Quantum theory, deformation and integrability

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 em

QUANTUM THEORY, DEFORMATION AND INTEGRABILITY

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

Editor: Saul LUBKIN

QUANTUM THEORY, DEFORMATION

ELSEVIER SCIENCE B.V

Sara Burgerhartstraat 25

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Contents

1.1 ALGEBRAIC AND GEOMETRIC METHODS 1

1.1.1 Remarks on integrability classical KP 4

1.1.2 Dispersonless theory for KdV 9

1.1.3 Toda and dToda 11

1.1.4 Remarks on integrability 18

1.1.5 Hirota bilinear difference equation 19

1.1.6 Quantization and integrability 21

1.2 VERTEX OPERATORS AND COHERENT STATES 23

1.2.1 Background 23

1.2.2 Heuristics for QM and vertex operators 31

1.2.3 Connections to ( X , $J) duality 36

1.3 REMARKS ON THE OLAVO THEORY 45

1.4 TRAJECTORY REPRESENTATIONS 49

1.4.1 The Faraggi-Matone theory 50

1.5 MISCELLANEOUS 56

1.5.1 Variations on Weyl-Wigner 5G 1.5.2 Hydrodynamics and Fisher information 57

2 GEOMETRY AND EMBEDDING 63 2.1 CURVES AND SURFACES 6 3 2.1.1 Background 63

2.1.2 The role of constraints 67

2.1.3 Surface evolution 68

2.1.4 On the embedding of strings 72

2.2 SURFACES IN R3 AND CONFORMAL IMMERSION 73

2.2.1 Comments on geometry and gravity 79

2.2.2 Formulas and relations 80

2.3 QUANTUM MECHANICS ON EMBEDDED OBJECTS 86

2.3.1 Thin elastic rod 87

2.3.2 Dirac field on the rod 88

2.3.3 The anomaly in R3 95

2.4 WILLMORE SURFACES, STRINGS, AND DIRAC 98

2.4.1 One loop effects 98

2.4.2 CMC surfaces and Dirac 100

2.4.3 Immersion anomaly 104

2.4.4 Quantized extrinsic string 108

V

vi CONTENTS

2.5 CONFORMAL MAPS AND CURVES 110

3 CLASSICAL AND QUANTUM INTEGRABILITY 113 3.1 BACKGROUND 113

3.1.1 Classical and quantum systems 114

3.2 R MATRICES AND PL STRUCTURES 119

3.3 QUANTIZATION AND QUANTUM GROUPS 124

3.3.1 Quantum matrix algebras 128

3.3.2 Quantized enveloping algebras 129

3.4 ALGEBRAIC BETHE ANSATZ 130

3.5 SEPARATION OF VARIABLES 136

3.5.1 r-matrix formalism 138

3.5.2 Quantization 140

3.5.3 XXX spin chain 141

3.6 HIROTA EQUATIONS 143

3.7 SOV AND HITCHIN SYSTEMS 147

3.8 DEFORMATION QUANTIZATION 149

3.8.1 Path integrals 151

3.8.2 Nambu mechanics 156

3.9 MISCELLANEOUS 160

3.9.1 Geometric quantization and Moyal 160

3.10 SUMMARY REMARKS 164

4 DISCRETE GEOMETRY AND MOYAL 167 4.1 INTRODUCTION 167

4.1.1 Phase space discretization 167

4.1.2 Discretization and K P 171

4.1.3 Discrete surfaces and K P 175

4.1.4 Multidimensional quadrilateral lattices 178

4.1.5 d methods 188

4.2 HIROTA, STRINGS, AND DISCRETE SURFACES 194

4.2.1 Some stringy connections 195

4.2.2 Discrete surfaces and Hirota 198

4.2.3 More on HBDE 201

4.2.4 Relations to Moyal (expansion) 204

4.2.5 Further enhancement 212

4.2.6 Matrix models and Moyal 221

4.2.7 Berezin star product and path integrals 223

4.3 A FEW SUMMARY REMARKS 227

4.3.1 Equations and ideas 227

4.4 MORE ON PHASE SPACE DISCRETIZATION 234

4.4.1 Review of Moyal-Weyl-Wigner 236

4.4.2 Various forms for difference operators 240

4.4.3 More on discrete phase spaces 246

5 WHITHAM THEORY 255 5.1 BACKGROUND 255

5.1.1 Riemann surfaces and BA functions 256

5.1.2 Hyperelliptic averaging 257

-

CONTENTS vii

5.1.3 Averaging with $J*+ 2G2

5.1.4 Dispersionless K P 264

5.2 ISOMONODROMY PROBLEMS 270

5.2.1 JMMS equations 275

5.2.2 Gaudin model and KZB equations 279

5.2.3 Isomonodromy and Hitchin systems 283

5.3 WHITHAM AND SEIBERG-WITTEN 293

5.3.1 Basic variables and equations 294

5.3.2 Other points of view 300

5.4 SOFT SUSY BREAKING AND WHITHAM 302

5.4.1 Remarks on susy 302

5.4.2 Soft susy breaking and spurion fields 306

5.5 RENORMALIZATION 309

5.5.1 Heuristic coupling space geometry 309

5.6 WHITHAM, W D W , AND PICARD-FUCHS 314

5.6.1 ADE and LG approach 314

5.6.2 F’robenius algebras and manifolds 319

5.6.3 Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations 321

6 GEOMETRY AND DEFORMATION QUANTIZATION 325 6.1 NONCOMMUTATIVE GEOMETRY 325

6.1.1 Background 326

6.1.2 Spectral triples 329

6.1.3 The noncommutative integral 332

6.2 GAUGE THEORIES 341

Background on noncommutative gauge theory 341

6.1.4 Quantization and the tangent groupoid 335

The Weyl bundle 343

6.2.3 Noncommutative gauge theories 349

6.2.4 A broader picture 353

6.3 BEREZIN TOEPLITZ QUANTIZATION 359

6.2.1

6.2.2

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P r e f a c e

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

t h a t 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 a t t e m p t 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 m a t h e m a t i c a l 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 per- haps the assumptions are more or less correct or, if not, perhaps other related assumptions

a n d / o r 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 publi- cations is worked into the text and references to publications before 1991 can be found in the books cited (cf also h t t p : / / w w w m a t h u i u c e d u / - r c a r r o l l / f o r 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 the- orem); 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

connection to the physics It seems desirable to discover and develop those m a t h e m a t i c a l 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 t h a t s y m m e t r y is the driving engine for physics, and the frequent t e m p t a t i o n to say t h a t mathematics i s 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

m o m e n t it is of primary concern for me at least to understand what is q u a n t u m 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 a t t e m p t to describe all the marvels now unfolding which relate to string (M) theory, q u a n t u m gravity, noncommutative geometry, etc (many of which are still in

a r a t h e r embryonic state) but will try to keep in touch with some of this Some themes will be pursued more diligently t h a n others and some historical m a t t e r s will be discussed

at some length because of their rich content and interaction with other areas The pace of development today is so rapid t h a t 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 repre- sentations 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 illus- trating 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, t h a t students may find it hard to cope with, let alone master, anything In

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 pos- sible 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 coher- ent 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, in- dex theorems, scaling, discretization, extremal principles, analyticity, scattering, moduli and various algebraic structures acting as glue and language It is in this spirit that we sug- gest 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 con- versation 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

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

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

The dynamical variables evolve classically via

so t h a t the q u a n t u m evolutions (16A) x(t) = U, 9 x , U , 1 and p(t) = U, , p , U, 1 t u r n 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 / / d x d p f ( x , p ) = l (1.7)

Consequently any W F can be written as

so t h a t by inserting U, 9 U,1 pairs

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

Consider now a generalized form of this from [278, 930] where the motivation arises in

q u a n t u m statistical mechanics with kinetic equations of Lindblad type The classical picture

o p e r a t o r s

position (t momentum general fI

defined via

flw" A(q, p) , •w(A)= fi =

and the Wigner transform of 4 is

From the identities

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#~)

(1.13) one has the Weyl ordering

n(:)

0

~ The 9 product is defined now via

f * g = f~wl(f~w(f)~w(g)); { f , g } u = -~#[f,g]M = -~p(f * g g * f ) (1.17)

A(q,p) - qnpm then (1.19) gives an ordering different from (1.16) Setting ( 1 6 E ) w ( z ) -

(1/2~) f d a e x p ( i a z ) ( 1 / f l ( a ) ) the inverse ~t transformation exists and is given by

- 2 i # / dqe -i~q < q + ip~lftlq - i#~ > (1.22)

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

We obtain then in analogy to (1.17) a nonabelian associative algebra structure

f *n g = f l - l ( f l ( f ) f l ( g ) ) ; ~_~ [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

Some further manipulation gives also

is required, then (1.27) is the only possibility; this is related to the uniqueness of the Moyal

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

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

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 o p e r a t o r L = 0-~-U2 0-1 -~- .

0 + F_,~ Un+l O-n (with u2 = u) and a gauge operator P = 1 + E ~ w~ O-n d e t e r m i n e d via

L = P O P -1 where /) ~ cox For wave functions ~ = Pexp(~) with ~ = E ~ Xn An where

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

The last equation is the K P 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

( O n - B n ) 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 gener- ally, following [559], given differential operators A, B in c9 one considers flows (&&) OAP =

- ( P A p - 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 d e t e r m i n e d via the u~ in L

(cf [147, 701]) Thus define LA = P A P -1 and note t h a t OAPP -1 + P i ) A P -1 - 0 to obtain

and [L, M ] - 1 (this operator is also discussed later) One can write also

Amn - l~ni)m; Lmn MaLta; ( i ) m n - L + m n ) P - P ( O m n - Amn); (OmnP) - (1.34)

T h e bilinear form of K P is included in (1.38) in the form (04 + 302 - 401i)3)~'.T = 0 (note

t h e yj are "free" p a r a m e t e r s in (1.38) and the coefficients are set equal to zero) Write now

r = Pexp(~) = (1 + ~ WnA-n)exp(~) = @exp(~) and r = ( P * ) - l e x p ( - ~ ) = tb*exp(-~)" note also r = X*(A)~-/T = (1/7)exp( ~)G+w T h e n (cf [701])

a n d u = 02log(T) Relations between the ui in L and the wi in P (resp the w~ in ( p , ) - l ) can

be o b t a i n e d directly from writing out L = POP -1 and its adjoint Using in addition some of

t h e o t h e r 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 ( 1 8 B ) 0 = L + E ~ a ) L - J

0 - / and t h e a~ can be c o m p u t e d directly via L = 0 + ~ u/+l 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

T h u s 0n(71 : O n O l o g ( r determines a conservation l a w 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_ j 1 The proof can be o b t a i n e d by recursion via t h e Schur polynomial formula

~ylkl) ~yk2~

Pn(Y) = ~ \-~lv [-~2v ] "'" (~-~jkj = n) (1.41) Finally one notes the i m p o r t a n t formulas

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 7

Using a subset of the Hirota equations one finds in fact

8n - ~ P n ( ) " T = ~ T 2 O O n _ I T " T - 0n_10 1U (1.43)

K4 = K = ~ u + 3uOu + -~O- 02u; KN = OSn

Next we extract a few formulas from [7] which are often useful From the Fay trisecant identity (cf [7, 147, 902]) one can derive

~ - c ~ "~-n-JwJn (T) The Wn3 are the generators of the W I + ~ algebra (see below) and from (1.45) results also

particular conformal symmetries) related to this framework are developed in [160, 163] for example (see also the references there)

Now with a view toward d K P one can think of fast and slow variables with ,ex = X and

etn = Tn so t h a t On * eO/OTn and u(x, tn) + (t(X, Tn) to obtain from the KP equation

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

L ~ = e O + E U n + l ( e , T ) ( e o ) - n ; M ~ = E n T n L n - i + E v n + l ( e , T ) L [ n - i (1.49)

8 C H A P T E R 1 Q UANTIZATION AND I N T E G R A B I L I T Y

L r = Ar

r [ l + O ( ~ ) ] e x p ( ~ T n A n ) = e x p ( 1 S ( T ' A ) + O ( I ) ) e

Me, and r = ~-(T- (1/n;~n))exp[E7 Tn:~n]/~-(r) P u t t i n g in the e and using On for O/OT,~

Bn = L n + ~ a2L-J) We list a few additional formulas which are easily obtained (cf

(Fmn/n), Vn+l - -nSn+l, and OAS = Ad Further

(9O

Sn = /~n -4- E OnSJ +l/~-j; OSn+l ~'~ OVn+l ~ OOnF (1.53)

We sketch next a few formulas from [555] (cf also [160] and Section 5.1.4) First it will be

T h e n

< S : )~A; 0In ~ { ~ n , ) ~ } ( ~ n - - ~ n ) ; (1.54)

O~n p = OQn = OqQn -I- O p ~ n O P ; Oq~nQm - O ~ Q n - { Q n , Q m }

Now t h i n k of (P, X, Tn~), n _> 2, as basic Hamiltonian variables with P = P ( X , T~) T h e n

- Q n (P, X, Tn ~ ) will serve as a Hamiltonian via

dT~n = OQn; fCn = dT~n = - 0 p Qn (1.55)

theory with action angle variables ( A , - ~ ) where

(v satisfies t h e m K d V e q u a t i o n ) C a n o n i c a l formulas would involve B ~ B3 - L 3 as

i n d i c a t e d below b u t we r e t a i n 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 G a l i l e a n i n v a r i a n t (x' - x - 6At, t' - t, u' - u + A) a n d c o n s e q u e n t l y one can consider

L + O 2 + q - ~ = (O + v ) ( O - v, q - )~ = - v x - v 2, v = ~bz/~, a n d - ~ x x / ~ 2 = q - )~ or

r 1 6 2 = /kr ( w i t h u' = u + / ~ ~ q' = q - A ) T h e v e q u a t i o n in (1.61) b e c o m e s 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 a n d w i t h 2 - A = - v ~ - v 2 one

10 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

(assuming for convenience t h a t there are no b o u n d states) Now for C 2 2 - - R L / T a n d c21 =

1 / T one has as k ~ - o o (.~k > 0) t h e 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

Hence f r = 0 a n d C2m+1 = - f r 2m+1 T h e C2n+l are related to Hamilto- nians H2n+l = anC2n+l as in [155, 145] and thus the conserved densities Vn ~ Cn give rise to H a m i l t o n i a n s Hn (n odd) T h e r e are action angle variables P = klog[T[ a n d

Q = 7 a r g ( R L / T ) with Poisson s t r u c t u r e { F , G } ~ f ( h F / h u ) O ( h G / h u ) d x (we omit the second Poisson s t r u c t u r e here)

Now look at t h e dispersionless t h e o r y based on k where A 2 ~ ( + i k ) 2 = - k 2 One obtains for P = S x , p 2 + q = _ k 2, and we write P = ( 1 / 2 ) P 2 + p = (1/2)(ik) 2 with q ~ 2p ~ 2u2

O n e has Ok/OT2n = {(ik)2n, k} = 0 a n d from ik = P ( 1 + qp-2)1/2 we o b t a i n

Note here some rescaling is needed since we want (02 + q)+ + + (3/4)qx = B3

i n s t e a d of our previous B3 -~ 403 + 6qO + 3qz T h u s we want Q3 = ( 1 / 3 ) P 3 + (1/2)qP to fit t h e n o t a t i o n above T h e 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

r n R e 8 ( _ k 2 ) n + ( 1 / 2 ) _ ( n + ( 1 / 2 ) ) qn+l _ (n + 1 / 2 ) ' ' ' (1/2 n+l

T h e inversion formula corresponding to (1.51) is P = ik - E ~ P j ( i k ) - J (again ik -, - i k

arises later) a n d one can write

0' 2n+l (p2 + q) = 0' 2n+l( k2); 0' 2n+lq = 2n + 1 2

2

~ O r n = 2n + 10qrnqx - q x r n - 1 (1.69) Note for e x a m p l e r0 = q/2, r l = 3q2/8, r2 = 5q3/16, and O~q = qxro = (1/2)qqx (scal- ing is n e e d e d here for comparison) Some further calculation gives for P = i k - ~ Pn(ik) -~

Pn "0 -Vn ~ - ~ - ; C2n+l = (' P 2 n + l ( Z ) d X

o o

(1.7o)

T h e d e v e l o p m e n t above actually gives a connection between inverse scattering a n d t h e d K d V

t h e o r y (cf [153, 158, 160, 168] for more on this)

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 11

1 1 3 T o d a a n d d T o d a

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

T h e dispersionless hierarchy arises as e, * 0 upon positing Un(e,, z, ~, s) = u n ,

along with similar expressions for Vn, ~tn, and ~)n To arrive at the limiting forms it is perhaps

clearest to introduce "gauge" operators

e A 0 l o g _~ =Enz" ~ An + S + E V n ) _ n ; ~ l o g ~ o0 cc (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)

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

(a corresponding replacement is also appropriate in (1.82) and (1.83)) Then some calcula- tion as in the d K P theory yields for

)k"'P-Jr - E ? t n ( Z , 2)p n; ~"" E U n ( Z , Z , 8 ) p n ; ~n ''~ ()in)>0; ~ n = (~ n)< I (1.86)

{1.79/

T h e theory is also developed in [902] for example in a slightly different m a n n e r and with

s o m e w h a t different notation and it will be expedient to indicate this Thus one starts with

14 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

This n o t a t i o n yields certain advantages in subsequent formulas and is accompanied by M exactly as in (1.74) with M = - E ~ ~ n t n L n + s + E ~ ~n L-n" Thus i, ~ ~ - 1 in these formulas Note also t h a t OnL = [Bn, L] ~ On(L -1) = [Bn, L-1] In this context the wave functions have the form (1.82) with r ~ ~, S S, and A ~ ~ -~ ~ T h e rest is

s t r a i g h t f o r w a r d and one notes t h a t the Sato t y p e equations (1.77) take a s o m e w h a t simpler form which will be exhibited if needed The embellished d T o d a hierarchy t h e n has the form

Vn - a t ~ ; v~ a ~ ; s0 = r = ~ s (1.98)

O t h e r formulas will also be displayed as needed

For completeness we add some remarks on T o d a and W h i t h a m following [87] (cf also Sections 5.3.1 and 5.3.2 for references and discussion) Thus consider T o d a equations in the form

These can be w r i t t e n as compatibility equations for the equations

where L = A + u0 + Ul A - 1 and B = A + u0 Here A is the shift o p e r a t o r A = exp(O/Ok)

so t h a t A f ( k , t ) = f ( k + 1, t) and we take u o ( k , t ) = ak(t) with u l ( k , t ) = bk(t) T h e n (1.99) becomes

where Fi - Fi(uo, ul) and Gi = Gi(uo, Ul) and some elementary calculation gives

~log = ~log A- ~ c ~ A -~ = (1.1o3)

0

1.1 ALGEBRAIC AND GEOMETRIC METHODS 15

k variable (withfk ~ ~ k ) so that (1.103) represents a system of conservation laws with conserved densities given via

log = l o g ( h - Z a~ h-~) = logh - X a o + - ; + (1.105)

Now it is known that all periodic solutions of the Lax equations for Toda systems of the form (1.99) will be of finite zone type, associated to a hyperelliptic Riemann surface Eg of genus g of the form

of integrals of motion) corresponds to quantization or quasiquantization (in the first W K B approximation) Indeed q u a n t u m wave functions appear from averaging along the classical trajectories, very much in the spirit of ergodicity theorems, and that corresponds to the averaging involved here (more on this later) Thus suppose F is a flux or density and write < F > for the spatial average over the fast variables of F (keeping the slow variables

many commuting flows which can be viewed as translations on the g-torus; generically each flow will cover the torus ergodically (assuming incommensurate characteristic frequencies)

so one can replace e.g spatial averages by

1

16 CHAPTER 1 QUANTIZATION AND I N T E G R A B I L I T Y

An interesting observation in [87] indicates that such an integral in 0i variables corresponds

to an integral over the cycles on the associated Riemann surface and thus one expects the averaged quantities to correspond to differentials on Eg One now breaks up the dynamics into fast and slow scales Let to and no denote the fast time and spatial variables and let

T = et and X = en be slow variables Then one can write

get the W h i t h a m equations

OT < Pi > = ~ < Fi > (1.110) and we now want to express these equations in terms of differentials on Eg as in Sections 5.1.2, 5.3.1, and 5.3.2 (cf [148, 338, 722])

Now in [87] one takes a Baker-Akhiezer (BA) function (or wave function)

here to cycles over (A2j, A2j+I) and the coefficients ai and 7i in (1.112) - (1.113) are uniquely determined by (:[:) Further one defines here

be developed later Note that r0 can be given an explicit (complicated) form which is not needed here Now from (1.111) we obtain

1.1 ALGEBRAIC AND GEOMETRIC METHODS 17

It is interesting to compare this with formulas for hyperelliptic situations with KdV and

(dF/F)(t) in t for example (via (1/2T) fT_r gives generically a bounded function log F

divided by factors of T ~ oc and hence there is no contribution On the other hand

dp ,,~ -i(dlog~2)x ,,~ d&o Now the integrability condition for (1.117) gives the Whitham equations

Od&o Odwl

Further one can regard (1.118) as an averaged form of the conservation laws (1.103) To see

l o g A - ~ PiA -i which gives

1

But from (1.111) one has

log = i d&o :=~ d log = id&o (1.120)

o

B = A - ~ FiA -i We can average directly now to get

Next from ( 1 1 1 2 ) - ( 1 1 1 3 ) o n e has (Ox ~ o~x)

Od&o _ ~-~g dr Oai ~- 2g+2x , d&o OAk

OT 1 A - a~ OT 2 ( A - Ak) OT (1.124)

18 CHAPTER 1 QUANTIZATION AND INTEGRABILITY

)

2g+2 &idA OxAk + - AgOxal + Ag-lcqx~l -+-" +- OXq/g

OXd~l E 2 ( t - hk) 2V~g

1

(1.107) Evaluating the residues in (1.124) at A = Ak one obtains then from (1.118)

1 1 4 R e m a r k s o n i n t e g r a b i l i t y

At this juncture it is appropriate to make a few more comments about integrability (cf

in particular [456, 615, 966]) There is some possible confusion in dealing with classically integrable systems of infinite dimension such as KdV which are phrased in terms of par- tial differential equations (PDE) How is this related (if at all) to quantization schemes

for example and this will be examined at various places (cf also Section 3.1.1) We recall several background facts First from the Groenwald-vanHove theorem (cf [426, 630]) there

space which realizes (.) in a general sense (beyond quadratic polynomials- cf also [122]) Now what about quantized integrability? Classical integrability for a Hamiltonian system involves e.g N commuting constants of motion In such that {H, I~} = 0 For infinite di- mensional systems such as KdV one expects N - ec (cf (1.4) for { , }1,2) Then one could anticipate that quantum integrability should involve N commuting quantum constants of motion such that [/~,/~n] = 0 and [/~n, Im] = 0 However one knows by a theorem of vonNeu- mann that for any number of commuting selfadjoint operators/~n there exists a selfadjoint

This shows in particular that it is impossible to fix unambiguously the number of elements comprising a commuting set of selfadjoint operators Thus one cannot use this definition

of quantum integrability to separate quantum systems into two classes of integrable and nonintegrable systems

Some attempts to deal with such problems occur in [127, 671, 507, 509, 82, 481, 455,

456, 615] for example where a main feature of standard versions of quantum integrabil- ity is to ignore the Hilbert space context (and the reliance on selfadjoint operators) and

(cf [127]) For example if a classically integrable system is specified by N algebraically

the context of algebraic geometry, differential Galois theory, 73 modules, etc as in [127] (cf also [82, 481]) In [615] (cf also [507, 509, 671]) one looks at the preservation of the

symbol multiplication as with pseudodifferential operators (PSDO - cf Section 1.1.1) One

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 19

and L m ,,~ L *m with Un - 1 and Un-1 = 0 (while Om "~ O/Otto); the ui are functions of q

(s+l)

rise to e deformed integrable soliton equations in standard situations, e.g with the Moyal product (8.6) (modulo a minus sign)

(ut 4- ( e 3 / 1 2 ) u ' ' - euu')x while KP itself is Uyy = (ut + (1/12)u'" - u u ' ) x Using e t -

T, ex = X , and ey = Y with 0t = e0T etc one obtains then dKP forms e2uyy -

~.(6.UT 4- ( 1 / 1 2 ) e 3 u x x x - e u u x ) x or u y y = (u T 4- ( 1 / 1 2 ) e 2 u x x x - u u x ) x -+ u y y = ( U T - UUX)X Note in [615] one is only scaling the x variable which corresponds to

u y y = e(UT + ( 1 / 1 2 ) e 3 u x x x e u u x ) x We see that (&) and (1.126) appear to be dif- ferent (in fact scaling of y and t would also be needed as e ~ 0) but regarded as a Ben-

tions such as ( 1 5 D ) satisfying ( 1 5 D D ) a b = ab for a , b E ,4, with p~ 9 pC' _ pe+e' and

ap t * bp ~' = abp ~+~' 4- O ( < t~ 4- t~ ~) parametrize realizations of integrable Hamiltonian systems (A is a differential Q algebra with derivation 0 ~ p)

In [456] one again avoids the problem of selfadjoint operators and the vonNeumann the- orem by using differential operators (no boundary conditions, no Hilbert space, no domain restrictions) and suggests that the transition from classical to quantum integrability seems

to be always possible if one adds correction terms to the Hamiltonian and the In proportional

to h 2 Situations of an excess of commuting (algebraically related) differential operators are discussed and many questions appear unanswered or unclear Relaxing "regularity" of the functions In apparently requires relaxing the idea of functional independence (polynomial relations should be allowed)

1 1 5 H i r o t a b i l i n e a r d i f f e r e n c e e q u a t i o n

We give some discussion here of the Hirota bilinear difference equation (HBDE) based on [7, 102, 154, 561, 740, 757, 830] (cf also Sections 3.6, 4.1.2, 4.1.3, and 4.1.4) The original form of Hirota was (cf [830])

af(A 4- I, #, ~,)f(A - 1, #, v) 4-/3f(A, # + 1, ~,)f(A, p - 1, ~,)+

where a + 3 + ~/= 1 It was shown by Miwa (cf [740]) that every KP tau function satisfies

20 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

To see the equivalence to (1.127) one writes (21C) f ( q + r + 1, p + r + 1, p + q + 1) = ~-(p, q, r)

with a = a ( b - c ) , 13 = b ( c - a ) and -~ = c ( a - b ) Thus solving to the HBDE (1.127) enables

which the KP hierarchy is defined Note here that (6.1) in Section 3.6 and (1.49) in Section 4.3.2 have the form (a +/3 + 7 = 0)

a g ( n , g + 1, m ) g ( n , e , m + 1) + ~g(n, e, m ) g ( n , e + 1, m + 1)+

+Tg(n + 1,i + 1, m ) g ( n - 1 , e , m + 1) - 0 (1.129) which is another variant on (1.127) (cf also [757] and we refer to [993] for a survey of

f o r m s - cf also remarks after (2.37) to connect (1.127) and (1.129)) Somewhat more general formulations can be developed via [102] and (1.120) in Section 4.1.2 Note with the normalization there X ~ 1 / ( A - #) one is working with Cauchy kernel type BA functions r

Note also that for b = 0 (1.131) takes the form

)~X()~, O, x) = (A - a)X(A, co, x - [a0]) Z(,X, x)

to the Fay trisecant formula)

1 1 6 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

In view of the Section 1.1.4 we go now to [374, 406, 891, 892, 910, 913] First in [891] the Moyal bracket deformation of KP is related to dKP The idea is to develop a deformed differential calculus based on an associative star product and obtain a description of the KP hierarchy in terms of vector fields and differential forms (rather that PSDO) In particular one wants a natural passage from KP with Lax structure to a dKP form, and moreover without scaling, in order to facilitate the study of geometry for KP Thus let M be a P manifold with a bilinear skew symmetric P bracket {u, v } = ~ i j ( O u / O x i ) ( O v / O x j ) satisfying the Jacobi identity in the form

{ { f , g } , h } + { { g , h } , f } + { { h , f } , g } = 0 (1.140)

(cf Chapter 3, (8.1)) The Moyal bracket is then defined as {u, V}M - - (ll, $ V - - V * lt)/t'C

The following properties are straightforward

lim,~ ,ou 9 v = uv; c 9 u = cu; lin~{u,_+ V } M = { U , V } (1.142)

identity (~ = - i h for Moyal, Fedosov, etc.) A different non-associative product is also introduced via

o 9

22 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

This product satisfies

d

u o v = v o u ; c o u = c u ; l i m ~ _ _ , o u o v = u v ; 2 - - ~ ( ~ u o v ) = u , v + v , u (1.144) This will lead to a deformed differential calculus satisfying the key property

dg = gxdx + gydy one has df A dg = (fx o gy - fy o gx)dx A dy = {f, g } M d x A dy Note t h a t

A A ( B A C ) ~ ( A A B ) A C The Lie derivative is defined via (15F) s = X I and ( s i =

x J o ( O Y i / O x j) - F J o (Oxi/OxJ); one writes s = [Z, Y] and d ( s - s for a p-form w The normal relations

[ X , Y ] f - X ( Y f ) + Y ( X f ) = O ; [ [ X , Y ] , Z ] + [ [ Y , Z ] , X ] + [ [ Z , Y ] , X ] = O (1.147)

do not hold for the deformed definitions but for (local) Hamiltonian vector fields ( 1 5 G ) X f =

wiJOifOj one has a version of this, namely

X l h = { f , h } M ; [ Z f , X g ] = Z{f,g}M; (1.148)

[xl, X~]h = X ~ ( X ~ h ) - X~(X~h), [[X~, X~], Xh] + [[X~, Xh], X~] + [[Xh, X~], X~] = 0

For two dimensional manifolds M such Hamiltonian vector fields generate the Lie algebra

S d i f f , ~ ( M ) of area preserving diffeomorphisms of M (area element dx A dy) and the compo- sition of two Hamiltonian vector fields is defined as the Lie bracket Explicitly X I generates infinitesimal transformations x + x - efy and y + y + efx (here e ~,, ~) The Kupershmidt- Manin bracket ( 1 5 B ) defines a bracket { f , g } ' - ( f 9 g - g , f ) / e and a corresponding o product is (e ~,, ~)

to the algebra of PSDO Thus for PSDO P = ~Nc~ aj(x)O j E • with ( 1 5 H ) i)ma -

aO m + ~ [ m ( m - 1 ) ( m - k + 1)/k!]OkaO m - k one has s y m ( P Q ) - s y m ( P ) *~=1 s y m ( Q )

for 9 as in ( 1 5 B ) so there is a natural relation to the standard KP hierarchy (cf [615] and earlier remarks)

Now let M = M | T where T ~,, times {tn} and for some 9 product on M write

u(x, t ) , v(x, y) = exp ( ~ z ij 0 0 ) u(x, t)v(2, t) (1.150)

(note X I = w ij ( O f ( x , t)/Oxi)(o/~)x j) will be time dependent) Let L = 0 + ~ Un(X, t)O -n

be the Lax operator for KP (cf now Section 1.1.1) One then applies the geometrical

1.2 V E R T E X O P E R A T O R S A N D C O H E R E N T S T A T E S 23

equivalent to the Sato hierarchy based on PSDO Similar calculations apply to Toda and

format w i t h / 2 , AzI etc as in Sections 1.1.1 - 1.1.3 It is in this geometrical context t h a t the o product and associated geometrical ideas are fruitful (details are o m i t t e d h e r e - cf also [910, 913]) It seems from this t h a t if one starts with d K P as a basic Hamiltonian

say is equivalent to K P (or n = 1/2 in [374]) The Bn (n) would perhaps have to be e x t r a c t e d from K P after establishing the isomorphism (cf [374]) and we t u r n briefly to this approach

d e t e r m i n e d by relating Vn and un in the form (n - 1/2)

n

Un " E 2-j V n _ j J (1.155)

0 where n = 0, 1 , - and vJ = O~vo

1.2 V E R T E X O P E R A T O R S A N D C O H E R E N T S T A T E S

We extract here from [153, 159] with considerable reorganization and clarification Notations ( A ) and ( A B ) are used

1 2 1 B a c k g r o u n d

We will give first a refinement of some ideas and formulas developed in [159] We begin by

24 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

a n d a t = z with e.g z E S 1 (but z E C is also permissible unless otherwise stated) T h e r e

is a v a c u u m I 0 > with hi0 > - 0 and one creates states in a Fock space via

In > - - ~ (at)nlO ; > aln > = x / ~ l n - 1 >; atl n > - v/n + l l n + 1 > (2.1)

with (A) [a, a t] = 1 and ataln = nln > while < rain > = 5mn For example one could choose

z e S 1 with a ~ Oz, a t = z, a n d I n > = zn/v/-~, with < h i m > = (1/2~ri) ~ z n 2 m ( d z / z ) T h e

o p e r a t o r s ~ a n d 15 can be defined via

1 ( a + a t i t = 1 ( a - a t ) ; [ih, O ] = - i ; ih~-iOq; (t"~q (2.2)

a n d to bring q u a n t u m mechanics (QM) into the picture we write (h ~ h = h/27r)

hA

qh v/hq; Ph v/-hP; Ph -iv/-hOq - i Oqh; ~h, O h ] - - i h (2.3) while coherent states are defined for z E C via

Note t h a t one can write

ah = - ~ ( ( t h + i~h) = a; = - ~ ( q h i~h) a t

(2.4)

(2.5)

while zath 5ah = (i/h)(pOh qPh) We set also (B) U(p, q) = e x p ( i / h ) ( p O h - qDh) "-' D ( z )

a n d I z > ~ IP, q >- Here we record also t h a t the coherent states I z > are defined via

alz > = ahlz > = zlz > and one has

D ( z ) = e-(1/2)lzl2eza~e-~ah = e (1/2)lzl2e-zaheza~ ; < zlz I > - e-(1/2)lzl2+2z'-(1/2)lz'j2

In a d d i t i o n (cf [591, 592]) one observes t h a t (2.4) results from writing Iz > = E ~ Cnl n >

a n d using ahlz > = zlz > to get v/n + lcn+l = ZCn from which Cn can be d e t e r m i n e d as in (2.4) Note also from (2.6) t h a t D(z)I0 > = exp[-(1/2)lzl2]exp(za~)lo > We see also from (2.6) t h a t coherent states are not orthogonal but t h e y are overcomplete with

U(p, q) ,.~ D ( z ) as in ( B ) with U* = U r ~ D t (z) = D ( - z ) ~ V ( - p , - q ) and set h - 1 / 2 l a > -

U(h-1/2a)lO > with V ( h - 1 / 2 a ) ~ D ( z ) and z = ( ~ + i T r ) / v / ~ T h e n consider (cf [159,452])

7r

E = < h - U 2 a ( q - ~ ) ( P - - - ~ ) 1 h - 1 / 2 a > = (2.11)

71"

<O,U*(h-1/2o~) ( q - - - ~ ) UU* I p - - ~ ) U[O>

where U ~ U ( h - U 2 a ) However from (2.5)-(2.6)

S U M M A R Y 1.1 T h u s in particular (cf (2.54) and ( A N ) ) the scaled variables (~

qh ~ ~ and P ~ ifih ~ 7r where ~, 7r correspond to classical values of ~,/~

We are trying to accomodate here a number of notations from [159, 452, 591,592, 796, 987] (cf also [392,^593, 594] for difference schemes) Another notation used is (C) (~ ~ qh

e 2 ~ h; this produced correct mathematical structures but the e 2 seemed misleading at times (and certain sections are improved with some reorganization) Thus in dispersionless

KP for example (cf [148, 149, 143, 153, 158, 160, 902]) one thinks of q -+ eq - Q and hence

,~ iOq o -ieOQ In [159] we took (D) a = [(Q/e) + eOQ]/X/'-2 = (Q + i [ ' ) / e v ~ - a(e)

where P = -ie2OQ One notes t h a t a(e) ~ ah for e 2 = h and this is consistent with

qh = v/-hq = eq and iPh = ix/~p = v/hOq = hOQ while a(e) = at, = a W h a t is slightly misleading is the fact t h a t in [902] for example one thinks of h as the scaling p a r a m e t e r

26 CHAPTER 1 Q UANTIZATION AND INTEGRABILITY

instead of e = v/-h; this obscures the nature of the quantum relations based on formulas

in QM to the classical coordinate ~ There have been many papers on various aspects of

[434, 435, 436, 548, 549, 723, 873, 987] for developments involving coherent states and cf also [315,345, 1007, 1008, 1009]) We are mainly concerned here with comparisons to scaling

in dispersionless soliton structures

In a more QM spirit one has position and m o m e n t u m representations via (cf [842]) (E) QIQ' >= Q'IQ' > and < Q'IQ >= 5(Q'-Q) with 1 = f d Q [Q > < Q[ and I(~ > -

<~ Q l a > = Ca(O); </31a > = f dQ </31Q > < Qla > = f dQ~z(Q)r ) (2.14) Also in general (F) </31Ala > - f f dQdQ ' - Cz(Q) < QIA] Q' > Ca(Q') For the m o m e n t u m

5 ( P - P " ) and 1 = f d R [P > < Pl while Is > = f d R [P > < Pla > and

In this context vacuum vectors 10 > such that hi0 > = 0 (~ ah]O > = 0) can be represented

If one starts out with Bose operators in a Fock space there is a priori no reference to

related to physical objects q and p while Q ,,~ x / ~ a n d / 5 = x/~i5 lead to classical objects and 7r as above Another approach to classicalization is described in [1007, 1008, 1009] Thus first one wants to avoid the often ambiguous h ~ 0 approach (see however [315, 345]) Secondly one should begin with q u a n t u m mechanics and reverse or retrace the geometrical quantization program where a symplectic geometry and classical phase space (hence clas- sical Hamiltonian dynamics) are posited in advance This leads to the approach of [987] where the geometric quantization theme is reversed and one obtains a large N limit of a

q u a n t u m structure as classical mechanics Note however the retracing technique of [987] can

1.2 V E R T E X O P E R A T O R S A N D C O H E R E N T S T A T E S 27

.~ is the associated Lie g r o u p - see below) The path integral formalism of [548] gives still another direct connection In any event coherent states seem to play a fundamental role in the quantum-classical correspondence and since the quantum picture appears more basic (cf [345]) one is led to the path of classicalization from quantum to classical and various forms of semiquantal mechanics (instead of semiclassical) The approach of [1007, 1008, 1009] starts from the geometrical structure of a quantum system determined by a basic Lie algebra ~

mathematical image of a quantum system is an operator algebra ~ in a Hilbert space T/and one can restrict to an irreducible representation (irrp) of G to describe dynamical proper- ties) Let Ir > be the lowest weight bound state of an irrp of G and let H be its maximal stability subgroup Roughly sketching from [1007, 1008, 1009] (with further detail to follow)

has an explicit symplectic form

w = i h O 2 ~ ( z ' Y.)OziO2j d zi A dhj; )~(z,2) = d e t ( I 4- ztz) +=- (2.18)

of G / H , find ~ = h | k is a standard Cartan decomposition (h = L i e ( G ) and [h, hi C

associated irrep space is finite or infinite dimensional) and = is called a quenching index

induced by the fixed point of Ir >) Given this structure one defines canonical coordinates

pear in the papers listed in [1007] and also with variations in [1008] (cf also [63, 845]) Now the quantum phase space can be either compact or noncompact depending on the

one says that a quantum system with M independent degrees of freedom (2M dimen- sional phase space) is integrable if and only if there exists M quantum constants of mo- tion related to the eigenvalues of M commuting nonfully degenerate (NFD) observables

Ai (i = 1 , - , M) Here an operator A is fully degenerate if Alr > = cI~Pi > for all ~p~ E ~

(h = dim(it) and k = dim([~)) The metric of G / H is