The connection between the Riemann-Hilbert transformation and the Poisson bracket for Lax-pair systems

16 February

1998

PHYSICS

EISEVIER

LETTERS

A

Physics Letters A 238 ( 1998) 344-348

The connection between the Riemann-Hilbert transformation and the Poisson bracket for Lax-pair systems Mo-Lin Ge”, Kang Xueavb a Cenfer for Advanced Study, &in&a University, Beijing looO94, China b Department of Physics, Northeast Normal University, Changchun, JiLin 130027. China Received

29 July 1997; revised manuscript

received 15 November 1997; accepted Communicated by CR. Doering

for publication

17 November

1997

Abstract

The Riemann-Hilbert transformation (RHT) for Lax-pair systems is shown to be connected with the Poisson bracket description. We set up the classically canonical formulation for the variation of 0(x, r; A) in terms of the Poisson bracket between the corresponding conserved quantities and @(x, t; A), where @(x, t; A) satisfies the Lax-pair condition and the variation preserves the equation of motion. @ 1998 Published by Elsevier Science B.V.

1. The canonical formulation is important before quantization. If S$ is the classical variation of a field quantity +, the canonical formulation can be set up through

where Q is a conserved quantity corresponding to the transformation SC/l and { , } stands for the Poisson bracket. Further, by using the correspondence principle ( 1/i) { , } = [ , ] (Fi = 1)) the quantization is established on the basis of the fundamental commutators. For linear theory S$ is linear, i.e. (Sti)i is a linear transformation of {+j} multiplied by infinitesimal parameters {a’}, for example, for isospin we have SJI; = (Y”( Z,)&jj, where Ia are generators of isospin. The conserved quantity is nothing but the field form of the isospin operator. The approach works well for linear models. However, in the case of nonlinear models, in general, there has not been such a formulation even for nonlinearly integrable models with Lax pairs, 0375-9601/98/$19.00 @ 1998 Published PIISO375-9601(97)00904-3

where @(x, t; A) was introduced to linearize ear equations through the Lax pair

nonlin-

a,@(& t; A) = V(x, t; A)@(x, t; A),

(2)

a,@( x, t; A) = M(n, t; A)@( x, t; A),

(3)

where A is the spectral parameter. A solution to the solution transformation S@ was introduced, i.e. S@ preserves the Lax pair (2) and (3) in such a way that Eq. (2) is subject to a gauge transformation [ l-5 ] @‘(A) =X(A)@(A),

(4)

V’(A) = X(A)V(A)(X(A))-’ - (&X(A))(X(A))-‘,

(5)

where for simplicity only the A-dependence is emphasized. Hereafter the x, t dependence of X(A) and O(A) should be understood. As pointed out in Refs. [2,3,6] the transformations (4) and (5) can be performed by the Riemann-Hilbert transforma-

by Elsevier Science B.V. All rights reserved

M.-L. Ge. K. Xue/Physics

tion (RHT) . The RHT consists of constructing X(A) by the analyticity and continuity on an integral contour [l-3,6] with the boundary requirement X(A=cQ)

=I.

(6)

The standard calculation gives the integral expression for X(A) which takes the form

V’(A)-V(A)&f - dp P-A x

GGG + 6zw

Letters A 238 (1998) 344-348

As is known, the RHT for Lax-pair systems only describes their geometric integrability. To our knowledge there has not been a general form of connection between the RHT and the canonical formulation ( 1) . In this Letter we shall set up such a connection for ultralocality models.

(7)

Ref. [ 93 and denoting

2. Following &A)

- VPu>~)

345

= V(x,A)

G31,

&II, A) = I @ V(x, A),

Poisson bracket at the same t reads

the fundamental

where CT)

=@(A)n(A)(@(A))-’

(8)

and A( A) must satisfy the two requirements &A( A) = 0,

(9)

A( A) belongs to the same group as Q(x) When the infinitesimal Eq. (7) becomes &V(A)

= & f

transformation

.

= [r(A-~),;(x,A)+:(4..~)lS(x-y),

(13)

where I( A) satisfies the classical Yang-Baxter equation [9,11]. In order to connect Eq. ( 13) with Eq. ( 11) let us rewrite Eq. ( 1) in terms of

(10)

is concerned

&@@I,

&I@@

in the form

~(a,G,(r)+[G,(t),v(r)l),

a,&

(11)

G&

(i=

1,2).

(14)

By making use of

where &{&(x,A),&,l~)} C,(A)

= @(A)MA)(+(A))-’ = {~(x,A),;(y.~)~(y,~)}

with the involution of infinitesimal parameters {a,}. Eq. ( 11) is valid for many ( 1 + 1 )-dimensional integrable models such as the Chiral model, the LandauLifshitz equation, and the sine-Gordon equations [ l-

81. Here we emphasize that Eq. (3) can be replaced by the fundamental Poisson bracket canonical equation [9,10],

v,+{H,v}

+

~(y.ru){di(*.A).~ty,~)},

(15)

we find the solution {~(x~A)&+)] =

h-4

sdy’t&‘,

pu)1-I

-L through H(Ac)

= trT~(Ao),

where

x

{B~~x,A),~(v’,IL)}~(Y’,~). (16)

which can be checked directly. Similarly i.e. the transfer matrix. This is why only the gauge transformation (5) for Eq. (2) is concerned.

&{&,A),&+)}.

by using

M-L.

346

Ge, K. Xue/Physics

we obtain

{&XV A),:(W)) X = C&X, A)

dx’(&(x’,

A))-’

J -L

x

(17)

{~(X’,A),:(Y,~)}~(XI,A).

For ultralocal models the fundamental Eq. ( 13) holds and we obtain

Poisson bracket

Letters A 238 (1998) 344-348

where TL( A) is given by Eq. (12). If @(x, t; A) does not have the uniform limit shown by Eq. ( 12) it should be redefined by reducing the “bad” asymptotic behavior. The main idea of the above calculations, in fact, follows those given in Ref. [ 91, though we have made some deformation for our later discussions. In order to express the right-hand side of Eq. ( 11) by the Poisson bracket we start from Eq. (18). Leftmultiplying Eq. ( 18) with (@I (x, A))-’ and fixing y ( x > y) , then putting x = L + 00 we obtain (~L(A))-'{~L(A),~(Y,~)}

(da(x,A).~(y,~)}=B(x-y)~(x,A)(~(y.A))-’ = (~(Y,A))-‘[~(A-)(L),;(~,A)

(18)

x [T(A-~),~(Y,A)+;(~,cL)I~(Y.A), where 0 (X - y) is the step function.

+ &Yddl

Noting that

(22)

&YA.

By making use of ~{(~(*,A))-‘r(l-~)~(r,A)~=

(&,A))-’ (~(y,h))-l[r(h-/l),~(~.A)l~(y,A)

x [r(A---),t(X,A)l&(x,A)

(i=

1,2)

=a,((~(y,A))-‘r(A-~)~(y,A)) (19)

(23)

it follows that

and on account of ~[(~(y,A))-‘(;l(y.~))-‘r(A-p) x ~(YJ&YJ~I = !~(y,p))-‘$[(~(y.A))-‘r(A Y

- p) which by left-multiplying with A’(A) trace over the first space leads to

x &Y, A) 1 &Y, EL) + (~(y,A))-‘gI(~(y.p))-‘r(A Y

- ru>

x ~~(YJJ)&Y~L

trl(~(A)(~L(A))-l{~L(A),~(y,l*)}) (20)

we have

= trr (a, &y,

d(xv A), &Y,P)} = 6(x - y) h(n, A) $(y,p) x$t(~(y,A))-‘(~(y,~))-‘r(A-~) ? (21)

Substituting Eq. (21) into Eq. ( 16), and taking x > y = L -+ 0~). we have the well-known result [9,10] = [r(A - ru), &.(A)&.(IL)~.

A) (25)

where a,A(A)

= 0

(26)

has been used and trr denotes the trace over the first space in the tensor product. Further, to find the connection between Eq. (25) and Eq. (11) we assume that the r(A) -matrix takes the rational form r(A-p)=-

{+L(A),&(cL)}

A)r(A - pu> + &y,

x lIr(A-~)~~(y~~)l)~

x ~(y.A)~(y,~)l(~(y.~L))-‘.

and taking the

h A-P

c

I, @ I,, D

(27)

M.-l.. Ge, K. Xue/Physics

where I,, are generators of a simple Lie algebra, h an arbitrary constant and the normalization is tr( I,Ib) = /3S,, (p = const). The form (27) works for many physical models. It is easy to prove that

Letters A 238 (1998) 344-348

AV(x,A) =

&

331

d~ftlU)(trTLtLL),V(X,h)} c

(34) When

f(P) = Pm? We thus from Eq. (25) obtain trl(,jl(A)(~r.(~))-‘{~L(r). = &(&,,A)

we have A(m)V(x,

$(Y,p))) +

(35)

A) = {T(“‘+~), V(x, A)},

(36)

where

&,ALhJ.PH). (29)

(37)

Taking the integral contour along c we get and

&

d/*tr,(~(~)(~L(l*))-‘{~L(~).~(Y,~)}) [?,?@)I

p’ c =

(30) which in comparison

to Eq. ( 11) yields

s, ~(*,A,= qg

where an infinitesimal transformation parameter is included in A’ (A). There are many possibilities to choose n(p). We only give some examples. (i)

n(p)

= (cu”Z,)~“* G CynAjlnt),

(32)

where LYOare infinitesimal parameters. As was shown in Refs. 1261 this gives rise to the loop algebra [ &nr) a(n) ‘I ’ h I= Cib8im+“). This is the usual case. (ii)

A(,u) = a@rf(p)T~(p),

8 = aA,

(33)

where rw is an infinitesimal transformation parameter and f(p) can be an arbitrary function of the spectral parameter p. Eq. (30) then becomes

= 0,

(38)

i.e. r(“‘) are conserved quantities. The superscript indices (m) indicate the variation arising from the different m in Eq. (34). Eq. (33) indicates that the A(p) is group-valued in contrast to example (i), where they are Lie-algebra-valued. Eq. (34) is a modelindependent general form. If we substitute the V(A) and Lax pair for the Landau-Lifshitz equation into Eq. (34) it gives the result shown in Ref. [ 121, where V(A) = iAS’Z, ( Sa are spin operators). 3. Through the above discussion we have seen that there is a connection between the RHT approach and the classical inverse scattering formulation through the fundamental Poisson bracket. The key point is the coincidence between the RHT approach ( 11) and the Poisson bracket (34) through Eq. ( 1) for the systems associated with the r-matrix shown by Eq. (27). Thus we reformulate the conclusion made in Ref. [ lo] from the point of view of the RHT approach. Furthermore, by substituting a particular form of V( x, t; A) and the corresponding conserved quantities r(“+‘) into Eq. (36), the r.h.s. of Eq. (36) can be calculated explicitly because both the V and r(““’ ) are formed by the basic field quantities whose Poisson brackets and first few conserved quantities are known. Eq. (36) takes the same form for either linear systems or nonlinear systems. However, for nonlinear models 6V is expressed in terms of the RHT that reflects the nonlinearity. The validity of Eq. (36) for nonlinear

348

M.-L. Ge. K. Xue/Physics

models may provide a new possibility of quantization for Lax-pair systems with ultralocality. It is interesting to extend the connection between the RHT (geometric integrability) and Hamiltonian integrability to the self-dual Yang-Mills (SDYM) equation. Following the Chau-Yamanaka theory [ 133 the SDYM belongs to the second class of constraint of Dirac [ 121. Hence, the Poisson bracket should be replaced by the Dirac bracket for SDYM. The calculation is lengthy and will be reported elsewhere. Of course, another interesting extension is to relate the works in Refs. [ 14,151 to the Poisson bracket approach through the method discussed in this Letter. This work was in part supported China.

by the NFS of

References [I] C.M. Cosgrov, J. Math. Phys. 23 (1982) 615. [2] Y. Nakamura, K. Ueno, Phys. Lett. B 107 (1983) 337. [3] Y.S. Wu, Commun. Math. Phys. 90 (1983) 461.

Letters A 238 (1998) 344-348 141 Y.S Wu, M.L. Ge, Vertex Operator in Mathematics and

Physics, eds. J. Lepowsky, S. Mandelstain (Springer, Berlin, 1984) pp. 329-352. ]5] L.-L. Chau, M.-L. Ge, Z. Popowitz, Phys. Rev. Lett. 52 (1984) 1940. ]6] L.-L. Chau, M.-L Ge. J. Math. Phys. 30 (1989) 166. ]7] L.-L. Chau, Geometrical integrability and equations of motion in physics: A unifying view. Nakai Workshop on Geometrical Integrability, 18-22 August, 1987; L. Dolan, Phys. Len Rev. 47 (1981) 1371; C. Devchand, D.B. Fairie, Nucl. Phys. 47 (1981) 1371; M.-L. Ge, Y.S. Wu, Phys. Lett. B 108 (1982) 411. ]8] H.J. de Vega, plenary talk, 14th ICGTMP, Seoul, 1985 (World Scientific, Singapore), and references therein. [9] L.D. Faddeev. L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer, Berlin, 1987). [IO] N. Yu Reshitikhin, L.D. Faddeev. Theor. Math. Phys. 56 ( 1983) 323 [in Russian]. [ 111 A.A. Belavin, V.G. Drinfeld, Triangle Equations and Simple Lie Algebras, Soviet Mathematical Reviews Sec. C, Vol. 4, ed. S.P. Novikov. [ 121 D. Bernard, Inter. I. Mod. Phys. B 7 (1993) 3517. [ 131 L.-L. Chau, Yamanaka, Phys. Rev. Len 68 (1992) 1807; 70 (1993) 1916. [ 141 1. Bakas, Nucl. Phys. B 428 (1994) 374. [ 151 A.D. Popov, CR. Preitschopf, Phys. Len. B 374 ( 1996) 71.