QUANTUM THEORY, DEFORMATION AND INTEGRABILITY


QUANTUM THEORY, DEFORMATION AND INTEGRABILITY

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NORTH-HOLLAND MATHEMATICS STUDIES 186
(Continuation of the Notas de Matem&tica)

Editor: Saul LUBKIN
University of Rochester
New York, U.S.A.

2000
ELSEVIER
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Q U A N T U M THEORY, DEFORMATION
AN D I NTEGRABILITY

Robert CARROLL

University of Illinois, Urbana, IL 61801

2000
ELSEVIER
Amsterdam

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Oxford

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Contents
Preface

ix

1 QUANTIZATION AND INTEGRABILITY
1.1 ALGEBRAIC AND GEOMETRIC METHODS . . . . . . . . . . . . . . . . .
1.1.1 Remarks on integrability .classical KP . . . . . . . . . . . . . . . . .
1.1.2 Dispersonless theory for KdV . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Toda and dToda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Remarks on integrability . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.5 Hirota bilinear difference equation . . . . . . . . . . . . . . . . . . . .
1.1.6 Quantization and integrability . . . . . . . . . . . . . . . . . . . . . .
1.2 VERTEX OPERATORS AND COHERENT STATES . . . . . . . . . . . . .
1.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Heuristics for QM and vertex operators . . . . . . . . . . . . . . . . .
1.2.3 Connections to ( X ,$J) duality . . . . . . . . . . . . . . . . . . . . . . .
1.3 REMARKS ON THE OLAVO THEORY . . . . . . . . . . . . . . . . . . . .
1.4 TRAJECTORY REPRESENTATIONS . . . . . . . . . . . . . . . . . . . . .
1.4.1 The Faraggi-Matone theory . . . . . . . . . . . . . . . . . . . . . . . .
1.5 MISCELLANEOUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Variations on Weyl-Wigner . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Hydrodynamics and Fisher information . . . . . . . . . . . . . . . . .

1
1
4
9
11
18

2

GEOMETRY AND EMBEDDING
2.1 CURVES AND SURFACES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 The role of constraints . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Surface evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 On the embedding of strings . . . . . . . . . . . . . . . . . . . . . . .
2.2 SURFACES IN R3 AND CONFORMAL IMMERSION . . . . . . . . . . . .
2.2.1 Comments on geometry and gravity . . . . . . . . . . . . . . . . . . .

2.2.2 Formulas and relations . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 QUANTUM MECHANICS ON EMBEDDED OBJECTS . . . . . . . . . . .
2.3.1 Thin elastic rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Dirac field on the rod . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 The anomaly in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 WILLMORE SURFACES, STRINGS, AND DIRAC . . . . . . . . . . . . . .
2.4.1 One loop effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 CMC surfaces and Dirac . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Immersion anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Quantized extrinsic string . . . . . . . . . . . . . . . . . . . . . . . . .
V

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19
21
23
23
31
36
45
49
50
56
5G
57

63
63
63


67
68
72
73
79
80
86
87
88
95
98
98
100
104
108


CONTENTS

vi

2.5 CONFORMAL MAPS AND CURVES . . . . . . . . . . . . . . . . . . . . . .

110

3 CLASSICAL AND QUANTUM INTEGRABILITY
113
3.1 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113

114
3.1.1 Classical and quantum systems . . . . . . . . . . . . . . . . . . . . . .
3.2 R MATRICES AND PL STRUCTURES . . . . . . . . . . . . . . . . . . . . .
119
3.3 QUANTIZATION AND QUANTUM GROUPS . . . . . . . . . . . . . . . . . 124
128
3.3.1 Quantum matrix algebras . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Quantized enveloping algebras . . . . . . . . . . . . . . . . . . . . . . 129
130
3.4 ALGEBRAIC BETHE ANSATZ . . . . . . . . . . . . . . . . . . . . . . . . .
136
3.5 SEPARATION OF VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . .
138
3.5.1 r-matrix formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
141
3.5.3 XXX spin chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 HIROTA EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
143
3.7 SOV AND HITCHIN SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . .
147
149
3.8 DEFORMATION QUANTIZATION . . . . . . . . . . . . . . . . . . . . . . .
151
3.8.1 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.2 Nambu mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
3.9 MISCELLANEOUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.9.1 Geometric quantization and Moyal . . . . . . . . . . . . . . . . . . . .
160
164

3.10 SUMMARY REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 DISCRETE GEOMETRY AND MOYAL

4.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Phase space discretization . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Discretization and K P . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Discrete surfaces and K P . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Multidimensional
quadrilateral lattices . . . . . . . . . . . . . . . . . .
4.1.5 d methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 HIROTA, STRINGS, AND DISCRETE SURFACES . . . . . . . . . . . . . .
4.2.1 Some stringy connections . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Discrete surfaces and Hirota . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 More on HBDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Relations to Moyal (expansion) . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Further enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.6 Matrix models and Moyal . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.7 Berezin star product and path integrals . . . . . . . . . . . . . . . . .
4.3 A FEW SUMMARY REMARKS . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Equations and ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 MORE ON PHASE SPACE DISCRETIZATION . . . . . . . . . . . . . . . .
4.4.1 Review of Moyal-Weyl-Wigner . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Various forms for difference operators . . . . . . . . . . . . . . . . . .
4.4.3 More on discrete phase spaces . . . . . . . . . . . . . . . . . . . . . . .

167

167
167
171


175
178
188
194
195
198
201
204
212
221
223
227
227
234
236
240
246

5 WHITHAM THEORY
255
5.1 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
255
5.1.1 Riemann surfaces and BA functions . . . . . . . . . . . . . . . . . . . 256
257
5.1.2 Hyperelliptic averaging . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.2

5.3

5.4

5.5
5.6

vii

5.1.3 Averaging with $J*+ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Dispersionless KP . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ISOMONODROMY PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 JMMS equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Gaudin model and KZB equations . . . . . . . . . . . . . . . . . . . .
5.2.3 Isomonodromy and Hitchin systems . . . . . . . . . . . . . . . . . . .
WHITHAM AND SEIBERG-WITTEN . . . . . . . . . . . . . . . . . . . . .
5.3.1 Basic variables and equations . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Other points of view . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SOFT SUSY BREAKING AND WHITHAM . . . . . . . . . . . . . . . . . .
5.4.1 Remarks on susy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Soft susy breaking and spurion fields . . . . . . . . . . . . . . . . . . .
RENORMALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Heuristic coupling space geometry . . . . . . . . . . . . . . . . . . . .
WHITHAM, W D W , AND PICARD-FUCHS . . . . . . . . . . . . . . . . . .
5.6.1 ADE and LG approach . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.2 F’robenius algebras and manifolds . . . . . . . . . . . . . . . . . . . . .

5.6.3 Witten-Dijkgraaf-Verlinde-Verlinde
(WDVV) equations . . . . . . . .

2G2
264
270
275
279
283
293
294
300
302
302
306
309
309
314
314
319
321

6 GEOMETRY AND DEFORMATION QUANTIZATION
325
6.1 NONCOMMUTATIVE GEOMETRY . . . . . . . . . . . . . . . . . . . . . .
325
6.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
326
329
6.1.2 Spectral triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1.3 The noncommutative integral . . . . . . . . . . . . . . . . . . . . . . .
332
6.1.4 Quantization and the tangent groupoid . . . . . . . . . . . . . . . . . 335
6.2 GAUGE THEORIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
341
6.2.1 Background on noncommutative gauge theory . . . . . . . . . . . . . . 341
343
6.2.2 The Weyl bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.3 Noncommutative gauge theories . . . . . . . . . . . . . . . . . . . . . 349
353
6.2.4 A broader picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
359
6.3 BEREZIN TOEPLITZ QUANTIZATION . . . . . . . . . . . . . . . . . . . .
Bibliography

363

Index

401

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Preface

Some years ago I wrote a book [143] called Mathematical Physics (not my choice of title)
which was designed mainly to be of service in teaching courses in this area to students
from engineering backgrounds. It also included some original work on inverse scattering
and transmutation following [144, 145, 146] (cf. also [147], which was aimed in somewhat
different directions). These books all contained an enormous amount of information in a
limited number of pages and the results were at times difficult to read. It has been said
that I tend to write for myself and in particular these books exhibit what I needed in order
to lecture on the subjects in question without recourse to outside reference, thus making it
easy to teach from them. Also I tried to make the books self contained and rich enough in
content to justify their purchase. I often attempt to establish meaning through computation
and via relations and analogy; this often requires working through the calculations in order
to find and appreciate the meaning. At the present time I am retired from teaching and my
motivation in writing is primarily to learn and understand, which is partially accomplished
by organizing and connecting various subjects, with occasional contributions inserted as
they may occur. The hope in this book is to produce a vdhicle to greater understanding
and a stimulus-guide for further research. We do not believe it is possible to include all of
the relevant background mathematics and leave room for anything else. Thus we will not
even try to develop mathematical topics axiomatically and will adopt style, definitions, and
notation from reference sources we have found to be especially illuminating (such references
are given as we go along). We feel that mathematics is basically easy (but very complicated)
and can be self taught (and often developed) quickly, but physics is by nature exploratory
and philosophical, and intuition therein seems to require more time (or perhaps discovery
of the appropriate mathematical setting). One sees in many places where a few physical
assumptions (possibly incorrect as such or suspicious) lead nevertheless to a mathematical
thread of reasoning of some elegance and beauty whose consequences are believable. In
such cases we do not probe the physical assumptions too carefully but suggest that perhaps the assumptions are more or less correct or, if not, perhaps other related assumptions
and/or techniques might lead to similar mathematical conclusions. Thus the mathematical
framework is given priority here. The theme of quantization via deformation of algebras is
emphasized throughout as are connections to integrability. Some material from my publications is worked into the text and references to publications before 1991 can be found in
the books cited (cf. also a home page listing). We

have adopted a writing style that lies close to the discussion format of physics (although
there is an occasional S U M M A R Y x.y remark which could often be regarded as a theorem); theorems are generally worked into the text either before or after a proof. This is
largely experimental at first writing and will probably not be easily describable in any event;
the main point seems to be to know what not to say. As models of good writing we think
immediately of [42, 54, 486] which are rich in verbal imagery. With the mathematics we try
to avoid excessive generality which might result in axiomatic "trash" without any visible

ix

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connection to the physics. It seems desirable to discover and develop those mathematical
structures which capture some physical behavior but it would seem silly (or premature) at
the present state of knowledge to identify "physics" with appropriate generalizations of such
structures (despite maxims to the effect that symmetry is the driving engine for physics, and
the frequent temptation to say that mathematics is physics - or perhaps nature). In this
spirit one might discover mathematics via physics as has occured at various times in history
(see e.g. [220] involving a "derivation" of K theory from M theory). In any event at the
moment it is of primary concern for me at least to understand what is quantum and what
is classical and how all this is logically connected (as one perhaps hopes it must be - only
a coherent emergent reality is requested, not an a priori given structure or grand design).
We will make no attempt to describe all the marvels now unfolding which relate to string
(M) theory, quantum gravity, noncommutative geometry, etc. (many of which are still in
a rather embryonic state) but will try to keep in touch with some of this. Some themes
will be pursued more diligently than others and some historical matters will be discussed
at some length because of their rich content and interaction with other areas. The pace of
development today is so rapid that one feels at times privileged to serve even in a reportorial
role.


About 4 years ago a prominent string theorist was quoted as saying that it might be
possible to understand quantum mechanics by the year 2000. Sometimes new mathematical
developments make such understanding appear possible and even close but on the other
hand increasing lack of experimental verification make it seem to be further distant. In any
event one seems to arrive at new revolutions in physics and mathematics every year. I hope
to be able to convey some of the excitement of this period here but will adopt a relatively
pedestrian approach designed to illuminate the relations between quantum and classical.
There will be some discussion of philosophical matters such as measurement, uncertainty,
decoherence, etc. but philosophy will not be emphasized; generally we want to enjoy the
fruits of computation based on the operator formulation of QM and quantum field theory. In
Chapter 1 connections of QM to deterministic behavior are exhibited in the trajectory representations of Faraggi-Matone [90, 315, 317] (these papers give a deep theoretical foundation
for such approaches and we have made some contact in [153, 155, 169]). Chapter 1 also
includes a review of KP theory and some preliminary remarks on coherent states, density
matrices, etc. and more on deterministic theory. We develop in Chapter 4 relations between
quantization and integrability based on Moyal brackets, discretizations, KP, strings, and
Hirota formulas and in Chapter 2 we study the QM of embedded curves and surfaces illustrating some QM effects of geometry. Chapter 3 is on quantum integrable systems, quantum
groups, and modern deformation quantization. Chapter 5 involves the Whitham equations
in various roles mediating between QM and classical behavior. In particular, connections to
Seiberg-Witten theory (arising in iV" = 2 supersymmetric (susy) Yang-Mills (YM) theory)
are discussed and and we would still like to understand more deeply what is going on (cf.
[148, 149, 150, 151, 152, 154] for some forays in this direction which often indicate some
things I don't know and would like to clarify). Thus in Chapter 5 we will try to give some
conceptual background for susy, gauge theories, renormalization, etc. from both a physical
and mathematical point of view. In Chapter 6 we continue the deformation quantization
theme by exhibiting material based on and related to noncommutative geometry and gauge
theory.
We would also like to say a few words about pedagogy in the information age. There is
so much new material arising today in mathematics and physics, much of it on electronic
bulletin boards, that students may find it hard to cope with, let alone master, anything. In


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xi
1995-1997 for example, before retiring from teaching, I found that, with a small group of
thesis level students (from mathematics, physics, and engineering departments), it was possible in a course on mathematical physics to build some foundations and then to go directly
to the net where I would select current papers and work through them in class. I also wrote
out lecture notes from such papers, some of which are incorporated into this book, and this
accounts in part for the style of the book. I believe that what made this approach possible.
and reasonably successful, was the presence in class of two or three students with ample
chutzpah and sufficient understanding to push me to explain (when I could) so that everyone
benefits. This is probably a classical recipe for transmission and expansion of knowledge of
course but in any case I am pleased to report that it seems to work at a distance using the
net. In any event the net has made it possible for people in the provinces to be much closer
to the main centers of research activity and one can use this material in a pedagogical spirit
as indicated, in addition to enhancing personal research activity.
A few musings may not perhaps be inappropriate. Thus we do not present here a coherent deductive structure of any physical theory. Rather, topics have been selected to describe
many interacting aspects of quantum mechanics, integrability, noncommutative geometry.
deformation quantization, classical mechanics, string theory, etc., for which any complete
physical theory must account. There are many ideas and themes with general features such
as tau functions, Hirota formulas, prepotentials, deformations, gauge transformations, index theorems, scaling, discretization, extremal principles, analyticity, scattering, moduli.
and various algebraic structures acting as glue and language. It is in this spirit that we suggest the book's possible usefulness as a guide and stimulus for further research. One might
inquire into the development of discretization methods with deformations as a supplement to
noncommutative geometry for example (cf. however Section 4.4.3). Another theme possibly
devolves from the appearance of K theory and noncommutative algebraic geometry in recent
work on string, brane, and M theory (and from related ideas in noncommutative geometry);
namely a lot of apparently fundamental physics seems to be intimately connected with basic
mathematical objects such as natural numbers and various structures on finite (or infinite)
sets, albeit via category theory, schemes, modular functors, etc. Again discretization and
combinatorics, along with appropriate algebraic structure and deformations thereof, seem

to arise naturally, leading one to question calculus as an appropriate directive language for
quantum mechanics (or nanotechnology). One remarks also that discretization is compatible
with digital computing and algorithmic thinking which implements such algebra; however
we hope that the converse does not become prevalent in the guise of channeling discovery
through computability (we omit quantum computation from discussion here since it remains
to be realized and its realization should involve a lot of discovery).

I have given many invited talks (in Australia, Brazil, Canada, China (PRC), England,
France, Germany, Greece, India, Italy, Japan, Mexico, The Netherlands, Russia, Scotland.
South Africa, Taiwan, Turkey and the USA) and would like to acknowledge fruitful conversation and/or correspondence with many people. A list of names would seem excessive
so let me simply express thanks for opportunities and information. I am also very grateful
to my wife and traveling companion Joan for over 21 years of love, support, patience, and
rationality. The book is dedicated to grandchildren: Bradley, Christopher, Emilee, Geoffrey,
and Jonathan from Joan's side and Annette and Katherine from mine.

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Chapter 1

QUANTIZATION AND
INTEGRABILITY
1.1

ALGEBRAIC

AND GEOMETRIC


METHODS

Notations such as (15A), (16A), ( 1 S t ) , and (21A) are used. We begin with a more or
less classical treatment of some main features in the Moyal-Weyl-Wigner (MWW) theory.
More general material on deformation quantization appeaars in Section 3.8 for example.
We refer here to [12, 18, 31, 68, 186, 138, 218, 404, 410, 458, 522, 536, 537, 538, 539,
540, 611, 647, 750, 753, 931, 901, 987, 997] for perspective and will follow here mainly
[192, 203, 204, 308, 310, 311, 312, 313, 343, 638, 781, 930]. We will be essentially formal
here and exhibit various formulas from the classical theory without regard for domains of
validity; proofs will often be sketched, deferred, or omitted. Thus following [203, 204] for
SchrSdinger wave functions ~b(x), Wigner functions (WF) are defined via
1 /

f(x,p) = -~

( h )

dye2* x -

7y

When r is an energy eigenfunction (He = Er
H, f = f ,H=Ef;

f ( x , p ) 9g(x,p) - f

exp(-iyp)r

( h )


(1.1)

x + 7y

then f satisfies

,~exp

(O~Op-

OpO~

;

(1.2)

( x + -~irz~COp,p- -~i~0 x ) g(x,p)

The WF's are real and satisfy -(2/h) _< f < (2/h); they may be negative. Time dependence
for WF's is expressed via
i h O t f ( x , p , t ) = H 9f ( x , p , t ) - f ( x , p , t ) , H

(~.3)

As h ~ 0 this reduces to O t f + {f, H} = 0 (standard P=Poisson bracket). The 9exponential
is determined via
V . ( x , p , t ) = exp.

= 1 + -~g(x,p) +


H. H +...

(1.4)

and generally one has
f(x,p,t) = U.l(x,p,t)

9f ( x , p , O ) 9U . ( x , p , t )

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(1.5)


2

CHAPTER 1. Q UANTIZATION AND INTEGRABILITY

The dynamical variables evolve classically via

=

x,H-H,x
ih

= OpH; p =

p,H-H,p
ih


= -OxH

(1.6)

so that the quantum evolutions (16A) x(t) = U, 9x , U, 1 and p(t) = U, , p , U, 1 turn out
to flow along classical trajectories. We note also that given an operator A(2, i5) one has
< .4 > = / / d x d p f ( x , p ) A ( x , p ) w h e n

/ /dxdpf(x,p)=l

(1.7)

Now one can always write (for suitable f ) (16B) f(x,p) = f f dadb]exp(iax)exp(ibp) and
further exp(iax)exp(ibp) = exp,(iax)exp,(ibp) with

9 = exp, [X + - ~i~ -'~p] ez,bp
9 = e _iax
_iax 9ez,bp
e,
, ei,bPe_ihab/2

(1.8)

Consequently any W F can be written as

f (x, p) = f ] dadb](a , b)eihab/2 e,iax e,ibp

(1.9)


so that by inserting U, 9U,1 pairs

f ( x , p , t ) - / f dadbeiabh/2eiau*l*x*U* *e ibU*l*p*U* -

-- / /dadb](a, b)eiabh/2eiax(-t) , ei,bp(-t)

(1.10)

The steps cannot generally be reversed to yield an integrand ](a, b)exp(iax(-t))exp(ibp(-t)).
Consider now a generalized form of this from [278, 930] where the motivation arises in
quantum statistical mechanics with kinetic equations of Lindblad type. The classical picture
for phase space functions A(q, p) is
operators

matrix elements

position (t
< xl~lx' > = ~ ( x - ~')
momentum
< xl~l~' > = -i~Ox~(X- x')
general fI
< xlAIx' >= A(x,x')
density matrix ~ < xl~lx' > = p(x,x') =< x,x'l~ >

(1.11)

with Fourier transforms

1/ dqdpe-i(rlq+(P)A(q,p); A(q,p) = ~1/ drld~ei(rlq+(P)ft(rl,~)


.7t(rl,~) -- ~

One will write z = (q,p), ~, = (0,/5), and o" = (rl,~).
defined via

(1.12)

The Weyl transform of A(q,p) is

flw" A(q, p)--, •w(A)= fi = 1 f daft(a)ei~ ~

(1.13)

and the Wigner transform of .4 is

W" A ~ W(A) = A(q, p) = 2 / dte -pt/2 < q -

tlAlq + t >

(1.14)

From the identities

e i(v4+~) -- ei~O/2ei~peiT?O/2;

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(1.15)



1.1. A L G E B R A I C A N D G E O M E T R I C M E T H O D S

3

< xlei('70+~P)lx' > = ei'7(x-it~)5(x ' - x + 27ri#~)
(for # = ih/2) one obtains (16C) W(f~w(A)) = A and f~w(W(J)) = J so W ~ f~wI From
(1.13) one has the Weyl ordering

n(:)

aw(qnp m) = ~

ok~mo n-k

(1.16)

0
whereas (1.14) implies that for any t5 and J (16D) T r ( ~ J ) = 1 f pAdpdq so p ~ ~wl(t))
and A = f~wl(j.). In fact p/27rh is the Wigner function associated with the density matrix
~. The 9 product is defined now via
1

1

f * g = f~wl(f~w(f)~w(g)); { f , g } u = -~#[f,g]M = -~p(f * g -- g * f)

(1.17)

where { , }M Moyal bracket. The corresponding phase space formulation of quantum statistical mechanics follows from the above via (16D). Thus (z* = 5 and flwl(A) = Aw(q,p))
~


9 (i) J = J t

~

Aw=A~

9 Tr(A)= 1 r

f Awdqdp= 27rh

. fl;~(Ok) = qk, ~ ; l ( $ k ) = pk
9 If J = ~ = Ir > < @l is a pure state then

/,.(q..)n. = 2nlr

(1.18)

=

A substantial and natural generalization of (I.13), which includes many useful operator
orderings distinct from (I.16), arises from

1/

f~" A(q,p) - , ~t(A) = ~

dafl(a)J(a)e i ~

(1.19)


One assumes the weight function fl to be an entire analytic function with no zeros. If e.g.

A(q,p) - qnpm then (1.19) gives an ordering different from (1.16). Setting ( 1 6 E ) w ( z ) (1/2~) f daexp(iaz)(1/fl(a)) the inverse ~t transformation exists and is given by
fl-1.

1
j __. A(q, p) = -~
f dq'dp'dtw(q - q', p - p')e -(p't/t~) < q' - tlA]q' + t >

(1.20)

The proof is straightforward and by (1.14) one obtains

fl-l(J)

1 /

dq'dp'w(q-

= ~

q,

1

, p - p ' ) A w ( q ' , p ' ) = -~-~w@ Aw

(1.21)


where | ~ convolution (in fact any linear transformation of quantum operators to phase
space functions which is phase space translation invariant is of this form). Consequently
f | g = (27rfg) and (16F)
= A(q,p) =~ ft(a) = (1/ft(a))ftw(a) with

ft-l(J)
I

- 2 i # / dqe -i~q < q + ip~lftlq - i#~ >

(1.22)

This leads then to

f~-l(.j,.)

= _itt

dq, drld~

ei[~(q-q')+(p]

fl(~,~)

< q, + i#~lJlq'

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ip~ >


(1.23)


4

CHAPTER 1. Q UANTIZATION AND INTEGRABILITY

We obtain then in analogy to (1.17) a nonabelian associative algebra structure
f *n g = fl-l(fl(f)fl(g));

1
1
~_~[f, g] = ~_~(f *a g - g *n f)

(1.24)

This is called a generalized Moyal product corresponding to the associated generalized Weyl
and Wigner transformations.
Now

(1.24)

can be given an explicit form as follows. From (1.19) one has

1 fdada,f(a)~(a,)f~(a)n(a,)e(i~)e(i~,~)
f~(f)f~(g) = (27r)2

(1.25)

Using (1.15) one obtains (16G) exp(ia~,)exp(ia'~,) = exp[p(a' A a)]exp[i(a + or')2] where

a' A a = (~', ~') A (rl, ~) = rl'~ -- ~?~'. This leads to

(f , a g)(z) = (2 1)2 f dada'](a)[7(a')f~(a)f~(a')eU(G'A~)f~-l[ei(a+~')~]

(1.26)

However by (1.19) (16H) f~[exp(iaz)] = exp(ia~,)f~(a) and hence

(f .~ g)(z) =

1 / dada' f(a)(?(a') f~(a)f~(a') eU(a,A~)ei(a+a,)z
(2 )2
+

(1.27)

Some further manipulation gives also
--21 g](z) = 1 f dada' f~(a)a(a') sinh[#(a' A a)]
2 , [f'
(27r) 2
a(o" + a')
#

(1.28)

One can then show that from (16I) Uf(z) = f do~(o)f(o')exp(ioz) there follows U(f,ng) =
Uf 9Ug. This shows that the 9 and ,n products define isomorphic algebras. In fact if in
(1.27) the kernel [f~(a)f~(a')/f~(cr+~')]exp[p(a'Aa)] is replaced by B(a, a'), and associativity
is required, then (1.27) is the only possibility; this is related to the uniqueness of the Moyal
algebra (cf. [343]). In particular one has now (16J) (1/2#) [f, g] = U -1 ((1/2#)[UI, Ug]M)

and some further calculation shows that if f~ is # independent then the Moyal bracket is the
only *n product whose associated Lie bracket tends to the P bracket as # ~ 0. We note
finally the generalization of (16D) in the form

Tr(a(f)a(g)) = ~a(O)
h f f *a gdz
1.1.1

Remarks

on integrability-

(1.29)

classical KP

We will be concerned with various aspects of classical and quantum integrability with special
interest in relations between dispersionless hierarchies and deformation quantization. Some
references of interest here are in particular [22, 134, 158, 160, 175, 308, 309, 374, 409, 422,
423, 424, 536, 537, 615, 617, 695, 696, 752, 891, 892, 902, 910, 911, 912, 913, 973] (let us
cite also [57, 82, 127, 481, 527, 528, 623, 649]). We go first to a review of dKP following
[158, 160, 161, 555, 902]. A brief sketch of this appears in Section 5.1.4 but we want to
include now the "twistor" formulation of Takasaki-Takebe (cf. [160, 902, 911,912]), a more
detailed examination of the dKdV situation as in [160, 153], and a treatment of dispersionless
Toda (dToda) as in [32, 87, 760, 902, 911,916]. Thus there will be some overlap with Section
5.1.4 but each section is self contained.

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1.1. A L G E B R A I C A N D G E O M E T R I C M E T H O D S

5

We follow here [147, 163] at first and simply list a number of formulas arising in KP
theory; the philosophy is discussed at greater length in [147] for example or in other books
such as [222, 558] (cf. also [450, 639]). One begins with a Lax operator L
0-~-U20-1 -~0 + F_,~ Un+l O-n (with u2 = u) and a gauge operator P = 1 + E ~ w~ O-n determined via
L = P O P -1 where /) ~ cox. For wave functions ~ = Pexp(~) with ~ = E ~ Xn An where
x ,,~ Xl one has ( . ) L r = )~r with Om~2 = B m r for B m = L'~ (here + ~ En>0 and
"~ ~ n < 0 in an obvious notation where ~ means "corresponds to"). One knows that
(8) OnBm- OmBn = [Bn, Bm] (Zakharov-Shabat (ZS)equations) and O n L - [Bn, L] (Lax
equations). The symbol 0 "1 can mean e.g. f__x , _ fxoo, or ( 1 / 2 ) ( f x ~ - fx~ in particular
contexts but generally we think of it algebraically as simply f. In particular
=

.

.

.

.

-

B2=0 2+2u;

B3=0 a+3u0+~ux+


0 -102u;

03u = 403u + 3uux + 3 0 - l O ~ u

(1.30)

The last equation is the KP equation for 02 "~ Oy and 03 ~ Or. One writes also (t b) ~P* =
( P * ) - l e x p ( - ~ ) with L*r = Ar and One* = - B n r
where B n = (L*)+ n and 0* ~ - 0 .
The operators Bn can also be considered as arising from "dressing" procedures
(On- Bn)P-

P(On

-on);

(i)nP)P -1 = B n -

p i ) n p -1 = - L n_

(1.31)

(cf. [147, 805]) and the equation ( . . ) OuR = -Ln___P is called the Sato equation. More generally, following [559], given differential operators A, B in c9 one considers flows (&&) OAP =
-(PAp-1)_P
or equivalently for W = p - l , OAW -- W ( W - 1 A W ) - .
Here OA ~ i)/OXA
and one thinks of L and P as fixed with the coefficients wi of P determined via the u~ in L
(cf. [147, 701]). Thus define LA = P A P -1 and note that OAPP -1 + Pi)AP -1 - 0 to obtain
i)AL B = [L~, LB] + P ( O A B In particular for r


(1.32)

Pexp(~) one has (r ~ Baker-Akhiezer = BA function)

oo

r

[A, B ] ) P -1

(x)

= P(~-~ kxk)~k-1)exp(~) = M e ; M = P(~"~, k x k O k - 1 ) P -1 = p M p - 1
1

and [L, M ] -

(1.33)

1

1 (this operator is also discussed later). One can write also

Amn - l~ni)m; Lmn -- MaLta; ( i ) m n - L+mn)P- P ( O m n - Amn); (OmnP) ---

(1.34)

oo

= -(MnLm)-P;


Mlm

:

Am+l,1

--

]~0m+

1

kxk Oqk + m ; PMlm P - 1

= ~

-

MLm+I

1

One notes that (tbtb) OnMlm = [0n, Mlm] = nO n+m and hence (1.32) becomes for LB = L
and LA = 0m+l,1 ~ A = Mlm
Om+l,lL = [(MLm+I)+,L] + L m+l

(1.35)

where one takes Om+l,l(i)) = 0 with which it can be seen also that the flows based on/)~

and 0m+1,1 commute.
Now for some background on the tau function one recalls the vertex operator equation

(VOE)
r

A) = X()QT = _e~G_(~)
1
- =
T = e~(Z,~) -7_
T

T

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T

(1.36)


6

CHAPTER 1. Q UANTIZATION AND I N T E G R A B I L I T Y
- - ~

(

; G-t-(A)--exp(i~
T


A-I,~

)

; ~=

and ~(A -1, c5) = ~ Oj/jA j. The Hirota bilinear identity (18A) 0 = f c r
A)r
A)dA
leads to various Hirota equations which describe tau functions via bilinear formulas, where
the notation

Ot}

a(t,b(t')

=

(1.37)

t=t'

=

~s~

a(tj + s j ) b ( t j - sj)
sj=O


is used. In particular, with pj ,.. Schur polynomials, one has

Epn(--2y)pn+l(~)exp

yjOj T " T = 0

(1.38)

0

The bilinear form of K P is included in (1.38) in the form
the yj are "free" parameters in (1.38) and the coefficients
r = Pexp(~) = (1 + ~ WnA-n)exp(~) = @exp(~) and r
note also r = X*(A)~-/T = (1/7)exp(--~)G+w. Then (cf.
-

wj =

,

(04 + 302 - 401i)3)~'.T = 0 (note
are set equal to zero). Write now
= (P*)-lexp(-~) = tb*exp(-~)"
[701])

1

pj(-O)T; wj = Tpj(O)T

(1.39)


and u = 02log(T). Relations between the ui in L and the wi in P (resp. the w~ in ( p , ) - l ) can
be obtained directly from writing out L = POP -1 and its adjoint. Using in addition some of
the other relations above one can also obtain formulas of the form u3 = (1/2)(O02--03)log(T)
etc.
We list some further formulas of interest now. In particular (18B) 0 = L + E ~ a)L-J
and the a~ can be computed directly via L = 0 + ~
u/+l 0 - / or via 7 as indicated below.
Using L r = Ar with ( 1 8 B ) one obtains ( 1 8 C ) 0 r = Ar + E ~ ( a l / A J ) r so Olog(r A + a ~ a 1/Aj). In this spirit one obtains a l s o
oo

Bm = L M = L m + ~ a~L-J; Omr = Amr + amr

(1.40)

1

oo
1
0n(71 : O n O l o g ( r
determines a conservation law via ( . . . )
fo_~ O(Onlog(r
On $ a l d x = 0 (given suitable b o u n d a r y conditions). Further from [701] ( 1 8 D ) 0 7 =
OmPj(--O)log(T) and OnOIOg(T) = -- naln -- E ~ - - I Ojan_
1 j. The proof can be obtained by

Thus

recursion via the Schur polynomial formula


~ylkl) ~yk2~
Pn(Y) = ~

\-~lv. [-~2v.] "'" (~-~jkj = n)

(1.41)

Finally one notes the important formulas

(X)
r

" --

E

0

n
8n A - n ; 8n --

E

0

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WjWn_ j

*


(1.42)


1.1. A L G E B R A I C AND GEOMETRIC METHODS

7

Using a subset of the Hirota equations one finds in fact

1

8n -- - ~ P n (

c~)

1

T --- 0n_10--1U

" T = ~T2OOn_IT"

(1.43)

since On-lOlog(T) = On-lOT" T/2T 2 and u = 02log(T). The Sn represent conserved densities
since Sn = On-lOU and the basic symmetries for KP can be exhibited via the sn via
K0 = O(1) = 0; K1 = 0(0) = 0; K2 = Ou; K3 = O2u;

(1.44)


3 i 02u; ... KN = OSn
K4 = K = ~1 03 u + 3uOu + -~Oleading to flows On-l U = OSn = Kn.
Next we extract a few formulas from [7] which are often useful. From the Fay trisecant
identity (cf. [7, 147, 902]) one can derive

r

1

=,
=

(._

o(X(x,)~,p)T(x))

j!

=

0

(1.45)

1

T

where X ( x , )~, #)T(X) = ezp[~ Zn(.n--)~)]exp[E()~-~--.-n)O~/n]r(x) and (18E) W~ (T) =
~ - c ~ "~-n-JwJn (T). The Wn3 are the generators of the W I + ~ algebra (see below) and from

(1.45) results also
V~2*(X' A ) 0 ; - I ~ ( x ' ~ ) : 0 ( ~ - ~ - ~ ) ~ - n - u W) ~ ( TT)

(1.46)

Here one can write (n > 0)
n

c~

W 1 = On = j1; W2n = j2n _ (n + 1)J 1 = y ~ C~jGgn_j -}- 2 E J X j t ~ n + j - 0

1

- ( n + 1)On = 2nn - ( n + 1)On

(1.47)

where Ln are the Virasoro generators with [Ln, Lm] = ( n - m)Ln+m for n, m > 0 (we will
disregard On and tn for n < 0 most of the time). In particular this leads to (cf. [163])
r162 * = ~--~P~+l,1)~-n-2; Rn+l,1 = ~ 0

T

;Kn+l =

T

(1.48)


Further one notes that ( 1 8 F ) S n + l = O(WI(T)/T) (n >_ --1). Additional symmetries (in
particular conformal symmetries) related to this framework are developed in [160, 163] for
example (see also the references there).
Now with a view toward dKP one can think of fast and slow variables with ,ex = X and

etn = Tn so that On --* eO/OTn and u(x, tn) --+ (t(X, Tn) to obtain from the KP equation
(1/4)uxxx + 3uux + (3/4)0 -102u = 0 the equation OT5 = 3(tOX5 + (3/4)0 -1 (02(t/OT~) when
e ~ 0 (0 -1 --~ (1/e)0-1). In terms of hierarchies the theory can be built around the pair
(L, M) in the spirit of [160, 163, 902]. Thus writing (tn) for (x, tn) (i.e. x ~ tl here) consider
(x)

L~=eO+EUn+l(e,T)(eo)-n;
1

oo

oo

M~=EnTnLn-i+Evn+l(e,T)L[n-i
1

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1

(1.49)


8


CHAPTER 1. Q UANTIZATION AND I N T E G R A B I L I T Y

Here L is the Lax operator L = 0 + ~
Un+lO - n and M is the Orlov-Schulman operator
defined via r = M e . Now one assumes Un(e,T) = Un(T)+ O(e), etc. and sets (recall
L r = Ar

r

[l+O(~)]exp(~TnAn)

= e x p ( 1 Se ( T ' A ) + O ( I ) )

T=exp(~F(T)+O(1))

(1.50)

We recall that OnL = [Bn, L], Bn = L~_, O n M = [Bn, M], [L,M] = 1, L r = Ar Ox~ Me, and r = ~-(T- (1/n;~n))exp[E7 Tn:~n]/~-(r). Putting in the e and using On for O/OT,~
now, with P = Sx, one obtains
(X)
)~ -" P + E

(X)
Un+lp-n;

1

CXD

(1.51)


P = )~ - E Wi)~-l;
1

(X)

.h,4 = E nTnAn-1 + E Vn+lA-n-1; OnS = Bn(P) =:~OnP = 9Bn(P)
1

1

where 0 ~ Ox + (OP/OX)Op and M
~ + I ( T ) + O(e)). Further for Bn = ~
Bn = L n + ~ a2L-J). We list a few
[160]); thus, writing {A, B} = O p A O A -

+ Ad (note that one assumes also Vi+l(e,T) bnmOTM one has ~n -- ~-~ bnmP m ~ A~_ (note also
additional formulas which are easily obtained (cf.
OAOpB one has

OnA = {Bn, A}; OnAd = {Bn,A/I}; {A, A J } - 1

(1.52)

Now we can write S = ~ T n A n + ~
Sj+IA-J with Sn+l = -(OnF/n), OmSn+l =
(Fmn/n), Vn+l - -nSn+l, and OAS = Ad. Further

Sn


(9O
= /~n -4- E OnSJ +l/~-j; OSn+l ~'~
1

OVn+l ~

OOnF
n

n

(1.53)

We sketch next a few formulas from [555] (cf. also [160] and Section 5.1.4). First it will be
i m p o r t a n t to rescale the Tn variables and write t ' = ntn, T~n = nTn, On = nO~n - n(O/OT~n).
Then

)~A; 0In ~ -- {~n,)~ } ( ~ n - n

~n);

(1.54)

n

O~np = OQn = OqQn -I- O p ~ n O P ; Oq~nQm- O ~ Q n - {Qn, Q m }
Now think of (P, X, Tn~), n _> 2, as basic Hamiltonian variables with P = P(X, T~). Then
- Q n (P, X, Tn~) will serve as a Hamiltonian via

ptn = dP I
dX
dT~n = OQn; fCn = dT~n = - 0 p Qn

(1.55)

(recall the classical theory for variables (q, p) involves 0 = OH/Op and/5 = -OH/Oq). The
function S(A, X, Tn) plays the role of part of a generating function S for the Hamilton-Jacobi
theory with action angle variables (A,-~) where

P d X + QndT~n = -~dA - KndT~n + dS; Kn = - R n = - - - ;
d~
d%

d~
= 2n =

= 0;

~n-1
= ?n =

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=

-

(1.56)

1.1.

ALGEBRAIC

AND GEOMETRIC

METHODS

9

(note t h a t J~" = 0 ~ 0"A = { Qn, A}). To see how all this fits t o g e t h e r we write
OP d X
d P = O ' P -~. . . .
dT"
O X dT"

OQn +

OP _. ,
OQn + OPOpQn + O F f { ,
-'~Xn =
n

(i.57)

This is c o m p a t i b l e with (1.55) and H a m i l t o n i a n s - Q n . F u r t h e r m o r e one has
SA = ~; S x = P; 0 " S

:

~n

-

(1.58)

Rn

a n d from (1.56) one has
I

P d X + Q n d T n = -~d)~ + R n d T " + S x d X

/

"

!

+ S;~d)~ + O n S d T n

(1.59)

which checks. We note t h a t O ' S = ~ n : ~ n / n and S x = P by c o n s t r u c t i o n s and definitions.
Consider S = S - ~
AnT'n/n. T h e n S x = S x = P and S'n = S ' n - Rn = Q n - Rn as desired
with ~ - S), = S ) , - ~
T" )~n - i . It follows t h a t ~ ~ A / / - ~
T'n)~n - i = X + ~

Vi+IA - i - 1
If W is t h e gauge o p e r a t o r such t h a t L = W O W -1 one sees easily t h a t
M~2=W

kxkO k-1

W-Ir

"-

(

G+EkXk/~
2

k-1

)

~

(1.60)

from which follows t h a t G - W x W - i ---+~. This shows t h a t G is a very f u n d a m e n t a l object
a n d this is e n c o u n t e r e d in various places in t h e general t h e o r y (cf. [160, 163]).
1.1.2

Dispersonless

theory

for KdV

Following [147, 160, 165] we write
L 2 = L~_ = 02 + q = 02 - u (q = - u = 2u2); qt - 6qqx - qxxx = 0;
B=403+6q0+3qx;

L2

[B, L2]; q = - v

(i.6i)

2-vz~vt+6v2vx+vxxx=O

(v satisfies t h e m K d V equation). Canonical formulas would involve B ~ B3 - L 3 as
i n d i c a t e d below b u t we retain t h e B m o m e n t a r i l y for c o m p a r i s o n to o t h e r sources. K d V
is Galilean invariant (x' - x - 6At, t' - t, u' - u + A) and c o n s e q u e n t l y one can consider
L + O2 + q - ~ = (O + v ) ( O - v, q - )~ = - v x - v 2, v = ~bz/~, and - ~ x x / ~ 2 = q - )~ or
r162
= /kr (with u' = u + / ~ ~ q' = q - A ) .
T h e v e q u a t i o n in (1.61) becomes t h e n
vt = 0 ( - 6 ) w + 2v 3 - vxx) a n d for A = - k 2 one e x p a n d s for .~k > 0, Ikl - , oc to get
( . ) v ~ ik + Y : . ~ ( V n / ( i k ) n ) . T h e Vn are conserved densities and with 2 - A = - v ~ - v 2 one
obtains
n-1

p

=

-2Vl;

2Vn+l

=

-

Vn-mVm-

E

!

!

Vn; 2 v 2 - - - v 1

(1.62)

1

N e x t for r 1 6 2
= - k 2 r write r ~ e x p ( i i k x ) as x ---+ •
Recall also t h e t r a n s m i s s i o n
a n d reflection coefficient formulas (c.f. [147]) T ( k ) r
- R(k)r
+ r
and T~+ RLr

+ r
W r i t i n g e.g. r
= exp(ikx + r
with r
oc) - 0 one has
r + 2 i k r + (r
= u. T h e n r 1 6 2
= ik + r = v with q - A - - v x - v 2. Take t h e n

r

r

1

(2ik) ;

r
~ ik +

v.
=

+ Z

(ik)

=> Cn = 2nvn

(1.63)

F u r t h e r m o r e one knows
oo

l o g T = - y~. C2n+i
1 / _ ~ k2nlog(1 _ iRi2)d k
0 k2n+l; C2n+1 = ~
e~

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(1.64)


C H A P T E R 1. Q U A N T I Z A T I O N A N D I N T E G R A B I L I T Y

10

(assuming for convenience t h a t there are no b o u n d states). Now for C 2 2 - - R L / T and c21 =
1 / T one has as k ~ - o o (.~k > 0) the behavior r
---, c22exp(-2ikx)+c21 ---+c21.
Hence exp(r ---, c21 as x ---, - o o or r 1 6 2
= - l o g T which implies

?

r

= logT =

517

(2ik)n

(1.65)

Hence f r
= 0 and C2m+1 = - f r
2m+1. The C2n+l are related to Hamiltonians H2n+l = anC2n+l as in [155, 145] and thus the conserved densities Vn ~ Cn give
rise to Hamiltonians Hn (n odd). There are action angle variables P = klog[T[ and
Q = 7 a r g ( R L / T ) with Poisson structure { F , G } ~ f(hF/hu)O(hG/hu)dx (we omit the
second Poisson structure here).
Now look at the dispersionless theory based on k where A2 ~ (+ik) 2 = - k 2. One obtains
for P = S x , p2 + q = _ k 2, and we write P = ( 1 / 2 ) P 2 + p = (1/2)(ik) 2 with q ~ 2p ~ 2u2.
One has Ok/OT2n = {(ik)2n, k} = 0 and from ik = P(1 + qp-2)1/2 we obtain

ik = P

1+

qmp-2m
1

)

(1.66)

(cf. (1.51) with u2 = q/2 - we use +ik for convenience here but - i k will be needed later).
T h e flow equations become then

02n+lP = 0~2n+1; 02n+l(ik) = {~2n+1, ik}

(1.67)

_,3/2 = 03 + (3/2)q0 + (3/4)qx = B3
Note here some rescaling is needed since we want (02 + q)+
instead of our previous B3 .-~ 403 + 6qO + 3qz. Thus we want Q3 = ( 1 / 3 ) P 3 + (1/2)qP to
fit the n o t a t i o n above. The Gelfand-Dickey resolvant coefficients are defined via Rs(u) =
(1/2)Res(O 2 - u) s-(1/2) and in the dispersionless picture Rs(u) ~ ( 1 / 2 ) r s _ l ( - u / 2 ) (cf.
[160]) where
rn

__ R e 8 ( _ k 2 ) n + ( 1 / 2

) _

--

( n + (1/2))
n + 1

qn+l _ (n + 1 / 2 ) ' ' ' (1/2 n+l
-(n + iii
q
;

20qrn = (2n + 1)rn-1

(1.68)

T h e inversion formula corresponding to (1.51) is P = ik - E ~ Pj(ik)-J (again ik ---, - i k
arises later) and one can write
0'

2n+l (p2

+ q) = 0'

2n+l(--k2);

0'
2r n
2n+lq = ~2nO +
1

2

= 2n + 10qrnqx - q x r n - 1

(1.69)

Note for example r0 = q/2, rl = 3q2/8, r2 = 5q3/16, ... and O~q = qxro = (1/2)qqx (scaling is needed here for comparison). Some further calculation gives for P = i k - ~
Pn(ik) -~

Pn "0 -Vn ~ - ~Cn
-;

C2n+l = ('-- 1)n+l . I ?

P2n+l(Z)dX

(1.7o)

oo

T h e development above actually gives a connection between inverse scattering and the d K d V
t h e o r y (cf. [153, 158, 160, 168] for more on this).

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1.1. A L G E B R A I C

1.1.3

Toda

AND GEOMETRIC METHODS

11

and dToda

We will follow the formulation of [902, 916] (dToda can also be subsumed in a general
formulation of d K P as in [32]); thus one exploits the Orlov-Schulman operator M --. A4
directly in creating the embellished d K P hierarchy which will have an extension to dToda
(indicated below); this provides a richer structure from the beginning. Let us begin with
[916] and write the ordinary Toda hierarchy in the language of difference operators in a
continuous variable s with spacing unit e. Thus
OL

OL = [Bn L] e

a t = [B., t]

e-~Zn

where n -

;

~

= [/~n L]

(1.71)

at = [B t]

O~n

'

1, 2 , . - . and
L=e

(X)

(2

o

1

e__0
o, _~_EUn(e,,z,~;,8)e-ne~ t - - E ~ t n ( e , , Z , ~ , 8 ) e n e ~

(1.72)

Here Bn = (Ln)>o and /~n = ( L - n ) < - 1 where one is thinking of projections onto linear
combinations of exp[ne,(O/Os)] with n >_ 0 or n < - 1 . We note that exp[ne,(O/Os)]f(s)
^
f ( s + ne,). One will also have Orlov-Schulman type operators M and M satisfying
0M
0M
e,-~ZnZ
n = [Sn, M]; e,~n-n = [J~n, M];
i) M = [B n 1~/I] e,
~~z~

'

0M

(1.73)

[/3n, A~]

=

'

for n = 1, 2 , . . - w h e r e [n, M] = E,L, [L, A?/] = E,L and
(X)

M-

(X)

EnznLn

+ s

1

+ EVn(e,,z,z, 8)n-n;
1

O0

(X)

= --EnZn
1

(1.74)

~ - n at- 8 -~- E ~ ) n ( e , , z , fi.,s)Ln
1

The dispersionless hierarchy arises as e, --* 0 upon positing Un(e,, z, ~, s) = u no (z , e, ~) + o(~)
along with similar expressions for Vn, ~tn, and ~)n. To arrive at the limiting forms it is perhaps
clearest to introduce "gauge" operators
O0

W=

(X)

1 + EWn(e.,z,~,s)e-he~

~r=

(1.75)

E(Vn(e.,z,~,s)enE~

1

o

where the Wn and Wn are singular as e, ~ 0 (unlike the Un, Vn, etc.). Then one has
L = W.e

M = W

nz~o

~o

e__0

0~ 9W - l ; L = 12d.e 0~ . l ~ -1",

+ ~ w_l; M = W

_ ~n~n

_~

1

(1.76)

)

+ ~ W-1

The Lax equations can be converted into Sato type equations
OW
We~ ~
OW
e,-~ZnZn -- Bn W ; e,-~ZnZn : JBn W ;

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(1.77)


CHAPTER

12

0w

1. Q U A N T I Z A T I O N

0w

AND INTEGRABILITY

w~-n~

One then introduces WKB type wave functions via

r =

1 + ~-~wn(e,z,2, s)A -n
1

ZnA n and 2(A -1) = ~

where z(A) = ~

e x p [ e - l ( z ( A ) + slogA)];

(1.78)

~,nA-n. The equations (1.76) then imply that
{1.79/

Further

O~b

a~b = Bn%b; e a~b = Bn~b; E--=O~

/)n~

(1.80)

and we note that the coefficients Vn and Vn of M and M can be read off via

e A 0 l o g_______~ ~

=Enz"An + S + E V~n ) _ n ;
1

1

l og ~

o0

cc

C~A

1

1

(1.81)

Now one thinks of asymptotic forms as e --* 0

~) -- eXF[E-1S(z,z.,,8,)k) + O(1)]; ~ - eXp[E-1S(z, 2,8,)k) + O(1)]

(1.82)

where S = z(A) + slogA + Ec~ Sn(z, 2, s)A -n and S = 2(A -1) + slogA + Ecff Sn(z, 2, s)A n.
Similarly tau functions are defined via

T(E, Z -- E[,~--I], 2, 8)
T-~,7~;8 i
eXp[E--I(z()k) + slogA)] = r

z, 2, A);

(1.83)

T(E,Z,Z.- E[)k], 8 -~-E)eXp[E_l(~()k_l)-~- 810g~)] = ~(E, Z, 2, )k)

7-(E,z,~,s)

where [A] = (A, A2/2, .-. , An~n, .. .). One sees easily that log'r(e, z, 2, s) = e - 2 F ( z , ~,, s) +
O(e -1) as e ~ 0 for some function F and it is immediate that

Sn =

1 OF
1 OF
OF
--nOz----n ; ~n . . . n. 02n ; S 0 = as

(1.84)

Hence in analogy with the KP situation one refers to F as a "free energy" and writes
F = logTdToda.
It will be convenient to distinguish between the A's in S and S now and write

.OS(z,

exp[

-O-s

] = p = exp[

bs

]

(1.85)

(a corresponding replacement is also appropriate in (1.82) and (1.83)). Then some calculation as in the dKP theory yields for
(3O
OO
)k"'P-Jr - E?.tn(Z, 2)p--n; ~"" E U n ( Z , Z . , 8 ) p n ; ~n ''~ ()in)>0; ~n = (~--n)<--I
(1.86)

0
1

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