Methods of Noncommutative Analysis: Theory and Applications 9783110813548, 9783110146325

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de Gruyter 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. Α. Klingenberg 2 Semimartingales, Michel Metivier 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 Krengel 7 Mathematical Theory of Statistics, Helmut Strasser 8 Transformation Groups, Tammo torn Dieck 9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii 10 Analyticity in Infinite Dimensional Spaces, Michel Hervi 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, lipo Laine 16 Rational Iteration, Norbert Steinmetz 17 Korovkin-type Approximation Theory and its Applications, Francesco Altomare and Michele Campiti 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 Hey er 21 Potential Theory on Infinite-Dimensional Abelian Groups, Alexander Bendikov

Vladimir Ε. Nazaikinskii · Victor Ε. Shatalov · Boris Yu. Sternin

Methods of Noncommutative Analysis Theory and Applications

w

Walter de Gruyter G Berlin · New York 1996 DE

Authors:

Vladimir Ε. Nazaikinskii Moscow State Institute of Electronics and Mathematics Technical University 3/12 B. Vuzovskii per Moscow 109028, Russia Series Editors Heinz Bauer Mathematisches Institut der Universität Bismarckstraße 1Ά D-91054 Erlangen, F R G

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

Jerry L. Kazdan Department of Mathematics University of Pennsylvania 209 South 33rd Street Philadelphia, PA 19104-6395, USA

Eduard Zehnder ETH-Zentrum/Mathematik Rämistraße 101 CH-8092 Zürich 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 ©

Printed on acid-free paper which falls within the guidelines of the A N S I to ensure permanence and durability.

Library of Congress Cataloging-in-Publication

Data

Methods of noncommutative analysis : theory and applications / Vladimir E. Nazaikinskii, Viktor E. Shatalov, Boris Yu. Sternin 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. Sternin, 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. Sternin. Berlin ; New York : de Gruyter, 1996 (De Gruyter studies in mathematics ; 22) ISBN 3-11-014632-0 NE: Satalov, Viktor E.:; Sternin, Boris J.:; GT

© 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 für Mathematik, Universität 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

ν

Elementary Notions of Noncommutative Analysis

1

1

1

2

3

4

5

Some Situations where Functions of Noncommuting Operators Arise . 1.1 Nonautonomous Linear Differential Equations of First Order. Γ-Exponentials 1.2 Operators of Quantum Mechanics. Creation and Annihilation Operators 1.3 Differential and Integral Operators 1.4 Problems of Perturbation Theory 1.5 Multiplication Law in Lie Groups 1.6 Eigenfunctions and Eigenvalues of the Quantum Oscillator . . 1.7 Γ-Exponentials, Trotter Formulas, and Path Integrals Functions of Noncommuting Operators: the Construction and Main Properties 2.1 Motivations 2.2 The Definition and the Uniqueness Theorem 2.3 Basic Properties 2.4 Tempered Symbols and Generators of Tempered Groups . . . 2.5 The Influence of the Symbol Classes on the Properties of Generators 2.6 Weyl Quantization Noncommutative Differential Calculus 3.1 The Derivation Formula 3.2 The Daletskii-Krein Formula 3.3 Higher-Order Expansions 3.4 Permutation of Feynman Indices 3.5 The Composite Function Formula The Campbell-Hausdorff Theorem and Dynkin's Formula 4.1 Statement of the Problem 4.2 The Commutation Operation 4.3 A Closed Formula for In (eBeA) 4.4 A Closed Formula for the Logarithm of a Γ-Exponential . . . Summary: Rules of "Operator Arithmetic" and Some Standard Techniques

1 4 7 10 14 16 20 23 23 26 33 42 45 48 51 52 54 55 60 66 70 70 71 74 77 84

viii

CONTENTS 5.1 5.2 5.3

Notation Rules Standard Techniques

II Method of Ordered Representation 1

2

3

4

5 6

94

Ordered Representation: Definition and Main Property 94 1.1 Wick Normal Form 94 1.2 Ordered Representation and Theorem on Products 97 1.3 Reduction to Normal Form 99 Some Examples 104 2.1 Functions of the Operators χ and —ihd/dx 105 2.2 Perturbed Heisenberg Relations 107 2.3 Examples of Nonlinear Commutation Relations 108 2.4 Lie Commutation Relations 110 2.5 Graded Lie Algebras 115 Evaluation of the Ordered Representation Operators 117 3.1 Equations for the Ordered Representation Operators 117 3.2 How to Obtain the Solution 121 3.3 Semilinear Commutation Relations 125 The Jacobi Condition and Poincare-Birkhoff-Witt Theorem 132 4.1 Ordered Representation of Relation Systems and the Jacobi Condition 133 4.2 The Poincare-Birkhoff-Witt Theorem 138 4.3 Verification of the Jacobi Condition: Two Examples 142 The Ordered Representations, Jacobi Condition, and the Yang-Baxter Equation 144 Representations of Lie Groups and Functions of Their Generators . . . 1 5 6 6.1 Conditions on the Representation 156 6.2 Hilbert Scales 158 6.3 Symbol Spaces 161 6.4 Symbol Classes: More Suitable for Asymptotic Problems . . . 1 6 7

III Noncommutative Analysis and Differential Equations 1

85 86 87

171

Preliminaries 171 1.1 Heaviside's Operator Method for Differential Equations with Constant Coefficients 174 1.2 Nonstandard Characteristics and Asymptotic Expansions . . . 1 7 9 1.3 Asymptotic Expansions: Smoothness vs Parameter 182 1.4 Asymptotic Expansions with Respect to an Ordered Tuple of Operators 185 1.5 Reduction to Pseudodifferential Equations 186 1.6 Commutation of an h~l-Pseudodifferential Operator with an Exponential 189

CONTENTS

2

3

4

1.7 Summary: the General Scheme 191 Difference and Difference-Differential Equations 193 2.1 Difference Approximations as Pseudodifferential Equations . . 1 9 4 2.2 Difference Approximations as Functions o f χ and 196 2.3 Another Approach to Difference Approximations Propagation of Electromagnetic Waves in Plasma 3.1 Statement of the Problem 3.2 The Construction of the Asymptotic Expansion

198 200 200 202

3.3 Analysis of the Asymptotic Solution Equations with Symbols Growing at Infinity

205 208

4.1 4.2 5 6

Statement of the Problem and its Operator Interpretation Asymptotic Solution of the Symbolic Equation

. . . 208 210

4.3 Equations with Fractional Powers o f χ in the Coefficients . . . 2 1 2 Geostrophic Wind Equations 216 Degenerate Equations 224 6.1 6.2 6.3 6.4

7

ix

Statement of the Problem Localization of the Right-Hand Side Solving the Equation with Localized Right-Hand Side The Asymptotic Solution in the General Case

224 225 229 232

Microlocal Asymptotic Solutions for an Operator with Double Characteristics

233

IV Functional-Analytic Background of Noncommutative Analysis 1

2

3

Topics on Convergence 1.1 What Is Actually Needed? 1.2 Polynormed Spaces and Algebras 1.3 Tensor Products Symbol Spaces and Generators 2.1 Definitions 2.2 S°° Is a Proper Symbol Space 2.3 5°°-Generators Functions of Operators in Scales of Spaces 3.1 Banach Scales 3.2 S 0 0 -Generators in Banach Scales 3.3 Functions of Feynman-Ordered Selfadjoint Operators

Appendix A. Representation of Lie Algebras and Lie Groups 1

Lie Algebras and Their Representations 1.1 Lie Algebras, Bases, Structure Constants, Subalgebras . . . . 1.2 Examples of Lie Algebras 1.3 Homomorphisms, Ideals, Quotient Algebras 1.4 Representations 1.5 The Associated Representation ad. The Center of a Lie Algebra

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

287 287 287 288 289 290 291

x

CONTENTS

2

3

4

1.6 The Ado Theorem 1.7 Nilpotent Lie Algebras Lie Groups and Their Representations 2.1 Lie Groups, Subgroups, the Gleason-Montgomery-Zippin Theorem 2.2 Examples of Lie Groups 2.3 Local Lie Groups 2.4 Homomorphisms of Lie Groups, Normal Subgroups, Quotient Groups Left and Right Translations. The Haar Measure 3.1 Left and Right Regular Representations 3.2 Representations of Lie Groups The Relationship between Lie Groups and Lie Algebras 4.1 The Lie Algebra of a Lie Group 4.2 Examples 4.3 The Exponential Mapping, One-Parameter Subgroups, Coordinates of I and II Genera 4.4 Evaluating the Commutator with the Help of the Mapping exp 4.5 Derived Homomorphisms 4.6 Derived Representation 4.7 The Lie Group Corresponding to a Lie Algebra 4.8 The Krein-Shikhvatov Theorem

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

Pseudodifferential Operators

291 292 292 292 293 294 294 295 295 296 297 297 298 299 301 302 303 305 307 311 311 317 321

Glossary

327

Bibliographical Remarks

351

Bibliography

357

Index

371

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. Γ-Exponentials The equation χ = A(t)x can easily be integrated in quadratures if χ e K 1 and A(t) is a given continuous function: (1.1) No more complicated is the case in which χ e M" but the η χ η matrices A{t) and A(t') commute with each other: [A(r), A(t')] for any t,t'.

def

= A(t)A(t')

- A(t')A(t)

= 0,

(1.2)

2

I. Elementary Notions o f Noncommutative Analysis

The solution is expressed by the same formula, where exp stands for the exponential, defined as the sum of the convergent series 00

71=0

matrix

βΠ

η

·

Indeed, under condition (1.2) we have

CT

exp I

I

\

(f

A(t) dt I = exp I

\ CT

/ A(r)c/tlexp|

/

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 eaeb = ea+b valid for the case in which a and b are numbers). Since ' t+At exp I

I

\ 1 = 1 +

A{t)dt

ο it is easy to differentiate x(t)

t+At I

/

A{t)dt

+

0(At2),

t

with respect to t and prove that jc(i) 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(AjvAfw)exp(Aw-^/N-i) · · ·εχρ(ΑιΔίι)*(0),

Ν-κχ> max Δί,· —•Ο i

where 0

=

to
F is 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 χ and —ih 3/3jc' 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 Η is a Hamiltonian of the form Ν Η { χ , ρ )

=

2

Σ ± £ -

+

ν { χ ) ,

i=l then, due to its additive structure, the choice of Feynman indices of and

χ

—

ih

d — r dxl

is unimportant, and the energy operator N

Η

=

H ( x ,

ρ)

=

-

h

Σ

2

^ Δ , ·

+

V ( x )

i'=l

(where Δ, is the Laplacian w.r.t. the coordinates of the ith 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 η distinct basis states 2 . Let us represent the state space Η as the Hilbert sum of one-dimensional subspaces Ή

=

Ή]\ J\

1

Jn> ο

The picture would be much more complicated should we consider the quantization in "generalized" coordinates and momenta. 2 In realistic systems the number of possible states, as a rule, is infinite.

6

I. Elementary Notions of Noncommutative Analysis

where Ti.jl jn is the space of states such that exactly jk particles occupy the Jtth basis state, k — I,... ,n. The numbers ji,..., j„ are referred to as occupation numbers. The space Ho = Ho ο 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 j\,..., j„, whereas in the second case the sum is finite and extends over y, € {0,1}. According to the type of statistics, the particles are referred to as bosons or fermions. To each of the η basis states there corresponds a creation operator and an annihilation operator άζ, the adjoint of These operators "create" and "destroy" particles in the fcth basis state, that is,

°k tyl-jk-jn

C

tyl-jk-l-jn

(it is assumed that Hjl ...jn = {0} provided that at least one of the indices jk is negative or, in the case of fermions, greater than 1). The operators a£ and a^ satisfy the following commutation relations: [af > α / + ]± = iak> α Π ± = [at, af]± = Ski I· Here 8u is the Kronecker delta, I the identity operator, and [A,B]±

= AB ± BA

the (antiCommutator of operators A and Β (the upper sign is taken for fermions and the lower for bosons). The Wick normal form of an operator A acting on the space Η is its representation in the form ßM)ax • • •( e i r w '

a = Ε (α,β)

1

··· ( Ό * .

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 Ψο € Ho, ||Ψο|| = 1). Indeed, we have (Ψο, ΑΨο)

=

Σ>«ι (α ,β)

—

C0...0,

αηβι

ß „ ( ( a t ) a i • · · ( α η) α η ^ο» {α^γ χ . . . ( α ~ γ η Ψο)

1. Some Situations where Functions of Noncommuting Operators Arise

7

since all terms with (α, β) φ (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

Λ = (f(af,...,

1

1

e f , . . . , ap

def

=

2

1

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 /(Z, w) = /(Z,, . . . , Zn, wlt . . . , W„) = Σ cau_anßl (ct,ß)

ßnz[l . . . Z ^ w f 1 . . . U)f"

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: 2η n+1 η 1 A = f ( a f , . . . , a+, a f , . . . , a~). The problem of calculating the Wick normal form of an operator acting on Η 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| 0 the frequency of the oscillator, and χ e R . A 1 Consider the eigenvalue problem for Η in the space Z,2(K ): # Ψ ( ; 0 = £Ψ(χ),

Ψ G Ι2(Μ').

One should find the values of Ε for which this equation has nontrivial (nonzero) solutions Ψ. It is easy to check that Η can be represented in the form Η = ha)(a+a~ + 1/2), where a± = (2M"1/2

±

are the creation and annihilation operators (it is not mere chance that this term coincides with the one already introduced in Subsection 1.2; 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 2 1 a+a =a+a : 2 ι a a ~ Ψ (χ) = λ Ψ ( χ ) . We seek the solution in the form 2 ι ι Ψ(*) = G(a+, a~a~)v(x),

(1.8)

where v(x) is an arbitrary function and G(£, y) is an unknown symbol to be defined. 2

1

2

1

It suffices to require that the product of the operators a+a~ and G(a+, a~) be equal 2 1 + to kG(a ,a~). Using the introduced notation, we can rewrite the cited product in any of the following forms: 2

1

+

Ια α"Ά 3

2 +

lG(a , 2 ι

2

= a+a-lG(a+,a~)l

1

4

3

+

2 +

a " ) ] = a a~G(a , 1

=

BA,

1

2 +

a~) = a+a~lG{a ,

1

a")]]

1. Some Situations where Functions of Noncommuting Operators Arise

17

2 1 where Β = a+a~ and Λ = 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~ft_G(a+, α~)]| = XG(a+, a~)

(1.9)

2 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 + _, def + _ — -4« [a, a J = a a —a a =— 1, r

which enables us to compute the symbol of the product on the left-hand side of (1.9). 2 ι Compute first the product W- = a~^G(a+, a _ ) J . Here we apply a trick standard in "operator arithmetic". Clearly, we have 3

2

ι

=a~G(a+,a-), 2

3

and the problem is to permute a+ to the last place so as to identify the arguments a~ ι 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 2

ι

3

a+(a~ -a~)

= 1.

If G(£, y) is a polynomial in ξ, 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

3

2

1

4

1

a-G(a+,a-)+a-(G(a+,a~)

=

— 4- — — 4-44- 4- — a G{cr,a )+a (a — α ) — — ( a , a , a ), δξ

3

where

1

=

4

SG 8ξ

1

^ y) =

3 2

G(a+,a~))

4

GG,y)-GÜ,y) ξζ - η

2

4

1

18

I. Elementary Notions of Noncommutative Analysis

is the difference derivative of G(£, jy) by ξ. The first term is already in the desired form, and we transform the second one by changing the Feynman indices: 3 2 +

a~(a -

* SH 2 4 a~)-—(a+,a+,a~) όξ

1

3

2

4

= =

5 +

a~(a 3 3 +2 +

1

SH

+

0 +

a~)—(a ,a ,a~) όξ 4

la-(a -a n

1

SH — οξ

5 +

0 +

(a ,a ,a~)

(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 — ( α ^ , α ^ , α ) = — (α , α , α ) = — (tf ,α ) όξ όξ όξ (the difference derivative becomes the usual derivative on the diagonal ξ = η). Finally, we obtain 2

2

H(a+,a~),

W- = where

όξ We can now compute the product 2

W+ = a+a~lG(a+,

1

2

a1)! = a+ W- =

1

a+lH(a+,

In fact, this is trivial: 2

W+ =α+ΙΗ(α+,α~)Ή

1

3

2

1

= a+H(a+,a~)

2

2

=a+H(a+,a~)

1

2

=

1

H+(a+,a~),

where Η+(ξ,γ)

=

ξΗ(ξ,γ).

(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

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~^c(y),

1. Some Situations where Functions of Noncommuting Operators Arise

19

2 1 where c(>) is an arbitrary function. We substitute a+ instead of ξ and a~ instead of y into G(£, >>) and substitute the resulting operator in (1.8): 2

V(x)=e~a

1_ a

2

1

(a+)xc(a~Mx).

Extracting ( α + ) λ , we obtain (α+)λφ(χ),

vf(jc) = where ßo all hß e Hk and hß -»· h in J/*. A linear operator A on Hoo is said to be bounded (of order r) in the scale {//5} if there exists an integer r such that for any s one has AHS

C

Hs+r

and the operator MHs

'· Hs

^s+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 {ifj} is said to be the generator of a semigroup of tempered growth (or simply a tempered generator) if the Cauchy problem D U

- i — at

Λ

= Au,

I

u\t=0

=

uo,

has a unique solution for any UQ e HQQ, and there exists an I such that for any s e Ζ the inclusion UQ e Hs implies that u(t) e H S + I for all t, is different!able in HS+I, and i w o i u / s c a + i'iyVoii,,

44

I. Elementary Notions of Noncommutative Analysis

where ||/i|U is the norm of h in Hk and the constants C and Ν depend on s but are independent of «ο £ H s . Thus, a tempered semigroup in a Hilbert scale grows at most polynomially as t oo. We denote u(t) = exp(i'Af)«o· (c) The Functional Calculus. Let A be the algebra of bounded operators in the scale {Hs}, and let A e A be a tempered generator in {//,}. Let us define the mapping μ,a : S°° (R 1 )

A

by setting oo

(i 4- A)m Γ = ' / exp(iAt)g(t)dt,

ßA(f)

-Jini

J

00

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

=

/OO (y + i)m'

00 g(t) =

- J L =

J

e-'VgWy

—00 is the Fourier transform of g(y), and the number m is chosen large enough to ensure that g(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 {//y}. Moreover, μ ^ takes y into A and is an algebra homomorphism; let us give a formal calculation proving the latter statement: if 00 mi

(i + A)

f

V—2 πι —oo J

where k(t) is the Fourier transform of k(y) = (t + y)~mih(y), then

μΑ(/)μΑ(Η)

=

(i + A ) m + m i

i

^

(ι + A ) m + m i λ/ΪτιΪ

00 00 f f

Ι

I exp(iA(i + T))g{t)k{r) άτ

00 —oo 00 —00

00 —00

y/2 ni

άη.

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

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 HA(f)ßA(h)

= tiA(fh),

or f(A)h(A)

=

fh(A),

as desired.

ι η We can now define f(A\,..., An) for a function f e S°° (R n ) and a Feynman tuple ι η ( Α ι , . . . , Α η ) of tempered generators in a usual way (see Remark 1.3). Obviously, we get l ( 1 \"/2 f _ n f(Au...,An) = I— 1 / /(*!,..., tn)exp(iAntn)...exp(iAiti)dti...dtn, where /(*ι>···>*η)=

( j ^ j

j f(yv

•••.y«)exp(-i/i)'i

itnyn)dy\...dyn

is the Fourier transform of / 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 Τ 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 Τ is closely related to the possible spectrum of a generator. For instance, if the symbol f(x) = (x-

a)~l

belongs to Τ, then necessarily a & σ (A) for any generator A (here σ (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 (JC — a ) - 1 G S 0 0 ^ 1 ) for any nonreal a. Therefore, the spectrum of each generator lies

46

I. Elementary Notions of Noncommutative Analysis

completely on the real axis, so that the polynomial estimates for the growth of exp(t 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(i'fA); some estimates of the resolvent of A are also needed; however, these estimates can also be derived from the fact that μ-Α-f

/(A)

is defined and continuous on S 0 0 ^ 1 ) . Example 1.4 This example is somewhat less trivial. Suppose that the symbol class T i contains the function Suppose also that operators A and Β satisfy [A, Β] =

-i

(for example, A = — i g j and Β = χ). Then at least one of the operators A and Β is not an ^"-generator. Indeed, we have, by Theorem 1.10, 4 f 1 3 5 2 6 + [A,B]—{-(Α, Β, B, A, A) oyioyi 1 3 2 δ df 1 3 2 4 / ( A , B, A) — i-——^—(A, Β, Α, A ) , oy2

1 2 3

f(A,

1 3 2

Β, Λ)

f(A,B,A)

=

=

since the commutator [A, Β] commutes with B. Furthermore, δ

df

τ— -r— (yi,y2,y3,y*) , · ] of 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, [D, f(LA)] =ox [D,LA]S-/-(LA,lA), By Theorem 1.2,

f(LA) = Lf(A),

and so, by Lemma 1.1, we obtain 2 5 / 1 3 Ld(/(A)) = (La, LA) = L 2 °x LD(A) —D(A)%(A,A) (we have applied Theorem 1.2 one more time). Since L is a faithful representation, we obtain 2 5 / 1 3 D(/(A)) = D ( A ) / ( A , A ) ,

ox

as desired. The proof is complete.

•

3.2 The Daletskii-Krein Formula Let us now return to our original problem of computing the coefficient Cι defined in (1.16). To this end, consider the algebra A{t\ whose elements are (infinitely differentiable) families of elements of A depending on a numerical parameter t. Clearly, the mapping

— : A{,) A{t) taking each family A(f) € A[t] into its i-derivative is a continuous derivation of the algebra A{By Theorem 1.7 we obtain

d

2

, 8f 1 3

ι3

3. Noncommutative Differential Calculus

55

This is the famous Daletskii-Kreinformula, 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 (1.16). Namely, take Α(ε) = A + εΒ\ then Α'{ε) == Β, and we obtain Ci

25/ ι 3 =B-f-(A,A), ox

i.e., / ( Α + εΒ) = / ( A ) +

δχ

A) +

0(ε2).

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 /(A + εΒ) = /(A) + εΒ^-(Α) οχ

+

0(ε2).

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 / ( A ) , that is, the linear part of the increment / ( C ) — / ( A ) w.r.t. the difference Β — C — A. It is given by the Daletskii-Krein formula, which can be rewritten as df(A;B)

def Γ d Ή = / ( A + eB)J|

=

2^/13 B-±(A,A)

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 / ( C ) — / (A) more comprehensively. Assume that C - A — εΒ, where ε, 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 / ( C ) — / ( A ) = / ( A + εΒ) — / ( A )

56

I. Elementary Notions of Noncommutative Analysis

in powers of ε is to apply the Daletskii-Krein formula successively. Namely, we can write down the usual MacLaurin expansion f(A + eB)~

Cksk

f (Α)

κ

k=ι

·

where the coefficient C* of ek is the kth ε-derivative of / ( A + εΒ) at ε = 0. By the way, the Daletskii-Krein formula holds for ε φ 0 as well: j 2 s/ ι — / ( A + εΒ) = αε ox

3 + εΒ]|, U + e f i l ) .

Thus we can differentiate it with respect to ε once more. There will clearly be two terms arising from the differentiation in the first and in the second argument of 8f/8x\ and the formula itself applies to each of these terms. Hence we obtain d2

2 2 4 s f ι /(Λ + εΒ) = 2BBj±(U

3

5

+ eB], [[A + ε BY U + εΒ]|)

(the factor 2 is due to the presence of two terms, which are equal to each other since 8f/8x(x, y) is symmetric, 8x 8x 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 24 - ^ / ( A + εΒ) = k\BB...

2 k f

1

3 2k+l + εΒΥ |[A + e B J , . . . , [[ A +

εΒ]).

Thus the factor 1 / k! in the MacLaurin expansion cancels out, and we obtain the Newton formula k

0°

.24

f ( A + £B)-f(A)^J2£ BB...B-^(A,A k=i

2kS k f

1 3

2Jt+l

A) x

°

or, forgetting about ε, °° 2 4 2k Xk f 1 3 2k+\ / ( C ) - / ( A ) = y ] | [ C - A ] l | [ C - A ] l . . . | [ C - A I T ^ k( A , A , . . . , A ) . · ' 8x , ' k copies

k+1 arguments

However, these formulas are still of little practical importance, chiefly because of the mysterious sign " = " 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

3. Noncommutative Differential Calculus

Theorem 1.8 (Newton's formula with remainder) Let A and C be generators. Then for any symbol /(x) and for any positive integer Ν we have Ν—1 2 2k S^ f 1 3 /(C)-/(A)= £IC-AJ...IC-AJ-4(A,A,..., k=\ °x

2/fc+l A ) + RN,

where the remainder RN is given by the formula 2 2Ν SN f 1 3 2ΛΗ-1 Rn = IC — A J . . . I C — A,..., A ) . Proof We proceed by induction on iV. Clearly, ι 3 1 3 1 3 / ( C ) - /(A) / ( C ) - / ( A ) = / ( C ) - / ( A ) = (C - A) [ C-A We can move the indices apart and then isolate the factor C — A, · A, i.e., an element of £(A). Having this in mind, we can represent the Daletskii-Krein formula as follows: M A ) =

y-(L ox

a

,R

a

).

This formula is understood in the sense that d f ( A , Β)

= MA)(B)

= M A ) ( B ) = ¥-(L ÖX

A

, RA)B

=

A). ox

We have considered various expansions for / ( A + Δ A). 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 / ( A , B) 1 2

2

in two different ways using Feynman's approach: one can take f ( A , B ) o r f ( A , How different will the results be? The answer is given by the following theorem.

1 B ) .

61

3. Noncommutative Differential Calculus

Theorem 1.10 (Index permutation formula) 12

f(A,B)

21

=

f(A,B)

=

f(A,B)

3

+

2 1

S2f 15 2 4 [B,A]-^r(A,A,B,B) SxSy S2 f 2 4 1 5 [B,A]-^-(A,A,B,B). SxSy

3

+

1 2

2

1

Proof. Let us consider the difference [[/(A, J?)] — I[/(A, 5 ) 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 |[/(A, B)I - Uik

5 ) 1 = /(Α, Β) - /(Α, Β)·

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 /(Α, Β ) - /(Α, Β ) = (A — A)

^

ι

= (A - A ) — / ( A , A, Β ) .

3

OX

A-A Strange as it may seem, this computation is perfectly rigorous; in fact, it means nothing other than that we take the formula /• / \ fix, y) ~

r/

fiz, y)

= ix/ -

\

, fix, y) - fjz, y) Sf z) = ix - z)— fix, χ—ζ ox

z; y)

1 2 3

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

12 2 1 1 3 Xf 1 3 2 /(A, Β) - /(A, B) = (A — A ) f - ( A , A, B). ox 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

fiA,B)~

21

fiA,B)

=

1 Xf 1 3 2 ox

3 Sf

3 Xf 1 3 2 A-fiA,A,B)-A^-iA,A-B) ox

1 5 4

3 ^ / 1 5 2

=

A-^-(A, A, Β) — A-^-(A, A; B) ox ox

=

A[-^-(A, A; B) — γ-iA, Sx Sx

3 ^ / 1 5 4

Sf

1 5 2

A; B)].

62

I. Elementary Notions of Noncommutative Analysis

We transform the right-hand side of this relation in a similar way, by introducing yet another difference derivative, and obtain 1 2

2

1

3 4

2

\

5

2

4

/ ( Α , Β) - / ( A , Β) = A(B - B ) - ^ ( A , A; 5 , B), SxSy

or, by (i), 1

2

2

1

2,5 S2f

3 3,5

/ ( A , B ) - / ( A , B ) = A(B 3 3,5

1

oxdy

5 2 4 B)-j-(A,A;B,B).

2,5

We can now extract the linear factor A{B — B): 1 2

2

1

3 3,5

2,5

8^ f

1 5

/ ( Α , β) - / ( Α , Β) = I A ( 5 -

Α;

2 Β,

4 B).

8x8y

This is just the first variant of the index permutation formula since 3 3 5 - 2B)5 = BA - AB = [Β, A(B

A].

The proof of the second variant of the formula is left to the reader. The theorem is proved.

•

Remark 1.10 Let us point out (though this is trivial) that expressions of the form '1

«2

h

(A-A)f(B

}n u

...,B

n

)

ι

generally do not vanish if [2?, , Α] φ 0 for at least one I such that i'i < I < 12. Remark 1.11 By introducing additional operator arguments we find that 1 2 Μ ik 2 1 ii ik f(A, B,C\,.Cjt)

=

/ ( A , B,C\,..., 1,5

+

Ck) 1 2 1,25 1,75 i 1

ik

[Β, Α]δ? Β"; C , , . . . , Ck), 2(Α, Α; Β, *12
- d~x>~ Vx

1

·

64

I. Elementary Notions of Noncommutative Analysis

Since

~ihi]=ih>

[*· we see that ο . 1

=

/i Pix, ι

S2P

3 \

11 2

3 \ Γ a 8P' -ih— \+ih\ aχ ι idxSp

n

a

.,3 1

3

x,-ih—,-ih ox

— dx

(the last equality is due to the fact that 8f/8x(x, λ ) = fix)). ι 2 Let us again permute the operators —ihd/dx and x. We get n . 2 P\x,-ih—

=

dx ι

„ Λ P\x,-ih— I

d2P (\ he, dpdx \

3 \ \+ih dx I

(ihi

ι 3 a a x, —ih—, —ih — , —ih— dx dx dx

ΐ )

°

a -ih— dx

One can proceed with these manipulations and obtain the order changing formula modulo any power of h. Note that with this method we obtain an exact formula for the remainder. Example 1.9 (Product formula) Suppose that two Λ - 1 -pseudodifferential operators Ρ ^x, —ihd/dx^

and Q ^x, —ihd/dx^

are given. Let us compute their product. We

have

if 11

(*·-

ί Α

έ)

1

-

p

( * · - ' * £ )

e

(*· -

, k

h ) •

In order to reduce this expression to the standard ordering, we must permute the oper3 2 ators —ihd/dx and x. By Theorem 1.10, we have

IP

-ih-^

j n e

9 8P i , ( +[x, -'/i—]— dx dp \

-ih^

η

j j = Ρ

-ih^

j

Q

·, 3 .t d ) SQ f2 6 a . -ih—, ~ih— — \x,x, ~ih— . dx dx I dx 1 dx

-ihj-

3. Noncommutative Differential Calculus

65

Since [ ~

i h

i '

x

]

=

-

i h

'

we obtain ι ρ

(i

11 - i ^ M Q

•u8P

i

—i/i— op

1

u*

- i J

Ι x, —ih—, \ dx

ρ ζι

=

^

(

—in— I — dx J οχ

2

Α ]

6

I x,x, \

- J

(ι

β

.

—ih—

I . dx

Taking into account that >P

SP

-iJ^

l·

d P

-1 '

i

1

J

,

we get [ Ρ 11.

- i h ±

j n e (l.

3P / s

—ih— dp

3 \ iß (l 4

I x, —in— I — I dx j 8x

After permuting formula lP

2 χ

dp

and

,

. χ2

—in—

3

I .

dx

in the last term, we obtain the following product

—ihd/dx

j i = /> (ι

(i,

fx - i f c A ^ I

I x.x, I

e

3

, ι -iH^MQ

ih—

j , , , (i,

dx J

Λ

(ι dx

β [ι -ih±.

-ih—

I

9 \

2

(i 4 4

.,3

Obviously, this process can be continued so as to obtain the subsequent terms of the expansion. Remark 1.12 In both preceding examples the more elegant way to get the answer would be to use the left ordered representation of the tuple (λ, —ihd/dx) (see Chapter II).

66

I. Elementary Notions of Noncommutative Analysis

3.5 The Composite Function Formula In this subsection we will consider the problem of rewriting the composite function 12 12 / ( l £ ( A , 5)1) via functions of A, B, and their commutators. Recall that f(g(A, B)) 12 12 means ( / ο g)(A, Β), as distinct from /|[g(A, 5)J), which means " / ( C ) with C = 12 12 g(A, B)". We will derive a formula for /([[g(A, 5)]]) whose leading term coincides ι 2 with f(g(A, B)). First let us consider the relatively simple case in which g(x, y) = x

+

y.

Theorem 1.11 A function / ( I A +

/?]])

=

of the

sum

of two

1

2

3

f ( A

+

B )

+

operators

admits

the following

expansion:

3".

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

4

3

tively with Β and [A, B]:

3 . 5 6 1 2 1 4 3 ^ 4 6 1 2 1 5 [A, B]S2f(A + Β, A + Β, A + B) = [A, B]-±(A + B,A + B,A + B) öy* 4 4 3 6 δ /" 98 91 21 51 7 + [A, B][A, + Β, A + Β, A + Β, A + Β, A + B) 4 Sy

68

I. Elementary Notions of Noncommutative Analysis S2f

4

3

= [A, B]^r(A1

6 1

2 1

+ B,A

Sy

4 +[Λ,

[A, B]]^r(A Sy3

3

f

B][A,

+ B,A

+ B)

75

71

21

6

+ Β, A + Β, A + Β, A + B)

6

+[A,

5

4

98

91

21

51

7

B ] ^ j ( A + Β, A + Β, A + Β, A + Β, A +

B).

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 /(|A +

fll)

=

4

g2f

3

f ( A + B) + [A,B]-4r(A L

Sy

3

2 +[A,

1

Λ f [A,

6 I + B,A

3

1

2 1

5

+ B, A + B)

3

1

3

+ B, A + B, A + B)

. Λ4 f

2 +2\[[A +

3

1

3

1

3

1

3

1

3

1

+ B, A + B, A + B, A + B, A + B).

Sy4

2 3 4 In the second term on the right-hand side of this equation we permute Β with A and [A, fl], Similar calculations yield 1 /(ΙΙΛ

+ Β]])

3S

2

4

1

f { A + B) + {A,B}-+r{A dy·*·

3 1 + B,A

S3f

2

3 1

3

+ B, A + B)

1

+M.A, [A, fl]] + [[Α, Β],

3 1

3 1

3 1

3

+ B, A + B, A + B, A + B)

3 3 3 2 S4f 1 3 1 1 3 1 1 +3[[Λ + Ä] 2 3—4-(Λ + Β, A + B, A + B, A + B, A + B).

Sy*

The last formula proves the corollary, since S

k

f / S x

k

( x ^ •

•

Let us now proceed to the general case. T h e o r e m 1 . 1 2 The following

/(Ι*(λ

B)l) •So

composite

= 3

+ [ A , B]-2-(A, 8x2

function

formula

holds:

B)) 4

6

2

Β,

7

8 Λ2/1 1

1 2

2 8

A ; B ) - ± ( . l g ( A , B)J, Sy1

oxi

g(A,

7 8

Β),

Proof. We have

f(lg(A,

- f(g(A,

1 1 2 = (Ilg(A,

2 1 2 = (Ig(A,

h ) = f i g { A , B)J) - f{g(A, Xf 1 1 2 2 3 - g(A, B))^-dg(A, fl)l, g(A, B)) dy

2 3

4 5

if

1

B)1 - g(A, B))-f(lg(A, Sy

1 2

36 g(A,

B)).

B))

g(A,

B)).

3. Noncommutative Differential Calculus 3

2

69

12

By permuting the operators A and [g(A, fi)J in the last expression, we obtain

12 f(lg(A,

12

Β)Β - f(g(A,

3

3 1 2

45

Β)) = (|[g(A,

12

ι

1 2

+[A,

if 1 1 2

- g(A, B))-J-(lg(A, Sy

25

26

B)J, g(A,

Β))

4 5 Xg 2 4 5

Ä)J, g(A, Β), g(A,

*)·

Here the first term on the right-hand side vanishes, which can easily be proved by index manipulations (see Proposition 1.1)

12

3

Xa 1 2 4 OX 2

The theorem is proved.

•

Let us now apply the obtained formula to pseudodifferential operators. Example 1.10 Let function f{z) /(|[P 0(h2).

Ρ

be an h~l-pseudodifferential operator, and let a

^Jc, —ihd/dx^j

of a single variable ζ be given.

We will rewrite the operator

in the form of an h~1 -pseudodifferential operator modulo

—ihd/dx^J)

Due to Theorem 1.12 we have f ( l P Iχ,

5 +i[

I ) = ( / ο Ρ)

. , 3 — ih —, dx

SP x\— Sx

(4 6 . \x,x,—ih— 1

-ί A

L

d \ s P I — dx J Sp

A

is a I x, — in — 1 dx

Since [—ihd/dx, 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 — (χ,χ,ρ) = ox

dP(x,p) , dx

we get

JHP

I I

-

i h

dP 13 —ih— dx

± . j j ) = { f o P )

a W f

I x, —ih— I dx

I — I Sp

5

- i h A

a

I x, —ih —, \ dx

a —ih

— dx

70

I. Elementary Notions of Noncommutative Analysis

Computing up to terms of order A2, we can change arbitrarily the indices over operators in the second term on the right-hand side of the last formula. Hence,

f(V

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]] = [[a, b], c] + [b, [a, c]] (Jacobi identity). For our aims it suffices to assume that L is realized by square matrices of size m χ m, with the usual matrix multiplication1. 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. 'By Ado's theorem this is the general case.

4. The Campbell-Hausdorff Theorem and Dynkin'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 TeG 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(f)} be the one-parameter subgroup of G defined by the condition g(0) = X. Then we set exp(X) = g( 1). Thus we obtain a coordinate system in the vicinity of the neutral element e 6 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 - Β = A + B. If G is not commutative, there appear correction terms on the right-hand side of the last equation, namely, A · Β = A + Β + )-[A, B] + · · · Ζ

The first correction is equal to j [ 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. Ε. 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 \n(eAeB) in the form of an integral rather than a series. Then we use the conventional Fourier expansion so as to obtain the expression for \n(eAeB) 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

I. Elementary Notions of Noncommutative Analysis

Let A be an algebra and Β £ A some element. Denote by ad^ : A —• A the linear operator that takes each element A E A into its commutator with the element B : ADFL :

A

A

A

h* adß(A) =

[B,A].

Clearly, adß(A) : BA - AB = LB(A) - R ß (A), that is, ADß = Lß

—

RB,

where L Β and RB are the operators of left and right regular representations introduced in Subsection 2.3. The powers of ad5 give rise to multiple commutators with Β:

k times

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

[B, A] = BA-

l

AB = (Β -

2

B)A.

Similarly, ( a d B ) K ( A ) = (LB

- RB)K(A)

= (B -

B)KA,

according to the description of functions of the operators LB and Rß given in Subsection 2.3. Moreover, for any symbol fix) we have /(adß)(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 / suffices to consider the case fix) = (λ — jc) -1 . Let f ( x ) =

/ ( A ) is continuous, it

( k - x ) - \

where λ £ σ (Β)—σ(Β), that is, λ cannot be represented in the form λ = λχ—λ2, where λ, € er (Β), i = 1,2. Then the function (λ—x+;y) - 1 is analytic in the neighborhood of

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

73

3 t h e p r o d u c t a (Β) χσ (Β) c C , a n d h e n c e t h e o p e r a t o r (λ—Β+Β)

1

2 A is w e l l - d e f i n e d .

l

We have ί

I

A -A

(λ - a d ß ) \ λ - Β

=

4 0 ( λ - Β + Β)

+ Β 2 A

3 12 = (λ - Β + 5 ) [ [

)

τ

3 1 = ( λ - Β + Β)

λ - Β + Β

A i

Γ

λ - Β 2 Λ

+ Β

Τ

1

= Α

λ - Β + Β

( w e u s e d extraction of a linear f a c t o r a n d m o v i n g i n d i c e s apart). H e n c e t h e o p e r a t o r λ — ad β is t h e l e f t inverse of the operator

1

i .

3

1 λ - Β + Β

Similarly, it can b e p r o v e d that t h e operator λ — a d s is t h e right inverse. C o n s e q u e n t l y ,

/?x(adfi)(A) =

i

r

A.

λ - Β + Β

Β. The Fourier transform formula. Let 3

U(t)A

1

2 e ~ A. ,t(B B)

=

W e intend to s h o w that U(t) I n d e e d , let us d i f f e r e n t i a t e U(t)A d — U(t)A at

ei,adB.

by t. W e obtain

3 3 ι i(B - B)elt{B~B)A

1

2

=

1

=

3 4 0 2 it{B B) (B — B)le ~

2 A\I

=

3 1 2 i(B - B)U(t)A

M o r e o v e r , f o r t = 0 w e h a v e U(0)A U(0) W e s e e that U(t)

=

a n d el,adB

4 ο = i(B -

= i

3 1 2 B)ei,{B~B)A

adB(U(t)A).

= A, that is,

= id =

eitadB

f =o

satisfy the s a m e C a u c h y p r o b l e m a n d h e n c e are e q u a l ,

74

I. Elementary Notions of Noncommutative Analysis

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 € A are elements such that

Then Proof.

gadc

_ gadiJead,4

We have

e^'e^iD) But

=

312

eade(eA"AD)

=

3 4

3

=

4

eA+B

2

1 0

De~A~B.

10

Thus,

32

eadÄeadA(Z))

4 0 3 12

eB~BeA~AD

=

eCDg-C

1 =

eadC(D)

as desired.

4.3 A Closed Formula for In ( rij for any i and j, then ni nt pi pi ri rm / ( A i , . . . , Ak)g(B 1 , . . . , Bi, C i , . . . , Cm) = 0 for any generators Bv...,

Βl and Cv ...

,Cm.

(Γι) (Continuity axiom I). The mapping «ι

/(*,,...,Hf

n* f(A\,...,Ak)

is continuous.

η1 m (T2) (Continuity axiom II). If fa(A\,..., Ak) —• 0 and the indices p\,..., r\,... ,rm satisfy the conditions of axiom (μβ), then «1 n* fa(Ai,...,Ak)g(Bi for any generators Bl,...,B[

Pi

PI η B[,Ci,...,

and Cl,...,

Cm.

rm Cm)

pi,

0

Second, there is a group of techniques based on the naturality properties of the mapping symbol ι-* operator, as stated in Theorems 1.4 and 1.5. Actually, the main naturality property is that expressed by item 1° in the theorems cited, namely, that if ^; [LA, LB]

= [LA, a d s ] = [LA, an Ä ] =

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, Β an algebra of operators on W, Q an {A, B)bimodule of operators from W into V, and T> a (B, .4)-bimodule of operators from V into W (we do not exclude the case in which V = W and A = Β = Q =V). We may consider the algebra U of matrix operators

acting on V φ 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 Β € Β be generators. Then for any C G Q the operator C ^ is a generator in U, and Β

for any symbol f € T.

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

/A C \

Proof. The above formula clearly defines a continuous mapping Τ mapping takes 1 ι—1 and y ι-»· I ^

^

91

U. Next, this

J since Sy/Sy = 1. It remains to check that

the mapping is a homomorphism. We have

((iSMii)) 8(A) C&(A,B) = f /(A) c¥(A,B)){ + = ( f(A)g{A) 0 f(B)g(B)

|cf£(A,B)]*(5)

/(A)[c|*(A,Ä)l

The diagonal entries have the desired form; the superdiagonal entry can be transformed as follows:

B)lg(B) = CF(A,

2 p 3 1 2 t/ 3 1 / ( A ) | [ C / ( A , B)J + I [ C / ( A , oy oy

2

31

where

f(x)g(x) - f(x)g(y) + g(y)f(x) - g(y)f(y) 8(fg)f Sy

=

Sy

x

= ——(*,?)· Sy

y

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 we can write out this identity as ( a \ 0

b J \ 0

£ Λ where a and b are generators. In matrix form,

c W 0 J

or AC = where

= (o

t) -

I \ 0

c \ ( a 0 ) \ 0

d \ b )'

CA, c=(ö

ο)·

5),

92

I. Elementary Notions of Noncommutative Analysis

By Theorem 1.4, 2°, we obtain f(A)C

=

Cf(A)

for any symbol / , or, according to Lemma 1.4, m 0

V i m

c \

c

/ I

J v o o j - t o

0

0

f(b)

Multiplying the matrices on both sides of the last equation, we get 2 3/

3

1

8y

that is, we have obtained yet another proof of Theorem 1.6. (b) Our second example pertains to Lie algebras and employs η χ η rather than 2 x 2 matrices. Operators Al,..., An G Λ are said to form a representation of a Lie algebra if η [Aj,Ak}

=

J^kljkAi, 1=1

where kljk are 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

( Μ

\

Μ

0

\

Λ,· =

A =

0

( Aj

An

\

j

o

andA, = | | A ^ > / = 1 . With this notation, the Lie commutation relations take the form AAj = (Aj +

Aj)A,

which permits one to obtain various permutation formulas easily; for example, we have, by Theorem 1.4, 3° Af(Aj)

= f(Aj

+

Aj)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.

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

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: 2

- 8 f , \

*

Sy

a t

(Daletskii-Krein formula); N—l 2 2k fib f 1 2Jfc+l /(C) = / ( A ) + £ [ C - A ] | . . . | [ C - A ] ] - f ( A , . . . , A ) y

k = \

1 2 2Ν S f 3 ir/l Α ΤΙ IT Γ Ι Α "Π J / A +IC-AJ... |C-AJ^-(C,A,... N

f

2 N + 1 A

A )

(Newton formula); /W(A) / ( C )

=

/ ( A )

+^

£

1

L

A!

N — l

2 -

A

)

k

l

,

2 +

K

1 C

2 -

y N

8y

1

3

3

A , . . . ,

A )

(Taylor formula); 12 21 3 S2f 15 2 4 / ( A , Β) = / ( A , 5 ) + [5, A ] - — j — ( A , A; 2?) (Index permutation formula); /(IA +

fll)

=

/ ( A + B ) + [ A , Ä ] ^ ( [ A + B]|,iA + B]|,A +

fi)

1 2 3 52/ 5 1 2 1 4 / ( A + Ä ) + [A, B l - r y ( [ A + B ^ , A + B , A + Β),

/(|[*(A,fi)l) =

/(*(ÄJ)) 5

Λο

2 7 8

6 > ί

s2f

Λα 8 y

ι

1 2

3 4 6

2

2 8

Y y i

(Composite function formulas); / ( a d Ä ) ( A ) = f(B (functions of adß).

-

B)A

7 8

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 Η and two operators, a+ and a - , on Η satisfying the commutation relation [ α + , a~] = a + a ~ — a ~ a + = —I,

where I is the identity operator on Ή\ the operators a+ and a~ will be referred to as the creation and the annihilation

operator,

respectively.

1

For example, we can take Η = Z^iR ) and 3 \ = (2Αω)" 1 / 2 ( —ih— ± ϊωχ I, dx

J

where h and ω 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 Η is a representation of A in the 2 1 form of a function of the Feynman tuple (a+, a~), namely, 2 A =

ι +

f(a ,a-)

where f ( z , W) =

Σ α,β>0

c

aßZaU)ß

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 2 1 f(a ,a~)= +

c

oß(a+)a(a~)ß,

Σ α,β> 0

where the series is assumed to converge, say in the weak sense, on some dense subset of H). Given an operator A on H, 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 = A1A2, where A\ and A2 are already in normal form, 2 1 . Μ = fi(a , a~), +

2 1 A2 = fi(a , a~). +

In a particular case (A2 is arbitrary and A\ = 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) = ζ or f\(z, w) = w. For the former, we have a+if2(a+,

2 2 O l

3 2 1 2 2 1 2 1 = a+f2(a+, a~) = a+f2(a+, a~) = g(a+, a~),

where g(z, w) = zfi(,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 +

a~U2(a ,

1 3 ξχ• 4 2 a~) + a~, a+]^(a+, a+, Sz

1 3 2 1 1 2 + a")]] = a-f2(a , a~) = a~f2(a+,

1 a~)·

However, [a~,a+]

= I,

and so we can continue this chain of equations by identifying the first and the third arguments in 8f2/8z: a~U2(a+,

2 1 O I

1 2 1 = a~f2{a+, a~) +

9

f

3ζ

2

ι 2 1 a~) = h(a+, a~),

96

Π. Method of Ordered Representation

where 3/2 h(z, w) = wf2(z, w) - —— (z, tu). dz Denote by l+ and l~ the operators taking fa into g and h, respectively, i.e.,

oz Thus, we have 2 1 a+if(a+,a-)l 2

2 1 (l+f)(a+,a~),

= 1

2

a ~ l f { a + = (l~

1

f){a+,a~)·

The operators Z+ and l~ characterized by this property are called the left ordered 2 1 representation operators for the Feynman tuple (a+,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 J2b°'ßzawß-

f\(z, w) = C.ß Then Ι/ΐ(α+,α-)Μ/2(α+,0]| =

Y^ba,ß{a+)a{a~)ßlf2{a+,a~)J.

We can split off the operators a~ and a+ in the products (α+)α{α~γ one by one. As a result, each of them will be replaced by the corresponding operator l + or l~ acting on /2, and we obtain 2 1 2 1 Ι/ΐ(α+,ΟΜ/2(α+,α-)]]

=

2

=

2 1 f2){a+,a~)

Y^ba,ß{{l+)a(l~f ".β 1 +

2

1 +

[/ΐ(/ ,Π)/2](α ,0

(we do not discuss the convergence of the series). Thus, the symbol of the product 2 1 2 1 Ε/ι(α+,ΟΜ/2(α+,α-)1 can be calculated according to the following recipe:

(II.1)

97

1. Ordered Representation: Definition and Main Property

2 1 Take the left ordered representation operators l+ and l~ and substitute them into the symbol f\. Then apply the obtained operator to the symbol fz. 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.

1.2 Ordered Representation and Theorem on Products Let

1 Α =

η (Αι,...,Αη)

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 B = f(Ai

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 /(|[g(A)J). 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 Τ of unary symbols is fixed and T n = T ® n . 1 η Let A = ( A i , . . . , A„) be a Feynman tuple in an algebra A and suppose that each A, is an ^"-generator in A. ι η Definition II. 1 A Feynman tuple I = ( / i , . . . , / „ ) of operators lj '· Ffi -*• Tn,

j = 1,...,

n,

1 η is called the left ordered representation of the tuple A = ( A i , . . . , A n ) if the following two conditions hold: i) for any / 6 Tn and j = 1 , . . . , η we have AjUiAi,An)J

= ( l j f ) ( A u ...,

An);

Π. Method of Ordered Representation

98 ii)

if a symbol / e Tn does not depend on yj+l ,...,yn,f

= f ( y

y j ) , then

l

(II.2)

l j f = yjf Condition ii) will be referred to as the regularity condition. condition is quite natural; indeed, 1

AjlfiAi

7

7+1

ι

Aj)1 = A jf{A\,...,

7

7

In view of i), this

ι

Aj) = Ajf{A\,...,

7

Aj),

so that the symbol of this product can be chosen equal to yj f (y l,..., y.), and condition ii) simply says that the operators of left regular representation are consistent with this natural choice 1 . We intend to establish a theorem generalizing (II. 1) to the "abstract" situation of Definition II. 1. Since we have no reason to assume that it is possible to insert the operators Lj into arbitrary symbols / e J-n, we need to be extremely careful in our statement. Let Τ C Τ be a continuously embedded subalgebra (we do not exclude the case Τ = Τ). Assume that the operators Lj are ^"-generators in C(Tn)· Such a subalgebra Τ can always be found; in the worst case, it still contains all polynomials. Clearly, Tn = is a continuously embedded subspace in the symbol space Tn. Theorem II. 1 Under the above assumptions any product U(Ai

A„)l E g ( I i , . . . , A n ) ] | f

where f G Tn and g € Tn, can be reduced to normal form. Namely, U(Al, · · · , in)l l g ( A i A

n

) \ I = [/(/!,

...,

n

in)g] (Al

An).

Proof. In the following we will widely use the more convenient notation f(A) 1 η f(A\,...,An)\ with this notation, the preceding equation reads

for

I/(A)Ms(A)l = [/(/)*](A)f or even /(A) ο g(A) = / ( / ) 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 1

It is essential to impose this requirement since the uniqueness of the symbol of a given operator is not assumed anywhere.

1. Ordered Representation: Definition and Main Property

Τη

99

A

h

LAJ

μ

A,

ι

where μ = μ a is the linear mapping sending each symbol f(x l,... ,xn) into the correl η sponding operator f(A i , A LAJΒ

= AjB

for any Β e

n

) and LAJ is the operator of left regular representation,

A.

The operators lj and LAJ are ^"-generators in C{Tn) and C(A), respectively, and the continuous mapping μ intertwines the tuples / = ( / j , . . . , /„) and LA = (L A j , . . . , L A J , as expressed by the diagram. By Theorem 1.4, we have

for any symbol / € T n . Let us apply both sides of the last equation to an arbitrary symbol We obtain f(LA)(g(A))

=

(f(l)g)(A),

whence the statement of the theorem follows immediately, since f(LA)(g(A))

=

f(A)g(A)

(see Subsection 2.3). The proof is complete.

•

1.3 Reduction to Normal Form We now consider the case Τ = Τ. In this case the statement of Theorem II. 1 can be given the following interpretation. We introduce a bilinear operation * on T n by setting f*g

=

fd)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 II. 1 states that the mapping μ : (5"n,*)

A

(where A is equipped with the usual operator multiplication) is an algebra homomorphism. Next, assuming that Τ = Τ , we can give a somewhat more elegant statement of condition ii) in Definition II. 1.

100

Π. Method of Ordered Representation

Lemma II. 1 If all lj are Τ-generators in Tn, then condition ii) in Definition II. 1 is equivalent to the following condition: ii') for any g G Tn one has l

η

g(h,...,ln)l

= g(yx,...,

yn),

where the operator on the left is applied to the symbol identically equal to 1. Proof ii')

ii). Let f{yl,...,

y^ € Tn\ set •••,yj)

= yjfiyv

.·.,yj)·

By condition ii'), we have yjf(yi,---,yj)

=

W

=

i)i

=

ij(f(h,.-.,ij)i) = ijf(yl,...,yj),

the last equality also due to ii')· ii) => ii'). It suffices to prove ii') for decomposable symbols of the form g(yv^y

n

)=

gi(yi)---8n(yn)·

Condition ii) implies that the subspace Tj C Tn consisting of symbols independent of yj+ρ • ••

where the horizontal arrows are the embeddings. Furthermore, lj is an ^"-generator in T n by assumption and yj is an ^-generator in F j (for any / € Τ the corresponding function of yj is merely the operator of multiplication by f(yj)). Hence the embedding Tj Tn intertwines the ^"-generators yj and lj, and by Theorem 1.4 we have /(/,) L for any / e Τ .

=fiyj)

1. Ordered Representation: Definition and Main Property We now apply this identity successively for j g(il

?„) 1

101

1,2,...,

=

n,

and obtain

=

*„(/„)...*2(/2)*l(/l)l

=

gndn)

• · • g l ( h ) g \ ( y \ )

=

gnVn)

· • •g 3 ( h ) g i ( y 2 ) g i ( y i )

=

g ( y i > - - . , y

) ·

n

This equation extends by continuity to the entire symbol space T n . The lemma is proved. • We have considered the problem of reduction to normal form for the product Cι C2 of two operators in normal form. Let us now consider the general case. Let i j , . . . , i and j , . . . , j be two sequences of indices such that ji φ j whenever [ A j , Aj ] φ 0. Let