A new nonperturbative approach for nonlocal conserved currents

6 October 1994 PHYSICS LETTERS B

ELSEVIER

Physics LettersB 337 (1994) 95-101

A new nonperturbative approach for nonlocal conserved currents Huan-Xiong Yang a, Yi-Xin Chen a,u "Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 31002Z China b Institute of Theoretical Physics, Academia Sinica, Beijing 100080, China

Received 14 June 1994 Editor: H. Georgi

Abstract

Based on the chiral quantization, we develop a nonperturbative method for nonlocal conserved currents for Toda-type integrable systems. The sine-Gordon model and the ZMS model are dealt with uniformly in this scheme, just as these two models are treated in the perturbed CFT framework. Although there only exist two nonlocal conserved charges, this scheme can provide all the useful data of physics concerned, e.g., the braiding relations, the spectra-dependent S-matrices of the solitons, etc.

In our previous paper [ 1 ], it has been shown that the Chang and Rajaraman's (CR's) nonperturbative method [2] in the traditional canonical quantization scheme is not universally appropriate for studying, e.g. the quantum group symmetries, factorizable S-matrices and the integrabilities of nonsimply laced affine Toda systems of the famous ZMS model. CR's nonperturbative framework can merely display the partial quantum group symmetries of nonsimply laced affine Toda theories, which are not enough to determine the physical soliton S-matrices for such systems. Taking account of this fact, it is necessary to pursue new nonperturbative methods to evaluate nonlocal conserved currents, which should be applicable to the investigation of the quantum integrabilities of both simply laced affine Toda systems and their nonsimply laced analogues. In a sense, the present paper is a primary attempt in this direction. In the following, we suggest a candidate for the desired nonperturbative nonlocai current approach. Our approach is based on the chiral quantization prescription, i.e., the light-cone coordinate x_ = - (1/~/2) X (X 0 -- X 1) or X+ ~- ( 1//V~) ( x O + x 1) being characterized as the time-evolution parameter. This quantization prescription is completely different from both the radial quantization used by Bernard and LeClair to study nonlocal currents in the perturbation theories of the conformal field theories (perturbed CFT) [3] and the canonical quantization used in CR's nonperturbative method. As a matter of fact, it is well known that in the conformal field theories (CFT) the radial quantization is intrinsically equivalent to the traditional canonical quantization. Below, we will see that the employment of the chiral quantization will lead to constructing in a nonperturbative way two nonlocal conserved currents for affine Toda systems, regardless whether they are simply laced or not, and that these two currents will cover most of the physical information about these systems. For definiteness let us consider the sine-Gordon and the ZMS models concretely. These two models are the most typical representatives of the simply laced Toda and nonsimply laced Toda systems, respectively, of which the classical actions can be written into the following unified form

S=

d2x - - - 0 . q ~ 0 ~ b + ~ 1 ( e i ~ + e - i ' ~ )

.

0370-2693/94/$07.00 © 1994 ElsevierScience B.V. All rights reserved SSD10370-2693(94)00973-2

(1)

96

H.-X. Yang, E-X. Chen /Physics Letters B 337 (1994) 95-101

where/3 is a real constant and the parameter " s " is assumed to be among the set { 1, ~, l 2}. When " s " is taken as one, the action (1) describes the sine-Gordon model. Otherwise (1) describes the ZMS model. It can easily be seen

from ( 1) that the equations of motion in laboratory coordinates is O~c3~ 4 ' - 2"n-/A/3(e i ~ e k

-

se -isle,) = 0.

(2)

By using the light-cone coordinates x±, Eq. (2) can alternatively be expressed as O+ O_ 4 ' + "rri)t/3( e i~e" - s e - / s ~ ) = 0 .

(3)

The purpose of this paper is to investigate the nonlocal currents of the quantum versions of systems (1) in the lightcone quantization framework and to study the integrabilities of them. For a more transparent treatment, we choose the light-cone coordinate x_ as the time-evolution parameter. Then it is apparent that (3) is the Euler-Lagrange equation from the following Lagrangian density ~=

~

1

(4)

O+ 4"0_ 4 ' - ½A( e i~4~ + e -i,~4,) .

It deserves to be pointed out that the chiral quantization scheme does not simply describe the results of the conventional canonical quantization in the light-cone coordinate system because the light-cone coordinate x_ (rather than x °) will be regarded as " t i m e " variable throughout the chiral quantization scheme we work. Keeping this fact in mind, we see from the definition of the canonical energy-momentum tensor that T ~, ~ = c3( c3~"4' ) O ~4' - .Z~ag u ~,

(5)

tz, v = + , - ,

of which the off-diagonal components are T±. :~ = 1/4~-(0 :~4') 2. This means that the energy-momentum tensor for system (4) in the chiral scheme is not symmetric and that the angular momentum which generates the Lorentz transformation will not be conserved with respect to x _ - " t i m e " . Hence T,~(4) is the non-physical energymomentum tensor of the system (4) since both the sine-Gordon field theory and the ZMS theory are Lorentz invariant in the usual space-time. Now we perform the chiral quantization for the considered system (4). In order to do so, let us first describe the classical dynamics in Hamiltonian formalism. According to the light-cone time x_, the canonical momentum conjugate to 4' is ~'4,= ( 1/4~r) 0 + 4'. Thus the "velocity" 0_ 4' can not be solved in terms of the momentum 7r4,, which means that there exist an infinite number of constraints with respect to the spatial points ~'(x) = Try(x)

1

~

O+ 4'(x) = 0 .

(6)

Recalling the equal-light-cone Poisson bracket, defined by {A(x),B(y)}x_=y

- = f d~+ [ S A ( x )

~B(x__..__.~) _

La4'(O a~r~(O

6B(x)]

6a(x) a~-~(O ~-~-~J '

we get the Poisson bracket between the constraints {~(x), ~ ( Y ) l x _ = y _ = - 1/27r0+a(x+ - y + ) ,

(7)

which turns out to be a kernel rather than a matrix. Following Dirac's quantization prescription for the system involves second-class constraints [ 4], and the Poisson bracket must be modified for the system under consideration. Note that the inverse of the kernel (8), which is defined by f d,~+ ZI(X, st) {~(~), ~ ( y ) } (

=y_ =~(X+ - - y + ) ,

H.-X. Yang, Y.-X. Chen/ PhysicsLettersB 337 (1994) 95-101

97

is A(x, y) = - zre(x+ - y + ) ,

(8)

where E(x) denotes sign(x). Then the expected new Poisson bracket, usually called the Dirac bracket, for the considered system (4) is defined as {A(x), B ( y ) }~* =y_ /o

= {Z(x), B(y)}x_=y - +TrJ j d~+ dr+ [{a(x), ~(~:) Ix_ =~_ "(~+ - ( + ) { ~ ( ~ ' ) , B(y) lc+=y+] • In particular, we have { ~ b ( x ) , ~b(y) Ix*_ = , _

= - ~-e(x+

-y+)

.

(9)

With this Dirac bracket replacing the naive Poisson bracket, we acquire a well-defined Hamiltonian description for the system (4) in the light-cone coordinate framework. The evolution of our system in light-cone "time" x_ will be governed by the above Dirac bracket and the following Hamiltonian quantity H = ½h J dx+ ( e i Z ~ + e - i s ~ ) .

(10)

The chiral quantization for system ( 1 ) is carried out by regarding the field ~b(x) and its conjugate momentum 7r,~(x) as Hermitian operators in Hilbert space, and postulating that these operators obey the following equal-lightcone commutation relation [~(x), ¢~(Y) ]x_ =y_ = - i T r e ( x + - y + ) ,

(ll)

instead of the Dirac bracket (8). [ For convenience we will suppress the subscript x_ = y _ . ] For the same reason as that indicated in our previous paper [ 1 ], the field q~(x) cannot be expanded in terms of plane wave modes. Nevertheless, it can be expanded at an arbitrary given "time", called x_, in terms of its Fourier components. In this sense, we divide ~b(x) into its annihilation and creation parts as follows:

4,(x) - 4'+ (x) + 4'- ( x ) , where ~b+ (x) and ~b_ (x) are nonlocally dependent upon the positive light-cone coordinate x+

f dkat:[¢---L~eia+-ilvr~f

+~

4,+(x)--½

x+

dy+eiky+],

--~

--oo

~b-(X)=21- f dk a ~ [ ~-~'~ e -ikx+ + i ]~-~ f dy+ e -ikY+ ] . --~

(12)

--oo

In accordance with the commutator ( 11 ), the annihilation operator at: and creation operator a~ satisfy the standard commutation relations [at:, a~, ] = 6 ( k - k ' ) , [ak, at:, ] = [akt, ak*, ] =0. Hence it follows directly from (12) that [~b+(x), ~b_ (y)] = - l n [ i k o ( x + - y + - i e ) ] ,

[4,± (x), ~± (y)] = 0 .

(13)

In (13) the factor ko (/co--*0) comes from introducing an infra-red cut-off koe -3, into k-integrals, where 3; is the Euler constant. An important ingredient of the method for nonperturbatively studying the nonloeal conserved currents of Todatype systems is the concept of so-called vertex operators. In our case such operators are defined as

Aa(X) =-- : e ia~(x) : =eiaq~-(X)e iaq~+(x) . It is easily deduced that the above vertex operators satisfy the following operator product arithmetic

(14)

H.-X. Yang, Y.-X. Chen / Physics Letters B 337 (1994) 95-101

98

Aa( X ) A b ( y ) = iab[ ko( X + -- y + -- ie) ],b : eia~(x) +ibqS(y)

, ,

(15)

and commutation relation [ ( 0 - ~ + ) , A a ( x ) ] =2~raAa(x)~(x+-y+).

(16)

As a fundamental hypothesis, the Heisenberg equation of motion of vertex operator Aa(x) is assumed as iO_Aa(x) = [Aa(x), HI in our chiral-quantization scheme. Due to this equation and (10) and ( I 5 ) , we see that the evolution of Aa(X) in " x _ - t i m e " is governed by

iO_A,(x) = [A~(x), H] = i,~ A2k'~a f dy+ [ (x+ - y + - ie) ' ~ - (y+ - x + - ie) ~ ] : e iaek(x)+i/:/4,(y) + i - : s t ~ koa~ f dy+ [ (x+ - y + - i e ) - : ~ - ( y +

{

•

-x+ -ie) -as'~] : ei"o(~)+iao(Y) : .

(17)

Eq. (17) is very enlightening for constructing nonlocal conserved currents for system (1). Taking account of ~, 1, 2} and the mathematical formula

S -- !

lira [ ( x l - - y l - - i e ) - " - - ( x l - - y ~ + i e ) ,-~o

-'] =27r/

(-1)"-'

-

-

(n- l)!

8(n-l)(xl--yl),

n = l , 2, 3 . . . . .

(18)

we find that Eq. (17) will become the equation of current conservation

O_j+(x) +O+j_(x) = 0 , when the parameter a is taken to be - 2//3 or 2/s/3. In such a way we can obtain two nonlocal conserved currents both for sine-Gordon and ZMS models:

j(+°)(x) =A_2/#(x),

j(_°)(x) =

/3 2

kZo 2 - [~2Ac~-2/~(X)

(19)

and ~ra

j (~1)(X) = A 2 / s t ~ ( x ) ,

j ~__1)(x) =

k2

(s/3) 2 2_(s~)za2/,:_sB(x).

(20)

These conserved nonlocal currents do exist for all real coupling constant/3 on which the system (1) is well defined, which is just the reason why we develop a nonperturbative method in the chiral-quantization scheme. The price that has to be paid is that the above nonlocal currents have no longer the Lorentz covariance, based on the fact that the action ( 1 ) does not possess the "Lorentz" invariance in view of the light-cone coordinate x_ being " t i m e " . It is worthwhile to stress that although there are only two nonlocal conserved currents acquired from the nonperturbative scheme based on the chiral quantization, rather than the four currents appearing in the perturbed CFI" framework, the main physical information is not lost and the quantum integrability of the system ( 1 ) can still be displayed. In order to make this argument transparent, let us focus our attention on the case of s = 2, i.e. the ZMS model firstly. Following (19) and (20), the two nonlocal charges of this system read Qo =

el =

:

dx+j~(x)

=

dx+j~)(x) =

/

dx+ :exp

-i~d~(x)

dx+ :exp i--;d~(x)

:, :.

(21)

H.-X. Yang, Y.-X. Chen / Physics Letters B 337 (1994) 95-101

99

Moreover, there are two conserved topological charges To and Tt for the ZMS system and

Then it is a consequence of the commutator (16) that the nonlocal charges defined above are subject to the algebra

[Ti, Qj]=aoQj, i , j = 0 , 1,

(23)

where [a0] = [ 8 -4

2 4]

This tells us that the conserved charges frame the affine Lie algebra,a- 2~2) with zero center, which is a subalgebra of the quantum loop algebra A q~2 z) which is the symmetric algebra of the ZMS system as well [ 5 ]. To confirm that the dominant data about the physical S-matrix of the ZMS solitons are preserved, we have to define the soliton fields also. These fields will create the ZMS solitons when they act on the vacuum. In the procedure of constructing such fields, one should keep the following requirements in mind: since the fundamental representation of the zero-center A~ 2~ algebra is three-dimensional, the fundamental soliton fields must appear as a triplet and carry topological charges - 2, 0, 2, respectively. Such fields are found to be ~:~(x) = : exp

+i~

4fix)

:,

~o(x) = : 0+

exp(ia4)(x)) :: e x p ( - i a 4 ) ( x ) ) :

(24)

where a is an arbitrary real constant. It is a trivial thing to check that the soliton fields (24) turn out to be the eigenvectors of the operator T~ with the eigenvalues _4-2, 0, respectively,

[T~, % ( x ) ] =~-1(o-)%(x),

o-= +, - , 0,

(25)

where ~'1( + ) = 2, ~'1( - ) = - 2 and ~'1(0) = 0. Thus, by acting with Q~ on these soliton fields one will find fields with the topological charge rt increased by 2 with respect to the initial ones. In fact, the Wick product formula (15) will result in a set of important braiding relations when one acts with the nonlocal conserved current components j ~ a n d j ~ ~ on the soliton fields, given by

j~(x)~,~(y) =q~°~')~,~(y)j~(x), where ¢o(O-) = - 2zl (o') and q --- exp( for the conserved charges

A(Qi) =Q~®I+qT'®Qi,

j~(x)~,~(y)=q~('~,~(y)j~)(x),

forx+
(26)

iTr/2~z). These braiding relations induce the following comultiplications

A(T~) =Ti®I+I®Ti

(27)

by the charges acting on the tensor products of two soliton fields. In (27) the second relation is deduced from the additivity of the topological charges T~. The commutators (23) tied with comultiplications (27) could set up the bridge between the charge algebra and the infinite dimensional " B o r e l " subalgebra of the quantum algebra .~(2) _kt.q2 It is not difficult to see that there is an isomorphism .

Qi = Eiq 14'/2,

Ti = / / i

(28)

between the charges and the generators o f A ~ ~. In (28), H~ and E~, i = 0 , 1, form the Chevalley basis of such a " B o r e l " subalgebra, which can explicitly be expressed as the 3 × 3 matrices Ho=-4

[i0o] 0 0

i

,

Hi=2

[i00] 0 0

0 , --1

Eo=2A

[!0i] 0 0

,

El=2

0 0

0]

--1 0

H.-X. Yang, Y.-X. Chen /Physics Letters B 337 (I994) 95-101

I00

in the fundamental representation o ~-(2) i ~ q2 • Note that an arbitrary A parameter appears in the generator Eo as the loop parameter of this zero-center -~ q2 (2) . With the generators/4- and Ei, the comultiplication laws (27) can alternately be recast as

A(Hi) =Hi®l + l®Hi, A ( E i ) =Ei®q,-ltl/2 +qHi/2®Ei, (29) where q~= qa,/2. Since the above "Borel" subalgebra has non-trivially carried the spectrum parameter A, it describes an infinite dimensional symmetry of the ZMS model. As a result, the physical S-matrix of the two-soliton states must be commutative with the comultiplications of the generators H i and Ei [S, A(Hi)] = [S,

A(Ei) ] = 0 .

(30)

Compared with the corresponding result obtained in the perturbed CFT approach [5], the commutators [S, A(Fi) ] = 0 (i = 0, 1) are suppressed here because there only exist two nonlocal conserved charges in our scheme. Nevertheless this outcome provides sufficient data for demonstrating the integrability of the ZMS model. In terms of the "proposition 2 " proved by Jimbo in Ref. [6], a solution of the second equation in (30) is also the solution of the first one in (30), as well as that of [ S, A (F~) ] = 0 ( i = 0, 1 ). This solution is unique, and is proportional to the well-known Izergin-Korepin R-matrix [ 7 ]. In view of this analysis, we see that the S-matrix of the ZMS model determined by Eq. (30) will be almost the same as that obtained by Efthimiou [5] except that the S-matrix will be associated with the so-called homogeneous gradation ,v rl , cax q2 (2) rather than with the spin gradation. Secondly, we discuss the application of our method to the sine-Gordon model for completeness. In this case, the nonperturbatively defined nonlocal charges and the corresponding topological charge are written as Qo =

dx+j(+°)(x) =

dx+ : e x p

-i-~b(x) ", 2

Q,= f dx+ j~(x)= f dx+ "exp(i~ch(x)) " ,

/3

T= -f~ f dx+ (~x$+),

(31)

which gives [T, Qo] = - 2 Q o ,

[T, QI] = 2 Q 1 .

(32)

Corresponding to this commutator algebra, the fundamental soliton fields should be among the families of operators with topological charge _+_1: [ T, ~+_ (x) ] = +__#_+ (x). These fields are defined as ~ ± (x) = : exp( + i( 1 //3) th(x) ) :, which possess the following braiding properties with the nonlocal currents

j(+O)(x)

~±(y)=q~:l

~±(y)j(+O)(x), j~)(x)

~±(y)=q±l

~±(y)j~)(x),

forx+
(33)

where q - e -2"~/t~2. Therefore, the comultiplication of the charges exhibits

A(Qo) =Qo®l+qT®Qo,

A(Q~)=Qt ®l+q-V®Q1,

A(T) = T ® I + I ® T .

(34)

Associating with~this comultiplication, we see that (32) is actually an infinite dimensional subalgebra of the qdeformation s/q(2) of the sl(2) Kac-Moody algebra. Let E;, Hi ( i = 0 , 1) denote the Chevalley basis for this centerless quantum algebra, we have

Qi=Ejq n~/2

( i = 0 , 1),

T= -/40 =H1.

(35)

This is in fact the isomorphic relation between the conserved charges and the generators of quantum loop algebra (29), one can see that the S-matrix determined by means of the symmetric algebra (32) and (34) is indeed in accord with that obtained by Bernard and LeClair in the perturbed CFT framework [ 3 ]. In summary, we have established a nonperturbative framework for nonlocal conserved currents in sine-Gordon and ZMS models based on the chiral quantization. Although the sine-Gordon and ZMS models considered as

slq(2"--~(or A ~ ~ with zero center). Along the lines of the discussions below Eq.

H.-X. Yang, Y.-X. Chen /Physics Letters B 337 (1994) 95-101

101

Hamiltonian systems in light-cone coordinates are quite dissimilar to those in laboratory coordinates, the equivalent amounts of information could be obtained from either. This fact was noticed in Ref. [ 8 ] earlier at the level of soliton solutions of the classical sine-Gordon equation, Now we have reached the same conclusion at the level of the multisoliton S-matrix for quantum ZMS and sine-Gordon fields. In contrast to the appearance of four nonlocal charges from the perturbed CFT in laboratory coordinates, in our scheme only two nonlocal charges arise and they are not Lorentz covariant quantities. Fortunately, such two charges together with the topological charges generate infinite dimensional subalgehras of the quantum algebrasA ~q2)and A ~q~),respectively, for the ZMS and sine-Gordon systems. In addition, these charges satisfy some nontrivial comultiplication laws which turn out to be the part ingredients of algebras ~,q2a ~2) and Aq~ ) . The soliton S-matrices obtained in this way are evidently in agreement with those given by BL for the sine-Gordon system and Efthimiou for the ZMS model. Relying on these facts, we have reason to regard the present method as a reasonable candidate for the nonperturbative counterpart of the Bernard and LeClair's approach. This work is partially supported by the NSF of China. Y.X. Chen is also grateful to the Fok Ying Tong Education Foundation in China for the support.

References [ 1] H.X. Yang and Y.X. Chen, in preprint ZIMP-94-05,Phys. Lett. B 337 (1994). [2] S.J. Chang and R. Rajaraman, Phys. Lett. B 313, (1993) 59. [3] D. Bernard and A. LeClair, Commun. Math. Phys. 142 (1991) 99. [4] A.P.M. Dirac, Lectures in quantum mechanics (Yeshiva University, New York, 1964). [5] C.J. Efthimiou, Nucl. Phys. B 398, (1993) 697. [6] M. Jimbo, Commun. Math. Phys. 102 (1986) 537. [7] A.G. Izergin and V.E. Korepin, Commun. Math. Phys. 79 (1981) 303. [ 8] O.J. Kaup and A.C. Newell, SIMP J. Appl. Math. 34 (1978) 37; F. Luud, Physica D 18 (1986) 420.