- Email: [email protected]
On the method of inverse scattering problem and Bäcklund transformations for supersymmetric equations
Volume 78B, number 4
PHYSICS LETTERS
9 October 1978
ON THE METHOD OF INVERSE SCATTERING PROBLEM AND BACKLUND TRANSFORMATIONS FOR SUPERSYMMETRIC EQUATIONS M. CHAICHIAN Research Institute for Theoretical Physics, University of Helsinki, Finland and P.P. KULISH Leningrad Division of the Steklov Mathematical Institute, Academy of Sciences, USSR Received 16 May 1978 Revised manuscript received 21 July 1978
Supersymmetric Liouville and sine-Gordon equations are studied. We write down for these models the system of linear equations for which the method of inverse scattering should be applicable. Expressions for an infinite set of conserved currents are explicitly given. Supersymmetric B/icklund transformations and generalized conservation laws are constructed.
Intensive development of the method of Inverse Scattering Problem (ISP) for solving nonlinear differential equations has enabled to investigate, with exact results, physically interesting such as sine-Gordon, massive Thirring and many other models (see, e.g. the review article by Faddeev and Korepin [1] ,1). Although in many of these models there is no particle production and the set o f initial momenta is conserved, in their quantum treatments such a remarkable result as the exact S-matrix has been obtained [2] ,1 We shall confine our consideration to the classical level. Almost in all works devoted to ISP, nonlinear equations have been solved for the functions ~0 with the values from the complex numbers. The classical massive Thirring model has also been investigated when the fields ~ ( x 1, x0), c~ = 1, 2 take values from the Grassmann algebra [3] and further it has been shown that in this case the equation can be solved with the method of ISP [4]. ,1 In this note we do not quote all the original references but refer only to the recent works where one can find the complete list of references on the subject.
Due to great interest in supersymmetry, there have been investigations on 2-dimensional ( s p a c e time) models which are invariant under supersymmetric transformations [2,5]. In a recent preprint [6] a recursion formula has been proposed for building up an infinite set o f conserved currents for the supersymmetric sine-Gordon theory. Since the existence o f an infinite number of conserved currents directly follows from the linear representation of a given nonlinear equation, an interesting question arises: how to find a suitable linear system for which the supersymmetric sine-Gordon equation is the integrability condition and then develop the ISP scheme for it. Let us mention that the existence of recursion relations for an infinite set of conserved currents for the classical equations of motion is often the starting point in proving such an infinite set of conserved currents in its quantum theory. In this work we show that the formalism of the inverse problem can be applied also to supersymmetric models. One needs to develop sufficiently the analysis with supersymmetric differential operators, since the representation of the linear equations in the usual (complex and Grassmann) components re413
Volume 78B, number 4
PHYSICS LETTERS
9 October 1978
quires systems with large rank matrices even in the case of the simplest nonlinear equations. Let us consider the supersymmetric Liouville equation. But before that we mention a few formulas concerning the usual Liouville equation, where
denote only the light-cone coordinates x -+ = ½(x 1 ± x 0) of ref. [6] by o =- x + and r -~ x - . Supersymmetric covariant derivatives are
soo'r = e2~,
where 01 and 02 are elements of the Grassmann algebra. They have the properties
(1)
1 1 where SO~r = 3oOrso, o = ~(x + xO), r = ½(x 1 - x0). There exists a formula which gives the general solution of eq. (1):
D 1 = (--002 + i0200),
D 1 D 1 = -iO o ,
(5)
D2=(3Ol+iO1Or),
D 2 D 2 = iO r ,
D 1 D 2 = - D 2 D 1 = (00100: + 010o20 T
~(o, r ) = ½ l n [ f o ( o ) & ( r ) / ( f ( o ) + g(r)) 2] ,
where f(a) and g(r) are functions with appropriate analytic properties. The ISP method was used for solving eq. (I) in ref. [7] (One ought to add an extra set of requirements on scattering data to what is given in ref. [7].) According to the ISP method, eq. (1) is the integrability condition for the following system of linear equations: if.~(o, r) = (¢o.4 + X B ) f ,
ifr(O, r) = ~ e2~'Cf,
(2)
where X is a spectral parameter, f(o, r) is a twocomponent complex function and A, B, Care 2 X 2 matrices of, e.g., the form A = io3, B = x/2 o 2, C = - 2 -3/2 (o 2 + iol) and o i the Pauli matrices. The asymptotic behaviour of SOis SO(o, r) -+ ~ ao + C* (r), for o -+ ± % with'a a positive constant. A detailed presentation of the direct and inverse scattering problem for the first of eqs. (2) with such asymptotic behaviour of SOis given in ref. [8]. For the study of nonlinear equations it is also useful to have the Bficklund transformations (see, e.g. ref. [9]). For eq. (1) one of such transformations has the following form: Or(SO -- ~ ) = ~: exp(so + ~'),
+ i023010o +
01020o0r)
(6)
.
Consider the scalar superfield (7)
cb(x) = SO(x) + i 0 4 + ½i-ffOF ,
where s0(x) and F(x) are the scalar fields (even elements of the Grassmann algebra), 4(x) is a twocomponent column Majorana spinor (odd elements of the Grassmann algebra), O=
(01) 02
'
~=0TT0=(0102
) ( ~ ,-
1) 0
"
From the action fd2xdOldO
( - ½ i ~ D 1 D 2 ~ + e*) ,
one obtains the supersymmetric Liouville equation D1D2d~ = - i e '~ .
(8)
Using the form (7) for q~ and independence of the basis elements 01 and 02, we can write eq. (8) in components: soot =e~°( e ~ - i 4 1 4 2 ) ,
F ( x ) = eSO,
~a(SO+ ~ ' ) = ~ s h ( s o - ~ ' ) ,
(3)
41r = es°42 ,
42o = e~° 41 •
(9)
where K is an arbitrary complex number. The Liouville equation for SOand ~" is the integrability condition for the system of nonlinear differential eqs. (3). From B/ickland transformation (3) (which is the transition from the known solution ~" of eq. (1) to a new solution SO), in particular, follows the generalized conservation law
The system possesses the discrete symmetry (o, r, 41, 42) -+ ( - r , - a , - 4 2 , 41), which in addition with (01, 02) -+ ( - 0 2 , 01) corresponds to the symmetry of eq. (8) with respect to (191, D2) -+ (--/92, D 1) . As a simple example we give here an infinite set of conservation laws:
ao(exp(so + ~')) = (2/K2)ar(ch(sO - ~')) .
Or(SOo° _ SO2 _ i4141~) = 0 .
(4)
To describe the supersymmetric generalization of Liouville eq. (1), we use the notation of ref. [6]. We 414
(10)
This is an infinite set since the given density in eq. (10) is independent of r.
Volume 78B, number 4
PHYSICS LETTERS
Let us find a linear representation for eq. (8), i.e. a system analogous to eqs. (2):
Dlf= (DIOA + XB)f,
D 2 f = (-i/X) exp(q~)Cf .(11)
In this system f is a column with three elements, and the 3 X 3 matrices A, B, C have the form A
=
1~(01 a3~l
/0 C=b~(1 1
l\ 1
-1
1\,,
0
al,
a
0j
-1
1",
0 0
Oi 0]
/b 1
b
,\0
0
B=lb
,
b2
0\\
0 b2]
b 1 + b2 •
These matrices are a special solution to the communication relations A = {B, C} and C = [A, C], because of which eq. (8) is the integrability condition for the system (11). One can write the system (11) in components as well. Then one obtains the usual differential equation with the only difference that part of the coefficients are elements of the Grassmann algebra. To such systems the ISP method can also be applied. However, in our case one gets a 12 X 12 matrix system, or when dividing into even and odd elements, a 24 X 24 one. Therefore, in order to preserve the compactness, one has to develop corresponding apparatus for solving equations with operators of the type D = 00 + i00,,. This operator has the same kernel as the operator of usual differentiation (i.e. Df = 0 -+ f = const) and the solution of Df = XBf has the form exp(XB0 - iX2B2o). Such a development will be given in a separate work. We present here two different B/icklund transformations for eq. (8):
(13)
from which an infinite set of conserved currents follows. Notice that our current in eqs. (13) is an even element of the Grassmann algebra, while the one of ref. [16] is odd. For this reason, the usual conserved current ]u' 3Uu(x) = O, which can be obtained from the supersymmetric currents of eq. (13) is odd. By exploiting f, one can also obtain from eqs. (12) conservation laws for odd supersymmetric currents:
Dl(fe(°+~)/2) = -(1/X2)D 2 [fsinh(½(cb - ~))1 . The following B/icklund transformation, however, connects the solution of eq. (8) for q5 with the solution ~ of the free equation D1D2~ = 0: Dl(q~ + ~) = (i/?Qfexp(½(q~ - ~)) , D2(O - ~) = (i/X)fexp(½(O + i~)) ,
Dlf =" - X exp(½ (~ - ~)) , D2f = X exp(½(~ + ~)) .
(14)
For the supersymmetric sine-Gordon equation [6],
D1D2¢, = irn sin d/, ,
(15)
in analogy with the usual scalar case [10], we choose the first equation in the same form as for the system (11). Then we obtain the ansatz
Dlf = (DI ~A +XB)f ,
D I ( + + ~) = (i/X)fcosh(½(O - ~)) ,
D2f= (i m/X) (cos q~ C+ sin q5 E)f,
D 2 ( ~ - ~) = iXfexp(½(~ + ~)) ,
(16)
where A, B, C and E are matrices, X is the spectral parameter. Using the integrability condition for the system (16), we obtain the following set of equations:
Dlf = -(2/X) sinh(½(O - ~)) , 1 D2f= 2Xexp(~(~ + ~)) .
separately. When ~ and ~ vanish, eqs. (8) and (12) are reduced to the corresponding formulas (1) and (3) of the usual Liouville equation. From system (12) we get the generalized conservation law Dl(e(*+~)/2) = (1/X2)D2(sinh(½(O - ;~)) ,
b
a-
9 October 1978
(12)
Because of supersymmetry and the oddness (with respect to elements of the Grassmann algebra) of the operators D~, we were forced to introduce in addition an odd function f. The conditions for the compatibility of eqs. (12) are eqs. (8) for both • and
DID2~A - i m cos • (B, C} - i m sin q5 {B, E} = 0 , cos q b D I ~ ( [ C , A ] + E ) : 0 , sin q5 DlqS([E , A ] - C) = 0 .
(17)
By choosing these matrices so that they satisfy 415
Volume 78B, number 4
C=[E,A],
PHYSICS LETTERS
E=[A,C],
A=a{B,C}=3{B,E},
with c~ and t3 real numbers, the set (17) is equivalent to the general (with phase) sine-Gordon equation. The particular form of these matrices which satisfies {B, C} = 0 with 13 = 1 gives exactly eq. (15):
o
u:
0 0) '\\ 0 0 Oj
-1
,
\0
o/
-b
-1,,
-1
b/
three of which are just the generators of the wellknown adjoint representation of SO(3). Now we construct a Bficklund transformation which connects the two solutions • and ~ of eq. (15):
9 October 1978
DI(2DI~sin~)=-X2D2(2D1"~
1
5-~ a) ,
i.e. D I ( D I ~ sin ~ ) = - ( 1 / i
m)D2(DI~DT~ ) ,
which is the first conservation law in ref. [6]. The p r o o f of the existence of an ir~finite number of conservation laws in the quantum theory has turned out to be crucial for obtaining the exact quantum S matrix. For renormalizable models such as the usual and the supersymmetric sine-Gordon, results of the classical theory are the starting point for the proof. However, in the case of essentially nonlinear fields as the usual and the supersymmetric o-model, the situation becomes more complicated [11] and there is no direct connection between the classical and quantum conservation laws.
nl(,I, + ~) = X fcos(½(~ - ~)),
One of us (P.P.K.) wishes to thank the Department of Theoretical Physics, University of Helsinki, for hospitality.
D 2 ( } - ;~) = (l/X)fcos(½(<:I, + ~ ) ) ,
References
Dlf= 2imXsin(½(,:I, - ~ ) ) , D2f = - 2 i(m/X) sfn( 1 ~b + ~ ) ) .
(19)
From the system (19) follows the generalized conservation law
DC~Jc,(x, 0) = 0 ,
(20)
where Jl(X, 0) = k2sin(½(O - ~)), J2(x, O) = sin(½ (eo + ~)). Since one of the two solutions, e.g. eo, can be considered, through the system (19), as a functional of the other solution and the parameter X, then by expanding it in an asymptotic series one obtains an infinite set o f usual conservation laws. For the odd conserved current, from the system (19) we have D 1 [fsin(½(cb + ~))] = - X 2 D 2 [fsin(½(qb - ~))] . (21) By expanding the solution ~I, in an asymptotic series in X- 2 , with coefficients depending on ~ , and substituting into eq. (21), we obtain the usual supersymmetric conservation laws for different powers of X-2: = ~ + X-Za + .... From eqs. (19) one has a = (Xlim)Dlf and 2D1~; = Xf, i.e. a = (21im)D2"~. Also from eqs; (19) we have 416
[1] L.D. Faddeev and V.E. Korepin, Quantization of solitons, Phys. Rep. 1978, to be published. [2] R. Shankar and E. Witten, Harvard Univ. preprint HUTP-77/A076 (Dec. 1977). [3] P.P. Kulish and E.R. Nissimov, Theor. Math. Phys. 29 (1976) 161 (in Russian); A.G. Izergin and J. Stehr, DESY preprint 76/70 (1976); H.C. Morris, J. Math. Phys. 19 (1978) 85. [4] A.G. Izergin and P.P. Kulish, Lett. Math. Phys. 1978, to be published. [5] P. Di Vecchia and S. Ferrara, Nucl. Phys. B130 (1977) 93; J. Hruby, Nucl. Phys. B131 (1977) 275. [6] S. Ferrara, L. Girardello and S. Sciuto, preprint TH. 2474-CERN (March 1978). [7] V.A. Andreev, Theor. Math. Phys. 29 (1976) 213. [8] V.S. Gerdjikov and P.P. Kufish, preprint, JINR, Dubna (1978). [9] G.L. Lamb Jr., J. Math. Phys. 15 (1974) 2157. [10] M. Ablowitz, D. Kaup, A. Newell and H. Segur, Phys. Rev. Lett. 30 (1973) 1262; L.A. Takhtadzhyan and L.D. Faddeev, Theor. Math. Phys. 21 (1974) 1046. [11] A.M. Polyakov, Phys. Lett. 59B (1977) 79; I. Ya. Aref'eva. P.P. Kulish, E.R. Nissimov and S.J. Pacheva, LOMI preprint E-1-1978.
PHYSICS LETTERS
9 October 1978
ON THE METHOD OF INVERSE SCATTERING PROBLEM AND BACKLUND TRANSFORMATIONS FOR SUPERSYMMETRIC EQUATIONS M. CHAICHIAN Research Institute for Theoretical Physics, University of Helsinki, Finland and P.P. KULISH Leningrad Division of the Steklov Mathematical Institute, Academy of Sciences, USSR Received 16 May 1978 Revised manuscript received 21 July 1978
Supersymmetric Liouville and sine-Gordon equations are studied. We write down for these models the system of linear equations for which the method of inverse scattering should be applicable. Expressions for an infinite set of conserved currents are explicitly given. Supersymmetric B/icklund transformations and generalized conservation laws are constructed.
Intensive development of the method of Inverse Scattering Problem (ISP) for solving nonlinear differential equations has enabled to investigate, with exact results, physically interesting such as sine-Gordon, massive Thirring and many other models (see, e.g. the review article by Faddeev and Korepin [1] ,1). Although in many of these models there is no particle production and the set o f initial momenta is conserved, in their quantum treatments such a remarkable result as the exact S-matrix has been obtained [2] ,1 We shall confine our consideration to the classical level. Almost in all works devoted to ISP, nonlinear equations have been solved for the functions ~0 with the values from the complex numbers. The classical massive Thirring model has also been investigated when the fields ~ ( x 1, x0), c~ = 1, 2 take values from the Grassmann algebra [3] and further it has been shown that in this case the equation can be solved with the method of ISP [4]. ,1 In this note we do not quote all the original references but refer only to the recent works where one can find the complete list of references on the subject.
Due to great interest in supersymmetry, there have been investigations on 2-dimensional ( s p a c e time) models which are invariant under supersymmetric transformations [2,5]. In a recent preprint [6] a recursion formula has been proposed for building up an infinite set o f conserved currents for the supersymmetric sine-Gordon theory. Since the existence o f an infinite number of conserved currents directly follows from the linear representation of a given nonlinear equation, an interesting question arises: how to find a suitable linear system for which the supersymmetric sine-Gordon equation is the integrability condition and then develop the ISP scheme for it. Let us mention that the existence of recursion relations for an infinite set of conserved currents for the classical equations of motion is often the starting point in proving such an infinite set of conserved currents in its quantum theory. In this work we show that the formalism of the inverse problem can be applied also to supersymmetric models. One needs to develop sufficiently the analysis with supersymmetric differential operators, since the representation of the linear equations in the usual (complex and Grassmann) components re413
Volume 78B, number 4
PHYSICS LETTERS
9 October 1978
quires systems with large rank matrices even in the case of the simplest nonlinear equations. Let us consider the supersymmetric Liouville equation. But before that we mention a few formulas concerning the usual Liouville equation, where
denote only the light-cone coordinates x -+ = ½(x 1 ± x 0) of ref. [6] by o =- x + and r -~ x - . Supersymmetric covariant derivatives are
soo'r = e2~,
where 01 and 02 are elements of the Grassmann algebra. They have the properties
(1)
1 1 where SO~r = 3oOrso, o = ~(x + xO), r = ½(x 1 - x0). There exists a formula which gives the general solution of eq. (1):
D 1 = (--002 + i0200),
D 1 D 1 = -iO o ,
(5)
D2=(3Ol+iO1Or),
D 2 D 2 = iO r ,
D 1 D 2 = - D 2 D 1 = (00100: + 010o20 T
~(o, r ) = ½ l n [ f o ( o ) & ( r ) / ( f ( o ) + g(r)) 2] ,
where f(a) and g(r) are functions with appropriate analytic properties. The ISP method was used for solving eq. (I) in ref. [7] (One ought to add an extra set of requirements on scattering data to what is given in ref. [7].) According to the ISP method, eq. (1) is the integrability condition for the following system of linear equations: if.~(o, r) = (¢o.4 + X B ) f ,
ifr(O, r) = ~ e2~'Cf,
(2)
where X is a spectral parameter, f(o, r) is a twocomponent complex function and A, B, Care 2 X 2 matrices of, e.g., the form A = io3, B = x/2 o 2, C = - 2 -3/2 (o 2 + iol) and o i the Pauli matrices. The asymptotic behaviour of SOis SO(o, r) -+ ~ ao + C* (r), for o -+ ± % with'a a positive constant. A detailed presentation of the direct and inverse scattering problem for the first of eqs. (2) with such asymptotic behaviour of SOis given in ref. [8]. For the study of nonlinear equations it is also useful to have the Bficklund transformations (see, e.g. ref. [9]). For eq. (1) one of such transformations has the following form: Or(SO -- ~ ) = ~: exp(so + ~'),
+ i023010o +
01020o0r)
(6)
.
Consider the scalar superfield (7)
cb(x) = SO(x) + i 0 4 + ½i-ffOF ,
where s0(x) and F(x) are the scalar fields (even elements of the Grassmann algebra), 4(x) is a twocomponent column Majorana spinor (odd elements of the Grassmann algebra), O=
(01) 02
'
~=0TT0=(0102
) ( ~ ,-
1) 0
"
From the action fd2xdOldO
( - ½ i ~ D 1 D 2 ~ + e*) ,
one obtains the supersymmetric Liouville equation D1D2d~ = - i e '~ .
(8)
Using the form (7) for q~ and independence of the basis elements 01 and 02, we can write eq. (8) in components: soot =e~°( e ~ - i 4 1 4 2 ) ,
F ( x ) = eSO,
~a(SO+ ~ ' ) = ~ s h ( s o - ~ ' ) ,
(3)
41r = es°42 ,
42o = e~° 41 •
(9)
where K is an arbitrary complex number. The Liouville equation for SOand ~" is the integrability condition for the system of nonlinear differential eqs. (3). From B/ickland transformation (3) (which is the transition from the known solution ~" of eq. (1) to a new solution SO), in particular, follows the generalized conservation law
The system possesses the discrete symmetry (o, r, 41, 42) -+ ( - r , - a , - 4 2 , 41), which in addition with (01, 02) -+ ( - 0 2 , 01) corresponds to the symmetry of eq. (8) with respect to (191, D2) -+ (--/92, D 1) . As a simple example we give here an infinite set of conservation laws:
ao(exp(so + ~')) = (2/K2)ar(ch(sO - ~')) .
Or(SOo° _ SO2 _ i4141~) = 0 .
(4)
To describe the supersymmetric generalization of Liouville eq. (1), we use the notation of ref. [6]. We 414
(10)
This is an infinite set since the given density in eq. (10) is independent of r.
Volume 78B, number 4
PHYSICS LETTERS
Let us find a linear representation for eq. (8), i.e. a system analogous to eqs. (2):
Dlf= (DIOA + XB)f,
D 2 f = (-i/X) exp(q~)Cf .(11)
In this system f is a column with three elements, and the 3 X 3 matrices A, B, C have the form A
=
1~(01 a3~l
/0 C=b~(1 1
l\ 1
-1
1\,,
0
al,
a
0j
-1
1",
0 0
Oi 0]
/b 1
b
,\0
0
B=lb
,
b2
0\\
0 b2]
b 1 + b2 •
These matrices are a special solution to the communication relations A = {B, C} and C = [A, C], because of which eq. (8) is the integrability condition for the system (11). One can write the system (11) in components as well. Then one obtains the usual differential equation with the only difference that part of the coefficients are elements of the Grassmann algebra. To such systems the ISP method can also be applied. However, in our case one gets a 12 X 12 matrix system, or when dividing into even and odd elements, a 24 X 24 one. Therefore, in order to preserve the compactness, one has to develop corresponding apparatus for solving equations with operators of the type D = 00 + i00,,. This operator has the same kernel as the operator of usual differentiation (i.e. Df = 0 -+ f = const) and the solution of Df = XBf has the form exp(XB0 - iX2B2o). Such a development will be given in a separate work. We present here two different B/icklund transformations for eq. (8):
(13)
from which an infinite set of conserved currents follows. Notice that our current in eqs. (13) is an even element of the Grassmann algebra, while the one of ref. [16] is odd. For this reason, the usual conserved current ]u' 3Uu(x) = O, which can be obtained from the supersymmetric currents of eq. (13) is odd. By exploiting f, one can also obtain from eqs. (12) conservation laws for odd supersymmetric currents:
Dl(fe(°+~)/2) = -(1/X2)D 2 [fsinh(½(cb - ~))1 . The following B/icklund transformation, however, connects the solution of eq. (8) for q5 with the solution ~ of the free equation D1D2~ = 0: Dl(q~ + ~) = (i/?Qfexp(½(q~ - ~)) , D2(O - ~) = (i/X)fexp(½(O + i~)) ,
Dlf =" - X exp(½ (~ - ~)) , D2f = X exp(½(~ + ~)) .
(14)
For the supersymmetric sine-Gordon equation [6],
D1D2¢, = irn sin d/, ,
(15)
in analogy with the usual scalar case [10], we choose the first equation in the same form as for the system (11). Then we obtain the ansatz
Dlf = (DI ~A +XB)f ,
D I ( + + ~) = (i/X)fcosh(½(O - ~)) ,
D2f= (i m/X) (cos q~ C+ sin q5 E)f,
D 2 ( ~ - ~) = iXfexp(½(~ + ~)) ,
(16)
where A, B, C and E are matrices, X is the spectral parameter. Using the integrability condition for the system (16), we obtain the following set of equations:
Dlf = -(2/X) sinh(½(O - ~)) , 1 D2f= 2Xexp(~(~ + ~)) .
separately. When ~ and ~ vanish, eqs. (8) and (12) are reduced to the corresponding formulas (1) and (3) of the usual Liouville equation. From system (12) we get the generalized conservation law Dl(e(*+~)/2) = (1/X2)D2(sinh(½(O - ;~)) ,
b
a-
9 October 1978
(12)
Because of supersymmetry and the oddness (with respect to elements of the Grassmann algebra) of the operators D~, we were forced to introduce in addition an odd function f. The conditions for the compatibility of eqs. (12) are eqs. (8) for both • and
DID2~A - i m cos • (B, C} - i m sin q5 {B, E} = 0 , cos q b D I ~ ( [ C , A ] + E ) : 0 , sin q5 DlqS([E , A ] - C) = 0 .
(17)
By choosing these matrices so that they satisfy 415
Volume 78B, number 4
C=[E,A],
PHYSICS LETTERS
E=[A,C],
A=a{B,C}=3{B,E},
with c~ and t3 real numbers, the set (17) is equivalent to the general (with phase) sine-Gordon equation. The particular form of these matrices which satisfies {B, C} = 0 with 13 = 1 gives exactly eq. (15):
o
u:
0 0) '\\ 0 0 Oj
-1
,
\0
o/
-b
-1,,
-1
b/
three of which are just the generators of the wellknown adjoint representation of SO(3). Now we construct a Bficklund transformation which connects the two solutions • and ~ of eq. (15):
9 October 1978
DI(2DI~sin~)=-X2D2(2D1"~
1
5-~ a) ,
i.e. D I ( D I ~ sin ~ ) = - ( 1 / i
m)D2(DI~DT~ ) ,
which is the first conservation law in ref. [6]. The p r o o f of the existence of an ir~finite number of conservation laws in the quantum theory has turned out to be crucial for obtaining the exact quantum S matrix. For renormalizable models such as the usual and the supersymmetric sine-Gordon, results of the classical theory are the starting point for the proof. However, in the case of essentially nonlinear fields as the usual and the supersymmetric o-model, the situation becomes more complicated [11] and there is no direct connection between the classical and quantum conservation laws.
nl(,I, + ~) = X fcos(½(~ - ~)),
One of us (P.P.K.) wishes to thank the Department of Theoretical Physics, University of Helsinki, for hospitality.
D 2 ( } - ;~) = (l/X)fcos(½(<:I, + ~ ) ) ,
References
Dlf= 2imXsin(½(,:I, - ~ ) ) , D2f = - 2 i(m/X) sfn( 1 ~b + ~ ) ) .
(19)
From the system (19) follows the generalized conservation law
DC~Jc,(x, 0) = 0 ,
(20)
where Jl(X, 0) = k2sin(½(O - ~)), J2(x, O) = sin(½ (eo + ~)). Since one of the two solutions, e.g. eo, can be considered, through the system (19), as a functional of the other solution and the parameter X, then by expanding it in an asymptotic series one obtains an infinite set o f usual conservation laws. For the odd conserved current, from the system (19) we have D 1 [fsin(½(cb + ~))] = - X 2 D 2 [fsin(½(qb - ~))] . (21) By expanding the solution ~I, in an asymptotic series in X- 2 , with coefficients depending on ~ , and substituting into eq. (21), we obtain the usual supersymmetric conservation laws for different powers of X-2: = ~ + X-Za + .... From eqs. (19) one has a = (Xlim)Dlf and 2D1~; = Xf, i.e. a = (21im)D2"~. Also from eqs; (19) we have 416
[1] L.D. Faddeev and V.E. Korepin, Quantization of solitons, Phys. Rep. 1978, to be published. [2] R. Shankar and E. Witten, Harvard Univ. preprint HUTP-77/A076 (Dec. 1977). [3] P.P. Kulish and E.R. Nissimov, Theor. Math. Phys. 29 (1976) 161 (in Russian); A.G. Izergin and J. Stehr, DESY preprint 76/70 (1976); H.C. Morris, J. Math. Phys. 19 (1978) 85. [4] A.G. Izergin and P.P. Kulish, Lett. Math. Phys. 1978, to be published. [5] P. Di Vecchia and S. Ferrara, Nucl. Phys. B130 (1977) 93; J. Hruby, Nucl. Phys. B131 (1977) 275. [6] S. Ferrara, L. Girardello and S. Sciuto, preprint TH. 2474-CERN (March 1978). [7] V.A. Andreev, Theor. Math. Phys. 29 (1976) 213. [8] V.S. Gerdjikov and P.P. Kufish, preprint, JINR, Dubna (1978). [9] G.L. Lamb Jr., J. Math. Phys. 15 (1974) 2157. [10] M. Ablowitz, D. Kaup, A. Newell and H. Segur, Phys. Rev. Lett. 30 (1973) 1262; L.A. Takhtadzhyan and L.D. Faddeev, Theor. Math. Phys. 21 (1974) 1046. [11] A.M. Polyakov, Phys. Lett. 59B (1977) 79; I. Ya. Aref'eva. P.P. Kulish, E.R. Nissimov and S.J. Pacheva, LOMI preprint E-1-1978.