in fact the ratio of two amplitudes corresponding, respectively, to the rays of geometric optics and to the transient rays. The cited phenomenon (the appearance of transient rays) was discovered very long ago, but until very recently the diffraction coefficient was computed by a semiheuristic method based on model problems. The uniform asymptotic expansion with respect to smoothness and parameter provides rigorous justification for both the diffraction coefficient and the appearance of the transient part of the asymptotic expansion. Since it it just simultaneous asymptotic expansion (with respect to parameter and smoothness) that is needed, we are led to the use of noncommuting operators. Their usage provides uniform asymptotic expansions admitting analysis in physical terms. Our exposition is arranged as follows. In the first subsection we describe the statement of the problem and choose a family of noncommuting operators adequate to the type of asymptotic expansion required. The second subsection deals with the construction of the asymptotic expansion for the considered problem from the viewpoint of noncommutative analysis. The third (and the last) subsection is devoted to the analysis of the obtained asymptotic expansion in different zones of the physical space (the illuminated region, the umbral region, etc.) Of necessity, the exposition in this section is more technical than the authors would like. However, we tried to avoid clumsy calculations by considering the asymptotic solution of the Cauchy problem only for small values of t, which permits us to use solely the WIG3 method and to sidestep the much more complicated language of Maslov's canonical operator. Of course, with the help of the latter all constructions of this section can be carried out for arbitrary values of t. We refer the reader to [145] and [146] for further details.

3.1 Statement of the Problem We consider the Cauchy problem

1

a2 u

---,-, ± C 2 — 19 8. --7 214 — )1/4. 2 b2 (X)/4

atz

ult.o = — rtlt.0=0

=

3. Propagation of Electromagnetic Waves in Plasma

201

describing propagation of electromagnetic waves in plasma (see, e.g., Lewis [120]). Here Ab(x) is the plasma frequency and A the average plasma frequency considered as a large parameter. The function b(x) hence describes spatial inhomogeneity of the plasma and is dimensionless; it is assumed to be everywhere positive. The right-hand side of the equation describes a point source at the origin with amplitude r(t) and instantaneous frequency Xq / (t). Our intention is to study the asymptotic behaviour of the solutions as A —> oo. We begin with several remarks. First, we need to take smoothness into account when constructing asymptotic expansions with respect to A, since the right-hand side of the equation contains a distribution. Otherwise each subsequent term of the asymptotic expansion would be less smooth than the preceding one, which makes the expansion physically unjustified. Second, it will be more convenient to consider the Cauchy problem for a homogeneous equation. This can be achieved with the help of the classical Duhamel principle. Specifically, it is easy to see that the solution can be represented in the form r u(x, t) = — f v(x, t, r) dr,

(111. 14)

0 where v(x , t, r) is the solution of the following problem:

I

at-

(III. 1 5)

F (x , X, r). vit=r =°' 4 ) 1t=r =

Here we use the following notation. By il we denote the operator

/4

= e2

02

_ x 2b2(x ) ,

ax2

and F = XS (x)r(t)e —ix q (t)

is the right-hand side of the original equation. Having in mind that the solutions of (III. 1 5) and the original problem are related, we see that it suffices to construct the simultaneous asymptotic expansion of a solution to (111.1 5) with respect to smoothness and the large parameter A. This being done, the Duhamel formula yields the asymptotic expansion of the same type for the solution to the original problem. We consider the following set of operators: 1 A1 =

a

—i—, Ox

1 A2 = A.,

2

B = x,

acting on the Hilbert space Ho = L2(Rx2 X [1, oo)x). Only the operators A1 and A2 are "large parameters" of the asymptotic expansion; consequently, the scale H 1 (A) is

202


defined by the following sequence of norms:

M u Il i = (1 + A

± Absi2u L 2 = 11

.

0 - A + x2 ).v/2 u L2

In Subsection 3.2 we construct the asymptotic solution of problem (III.15) in the scale W (A). The asymptotic behaviour of the solution u(x , t) to the original problem is carried out in Subsection 3.3.

3.2 The Construction of the Asymptotic Expansion We begin with an operator interpretation of the Cauchy problem under investigation. First of all, we represent .fi as a function of the operator tuple (A1, A2, B), ,.,1 2

1 2 ,, 2

H = —c`A i — A 21)` (B).

Next, we seek the solution to (III.15) in the form 1

V(X,

t, r) = (I)(Ai,

1 2 A2, B, t, r)

F = 43F,

(III.16)

where 4) is the operator with unknown symbol (1(p, A, x, t, r). Here we use the correspondence p Ai, )k. oc. Operators corresponding to such symbols are of large negative order in the scale HS (A). It is therefore natural to consider ,u = -NA2 + p2 as a large parameter and to seek the asymptotic expansion of the solution as ti. —)- oc. We set wi = p Iti, and co2 = A.111,; the "angular parameters" col and co2 range over the unit sphere co? + 4 = 1. To obtain the asymptotic solution of (111.1 7) as p, > oc for large (but finite) values of t one should use Maslov's canonical operator method. We have no intention of describing this theory here (see, e.g. [1371) and limit ourselves to obtaining an asymptotic expansion of the solution for small t. For these t, the WKB approximation can be used: —

, t, = e ips(0),x,r) (a0(a), x, t) + it —l ai (w, x, t) + • • .)• Moreover, here we only find the leading term of the asymptotic expansion, via the functions S(o), x, t) and ao(w, x, t). Following the standard procedure of the WKB method, we insert 4) into (111.1 7), collect similar terms, and equate the coefficients of powers of 12, to zero. This procedure yields the Hamilton—Jacobi equation

3 s) 2 2 ( 485) 2 \ = Vn -I- C (01 + — 1- , 0) 2L 2 /X) at ax

— (—

211

for S (w , x, t) and a transport equation for ao (co, x, t). The Cauchy data in (111.1 7) induce the initial data

Sit=o = 0 for the Hamilton—Jacobi equation. Clearly, the latter splits into two equations:

a s±

at

a s±

2

± c2 (0)1 + —) + coib2 (x) = 0,

ax

and hence we obtain two solutions S± = S±(w, x, t). These solutions can be evaluated explicitly by the standard method of characteristics. Specifically, denote by 7- ( A, x, q) the Hamiltonian function 'H(w, x, q ) = .1c2(coi + p)2 + 04b2 (.0

204


(we consider the "+" sign). The corresponding Hamiltonian system for the characteristics has the form

J •i = 'MI (co , x, q) = cr (col + q)17-t,

I. 4= —Rx(w,x,q) = —24b(x)1/ (x)I7 i,

with the initial data

J x(0) = xo , 1 q(0) = 0. Let

X = x + (xo, t), q = q + (xo, t)

be its solution. Then, for small t, the function S1 (co, x, t) can be calculated according to the formula

[r f c2 0)1((oi + q) + b2 (x)(03 S+ (co , X , t) = 7-I o

1

x=x+ (x0,t),q=q+ (xo,t)

where the integral is in fact taken along the trajectories of the Hamiltonian system and xo = x ( 3I - (x , t) is the solution of the equation x = x+ (xo, t) for xo. Similarly, the function S(co, x, t) is given by the integral

f r

S_ ( co, x, 0

=[

—

c2 (01 (Col ±

o

q) + b2 (x)o) 2 7-1 x=x - (xo,t) ,q=q - (xo ,t)

4

where the functions x — , q — , and x o— are defined by analogy with x +, q+,and from the Hamiltonian system corresponding to the "—" sign in the Hamilton—Jacobi equation. Since, on the one hand, there are two functions S±, the solutions of the Hamilton— Jacobi equation, and, on the other hand, there are two initial conditions in (III.17), the asymptotics of the solution to problem (111.1 7) is represented as the sum (1) = e ips+(x,x,r) ao+(co, x, t) + e ips_((0,x,r) acT(0), x, t) (recall that we only construct the leading term of the asymptotic expansion). The initial conditions in (111.1 7) lead to the initial data for the functions cisc ; each of these functions is found from the transport equation associated with the corresponding function S+ (w, x, t). We shall not dwell upon the computational aspect and merely write out the leading term of the asymptotic expansion of the symbol 41)(x, p, ). , t) with respect to the parameter tr, = ../A2 ± p2 . It has the form

cr. (x,

E

i p, )k., t) = — jeioi 2 je(+,—)

t)

c,2

p ±

asi ax

2

]-1/2

+ b2 (OA? ji

,

3. Propagation of Electromagnetic Waves in Plasma where

ax • ax o

205

Ji = det [--1-(xo, t)] x0 .x 0 ,x,t,

is the corresponding Jacobian. Now the simultaneous asymptotic expansion of the solution to the original problem with respect to smoothness and parameter is given by (III.14) and (III.16). On substituting the expression obtained for (I), into (III.16) we obtain an explicit expression

u(x, t, A) L' u ±(x, t, A) + u_(x, t, A) for the solution u(x, t, A) of the original problem, where

t u±(x, t, A) = ± iX f if ea±(p,x, A, t, r) dp ch. , 2(27r) n

(1II.18)

0 RP

and the phase and the amplitude of the integrand are given by the formulas

ço+ = px ± S±(p, x, X, t — r) — Xq(t), a± = [4 (2 (4+ )2 + x 2b2 (0)1 ax

-1/2 r(r).

Of course, the integral over Rnp is in the sense of the theory of distributions.

3.3 Analysis of the Asymptotic Solution First, consider the simplest case in which b(x) = b = const. Then

J+ = J._ = 1, s± = ±.1c.2 p2 + x2b2 t. Straightforward computations show that

- \ Ic 2 p2 ± x2b2(t — r) — Xq(r) = X(k x ±- \ Ic2 k2 ± b2 (t — r) — q (r)), c1)± = px + where k = p /X, and

ix2-n ft f r(t)e" u± = + ,s /c 2 k2 ± b2 dr dk 2 (2nin i 0Rk

(see R. Lewis [120], p. 848). Note in particular that the method described gives a precise solution for constant coefficients since the right-hand sides vanish for higher transport equations.

Ill Noncommutative Analysis and Differential Equations

206

Let us now study the general case b = b(x). The asymptotic expansion of the integral (111.1 8) can be computed with the help of the stationary phase method. The stationary point equations for the phase function have the form

I x+

fis)(p, A., x , t — r) = 0,

(p, A., x , t — T. ) + A.q' (r) = 0

(111.1 9)

(here we consider only the function u+; the function u_ can be considered in a similar way).

However, the integration domain in (111.1 8) has boundary points, and henceforth we must allow for the contribution of boundary stationary points at r = 0 and r = t. The equations of these stationary points are as follows:

I

(III.20) r=0

for the boundary point r = 0 and

x = 0, r = t

(111.2 1)

for the boundary point r = t.

We shall not analyze asymptotic expansions in the neighborhood of the point x = 0 (i.e., of the source). Hence we do not have to take the stationary points (111.2 1) into consideration. By the change of variables p = Al we reduce MIA 8) to the form ix2—n U

2(27r)n

ff

eiXcNk,x,l,t,t)r(r) dk dr

o lizz

I/ (c2 (Px )2 + b2 (x))

t

The stationary point equations can be obtained from (III. 1 9) and (III.20) by the substitution p = k, A = 1. In order to apply the stationary phase method correctly, we should require that the stationary points flc(x , t), r (x, t)} of system (III. 1 9) and k' (x, t) of (III.20) be located in a bounded domain RI < R < oo of the space R. In that case we can use a partition of unity {ei (RD, e2(IkI)} such that ei(11(1) + e2(1k1) = 1 and e2 (z) = 0 for z < R. The integral in question reduces to a sum of two terms, of which the second one can be shown to be 0 (A—m) uniformly with respect to smoothness. This can be performed by integration by parts with respect to k. The first term is an integral over a compact domain in Rik% so one can apply the stationary phase method to obtain its asymptotic expansion. The boundedness of the set of stationary points is equivalent to the nontrivial solvability of systems (111.1 9) and (III.20) for p at A = 0.

3. Propagation of Electromagnetic Waves in Plasma The relations

I

207

S(p, 0, x, t — r) = cp(t — r), _wi as (p, 0, c, t — r) = c(t — r),

vas, (p, 0, x, t — r) = cp

imply that for X = 0 these systems can be written as x + c(t — v) = 0,

cp = 0,

and x + ct = 0,

(111.22)

respectively. Since the first system implies that p = 0, we see that system (III. 1 9) does not have any "bad" solutions, and (111.22) shows that neither does system (III.20), unless the point (x, t) lies on the characteristic cone. Since we wish to obtain a uniform asymptotic expansion, we have to use uniform asymptotic expansions given by the stationary phase method for the case in which the boundary stationary point (i.e., the solution of equation (III.20)) coincides with the interior stationary point (III. 1 9). This case corresponds to the points (x, t) lying on the so-called shadow boundary. We obtain the following results by using the stationary phase method: A. If the integral has interior stationary points (the illuminated region), the leading term of the asymptotic expansion has the form

u , A, 2 - 3n/2 e o.s(x,r) (1)(x,

t),

which coincides with the approximation given for the original problem by geometric optics. B. If the integral has no interior stationary points, but rather has stationary points at the boundary, then the asymptotic expansion of the solution has the form u—

e iÀS(x ' t) ço(x, t)

aacl: (k, x, 1, t, 0) (see [48], p. 141). This formula corresponds to the so-called transient rays of geometric optics. The factor 1

I

at (k, x , 1, t , 0) = x — q i (r)

ail)

r=0

VC2 k2 ± b2 (0) — q'(0)

(.D s

—

WO

is known as the diffraction coefficient. Here coo is the frequency of the source, and cos is the instantaneous frequency on a given transient ray. We do not compute the asymptotic expansion for the case in which an interior stationary point coincides with a boundary stationary point. This can be done with the help of an appropriate version of the stationary phase method (see, e.g., [48 1 ).


208

4 Equations with Symbols Growing at Infinity In this section we consider the technique of noncommuting operators on the example of constructing asymptotic solutions to differential equations with respect to smoothness and decay at infinity. We only consider a model example that displays all the main features of the theory. We refer the reader to [134] for more detailed information.

4.1 Statement of the Problem and its Operator Interpretation We consider the Cauchy problem

1

2 u _ a 2 u _ x c (x ) u + f (x, t), 21

at

8u It

au at

.° =

(111.23) t=0 = ui(x),

where t R, x e R, and the function c(x) is bounded together with all its derivatives and satisfies the condition c(x) > co > 0. In order to construct the simultaneous asymptotic solution with respect to smoothness and growth as x > oo we define the scale Hs (A, B), where — -

a

A = —i— ax

and B

x.

The norm in Hs (A, B) is given by the expression

u (.0 1,52 =

+ A2 + B2y/2u(x) L2(R1)

•

However, even at first glance it is clear that the role of the variable x is quite different in the factor x 21 than in the coefficient c(x). Whereas x21 increases at infinity, c(x) is uniformly bounded. Recall that a necessary condition for success of our method is that the operator in question be homogeneous with respect to the operators defining the scale Hs (A, B). To this end, we introduce three rather than two operators, forming a Feynman tuple: 1 a 2 2 A= , Bi = x, B2 = x

ax

2

2

and use Bi for "homogeneous" factors and B2 for "well-behaved" bounded factors. We can rewrite the Cauchy problem in the form

I

a2 u _ 2

[A ± Bfl c(B2)] u f (x, t),

, uit=0 = uo(x),

au I at it..0= ul(x).

(111.24)

4. Equations with Symbols Growing at Infinity

209

Here only the operators A and B1 occur in the definition of the scale H' (i.e., one should replace B by B1, and the operator B2 is, in a sense, an "operator parameter". We can now present an operator setting for the Cauchy problem (111.24). Clearly, in order to find the solution of (111.24), it suffices to construct the Green operator of this problem, i.e., an operator it(t) depending on the parameter t and satisfying the following operator Cauchy problem:

1

02 h(t) _ —i-A2 ± B.12./c(B2)] h(t), at2 —

L

(111.25)

a h 0) — 1 = 0, TT

/03)

-

'

Indeed, by Duhamel's principle, the solution of (111.24) can be expressed via h(t) by the formula

Itoc, t) =

t

0k(t)

., uo + R(ou i + f h(t — r) f (x , r) ch. at 0

(we take into account that the coefficients of the operator do not depend on t). Moreover, it is clear that for constructing the asymptotic expansion of the solution to (111.24) in the scale of spaces H' (A, B) it suffices to construct the asymptotic solution of (111.25) modulo operators of low negative order in this scale. Taking all this into account, we seek the operator h(t) (or, more precisely, its asymptotic expansion) in the form 122

R(t)

= (1)(t, A, B1, B2)

(the choice of Feynman indices reflects the fact that B1 and B2 commute with each other). Let us denote by z, wi , and w2 the arguments of the function (1) corresponding to the operators A, B1, and B2, respectively. The left ordered representation operators 12

2

for the tuple (A, B1, B2) can be computed easily. They are lA

a + Z - i awi

i

a aw2

1,31 --- W1,

and /B2 =

2•

(Numerous examples of similar computations are given in Chapter II; we leave detailed computations to the reader.) On substituting h (0 into the operator equation (111.25), we obtain the following Cauchy problem for the symbol 0 (t, z, wr , w2):

18

. a 2 43 --T +[(z — i awi at olt=o0, "1

. a \2 21 =0, l a't) 2) ± w l c(w2) ] 4)

(111.26)

210


We should construct the asymptotic expansion of the solution (1) to the Cauchy problem (111.26) as (Z, WO -> oc. This expansion should be uniform with respect to tv2. This difference between the roles played by wi and w2 clarifies the reasons for "duplicating" the operators B1 and B2 in the expression for the Green operator ft(t).

4.2 Asymptotic Solution of the Symbolic Equation We shall now construct the required asymptotic expansion of the solution to (111.26). As was mentioned above, the problem of constructing such asymptotic expansions is not an intrinsic problem of noncommutative analysis. However, the difficulties encountered in constructing asymptotic solutions to (111.26) are quite characteristic of the problems arising in the applications of the theory of noncommuting operators in the asymptotic theory of differential equations. Therefore, we devote some space to the construction of the asymptotic solution to problem (111.26). In order to spare the reader superfluous technical details we restrict ourselves to constructing the asymptotic expansion of the solution for small values of t. The solution of (111.26) will be sought in the form

where

0± .

eis±o w) a±(t, z, w).

By substituting the components 41+ of cl) into equation (111.26) we obtain the following equation (the ± sign is omitted):

[_ (aast ) 2 ± (z + aaws ± aaws ) 2 + w?i c (w2) 1

a(t, z, w) — ifila(t, z, w)

+132a(t, z, w) = 0,

(111.27)

where i3i are operators of order j with coefficients depending on derivatives of S; the explicit form of these coefficients is inessential to us as yet. Let us focus our attention on the first term on the left-hand side of this equation; in the following we shall see that this term gives the Hamilton—Jacobi equation. The construction of asymptotic expansions of such a form usually employs homogeneous functions (thus, the action S is usually a homogeneous function of order 1 of the variables with respect to which the asymptotic expansion is constructed; in our case, these are z and w 1 ). However, since different powers of z and w I occur there, it is natural to use generalized homogeneity. Specifically, we assume that S is a homogeneous function of Z and wi of the form

S(t, Xl z, Xwi, w2) = XI S(t, z, wi, w2),


211

and we seek the amplitude a in the form of a sum of homogeneous functions of the same type and decreasing orders of homogeneity. The orders of homogeneity of the functions occurring in the first term on the lefthand side are as follows:

ord z = /; ord wi = 1. Hence, the terms of the highest homogeneity degree 2 1 give

E

( sas

2 (

Lm+.

as \ 2

1 ,2 ) + wi c(w2)1 ao, z-F-a-;---

-

where ao is the zero-order homogeneous component of a. The next order of homogeneity present is (2/ — 1). The corresponding terms have the following forms:

2 a.v2) (z + S\ as ao. awi

(111.28)

Evidently, following the standard method, we should equate both quantities to zero, thus obtaining two different Hamilton—Jacobi equations for S. This is because the functions occurring in the second summand in expression (111.27),

as as as ) aa as ) a a 2 (z + -5—w7 + -5-11) a wi +(z+ $7,-Di- + ,9,2 Tt. 'Pia = —2 aw2 2 s a2 s 2s 02 5 as aa +

■

a

2 awlaw2 ± a8 tiiat2+w?

[

a,

all have homogeneity degrees not exceeding 1, and therefore do not occur in the second term in (111.28) provided that 1 > 1. The remedy is to include all terms homogeneous of degrees from l + 1 to 2 1 into the leading term. This gives the Hamilton—Jacobi equation

as )2 w 2/ (as\ 2 l as at )=z + — +aw2 — + 1 COD 2 ) , 011/1

(111.29)


Slt=o = O. Note that the function S(t, z, w) thus determined is no longer a generalized homogeneous function of z and wi . However, it can be shown that S is an asymptotically generalized homogeneous function of these variables, which is more than sufficient for proving appropriate estimates. The same can be said of the components ao, a_i , . .. of the amplitude a(t , z, w). For these components transport equations can be obtained in a standard way, and solving these equations leads to asymptotically generalized homogeneous functions of z and wi .

212


Thus, the Hamiltonian of problem (111.23) corresponding to the asymptotic expansion with respect to smoothness and growth at infinity is the function

(III.30)

(z + pi ± p2) 2 + w?l c(w2)•

However, it should be noted that the principal (generalized homogeneous) component of S(t, Z, w) is not defined by the Hamiltonian function, but rather by the Hamiltonian (111.3 1) (z + /3 2) 2 ± qc(w2). This is because the term asiawi is of degree lower than that of any other term. It is now clear that all conditions on the behaviour of the trajectories of the Hamiltonian system (usually occurring in stationary rather than Cauchy problems) should be imposed on the Hamiltonian (111.3 1), whereas S is a solution of problem (111.29) with the Hamiltonian (III.30). Thus two Hamiltonian functions have arisen in the problem! The second one (111.31) is referred to merely as the Hamiltonian, whereas the first one is called the essential Hamiltonian of problem (III.30). Such a situation is characteristic of problems related to generalized homogeneity (see [129]). The subsequent steps are not difficult and can be carried out by analogy with their counterparts in the preceding section. An interested reader can either do this himself or herself or look through the literature recommended above.

4.3 Equations with Fractional Powers of x in the Coefficients We have considered an example of an equation with coefficients growing at infinity as an integral power of x, so that the operator L occurring in the equation could be written in the form L = f (Ai, B1, B2), where f (Yi, y2, y3) is a quasihomogeneous function of (Yi, y2) and is smooth for n. 2 1- uY1+

Let us now consider a more complicated example in which the coefficients behave as a fractional power of x. For simplicity, we restrict ourselves to the first-order equation

1)

au au 2 a Lu —i— — i— ± c(x)Ixlau + b x, — i— u = 0 at ax ax

(111.32)


(111.33) where a > 0, c(x) is bounded together with all its derivatives, c(x) > co > 0 for 1x1 > 1, c(x)Ixl a is everywhere smooth, and b(x, ) = bi(x, x, 0, where bi (x, y, z) is an (a -1 , 1 )-quasihomogeneous function of order 0 of (y, z) for 1 y la ± Izi > 1, that uit.o = uo(x),

is, bi(x, X l ia y, Xz) = bi (x , y, z)


213

for A> land I yla + 1z1 1. Introduce the operators

a

Al= —i—,

(111.34) A2 = Vf(x), B x ax where 1r (x) is a real-valued function such that V, (x) E C(R), tr (x) = Ix I for Ix I > 2 and lif(x) > 1 everywhere. The operators (111.34) satisfy the commutation relations [A1, A2] =

[A2, B] = 0,

(B), [A1, 13] =

(111.35)

where x (x) is a smooth real-valued function, x (x) = sign x for Ix1 > 2. It is easy 1

2

2

to compute the left ordered representation operators for the tuple (Ai, A2, B). They have the form

lA i =Y1 — 1A 2 = Y2,

a

— x (Y3) —

43 1B =

(we assume the correspondence yi ÷-> A1, y2 ÷->- A2, Y3 ÷->. A3). Equation (111.32) takes the form

au at

1

22

21

Aiu c1(B)Au bi(B , A2, Ai)U

b2(B, Ai)u = 0,

where the function

ci (x) = c(x)

(x) is smooth and bounded together with all its derivatives, ci (x) > co > 0, and b2(x, is homogeneous of degree 0 in for large I I and is compactly supported with respect to x Let us seek the solution of problem (111.32) — (111.33) in the form 1

2

2

u = u(t) = (t)uo = g(Ai, A2, B, t)uo, then for g(yi, Y2, y3, t) we obtain the equation

[

a

a

a

—i— + yi — i,— — i X (Y3) — ± cl (Y3)31 at 42 on a a ) , 2 2 1 -Hoi (y3, Y2 , [[Y1 – i — , – ix (Y3)_ oy3

ay2

a a , 2 1 +102(y3 , ffyi — i — — i x (Y3) —II)] g(Yi , Y2, Y3, t) 0Y2 0 Y3 with the initial conditions g(Y1, Y2, Y3, 0) = 1 . The following assertion is valid.


214

Theorem 111.1 For any N there exists an N1 = N1 (N) such that the bounds

as T ayS

CO + lYti + 1y21) - " 1/31 ,

(yi, Y2, y3)

WI = 1, ... , N1, y2

112,

imply that the operator

a 6

—i—) XY

Ox

(

1 22 T(Ai, A2, B)

is bounded in L2 for y + 3 < N. Proof The

spectrum of the self-adjoint operator 1

hence the operator Ti,

22

A2, B)

A2 lies in the

domain y2 > 1, and

does not depend on the values of its symbol T (y) for

Y2 < 1. Let w(Y2) E C x (TR), q)(Y2)----- 1 for y2 1, (p(Y2) = 0 for y2 < 1/2. If T satisfies the estimates in the theorem for y2 > 1/2, then so does ç9(y2), and T (y) for

all y. On the other hand, as was mentioned above, 1 2 2 1 2 2 T(241, A2, B) = ((pT)(Ai, A2, B),

and the required assertion follows from the results of Section 3. The theorem is proved. 0

Let us now make the change of variables Y2 = A. 1/a X2, y3 = X3,

Y1 = XXI,

where A. is a large parameter, in equation (111.36). We obtain the equation [—iA, -1 — a + x1 — iA. -1 a —

at

ax3

2 2

1

-1-). —1 b1 X3, ( X2, 11X1 —

is x(x 3 p1/4. -1 -a + ci(x3)xY ax2

1

a

(111.38)

01/4. - — — iEX (X3)À 1:1 1

ax3

ax2

,

±X -1 102 (X3, 11X1 — iA. -- — ieX(X3P. - — 1)] G(X,

ax3

ax2

t, 8,

A) = 0,

where e = A -1 /a and G(x, t, E, A) = ey, 01 yi =xx i , y2 =e--1 x2 ,y3 =x3 •

It suffices to construct an asymptotic solution to equation (111.38) in the domain x2 > E, i.e., to find a function G(x,t, E, A.) such that the left-hand side in (111.38) is O (As) in x2 > E uniformly with respect to E. We seek G (x , t, E, A.) in the form G(x,t,

s E —k = e ixs(x,t,e) 8, )1/4.) A. R (x , t, E, A -1 ). ,

k=0

(111.39)


215

We insert (111.39) into (111.38) and obtain the following equations for S and R:

as at

as

as

+ x i + — +6x(x3) Ox 3 09X2

+c (x )xa = 0

I

3

(III.40)

2

(the Hamilton—Jacobi equation) and the transport equations

a Wk , ,(pk a , + , -._- + E X k-x3i Tx dt d X3

490c

+ i [ bi x3, x 2,

as

X i + -,

d x3

(111.41) — +8x(x3)— ,as ) ox 2

as

as

+b2(x3,x1+ ,_+sx(x3)— , )](pk = B(pk-1 ox 3 u.x 2

where the right-hand side of the latter equations is equal to BWk-1

, as as :, 1 2 3 rt = —ix{ v i (x3, X2, 11X1 ± — + 8 X (X3) -, — IA 0.,c3

—isx(x3)x-1

OX2

a 0.7C3

a — ax 211 ) as

—bi x3, ( x2, xl + :973 + EX (x3)

as) 2

/ 21 as 495 _ix_ 1 +b2(x3 ,pi + -$9.+Ex(x3) ax2

a ax3

—isx(x3)À-1_a 1) 99.x2

as + E X(X3) — , ) 1 Wk-1 02 X

as

—b2 (X3, XI + 19x3 —

(here q)_i .-- 0 by convention). To be able to use the equation (111.4 1) for constructing the asymptotic solution, one must verify that the right-hand side BR_i of this equation does not, increase as X -- -Foo. This fact can be verified directly from the explicit formula for BçOk_i with the help of the Newton series expansion. The corresponding simple calculation is left to the reader. It is now rather simple to write down the explicit formulas for the action S(x, t) and the amplitudes (pk(x, t) via integrals along trajectories of the vector field

V

a

a

a

t

8x3

ox2

= y + — ± EX (x3) — ,

occurring on in the left-hand side in (111.4 1). These formulas give an asymptotic solution of equation (111.38) with respect to powers of X -1 in the domain t

I x2 — 6 f X (X3 — r)dr > o


216

i.e., on substituting this solution into the equation we obtain the remainder Rs (x , t, on the right-hand side such that the following estimates are valid:

E,

))

r)dr > 612, and it can be shown that the constant

C

alaiRs (x, t, 8, X)

< cx-s-l+lal

aX"

for (x, t) E K, x2

—

t E f X(X3 —

o depends only on K and does not depend on E. Hence, we obtain a simultaneous asymptotic expansion of the solution to the equation considered with respect to smoothness and growth at infinity. The condition t X2

1 8 (2 + f X(X3 —

r)dr)

o imposes restrictions on the size of the interval [0, T]. Namely, since we seek the solution in the domain {x2 > e } , T can be defined from the condition T

sup f IX (x3 — -r)ldr > 1/2, X3 o e.g., one can set T = (2 sup x (x3)) -1 . To increase T, it suffices to choose ifr (x) so that inf * (x) > M; then one can take

T = (M— 1 / 2)/ suP IX (x3)I. X3

5 Geostrophic Wind Equations Let us consider an example of an asymptotic solution with simultaneous smoothnessgrowth at infinity for a model but still physically meaningful problem. We will consider the so-called geostrophic wind and study the evolution of small deviations of velocities and pressure from their geostrophic values in the equatorial zone. According to [153], the geostrophic wind is described as follows. The equations of the gradient wind are derived from the equations of atmospheric dynamics under the assumption that the motion is stationary and the trajectories of particles are isobars. In considering large-scale atmospheric phenomena it is often possible to neglect the nonlinear terms describing the acceleration of particles due to the curvature of the

5. Geostrophic Wind Equations

217

trajectories in the equations of motion. The expression for the components of the flow velocity obtained under this assumption in conjunction with the hypothesis that the flow is stationary is known as the geostrophic wind. Obukhov showed that a deviation from the geostrophic state results in the appearance of rapidly propagating waves which "dissolve" the perturbation in a short period of time. Namely, the pressure field adapts to the velocity field. His conclusions were based on the following argument. He considered the approximate system

I

DU + — au at ax +...2 ,Ati ay ± w au — 2coz t) =

az

DV

at

av , av ± n Wz LI -r- W aZ

au

± 14-T x + V aY

L

1 P2

= — —p1,-,aLy 5

—P

a z —-g

for the flow velocity ii = (u, v, w) and pressure p, where p is the density and coz the vertical component of the Earth's angular velocity, coz = coo sin 0 (here coo = 7.29 x 10-5 sec -1 and 0 is the latitude). Hence, a w/a t, the component Fz of the Coriolis force, and quantities such as cox w, coy w, etc. are neglected. Further, additional simplifying assumptions are made. 1) The surface of the Earth is assumed to be flat (i.e., a Cartesian coordinate system (x, y) is used); 2) coz is assumed to be independent of (x, y), 3) the terms quadratic in 76 and derivatives of 67 are discarded; 4) averaging over z is performed; 5) the barotropic scheme for the dependence of P on p is used. This results in the following averaged system:

lau -Tr — 2coz V =

—gH o aaxx

,

av- + 2,coz U = —g1-10 12277 ay ' aX__

at —

where

(au _,av) \ ax i ay ) '

00 00 z z U = — f pudz, V = — f pvdz, Po Po

o

and

x=

o

(po is the pressure averaged over the surface and Ho — 8 km). The geostrophic wind takes place for

au _ av ax

n

P — P0 Po

218


The adaptation of perturbations of the stationary solution goes as follows. There occurs a wave with velocity c = VP-17 280 m/sec, which "carries the perturbation away" in time t 2R/C, where R is the radius of the perturbed region. Here we study perturbations of the geostrophic state under different assumptions. We do not assume that wz is constant. Quite the opposite, we consider the flow in the equatorial zone. Since wz = wo sin 0, we see that co, = 0 on the equator. We assume that the zone in small enough to take sin 0 0, but the characteristic length of the perturbation is relatively small with respect to the size of the zone, so that the distance y from the equator can be considered as a large parameter. With these assumptions in mind, we will state the problem of construction of simultaneous asymptotic expansions with respect to smoothness and large y. Hence, if we denote the longitude by x and the latitude by y, the asymptotic expansion of the solution will be sought with respect to the tuple of operators

A1 =

a —

,

A2 =

a

- and ay,

A3 = y.

Here we derive the geostrophic wind equations in the situation described, pass to the ordered representation, and construct the leading term of the asymptotic expansion. We assume that the surface of the Earth is the unit sphere

x2 + y2 + z 2 = in space RV with coordinates (X, Y, Z) (not to be confused with the lower case letters x and y, whose meaning is quite different). The equator is given by the equation Z = 0 and the angular velocity of the Earth by the vector

= (0, 0, 1). The Coriolis force is

F = 2m[V, where

V =(X k,2), in what follows we assume that 2m = 1. Let w be the longitude and 0 the latitude. Then we obtain

X = sin w cos 0,

Y = cos w cos 0, Z = sin 0,

In the derivation of the equations, it suffices to consider an arbitrary value of w, say w = 0; then = çb cos 0, 1.7 = —6 sin 0, 2 = 6 cos O, and [V, Q] = (-6 sin 0, —cos0, 0).


219

The tangent plane at any point ((p, 0), = 0, is determined by the outer normal = (0, cos 0, sin 0). The projection of F on this plane has the form

— , [V , S-21) (-6 sin 0, —0 cos 0 sin 2 O, çbcos 2 0 sin 0).

Pr (F) = [V , S21

=

By neglecting second- and higher-order infinitesimals and denoting q) = x, 0 = y, = v, and O = w, we obtain

Fx = — yw, Fy = yv. The hydrodynamic equations for i; = (y, w) have the form

div(p0 = 0, %Pt=+P(P). 1

We linearize this system assuming that

p = 1 + p' ,

ap Op

and the quantities p' and y are small. The linearized system can be written in the form

= 0.

(111.42)

(

We assume the correspondence

a at

P0

a ax

P

P2

a

— ay

between the derivation operators and the corresponding arguments of symbols; we then obtain the following symbol matrix for this system: ( PO

iy

— 6' Pl PO P2

P1 P2

•

po

Its determinant is equal to det

(

po iy

— iy po

pi

pi

P2

po

P2

= PO(Psi —

Pi — Pi — y 2 ) ,


220

and it is clear intuitively that the system is "hyperbolic" with respect to the desired type of asymptotic expansions. We equip equation (111.42) with the initial data V

w P

(

( vo )

= )

(111.43)

WO PO

t=0

and seek for the solution of the Cauchy problem (111.42) — (111.43) in the form y

(2 2 ( ) = G x, y, —i—, —i—, t

w

Ox

P

(

V° wo ,

ay

(111.44)

Po

G is a 27-periodic matrix function with respect to the variable x and Glr=o = 1 .

The equation for G is readily obtained with the help of the left ordered representation I I 22\ of the tuple ( —i a/ ax , —i0/0y, x, y). Recall that this representation has the form

a

a

Lia/0x = Pt — i — , Lia/ay = P2 — i ax Ty

lx = x,

,

1y = y,

so that for G(x, y, pi, P2, 0 we obtain the system ac + _i ___

at

0

—iy

iy

0

P1 — i

4

pi

. a - /-w,,

p2 —

•

a

P2 — 1 7957

it

G=0

0

with the initial conditions G I t=0 = 1. To visualize the orders of the terms in the expansion, let us make the change of variables

x = xi,

y=

xx2,

pi

= xni,

P2 =

xn2,

where A is a large parameter (note that we have assigned A to exactly those variables that correspond to the operators —ia/ax, —i a/ay, and y, which define the type of the asymptotic expansion. We denote g(x, y, t,

X) = G(xi, Xx2, xni, xn 2 ,1-).

The problem for g reads

ag ± -ix -1 —

at

(

gl t .0 = 1 .

in

-

O

-ix 2

iX2

o

a 112 — iX -1 A a ix -1 axi ux2

ni - ix -1 ,ixi A 'j-

k

0

(111.45)


221

Clearly, g is independent of XI. We seek for a particular solution to the equation in this problem of the form g = ea S (X , 11 , t ) w (x , 77, 1 , 1.), where S and w are smooth functions of their arguments and S is real-valued. Substituting g into the system and collecting the coefficients of the powers of A-1 , we obtain the following system of equations:

_

0

..

(as (. ix2 at + ln

0 0

—ix

711 712,

o

1 a — ix - -

o

772

+

(

0

o

0

.as ;2

o

_A.2

0

O

J

i aax 2 )1ço(x, 77, t, A. --1 ) =o

o (we omit the derivatives a/axi since w and S are independent of xi, aS/axi = aw/axi = 0). Next, we equate to zero the coefficients of powers of A-1 in the equation obtained. First of all, we should have

o

[as

—

ix2

71 772 )1 7

0 712

—+ a-T ix2 7/1

W(X,

71, t, 0) = O,

0

but this equation can be satisfied for w (x, /7, t, 0) 0 0 if and only if

det

as —ix 2 in at ( i x 2 as at 772 ) = 0, as 711 7/ 2 at

that is,

as at

Rasy

2 2 2 =0. x2]

at

Also, we should have 51 t=0 = 0 (this follows from the initial data). There are three solutions satisfying these conditions:

So (x ?I, t)-=- 0 ,

and


222

Let us find the corresponding eigenvectors of the symbol matrix, of which ço(x, 77, t, 0) is a multiple. Lee t (a, b, c) be an eigenvector with eigenvalue zero. We have 0 —ix2 111 ) ( a cqi — ibx2 = 0= b i x2 0 iax2 ± c772 17 2 • c aqi ± bq2 0 172 1/1 We normalize the eigenvector by the condition c = 1; then we obtain (

a = i 772/x2,

b = —iqi /x2,

and the eigenvector normalized to unity has the form Xo =

t

(j

172, — i 171, x2)

2

2

2

The eigenvectors corresponding to S+ (x, 17, t) can be obtained in a similar way. Specifically, the system of equations for these vectors has the form —

iX2

J

171

±1q + id +4

12

7

11 2

a

b

0

C (

)

(one takes the lower or the upper sign in all three rows simultaneously). The normalized eigenvectors have the form

x±=

t(±71,17.1? +

Ili

+ 4 + i17ix2, ±772,V77? + qi + 4 - i111x2, rd ± 17 i)

,,r2Vq+74,417,7: + tii + xi

The orthogonal projections onto the one-dimensional eigenspaces have the form Po

=

2

77 1

+ 772 + x

— iril 772

2

—ix2771

An

1

P+ =

—1702

2 ( qi

1 2

2 (17 ? + /73

+4)

ni ill1x2

ix2 172

—1111x2

,

4

Al2 A13

A21 A22 A23 A 31 A32 A33

where

All

2 = rh.

+4,

A21 = 171r/2 + ix2V77? ±

q

+4

A32 = — i111x2 f i 172-1 11 + qi + 1 Here

the superscript

t

A33 = rii2 '7'1 ,,,2 ii2,

2 1 2 1122 = 77 2 -r- X2 ,

A

A31 = i172x2 f i 171 V17? 4-

4

stands for the transposed matrix.

273 + 4,


223

and the matrix II Aii II is Hermitian, that is, Au = Au (the bar stands for complex conjugation). Clearly, we have Po ± P± + P_ 1. We now seek the asymptotic solution to problem (111.45) (more precisely, we seek its leading term) in the form

eixs+q4 ± g(x, 77, A, t) = eixsowo 4_ eas- W-, with

çaj = ci (x , 71, OP» Wilt=o = P i ' j = 0, +' — . Substituting this representation of the solution into the equation, we obtain

_

o o o [ a +( o o i_--as ,,ox2) o i tS2 0 _

Wi = A(S)W.j,i,

j = 0, +, —,

where ço1,1 is the next term of the asymptotic expansion and

A necessary and sufficient condition for the solvability of this equation is that 00

0

0 0

i ks

as o ax2

0

or

ac,

as

1c

1 =o,

ooo

P ----' + i—c P 0 0 1 Pf = O. 3 8t 0x 2 3 ' o 1 0 However, it is easy to check that

Pi

(

00 0 0 0 1 ) Pi = 0, 0 1 0

j = 0, ±, —.


224

1 and we obtain, in the first approximation, the following expression Henceforth, ci for the symbol of the resolvent operator: (

G(y, px py , t) =

py2

1 2 2 . 2 Px ± Py -t- Y

PxPY

2 Px

— iYPx

iYPy

iY

y2

—

cos[t.ip + /3; + y 2 ] + P,

++2

Px2 + Y 2 PxPy

+ 13,

++2

PxPy py2 ± y2

0 Y — iPx

)

— iYPy iypx

— iYPx P x2 + Py2

ixPy

sin[t.ipi ± 13; + y 2 ]

— PxPy iYPY

—Y 0

— Px — ipy

— iPy

0

+ lower order terms. -

This expression provides the leading term of the asymptotic solution to the model geostrophic-wind problem (111.42) — (111.43) by formula (111.44).

6 Degenerate Equations In this section we consider a simple model example showing the application of noncommutative analysis to a problem of constructing the asymptotic expansion of the solution to a degenerate equation. The problem itself is probably of little interest mainly because it can be solved by applying an appropriate "quantized canonical transformation" that takes the operator in the equation into the Hamiltonian operator of the harmonic oscillator. However, it is the technique of noncommutative analysis that we intend to illustrate by considering this example; specifically, we first obtain a "weakened" asymptotic solution for localized right-hand sides. Then, using the known exact solution for some "standard" operator (in this particular example, this is the energy operator of the harmonic oscillator), we obtain asymptotic solutions for arbitrary right-hand sides.

6.1 Statement of the Problem Consider the ordinary differential operator 82

H = —a(x)— ± b(x)x 2 22 , x

ax 2

E

R,

where S-2 E R is a large parameter. We assume that a(x) and b(x) are smooth realvalued functions bounded from below by a positive constant. Moreover, we assume that a(x) and b(x) are bounded together with all their derivatives. Without loss of

6. Degenerate Equations

225

generality it can be assumed that a (0) = b (0) = 1, which can always be achieved by a linear change of variables followed by the multiplication of 11 by an appropriate constant. Under these conditions 1"/ can be defined by closure from C(R) as a closed operator on L2(R). Consider the following stationary problem for find the solution u E L2 (R) of the equation 8 2 u(x) ax 2 b(x)x 2 Q2 u(x) = v(x),

—a(x)

Hu(x)

(111.46)

where I) E L2(R) is a given element.

oo of problem (111.46) is a sequence Definition 111.3 An asymptotic solution as Q of operators 6N, N =1, 2, . . . , on L2 (R) such that HHGN — 1 .L2(R) _, Hk(N) (R)

C N•

—k(N)

Q>

oo and Hk (R) W (R) is the Sobolev space of order k.

where k(N) --+ oo as N

The construction of the asymptotic solution to (111.46) is carried out in three stages. Namely, at the first stage we reduce the problem to a similar one with localized righthand side. At the second stage we present the solution of this problem. Finally, at the third stage we construct the asymptotic solution for a general case.

6.2 Localization of the Right-Hand Side Near the degeneracy point x = 0 the operator LI is close to the operator a2

Q2 x2 ,

8X 2

which coincides with the energy operator of the quantum mechanical harmonic oscillator up to a constant factor. In this subsection we construct an operator 6 1 N such that the substitution u = 61 Ny yields an equation whose right-hand side is sufficiently smooth and, in some sense, is localized near the point x = O. This enables us to solve the resulting equation with the help of perturbation theory, using the known inverse We seek G 1 N in the form of a function of the following self-adjoint operators in

L2(R): Ai

=

a ax

,

A2 = S-2x, A3 =

1 22 2 61 N = GN 1 (A1, A2, A3, B).

B = x;


226

The operators A1, A2, A3 and B generate the Lie algebra with the commutation relations [A1, A2] = - i A3, [A1, B] = - i (all the other commutators are zero). According to the general theory presented in 1 2 2 2 product of two operators ii fi(Ai, A2, A3, B), i = 1, 2, can be Chapter II, the written as 22 2 1 0 = f3(A1, A2, A3, B), 12 /1 where the symbol f3(y , a), y e R 3 , a E JR, of the product

ii /2 is given by the formula

1 2 2 2 f3( )' , a) = fl(lili, 1 A2, /A3, 1 B)(f2(Y, a)),

where /A i , /A2 , /A 3 and /B are the left ordered representation operators,

a

a

1A 1 = yi - 1Y3, - 1, , 0)12 oa

1A 2 = Y2, /A3 = Y3,

13 =

CY.

In particular, since 212

212

H = a(B)A 1 ± b (B) A 2 , it follows that fidiN

=

1 + iI N -- 1 + R bi / (Ai , /-12, I13, L. ),

where 2 R NI (y , a) = a(a) (yi - iy3- - i - ± b(a)y3: G NI (y , a) - 1. ay2 ace a

a

)

(

Being solved exactly, the equation R N1 (y, a) = 0 for G NI (y , a) provides the exact solution of the original problem after the substitution of A1, A2, A3 and B for y 1 , yz, y3, and a. We will construct an asymptotic solution of that equation assuming that 1 /\/y 12: ± y3. and y3 I (y 12. ± y ) are small. Clearly,

a

a 2

a(a) (yi - 1y3- - i -) + b(a )yi = Po + P1 + P2, dY2

da

where P3 , j = 1, 2, 3, are differential operators of order j; more precisely, Po = a(a)Y? + b(a)A, Pi

= -2ia(a)yi

a

,

a

\

(y3yy 1- Tx ) '


227

and

a

a \2

( Y3W2

)

Set

1

0 g (y ce)

g 1 (y a)

fa(a)y

b(a)y}

and define the functions g (y, a), k = 1, 2, . , by the iterative formulas 1

gk (Y, a) =

a (a) yf b (a) yi

(Pi gic -

a) ± P2g1c--2(Y , a))

(note that a (a)y? b(a)yq 0 0 for y i2. + y3. 0 0 by virtue of the conditions imposed on a (a) and b(a)). Next, choose a function V/ E Cceic (R) such that iii (Z) =- 1 in the neighborhood of zero. Set' GN I (y , a)

= *

)7 2 43_ ( - • - 2 k=0

v

Proposition 111.2 For this choice of the symbol G N1(y, a) the remainder R N1 (y, a) is bounded along with all its derivatives and satisfies the estimates

a lpi+y Pl

(y , a)

)62

ayi aY2 aaY

101, Y

= 0, 1,

Proof First let us show that the following assertion is valid.

l

Lemma MA The functions g (y, a), k = —1, 0, 1, 2, ..., have the form

6k

k/

=

Ecrik(yi, Y2, 003'3 ,

i=1 where the functions crik(yi,y2,a) are smooth for (Yi, y2) of degree (k j + bounded uniformly in a E R.

y2i

y3,'

0 and homogeneous with respect to

2). On the sphere y? + y = 1 all derivatives of these functions are

This lemma can easily be proved by induction on k. 1 The function G 1 thus defined has a singularity. However, we are only interested in the domain

y3> 1, where no singularities occur (y3 is replaced by 0)). Since in this case the function lit vanishes for yi2 ± )1, sufficiently small.

228


Let us now proceed to the proof of Proposition 111.2. The formula for R N1 (y, a) can be rewritten in the form R N ' a) (P0 + P1 + P2) * ,2

2_, gk(y , a) — 1

,2)

k=0

N =[P1 + P2, ( 2 Y3 2 Yi + Y2 )] k=0

g k (Y , a ) N

-Flk ( , Y3 ,) (P0gP(Y , a) —1) +

-FVF ( 9 y3 9 (PlgN i (Y

Evogl (Y, a) + Pigl_i (Y, a)P2glic _2(Y, a))

P2 g N 1 -1(Y , a ) + P2 g N1 (Y'' "

yr + Yi

The expression in curly brackets is equal to zero, which follows from the construction of g l (y, a). Let us estimate the remaining terms. (a) The derivatives of * (y3 /(y? + y3')) of order II > 1 satisfy the bound

a II

/

0 1 /32 aYi aY2

)

2

2

(

Y3

5 -

Yi 4- Y2

C

)10112

1

m13 ( 2 2 YI + Y2 )

+

for any m, which can be checked by straightforward derivation (one should take into account that the supports of all derivatives of *(z) lie in the domain 1/C < z < C for some C > 0). (b) The commutator [P1, VT] has the form 2

y3

Y3

Y3

P1 ' ( 2 2) YI + Y2

a (a ) Y1 Y2

(

2

2) YI + Y2

( 2

2)

YI + Y2

It follows immediately from Lemma MA on glic and from the bounds for 1î that [P1,

E g satisfy k=0

the needed estimates. Similarly we can estimate the term containing [P2, In (c) Let us estimate Pi g N1 (y , a): g Al , (y , a)

—2ia (a)

E ( y3

i +i

yl

J.°

a:iN o

ao-iN

+ Y 3 yi — •

0Y2

The derivative alPHY la4laylPae of this expression is a sum of terms of the form 3

Y3 6a(Y1, Y2, a)

where 0 s N We have, for y?' +

1 and o- (yi , y2 , a) is a function homogeneous of order r < — (N + fil I +s + 1). > c > 0, y3 > 1, Y'3a(Y1, Y2, a) < const Y(Y?' + YD- 1/31/2(y? ( Y3 2 2 Yl + Y2

s

) (N+1)/2 2 —I/3I/2 (YI + Y2 )

Y3

2

2

2)

YI + Y2

'

In analogy with the above, we can estimate P2gN 1 (y, a) and P2g h1r1 (y, a). In conjunction with the estimates (a), these estimates imply Proposition 111.2, since y3 Ay?

yD and 4y? + y are

bounded on supp Vf (owing to the condition y3 > 1). Proposition 111.2 is proved.


229

Now let v(x, Q) E L2(R) be a function such that the norm I IvIlL2 (R) is bounded uniformly in Q. In the equation flu = v we replace u by uN = 6 N 1 v and obtain the following equation for the difference (u — 14N): fi(u — UN) = — 14v -=- rN(x, S2). As was already mentioned at the beginning of this section, the right-hand side r N (x Q) of this equation is localized in a neighbourhood of the degeneracy point x = O. We will now give a precise meaning to this assertion. Introduce the new variable 4 by setting x = 4. 0 -1 /2 , Q) f (x, S2) for any function f E L2 (Rx ) depending on the parameter Q.

Q) 14 -f

Proposition 111.3 The function 7-N( , S2) lies in the domain of (4'2 — 82/n2) (N+ 1)/2 and

(4.2

82 ) (N+1)/2 < c s2 1/4 11v( x,

r N(',Ç2)

a4.2

iL2 (R)

L2 (1ik)

where the constant C is independent of Q.

This proposition can be proved by the standard technique of L2-estimates. We omit this proof here since such estimates are not the topic of the present book. The reader can carry them out for himself or herself.

Let us now proceed to the second stage of the solution.

6.3 Solving the Equation with Localized Right-Hand Side Let

E

denote the small parameter s = Q -1 /2 • We again use the change of variables

X

For brevity, we denote the difference u — UN by the same letter u. The equation for u reads ,9 2 u( 8)

Hu

where

a(8)

E) = E2 FN (4, 6 -2 )

a •

2

b(s)4 2u(, s) =

Thus, the right-hand side

0 2 )(N+1)/2 11(,

3;2

s), s) satisfies the estimate

< c E 3/2

s)

(111.47)

L2(R4 )

where the constant is independent of E. The function u( 9 s) will be calculated with the help of perturbation theory, by expanding in E. To this end, let us choose a positive integer 1 (the precise value of 1 will be fixed later) and rewrite the equation for u in the following form using the Taylor expansions of a and b:

/ + E e k iik ± 8/±1 hi±i (8) u = 1), ( fiose k=1


230 where

82 fiosc — 2 82

is the energy operator of the quantum oscillator, a2

1 (bk e+2 _ fik = _ k!

02 /+1 _ b/+1 (eM 1+3 — cii-Fi (EM n2 ) '

1 = — h/+1 ( 6) /!

ak, bk are the MacLaurin coefficients of a(x) and b(x), and

1 a i+l a ai+i (x) = f ax1+1 (rx)(1 o

f1 a i+lb -t- )1 dr, bi±i(x) = o

dx 1+1

(rx)(1 —

Let us seek u(x, E) in the form 1 14(X, s) =

i \ ES k /4kkX, s).

k=0

For uk (x, s) we obtain the system of equations

floscuo(x, E) = v(x,

£),

k

floscUk(X , e)

= — E fli uk_i (x, 8 ) , i=1

Assuming this system to be satisfied we obtain the following remainders on the right-hand side: [fiu(x, s) — v(x, s)]

= E1+1 k=1

j=0

j=0

Let us solve the system for uk (x, E). The operator fiosc is boundedly invertible in L2 (TR) . The inverse has the form

.

fio—s ci =

n=0

where A.„ = 2n -I- 1 is the nth eigenvalue of !lose and Pn the orthogonal projector on the corresponding (one-dimensional) eigenspace. Thus, formally we obtain k

u0 = Its cl v ,

Uk = — fio-sci

E fiiuk_i, i=1

k = 1, 2, ... , d.


231

In order to prove that the above formulas indeed define an asymptotic solution of the problem flu = y as s ---> 0, we should show that the functions uk successively obtained lie in the domains of flow , iii, i = 1, . .. , 1 and hi± i (s). Also, it is necessary to estimate the remainder. We shall prove the following assertion.

Proposition 111.4 Under the above assumptions on .v(, 8) the formulas constructed define correctly for eachl < N +1 an element u of L2 (1R) such that u lies in the domain of fI. For each k = 0, . . . , 1 the element u lies in the domain of (flosc) (N +3—k) /2 ; and the remainder satisfies the following estimates for 1 < 7r/2:

_ Cs I 1 fioNsZ2-1 (fiu — 01 1L 2 (Re )
Hk(Rx) _
. T * 11 2n

is the mapping taking any vector to the form d2 L2(, .); 0) -1 0 d2 L2 is restricted to the skew-orthogonal complement of TS prior to evaluating the determinant. Note that the function H1 (x', p') is determined uniquely up to a symplectic transformation in the space lexr,' p,. The proof is based on the technique of microlocal reduction to a normal form (e.g., see [172], [124], [143], [144]). This technique has nothing to do with noncommutative analysis, and so we omit the proof and deal directly with the reduced operator in what follows. Hence we consider the equation

flu = f, where

il is the pseudodifferential operator with symbol —00 H(x, p) = pi' ± xiip'1 2

—

i Hi (x' , p') + E (_i )iiii(x' , p'). j=0

Suppose for the moment that x' does not occur in the equation, that is, the operator H does not depend on x'. Then, by applying the Fourier transform with respect to x', we arrive at the equation a2

[

—

ax?

—oo ± IP/ 1 2 X? — D — i) i Hi (P) 171- = f

.J =1

where the tilde stands for the Fourier transform. The task is to obtain the asymptotic expansion of the solution as I pl i —> oc. This is pretty easy. Denoting = A.,

...fi.xi = Y,

w=

13 '

Ip'l

7. Microlocal Asymptotic Solutions for an Operator with Double Characteristics

237

we rewrite the equation as [

—oo

a2

-

ay 2

+ y2 - Hi (w) + z xi - ' ( - i) i Hi (w) 1'4 = x -1 f. j=0

asymptotics as A. ---> oo of the solution to the last equation can be obtained by regular perturbation theory provided that H1 (co) is not an eigenvalue of the operator The

— 02/8 y 2 ± y2 for any co E

Hence, the idea was to consider p' as parameter thus reducing the problem to the familiar quantum-mechanical oscillator. Let us now return to the general case, in which x' does occur in the equation. Then the Fourier transform does not help us make the primed variables into parameters. However, still there is an appropriate technique, namely, that of operator-valued

symbols2 . Let us say a few words as to what an operator-valued symbol is. Let A be an operator algebra and .F a proper symbol space. The usual definition of the spaces of n-ary symbols reads

.Fn = .F6 • - • 6.F . ,.._...„.,_., n copies

We consider the space

frn = A6.F6 • • • 6.F = A® T n . The elements of

.frn are called operator-valued symbols. If A 1 , . . . , An are .F-

generators in A and f E frn an operator-valued symbol, then we can substitute An into f. However, in this case not only A1 , . . . , An , but also f itself should be equipped with Feynman indices. If f = a 0 g, a E A, a E Tn , then we set

su in in s il f (Ai, . . . , An ) = ag(Ai, ... , A n ). The definition is extended to the entire .fro by linearity and continuity. The calculus of operator-valued symbols includes many useful counterparts of formulas valid for ordinary symbols; they are usually easy to prove by considering factorable elements first. Let us give a few examples: (a) The conjugation formula , s ll

in

3

.11

in

U — ifff (Ai, . . . , A n AU = U -1 fU (U -1 AiU , . . . , U -1 AnU), 2 This technique, which originally arose in context of differential equations, is now well-integrated into noncommutative analysis.

238


(b) The left multiplication formula s

L s :11 in f(A1 ,...,An)

in

il

= Lf(LAi, • • • , L An)•

Returning to our example, the idea we have in mind is to represent fi as a function with operator-valued symbol that can be considered as a perturbation of the quantum oscillator. Namely, we write 2

1a ' H=O (x', —i—) ,

ax/

where

0 (x' , p') = '41p 12 —

00

a

+ —i Hi (x' , p') +E(-0— i Ili (x' , p') I,

494

i=0

(I is the identity operator). Note that the symbol 0(x', p') satisfies the inequalities aa+P (ax')°€ (ap')fi 0 (x' , p')u

< Cap(1 +1111 2—IP l ) (1 + Xf L2

2 (

a.0 2 )8 u

L2

Naturally, the regularizer will also be sought in the form 2 a f? = R (x' , —i l— ) ax/

with operator-valued symbol R(x' , p'). We should have where

"6 is a smoothing operator in the Sobolev scale; if so, then the operator i?'i = cb g iihiiii■ g*,

where 3g, Â, and h are the operators constructed in Theorem 111.3, is a microlocal right regularizer of L. In order to compute the product II h, we use the left ordered representation of the 1 2 tuple (x, —i a/ax'); this representation has the form (cf. Chapter II)

= x",

Lia/ax' = P' — ia lax' .

since both symbols il and h are operator-valued, care should be taken so as to guarantee that the usual product formula be valid. In our case, the symbol 0(x' , p') commutes

239


with each of the operators x' and —id/ax', and we shall require that the same be true of the symbol R(x' , p'). Then we have (1 2 (-

smbl(ilf?) = 0 x' , p —i

R(x' , p') = E

ax/

OM a 0 (x' , p') a R(x l , p')

a!

ce>o

(ax')a .

(ap')a

(III.50) We intend to solve the equation

smbl(iii?) = 1 asymptotically. For convenience, we make a certain transformation of the symbols 0 and R of the operators il and R. Our aim is to transform the principal part of 0 (x' , p') into the operator of the quantum oscillator. Let

Ul : L2(10)

L2(10),

be the operator that acts according to the formula Uil (Y) = f (Y

IX).

Let

U =_-- U(p) = U.,/71 . Clearly, we have 0 (x l , p') = U

1 [111 1{flosc — iG (x' , p')1 ± N (x' ,

p')]U,

where _00

G(x' , p') = 111(x' , 11 )111/1,

N (x' , p')

=E(- 0 -iHi(x',

p'),

i=o and

ti

OSC

a )2 (_ i_ axi

is the operator of the quantum oscillator. We seek R in the form R=U-1 CU. We equate the right-hand side of (III.50) to 1 term-by-term and solve the resulting infinite system of equations recursively. The difficulty is that U does not commute with a/ap', and this leads to additional terms in the transformed equation (III.50).


240

To this end, we should be able to compute the derivatives — i a I api(U-1 AU) and —ia/axi (U -1 AU) for a given symbol A. Clearly, we have

au f ] (y) = a fJ H , api

y af 1 1 „II -312 Pi y )_ Gr pTi - - II P I . - ( Y - ) (— ,-- ) 114 ay -VIM

ape

1 pi

y

2 Ip i i 2

af (

y ) _ 1 pi

( ±) f ( y ) .

aY j/71 — 2 1p'1 2

Hence

au

1 pi

8Y )

VT'l

a

api = 21p1 12Y ay u. On the other hand,

au-1 ay _ 1= 1 , = _rj— l —ri 2 opi

a pi

pi

F19, 12 U

_1

a

3' Ty .

These formulas yield

1 pi r a (u-1 Au) = (J-1 I a A 21p'12 Ly ,, , A1 l u. api tapi +

a

The formula for (a/axi)(U-1 AU) is evident,

a

axi

,

(U — ' AU) = U—1 aA y

axi .

We obtain the following equation for the symbol C: smbl(fift) = U -1 [i/Il Posc — iG(x / , 11)1 ± N(x' , ',

+E Icell

( -

i)°` u-1

a!

( a +1 p'

ap'

2 1 p'1 2

ICU

(111.51)

la CU = 1.

ady alay ) " (— ax

Let C = Co — iCi — C2 + • • •

be the expansion of C by homogeneity degree. Equation (111.5 1) results in the following system of equations:

—iG(x' , p')) Co = 1, —iG (x' , p')) C1 = —iG (x' , p')) C2 = -F2(Co, CO,


241

where

F(C0), F(Co, C1), . . . are expressions that depend only on the symbols determinable from the preceding equations. The symbols Ci (x', p') can be computed from this system if the operator /lose — iG(x' , p') is invertible, that is, if iG(x / , p') does not assume values in the spectrum of !lose . Let us write down the expressions for the Co, C1: C1: Co (x i , /1 )

1 { flow — iG(x / , 11)1 II

Ci(x' , p') =

114

{fl ow — iG(x i , Pi)1 1 I N (xi 13') {flow — iG(x i , 11)1 IP'l {,

+

ip I

i=2

aG(X l ?

PI

)

Pi tio_se _L

_L 1

where

1Pi IPf l °sc1

aPi

111 osc — iG(x / 191)1

Ip'l

fi

—2

aG

P')1,

axi

= y2 ± 82/42 .

fiosc We have taken into account the fact that 2 7 y2 y a , y2 ± 32

L

a2

ay

ay 2 0 y2] Theorem 111.4 If the function iG(x' , p') does not assume values belonging to the spectrum of the operator flow , the microlocal regularizer of the operator H exists and has the form

it

= (ucu) -1

1

2

a

(x' —i— •

'

axi)

'

where the symbol C is the solution of equation (III.5 1 ). This equation possesses an (asymptotically) unique solution, whose first two components are given by the above formulas. In closing, let us point out that this is the simplest example, which indicates two main features of the approach: 1) The operator-valued symbol, in conjunction with operator transformations of the symbol, is used in such a manner that reduces the principal part of the equation to a simple (though infinite-dimensional) operator. 2) Ordered representations are used to compute products of operators with operatorvalued symbols. In that case certain commutativity conditions should be imposed in order that the usual product formulas be valid.

Chapter IV

Functional-Analytic Background of Noncommutative Analysis

1 Topics on Convergence Functions of several Feynman-ordered operators were introduced in Chapter I. Their definition and the proofs of theorems concerning their properties in fact rely heavily upon the notions of continuity and convergence, completion of tensor products, etc. However, we were not too rigorous about all these concepts and abandoned precise statements in favour of brief and clear exposition based primarily on common sense. The present chapter is intended to justify the definitions and assertions made earlier in this book; we begin with locating the bottlenecks of our argument.

1.1 What Is Actually Needed? Let us briefly look through Section 2 of Chapter! and find out exactly in which places the convergence and related notions are used and in what context (that section is a crucial one, since the sections following it are based on its results rather on the properties of convergence itself). There are quite a few such places where some mapping is required or proved to be continuous, but the most important are as follows: (a) In Definition 1.2 the Feynman quantization mapping

AA f (yi, • • • ,

: Tn.

—>

A

Yn)

1--

f (A)

1

n

for some Feynman tuple of operators A = (A1, .. . , A n is described as the unique extension by continuity from the subspace .7. 0 • • • 0 .F, where .T. is the space of unary symbols; moreover, the space .Fn of n-ary symbols is defined as the completion .').F of the tensor product T 0 • • • 0 T. However, nothing was said Fn = .F6 • • • (g about the topology chosen on .F 0 • • • 0 .F, so that this point needs clarification. )

1. Topics on Convergence

243

(b) The main hypothesis in the uniqueness theorem (Theorem 1.1) is that the difference derivative is a continuous mapping

s

3

: .7. --->- .F6.F;

-3

the same type of objection applies here. (c) In the statements and proofs of the main theorems in Section 2 the passage from an operator algebra A to the algebra G(A) of continuous linear operators on A and to (at least 2 x 2) matrix algebras over A is freely used, and, for a linear space E, algebras of continuous operators on E are considered. Hence, we should equip the newly arising spaces and algebras with a natural notion of convergence. Let us examine the question about tensor products more closely. Given a space ,F of unary symbols with some topology, what topology is actually needed on .T. 0 ,F? The answer becomes clear if we look at how this tensor product is used in our definitions. Suppose we are given two generators A, B e A and, hence, two continuous mappings

AA : Y . —>- A

and

,uB : ..T —> A.

We define the mapping Ii 2 : A,B

TOT-±A

by setting

1-t 1 2 ( f

0

g) = AB (g) p, A( f ) = g(B) f (A)

A,B

the mapping is extended to the whole TOT by linearity and to .F(g).F, the completion of .F 0 F, by continuity. We see that our definitions must ensure the continuity of P' 1 2 on .F 0 T, otherwise the construction fails. A,B

Suppose momentarily that .F and A are Banach algebras; we can define a mapping

v:TOT ---> A by setting

v(f, g) = g(B) f (A); clearly, y is a continuous bilinear mapping, that is,

II v(f, g )II

clIf I 110,

where C = II ILA II • II AB ll . Consider the following projective norm on .F 0 .F:

II(PII = infi tII fill II gill, i.i

IV. Functional-Analytic Background of Noncommutative Analysis

244

where the infimum is taken over all representations N (P

=

E fi 0 gi i=1

(where N finite is not fixed). We claim that u extends by linearity to a continuous mapping .F 0 .F --> A. Indeed, denote the extension by T) and let ço ETOT. For each E > 0 there exists a positive integer N and elements f, gi E .T such that n

cp =EfiOgiand i=1

t II.fi I Ilgill i=1

11Ç911 +E.

Then IF) ( q)) 11 =

0 Ê gi(B) fi(A)0

C l'

i=1

Ilfi I Ilgi I

calsoll +0.

i=1

Since E is arbitrary, we obtain

Ilf( (p )I I

c (ho l l).

The completion FF of T 0 Y with respect to the chosen norm is called the projective tensor product of .F and .F and is characterized by the following property: for any continuous bilinear mapping u : F (8) ..T . --÷ A into a Banach space A there exists a unique continuous mapping 13: FF. 6 .—> T A such that the diagram v

YxY

A

.T.6.F commutes, where the left arrow is the natural embedding (a, b) 1—> a 0 b. Let us consider a simple example. Suppose that .7- is the Banach space of continuous functions on the unit circle, T = C (S 1 ). Then the space .F2 = FFof binary symbols has a rather odd-looking structure. It is continuously embeddedl into C (S 1 x 5 1 ) = C(T 2 ) since if ço = E fi 0 gi E C(S 1 ) 0 C(S 1 ), then

II w 11 c(T 2)

=

max I x, y ET2

fi (x) max gi (y) E fi (x)gi wi ..-.. E max .x

= E lifi Ilc(si)llgi 11c(si), 1 We omit the proof of injectivity.

Y


245

and we can pass to the infimum over all possible representations of ça. However, the norm on .T"(3:‘).7 is not the one induced from C(T 2). Indeed, let 41 (x ) be a continuous function with C-norm 1 and support in [0, 27r]. Set N

N(x, y)

=

E W(Nx —

27rk)xli(Ny — 2nk);

k=1

then

Ilç oN (x , Y)lic(T2) = 1 and Ilft/(x, .Y)11c(s1)6c(s1) > (the latter inequality is easy to prove since the supports of the terms in the sum defining cioN (x, y) do not intersect one another). Hence,

C(S 1 x S 1 ) 0 C(S 1 )(i?) C(S 1 ). Fortunately, the situation is quite different for infinitely differentiable symbols, which, by Remark 1.6, are of primary interest to us. Let us take .T. = C°°(S 1 ) and prove that

.F3‘).T. = C° (T 2) (the symbol space C°°(S 1 ) is not used anywhere in this book; however, we think it is useful to consider this example, because the idea of the proof, which is quite obvious here, remains essentially the same for the symbol spaces S"(R), but the proof itself becomes rather cumbersome). We should first equip C 0 (S 1 ) and C"(T 2) with some topology. We consider C°°(S 1 ) as the Fréchet space with topology defined by the system of seminorms k

11111k = max E if (i) ( x)I ,

k = 0, 1, 2, ...

xES 1 j .0

Similarly, the system of seminorms

111fIllk = max

E ifo , i)(x,y)l,

(x,y)€T2 i+ j 0 is arbitrarily small. Set

Then 0

N - ft 0 ki), i=1

and we have

o çoIllm Mie, 0„ Il q) — 14:),0 5. _ M2s,

119

—

where M1 and M2 depend only on N, maxi II fi Il m , and maxi Ilgi Il m • Since 6 > 0 is arbitrary, we see that it suffices to prove (IV. 1) for trigonometric polynomials. Thus, we can assume that R

ço(x, y)

=

E k,1=-R

me ikx e ily ,


248

and, consequently, R

hou6 ,05_ E

k9kii.

k,1=-R

Take m = 2. We have R

R

199kil = E iodiv(1 +k2)(1 + /2 ) V,(1 + k 21) (1 ± 12 ) E k,1=-R k,1=-R < { R

E

-

E

igokii2(1+k2)(1 + / 2 )

k,1=- R

1

kR (1 + k2)(1

R

1/2

+12)

by the Cauchy—Schwarz—Bunyakovskii inequality. The second factor in the braces is bounded uniformly with respect to R, since the series 00 1

E

k=-00 1+ k2

is convergent. The first factor can be rewritten as R

E

iodi2(1+k2)(1 +12)

k,1=-R 1

ff

42 i=

JT2

82

(

i

ax2)

82 çO(x, y) (1 — ay2 )

(x, 31 ) dx dY 5_ Ill(Pill2,

for the same reason. Hence, we arrive at inequality (IV.1), and the desired assertion is proved. Let us now outline the requirements on the class of spaces and algebras to be used in the constructions of noncommutative analysis. Partially, these requirements follow from the preceding considerations, and partially they are implied by general mathematical principles. The spaces and algebras should be endowed with the notion of convergence, and notions such as "Cauchy sequence", "completion", "dense subspace" should be welldefined. If El and E2 belong to the considered category, then the same should be true of E1 ED E2 and of the space L(E1, E2) of continuous linear mappings from El to E2. Moreover, the projective tensor product should be well-defined (as the universal object for polylinear mappings). From the practical viewpoint, we should have the possibility to deal with unbounded operators (such as differential operators), either by involving the technique of scales of spaces, or otherwise. However, it seems impossible to devise a consistent approach to the construction of symbol spaces and operator algebras within the framework of topological linear spaces. The most adequate apparatus is probably supplied by the theory of polynormed spaces and algebras, some elements of which are presented below.


249

1.2 Polynormed Spaces and Algebras The exposition in this and the following section is neither complete nor self-contained owing to space limitations. We only give the necessary definitions and provide examples making these definitions understandable; most of the proofs are omitted and can be found elsewhere. Following Moore and Smith, we describe convergence in terms of generalized sequences, and so it is appropriate to say a few words on that topic. Let I be a poset (a partially ordered set). The set / is said to be directed if for any a E / and /3 E / there exists a y E / such that y > a and y > fi. Definition IV.1 A generalized sequence in a set X is a mapping I ---> X, a } — xa , where / is a directed set. One says that {x ot } is a (generalized) sequence indexed by (elements of) /; the words in parentheses may be omitted. Suppose that X is a topological space. A generalized sequence {X a }ote i in X is said to be convergent to an element x E X if for any neighborhood U C X of the point x there exists an ao G / such that xoe E U for all a > ao. A topology on X is uniquely determined by the class of convergent generalized sequences in X. Next, we need the notion of a filter. Definition IV.2 Let X be a nonempty set. A family F of subsets of X is called a filter in X if the following conditions are satisfied: i) 0 V F; ii) ifAEFandBEF,thenAnBE F; iii) if A E F, then for any subset B D A of X we have B E F. Let F be a filter in X. A subset Fo c F is called afilter base if each A E F includes some Ao E Fo. Note that if Fo is a filter base, then it necessarily has the following property: for any A, B E F0 there exists a C E F0 such that C c An B. Indeed, AnBEF since F is a filter, and the existence of C follows since Fo is a base of F. Definition IV.3 A bidirected set is a poset / such that both / and f are directed sets (here /' is the poset obtained from / by inverting the ordering relation). Hence, a poset / is a bidirected set if for any a, 0 E / there exist y, 8 E / such that y > a, y > 13 and 8 < a,8 < 0. Definition IV.4 Let / be a (bi)directed set. A section of / is a nonempty subset S c I such that if a E S, then each 8 E I satisfying (5 > a belongs to S. We can now give the definition of a polynormed space. Let H be a linear space over the field C (spaces over Ek are considered in the same way).

250


Definition IV.5 A polynorm on

H

is a mapping

p:Hx1-±1R± = [0, co) U {oo}, where / is a bidirected set, such that the following properties hold: 1) px , a) = P.Ip(x, a) for any ),. E C, X E H , and a E l; 2) p(x -I- y , a) < p(x , a) + p(y , a) for any x,y E H and a E I; 3) p(x, a) < p(x , 13) whenever a > 13 for any x E H; 4) for any x E H there exists an a E / such that p(x , a) is finite. Consider a linear space H equipped with a polynorm H x / --> 14 . Note that property 1) yields p(0, a) = 0 for any a; property 3) means that p(x , a) is a nonincreasing function of a for any fixed x. Properties 1) and 2) imply that for each a the set Ha c H of elements x with finite p(x , a) is a linear space and that p(, a) is a seminorm on Ha . Hence, Ha can be considered as a topological linear space with topology determined by the seminorm p(. , a). As follows from property 3), there is a continuous embedding Hp C 11, for any a > Let x E H. It follows from property 4) that the subset I (x) c I of indices a such that p(x , a) is finite is nonempty. By property 3), if ao G 1(x), then a G / (X) for any a > ao, so that 1(x) is a section of I in the sense of Definition IV.4. 1(x) will be referred to as the finiteness set for x. Let {x i, . .. , x,} be a finite subset of H. The intersection I(xi) n • • • n I(x) is nonempty; indeed, if al E 1(xi) and a2 E 1(x2), then there exists a y E / such that y > ai and y > a2; all such y belong to I(xi) n / (x2), since l(x) and I(x2) are sections of I; induction over n accomplishes the proof. Moreover, we see that I(xi) n • • • n l(x) is a section of 1. The finite intersections of the sets 1(x), x E H, thus clearly form a filter base for some filter Ao in 1; namely, Ao contains a subset A if and only if A D 1(xi) n • • • n I(x) for some xi, ... , x n E H. The filter AN is said to be a filter of sections of I since it has a base consisting of sections. Let A be an arbitrary filter of sections of I such that A D A0 (in this case one says that A majorizes Ao; we do not exclude the case A = A0). Then the following representation is valid:

u(nHa).

H==

AEA

(IV.2)

ctEA

Indeed, to verify this equation we need only to prove that for any x E H one can find a set A E A such that X E Ha for all a e A, or, equivalently, p(x , a) is finite for any a E A. Since A D Ao, it suffices to find such a set in Ao. This, however, is trivial, because Ao contains 1(x) and we can set A = 1(x). Denote HA

=n Ha' (YEA


251

Each HA is a locally convex space with respect to the topology determined by the system of seminorms {P( . , a)}aEA • We introduce a convergence in H as follows. A generalized sequence {x i, } I,, E r in H is said to be convergent to zero in H if there exists an A E A and A0 E r such that x t, E HA for all IL> po and x t, converges to zero in H A (in other words, p(x 4 , a) —> 0 for each a E A). Accordingly, we say that a generalized sequence {x i, } iE r in H converges to x E H x —> 0, that is, for some A E A and po E r one has x E HA , x if x H A for and x i, —> x in HA .

Definition IV.6 The space H equipped with the convergence described above is called a polynormed space over the filter A. Clearly, linear operators on H are continuous with respect to the convergence introduced. The usual method of constructing polynormed spaces in applications is as follows. Suppose that 1 is a bidirected set, and let a family {Va }a a of linear spaces be given such that Va c 17/3 whenever a < p. Moreover, suppose that each Va is equipped with a seminorm pa () in such a way that all these embeddings are continuous with norm < 1, pp (x) < pa (x) for a < fi and x E V. Now let A be an arbitrary filter of sections of 1. Set

H= = U(n ia) AEA

aEA

(note that this formula, though it is quite like (IV.2), is, in fact, essentially different, since Va irt H in general). For any x E H we define the finiteness set 1(x) by the formula 1(x) ={ceE/laEAandxE n

,

forsomeAEA)

yEA

and the polynorm E /(x), p(x , a) = { Pa(x) if a -Foe otherwise

(one should resist the temptation to define p(x, a) = pa (x) for all x E Va n H since then it could not be guaranteed that A majorizes the filter Ao generated by the finiteness sets). Such a definition makes H into a polynormed space over A, and representation (IV.2) is valid with Ha = ix EVa nHlae/(x)). Indeed, Ha is clearly a linear space; properties 1) — 4) in Definition IV.5 are obviously satisfied; Ha is the space of elements x E H for which p(x , a) is finite. It remains to show that A majorizes Ao. By the definition of a filter, it suffices to prove


252

that 1(x) E A for each x E H. But this is evident since the definition of 1(x) can be rewritten as follows: / (x) = A. U AEA:xE

n

II,

a EA

Thus, 1(x) contains a subset belonging to A and therefore belongs to A (recall that A is a filter). Let us now consider two examples of polynormed spaces. These examples play a crucial role in noncommutative analysis and its applications to differential equations.

Example IV.1 Let ST (Rn) denote the space of Ck -functions

1(Y)

= f (Yi, • • • ,

Yn)

on Rn satisfying the estimates k

(1+ lyirn III IIm,k = ysup E litn

E I f(a) (y)1 mo can be replaced by the intersection over all k with m = mo owing to the cited embeddings of the spaces skmo (Rn ) ) . srkti (Rn ) form an increasing sequence, and n kEz+

Furthermore, the spaces Sm (Rn) =

it is easy to see that for any f E Sm (Rn ) the finiteness set has the form

I (f) = {(1,k) 11 >m, k EZ). Hence, the convergence on S°°(W2 ) can be described as follows. A generalized sequence {fo lt, E r converges to f E Scx) (Rn ) if there exists a Ao E r and m E Z such that ft, E Sm (Rn ) for ,u > ,u0 and f — 4, —> 0 in Sm (Rn ), that is, k

E

11 f — f fillm,k = sup (1 + 1y1) — m 1 f (a) (x) — f, (f ) (x)1 —> 0, k = 0, 1, 2, ... yEiRn la1=0 In other words , starting from some index ,uo, the functions fi., and all their derivatives grow at infinity not faster than (1 ± I y Dm and /4/(1 + ly Dm ---> f/(1 + I y Ir with all derivatives in the uniform metric.

Example IV.2 Let K C Cn be a compact set. For any neighborhood U of K in Cn denote by Ou the locally convex space of holomorphic functions on U with the topology of uniform convergence on compact subsets of U. Next, denote by OK the space

°K

= U (91h where the union is taken over all open sets U containing K. The space OK can be interpreted as a polynormed space in the following manner. Denote by I the set of pairs (U, L), where U is an open neighborhood of K and L U a compact subset of U. We write (U, L) < (U', L') if tE c u and L' c L. Clearly, I is a poset. We denote by O(U, L) the space Ou equipped with the seminorm

11f 11 (u ,L) = sup I f (z)I; zEL

then, for (U, L) < (U', L) we have OU,L C 0UI,L 1 (the embedding is given by the restriction from U to U') and

11f 11(LP,P) = sup 1 f (z)1._- _ sup 1 f (z)1 = 11f 11 w ,o• zEL

zEL

Next, we take the filter A with base formed by the sets {(U, L)) with U fixed and L arbitrary. We may write 0K

=u( n ow,o) U LU

254


and define the convergence in OK by the following condition: a generalized sequence ft, converges to zero in OK is there exists a ,u0 and a neighborhood U of K such that for all ii, > p.o we have fii, E Ou and 4, converges to zero uniformly on compact subsets of U. A polynormed space H is called a Hausdorff space if each generalized sequence in H has at most one limit. Let {xiL } AE r be a generalized sequence in a polynormed space H. Consider the set r x P with the partial order given by (it, y) < (4a', y') .#. ti, < i.il

Clearly,

r xr is a directed set.

and y < y'.

Consider the generalized sequence {xt, — x v }i.., vi. V)ErXr •

One says that xii, is a Cauchy sequence in H if the sequence xi., — x v is convergent to zero.

Definition IV.7 A polynormed space H is called a complete polynormed space (or a poly-Banach space) if H is a Hausdorff space and each Cauchy sequence in H is convergent. In particular, if all HA, A E A, are complete Hausdorff locally convex spaces, as is the case in Examples IV.1 and IV.2 then H is a poly-Banach space. Suppose that a polynormed space H is not complete. Under certain conditions, it is possible to define the completion of H, which is a poly-Banach space. These conditions are summarized in the following statement.

Lemma TV.1 Let H be a Hausdorff polynormed space satisfying the following property: if A, B E A, A c B, {x i, } 1i Er is a Cauchy sequence in HB , and x i, --> 0 in HA, then x i, --> 0 in HB . Then the completion of H exists and is a poly-Banach space.

We omit the proof but mention the important particular case in which the polynormed space H

( rl

= AEA U aEA HO

satisfies the following property. For each a E I the seminorm p (. , a) is a norm on Ha and for any a < /3 e I the embedding Ha c Hp is consistent, i.e., if a generalized Cauchy sequence x i, E Ha is convergent to zero in Hp, then it is convergent to zero in Ha .

In this case the completion It of each Ha is a Banach space, there are continuous embeddings fia C tip for any a < p, and the completion of H is

AEA aEA


255

with the natural convergence: x i, —> x in ii if, for some A Xti E n fia for ,u, > ,uo and xi, --> x in each Ha , a E A.

E

A and some 1u0, we have

aEA

We now pass to the consideration of continuous linear mappings of polynormed spaces. Let

( n1-1,y) , H=u AEA aEA

nG/3)

G=U(

fiEB

BEE

be polynormed spaces indexed by I and J and equipped with polynorms p (. , a) and q (. , fi), respectively.

Definition IV.8 A linear mapping Ili

: H --> G

is said to be continuous if it takes convergent generalized sequences into convergent generalized sequences. Clearly, lif is continuous if for any A E A it maps the locally convex space H A continuously into some G B . Furthermore,

p

is continuous if and only if for any seminorm q • , fi), seminorm p (. , a), a E A, on HA such that q(*(x), /3) < Cp(x , a),

x

E

E B,

on GB there exists a

H.

Hence, the set L(H , G) of all continuous mappings from H to G can be described as

L(H, G) =

GP), nunuL(Ha, AEA BEE 13EB aEA

(IV.3)

where L(HŒ , G,) is the space of continuous mappings from Ha to G p. We introduce a seminorm in L(Ha , G p) by setting Pap(*) =

sup

q(*(x),

p).

xEHOE , p(x,a) ai and fi < pi . Clearly, IIPcip(*)11 ?. IIPaifii (*)Il whenever (a, fi) < (ai, fii), so that pap(*) is a polynorm on £(H, G). It remains to reduce the number of intersection and union signs in (IV.3) so as to give the "canonical" representation of £(H, G) as a polynormed space. Let Q be the

256


filter of sections of I' x J defined as follows. For any A E A and B e E denote by K (A, B) the set of all sections C of I' x J such that for any fi E B there exists an a E A such that (a, P) E C. The filter Q is generated by the family n H43 E 7.- K (A, B). AEA Then the filter Q majorizes the filter in I' x J generated by the finiteness sets for the polynorm loco, we have r(H, G) =

un

KES2 (a, fi)EK

L(HcoGp),

and L(H, G) is a polynormed space over the filter Q. It can be proved that L(H, G) is a Hausdorff space whenever G is so, and that if G is a poly-Banach space, then so is L(H, G). In what follows we always consider L(H, G) endowed with the structure of a poly-Banach space defined above. Let us now consider the particular case in which H = G. We will denote ,C(H, H) = C(H). The polynormed space L(H) is equipped with the natural multiplication defined by the composition of mappings. We have Pap(lif 40)

Pay (40 )

PYP(*)

whenever the right-hand side is finite. This suggests that the multiplication is continuous on L(H) x L(H), which is indeed true. In fact, for any K1, K2 E Q we can find a K E Q such the multiplication is continuous from G(H) K1 x r(H)K2 to L(H) K , that is, for any 3 = (a, 13) E K there exists a Si = (ai, pi ) E K1 and 32 = (a2, P2) E K2 such that P32 (W) Ps (1fr 49) const . Ps i (the constant, of course, can be taken equal to 1). Furthermore, L(H) contains the identity element represented by the identity mapping 1 of H, and 1 is separated from 0 in the sense that py (1) 0 0 for some y E / I X J (it suffices to choose y of the form y = (a, a)). We obtain the definition of a polynormed algebra by taking the abstract version of these properties.

Definition IV.9 An (associative) polynormed algebra (with 1) is a polynormed space A over a filter A, equipped with a bilinear associative operation (multiplication) such that (a) multiplication is continuous, i.e. for any A1, A2 E A there exists an A E A such that multiplication is continuous in the space

AA 1

x

AA 2

AA ;

in other words, for any a E A there exist ai E Ai, i = 1, 2, such that P(ctic1 2, a)

p(ai, a)P(a2, a)


257

for any ai and a2; (b) A is an algebra with 1; (c) 1 is separated from 0 by the polynorm, i.e., p(1, a) > 0 for some a. If a polynormed algebra is Hausdorff and complete, then it is called a poly-Banach algebra. Thus, if H is a poly-Banach space, then L(H) is a poly-Banach algebra. The following lemma is obvious.

Lemma IV.2 Let A be a polynormed algebra. The mappings L: A A

---> L(A) 1—> L A

and

R: A A

---> L(A) 1---> RA,

where L A and RA are the operators of left and right multiplication by A in A, are a continuous homomorphism and a continuous antihomomorphism of polynormed algebras, respectively.

In particular, each polynormed algebra can be realized as an algebra of continuous endomorphisms of a polynormed space; such a realization is given by the left regular representation A i--> L(A), which is faithful since LA(l) .A.1=A00ifA0 O.

Example IV.3 The polynormed spaces S°°(1[In ) and 0 1( described in Examples IV.1 and IV.2 are poly-Banach algebras with respect to pointwise multiplication. Remark IV.1 The convergence of a polynormed space H is defined by the triple (I, p, A), where / is the indexing set, p the polynorm, and A the filter involved in Definition IV.6. However, different triples may define the same convergence on H (in this case the identity mapping of H is two-sided continuous between these convergences). More generally, we can consider two polynormed spaces H1 and H2 with a continuous continuously invertible linear mapping i : H1 --> H2. In this case we say that H1 and H2 are equivalent and do not distinguish between them.

1.3 Tensor Products Let H1 and H2 be poly-Banach spaces over filters A1 and A2 with polynorms pi and 192 and indexing sets /1 and 12, respectively. We intend to define the tensor product of H1 and H2 in the category of poly-Banach spaces. Following the usual practice, we define the tensor product by the universal mapping property.

258


Definition IV.10 The tensor product of the poly-Banach spaces H1 and H2 is the polyBanach space H1 & H2 such that for any poly-Banach space H and any continuous bilinear mapping r : H1 X H2 ----> H there exists a unique mapping r : 1/1 & H2 —> H such that the diagram

H1 6 H2

Hi X H2

r

H

commutes. Here j is a predefined continuous bilinear mapping. 2

Theorem IV.1 The tensor product of poly-Banach spaces is well-defined, that is, it always exists and is unique up to a uniquely determined equivalence (see Remark IV.1). (The explicit structure of H1&H2 will be given below.)

Remark IV.2 The tensor product thus defined is usually referred to as the projective tensor product of H1 and H2. Proof Uniqueness. Suppose that Hi &H2 and H1 H2 are two different tensor products of H1 and H2. According to Definition IV. 10, there are unique morphisms f and re

such that the diagrams

Hi '6 1-11

Hi X H2

and

111 2 Purists

X

H2

i

Hi Hi

would probably object to our definition by saying that j should be included in the notion of the tensor product, i.e., the tensor product of H1 and H2 is the pair (Hi '0)H2, j : H1 X H2 —> Hi '0: b H2 ).


259

commute. Then so does the diagram Hi '0H2

y

< of

i

Hi X 112

H10‘111

=

since f 0 f is unique, we must have f- 0 f = id. Similarly, f o r = id, and that is, f is an isomorphism. Existence. Consider the algebraic tensor product H1 0 H2. Clearly,

U y n Hia) 0 ( n

Hi 0 H2 =

AEAI,BEA2 aEA

=

I

U

H2/6)}

/3EB

n

(A,B)EAl xA2(a,fi)EAxB

Hice 0 H2/31 .

We equip 1 1 x 12 with the product partial order and introduce a polynorm pap 0, (a, )3) E /1 x 12, on H1 0 H2 by setting Pap = inf E

where

Ç9

II fi iiallgillp,

E Hi°, 0 H2 13 and the infimum is taken over all finite presentations N

ça = E

® gi

.i =1

with fi e Hi, and g1 E H2,8 (here N is not fixed). The space H1 0 H2 is polynormed over the filter Ai x 42; its completion is a poly-Banach space, which will be denoted by H1 '6 H2 Let us prove that this space is the tensor product of H1 and H2. Ha is a poly-Banach space and Suppose H =

un

AEA

aEA tt : Hi X H2 ---> H

i

a continuous bilinear mapping. By the properties of the algebraic tensor product, there is a unique mapping ii : Hi 0 H2 --> H such that ii, = ri, o j, where j : H1 x H2 ---> Hi 0 H2 is the natural embedding,

j (x, y) = x 0 y. Since H1 H2 is the completion of H1 0 H2, the mapping ii can be extended to H10' H2 uniquely provided that /..i is continuous. Thus, it remains to prove the continuity of ri,.

260


Since p, is continuous, it is continuous in the spaces : Hia X H2 fi —>- Hy

for certain triples (a, p, y); the structure of the set of such triples agrees with the filters A1, A 2 , and A. Hence it suffices to prove that / : i is continuous in the normed spaces

ii : 1-11 a 0 H2p —> Hy for the same triples (a, /3, y). But the proof of this fact is essentially that used for Banach spaces (see the discussion at the beginning of Subsection 1.1). Thus, the 0 theorem is proved. In a similar way one can define tensor products H1 "6 H2 6 • • • 6H,1 of n > 2 polyBanach spaces. This definition involves no new ideas and is therefore omitted.

Remark IV.3 If H1 and H2 are poly-Banach algebras, then H16 H2 is also a polyBanach algebra.

2 Symbol Spaces and Generators Symbol spaces and their properties are a topic of interest in noncommutative analysis. They were widely used throughout the exposition, but at an intuitive level: we appealed to the notion of convergence in symbol spaces without knowing exactly what is meant. Now that we are aware of the notion of poly-Banach algebras, we are in a position to give precise definitions.

2.1 Definitions The main properties of symbol spaces are those used in Definitions 1.1 and 1.2 and in Theorems 1.1, 1.4, and 1.5. Here we collect these properties and express them in the language of poly-Banach spaces, which yields the desired definition. Let Z c C be a subset without isolated points, and let T be a linear subspace (over C) of the space of C-valued continuous functions on Z. We impose some conditions on T. Condition A. The space .T. is a poly-Banach algebra with 1 with respect to the pointwise multiplication. That is, ..F is equipped with a polynorm and is Hausdorff and complete with respect to this polynorm; moreover, the product of any two functions in .T. belongs to .F, and the multiplication is continuous; the function identically equal to one on Z belongs to T.

2. Symbol Spaces and Generators

261

Condition B. The function f (x) = x, x E Z, belongs to Y. Condition C. The poly-Banach convergence in Y implies (i.e., is stronger than) locally uniform convergence on Z. Consider the space Fz of all continuous functions f : Z ---* C (we assume that Z is equipped with the topology induced from C). The space Fz can be equipped the structure of a poly-Banach space as follows. Let I be the set of all subsets of Z (the powerset of Z). Let I be partially ordered by inclusion, a > 16 ..ciC/3,

a, 13 E I.

We set P(f; a) = sup I f (x)I, f X

E

FZ.

Ea

Clearly, p is a polynorm on Fz and p(f, a) is finite whenever a is a compact subset of Z. Let A be the filter in I determined by the condition that each A E A contains all compact subsets of Z. Then, since each finiteness set satisfies the same property, it follows that A majorizes the filter Ao generated by finiteness sets, and we can equip Fz with the structure of a polynormed space over the filter A. It is easy to see that Fz with this convergence is a poly-Banach space and the convergence in F is just the locally uniform convergence. Condition C means that the embedding Y c Fz is a continuous mapping. Let Fz x z be the poly-Banach space of all continuous functions on Z x Z constructed by analogy with F. There is a sequence of continuous mappings

.Fx.F --›- FzxFz—>Fzxz ( f, g) 1-* (f, g) 1-* f (x)g(y). The composite mapping is bilinear and, passing to the projective tensor product, we obtain the natural mapping FF —)- Fzx z • Condition D. The natural mapping .F&F --> Fz x z is an embedding. The meaning of Condition D is that the elements of .F&F can be unambiguously identified with some continuous functions on Z x Z. Technically, no extension of this condition is really necessary to construct functions of operators. However, in all practical examples the following condition is satisfied: Condition D'. The natural mapping

n copies

n factors

is an embedding. Conditions A—D are routine ones. Finally, we introduce the most important condition.


262

Condition E. For any f E .F the difference derivative { f (x) — f (Y) x-y ' — (x, Y) = 8x lim f (v)

x E Z, y E Z, x 0 y,

8f

v-)...t, vEz

- f (Y ) , x=yEZ

exists and is continuous. Moreover, 6f/6x E .F6.T. (this assertion makes sense in view of Condition D) and the mapping 3 — : .F --> .F6T

Sx

is a continuous mapping of poly-Banach algebras.

Definition IV.11 The space Y is called a (proper unary) symbol space if it satisfies Conditions A-D. 3 Given a symbol space .F, we define the spaces of n-ary symbols as the projective tensor products

Fn = F6F6 • • • 6.7' = .F6 n . ..._...__v____—, n copies Each Fn is a poly-Banach algebra. For n = 2 elements of Y are naturally interpreted as functions of two arguments yi , y2 E Z, and the same is true for arbitrary n provided that Condition D' is satisfied. Let A be a poly-Banach algebra and Y a symbol space.

Definition IV.12 An element A e A is called an F-generator if there exists a continuous homomorphism ktA : .7' --›

-A

of poly-Banach algebras such that AA

(x) = A

(here x is the function taking each point x

E

Z into x E C). We denote tiA Cf ) = f (A).

Let A1, .. . , A n E A be Y-generators. The mapping Yx•••x.F --> (fi, • • • , fn) i -->

A fn(An) • • • fi (A 1)

factors through Fn = Y 6' n, and we denote the associated mapping by AA : Fn -- A

1

n

where A = (A I , .. • , An). 3 The words "proper" and "unary" will usually be omitted.


263

n Definition IV.13 A tuple A = (Ai, . .. , A n ) of .F -generators equipped with Feynman indices is called a Feynman tuple. For any f E .Fn we define the function of A with symbol f by setting 1 n (Ai, --:- f f (A) =An) = FLA(f)• 1

.

.

.

,

We have reproduced all definitions given in Subsection 1.2 in the rigorous context of polynormed spaces. As to propositions, lemmas, theorems, corollaries, etc. concerning functions of Feynman tuples, we point out that no one can object to the rigor of the argument given in their proofs, with the understanding that convergence, continuity, etc. is considered in poly-Banach spaces. For this reason, we do not dwell on these assertions here and leave the subject.

Remark IV.4 Our definitions are flexible enough to cover most of the applications. Nevertheless, there is an important case that does not match the definitions literally. We mean functions of matrices. Indeed, given a matrix A, the natural symbol space for functions of A is comprised by functions on the spectrum a (A) together with their derivatives up to a certain order (depending on the size of Jordan blocks of A). The set a (A) is discrete, so that the derivatives of a symbol cannot be reconstructed from its values but must be determined separately. Hence we can take Z = a (A), but in this case jets should be considered instead of functions (the order of the jet may vary from point to point). Having finished with the general theory, we proceed to the consideration of concrete symbol classes. The symbol classes most frequently used are S" and OK. The former is used to define functions of tempered generators and the latter to define functions of bounded operators in a Banach space. We consider S" in some detail and leave the consideration of 0K as an exercise.

2.2 S

Is a Proper Symbol Space

Theorem IV.2 The poly-Banach space S"(IR 1 ) satisfies Conditions A—D and D' of the preceding subsection. Moreover,

where the._.^_ stands for the isomorphism of poly-Banach spaces and is given by the natural interpretation of elements of [S"(11I 1 )] . 'n as functions of n variables. Proof The validity of Conditions A,B,C, and E follows directly from the definition of

S' (R1 ). The main difficulty is, of course, to prove D and D'. We carry out the proof only for n = 2 (Condition D), since the argument for n > 2 (Condition D') involves no additional ideas. Following the argument used in Subsection 1.3 to prove the equality

IV Functional-Analytic Background of Noncommutative Analysis

264

C"(R 1 )6C" (R 1 ) = Cœ) (T 2 ), we subdivide the proof into three stages expressed by the following lemmas.

Lemma IV.3 There is a natural embedding j:

(R 1 ) 0 S' (R 1 ) c

(R2 ),

and this embedding is continuous with respect to the projective inf-seminorm on S" (1R 1 ) S" (1R 1 ).

Lemma IV.4 The space S 00 (R. 1 ) S' (111 1 ) is dense in S' (R 2 ). Lemma IV.5 The embedding j has a continuous inverse j

(S00 (R1) 0 s co (R)) _>. sco(Rl) s oc (R i )

on its range (the range j (Se° (R1 ) 0 S' (R 1 ) is equipped with the polynorm inherited from S 00 (R2 )). Combining these lemmas, we readily obtain the assertion of the theorem. Indeed, Lemmas IV.4 and IV.5 imply that j -1 can be extended by continuity to the entire space S(112 ) (of course, the range of the extension j -1 will be in the completion S 00 (R 1 ))S 00 (111 1 ) of Sœ)(Ezi) 0 soorb itt ). Furthermore j extends by continuity to a mapping j : s00(R1)(g)s00(Ri) --± S 00 (R2 ). Since fr i = j 1 j = id,

the same is true of their extensions by continuity: J J-1 = J-17 = id ,

so that

i --1 = CO-1 Thus,

.

7= s00mi)(S00(:0)_, s o0 (R 2 )

is an isomorphism of polynormed spaces, and the theorem is proved. It remains to prove Lemmas IV.3 — IV.5. Proof of Lemma IV.3. Obvious, since if the functions f (x) and g(y) grow, together with all their derivatives, not faster than some polynomial, then the same is true of f (x)g(y). Proof of Lemma IV.4. As with C 00 (S 1 ), trigonometric polynomials are dense in S' (le) , k = 1, 2.4 However the periods of the exponentials are not exhausted by 4 And

for k> 2 as well.


265

multiples of 27r. Let us prove this assertion. Recall that S"(Illlk ) is the union S c) ° (Ile) = U Sm (iIR k ) meZ

of the Fréchet spaces Sm (Iilk ) with locally convex topology determined by the system of seminorms /

IlfIlm,/ =

sup (1 + IYI) —m yeRk

E

If(x)I.

ai=a We need to prove that any f E Sm (Rk ) can be approximated by trigonometric polynomials in the convergence of S'(IR k ) (not in the topology of Sm (IRk ); this is exactly the point where the apparatus of polynormed spaces works). In fact, such an approximation already exists in ST (IlR k ), where r = max{m, 1}. Indeed, let yo(y) be a smooth compactly supported function of y E Rk equal to 1 in the neighborhood of the origin. Then the sequence fn(Y) = (P(Y / n) f 00

of smooth compactly supported functions converges to f (y) in S r (Ile) as n —> oo, since the norm of the "tails" will be killed by the extra factor (1 + IYI) -1 ^1 n -1 in the definition of the norm. Hence, it suffices to obtain trigonometric approximations for smooth compactly supported functions. However, this is trivial: any f E qT(1111`) can be expanded into the Fourier integral i \k12 f

( 27 )

eipy f (p) dp

le

f(Y)=

with smooth rapidly decaying Fourier transform f (p), this integral converges absolutely in any Sr (IRk ) with r > 0, and the desired approximations can be taken in the form of finite Riemann sums f (y) '_-_-

E caeipay.

Take k = 2. Then e iPaY = e iPalYl eiPa2Y2 E S 00 (R 1 ) 0 S 0° (R 1 ).

The lemma is proved.

CI

As was shown just a few lines above, smooth compactly supported functions are dense in S'(IRk ) for any k and hence in S'(1R2 ) and in S°°(1R 1 ) 0 S 00 (11Z 1 ). Consequently, it suffices to check the boundedness of j -1 on the subset consisting of finite sums

Proof of Lemma IV.5.

ço(x,

y)

= Es fs(x)gs(y),

(IV.4)


266

where fs , gs E C(NII1 1 ). Let q) be such a function. Let us prove that for any m' >

m > 0, const holl s2,.-2 (R2) -=- const MOH.

il(Pii =----= Wil sgli (Ri )oscn Ri )

(IV.5)

Assuming that (IV.5) is valid, it is easy to prove the inequalities Il 4° 114n' oft 1 )04m i (RI) < const II ço II r±-,2±2(R2) for all r, t > 0 by induction on r + t. The collection of these inequalities implies that for any m' > m > 0 the mapping j -1 is continuous between the spaces

J -1 : Sm -2 (11R 2 ) _÷. s m' (IRt) 0 on the range of j, as desired. We will prove inequality (IV.5) and omit the trivial induction step. We have

sup

al±kkx , y)

E

axiayk

(x,y)cR2 1+k2

inf E

ci ± ix' ± iy1) — m,

hodisgi (Ri ) oh ii sg: (RI) ,

I

where

II* Ils-o ( R i) = su p (1 + ixi) —m lik(x)i X

ER 1

and the infimum is taken over all finite representations ço = Ei çoi 0 *1 (without the assumption that p and *1 are compactly supported). Let us represent q)(x, y) defined in (IV.4) by the Fourier integral

' y) =

i f f e i(Px+gY) E is (p)k-s (q) dp dq,

Tr q)(x

R2

s

where is and ks are the Fourier transforms of f and g, respectively. Denote fsm x (

)

= ( 1 +x2 ) —m/2 fs(x), gsm(Y) = ( 1 +

Then we can write i (19(x 'Y)= Tr

f

f 1 R2 /

± x 2r /2 0 ± y 2 ) 170.

fsm(p )k sm(q)e i(px+gy) dp dq. tsi


267

The last integral can be approximated by Riemann sums in S'(1) 0 S'on'(IR1 ) for any m' > m:

E [ f (1 + x2ri2fsm(p)eiPx dp][ fo ± y 2r1'2g5m (q ) e i1 x dq ]

27r yo (x , y) = i s

. E iim x2[ E (1 ± x2r12 eakx fsm(XIC)11i111[ E ( 1 ± y 2 ) ,./2 e wygsm( ,01 s

=

hr. ),.2

iki str,' (TR 2 ) 0 (TR 1 ) is continuous. Here we used the notation 0m - (p, q) = fs m( p)ksm(q) .

se S'(11)

se

E s

It follows that 1 1160 11 = — 27r lim X2 II

0m( A1, Al).

0 ± x 2 r/ 2 (l ± y2 ) InI2e a(kx+1Y) Ç

E 1k1, 111 y = (yl, ... i.e., Aix —›- yi, j = 1, • .., n, then yi = 0 because Ai is closed. Furthermore, C2 generates a strongly continuous one-parameter group in B ORn, namely,

e te2t =-- e f(e0I—I01)t = e t 0 e —iAt (the reader is advised to check this identity by direct computation). Thus we may conclude that Ae ici t x = e_i C2t ti X, X E D, A

or, in more detail,

n A. iCt x = et Je

E exp[—AdikAkx. k=1

Let us now estimate e ict in Bi . For x E D, we have n

ileict xiii

=iieict xllo ± E liAi eict xlbo j=1 n

0), the matrix A has purely imaginary eigenvalues, and the operator C satisfies the conditions of Theorem IV.4, then C is a

tempered generator in L(B oo ) =

L r-1s Bs). (

Proof If the spectrum of A is pure imaginary, then

Ile -At II - O. Passing to the limit as s —> oo, we complete the proof. Theorem IV.5 follows from Lemmas IV.9 and IV.10. , A n be a tuple of selfadjoint operators on a Hilbert space 7-t. We Now let A1, assume that these operators satisfy the conditions stated at the beginning of this subsection and also the following conditions: (a) The operators Ak+i, ..., A n are pairwise commuting (k < n is fixed). (b) The left ordered representation /1, . , in of the Feynman tuple (A1, .. . , A n ) in S°°(Rn) exists, and the operators /1, , In are S°°-generators in S". In other (Rn) we have words, for any two symbols f, g E Lf

An)] = h(A1,

An)Hg(Al, •

on D, where

1

An)

n

h(Y) = f(ii, . . . , 171)(g (Y)), 1 • . . ,ln are linear operators on S"(IEV), and the operator f (li, . . . , l n ) is defined

on S" by the formula 1

f

n

, ln) = f f (ti,

, tn)exp(ilntn)

Condition (b) is satisfied e.g. if A1, , An form a representation of nilpotent Lie algebra and satisfy the conditions of the Krein—Shikhvatov theorem. Denote yk),

Y = (Y' , Y ") , y' = (Yi,

y' = (Yk+1,

yn),

and introduce similar notation for multi indices a = (ai, • • • , an ). Let -

spm (R k , R n—k ) s mp

M E R,

p > 0,


282

be the space of functions f (y)

E

C°°(Rn) such that

alalf aya

(y) _ ads ,

where ad, (y) = [x, y]. The mapping (A.12) is necessarily a representation since by the Jacobi identity we have adv y i(z) = [[x, y ] , z] = — [[y, z], x] — [[z, x], y] = [x, [y, z]] — [y, [x, z]] = [ad s , ad] (z).

(A.13)

The kernel Z c L of the representation ad consists of all x E L satisfying the property [x, y] = 0 for all y e L. Z is called the center of the Lie algebra L. Let L be a Lie algebra of finite dimension n. Let us fix some basis {al, . . . , an } in L. Let x = x i ai. The operator ads in this basis is given by the matrix ads where

(A s )/1 = Xski , and Xski are the structure constants of L with respect to the basis {a 1 , . . . , an I.

1.6 The Ado Theorem The representation ad of a finite-dimensional Lie algebra L is a representation of L in a finite-dimensional space. Generally, this representation is not faithful (unless the center of L is trivial). However, the following theorem is valid. Theorem A.1 Any finite-dimensional Lie algebra possesses faithful finite-dimensional representations.

Thus we may consider any finite-dimensional Lie algebra as a matrix Lie algebra (though the size of the matrices may be very large). The proof of the Ado theorem is very complicated, and we do not consider it here.

292

Appendix A: Representation of Lie Algebras and Lie Groups

1.7 Nilpotent Lie Algebras A Lie algebra L is said to be nilpotent if for any x e L the operator ad x is nilpotent (that is,

(adx ) N = 0 for some N). The Engel theorem says that a Lie algebra is nilpotent if and only if there exists a number N such that for xi , x2, .. . , xN e L one has [. • • [xi, x2 1 , x 3], . . . , x N] = O. [

The minimal N with this property is called the nilpotency rank of L. Let L be a nilpotent Lie algebra. Let us define a decreasing sequence of ideals in L by setting

Ln = L, L (1) = [L, Ln], L (2) = [L, L (1) ], ... , L (k) = [L, We have

L = Lo ) D L (1) D L(2) D ... D L (k — i) D L (k) D . .. .

It is easy to see that L (N) = {0}, where N is the nilpotency rank of L. Let dim LR ) = rk, ro =n. Let us choose a basis in L, by using the following procedure: choose some basis [al, ... , arN _ 1 1 in L (N-1) , extend it to a basis [al, . .. , arN_2 ) in and so on, until a basis {al, . .. , an } in L is obtained. Now set bi = an+i—j, i = 1, ... , n. It is clear that [bi , bid

= E x/ikbi ; 1>max(j,k)

thus, )..ii i, = 0 for k > 1. In other words, all matrices A1, j = I, . . . , n, of the associated representation are strictly upper-triangular in the basis {b1, . • • , bnl.

2 Lie Groups and Their Representations 2.1 Lie Groups, Subgroups, the Gleason—Montgomery—Zippin Theorem A Lie group is a smooth manifold G together with a group structure such that the group operations G x G ---)- G, (x, y)i--> xy; G --> G, x 1--> x-1

Lie Groups and Their Representations

293

are smooth mappings of manifolds. The Gleason—Montgomery—Zippin theorem states that it suffices to require that the manifold and the group operations be of class C0 in the above definition. Under this condition, one may define the structure of a real-analytic manifold on G such that the group operations are real-analytic mappings. Let G be a Lie group. A Lie subgroup in G is an embedded submanifold N c G which is at the same time a subgroup of G.

2.2 Examples of Lie Groups The simplest example of a Lie group is the real line R with the multiplication (x, y) 1—›—x. It is easy to check all the conditions of the x ± y and the inverse element x definition. Another example is the group T of unimodular complex numbers with the usual multiplication. The mapping ço eiç° (A.14) introduces the (local) coordinate ço on T, and the group operations may be written in the form (p1

Çoi +

ço

492 (mod 2n- )

—

the coordinates y 1 ,. Let us now consider the space R2" c and define the multiplication law by setting

•Y n 9 Z1

9•••9

Zn,

(y (1) , z (1) , c (1)) (y (2) , z (2) , c (2)) = ( y (1) ± y (2) , z (1)

z (2) , c (1)

c (2)

(A.15)

where z y = ziy i . We leave to the reader the easy verification of the fact that the multiplication law (A.15) is associative. Further, this law possesses the neutral element (0, 0, 0), the inverse element always exists and is given by (y, z, c) -1 = (—y, —z, —c — zy). Thus, R2n+ 1 with this operation is a Lie group. It will be denoted by HB(n, R) and called the Heisenberg group. Consider now the set of nondegenerate n x n matrices with real elements. (Of course, they can be identified with nondegenerate linear operators in R n .) It is clear that this set is in fact a Lie group (the product is the conventional matrix product, whereas the smooth structure is inherited from the space Rn 2 , in which nondegenerate matrices form an open subset). This Lie group is denoted by gl(n , R) and called the full linear group. The group gl(n , R) and its subgroups are called matrix Lie groups. As an example, consider the group SO(n, R) c gl(n, R) of matrices A such that det A = 1, tA

A= E

294


(E is the identity matrix). Thus, SO(n, R) consists of orthogonal matrices with unit determinant. The group SO(n, R) is called the special orthogonal group.

2.3 Local Lie Groups Let (U, q) : U —›- Rn) be a coordinate chart on a Lie group G in a neighborhood of the unit element e E G. We assume that yo(e) = 0. Then the group operations are represented by the mappings

Vt : V X V --->- Rn , s: V -->. Rn

(A.16)

in some neighborhood V c ço(u) of the point 0 E R'1 , and lif(0, x) = lif(x, 0) = x, E(0) = 0, Ifr(x, Ifr(Y, z)) = ifr(*(x, y), z), Vf(x, E(x)) = lif(E(x), x) =0

(A.17)

(these equalities are satisfied whenever all the expressions involved are defined). A neighborhood of zero V c Rn together with a pair of mappings (A.16) satisfying (A.17) is called a local Lie group. Thus, any Lie group gives a local Lie group via the above construction. It turns out that, vice versa, any local Lie group gives rise to a Lie group. We do not prove the latter statement here.

2.4 Homomorphisms of Lie Groups, Normal Subgroups, Quotient Groups Let G1 and G2 be Lie groups. The mapping

is called a homomorphism of Lie groups, if ço is a smooth mapping of manifolds an a group homomorphism at the same time. If go : G1 —>- G2 is a Lie group homomorphism, its kernel N = Ker q) = Ix e GiWx) = el

is a Lie subgroup in G1. It is a normal subgroup, that is, x E N implies yxy -1 E N for any y E G. Indeed, if x E N then 49 (YxY -1 ) = W(Y)§ 0 (x)49 (Y -1 ) = 40 (Y) e4o(Y -1 ) = 49 (Y)W(Y -1 ) = 40 (YY -1 ) = 49 (e) = e. The quotient group GIN clearly possesses the structure of a smooth manifold concordant with the group structure, so that GIN is a Lie group.

3. Left and Right Translations. The Haar Measure

295

3 Left and Right Tr anslations. The Haar Measure Let G be a Lie group. Then for any g E G the following mappings are defined: Lg .

G --->- G, h 1--> gh

(left translation), and Rg :

G -->- G,

h i-->- hg (right translation). Both mappings are diffeomorphisms. The mappings (A.10), (A.11) induce the corresponding mappings of tangent and cotangent spaces, L g* : ThG —>- TghG, R g* : ThG --->- Thg G

and L; : Tg*h G --->- Th* G , R• g*•T*hgG -->. Th*G

for any h E G. A nondegenerate volume form dw on G is called a right (left) Haar measure on G, if it is right- (left-)invariant, that is, R* g (dw) = dw,

or L*g (dw) = dw,

respectively, for any g E G. Right and left Haar measures on a Lie group G always exist and are defined up to a nonzero factor. Indeed, we have (dr g) h = (R) 1 (drg) e or

(dig) h = WO 1 (di g)e .

Here dr is a right Haar measure, dig is a left Haar measure, e is the unit element of the group.

3.1 Left and Right Regular Representations Let G be a Lie group, C' the space of smooth functions of G. For any g define the operators Lg and R.g in C°°(G) by setting (Cg f) (h) = f (g -1 h),

(7Zg f) (h) = f (hg)

E

G let us


296

for any h e G. The introduced operators possess the following properties: L e = 'Re = id; (Lg1rg2 f) (h) =

(Lgi f) (g -1 h) = f

h)

= f ((g1g2) -1 h) = (Lg i g2 f) (h), (7Zg i 1Zg2 f) (h) = (R gi f) (hg2) = f (hgig2) = (R-g1g2f) (h).

Thus, the mappings

R, g

Rg

are homomorphisms of G into the group Aut(Cm(G)) of automorphisms of the linear space C"(G). These homomorphisms are called the left and the right regular representation, respectively.

3.2 Representations of Lie Groups The homomorphisms considered in the preceding subsection are nothing other than special examples of representations of Lie groups. A representation of a Lie group G in a linear space V is a homomorphism TG

Aut(V)

from the group G to the group of automorphisms of V. We assume that V is a topological linear space and that Aut(V) is the set of linear continuous operators on V with continuous inverses. Further, we introduce a certain topology in the set Aut( V), and the mapping T is assumed to be continuous in this topology. The following situation is of particular interest for us. Let V be a Banach space, Aut(V) be a set of bounded operators in V with bounded inverses, with the topology of strong convergence. The corresponding representations are called strongly continuous representations of the group G in the Banach space V. In particular, (1) the operators T (g) are bounded operators in V for all g e G; (2) T(e) = idv; (gig2) = (gi)T (g2) for any gi, g2 E G; (3) for any y e V the V-valued vector function g T (g) v on G is continuous on G with respect to the norm in V. A representation T is said to be faithful if T (g) 0 id v for any g 0 e.

4. The Relationship between Lie Groups and Lie Algebras

297

Examples of Representations. We consider only two examples here. A. Any matrix group acts in RV by left multiplications. B. Consider the Heisenberg group HB(n, R). We define the mapping TA : HB(n, R) —)- Aut(L 2 (Rn ))

(A.18)

(71(y, z, Of ) (x) = ei X(Y x +c) f (X ± z) ,

(A.19)

by the formulas where X E R is a fixed number. The mapping TA is a strongly continuous representation of the group HB(n, R) in the space L 2 (Rn). Indeed, we have (A.20)

IITA.(y , Z , C)11L2(litn)—>L2(1/0) = 1

for any (y, z, c) E HB(n, R). Further, if f (x) is a smooth finite function, it is obvious that formula (A.19) defines an element of the space L 2 (R) which depends on (y, z, c) E HB(n, R) continuously. Since C;cr) (1Eln ) is dense in L2 (Rn ) , and the norms of the operators (A.18) are uniformly bounded (A.20), one may conclude that the element (A.19) of L2 (11In ) depends on (y, z, c) E HB(n, R) continuously for any f E L2(Rn) . Finally, TA, (0, 0, 0) is the identity operator in f E L2 (R'1 ), and Tx(g) Tx(h) = TA,(gh),

as one can easily verify. Thus our statement is proved.

4 The Relationship between Lie Groups and Lie Algebras 4.1 The Lie Algebra of a Lie Group Let G be a Lie group. A vector field X e Vect(G) is said to be right invariant if -

R g* Xh = Xgh

(A.21)

for any g, h e G. It is easy to see that right-invariant vector fields form a finitedimensional linear space naturally isomorphic to the tangent space Te G of the group G at the point e. Indeed, (A.21) implies X g = R g* X e for any g E G, thus proving the claimed isomorphism.

298


The commutator of right-invariant vector fields is itself a right-invariant vector field. Indeed, let us consider any field X as an operator X : C°°(G) -- C'(G). The fact that X is right-invariant means exactly that X 'R, g = 'RI X

for any g E G. Now let Xi, X2 be right-invariant vector fields. We have [X1, X21R. g = XiX21, g - X2X17Z g = Rg X1X2 - 7Z.gX2X1 = Rg [Xl, X2], so that [Xi, X2] is also right-invariant. Thus, right-invariant vector fields on G form an n-dimensional Lie algebra Ç. It will be called the Lie algebra of the Lie group G. We use the above isomorphism to identify g with the tangent space Te G.

4.2 Examples A. Let us construct the Lie algebra of the group R. Any vector field on R has the form d X = a(x) Tx . Right translations act in R according to the formula Ry x = x ± y. Hence

d

(R g ,K X) x+y = and the right-invariance condition means that a(x ± y) =

that is, a(x) is a constant. We see that any element of the considered Lie algebra has the form d X = a— , a = const, dx and the Lie algebra is one-dimensional, with the trivial Lie bracket. B. Let us now consider the Lie algebra of the group HB(1 , R). Recall that the standard coordinates on HB (1, R) were denoted by (y, z, c). Any vector field on HB (1, R) has the form a a a X = ai (y, z, c)— + a2(y, z, c)— + a3 (y , z, c)— .

ay

az

ac

The Relationship between Lie Groups and Lie Algebras

299

Right translations on HB (1, R) are given by

5,z + 2, c + E + z3-7),

R(5.,,i,o(Y, z, c) = (y +

so that the matrix of the operator R6o,, has the form

1 0 0

= (

0 0 1 0 55 1

)

.

.

Thus, (R(-,,) * X),ky ±y . -, z+i,c+e+zji) =

= ai (y, z, c)

a

a

,

a

+ a2(Y, z, c)— + (a3(Y, z, c) + 5-7 a2(Y, z, c)) -5--- • Ty c az

The right-invariance conditions imply ai(y ± 5, z + 2, c

+ a. + z,S7- ) = ai(Y, z, c),

a2(y + j;, z + i, c + E + z53 ) = a2(Y, z, 0, a3 (y + 5;, z + .2, c + E + z5;) = a3 (37 , z, c) + Sja2(Y, z, 0, and so ai = const, a2 = const, a3 = ya2 + b, b = const . Thus, any right-invariant vector field X on HB(1, R) has the form

aa a 0\ X = ai— d - a2(— + y— -1 - b— ay ac ac az with some constants ai, a2 and b. Example A.1 The Lie algebra of HB(1, R) is isomorphic to hb(1, R).

4.3 The Exponential Mapping, One-Parameter Subgroups, Coordinates of I and II Genera Let G be a Lie group, g its Lie algebra, X e g an arbitrary element, considered as a right-invariant vector field on G. Consider an ordinary differential equation on G,

k=

xg

(A.22)

with the initial data g(0) = e.

(A.23)

300


Lemma A.1 The Cauchy problem (A.22) defined for all r e R and satisfying

(A.23) possesses a unique solution g(t)

-

(A.24)

g(r) g(t) = g(t + r) for all t, r c R.

Proof. The existence theorem for ordinary differential equations guarantees that the solution of (A.22) - (A.23) exists on the interval r e ( - E, E). The curve g(t+r) = k (r)

satisfies the equation (A.22) with the initial condition g (0) = g(t). Since X is rightinvariant, the curve g-: (r) = g(r) g(t) = 'Tg (t)g(T) satisfies the same equation, and the same initial condition. By the uniqueness theorem, , ' rI1 < E. k r) = g(r) on their common domain. Thus, (A.24) is valid for Itl,Irl,, I,t -IThe identity (A.24) allows us to extend the definition of g(t) for all r E R, by setting (

g(T)

= (g (i))N

for sufficiently large integer N. It is easy to verify that (A.22) remains valid, the lemma being thereby proved. A curve g : R -›- G satisfying the conditions (A.23) - (A.24) is called a oneparameter subgroup of G, the one-parameter subgroup constructed in the above lemma is said to correspond to the element X c Ç. It will be denoted by Ux (t). Let us now define the exponential mapping

exp : g —> G, by setting

exp(X) = Lix (1). It is clear that exp is a smooth mapping. Let us show that this mapping is nondegenerate in a neighborhood of zero. To do this, let us compute the derivative exp. (0) : To g

----›- Te G .

The tangent space To g may be identified with

X

E

g = Te G .

Then, for any

Te G ,

we have d

exp,,,, (0)X = --d-i exp(tX)

d

= —Utx (1) dt t =0 t=0

= t =0

= 1:6( (0) = X,

301


so that exp. (0) is the identity operator. Thus, exp is a diffeomorphism from a neighborhood of zero in g to a neighborhood of e in G. Special coordinate systems in G are related to the mapping exp. Let {al, . • . , an} be a basis in Ç. The coordinates of the I (first) genus are defined in the following way: the element exp(x i ai) e G is considered as having the coordinates x = (x1, .. . , x n ). Since exp is a nondegenerate mapping, we obtain a coordinate system in a neighborhood of e c G. Next we define the coordinates of the II (second) genus. Consider the mapping

exp2 :

G

Rn ----›-

x

1--->

(A.25)

exp(x n an ) .. . exP(x l ai ).

The derivative exp2 (0),,, is nondegenerate since it takes the vectors

a

a

axi' • • • ' axn

to the vectors ai, .. . , an which are linearly independent. Consequently, the mapping (A.25) defines a coordinate system in a neighborhood of the point e = exp2 (0). We point out that the coordinates of both I and II genera on the group G depend on the choice of the basis {ai, . . . , an } in its Lie algebra Ç.

4.4 Evaluating the Commutator with the Help of the Mapping exp Let G be a Lie group. Let X, Y e g be elements of the corresponding Lie algebra. Then the following formula for the commutator is valid: d [X, Y] = (— exP(sfi X) exP(..fin exp(-- ■fiX)exp(--,./iY)) dt

.

(A.26)

t=0

Indeed, let us consider X, Y as vector fields on G, and consider the following linear operator in C" (G) B(r)

= e rX e r Y e -rX e -T Y .

Here erx is the operator of left translation in C" (G), (e-rx f,) (g) = f (exp(tX)g),

g c G

and consequently, we have d _ex = Xe rx . dr

Furthermore, we have d B(t) = x e rx e TY e —rx e —ty + e rXy e rY e —rX e —r dr —e r x e rY Xe —rx C rY — e rx erY e—rxy e— T Y ;

302


and hence d B(r) dr r =0

= 0.

Differentiating once more with respect to r we obtain in a similar way

d2 B(r) = 2[X, Y]. dr2 r =0 Finally, we have proved that

,

d

dt

= [X, Y], t=0

which implies (A.26).

4.5 Derived Homomorphisms Let w : G 1 —>- G2

be a homomorphism of Lie groups. Consider the corresponding mapping def W* = W* (0) : g1 ---- g2

(A.27)

of Lie algebras. We claim that the mapping (A.27) is a homomorphism. To prove this, consider the following diagram. W

G2

Î exp

W*

g2

It is commutative. Indeed, if X c g1 and 14(0 = exp (t X) is the corresponding one-parameter subgroup, then go(14x(t)) is a one-parameter subgroup. Next, d

ço(14x (t)) w(exp x) = Obix (1)) = exP ( — dt

) t =0


303

as desired. Let us now use formula (A.26). We have, for X, Y e gi , ,0([x, I']) d = ( --cr- t exp(/iX) exp(ViY) exp(-- N/iX) exp(---NfiY)

) t=0

d = ço (exp(NfiX) exp(fiY) exp(—V i-- X) exp(---,,fiY)) d =— dt exp(4/iço * X) exp(/igo* Y) exp(---Nfi(p * X) exp(--Viço * Y) t=0

= [q)*X, q)* 11 The mapping g)* is called the Lie algebra homomorphism corresponding to the homomorphism .

4.6 Derived Representation Let G be a Lie group, T : G ----± Aut(V) be a representation of G in a vector space V. We wish to construct the corresponding representation of the Lie algebra Ç. We consider two cases. A. The space V is finite-dimensional, dim V = n. In this case Aut(V)'....j Gl(n, R) is a Lie group, and T is a homomorphism of Lie groups, T: G ----* Gl(n, R). The corresponding representation of the Lie algebra is defined as

g —> gl(n, R)-r_'-_ End(V) (see the preceding subsection to verify that T* is a representation of the Lie algebra). B. V is a Banach space,

T : G ----* Aut(V) is a strongly continuous representation of the group G. Consider the subspace V"" c V consisting of all vectors y e V such that the vector-function g i-->- T(g)v on G is infinitely smooth. The space V" is called the Gen-ding space of the representation T. The subspace V" is dense in V. Indeed, let ço (g) be a smooth finite function on G, and y e V be an arbitrary element. Consider the element i5 = f q)(h) T (h) vdih G


304 then

T (g) .5 = f ço(h) T (g) T (h) v dih = f ço(h) T (gh) v dih = f ço(g -1 h) T (h) v dih G

G

G

(here dih is a left Haar measure; see Section 3 above) and, since T (h)v is continuous, T (g)i3 is a smooth function on G. Let us now take ço converging to the 3-function. Under these conditions, i) —>- y, so we see that V" is dense in V. Define the mapping T,,, : g --* End( V °° ) by setting d (LX) x = — T (exptX)x dt

t=0

for v e V°°, X E g. As in Subsection 4.5, one can verify that T,,, is a homomorphism of Lie algebras. 7',,, is called the derived representation of g (associated with T). Let X E g, A = T* X. It is evident that A may be considered as an (unbounded) operator in V with domain V°°. Consider the representation U(t) = T(exp tX)

of the one-parameter semigroup corresponding to the element X. By the properties of T, U (t) is a strongly continuous one-parameter group of bounded linear operators in V (that is, U(t) are bounded operators). U(t) y is a continuous (with respect to norm) function of t E R for any y E V, and U(t) U(r) = U(t + r), t, r E R). By the Gelfand theorem, the generator Â of the group U(t) is an unbounded operator on V of the form d A- v = — (U(t)v) dt r=0 with the domain DA consisting of the vectors y E V for which U(t) y is differentiable with respect to t. The operator A is a closed densely defined operator whose resolvent 1?), (A) satisfies, for some M > 0, w > 0, the estimates

li RA. Ciir

i

M (Re?. — tor '

m=

1, ...,ReX > co.

(And, vice versa, each closed densely defined operator satisfying such estimates is a generator of a strongly continuous group in V.) It is clear that Â D A (that is, DA D DA and Ay = Ay for y E DA). In fact, A = A (the closure of the operator A). This is a consequence of the following lemma.

Lemma A.2 Let A be a generator of a strongly continuous one-parameter group of bounded operators U(t) = eAr in a Banach space V, and let D c DA be a dense subspace of V such that eAt C D for all t. Then A = (AID).

305


Proof Denote DA = (A — ),.E) D. Let us show that DA is dense in D for Re). > co. (This implies the desired result immediately since we have RA. (A) = (A) I DÂ and, consequently, A — AE = [RA(A)] -1 = RA(A) -1 1 D = (A —

Let

y .E

D.)

D. Consider the integral

=fe 00

w

—Xt e At x dt.

0

Since Re A > co and I le m II < Me t , this integral converges; furthermore, we have 00

(A

—

w=

(A

—

X)e —xt eAt v dt

o oo

d ee At v dt = —v. —

J dt o

Thus, the element u may be approximated by integral sums of the form

E (A

—

A.)e —Àti e Ati Ativ

E

DA.

Since D is dense in V, the same is true of D. The lemma is proved.

4.7 The Lie Group Corresponding to a Lie Algebra Let

r be an n-dimensional Lie algebra with basis ai , [ai, ail =

E 4aic,

, an , so that

j = 1, . . . , n,

k=1

, an . where A. / . are the structure constants of r with respect to the basis ai, By the Campbell—Hausdorff theorem, there is a unique, up to isomorphism, local Lie group with Lie algebra r, and by the Cartan—Levi—Maltsev theorem, there is a unique connected simply connected Lie group with Lie algebra r. Let G be a Lie group with Lie algebra r. We shall use the canonical coordinates of second genus in the neighborhood of the neutral element e E G: if x = (xi, . • • , xn) lies in a neighborhood of zero in ilk", then we set g(x) = gn (xn )gn _i(x n _i) . . .

(xi) E G,

306


where gi (t) is the one-parameter subgroup of G corresponding to a e position law in the coordinate system (xi, , xn ) has the form

r. The com(A.28)

g(x)g(y) = g(/î(x, y)),

where * is a smooth mapping of a neighborhood of the origin in Rn x likn into Rn, *(y , 0) = (0, y) E-.2 y. The mapping * can be expressed explicitly via the CampbellHausdorff—Dynkin formula. However, this expression is not needed for our aims, and we omit it. It is easy to calculate the derivative (avilax)(x, y). Since our consideration is local we may assume, by the Ado theorem, that r is realized as a matrix Lie algebra and the neigbborhood of e in G is that in a matrix Lie group. Then g(x) has the form _i g(x) = e xnan ex„_ian

where

exa

is the usual matrix exponent. (0/8*)g(*). We have

a

alp;

.•e

x i ai ,

We now calculate the derivative

e*iai

ajelbiai

g(*) =

Since

e tbae—tb = e t adb (a),

where adb = [h, • ], we obtain

a

adai (ai)i g(*).

g(*) = [e'frn ad an •

To simplify the expression in the square brackets, we note that in the basis (ai, . • • , an) the operator adas has the matrix As with entries (As) pq = 4q and, consequently, the operator exp(*n adan ) • • • • exp(*i ad ai ) is represented by the matrix exp(* n An) • • • • • exp (*i Ai ). Thus

a

avf; g

=

E [exp(ifnAn).... • exp(Vii

A.01pjapg (*)

p=1 =

Bp j (11I)C p) g (*) 19=1

the matrix

B(*) = B(Vii, • • • , Ifrn) =

(13 pq (

Bpq(*) = [exP(111nAn)

P)) being equal to exp(*q Aq )lpq .

In particular, B(0) = / (the identity matrix), so that the inverse matrix is defined when * is close to zero.

C(*) = B -1 (*)


307

Calculating the derivative with respect to x on both sides of (A.28), we obtain n

D

axi g(*(x, y)) = =

0

E

,fr.

IB

p,j=1

aX i Pi

• ()a g(*(x, y)), P n

a

E Bpi(x)ap g(x)g(y) axi (g(x)g(y)) = p=1 n

= P=1

The matrices ap , p = 1, ...,n, are linearly independent and so are ap g(*(x, y)), since g(*) is invertible. Thus we obtain n

Bp i(x) =

E Bpi(*) alp)

(A.29)

whence it follows that n

alp;

axi or simply

(X,

Y) =

E ciik (fr (x, Y))Bki (x) k=1

avi $7,- = C(i)B(x) = 13 -1 (*)B (x).

4.8 The Krein—Shikhvatov Theorem Assume now that a representation of the Lie algebra r is given in a Banach space. Under what conditions is it integrable, i.e., gives rise to a strongly continuous representation of the Lie group G? The answer is given by the Krein—Shikhvatov theorem for strongly continuous representations in Banach spaces and by the Nelson theorem for unitary representations in Hilbert spaces. Let us prove a version of the Krein—Shikhvatov theorem suitable for our purposes.

Theorem A.2 Assume that Ai, .. . , A n are the generators of strongly continuous groups of bounded linear operators in a Banach space B. Let D C B be a dense linear subset such that: (a) D c D A i CI • • • n DA n andD is invariant under the operators Ai, their resolvents 1?),(Ai), and the groups e itA i , j = 1, . . . ,n. (b) The operators i Ai form a representation of r in D, i.e., n

[Aj, Adh = — i

E k A s h, s=1

h E D, j, k = 1, . . . ,n.

308

Appendix A: Representation of Lie Algebras and Lie Groups Then there exists a representation T of G in B such that Ai are its generators, Ai = T(a).

Proof Since G is connected and simply connected, it suffices to construct T (g) for g in a neighborhood of e, namely in the neighborhood covered by a canonical coordinate system. For such g we set def T (g(x)) = T (x) = e iA n x n

Then T (x) is a bounded strongly continuous function, and moreover,

Tai (t) = exp(iAit) is a strongly continuous group with generator Ai, so to prove that the operators T (x) satisfy the group law in the neighborhood of zero, i.e., T (x)T (y) = T (*(x, y))

for x, y small enough, it suffices to check this identity on the dense subset D C B. Note that D is a core of each Ai . Lemma A.3 For h E D we have e iA s t Akh = E[exp(tAs)]pkApeiAsth, s, k = 1, . . . , n.

P=1 Proof Consider the Banach space

Y

= 13Ø C '= B EB J3 e3.. .EB l3. n summands

The operators in Y may be represented as matrices with operators in B as their elements. We introduce the operator B : B Y which is the closure l from D of the operator h —> (Aih, . . . , A n h) c Y, and the operator Cs : Y Y, which has the form C, (A,

0 )

• ••

+i 0

As

• ••

Dcs = DA, ED • • • ED DA, , is closable since hn 0, Bhn = 0, since Ai are closed operators. 1B

= (hi

, iin ) implies Ah n

i = 1, . . . , n, so that

309


where A's is the transpose of A. Then Gs is a generator of the strongly continuous group exp(iCs t) in the space Y. Indeed, direct computation shows that exp(iCs t) = exp(iA s t) 0 exp(— At).

Since Xs.jk are antisymmetric with respect to the subscripts j, k, it follows that B A s h = (AlA s h, . . . , A n A s h) = (A s Aih, . . . , A s A n h) n —i(EA. lis Aih, 1=1

n

. . . , E x`nsitio = (A s 0 I)Bh ± i(I x A's )Bh = Cs Bh, h

E

D.

1=1

Since V is invariant under the resolvent Rx(As ), we can apply Lemma IV.7 and obtain h E DB D D.

B exp(i As t)h = exp(iCs t)Bh,

This identity can be written in the form A p exp(As t)h

=

n exP(iAst) E[exp(—A sI t)]pkAkh k=1

n = exp(iA s t)

E[exp (—As t)]kpAkh. k=1

Since exp(—As t) is the inverse of exp(As t), the lemma is proved.

0

Lemma A.4 For any h e D the function T(A) is differentiable with respect to X E Rn and n a — i — T (A)h = E Bpj(X)ApT ()Oh.

ax• J

p=1

Proof Set h(X) = T (A.)h . For any c = (ci, . . . , sn ) E R', we have

n

—h(A1, . • • , Ai , X j±i ± Ei+i, .. . , An ± En )] = i

E sie iAn(xn-i-co ... j=1

1

X e iAi+10,..H..1+ei+1)

dreiAA-Frei)

.„iAi_ili....1 ..... e iApk ih

A

"Ic f

0

=

n i EE.J e iAn l "• j=1

...

iA• II• • J -1 • e iA.A.. J i A. J e -7 -

• • • • e iA1X1

h + 0 (II s II)


310

as Ms II = (s?+• • •±8n2 ) 1 /2 --)- 0. Here we used the strong continuity of exp(i A n An ) . . . exp(i Ai Ai) and the invariance of D under exp(i Ai t). Thus h(X) is differentiable and

a hoo = e iAn4 . .... _i _ a x•J

e iApj

AJ.eiA -i - ' xi- i • . . . e iA0. 1h.

Successive applications of Lemma A.3 yield —i

a

n

— h (X) =E[exp(X n A n ) • . . . • exp(XiAjApjAp h(X).

ax•J

p=1

The lemma is proved. For h

E

0

D, set 111(x , Y) = 7' (x)T (Y)h, h2(x,

y) = T(1/î(x, Y))h•

We have hi (0, y) = h2(0, y) = T (Y)h .

Be Lemma A.4 n = E Bpj(x)Aphi(x, p=1

= '2..., a*s (x ' Y) s=1 axi

y),

EnBps(lk(x, Y))Aph2(x , Y)

p=1

n

= E Bpi (x)Aph2(x, y) p=1

by (A.29). We note that Bpj (Xl, . . . , Xj , 0, ...

,0) = [exp(xi A

)]

R]

=

since the j-th column of Ai consists of zeros 01/4.ïi -___ 0), and so we have

0h —i — (xi, . . . , xi, 0, ... , 0) = Aihi(xi, ... , xi, 0, ... , 0, y), ax J• i = 1, 2,

j = 1, ... , n.

We can prove that hi (xi , • • • , xi, 0, • • • , 0, y) = h2(xi, • • • , xj, 0, • • • , 0, y), by induction on j. This is valid for j = jo; we see that for j = jo ± 1 both hl and h2 satisfy the same Cauchy data at xi = 0. Since Ai is the generator of a strongly continuous semigroup, the solution of the Cauchy problem is unique, and we obtain T (x)T (y)h = T (*(x, y))h. The theorem is proved. 0

Appendix B

Pseudodifferential Operators

Pseudodifferential operators, that is, functions of x and —i0/3x, are widely used throughout this book. However, we do not assume that the readers are familiar with the topic, and this appendix is intended to make the book self-contained by presenting the definitions and main properties of pseudodifferential operators. Pseudodifferential operators, originally known as "singular integral operators" in the theory of elliptic equations, have a long and intricate history. Accordingly, there are quite a few expositions of this theory in the literature, starting with [109] and followed by [84], [175], [178], and others. However, we follow neither of the cited papers here, and after an elementary introduction, where we consider the classical Kohn—Nirenberg pseudodifferential operators (*DO), we present an exposition in the spirit of noncommutative analysis of the theory of *DO's which may have rapidly oscillating symbols. Here we mainly follow the paper [103].

1 Elementary Introduction The aim of the present section is to present an elementary introduction to the theory of pseudodifferential operators (*DO's in the sequel). Thus, in this section the reader will find the motivation for the appearence of the notion of pseudodifferential operator and the main definitions and theorems of the theory of *DO's rather than an accurate description of function spaces and precise statements and proofs of the theorems. We hope, however, that this section will be of use for the beginner in the *DO theory. The reader who is interested in precise statements and proofs can find them in the subsequent sections. One of the themes that lead to the notion of pseudodifferential operators is the problem of constructing a parametrix for an elliptic differential operator. For simplicity we shall carry out our considerations in the Cartesian space Rn. To recall the statement of this problem let us consider a differential equation of the form

flu = f

(B.30)

Appendix B: Pseudodifferential Operators

312 where

1) fi = H (L _i axa = E

aa (x) (—i — a V

lai..in

ax )

(13.31)

is an elliptic differential operator in Rn with smooth coefficients aa (x) and f = f (x) is a (in general nonsmooth5 ) function of variables x = (x 1 , . . . , xn) E Rn. The upper indices in (B.31) determine the order of action of the operators included in the latter relation. The ellipticity of the operator (B.31) means that for any nonzero vector P = (P1, • • • , Pn) ERn

of the Cartesian space Rn (which is dual to the space Rn) the principal symbol of the operator ii

Hm (x, p) =

E

aa (x) pa

lal=m

is not equal to zero: Hm (x, p) 0 O.

(B.32)

The problem we shall now consider is to solve equation (B.30) up to sufficiently smooth terms. To formulate precisely the meaning of the latter phrase one must have some tools measuring the smoothness of the functions. This can be done, for example, with the help of Sobolev spaces. The idea of introducing these spaces is based on the fact that, due to the formula f (x) =

f e iPx f (p) dp,

representing the function f (x) via its Fourier transform f (p), the smoothness of the function f (x) is determined by the decay of f (p) as f pi --›- oo. Actually, the faster the function f(p) decreases at infinity, the smoother is the function f (x). Hence, it makes sense to introduce the space Hs (Rn) with the norm

11f e = f (1 + I p12) 1 If(p)1 2 dp. Clearly, smoother functions belong to the space Hs (Rn) for larger values of s. The spaces Hs (Rn) are called Sobolev spaces. We remark that the family of Sobolev spaces parametrized by s forms a decreasing scale of spaces, that is that

Hs (Rn ) C Hs' (Rn ) for any s' < s. reader familiar with the theory of L. Schwartz's distributions can view f (x) as a distribution, that is, as an element of the space S''(Rn). 5 The

313

Elementary Introduction

It can be easily shown that the operator (B.31) acts in Sobolev spaces as follows:

H s ( r)

_÷.

Hs—m

(R)

for any real value of s. Thus, the problem of finding a solution to equation (B.30) up to sufficiently smooth terms can be formulated as follows: Find an operator RN : Hs—m (Ir) ---> Hs (WI) such that the following relation holds: HR N = 1 ± (' -- N

(B.33)

where 6—Ar is a smoothing operator of order —N for sufficiently large values of N, that is, the operator

6 N : Hs ( r)

__>. H " (R")

is continuous. The operator hN satisfying relation (B.33) is called a parametrix of the operator H. If the parametrix for the operator il is constructed, then the solution to equation (B.30) up to elements of the space Hs+N (Rn ) is given by u = RN f; we note that the larger values of N we choose, the smoother are the functions from H" (W). Let us begin with the construction of a parametrix. To do this, we represent equation (B.30) in the form 11 u(x) = (1 ) 1 f e iPx f (p) dp.

27r The solution of this equation (parametrix) can be found in the form

f (p) dp

(B.34)

(x, p) = ( 1 ) 1eiPx . fIG

(B.35)

u(x) = f G(x , p) if the function G(x, p) satisfies the equation

27r

Moreover, to construct a solution to equation (B.30) up to sufficiently smooth terms one has to solve equation (B.35) up to functions with sufficiently strong decay at infinity. Thus, we have introduced in our problem a large numerical parameter I pi which corresponds to asymptotic expansions with respect to smoothness. As we shall see below, the asymptotic expansions for solutions to equation (B.35) can be constructed in terms of homogeneous functions in the variables p of decreasing order of homogeneity. Let us try to find a solution to equation (B.35) in the form —oc

G(x, p) = eiPx

E gi (x, p)

.i=ko

(B.36)


314

where gi (x, p) are homogeneous functions of order j with respect to the variables p (we do not yet know the value of ko, that is, the order of homogeneity of the principal term of the expansion (B.36)). Substituting the latter equality in equation (B.35) we obtain _0. i Px gi (x, p)]= (1 ) e iPx . (B.37) il [ei

E

27t

.i=ko

Now we must commute the factor eiPx with the differential operator if. To do this, we note that, due to the relation

a

a [etPx • —i -a-g(x, p)] = eiPx (-i— axk + Pk) ex, p) xic

the following commutation formula is valid: 1 ) H

(2

.

1

a

a

2

x,—t—

[eiPx g(x, pd= eiPx H (x, p — i Tc ) g( x, p).

ax

Using this relation, we rewrite (B.37) in the form 1

eiPx H

a

2

) — E gf• (x' -p)— 27

x, p — i—

ax

7, 11 .

'Px

e •

Cancelling out the factor eiPx we arrive at the equality 1

in-1

[Hm (x, p) -1--E Hk k=0 2

2 X,

p, —

a ax i—

/ 1 \

2 gi (X' P) = j =k0

27r. )

I

(B.38)

1

where Hk (x, p, — i P,

are differential operators of order m — k with coefficients

homogeneous in p of order k. Equating terms of equal order of homogeneity in the right- and left-hand sides of equality (B.38) we arrive at the following recurrence system for the unknown functions g) (x, p) (this equality also shows that the order /co of the leading term of the expansion (B.36) equals — m):

Hm (x, p)g,(x, p) = Hm (x, p)gi(x, p) = —

1

E 1=1-j

I (B.39) 2 a Hin _i_i x, p, —i — aX (

In the latter formula we set gi (x, p) = 0 for j >

—

m.

gi(x, p), j 5_ —m — 1.

Elementary Introduction

315

Due to the ellipticity condition (B.32) the latter system of equations has the unique solution

1

1)I

g—m (x , P) = ( 2 7r

Hm (x, p) .

(2

a ax

g—m-1(x, p) = — Hm—i x, p, —1— g_ m (x, p)

(B.40)

The functions gi(x, p) are homogeneous functions in p of order — j. Thus, to construct a parametrix for equation (B.30) one should truncate the expansion (B.36) at a sufficiently high level. Actually, substituting the truncated expansion in (B.34) one has: -N-m U(X) =

RN f(x) =

E

f eiPx g;(x, p) f (p)dp,

(B.41)

where the gi (x, p) are determined from (B.40) and N is a sufficiently large number. Unfortunately, the functions gi(x, p), being homogeneous functions of large negative order, may have a nonintegrable singularity at the origin and, hence, the integrals involved in the right-hand side of (B.41) are, in general, divergent. To overcome this difficulty, we use a smooth cut-off function x(p) which is equal to zero in some neighbourhood of the origin and equals unity outside a compact set of Rn . Thus, we obtain the formula for the required parametrix -N-m f

E

RN f(x) =

j el/3 'g' (x, P)f(P)X(P)dp.

(B.42)

i----ni

It is easy to see that the latter expression does not depend on the choice of the cut-off function x(p) up to arbitrary smooth terms, that is, up to terms from the Sobolev space Hs (Rn) for arbitrary values of s. Remark B.1 In the case when the homogeneity order of the function g1 (x, p) in (B.42) is positive, then the use of the cut-off function in this formula is not necessary. 2

1 . a

Remark B.2 Any differential operator H(x,—I T,) can be written in a form similar to the right-hand side of (B.42):

H

a

2 (1 x,—i— =

ax

m

E f eiPx H., (x, p) f (p)dp

j=0

(B.43)


316 where Hi (x , p) =

E aa (x) pa , j = 0, ... , m lal=i

are the components of the complete symbol H (x, p) of the operator il of order j. Here we have omitted the cut-off function x(p) in accordance with the previous remark. Thus, we arrive at the notion of pseudodifferential operator using the operator on the right in (B.42) as a model. Definition B.1 The operator (

2

1' .

a

(B.44)

P x, —t— = f e iPx P (x, P)f(p)x(P)dp ax

is called a pseudodifferential operator with symbol P(x , p). The only question which we must discuss to complete the definition of a pseudodifferential operator is the class of symbols used in Definition B.1. If we consider in (B.44) only polynomial (with respect to the variables p) symbols, then the class of pseudodifferential operators will coincide (due to (B.43)) with the class of differential operators. Certainly, this symbol class is too small; it does not even contain parametrices of elliptic differential operators. The smallest suitable class of symbols, which was in essence introduced in the original paper by J. J. Kohn and L. Nirenberg [109], is the class of so-called classical symbols. We say that the smooth function P(x, p) is a classical symbol of order m if for any multi-indices a, p the estimate

( a a— ( aX)a

Y P(alx

p)

Cap (1 ± 1191 2) 1146

"

with some positive constant C. We denote by Sm the set of classical symbols6 of order m. The set of classical symbols is suitable for constructing parametrices for elliptic differential operators. However, for other problems this class can turn out to be too small. A rather general symbol class will be considered in the subsequent sections; here we restrict ourselves to consideration of the class Sm. Let us formulate the two main theorems of the theory of pseudodifferential operators. The first of them is concerned with the action of a pseudodifferential operator in the Sobolev spaces Hs (Rn), and the second treats the composition of pseudodifferential operators. 6 For

a rigorous definition one should impose some conditions on the behavior of symbols as 1 x1 -oc. Exact conditions of this kind are presented, for example, in the cited paper. For simplicity, the reader can assume that all symbols considered have compact support with respect to the variable x, or that these symbols do not depend on x outside a compact set (which can itself depend on the symbol).


317

Theorem B.1 Let P (x, p) be a classical symbol of order m. Then the corresponding *DO (2

vs

a

.

H s (Rn)

r x, —t—

Hs—m (Rn)

aX

is a continuous operator for any s E Rn .

Theorem B.2 Let P(x, p) and Q(x , p) be classical symbols of orders m p and m Q, respectively. Then the relation 1 )

2

P (x,

1 )

a Q x, -t- -=

a —t— Ox .

2

.

ax

1 )

a

2

R x, --i—

ax

holds modulo operators of order —oo in the Sobolev scale. The symbol R(x, p) of the

_

2

1

operator K (x,-i

tc has the asymptotic expansion

R(x,

(-01al alalp(x , p) alai Q(x , p) iceio

as ipl

a /.

apa

axot

oo.

We shall not present the proofs of these theorems here. The reader can find them in many books and papers concerned with this topic (see, for example, [84], [175] and others). We also remark that we have omitted from the framework of this short presentation such important questions as the behavior of pseudodifferential operators under change of variables, the theory of pseudodifferential operators on smooth manifolds, and so on. These topics can also be found in the literature cited above.

2 Symbol Spaces and Generators The pseudodifferential operators used in this book usually act in symbol spaces. Technically, various symbol spaces may be considered; however, the most important in applications is S°° (Rn ), the poly-Banach algebra of symbols of tempered growth introduced in Chapter III. For this reason, we shall define and study pseudodifferential operators in S°° (Rn ). In what follows we use the abbreviation *DO for "pseudodifferential operator". Recall that S°° (Etn) is defined as the union of intersections soo

(Rn ) =

Un sit(Rn) k

(B.45)


318

where Sit (Rn) is the space of Ck functions on Rn with finite norm

, 1 11f 11 sti oz n ) = sup (1_ + lx1 2 ) 7 (E

If(a) (A,

lalk

and is endowed with the corresponding convergence (see Chapter III). The space S /k (Rn ) is not a Hilbert space, and in order to make our exposition as elementary as possible (and almost independent of Chapter III) we consider the following representation of S°° (Rn) in terms of Hilbert spaces: (B.46)

soo(Rn)=unHic(Rn), 1 k where

HI' (Rn) is the completion of C7 with respect to the Hilbert norm p u p Hik (R„ ) =

(f (1 + 1x1 2) 1 (1 l

—

A)

u(x)

1

2

i

dx

)/

.

(As an exercise, one can easily check that (B.45) and (B.46) give the same result by using Sobolev's embedding theorems). The convergence in S°° (Rn) can be described as follows: a generalized sequence {fa) C V ° (Rn) is said to converge to zero if there exists an 1 such that for any k we have fa E Hi' for sufficiently large a and II h II k ---> O. H1 Pseudodifferential operators are elements of the algebra L (S°°, S") of all continuous linear operators on S°° (Rn ). For Hilbert spaces B1, B2 let L (B1, B2) denote the Banach space of all continuous linear operators A : B1 --›- B2 equipped with the operator norm II A il L(Bi , B2 ) =

sup xEB I ,x 00

I AxIl B 2 . „ I lx 11Bi

The algebra L (S°°, S°°) has the form L

(s" , 5 00 ) = nunuL(Hik,H;), 1

r

(B.47)

sk

that is, each operator T E L (S°°, V') has the following property: V/ 3r Vs 3k such that T extends by continuity to a bounded linear operator from Hi' to IV The convergence on L (500, , S") is introduced as follows: a generalized sequence IL) E L (S°°, S") is said to converge to zero if V/ 3r Vs 3k such that L E L (Hi', 1.0 for sufficiently large a and I TŒ I L (Hik 914

9

Both S°° and L (5' , S') are poly-Banach algebras with respect to the convergence described above.

Symbol Spaces and Generators

319

We are interested in r°-generators in G (S°°, S'). Recall that an operator A E G (S° ° , S°°) is an S°°-generator if there exists a (necessarily unique) continuous homomorphism

p. : S" (IR 1 ) —> G (S" , S") such that p(y) = A where y is a coordinate on R1 .

Lemma B.1 An operator A is an S'-generator if and only if there exists a oneparameter subgroup {e t , t E R} C L (S', rip) such that i) et is differentiable with respect to t and

der —i — = A; dt ii)

the subgroup {et } is of tempered growth in G (S", 5 00 ) as t —> c>o, that is,

V/ 3r Vs 3k 3/3 I lierllifik_* H: 5- C (1 + ItI)P 1 .

(B.48)

Proof We shall not prove the "only if" part of the lemma since it is not used in

our subsequent considerations. Let us prove the "if" part, that is, the sufficiency of conditions i) and ii). Let f E S" (R 1 ). We intend to define II( f) = f (A). This can be done as follows. For some 1 we have

f

E

n

Hik

(

le)

k

whence it follows that the Fourier transform of p(y) = (1 + y 2) —N f (y) is continuous and decays rapidly as IA —> oo (it suffices to take N > 1+ 1/2). Consider the integral

1=

1 ri f _ W(t)er dt ,

(B.49)

where Cp. (t) is the Fourier transform of ço (y). Since if) (t) decays faster than any negative power of t as t —> oo, it follows from (B.48) that V/ 3r Vs 3k such that the integral in (B.49) converges in L(Hik , Hi). By the definition of convergence in G(S" , S°°), this integral converges in G(S" , S"). Hence, we can set f (A) = (1 ± A2)

N

1

ti .57—

f Cp(t)e t dt.

Trivial computations show that the result does not depend on the choice of admissible N and that the mapping f 1--> f (A) is a homomorphism, obviously continuous.


320 Furthermore, for f (y) = y we have f (A)

=

( Y 1 ri fF (1 - F A 2) fs— y-->i 1+ y2 ) et dt

=

y \ / F ri f Y—t f 1 + y2 s/Ir

)Y-4

1

,

02, ) er dt,

since it follows easily from condition i) that —iae t /at = Ae t for all t e R. Here Fy , t is the Fourier transform. Continuing the computations, we obtain f (A)

=

f

1

,s/.7ri J et

(1

02 ) Fy--›.t ( at2 1 1 : 2) dt

.,./7r.i 1 f et Ft (y) dt

— f e t 3' (0 dt

at as desired (we have used elementary properties of the Fourier transform). The proof 0 is complete. Using the Lemma, we can easily show that the following operators are S"-generators in L(Sc c ' , Sc°): a)

The operator of multiplication by the coordinate xi for each j = 1, .. . , n (this operator will be denoted by the same symbol xi).

b)

The differentiation operator Pi = —ialaxi, j =1, ...,n.

c)

The operator Pi + (x), where q)(x) is a smooth real function on Rn all of whose derivatives are bounded.

Indeed, we have only to construct the corresponding one-parameter groups and to check that they are of tempered growth. In case a) we have

et

exp (i xi t) = eix; r

(the operator of multiplication by etxi t ); we obviously have < C (1 +

3. Pseudodifferential Operators

321

whence it follows that Her Il iiik_ *Hik 5_ C (1 + iti)lkl . In case b) we have

a

et = exp(t—)

ax;

and et 41 (x) = kli(xi, ... , xi_i, xi + t, xi+i, • • • , x,); it is easy to observe that IlerIlHik,Hik 5_ C(1 + iti) 1/1.

Finally, in case c) we have t et (Pi + (x)) = exP ( i f q)(xl, • • • , xi + hr, ••• , x n )dr 0

exp {iti3i} ,

which is a one-parameter group of tempered growth in L(Sc ° , S').

3 Pseudodifferential Operators Pseudodifferential operators are functions of the multiplication and differentiation operators

x = (x i , ..., x n ) and

—

O

i— =

OX

(

a

a

-- i—, . . , --i— , aX n aX1

which satisfy the commutation relations

a1 [xk,—i— ax; Now let f

E

Se°(1K x In).

Definition B.2 The operator

(2

1 )

a

ax

(2

x i,

1 2 • . • ,xn,

a

1 )

a

. — 1 —, , • • • , — i —

ta xi

is called the pseudodifferential operator with symbol f.

axn

(B.50)

322


It is easy to show that the operator (B.50) acts on an arbitrary function u(x) E C(W2 ) according to the formula (

1 2 x,

Ox

„ ;.-, ,„,„ p)ru(y), =

(B.51)

)

where

1 p q)(y) = 27ti F I f e —i PY w(y)dy and frp--X 111 (P) =

(1

f

27r i

e—i PYT (p) dp

are the direct and the inverse Fourier transforms, respectively. Consider the function ep (x) = exp(ipx), which belongs to S' (Rn ) for any p E Rn . Let T E L(S", S") be an arbitrary operator. With T we associate a function of the variables p and x E Rn by setting

smb {T} (x, p) = e_(x)Te(x).

(B.52)

Definition B.3 The function smb {T} defined in (B.52) is called the symbol of the operator T. This definition is justified by the following lemma.

Lemma B.2 If 1

a

2

T = f (x, —i— , f E S(Rrxi X Rnp )

aX

is a *DO, then

smb {T} = f.

(B.53)

Proof To prove (B.53) we use (B.51). We have 1 2

Tep (x) = f x,

a

ax f

eiPx

OPY--le iPY

71) (27rO n/2 3(i1 — P)) 4—*x{f(x, -

fril—>x { f (x, p) (270'112 3 (77 — 10)) f (x, p)e iPx = f(x, p)ep (x).

(B.54)


323

On multiplying both sides in (B.54) by e_p (x), we obtain the desired identity.

ill

Our next step is to widen the class of symbols for which pseudodifferential operators are defined so as to make an arbitrary element of G(S", S") be a "pseudodifferential operator" in the sense that 1 2

a

T = smb {T} (x, —i —) I. ax

(B.55)

Let lirsil:rs22, (Rxn x ii) denote the completion of q° (Rxn x Rpn) with respect to the norm Il f ll Hsi .s2 1- 1 ,r 2

=

if f

(1 + IPI 2 ) r2

X 1(1 — Ap)

s2/2

0+

1x1 2

r

(1 - Ax ) s1 /2 f (x, p)

2

11/2

dx dp

Consider the poly-Banach space

L" (R x7l x R pn ) =

nunu S2

ri sl r2

1

H 1:;22: (Rixl x

This is a poly-Banach algebra with respect to pointwise multiplication. Obviously, we have

S" (IK x R np ) c G" (117ix x R np ) , and the embedding is continuous. Let T E r° (Rxn ,Rpn). Then the function smb {T} defined in (B.52) obviously

belongs to .03 (Mix x ii).

Theorem B.3 The mapping

2 1a f I--> f ( x, —i — ax

)

can be extended to a linear homeomorphism

p, : G" (i K' x Rpn) —> G" (S" , s°° ). The inverse mapping is given by the formula

= smb {T} .


324 Proof It is well-known that for any f

E

L 2 (Rxn x lielp ) the operator 1 2

a

f = f ( , —i — ) ax is a Hilbert—Schmidt operator in L 2 (Rn), and hence we have

1 f)] 112 — [Tr (f * — (270nI2 IIIflL2(Rn)i} f II 0 (R,x --(Rn) 5_

,

xR

)•

(B.56)

From the last estimate, by multiplying f by (1 — 6.)s and (1 + x 2)k with appropriate k and s, we obtain the following estimate 11,111Hir+k (lltn)_iliir+s(Rn) 5_

for any f

E

Crlsk

(B.57)

'I f11 H --sr:ki (R'; x15)

Hsr'ki (Rxn x Rpn).

In P° (litnx x Rpn) consider the convergence inherited from L" (111xn x Rpn). Then L" is a completion of S" in this convergence; it follows from the estimate (B.57) that the mapping ,u : S" x Ift pn) —> C (S" , S")

OK

takes generalized Cauchy sequences into generalized Cauchy sequences and hence extends to be a continuous linear mapping from C° (Rnx x Rpn) to C (S" , S"). Let us show that u is a homomorphism, that is, the inverse mapping

,u-1 = smb : r (S" , V') —> G" (I[exi x lik pn) is continuous. To this end we establish the following estimate on the symbol of an operator T in C (S" , S"): Vi E 2Z+ 3s E R Vr E 2Z+ 3k E R such that IISMb {T}II

,.. (41)

"r,1 (lie Rn\ ---' -- s+1,r—k k x x pi

■s

Hp ii ` -" risk II I II 'Irk:: (Rn) _), H :(Rn) •

(B.58)

Here 2Z+ is the set of nonnegative even integers. The estimate (B.58) is equivalent to the continuity of the mapping

smb : C (S" , S") —> .0° (UK' x R). Let us now prove (B.58). Since T Vr E 2Z+ 3k E R such that

E

G(S" , S"), it follows that V/

Hr < 00 • IIT li Hk-Fn s r+n

E

2Z+ 3s

E

R


325

Set smb {T} = f; then

1 T = f(—i

— a

ax

=f .

It follows from (B.56) that II f 11 2H rd

=

(27r) 2 Tr (fi A*) ,

s+1,r—k

where fi(x,

p) = (1 + lx12) 41 = (1 + Ip12)

. (1 — Ax)i (1— Ap ) f (x, p).

Consequently, we have (2) < CIrks gr,1 _ k — fi (1 — Mi (1 + lx1 2) 5 11 f 11 ..._s+1,r

L2 L2

in the scale {fl ik ). This is easy to carry out by using the formula that relates fi to f (we have purposely chosen / E 2Z+ , so that fi is expressed in terms of the derivatives of f of integral order). As a result, we obtain

Ilfll —s+1,r—k led

k+n Clirskil f 11Hr+n

fr

--,

Hsr •


Corollary f

E

B.1

El 1

2

An operator f (x ,—i »-ax 2-

is continuous in S°° (Rn) if and only if

C° (Rix' x R) .

We observe that the linear structure and the convergence structure of the spaces G" (Rxn x RnP ) and G (S°° ( Rn) , Soo (Rn)) are the same. However, these spaces have different algebraic structures. By using the homomorphism p, we can transfer the multiplication from G (S", S") to Lc° ( Rnx x R), thus obtaining a noncommutative multiplication in the symbol space G* (Rxn x ii). This multiplication is given by the formula 1

f * g = smb

a A f f (x, —1—) ax [ 2

.

ffg (x2 , —i l — a )]] 1

ax

(B.59)

where [H] are autonomous brackets (see Chapter I). An explicit formula for the symbol of the product in (B.59) is given by the following theorem.


326

Theorem B.4 The following relation holds: 1 f * g = f (

whenever g

E

2

ax

S" (R xn x Ri).

The proof follows from the results of Chapter II for the case in which both f and g lie in S°° (R xn x Rpn). The general result (f c G " (R xn x g c S°° (R xn x Rnp ) then follows by continuity.

Glossary

Ado's theorem A theorem stating that each finite-dimensional Lie algebra can be represented as a subalgebra of a matrix Lie algebra. asymptotic expansion If H is a linear space equipped with a decreasing filtration,

H= 110

D H1 D H2 D • • • ,

then an asymptotic expansion of an element h e H with respect to the filtration is a sequence {N} of elements of H such that h — hn E Hn ,

n = 1, 2, 3, .. .

Depending on the choice of the space and the filtration, various particular types of asymptotic expansions can be obtained. asymptotic expansion with respect to parameter A type of asymptotic expansion in which H is a space of bounded mappings f : (0, oo) —> W, lt. }--> f (?), where W is a Banach space, and the filtration is determined by the condition f E Hn 44. Xn f E H

(asymptotic expansion as À —> oc)

or f E Hn 0).

asymptotic expansion with respect to smoothness An asymptotic expansion in which H is a function space and the filtration is determined by the condition f E Hn 0,1;>0,mi>0

Willi! • • • lk!mk!

'

where the bracket in the numerator stands for the commutator (adA)ml ... (adB) 4 (adA)mk (B) [Bu iAtm' . . . B ikA ni kBi = (adB) I '

11 ± • • • ± lk + 1

Can be obtained by developing into a usual Taylor series in powers of t from the closed formula in(e tB e A ) = A +

t f ln(et adB eadA)

et adB e adA _ 1 (B) dt

o provided by noncommutative analysis. Campbell—Hausdorff theorem A theorem stating that in a local Lie group G the multiplication law is completely defined by the Lie bracket in its Lie algebra Ç. In the exponential coordinates of first genus,

1 A•B=A-1-B-F—[A,B1+• 2

where A, B c g, the neighborhood of zero in g is identified with that in G via the exponential mapping, and the dots stand for commutators of order > 2. 1

n

characteristics Let A be an operator algebra, and let A = (A1, . . . , An ) be a Feynman I n tuple possessing a left ordered representation /1, . . . , ln . If B = f (Ai, . . . , A n ) is a function of A, then one may consider the pseudodifferential operator

1 a iB = f((li, . • . , inn ) —= H (y, —i —) . ay The zeroes of its principal symbol (Hamiltonian) are called the characteristics of B. In fact, the definition depends on the type of homogeneity considered, which, in turn, is closely related to which asymptotic expansions are to be obtained for solutions to the equation Bu = u.

Glossary

330 commutation formula The formula

sf 1 3 2 [A, f (B)] = [A, B]—(B, B). Sy commutator The commutator [A, B] of two elements A, B of an associative algebra A is defined as [A, B] = AB — BA. commutation relations Let A be an algebra determined by a set of generators A1, .. . , An and relations il

ilc

[co(Ai i , ... , Ai k ) = 010)E E.

The system of relations E is called a system of commutation relations if it possesses a left ordered representation, that is, there exist operators

L i , ... , L n on the space of n-ary symbols such that

1 (Idj f)(Al , • • • ,

An)

n

1

n = Ajfff

(Al,

•••,

An)]

for any n-ary symbol f. In other words, the system of relations is rich enough to guarantee that in any product we can rearrange the operators A 1 , ... , An in the order prescribed by their indices. composite function formula A formula expressing 12

fief

f a[g(A, B)]]) 12

where C = g(A, B). It reads 12

12

f ([[(g (A , B)JI) = f (g (A , B)) 28 78 sg 2 7 8 8 2 f 1 12 A, B)—([[g(A, BA, g(A, B), g(A, B)). ±[A, B]—(A, B, B)—(A, 3),2 5

Sg

3 4 6

8 Y2 8 Y1 For the particular case in which eyi, y2) = yi + y2 one has, say, 1 2

1

8 2 f1

1

21

4

f IA ± B]]) = f (A, B) ± [A, B]-, ([[A ± B]], A ± B, A ± B), Sy L and there is a variety of similar formulas.

Glossary

331

Daletskii-Krein formula A formula expressing the derivative with respect to a parameter of a function of an operator depending on this parameter. It reads 3 — f (A(t)) = A' (t)-(A(t), A(t)), dt Sx 2

d

3f1

where 8f, f (x) - f (y) , tx, y) = x-y

— 8x

is the difference derivative of f (x). It can also be arranged as 8 df (A) =—f (RA, LA). Sx Here d f (A) is the differential of the function f, considered as a mapping f : A -> A A }--> f (A) at the point A, and RA and L A are the operators of right and left multiplication by A, respectively. However, the latter formula seems less justified in case f is only partially defined on A (recall that f (A) is meaningless if A is not a generator). derivation formula The formula

_ L 6ff 1 3A), D(f (A)) = D A (A, Sy

where D is any derivation of the operator algebra. difference derivatives The difference derivative of a function f (y), y E Ilk, is a function of two variables defined by 6f (Yi , Y2) = f Sy

f (Yi) - f (Y2)

Y 1 - Y2 .r (Y1),

'

Y1 0 Y2, Y1 = Y2.

Difference derivatives of higher order are defined inductively; to find S k f 1Sy k one should fix all but one argument of 8k-1 My" and apply 818y with respect to the remaining argument. Here are some useful properties of difference derivatives. a) S k f 1Sy k is a symmetric function of its k + 1 arguments; if f is smooth, then so is S k f 1 Sy k for any k.

Glossary

332 b) 8k f

sy k (Y1 , • • • , Yk+i )

Jdi

f

0

0

—

1.1 1 —• — 114- I

dp,2

f (k) (,U1Y1

+ • • + tikyk + (1— E kti) Yk+i) (here f (k) is the kth derivative of f); c) k+1

sk f syk (Yi, • • • Yk+i) =

if yi

yi for i

ji (y.; - yi) -1

f (Y.; ) j=1

i=i,i0i

j;

d)

Sk f

sy k (Y

1 Y ) = -/j f ("(y),

7•••7

e) 1

skf

sy

...,

k

.7C)

(

=

f

1

k — 1)!

.t.)k—lf(k)( ty

±

_ r)x)dr;

o

0 (

Ra y ) (

a

) ai

•••

al! • • • Œk+1!

(k

lai)!

a )

Œk+1 s k

f

Sy (Yi , • • • Yk+i)

ayk+i

YI ="'=Yk+1

=Y

f (k+Ial) (Y).

difference-differential equations An equation containing both derivatives and finite differences. Such an equation can always be considered as a pseudodifferential equation since the difference derivatives can be expressed via the derivation operator; e.g. f(x h) — f (x) f(x) 8h h is expressed as e1 '

Sh =

etc.

—

h

1

Glossary

333

differential of an operator function The linear part of the increment f (A ± B) — f (A)

considered as a function of B. The differential is the element of L(A), the algebra of linear operators in the operator algebra A, and is given by the formula d[f (A)] =

f

(L A, RA),

Sy

where L A and RA are the operators of left and right multiplication by A, respectively. intertwining operators Let A = (A 1 , . . . , An ) and B = (B1 , . . . , Bn ) be linear operators on the spaces E and F, respectively, and let p, : E --)- F be a linear operator. Then say that p, is an intertwining operator for the tuples A and B, or merely that p, intertwines A and B, if the diagram A•i

E

A

E

i

I Bj

F

F

commutes for any j = 1, . . . , n.

factor extracting rule The rule of noncommutative calculus stating that in the operator expression J! f

il

(Ai,

••,

km

k1

in

An) g(B1, • • • , B m)

in

the factor f (Ai, ... , An ) can be isolated by enclosing it into autonomous provided that the indices k 1 , . .. , km all fall within two categories: either ki > max( j 1 , ... ,in),

or ki < min( ji , ... , jn ), i = 1, ... , m. This being true, we have Ji f (Ai, • • • ,

in

k1

km

il

il

in

k1

km

An) g( 13 1, • • • , B rn) = Lf Mi, • • • , A0118'(B1, • • • , B m).

Glossary

334

Feynman indices Indices in operator expressions showing the order in which operators act (the arrangement of operators), the greater is an index, the closer to the left the corresponding operator stands. In writing Feynman indices they are placed just over the corresponding operator argument or over the left autonomous bracket, e.g., 1

2

1

2

2

(A ± B) (A — B) = A— BA + BA ± B 2 = A 2 ± B 2 , 7 2 11 3 1[AB — ClD = D(B A — C), etc. The value of an operator expressions depends on the order relation between Feynman indices rather than on the indices themselves. Also indices over operator arguments commuting with all the other arguments in an operator expression are irrelevant and may be omitted by convention.

Feynman quantization Here quantization is understood as a rule taking each symbol f (y 1 , . .. , yn ) into the operator f (A i , . .. , A n ) (for a prescribed tuple A1,. A n ). Different quantizations use different ways to define

f(A i , ..., An ). The term "Feynman quantization" refers to the case in which one sets

1 n (A An), 1, .. . , f (Y 1, • • • , Y„) = f i.e., Feynman indices are used to remove the ambiguity in the definition of f (A i , . . . , A n ). il

in

Feynman tuple A collection A = (Ai, . . . , A n ) of generators in an algebra A, equipped with Feynman indices j, . .. , jn . If f (y 1 , . .. , yn ) is an n-ary symbol then we can define the function of A by setting f (A)

=

in

h

def f

(Ai, • • • , An).

Feynman's extraction formula The formula

t t t T- exp [ f (A(r) ± B (r)) dr] = T- exp ( f A(r) dr) T- exp ( f C (r) dr), 0

0

0

where

t t C(t) = [T-exp ( f A(r) dr)] 1 B(t) T- exp ( f A(r) dr), 0

0

335

Glossary t

allowing one to extract the factor T- exp ( J A (v) dr) from the T-exponential of

0

A(r) + B(r).

2.

1

Fourier integral operators Functions of (x, —t 0/8x) with symbols given by oscillatory integrals and naturally occurring as parametrices of the Cauchy problem (pseudo)differential equations of hyperbolic type.

generators A generator is an element of an operator algebraon which functions can be defined. Specifically, let Y be a symbol space. An element A of an operator algebra A is called an Y-generator (or simply a generator, if Y is clear from the context) if there exists a continuous algebra morphism

such that ILA (Y) = A. The morphism it,A, if it exists at all, is always unique. Notation: p, A( f) = f (A). The element f (A) of the algebra A is called the function of A with symbol f (y).

graded Lie algebras Let G be an abelian group, and let x:GxG-÷C\{0} be a mapping such that i) x (g, .) is a homomorphism of groups for each fixed g

E

G;

ii) x (g , h)x(h, g) --_a 1. A G-graded Lie algebra of colour x is a g-graded vector space L equipped with the bilinear graded Lie bracket [., .] such that [A, 13 ] =

=

EDg EGL g

-x ( I A I, I B 1)[B, A ]

(graded antisymmetry) [A, [B, C ] l = [[A, Bl, C] ± X GAL 1/3 1)[B, [A, C ] l (graded Jacobi identity) (here I A I is the gradation of A, etc.). Any G-graded associative algebra A = ED gEG Ag can be made into a graded Lie algebra by setting [A, B] = AB — x(IAI, IBDB A. In applications to particle physics (supersymmetries) one mostly deals with the case

x x, y) = ( -1 ) xY . (

Hamiltonian See characteristics.

Glossary

336 Hamilton—Jacobi equation The equation

H x, aS(x) ) = 0,

ax

(

satisfied by the phase S(x) in the WKB-approximation

ifr(x) = e (i/h)s(x) g)(x) to the solution of the Schr6dinger equation 1 2

a

H x, ( —ih— * = 0 ax with Hamiltonian H(x, p), and various similar equations arising in the context of Maslov's canonical operator, Fourier integral operators, etc.

Heisenberg algebra The Lie algebra R2n+1 (x,

y, ) = (xi, • • • , xn,

yi, • • • ,

Yn, )

with the bracket (commutation relations) [xi, yk] = iOik, [xi, xk] =[Yi, Yk] = [xi, 0 = [Yi , 0 =

0,

where Sik is the Kronecker delta. It has the representation x 1. I--> X .J

(multiplication operators),

a

1--> —i—, ax;

Yi 4-

in L2 (Rn ), widely used in quantum mechanics and the theory of differential, pseudodifferential, and Fourier integral operators. The left ordered representa2 21 i 3 tion for the tuple (xi , x,, yi, . . . , yn , ) is ,

...

ly;

= yj — i

a

—. axi

Heisenberg commutation relations See Heisenberg algebra. 12

index permutation formula Any of the formulas permitting one to pass from f (A, B) 21 to

f (A, B), such as 1 2 3 12 82 f 1 5 2 4 f (A, B) = f (A, B) ± [B, A]— (A, A, B, B). Bx8y

337

Glossary

Jacobi condition The Jacobi condition is a condition on the operators of the left or1 n dered representation. Let (A1, .. . , A n ) be a Feynman tuple, and suppose that the operators A 1 , . . . , A n satisfy a system of alglebraic relations, say, {

ii is o)(Ai i , ..., AO = 01

, co€S2

where the sequences i l , . . . , is and j1 , . . . , is are, in general, chosen differently for different a) E Q. Suppose also that there exists a left ordered representation (i v . . . , In ) of the tuple (A i , . . . , A n ). Then the Jacobi condition requires that fi

for all a) E

,

is

l is ) = 0

a.

In fact, the Jacobi condition guarantees monomorphy of the mapping 1 f (x i , . . . , xn ) ---> f (A, . . . ,

n

A)

provided that the system 0 and its consequences exhaust all possible relations between the operators A 1 , .. . , A n . In the context of Lie algebras, this leads directly to the Poincaré—Birkhoff—Witt theorem stating that the ordered monomials Aan n , . . . , Ar form a basis in the enveloping algebra of the Lie algebra with basis A 1 , ... , A n . The system of equations given by the Jacobi condition can be viewed as a system for finding / 1 , . . . , in . Being equipped with the additional regularity condition 1

n

f( 1 1,•••,ln)( 1 ) = f(Yi, • • •, Yn) ,

it may well serve as such. Jacobi identity The identity [A, [B, C]] = [[A, B], C] ± [B, [A, C]] for the commutation in a Lie algebra, or the identity [A, [B, C ] l = RA, Bl, Cl + X(1A1, 1B D[B , [A, C ] l for the graded x -commutator on a graded Lie algebra.

338

Glossary

Leibniz rule The identity D(AB) = D(A)B ± AD(B)

to be satisfied by any derivation D of an algebra.

Nelson's condition A condition on the generators A 1 , .. . , An of a strongly continuous unitary representation of an n-dimensional Lie group G. Nelson's condition states that A 1 , .. . , A„ must be essentially self-adjoint on some dense subset D in the representation space H together with the operator A = —(Ai ± Ai ± • • • ± A 2n ).

Newton formula The expansion of f (C) N1 2 f (C) — f

(A) =

—

f (A) 3

sk f 1

2k

E ffc - Al... [[c - A]1— esxk (A, A, ... ,

2k+1

A

) + RN

k=1

with the remainder 2

2N

RN = IC — Al . . . E[ C —

esk f

1 3 2k+1 A). — . , (C, A, Al — sxk

Lie algebra A Lie algebra L is a linear space L equipped with a bilinear operation [. , .1 (Lie bracket) such that [A, B]±[B, A] = 0 for any A, B e L,

(skew-symmetry)

[[A, B], C] ± [[B, C], A] + [[C, A], B] = 0 (Jacobi identity). 1

n

normal form Let A be an operator algebra and A = (A 1 , . . . , A n ) a Feynman tuple of Y-generators on A (here Y is some fixed symbol class). A normal form of an operator B E A is its representation as 1 n B = f (Ai, • • • , An)

with some f E F,1 . The normal form neither exists nor is unique in general. However, if the tuple A has the left ordered representation (h . , . . . , In ) consisting of Y-generators in Y, then any operator of the form s 1 B = q)(Aj i , . . . , Ajs ), çoeFs

can be reduced to a normal form by setting s I f (Y 1 , • • • , Yn) = q)(lii, • • • , lis)(1).

The normal form is unique if the left ordered representation operators satisfy the generalized Jacobi condition.

Glossary

339

operator algebras In this book, by an "operator algebra" we mean an associative algebra with identity element equipped with an appropriate convergence (see Chapter IV for details on this point). That is, an operator algebra A is a linear space A over C on which a bilinear operation (multiplication) A, B 1----> AB is defined such that (i) (AB)C = A(BC) for any A, B, C E A;

(ii) there exists an element 1

E

A (the two-sided identity element) such that

A1= lA = A for any A E A;

(iii) the linear operations and the multiplication are continuous, e.g., if Ace and Ba are two (generalized) sequences in A converging, respectively, to A and B, then A a Ba is convergent to AB. Typical examples of operator algebras are supplied by algebras of continuous linear operators in some linear space. What is more, any operator algebra can be realized in this way by using the actual embedding A ---÷ L (A) , where £ (A) is the algebra of linear operators in A (see regular representations).

operator-valued symbol Let W and V be operator algebras, let T be a symbol class, and let f be a W-valued function of (yi, • • • , yn) such that f A1, .. . , A n are T-generators in V, then the operator 1 B = f (Ai, . . . ,

E Tn 6W .

If

itz ) E 17 (W

I n is well-defined (the mapping f 1-4- f (A 1 , .. . , A n ) extends by continuity from Fn 0 W, where its definition is obvious, to Fn W). The operator B is called a 1

n

function of A1, . . . , An with operator-valued symbol f. A slightly more complicated situation occurs if f takes values in the algebra itself where A1, . . . , An lie, f e .Tn 6VV. The values of f need not commute with A1, . .. , An , so that one must be careful. We define k

1

n

B = f (Ai, • • • , An),

where k, the Feynman index of the symbol, is distinct from 1, . . . , n, as follows. Let f =EfiOvi ETn OV. We set k

1

n

1

It

k

f (Ai, .

(this is well-defined since vi occur only linearly, and we need not assume that vi is a generator). This definition extends to the entire .T.n 6V by continuity.

Glossary

340 1

n

ordered representations Let F be a fixed symbol space, A = (Ai, ... , A n ) a Feynman tuple in an operator algebra A. The left ordered representation of A on Fn 1

n

is the Feynman tuple 1 = (lie ... ,1 n ) of operators determined by the following properties: (i) (li f )(A) = Ailf (A)]I for any i = 1, . . . , n and f (ii) (li f)(y) = yi f (y) whenever f is independent of

E

Fn;

Under the assumption that / 1 , ... , in are F-generators in portant property holds for these operators:

Fn , the following im-

[[ f (A)]1E g (A)] = [ f (Og](A)

for any f, g

E

Fn .

The right ordered representation is defined similarly; we replace the conditions (i) and (ii) by the conditions (i') (ri f)(A) = [f (A)1A for any i = 1, . . . ,n and f

E Fn;

(ii') (ri f)(y) = yi f (y) whenever f is independent of y 1 , ... , yi _ 1 .

Then If (A)1[g(A)]] = [g(r) f](A)

for any f, g

E

.F, .

path integral Also Feynman's path integral. The expression

t i f (pq — H) dr ifro(xo)dx0

f f [dx] [— di ] exp

=

27r

0 NI

def

=

iE ( Pk (xk+i —x0-1-1(xk+1,Pk,ric)Ark

f Ilin N--*oo, max Ati--0

e

k° =

)

R2N

--r dxkdpk I I 27

N-1 1 X

ifro(xo)

k=0

for the solution to the Schrtidinger equation

1

)

.8* a 2 —1— + H (x, —i — ifr = 0 ax at with the initial data

Ifr(x, 0) = *0(x).

Glossary

341

Poincaré---Birkhoff—Witt theorem A theorem stating that in the enveloping algelbra U(L) of a given Lie algebra L the elements 41 , ... , aai n form a basis, where (a1 , .. . , an ) is a basis of L.

Poisson algebra A space (13 of smooth functions on a manifold M, equipped with a Poisson bracket. A Poisson bracket is a bilinear operation {•, •} on 4) such that rsa

if

'

gl

af ag _ Ewik(x)_____ = j,k 8x, a Xk

in local coordinates. For this operation the skew-symmetry property

tf, 81 = —fg, fl, and the Jacobi identity (f, fg,

10) ± VI, (f, gll +

( g, fh, fl) = 0

are satisfied.

poly-Banach space A linear space B equipped with a polynorm p:BxI-->R + U(oo)

satisfying the axioms given below and with convergence defined as follows: a generalized sequence {x4 ) is convergent in B if for some A E A (where A is a given filter of sections of the bidirected set I) and for any a E Ap (xp,, a) < co and xt, is convergent in the seminorm X., a). Axioms of polynorm: 1) For any x E B there exists an a such that p(x , a) < 00. 2) p (. , a) is a seminorm on 13„ = fx E B I p(x , a) < oo). 3) p(x, a) < p(x, 13) whenever a > p. 1

n

product theorem A theorem stating that if a Feynman tuple A = (A1, . .. , An ) possesses a left ordered representation, then for any two symbols f, g E Fn the 1

n

1

n

product If (Ai, . .. , A n )r[g(iii, ... , A n )] can be reduced to a normal form,

1

n 1 Li. (Al, • • • , An)i1i[g(Al, • • • ,

n

1

n

1

n

An)] = U(11, • • • ,ln)gliAi, • • • , An)

provided that /1, . . . , in can be substituted into f (e.g., f is a polynomial or /1, . .. , in are .F-generators in Fn ). Similarly, 1 1 1 n n n —n —1 [[ f (Ai, . . . , A n )Hg(Ai, .. . , A n )]] = [g(ri , . . . , rn )f](211, ... where ri, .. . , rn are the operators of the right ordered representation. We use here the negative indices over the operators ri, . . . , rn.

342

Glossary

pseudodifferential operators Functions of the operators 2

1

—ia/aX)

with symbols f (x, p) that are sums of homogeneous functions with respect to p (the classical pseudodifferential operators) or with symbol f (x, p) satisfying the estimates aa+fi f

< co ± 1 p wn —pla1+81/31,

axaapfi (x, 13)

lai, L81

= 0, 1, 2,

when x lies in a compact set; here 0 < 8 < p < 1 (Htirmander's classes of 1 pseudodifferential operators),In the broad sense, any function of —i8/8x) may be referred to as a pseudodifferential operator. quantization This word has quite a few different meanings. In this book it refers to the choice of ordering of operators in substituting them for (commuting) numerical arguments of a function. See Feynman quantization, Weyl quantization. quantum oscillator The quantum system described by the Schrtidinger equation

ax

=

with quadratic Hamiltonian H(x, p) = P2 2m

(02x2.

quantum groups Quantum groups were introduced in [114], [166], and [47] as a formal object useful in the quantum method of the inverse scattering problem. The theory of quantum groups has been developing intensively since then; we refer the reader to [38], [170], [32], and the paper cited therein for detailed information (one should also mention the very important paper [152]). Here we only give some brief remark, following [38] and [170]. First, let us recall the notion of a Poisson algebra. Let M be a smooth manifold and A = F (M) a space of functions on M. Suppose that A ia equipped with a bilinear operation (a, b) {a, b) that is antisymmetric and satisfies the Jacobi identity. Then we say that A is a Poisson algebra and refer to { , ) as the Poisson bracket. A quantization of a Poisson algebra A on M is an algebra Ah over the algebra C[[h]] of formal power series such that i) there is an isomorphism Ah/ hAh

A;

Glossary

343

ii) [a, b] mod h = (a, b) under this isomorphism; iii) Ah is a topologically free Cah11-module.

Now let G be a Lie group and H = F(G) a space of functions on G. Suppose that H is equipped with a Poisson bracket such that the multiplication map

is a morphism of Poisson manifolds. Note that H is a Hopf algebra; namely, the mappings A :H---+HOH

(comultiplication),

S: H

H (antipode), and

e:H

C (counit)

are defined by duality to the multiplication, inversion x 1--> x -1 , and unit ( e) ---> G, respectively, and these mappings satisfy the axioms of a Hopf algebra. A quantum group is a quantization Hq of H in the class of Poisson Hopf algebras (here q is the tuple of quantization parameters). Thus, in the spirit of algebraic geometry, we consider a ring H of functions on G as the primary object and quantize H rather that G itself. Note that G can be reconstructed from H by applying the functor Spec; in contrast, Hq is usually not commutative, so applying Spec to Hq would make little sense. Here by Spec H we denote the space of maximal ideals in the ring H. It is convenient to describe quantum groups in dual terms. Let H be the Hopf algebra of regular functions on G. Then the universal enveloping algebra U(Ç) of the corresponding Lie algebra g is contained in H*, and H is known to be isomorphic to a Hopf algebra consisting of matrix elements of finite-dimentional representations of u (g). The quantization u (g) of U(Ç) is easy to describe (see the example below). Then Hq can be defined as the Hopf algebra generated by matrix elements of finite-dimensional representations of u (g). Consider a simple example (quantization of SL2(C)). Let

e

1 01 0)

01 1

o

)

'

h

=

f _(

0 )

0

be the standard Chevalley basis of the Lie algebra s12 = s12 (C). We have the commutation relations

[e, f]= h, [h, e] =2e, [h, f]= —2f.

(B.60)

The universal enveloping algebra U(s12) is the algebra with generators e, f, h, determined by the relations (B.60).

Glossary

344

0, ±1 be a complex number. Consider the associative algebra Now let q Ug (s/2) with generators e, f, a, a -1 and defining relations

aa-1 = = 1, ae=q2ea, af = 47 -2 fa, [e, f]= (a — a 1 )/(q — q -1 ). The algebra U(s12) can be obtained as the limit of Uq (s 1 2) as q —›- 1 in the following way. Let a = 1 + Eh + • • • , E

q = 1 + E,

0.

Then we obtain

e + Ehe = (1 +28)(e + eeh) + • • • , or

Elh,e]=2se + • • , and [h, e] 2e as E ---> 0. Similarly, [h, f] —2f and [e, f] h. We omit the verification of the fact that Ug (s12) has the required Hopf algebra structure. Hence, we have described the quantization of the enveloping algebra U(s12)• regular representations Representations of an algebra on itself by left and right multiplications. Let A be an associative algebra with identity element. For any A E A denote by LA and RA two linear operators acting on the linear space A by the formula LAB = AB, RAB = BA, B E A. The linear mappings L: A

R: A

G(A)

A(A)

and AI-4 LA

A— RA,

where L(A) is the algebra of linear operators on A, are called the left and the right regular representations of A. In fact, L is a representation, but R is an anti representation since it reverses the order of factors, RARB = RBA.

Both L and R are faithful representations, and they commute with each other, LARB = LBRA

for any A, B

E

A.

345

Glossary

representations A representation of an associative algebra A on a linear space E is a homomorphism of A into the algebra End(()E) of linear operators on E. A representation of a Lie algebra is defined in the same way, but the algebra End(OE) should be considered as a Lie algebra with Lie bracket [A, B]tf AB — BA, A, B E End(E). Sometimes a representation is understood as a morphism into an arbitrary algeEnd(E) is said to be faithful if Ker ço = 0, i.e. bra. A representation ço : A q)(a) = 0 implies a = O.

resolvent formulas The identites for the resolvent of a linear operator A: R(A) — R A (A) = Rx(A)R 1,(A) — Â)

(first resolvent identity);

R(A)

—

R(B) = Rx(A)(A — B)Rx(B)

(second resolvent identity).

semigroup property A family of linear operators T (t), t

E

114, is said to satisfy the

semigroup property if T (t

r) = T (t)T (r), t, r

E

R.

stationary phase method A method to obtain the asymptotic expansion as h -->- 0 of the integral i

) n/2 f

I (h) = 27th

eis(x)1 h w(x) dx , lin

where h E (0, 1], S(X) is smooth and real-valued, and (x) is smooth and compactly supported. Suppose that S(x) has a unique stationary point xo on supp g)(x), that is,

as ax

—(X0) = 0,

as ax

---- (x) 0 0

for x

xo,

x E supp

Furthermore, suppose that the stationary point is nondegenerate in the sense that J = det

a2S(x0))

axi axi

0 O.

Then the integral I (h) can be expanded into the asymptotic series

Glossary

346 where 10

e iS (x 0) / h w (x0 )

=

1

and the argument of J is determined by n

arg J =

E arg

Xk,

k=1

where 4 are the eigenvalues of the matrix a2 S(x0)

axi ax;

,

and arg 4 are chosen in the interval 3n-

< arg A.k
C is a (continuous) bilinear mapping, then there exists a unique (continuous) linear mapping if : A 0 B —> C such that Ilf = :lif 0i, where i: AxB —>

(a, b) i--->-

A0B,

a0b

is the natural bilinear mapping A.

T exponentials Let A(t) be a family of operators with parameter t E IR. Consider the Cauchy problem -

dU(t) dt

= A(t)U, U(0) = I

(identity operator).

Were the operators U(t) commuting with one another, we would have t U(t) = exp (f A(t)dt) .

o

349

Glossary In lack of commutativity, we denote

r U(t) = T- exp (f A(t)dt) . 0 The prefix T indicates that the operators A (t) in that expression are ordered by their arguments t. Using Feynman indices, we can write r

U(t) = exp (f A (t) dt) 0r t

i.e., U(t) is a function of the "infinite Feynman tuple" I A (t)

. Accordingly,

ret the symbol also depends on infinitely many arguments, i.e., is a functional:

r U (t) = f UAW)), where

t f (ty (OD = exp (f y(t) clt) 0 Either of these expressions is referred to as the T-exponential. Trotter formula Any of the formulas like e [WEB] = iim e Aln e Bln

e Aln e Bln n—).co'----:-.,:-:--- 1 2n factors

expressing the exponential of a sum via the exponentials of the summands. Trotter-type formulas Any of the formulas like t

t

r r-fo T-exp(f [[ f (A, B , r)]] ch) = T - exp( f f (A, B , t)ch) t

12

which allows one to remove the autonomous brackets in a T-exponential and further to pass from the T-exponential to the Feynman path integral. twisted product The product * in the symbol space introduced via the operators of the left ordered representation:

f * g = f (l)(g). With this product, the mapping symbol -÷ operator is a homomorphism of algebras. The associativity of the *-product is generated by the Jacobi condition.

Glossary

350

Weyl quantization A method of quantization in which all operators are assumed to act "simultaneously." More precisely, if f (y i , . . . , yn ) is a power of a linear function and A = (A 1 , ... , An ) is a tuple of operators, then the Weyl-quantized function fw (A 1 , . . . , A n ) is defined as the power of the given linear function of A p ••• , A n.• if f (y 1 , • • • , yn ) = (ceiYi + • • • ± anYn) tm ,

then fw(Ai, • • • , A n ) = Pei Ai ± • • • ± anAnr•

By linearity, this definition extends to arbitrary polynomials. As for wider symbol classes, the definition for, say, Fourier-representable functions can be given as follows:

—27ri y 1

fw (A 1 , • • • , An) = (

7

f f (ti, • • • , tn ) exP(iffti Ai + • • • ± tn A n 1) dt, Rn

where f (t) is the Fourier transform of f (y). The Weyl quantization is a more delicate issue then the Feynman quantization both in that it imposes stronger analytic requirecounts and in that the calculus is less evident with the Weyl quantization.

Wick normal form The representation of an operator A in the quantum state space as a function of the creation a+ and annihilation a– operators in which all a– act first and all a+ act last:


Chapter I Ordering operators by indices, one of the main ideas in noncommutative analysis, was first introduced by Feynman [54], but its systematic use in conjunction with other tools such as autonomous brackets was inspired only by Maslov's book [129]. Attempts to define functions of several noncommuting operators necessarily involve some kind of ordering. The Feynman ordering is apparently the most convenient one and is the only ordering used in this book. However, there exist several other kinds of ordering, of which the most widespread and familiar is undoubtedly the Weyl ordering (Subsection 2.6). It originally arose in connexion with problems of quantum mechanics. We refer the reader to [191], [2], [3], [4], [151] and [101] for details concerning the Weyl ordering. Almost not touched upon in this book are problems pertaining to functions of several commuting operators. Not that we consider them too simple; they merely lie off the mainstream of our exposition, and we give only the information absolutely necessary to maintain the desired level of rigor in dealing with the noncommutative case. The reader interested in problems specific to commutative functional calculus may refer to either textbooks on functional analysis such as [78], [105], and [197] or to more special treatments such as [14] and [174]. T-exponentials (Subsection 1.1) were introduced by Feynman [54], who also discovered a striking relationship between T-exponentials and path integrals (cf. [55], [150]). We describe this relationship in Subsection 1.7 following the paper [135]. Various functional-analytic subtleties, which are beyond the framework of an elementary textbook, arise in connexion with the T-exponentials; e.g., see [105], [135], [179], and [180]. The idea of representing operators of quantum mechanics as functions of creation and annihilation operators (Subsection 1.2) is very old. Perhaps the most rigorous treatment of the subject from the mathematical viewpoint can be found in [10] and [11]. The fact that (pseudo)differential operators are functions of operators of multiplication and differentiation by the coordinates was clearly understood very long ago. For example, the treatment of pseudodifferential operators in [109] in fact uses the Feynman ordering (however, this use is implicit since Feynman indices never occur in the paper). From the very beginning of the systematic development of noncommutative analysis it was realized that the emphasis should be put on common algebraic properties rather

352


than on specific details of the definition of functions of noncommuting operators. At first glance, the most attractive idea is to present these properties in the form of a set of axioms; then for each specific definition one will only have to check these axioms. This was the approach adopted in [129]. However, there is the disadvantage that serious technical difficulties occur whenever one tries to consider morphisms of operator algebras and find what happens to functions of noncommuting operators under these morphisms. That is why in the subsequent publications [99] and [101] the set of axioms was replaced by another set, which deals with the properties of the symbol classes and the mapping symbol --> operator rather than with the purely algebraic properties and their behavior under transformations. We hope that in the present book the definitions (see Section 2) have assumed their final form. In any case they have become simple, universal, and more convenient than before. The fundamental formulas of noncommutative differential calculus presented in Section 3 come from a number of sources. The Daletskii—ICrein formula expressing the derivative with respect to a parameter of a function of an operator depending on the parameter was first obtained in [29] and [30]. It was extended to a general derivation formula in [96] and [97]. Permutation formulas (Subsection 3.4) and the composite function formula (Subsection 3.5) can be found in [96], [97], [98], and [129]. We should point out that only a small part of a broad variety of formulas of noncommutative differential calculus is presented in the book. A great many of these formulas were obtained by Karasev. Here we cite only a few of his main papers; the complete list of references can be found in [98] and [101]. The Campbell—Hausdorff theorem and Dynkin's formula (Section 4) have a long and intricate history. Surprisingly, they have a number of important practical applications not limited to the reconstruction of the multiplication law in a Lie group [19], [53], [104], [117], [190]. There have been different approaches to the proof of Dynkin's formula [43]; e.g., see [21], [34], [53], [56], and [190]. The proof presented in Section 4 was obtained in [139], and the generalization to T-exponentials in [102].

Chapter II The method of ordered representation is the most powerful tool in special noncommutative analysis (that is, noncommutative analysis that deals with functions of a fixed Feynman tuple of operators). There is a substantial body of literature devoted, in essence, to functions of a tuple of operators satisfying a special class of relations. Primarily, these are Lie relations or graded Lie relations (see [12], [26], [28], [61], [62], [110], [119], [140], etc). The ordered representation (Section 1) was introduced in [129], and in [130] the essential role of the generalized Jacobi condition (Section 4) (which consists in the requirement that the operators of ordered representation should satisfy the same relations as the original operators) was announced, and was later studied in [133], [141], and


353

[134]. The ordered representation permits one to compute products in terms of symbols and leads (under the condition that the generalized Jacobi condition be satisfied) to the notion of a hypergroup (see [130] and cf. [122], [119]). The Jacobi condition for the ordered representation leads immediately to the Poincaré-Birkhoff-Witt (PBW) theorem (see [86], [162 ] ), which states the PBW property for enveloping algebras of Lie algebras. Far more interesting results are obtained if the Jacobi condition is applied to non-Lie commutation relation involving parameters. This results in a certain finite number of equations in these parameters. (Numerous examples of this sort were considered in [130], and some of them are included in Section 2 and Section 4.) These equations are closely related to the Yang-Baxter equations [7], [196] heavily used in the theory of quantum groups and quadratic algebras, which has been recently intensively under development (e.g., see [32], [35], [36], [37], [38], [47], [46], [70], [71],

[76], [90], [91], [92], [113], [114], [115], [116], [121], [122], [152], [158], [161], [166], [167], [168], [169], [170], [181], [182], [192], [193], [194], [195], and [198]). The situation is as follows: the Yang-Baxter equations are easier to verify but only guarantee that the PBW property is valid on polynomials of degree < 3, whereas the generalized Jacobi condition is much harder to check but ensures the PBW property for polynomials of arbitrary degree. The difficulty lies, of course, in how to compute the ordered representation operators. Some methods for this computation for certain classes of relations not restricted to Lie relations were proposed in [133], [134], [141], and [101] (see also references cited therein). Note that there is a very important branch of special noncommutative analysis not included in the present textbook, since it goes far beyond the elementary level. We mean asymptotic noncommutative analysis, which deals with commutation relations involving a small parameter and provides ordered representations, composition formulas, etc., in the form of asymptotic expansions with respect to this parameter. The small parameter being interpreted as Planck's constant, we arrive at the old, familiar quantization problem. Its solution leads to rich geometric constructions involving symplectic geometry (cf. [75]). We refer the reader to [99], [100], and [101], where a relatively complete bibliography on the topic can also be found. However, there is a case in which asymptotic and exact noncommutative analysis coincide, i.e., they give the same results. We mean the case in which the Feynman tuple in question satisfies Lie commutation relations. In this case the noncommutative analysis of functions of this tuple is closely related to the analysis on the corresponding Lie group. The relationship is considered in Section 6. For the reader's convenience, simple basic facts concerning Lie algebras, Lie groups, and their representations are collected in the Appendix at the end of the book. They can also be found in any standard textbook on the topic, e.g. [86], [162], [184] and others. The functional-analytic conditions on the generators of a representation of a Lie group such as the Krein-Shikhvatov and Nelson theorems are discussed in numerous works, e.g., [42], [58], [69], [78], [93], [95], [105], [107], [138], [149], [164], and [197]. Analysis in Hilbert (or Banach) scales (e.g., see [111]) is a natural framework

354


for our considerations in Section 6. The symbol classes introduced there are suitable for solving asymptotic problems arising in the theory of differential equations.

Chapter III This chapter comprises several examples in which noncommutative analysis is applied to obtain some kind of asymptotic solution to a problem for a differential equation. For standard settings of asymptotic problems for differential equations and related topics we refer the reader to [41], [79], [80], [81], [82], [84], [108], [109], [137], [142], [143], [165], [175], and [178]. However, the applications we cover also include nonstandard settings of asymptotic problems and our exposition chiefly follows [131], [132], and

[134]. The approaches to difference and differential-difference equations in Section 2 are taken from [129] and [131]. The problem on propagation of electromagnetic waves in plasma (Section 3) first considered by Lewis [120]. The rigorous mathematical treatment of this problem was carried out in [145], [146], [147], and [148] (cf. also [134]). The problem of obtaining synchronous asymptotics for equations with coefficients growing at infinity (Section 4) has been considered by many authors (let us mention, e.g., the papers [52], [163], and the literature cited therein). Here we include the results concerning coefficients of polynomial growth and obtained in [141]; these results were published in [133] and [134]. In Section 5 we consider a model example based on the geostrophic wind equations introduced in [153]. In Section 6 we demonstrate the method of re-expansions using the exact solution of some standard operator [129] on the example of a degenerate equation taken from [141]. Here the standard operator is just the operator of the quantum oscillator. Finally, in Section 7 we consider a problem involving an operator with double characteristics. This problem is well-studied (e.g., see [15], [16], [74], [80], [154], [155], and [157]) and we present no new results. Our aim is merely to demonstrate the use of operator-valued symbols [129] in problems of such sort in the context of noncommutative analysis. To this end, we first reduce the problem to a simple normal form in the spirit of [172], [124], and [1].

Chapter W The general definitions of noncommutative analysis given in Section 2 require an appropriate functional-analytic framework. This is provided by the language of polynormed


355

spaces. The theory of polynormed spaces (and, more generally, of spaces with convergence) has been developed intensively for a long time, see [13], [22], [23], [24], [31], [68], [72], [77], [87], [88], [89], [98], [126], and [138]. We reproduce the results necessary for our purposes in Section 1; the results of Section 2 are apparently new; at least we could not find them elsewhere in the literature. The exposition in Section 3 chiefly follows [134] and [141].

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Index

abelian algebra, 288 Ado theorem, 291 asymptotic expansion, 179, 327 —w.r.t growth at infinity, 327 —w.r.t ordered tuple, 185 —w.r.t parameter, 182, 327 —w.r.t smoothness, 182, 327 —, synchronous, 347 autonomous brackets, 15, 85

Banach scale, 328 Bose—Einstein statistics, 6 boundedness theorems, 170

Campbell—Hausdorf theorem, 70, 329 Cauchy integral formula, 72 characteristics, 179, 329 commutation relations —, linear, 156

—, semilinear, 125 commutator, 287, 330 coordinates of I and II genus, 301

Daletskii—Krein formula, 60 derivation, 289 derived

—homomorphism, 302 —representation, 304

—,difference-differential, 193, 332 —of geostrophic wind, 216 —, (pseudo)differential, 167 —with fractional powers of x, 212 —with growing symbols, 208 expectation value, 4 exponential mapping, 300 Faddeev—Zamolodchikov algebra, 152 Fermi—Dirac statistics, 6 Feynman indices, 3, 28, 85 Feynman tuple, 28 filter, 249 formula

—, commutation, 62, 330 —for ln eB eA , 74 —composite function, 66, 330 —derivation, 52, 331 —extraction, 334 —index permutation, 61, 336 —product, 64, 341 Fourier transform, 73, 166 —, group, 164, 166 —integral operator, 170, 335 functions of operators, 25, 26, 269

difference —approximation, 194, 196 —derivative, 18 Dynkin formula, 70

Ghrding space, 303 Gel'fand theorem, 304 generator, 24, 260, 268, 273, 317 Gleason—Montgomery—Zippin theorem, 293

Engel theorem, 292 enveloping algebra, 138 equation —, degenerate, 224 —, difference, 193

group —algebra, 164 —representation, 164 —, full linear, 293 —mollification, 163 —, special orthogonal, 294

372

Index

Haar measure, 158, 163, 295 Hamilton—Jacobi equation, 336 Heaviside operator method, 174 Heisenberg —algebra, 289, 336 —, representation, 290 —commutation relations, 336 —group, 293 —uncertainty principle, 4 Hilbert scale, 158

Newton formula, 57, 338 nilpotency rank, 157, 292 normal form, 338

inversion problem, 169

parametrix, 315 path integral, 340 Poincaré—Birkhoff—Witt —basis, 146 —property, 146 —theorem, 138 Poisson algebra, 341 poly-Banach space, 254, 341 polynorm, 250 polynormed —algebra, 249, 256 —space, 249 —space over filter, 251 product theorem, 341 pseudodifferential operators, 316, 321, 342 —, commutation with an exponent, 189 —, composition, 317, 325 —, continuous, 317, 324

Jacobi conditions, 133 —, generalized, 139 Jacobi identity, 287, 337

Krein—Shikhvatov theorem, 307 Leibniz rule, 289, 338 Lie algebra, 287 —, examples, 288 —, generators, 110,112 —, graded, 335 —, representation, 116

—, homomorphism, 289 —, nilpotent, 157, 291 —of Lie group, 298 —, quotient, 289 —, representation, 290 Lie bracket, 287 Lie commutation relations, 156 Lie group, 292 —corresponding to Lie algebra,

observable, 4 one-parameter subgroup, 300 operator —, annihilation, 6, 16, 94 —, creation, 6, 16, 94 —with double characteristics, 233

quantization procedure, 5 quantum oscillator, 16, 342

305 —, examples, 293 —, homomorphism, 294 —, local, 294 —, matrix, 293 —, multiplication law, 14 Lie subalgebra, 288 Lie subgroup, 293 Nelson condition, 157

representation, 290 —, associated, 291 —, derived, 304 —, examples, 290 —of Heisenberg algebra, 290 —, left-ordered, 97 —of Lie group, 296 —, regular (left, right), 33, 165

295

Index

373

—, right-ordered, 103 —, strongly continuous, 296 right-invariant vector field, 166 rule

—of changing Feynman indices,

86 —of deleting autonomous brackets, 87

semigroup property, 345 Sklyanin algebra, 153 Sobolev space, 312 state space, 4 stationary phase method, 345 structure constants, 287 symbol —classes, 42, 167 —, classical, 167 —of differential operator, 167, 174 —of pseudodifferential operator,

321 —, operator-valued, 339 —, principal, 167 —, subprincipal, 236 —space, 161,260

symplectic —form, 234 —space, 234

T-exponential, 20 —, extraction formula, 11 Taylor formula, 58, 347 tensor product, 257, 348 translation —, left, 295 —, right, 295 Trotter formula, 20, 349 Trotter-type formula, 349 twisted product, 165 uncertainty relation, 4

vacuum subspace, 6 wave function, 4

Weyl —functions of operators, 48 —quantization, 48 Wick —normal form, 6, 94, 350 —symbol, 7, 95 Yang—Baxter equation, 153, 156

Methods of Noncommutative Analysis: Theory and Applications (De Gruyter Studies in Mathematics)

Methods of Noncommutative Analysis: Theory and Applications

Methods of noncommutative analysis: theory and applications