21 June 2001

Physics Letters B 510 (2001) 264–270www.elsevier.nl/locate/npe

Properties of Moyal–Lax representation

Ashok Dasa, Ziemowit Popowiczb

a Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627-0171, USAb Institute of Theoretical Physics, University of Wrocław, 50-205 Wrocław, Poland

Received 9 March 2001; received in revised form 29 April 2001; accepted 3 May 2001Editor: M. Cvetic

Abstract

The properties of standard and the nonstandard Moyal–Lax representations are systematically investigated. It is shown thatthe Moyal–Lax equation can be interpreted as a Hamiltonian equation and can be derived from an action. We show that theparameter of non-commutativity, in this case, is related to the central charge of the second Hamiltonian structure of the system.The Moyal–Lax description leads in a natural manner to the dispersionless limit and provides the second Hamiltonian structureof dispersionless integrable models, which has been an open question for sometime. 2001 Elsevier Science B.V. All rightsreserved.

1. Introduction

Integrable models [1], both bosonic as well as su-persymmetric [2], have played important roles in thestudy of conformal field theories, strings, membranesas well as topological field theories. In recent years,it has become known that string (membrane) the-ories naturally lead to non-commutative field theo-ries [3], where usual multiplication of functions is re-placed by the star product due to Groenewold [4,5].It is interesting, therefore, to ask if integrable sys-tems can also be described in terms of star prod-ucts and Moyal brackets [5]. It appears that there aretwo possible approaches to this problem. In the firstmethod [6] one can formulate the soliton theory in thenon-commutative spacetime which is realized usingthe star product. In the second approach, the star prod-uct can be used directly in the Lax operator descrip-tion [7,8] or in the zero-curvature condition [9,10].

E-mail address:das@pas.rochester.edu (A. Das).

In this Letter, we follow the second approach andshow that it is possible to use the Moyal bracket asthe Poisson bracket in the phase space. Such an inter-pretation allows us to present a Moyal–Lax represen-tation for the soliton system and study such represen-tations systematically. So far, the Moyal bracket hasbeen used to construct only the standard Lax repre-sentation for bosonic integrable systems [7]. We showthat such brackets can be used to describe nonstan-dard representations as well. There are many inter-esting features that emerge from such a representa-tion and we argue that such a representation may, infact, be more desirable. Among the various interest-ing features that emerge, we note that the parameter ofnon-commutativity, in such an analysis, is directly re-lated to the central charge of the second Hamiltonianstructure of the system. Furthermore, in such a for-mulation, the Lax equation has a natural interpretationof a Hamiltonian equation and can be simply derivedfrom an action. Furthermore, such a description nat-urally leads to the dispersionless limit (in which themodels become related to membranes and topologi-

0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)00561-5

A. Das, Z. Popowicz / Physics Letters B 510 (2001) 264–270 265

cal field theories) and thereby allows us to derive theHamiltonian structures (first, second,. . . ), which hasbeen an open question for quite some time.

2. Basic definitions

Integrable systems are Hamiltonian systems and,therefore, are naturally defined on a phase space. Thestar product of two functions, on this space, is definedto be

A(x,p) � B(x,p)

(1)= eκ(∂x∂p−∂p∂x )A(x,p)B(x, p)∣∣x=x,p=p

.

The conventional Moyal bracket, then, follows to be

(2){A(x,p),B(x,p)

}κ

= 1

2κ(A � B −B �A).

Hereκ is the parameter of non-commutativity, which,as we will see, is directly related to the centralcharge of the second Hamiltonian structure in the caseof integrable models. From (1) and (2), it followsimmediately that

(3)limκ→0

{A,B}κ = {A,B},

where the last bracket in (3) stands for the standardcanonical Poisson bracket.

The star product gives the momentum an operatorcharacter. In particular, let us note that for any arbitraryintegerm,n (positive or negative)

pn � pm = pn+m,

(4)pn � f (x)=∑m=0

(n

m

)(−2κ)mf (m) � pn−m,

where(n

m

)= n(n− 1) · · · (n−m+ 1)

m! ,

(5)

(n

0

)= 1,

and f (m)(x) stands for themth derivative off (x)with respect tox. We note that these are preciselythe relations (up to normalizations) satisfied by thederivative operator.

With these, we can define two classes of Laxoperators on the phase space as

Ln = pn + u1(x) � pn−1 + u2(x) � p

n−2 + · · ·+ un(x),

Λn = pn + u1(x) � pn−1 + · · · + un(x)

(6)+ u(n+1) � p−1 + · · · ,

which we can identify, respectively, with the Lax op-erator for the generalized KdV hierarchy and the KPhierarchy [11]. In simple terms, we have replaced thespace of pseudo-differential operators by polynomialsin momentum, which nonetheless inherit an operatorstructure through the star product and define an alge-bra. We will call such an algebra the Moyal momen-tum algebra (Mm algebra).1 It is easy to check that allthe known properties of pseudo-differential operatorscarry through with suitable redefinitions. For example,for any two arbitrary operatorsA andB, which are el-ements of theMm algebra, the residue (the coefficientof thep−1 term with respect to the Moyal product) ofthe Moyal bracket can be checked to be a total deriva-tive, namely

(7)Res{A,B}κ = (∂xC).

Consequently, one can define

(8)TrA=∫

dx ResA,

which is unique (with the usual assumptions of asymp-totic fall off) and which satisfies cyclicity.

For a general Lax operatorΛn, it is straightforwardto show that

(9)∂Λn

∂tk= {

Λn,(Λ

k/nn

)�m

}κ, k = ln,

where k, l are integers, defines a consistent Laxequation, providedm = 0,1,2 and that the projectionsare defined with respect to the star product. Note herethatAk/n = A1/n � A1/n � · · · � A1/n involving k suchfactors and thenth root is determined formally ina recursive manner. The projection withm= 0 will bedenoted by()+ and the corresponding equation willbe known as the standard Moyal–Lax representation,

1 Notice that ourMm algebra is different from the concept ofpseudo-differential operators with the coefficients taken from theMoyal algebra introduced in [12].

266 A. Das, Z. Popowicz / Physics Letters B 510 (2001) 264–270

while the other two cases will be known as non-standard representations. Let us note the importantproperty

(10)limκ→0

(Λ �Λ′)

�m= (

ΛΛ′)�m

,

where the factors on the right-hand side are functionson the phase space (not operators). One of the advan-tages of this method is now obvious, namely, one cango to the Lax representation of the model in the dis-persionless limit in a natural manner [13]. In fact, ifwe consider the limitκ → 0, in such a model, it leadsto

(11)∂Λn

∂t= {

Λn,(Λ

k/nn

)�m

},

where the bracket on the right-hand side is the stan-dard Poisson bracket. With these definitions, one canconstruct the conserved charges as

(12)Hk = TrΛk/nn , k = ln,

where k, l are integers, prove that different flowscommute and can define Hamiltonian structures ina straightforward manner [11]. Let us illustrate theseideas through some examples.

3. Examples

3.1. KdV hierarchy

Let us consider the Lax operator

(13)L = p2 + u(x),

then, it is straightforward to calculate (remember theprojection is with respect to the star product)

(14)(L3/2)

+ = p3 + 3

2u � p − 2κ

2u(1),

whereu(1) = ∂u/∂x.Therefore, the Moyal–Lax equation

(15)∂L

∂t= {

L,(L3/2)

+}κ,

gives

(16)∂u

∂t= −

(κu3 + 3

2uu(1)

),

which is the KdV equation and the connection be-tween the parameter of non-commutativity and thecentral charge begins to emerge. However, we will seea more direct relation when we calculate the Hamil-tonian structure. The conserved quantities can be de-termined in a straightforward manner and the first fewhave the forms

H1 = TrL1/2 =∫

dxu

2,

H2 = TrL3/2 =∫

dxu2

4,

(17)H3 = TrL5/2 =∫

dx(4κu(2)u+ u3).

The commutativity of the flows follows directly fromthe Moyal–Lax representation [11].

Let us next turn to the question of Hamiltonianstructures. First, we define the dual to the Lax operatorin (13) by

(18)Q= p−2 � q−2 + p−1 � q−1,

which allows us to define linear functionals as

FQ(L)= TrLQ =∫

dx uq−1,

(19)FV (L)= TrLV =∫

dx uv−1.

Then, one can define, in a standard manner, the firsttwo Hamiltonian structures of the system as{FQ(L),FV (L)

}1 = TrL � {Q,V }κ ,{

FQ(L),FV (L)}

2

= Tr(({L,Q}κ)+ � (L � V )+ (Q � L)+ � {L,V }κ

)

(20)+ 1

2

∫dx

( x∫Res{Q,L}κ

)(Res{V,L}κ).

A direct calculations yields

{u(x),u(y)

}1 = 2

∂

∂xδ(x − y),

{u(x),u(y)

}2 =

(u(x)

∂

∂x+ ∂

∂xu(x)+ 2κ2 ∂3

∂x3

)

(21)× δ(x − y),

which are indeed the two Hamiltonian structures ofKdV. Furthermore, the parameter of noncommutativi-ty, κ , is now seen to be directly related to the central

A. Das, Z. Popowicz / Physics Letters B 510 (2001) 264–270 267

charge of the second Hamiltonian structure, which isknown to be the Virasoro algebra.

3.2. Two boson hierarchy

Let us next consider the Lax operator

(22)L = p − J0 + p−1 � J1.

It is easy to calculate (projection with respect to thestar product)

(23)(L2)

�1 = p2 − 2J0 � p,

which, then yields from the nonstandard Moyal–Laxequation2

(24)∂L

∂t= {(

L2)�1,L

}κ,

the two boson equations

∂J0

∂t= (

2J0 + J 20 − 2κJ ′

0

)′,

(25)∂J1

∂t= (

2J0J1 + 2κJ ′1

)′,

where the prime denotes derivative with respect tox.Once again there is already a hint of the relationbetween the parameterκ and the central charge whichwe will see more explicitly soon. The conservedquantities of the system are defined as

(26)Hn = TrLn,

and the first few have the explicit forms

H1 =∫

dx J1,

H2 = −2∫

dx J0J1,

(27)H3 = 3∫

dx(J 2

1 + J 20 J1 − 2κJ ′

0J1).

To study the Hamiltonian structures we define the dualto L as

(28)Q = q0 + q−1 � p−1,

2 Notice that if we take the projection with respect to the usualproduct (and not the star product), then, the equation becomesinconsistent.

so that the linear functionals take the forms

(29)FQ(L)= TrLQ =∫

dx (q0J1 − q−1J0).

The first two Hamiltonian structures can now bedefined in a straightforward manner and have theforms

(30){FQ(L),FV (L)

}1 = TrL � {Q,V }κ ,{

FQ(L),FV (L)}

2

= Tr(({L,Q}κ)+ � (L � V )+ (Q � L)+ � {L,V }κ

)−

∫dx

(Res{Q,L}κ Res

(L � V � p−1)

− Res{V,L}κ Res(L �Q � p−1))

(31)+∫

dx

( x∫Res{Q,L}κ

)(Res{V,L}κ).

A straightforward calculation yields( {J0, J0}1 {J0, J1}1

{J1, J0}1 {J1, J1}1

)= −

(0 ∂

∂ 0

)δ(x − y),

( {J0, J0}2 {J0, J1}2

{J1, J0}2 {J1, J1}2

)

(32)=(

2∂ ∂J0 − 2κ∂2

J0∂ + 2κ∂2 ∂J1 + J1∂

)δ(x − y),

which are the usual Hamiltonian structures of the twoboson hierarchy. The second Hamiltonian structure, inparticular, is the bosonic limit of theN = 2 twistedsuperconformal algebra (one has to redefine the basisto make an exact identification) [14] and the relationbetweenκ and the central charge of the algebra is nowexplicit.

It is known that the two boson equation reducesto many other integrable models. Without going intodetails, let us note that if we identify

(33)J0 = −q ′

q, J1 = qq,

then the Lax operator in (21) can be rewritten as

(34)L= q−1 � L � q,

where

(35)L= p + q � p−1 � q.

In other words, the two Lax operatorsL and L arerelated through a gauge transformation. It is now easy

268 A. Das, Z. Popowicz / Physics Letters B 510 (2001) 264–270

to check that the standard Moyal–Lax equation

(36)∂L

∂t= {

(L)2+, L}κ,

leads to the non-linear Schrödinger equation while,with the identificationq = q , the equation

(37)∂L

∂t= {

(L)3+, L}κ,

yields the MKdV equation.

4. Moyal–Lax representation as a Hamiltonianequation

The conventional Lax equation (in the standardrepresentation)

(38)∂L

∂tk= [(

Lk/n)+,L

],

resembles a Hamiltonian equation with(Lk/n)+ rem-iniscent of the Hamiltonian. However, such a relationcannot be further quantified in the language of pseudo-differential operators. In contrast, we will show nowthat the Moyal–Lax representation has such a naturalinterpretation.

For concreteness, let us consider an arbitrary flowin the KdV hierarchy described by

(39)∂L

∂t= {

L,(L(2n+1)/2)

+}κ.

Let us next consider an action of the form

(40)S =∫

dt(p � x − (

L(2n+1)/2)+).

The important point to remember, in this, is thefact thatL = L(p,x), but does not depend on timeexplicitly. Thus, we can think of(L(2n+1)/2)+ as theHamiltonian on the phase space. That this is truefollows from the Euler–Lagrange equations of thesystem, namely

x = ∂(L(2n+1)/2)+∂p

= {x,

(L(2n+1)/2)

+}κ,

(41)p = −∂(L(2n+1)/2)+∂x

= {p,

(L(2n+1)/2)

+}κ.

These are indeed Hamiltonian equations with Moyalbrackets playing the role of Poisson brackets, pro-

vided we identify the Hamiltonian of the system with(L(2n+1)/2)+. It also follows now that, sinceL isa function on this phase space,

(42)∂L

∂t= {

L,(L(2n+1)/2)

+}κ.

Namely, the Moyal–Lax equation is indeed a Hamil-tonian equation with(L(2n+1)/2)+ playing the role ofthe Hamiltonian. Furthermore, the Moyal–Lax equa-tion, as we have shown, can be derived from an ac-tion. Furthermore, although we have shown this fora standard Moyal–Lax representation, it is clear thatthis derivation will go through for nonstandard repre-sentations as well.

5. Hamiltonian structures for dispersionlesssystems

The Moyal–Lax representation, of course, has thebuilt in advantage that one can go to the dispersionlesslimit of an integrable system by simply taking the limitκ → 0. While the Lax representations for various dis-persionless integrable models are known [15–17], thedetermination of the Hamiltonian structures (at leastthe second) from such a Lax function has remainedan open question. The Moyal–Lax representation pro-vides a solution to this problem in a natural way. Letus illustrate this with two examples.

First, let us consider the KdV hierarchy, which inthe dispersionless limit, goes over to the Riemannhierarchy [15]. With

L= p2 + u, Q= p−2q−2 + p−1q−1,

(43)FQ(L)= Tr LQ =∫

dx uq−1,

we note that the definition of the first two Hamiltonianstructures (19) reduces, in the dispersionless limit, to(κ → 0){FQ(L),FV (L)

}1 = TrL{Q,V },{

FQ(L),FV (L)}

2

= Tr(({L,Q})+LV + (QL)+{L,V })

(44)+ 1

2

∫dx

( x∫Res{Q,L}

)(Res{V,L}).

A straightforward calculation leads to

A. Das, Z. Popowicz / Physics Letters B 510 (2001) 264–270 269

{u(x),u(y)

}1 = 2

∂

∂xδ(x − y),

(45){u(x),u(y)

}2 = (

u(x)+ u(y)) ∂

∂xδ(x − y).

These are indeed the correct Hamiltonian structuresof the Riemann equation and while the first structurewas already constructed from the Lax function, theconstruction of the second structure, from the Laxdescription, was not known so far [15].

As a second example, let us consider polytropicgas [16] withγ = 2 which can be thought of as thedispersionless limit of the two boson hierarchy. In sucha case, the Lax function has the form

(46)L = p + u+ vp−1.

Defining the dual and the linear functional as

Q = q0 + q−1p−1,

(47)FQ(L) = TrLQ =∫

dx (uq−1 + vq0),

we note that the definition of the first two Hamiltonianstructures follows from the dispersionless limitκ → 0of Eqs. (30), (31) to be{FQ(L),FV (L)

}1 = TrL{Q,V },{

FQ(L),FV (L)}

2

= Tr(({L,Q})+LV + (QL)+{L,V })

−∫

dx(

Res{Q,L}Res(LVp−1)

− Res{V,L}Res(LQp−1))

(48)+ 1

2

∫dx

( x∫Res{Q,L}

)(Res{V,L}).

A simple calculation yields( {u,u}1 {u,v}1

{v,u}1 {v, v}1

)= −

(0 ∂

∂ 0

)δ(x − y),

(49)

( {u,u}2 {u,v}2

{v,u}2 {v, v}2

)=

(2∂ ∂u

u∂ ∂v + v∂

)δ(x − y).

These are indeed the correct Hamiltonian structuresof this model. We would like to emphasize that itwas not known so far how to derive the secondHamiltonian structure from the Lax description of thesystem [16]. We would also like to note here that

the Lax function for the polytropic gas with arbitrary,largeγ [16] is highly constrained. Consequently theformulae in (48) need to be modified further (even forthe first Hamiltonian structure) and we have not yetanalyzed this question.

To summarize, we have studied the properties ofMoyal–Lax representation systematically in this Let-ter. In addition to the fact that they allow a smoothpassage to the dispersionless models, we have shownthat the parameter of non-commutativity is related tothe central charge of the second Hamiltonian structureof the system. We have shown that the Moyal–Laxequation can be interpreted as a Hamiltonian equa-tion and can be derived from an action. We have alsoshown how the Moyal–Lax description leads in a nat-ural manner, in the dispersionless limit, to the Hamil-tonian structures of dispersionless integrable modelswhich has been an open question for sometime. Inmany ways, this alternate description of integrable sys-tems seems more desirable. Properties of the Moyal–Lax representation for supersymmetric integrable sys-tems will be described separately [18].

Note added

We would like to thank C. Zachos for pointing out tous that the star product is really due to Groenewold [4].We would also like to thank I. Strachan as well as thereferee for bringing [7], which has some overlap withour work, to our attention.

Acknowledgements

One of us (A.D.) would like to thank the organizersof the 37th Karpacz Winter School as well as the mem-bers of the Institute of Theoretical Physics, Wrocławfort hospitality, where this work was done. This workwas supported in part by US DOE Grant No. DE-FG02-91ER40685 and by NSF-INT

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