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Ffc?^o Kfef
COMMISSARIAT A L'ENERGIE ATOMIQUE
DIVISION DE LA PHYSIQUE
SERVICE DE PHYSIQUE THEORIQUE
INTEGRABLE MODELS IN 1 + 1 DIMENSIONAL QUANTUM FIELD THEORY
by
Ludvig Faddeev + Steklov Institute, Fontanka 27, Leningrad 191011 U.S.S.R.
and S.Ph.T., C.E.N. Saclay
Summer school on recent advances in f ield theory and stastical mechanics
t , A , H r p « Les Houches,France 2 Aug - 10 Sep 1982 Permanent Address CEA-CONF--6565
S.Ph.T/82/76
CEN-SACLAY - 91191 GIFsur-YVETTE CEDEX - FRANCE
1. Arguing in favor of the 1 + 1 dimensional models
There exist several reasons why (some) people are interested in such models :
i.l. Every exact (non perturbative) solution of a field-theoretical model can teach ..s about the ability of quantum field theory to describe particle spectrum and scattering.
i.2. Some 1+I-d-models have physical applications i.e. in the solid state theory.
i.3. The mathematics of the subject is quite beautiful.
2. Possible approaches
There are several vays to become acquainted with the methods of exactly soluble models :
i) Via classical statistical mechanics. This way is associated with L"he names of Onsager, Lieb, Baxter,...
ii) Via Bethe Ansatz. Besides Bethe himself important contributions were made by Hulthen, C.N. Yang,...
iii) Via inverse scattering method. This method is only 15 years old and was introduced by Kruskal and coworkers and developed by Lax, Zakharov,...
I shall use the last way because it is nearer to my own interests. However in the course of lectures it will become clear that all the ways are connected and lead to an essentially unique mathematical structure.
3. People involved
Several groups are actively working nowadays in the field of exactly soluble models with the application to quantum field theory in mind. The contributions of professors Thacker, Lowenstein, Honerkamp and their collaborators in Fermi. Lab, New-York University and University of Freiburg are discussed in their lectures.
The list of groups includes also University Paris VI (de Vega...), the Landau Institute (Belavin, Wiegman, ...) and Kyoto University (Sato, Miwa, Jimbo, . . . ) . In my lectures the point of view of the Leningrad
group will be presented. This group consists of Kulish, Korepin,
Takhtajan, Izergin, Reshetikhin, Semyonov- Tjan - Shansky, Reiman, Smirnov
•and me. Many groups working on the theory of solitons should also be
mentioned, but their interest is mainly associated with the classical
field theory.
Several survey articles [l]-[6] cover many parts of my lectures.
However the way of exposition as well as some technical details are new.
They reflect the attitude taken in the forthcoming monograph on classi
cal and quantum theory of solitons which I prepare in collaboration
with Takhtajan.
4. What is a completely integrable model ?
The term originates in the classical hamiltonian mechanics. The system
given by the phase space F- with coordinates Ç = (p. ,q ) , i =1,...,n and
Hamiltonian H(p,q) is called completely integrable if there exist n inde
pendent functions I.(p,q) such that
{H,I.} - o {I . , I ,} = o (<>
The I. are called the (commuting) integrals of motion or conserved
quantities. The Hamiltonian depends on p,q only through them,
H = H ( I ) (z)
Moreover one can in pr inc ip le find a change of variables
(r»i) - ( !>¥) W such that the following relations are true
and, in particular
In the new variables the equations of motion simplify
r , { H , I } *° CO
so that
is a solution.
The Op, I) are known as the "angle-action" variables. Finding chera explicitly could be difficult but a general theorem of mechanics says that they exist whenever the full system of conserved quantities is known.
In what follows we shall consider field theoretical models and so the number of degrees of freedom will be infinite.
5. A representative model
The general considerations of the inverse scattering method will be illustrated on the example of the nonlinear Schrodinger (N.S.) equation. We shall try to write most formulas in a model-independent way.
Canonical variables of N.S. model will be used in complex form tpGO, ty*W • The Poisson brackets
and the Hamiltonian L ( 4i it* + * I Y H ) «U L \ dx ax « '
lead to the equation of motion
* it i»
which has quite a few physical applications. In particular it is a Hartree-Fock equation for the system of non relativistic particles with 6-function interactions, g playing the role of the coupling constant. We are going to show that this model is completely integrable.
» - i
6. Zero curvature representation
Consider the pair of first order differential operators (covariant
derivatives)
where U and V are 2 x 2 matrices
T U =
§ 4%)
( •«)
0») r Ç 46c > - |-
v = "
parametrized by i|>*, ty anda Comdex parameter X. The (zero curvature)
condition r -, _
[L ,M] - o being fulfilled for all X is equivalent to the N.S. equation.
From the zero curvature condition it follows that for a closed con
tour y in the space-time plane the ordered integral
£? f Udn rVdt (\l)
is a unit matrix.
7. Conserved quantities
Suppose we have periodic boundary conditions.
<\(-L) . +{D , +*(-L) -- +*(t) (W)
Take the boundary of the r e c t a n g l e -L < x < L, 0 < t < T as y . With n o t a t i o n s
(«)
the following relation is true TL(>,T) = 5 > ) T t(>,°) i j j ) (10
so that trT(A,t) is independent of t ; X being arbitrary we have an infinite family of conserved quantities.
To show the complete integrability we have to answer several questions i) do these quantities commute; ii) do they commute with H;
iii) <lo we have enough of them.
The answers will take quite a time in what follows.
8. The fundamental Poisson brackets Let us introduce a convenient notation. For two 2 x 2 matrices A
and B which are functionals of ip and ip the symbol {A® B} denotes the M*h matrix of Poisson brackets of all their matrix elements. Explicitly it means that
{ A ? B } = { A a ) B t „ } (17)
In this notation the following relation can be easily checked
08)
Here r(X-y) i s a 4 x 4 mat r ix
iL 1'3) where P is a permutation matrix. In terms of Pauli matrices we have
and in a natural basis P looks as follows :
T - l o i o O J v '
In what follows the relation (18) will play a fundamental role so that we shall give it a special shortening:FPR (fundamental Poisson bracket relation).
9. Poisson brackets for the monodromy matrix
The matrix T (X) will be called the monodromy matrix because it describes the transport along the "circle" -L < x < L. We shall calculate the Poisson brackets of all its matrix elements using FPR.
We divide the interval -L < x < L into infinitesimal sub-intervals of length A and introduce the corresponding transport matrices
Lfi)-. £f J V(.,»a« , » I + f U60) <U + 0(Al)
I n t e g r a t i n g FPR we get the r e l a t i o n
In terms of L . (A) , the matr ix T (A) i s w r i t t e n i n the form 1 . L«
T(>) = U L;(« + 0 (A) Qu,)
Now use the general formula J A « K } = I ® B {A®C} + {Af3} I®C
to get
p-i where IT means product starting from i-p+1, II the same ending at
P+I i=p-I, etc. Using FPR we rewrite the RHS in the form
?
which is nothing but the commutator of II L.(A)®L.(u) and f(A-u) •
Thus in the limit A ->• 0 we get the desired equation
{!!(»> ? T( r>} = [r(i-rl , TJ>,) » Tjr)] <»> In particular it follows that the family of conserved quantities combined in the generating function tr T (A) commute among themselves. r _ , . , . * - , , . i _ /-,„\
[trTJV , tr TL(r)} =0 (28)
10 . The Hamiltonian is a member of the commuting family
We shall show that tr T T(A) can be represented in the form
*t T t(v) = 2 cos?t(a) L ^ 2 3 ) where the function P (A) (called quasi-momentum) can be written as a formal series
«si
C being local functional of ty* ,ty and their derivatives, n
Introduce the transport matrix T(x,y|X) for the interval (y,x)
It satisfies the equations
ax
A T(» , , | i ) - - T («,-,;» U(y,>)
>1 = l
where W (x) are local functions of i(i*(x)ij/(x) and their derivatives, n '
To check it, substitute this representation into the differential equation and separate diagonal and anti-diagonal parts to get
i w + w *1 = u0 + > v <U An
where we introduced the notation
(3 2)
The solution can be found in terms of the gauge transform of a diagonal matrix
T(x;vjU) a(l + W(«,») ex pZ.( Mm (T + W(j;>))" ^3)
when Z is diagonal and W anti-diagonal, admitting the formal series expansion
lis)
V, - ; £ (+*«;-+«--)
u. - i*
This allows a formal 1/À expansion, W. being given by
W. = 1 61 U 3 o t
and subsequent W are recursively obtained as functions of ij/* ,ijj and
their derivatives at the point x. In particular all W (x) are periodic
in x, W (-L) = W (L). n n
Moreover one can see that in general W can be represented in the
form
W = JZ (v* r+ + "W <r_)
with some scalar function i*r.
(.30
so that U is anti-diagonal and (J. is diagonal.
Eliminating Z from the first equation we get the Ricatti equation
for W
i W -r i> ci W - UL + W V0 W = o t57> Ac J o
(u)
(23)
Integrating the equation for Z(x,ylX) we get
Zt*n\1) = H (*-y)<r, + f U. W 4, M
Observing that tr Z = 0 because of the unimodularity of T(x,y|A) we
get finally the desired representation for tr T (X) with P (X) given
by the expression
It is instructive to go through the first steps of calculating W to
see that coefficients C , C_ and C. are proportional to the functionals
M, P and H where N and P are number of particles and momentum
(« _ V A* At )
-L -L
and H is the Hamiltonian.
N = [ V * i . , P= X j L ( i + M -^£^)^x {<
Thus the questions i) and ii) are answered affirmatively.
II. FPR substitutes for the zero curvature condition
Before proceeding to answer the question iii) let us step aside to show that with use of FPR the equations of motion for any conserved quantity taken as a hamiltonian can be written as a zero curvature condition. We shall show it for the equation
•* -- { * ( r > , + } ( « )
covering all possibilities in one formula. Given this equation, U(x,A) satisfies the equation
n - U(r>,v(«,>>} <**> At The RHS can be read off the more general expression
|w?"M! * {1,s)
where the definition of T(x,yiy) in terms of U(x,y) was used.
Using FPR the RHS here is rearranged as
T(L/ir)®I [r(r)),U(x;u)«ItI^M] T(«,-L| )«I
which with the help of the differential equations forT(x.ylA) can
be wri t t en in tLe form
iQ(*| i«i*) «• [Q(*ir i*)> I * u ( ^ > ] dx
where Q(x|y,A) i s given by r^ Q(x|^) = (TfL^i^)®!) r(r->) (T(xrL|ri*I) ^
Let tr. denote the trace over the first space in our tensor product. With this notation we have
{*Tt(r) > u M)} = j ; Q . ^ l ^ ) *[<*,(* 1Mb U ( l t 'H!
Now
and the first term gives an inessential contribution to Q : it is x-independent and proportional to the unit matrix, so it does not contribute to the zero curvature condition. Omitting it we get finally:
0>7)
where Q (x|y,.\) = tr. Q(x|y,.\). This is already a kind of zero curvature condition.
Let us now simplify the expression for Q.(x;y,.\)) with the help of the formula for T(x,y|y) in terms of W(x,y) and Z(x,y). Periodicity and cyclic symmetry of trace allcw to write C.(xjy,~) in the form:
Q, = tr {((I • W ^ r ) ) - ' fcZ>) (I + W(« l (*)) * l ) H>->)} ($J>
' - ( K )
s o*?M I * l ^ W °i ^ >
Q = l sin?( r) J _ ( I • Wfr, r>)" o; (X t V ( x , ^ ) C S 0 )
Here the property V(A®I)P = A i
was used. Note that V(x|y,A) - - j Q. <x|y,A)/sin P (p) is a local function of p*(x), ip(x).
Remember now that tr T (y) » 2 cas?t (y) so that
{ *> T u ( h ) , V («,>)} ,-2 si. P L( h) { ? u ( ^ , V ( K ; > ) } (SI)
Dividing by sin P. (u) we get a local zero curvature representation
, i V0cju;}) + [V(XJ,M,}) j fx,})]
the matrix V(x!u,A) playing the role of the generating function for all possible matrices V(x,X).
Thus the FPR implies the zero curvature representation which was so useful for introducing the conserved quantities. From now on we shall rely only on FPR and its generalizations.
12. The L -*• « limit
We are going to return to the question iii). It is difficult to count property the number of integrals of motion when it is infinite. Things simplify ir the limit U -*• °° where explicit expressions for the angle variables can be found. To show this we need more information on the solution of the equation, called sometimes an auxiliary linear problem
(i - IAVO i = ° (")
We shall consider the simplest case where ty* and ij; vanish when |x| -*• °°. Then the asymptotic behaviour of solutions is governed by the matrix
£(x;>) = «r(a«r,x) (») In particular the following limit exists
T(>) « I.», £ ( - L ; > ) T t ( a ) £ ( - L ; > ) Css) L-+°o
called a t rans i t ion matrix. I t can be writ ten in the form
(*6)
(t) corresponding to the sign of g. Having in mind that W (x) vanish n
when |x| -*• °° we have the relation
«H aft) = U .(Fft)-^L) + 0 ( 0 (57)
so that in a(A) plays the role of the generating function for the conser
ved quantities.
13. Simplifying the Poisson brackets
A) and B(X)
can be found explicitly . Begin with the already known formula
( W * W | = [ r ^ - r ) )T L(»aT L( r)] (27)
multiply it by E(-L,X) ® E(-L,y) from both sides and go to the limit
L -»• °°. In the RHS it is convenient to consider the two terms in the commu
tator separately. Consider the function 1/X-v* to be v.p./X-p for defini-
teness. Use the property
P A ^ B = B ® A P (s«)
of the permutation matrix. Then the following relation holds
{TO)®T-(r)J = r f a - r ) T(\)9T{p) - T(})®T( r ) n(;\- r) (fe
where
r_ (>-»*) = &* r(>-j*) E(-L, - r) <g> E (-L} JU->)
These l imi ts can be found exp l ic i t ly with the help of the formula
L» ÏÏL e t i U = ± , > S(x) (6.)
thus leading to the expressions
o]
14. Linearization of the equation of motion In particular we have the following relations
(as it must be) and
From the last one we get that the equations of motions, generated by £n a(y) are linear for ku)
i LOO* / - M r ) > M » } --%— t(M ^
In particular for the Hamiltonian, which coincides up to a factor with the third coefficient in the expansion
/ * * ( » = / £i fro) »i A
we get
A KA) = / H , ;(*)} - - ; a * t w (to
Thus the map
linearizes the equation of motion for the N.S. model. To answer the question iii) it remains to investigate to what extent this map is one to one.
15. Elements of scattering theory We continue collecting the information about the solutions of the
auxiliary linear problem. It is useful to realize that it is nothing but the scattering problem for t'.c stationary massless Dirac equation.
Breaking Tj (A) into the product
T (V) , T (i ,X[A) T(*,-LU) {62)
and taking the limit L **• °° we see that T(X) has a representation
where T (x,1) = &m r(x,?L|>) £(^L ;?0 (6S) • L — o
The matrices T+(x,X) are the so-called Jost solutions of the auxiliary linear problem. They could be defined by the boundary condition
Until now we were not specific about the range of parameter X. For L finite it could be any complex number^everything being entire functions. For L infinite one must be more careful.
All formulae above are certainly correct for real X. It turns out that the columns of T+(x,X) allow an analytical continuation in one of the half-planes ImX £ 0. Introduce special notations T , T for the columns, so that
Then T and T have a continuation onto Ira X> 0, whereas T and (1) + +
T are analytic in Im X < 0.
The relation r /\ T(x^) = T + (x ; » l(>) (60
for fixed x can be considered as a sort of boundary problem (Riemann-Hilbert problem) in the theory of analytic matrix-functions and used for the reconstruction of T+(x,X) in terms of T(X).
16. The Riemann-Hilbert problem
The simplest example is as follows : let two (matrix) functions G+(X) be analytic for t Im X > 0 and have thereneitherzeros nor essential singularities, for instance
GtC\) - I as 1*1 -*- (&0 For real X we are given the relation
&00 = G+(>) G-O) (?o)
Then if some conditionson the given G(\) are fulfilled G+(A) can be found. One necessary condition is that
uJ G(^) = in dit G(A) 111)
vanishes. In a scalar case this condition is also sufficient and solution G^(X) can be found through dispersion relations. In a matrix case a convenient sufficient condition is the positivity of one of the two matrices
Ke 6 , ! ( & + & * ) , I m G = X (Or-G-*) {71)
Representing G(A) and G+(X) in the forms
Ht ,, G(a) = I + [ F(t) ) e ai
<>* ^ (?3) Gj.00 = T 4. f F + W «' a t
•we reduce the Riemann-Hilbert problem to the Wiener-Hopf e q u a t i o n . Indeed,
in the r e l a t i o n
F W = F +(t) + F(t) + J F+(r) F(t-i) as ty)
the LHS vanishes for t > 0 leaving the linear integral equation for F+(t).
17. Inverse problem reduced to the Riemann-Hilbert problem The equation
T li,)) = ! > , » 'T(a) (7S) can be reduced to the normal Riemann-Hilbert problem. To achieve it let us introduce two new matrices
which are analytic in the lower and upper half-planes, respectively. The linear relation for real X assumes the form
SA*,*) = SA*>*) SCO ( 7 7 )
where S (A) can be expressed through a(A),b(A)
/ ' '^]\
Note now that a(A) is analytic in the Im A > 0 half plane ; indeed it can be expressed in the form
a(J) = a«t S+(>) (75) and can vanish at points A. whenever the solutions T and T are linearly dependent
Both sides vanishing exponentially for |x| -»• °°, this can happen when A. is a discrete eigenvalue of the auxiliary linear problem. For g > 0 this problem is equivalent to the self-adjoint one. and the discrete spectrum is absent. For g < 0 the position of zeros of a(A) can be essentially arbitrary. In what follows we require a(A) to have a finite number of simple zeros.
Let us first suppose that a(A) has no zeros. Then we can include I/a(A) into the analytic matrix S (A) without spoiling it. With new notations
G-.M = S.(*,T0 , G + (*,» s ftll^L (|0 GOO -- aO) S(>)
we have a linear relation
G.(x^) = G+U,*) G-(») (*2) which i s equivalent to the normal Riemann-Hilbert problem because for
JA| -* °° we have the following asymptotic behavior
/? (>) = I + «(I)
so that G + (x,X )E(-x,à ) have no essential singularities in their
half planes.
Now one can see that the matrix GÇX) + G*O0 is always positive*,
in the case g > 0 it is just a unit matrix^ in the case g < 0 we have
G = G*", however unimodularity of T(>)
1 * 1 * = ' - I M X
shows that |b| does not exceed 1 and so G is positive.
Thus when a(à ) has no zeros the matrices G + (x, À ) and G(?i) are
in one-to-one correspondence.
If we allow now a(,"\) to vanish at points "X = * ., the matrix
G C X ) has simple poles there, whereas G_(à) has simple zeros at
*« V. J
We can absorb these singularities into a left multiple of the form
BOO = TT 3-(M J J
where each factor has the form
B-(» = I • V^i F •\ - a j J
and P. is a one dimensional projector. We can uniquely find B(^) in
such a way that G + (x, \ ) will be represented in the form
G ± (x ; » = BO) £ t (x , } )
where G + (x, À) have neither zeros nor poles. The factors ~( . come
explicitly into P.. In the linear relation we just drop this factor
and remain once more with a regular Riemann-Hilbert problem.
18. Inverse problem investigated
Given a linear relation
with the prescribed above asymptotic behaviour foi large | Aj it is easy to show that the solutions G (x,A ) and G_(x,A ) satisfy the diffe- , rential equation of the auxiliary linear problem. Indeed, differentiating both sides we find
( i O &-' - ( i <*W ) <* t"> Thus the R.H.S. gives an analytic continuation of the L.H.S. into the upper half-plane. The essential singularity at infinity being cancelled the only possibility is that the L.H.S is a polynomial in \ .
From asymptotic behaviour we get
thus reconstructing the auxiliary linear problem with matrix U givsn by
V. * [ +- w ; n] It is gratifying to observe that this matrix is antidiagonal. Its self-adjoint or antiself-adjoint form must follow from the corresponding involution condition imposed on G('X).
It is less trivial to show that the matrix U (x,A) vanishes when | x [ -*- <>a and that the solutions G f (x, ) ) , in their linear combinations T + (x,^) have prescribed asymptotics. However it can be done , thus showing that the system of scattering data (b( ) ,b* ("X ), V , f . ) are in one-to-one correspondence with the initial data ( T (x), TCx)). The proof goes out of the scope of these lectures. Parallel considerations based on the Gelfand-Levitan-Marchenko equation can be found in the texts on the inverse problem in quantum theory of scatt-ring. 19. Angle-action variables
We have found a change of variables •4*00 ; 4(x) - * ( t p o , ^ ) , ^ , ^ )
\9o)
^ 0
(discrete variables appearing only in the attractive case g< o)which linearizes the equation of notion. The new variables can be interpreted as the angles and actions. Indeed the dispersion relation for tna(/\) is given by
U .W . -I- f i*('tlH">l') * 7 -«.(131 Thus the generating function for the conserved quantities depends only on the variables
a
— half as many as the set ( L . b*,7\."y). They are naturally called action variables. From the Poisson bracket relation
which is contained in the set of the commutation relations found above, one can show that the conjugate angle variables are ara L>(à) ,
•in J Y •! and araY., the first and the last being of the phase type.
The main conservation laws now can be written as follows
j -aO ''J
J
F exhibiting the spectrum of modes. The continuous spectrum describes a particle of mass m * 1/2 which is the only mode of perturbation theory. The discrete spectrum introduces a non-perturbative contribution in terms of particles of mass vim, »» ç playing the role of momentum and - — 0 that of internal energy. When quantized in the quasi-classical approximation vj takes integer values and discrete modes are nothing but bound states of the original particles.
In this way we see the soliton mechanism of spectrum generation in action. The term "soliton" is appropriate because j , f , corresponding to b= 0 and just one zero A., describe the classical soliton solution.
This concludes the discussion of complete integrability of the N.S. model. To collect more material for general considerations we shall present a few more integrable models.
20. Heisenberg ferromagnet
The field variables are given by a unit vector
s =(s*c«>, * » » , v , s 1 . i )
subject to commutation re la t ions
The Harailtonian is given by
H - f m * Equations of motion look as follows
I Q , ZS x S 7>t
far y
(57)
(a*)
{?>)
The auxiliary linear problem is defined through operator L
L « i - u(x,M , U(<;>) = ± i S V Q0fl; dx *
where <r are Pauli matrices. The fundamental Poisson brackets (l5J)
are sa t i s f ied with a r-matrix coinciding with that of N.S. model
l o i )
The local conserved quantities (and the Hamiltonian among them) are given by expanding the transition matrix T(x,y j A ) in the vicinity of % = 0.
21. Sine -Gordon model
The relativistic equation
• *f + i 1 si» & V = 0 0 oi) f r
is a zero curvature condition for a connection given by
L = i t ; r i W s - *, « ^ t &„ «.-. S r,l dx u 4 * * i - J
at L 4 x 3 • z ' ' "i z j
{loi)
if the vector k = (k ,k .) is on the mass shel l
V0 - K - m1 (lo«) Parametrizing i t by the rapid i ty A
4 o = H c U ? £, = ». s U (lOS)
we get the matrix U(x,>, ) which s a t i s f i e s FPR with the f-matrix
>'(*-[*) = -J ! i i- + _ L-L <T3<s><r2 \10Q)
which we shall call trigonometric to distinguish from the rational r-raatrix of the two previous models.
22. Mon-isotropic magnet (Landau-Lifshitz model )
The same vector variable S (x) can be used in yet another model with equation of motion
4 i s £ ? x S 4 J(s) x 5 (lo7) at dti
where JS is a vector with components (J,S., JjS,, J S-,)» JpJj.J, being given constants.
If we look for an appropriate connection of the form
u = I u (log)
v = Z * S ^ + 1 ^ ' KSl ~ r* V ; c
* 4x
where U ,V ,W are some coefficients we find that the zero curvature a a a condition leads to the following relations among them
w. - u. > v . 2 ; | i * u K i ; . VOS)
The last condition . ives two equations on three coefficients so that they can be parametrized by one variable A . Explicit expressions involve elliptic functions. The FPR are satisfied with the r-matrix given by
which is completely non-isotropic and will be called elliptic.
The list of models can be extended. This calls for some unification. In what follows we shall try to present a scheme general enough which will produce the integrable models mentioned above as particular examples. It is only natural that we shall discuss primarily the realisation of FPR.
23. Lie-algebraic Poisson brackets-
One can invent not so many natural Poisson brackets. One important family is generated by Lie algebras.
Let G be a Lie algebra with the generators X and structure cons-3 tants Cc, ab
[*. > xk] - C x. M
Consider a linear space G* with variable Ç and introduce the Poisson a
bracket
or for two arbitrary functions on G-*
{Hi),i(S)} - cu If | f ?. 0»») This Poisson bracket satisfies the Jacobi identity, however it is degenerate, namely there exist functions Z(Ç) which commute with arbitrary functions
This bracket is associated nowadays with the names of Kirillov, Kostant, SouriaujBerezin,— but apparently it was known to Lie himself.
To make the bracket non-degenerate one must restrict it on a sub-manifold. The natural candidates are "orbits" of the algebra action in G* , namely the integral manifold for the system of equations
where a runs through all values. Depending on the initial conditions these manifolds can have different dimensions, but it is always even and the restriction of the Lie-Poisson bracket on-the-orbit is nonrdegenerate.
It is worth mentioning that in Kirillov-Kostant's program the irreducible representations of G appear in the course of quantization of the classical systems defined on orbits.
24. FPR as Lie-Poisson brackets
We shall show that FPR appear naturally in connection with current algebras. Let us omit variable x for the time being and consider the Lie algebra C of functions O (/\ ) with values in a finite-dimensional Lie algebra G with generators X and structure constants C . . A natural basis of generators in C is supplied by
with the commutators
[x:,x;] = C x r ( " 7
We shall distinguish two "triangular" subalgebras : C is generated by X" for m > 0 ; C_ is its complement. After change m •* -l-a, the new index being non negative, the coranutation relations in C can be written as
[x;,x;] - c:,xr" -,..o,i.... cm-I )
Thus the dual spaces C*" and C* are supplied with the Poisson brackets
respectively. Let us introduce new the generating functions for the variables F > a
(llSx,
It is easy to calculate that these functions satisfy the following relations
The next trick is to saturate the index a. Let K , be a Cartan-Killinc ab fc
ab matrix corresponding to the basis X and K its inverse. In what follows we require that such inversion is possible,reducing the admissible Lie-algebras to semi-simple. Introducing the matrices
the following relation can be checked irrespective of the representation
for X a
Cl A'® A" = [ TT, A*» i ] = - [ TT, I «A'] 0 «
Using th is for
U(V> = I u . (« A' C'"' a.
we obtain the relation
(uw? (;(/.)} = t [IL. ,i;w®i + iaU(r)J o«)
Now we remember about x which we reintroduce as a second variable
in the current algebras C . . The generators now acquire an additional
index % and commutation relations look as follows
{ X . " G O , X ; ( , ) } = c ; t x;r*> 5 6.-y> o «
Repeating what we have done and keeping ;; untouched we end up with
the general forms of rational FPR
(117 .
25. Orbits for N.S. and H.M. models
Let us show that models with the rational .*—matrix described above
are particular orbits in C or C_ when C = SU(2). Consider first the
Heisenberg ferromagnet.
The condition in C
is pvidpnrly invariant with respect to the action of C defined through
equations
for all b and n. Moreover, only ç (y) acts non-trivially in this
subspace and À £„( x) >„(x) = ? (x) is conserved.
Choosing the orbit by the condition t
we get the matrix .0,
?V/ = Citant ('3o)
satisfying FPR and defining a non-degenerate Poisson bracket. It is exactly the matrix L(x,A) used for H.M. model.
The N.S. model is defined in an analogous way as the simplest non trivial orbit in C*
Ur>)
U < V ) = V. (*) * * U, (x)
when
Ue = V A* / 1 a
with the commutation relations
CUM
It is clear that the matrices U and U. used for N.S. model do satisfy o 1 J
these relations defining a non-degenerate orbit.
26. Trigonometric and elliptic r-matrices
Define a group of shifts acting in C
where a is an automorphism of G. The reduction of phase space C* with respect to this action can be achieved by using generating matrices
where A is a representation of a. Let us suppose that the following relation is true
[ FT , A ® A ] = o
Then it is easy to check that U(A) satisfies the relation
where
In particular the r-matrix for S.G. model can be obtained in this way in the case G = SU(2) and A = <T.
In some cases (but not for all G) one can repeat this trick once (and only once) again. The elliptic S-matrix is obtained thereby.
27. Lattice systems
We have used above lattice approximations to continuous models. The role of covariant derivative -= U(x, A) was played by the infinitesimal parallel transport L. with the approximate FPR
The full transport matrix was defined by the ordered product
Now for the lattice system we shall require these relations to be exact.
It is clear that the Poisson brackets for T (X)
are satisfied, so that the family of dynamical systems defined by
L(») = { A T w ( r ) } L.(^r} have an infinite number of conservation laws.
28. A representative example
Let us fix r-matrix to be rational
and suppose that ÎI plays the role of permutation, namely for any A from G the following relation holds
[x« x rr] =o Then i t is easy to checkthata L-matrix of the form
L;(>) -- i * i r x
satisfies the lattice FPR if the field variables S? satisfy a discrete l
version of the Poisson brackets used in C : +
This procedure works in particular for G = SU(N). For G = SU(2)
the matrix L- is 2 x 2 and given explicitly by
and explicit calculations give
à S, Ox) , x „ iû
(\<*
where we multiplied the previous L.(X) by A. The model associated
with this L.(A ) is a discrete version of the Heisenberg magnet.
To get local conservation laws from tr T,.(A) we observe that
2 If we fix the length S. to be independent of i (as was done also m
the continuous case) ^ • - i L
then for A = i S, the matrix L. ( A) is degenerate
We can use it to expand tr T ("\ ) in the vicinity of A = T \S !
in power series of ( * t |s|).
In particular we have
to. T„(±!si) = ÏÏ ( p ± ( s i t l ) , « t ( s ; i ) 0
A T„(|s|) . *. T„(-U|) = TT ( S ' * J > - y
Thus £n(tr T^( | S|) tr T (-)S|)) gives a near neighbour interaction
hamiltonian
Hz - £ ^ - ( /S | l + £;•,-*:) + 2 > ^ ( i s l - ^ i ) 0 which is a natural lattice approximation to the continuous Heisenberg
magnet after rescaling
29. N.S. as a limit of a spin lattice model
We can realize the Poisson brackets for S. using the ordinary
canonical variables • ., "r. *i ' l
as fields. The formulas
Ossl
<; - IL V Vf A ^ }
where 0
'/ i
? T
USb b /
i'lS?)
2 2 2 give such a realization for S = 4/g A •
We intentionally have split S into product to indicate the
scale of the continuous limit ù —*0, when V • and 'Y • are of order
In this limit L.(/\) has an approximate form
L- *
li
hû r\
\ 1L + <%A
(I * 8)
which coincides with the expression
L"S- = i + r ut»,*) <« after multiplication by «^ G", from the left and change gA -» i/1.
Additional matrices <T in the product TlL. can be combined to mul
tiply «aeh L. with odd i from both sides,
l
(isa
Now it is easy to see that
», Li (+) «i - L; 1-+) (ito)
All this shows that the continuous N.S. model can be obtained from the
lattice spin model in the scaling limit if the following convention is
used
X J, 4. = (_,y &'- +(x) , Y* , t-'V A - 4Îx) ('l6.'}
0 Si
In particular one gets the Hamiltonian of the N.S. model from the
Hamiltonian of the lattice spin model given above.
Thus just one lattice model serves for both dynamical systems we
have associated with the rational r-matrix.
30. Trigonometric lattice model
We can repeat the trick of averaging over the group of shifts.
However instead of summing shifted L.(Â) we must use a product.
Suppress the index i and define L(A ,n) as follows
Z . ( V ) = A* *-(* + "«") A"" where A is a representation of an automorphism of G as above. Define L(^)
to be an ordered product
LfV, = ÏT L (V-) We show that L('A) satisfies the lattice FPR with the trigonometric
r-matrix.
Indeed, from FPR for L. ( "X ) we get
( l ( V ) ® Up») } =[*"„.„ (*-r> ; L
t ' V> » >-(nn)] ("
r_(>-u) 4A*- 1"®!) rft-,,) ( A " - „ I ) Ot-
where
Furthermore, differentiating the product we have
••>-! 1-' (lU ^
+;1 t" 1 +» l
r and after using the FPR this is rewritten as
^\f" r 1 r « . . . v Making resummation by parts with the natural convention
ft L(V) = 1 t
4 -
(«60 ? * = - , .
we get the usual FPR for «.he matrices L ( A ) with the trigonometric r* -matrix
'k-h) s 21 «v (^-r) It will be interesting to evaluate explicitly the product L(/O in the case of the spin system. This requires the calculation of the products of the form
TT SJLi n + ju
where B is an arbitrary 2 x 2 matrix and U is some number. I don't know any direct means of calculating such products. Some indirect con-siderations show that L( /•) is given by
L -- / f j ^ U + S J 3 /
I S. ffc) I
(143^
( l7a)
(J7i)
where Z2.
Such matrix L(^) can be used to define a non-isotropic lattice spin system. It can serve also as a lattice version of the S-G model. Indeed after pararaetrization
S, = Cos <f S + l7T
Vs*-v s e
(172
(173)
r where P and 7t are real canonical variables
{ <f. . 7C. I = I., we get a L-matrix which after multiplication by 0"| from the left coincides with the matrix
\ » •
in the limit A —*0 (in some particular units for m and S which can i
be reintroduced).
Thus once more we see that there exists essentially only one lattice model for trigonometric r-matrix.
The elliptic case can be considered in an analogous way. The lattice version of completely nonisotropic magnet is obtained. Its realisation as an elliptic-sine-Gordon has noz been made explicit yet.
31. Quantum systems
It is easier to begin with the lattice case. Now the field variables are operators. The Hilbert space for each lattice site being denoted by
Y) . the full Hilbert space is
"1
/74)
% = I I ® *7 ; Of course all Y1 . are the same. • l
Here R(^-t-*) is a matrix in V ® V independent of the field variables.
These relations could be guessed through the correspondence prin-Jeed, they tutfn Into classical FPR if R(^-.w-) allows
R(a-n) = P ( i - ; 4 T r ( 7 i - r ) +o(* 1))
( r -H7s;
In terms of the field operators acting in Yj . we bave to construct a matrix L.(^\) in an auxiliary space V depending in A . The FPR relations in quantum version look as follows
ciples. Indeed, they turn into classical FPR if R(^-.w-) allows an expansion
077 where P = I + TT . However their main merit consists in the fact that they allow to get a simple form of the commutation relations for the
r matrix quantum transport
T„(a) = ÏÏ L.-(a) which is well defined because the field operators for different lattice
sites are supposed to commute • In what follows we shall call them fun
damental commutation relations - FCR.
Multiplying FCR over all i we get these relations in the form
R(>-r)(Tww®TM(r)) s ( T N O O « T V M ) ft (HO It follows, in particular, that the family of operators generated by
tr T (^) is commuting. Whenever we find an interesting Hamiltonian
in such a family, we have a quantum model with an infinite number of
conservation laws.
32. Rational example
Consider the quantum lattice model with field operators S. of i,a
Lie-algebraic origin, satisfying the commutation relations
[S- ; S "j r ^ C\ S- &., and let A be a representation of the algebra G (defined by C , ) such
that P = I+TT is a permutation
p x*«xfc = x f c«x* p 2 2 . .
and is normalized as P = o< I. For SU(N) algebras this representation is th-2 fundamental one.
! 7 S )
(173)
(l8o)
(\t\)
Then for
and
M») • r + | S, = I X**,.
+ P
On)
(.1*3)
r the FCR are satisfied. In the process of checking it, the non trivial terms generated by the first term in R(^-f*) are
* v (i 5®i + i - r ® s - - L 5 « i - i r « s )
2 where (T. are Pauli matrices acting in <j. = (£ defines the spin 1/2 quantum Heisenberg chain.
ilSO
Cj«r) whereas the second term produces the expression
-L f ? S«S - S®5 ? ) It is exactly the permutation property which allows those two contributions to cancel.
Indeed in a more explicit form we have
? S&S - 5®S P = 7 \ s ' fP Aa®Ak - A%a A' p)
s P I 5 J t (A*® A* - A1* A*) Q«)
= ? Z &h-hK) A*«AL
(U7
Now we have to use the field commutation relations and the following a
property of the matrices A
C t A*® A' = [ T T ; A®r] = - [ IT ; I * A] t o reduce ( IS5) to
•g 1 (s © i - r <a s )
so that the two contributions above indeed cancel each other .
In the particular case of G = SU(2) the L-operator
L. = * I + i I c; <r. (la*)
33. Comments on trigonometric and elliptic reductions
The natural generalization of the process of averaging over shifts
used above would be to define the trigonometric L- and R-matrix as
follows :
R(V) = ff ( A M « I ) R(a+«wï (A"w®i) 23)
Unfortunately I don't know any general proof that such L.( A ) and
R( A ) satisfy FCR. In the particular example of the spin 1/2 system
one can show that beginning with the rational (XXX) case one gets in
this way the L. operator for the trigonometric (XXZ) and elliptic
(XYZ) models. For these models the L. operator have the form
k = Kb) I ® 1 . •*• I w-to ° i ® v ( ( 3°'-a.
where index i means that I. and <r . acts in Vi the functions l a,i 'l
W ( A ) , W (}) are trigonometric or elliptic functions of A .
I hope that some way could be found to justify the process of ave
raging introduced here.
34. Comments on higher spins
The construction of L-oper^tor in rational case used above works
only for lowest dimensional representation of the Lie-algebra G. The
construction for any representation can be given through the use of
tensor products and their decomposition into irreducible representations.
In particular, for G = SU(2) one can use a L. of the form
v x )
for any representation S of SU(2) with the same r-matrix as above.
However if we want to have another representation for the auxiliar"
space (below we shall see when it is needed) the expression will be
more difficult. Moreover even with two dimensional auxiliary space
in the process of averaging we will obtain L-(/) of the form
I- = W,ft) I <*T (£) + 21 W (>) <rA a T(£-) (let
where T and T , a = 1,2,3 are non t r i v i a l functions of spin operators S in contrast with the spin 1/2 case, a
I t is in te res t ing to observe that the commutation re l a t ion for T
and T which can be read from FCR have the following general s t ruc ture
fc,Tj - K J (XTC +T7.) 'f t
where Y . are seme constants. These relations could be looked upon as those of deformations of the SU(2) Lie-algebra we begin with. For the spin 1/2 representation they reduce to the representation used above ( I l Î )
032.)
T = I
35. Concrete models
Let us collect several quantum lattice models associated with the SU(2) Lie-algebra which were already mentioned above. At first the corresponding L-operators will be given in two-diraensional auxiliary space.
35.1. Spin 1/2 XXX model
Field variables are Pauli matrices S, • = TT °" •. The L.(A) *; l 2 a, î î
matrix has the form
Lv -- ^ I&li. r ± J_ c ; ® < r ^ ( j 9 3 )
where we made a convenient change A —» i A The Hamiltonian N
can be obtained from T..(^) * | T L . ( ^ ) as follows N 1
H = CoMt. ± JU fr T{*)\ (195) The role of the point A= i/2 wil l be clearer l a t e r . Note that in the
c lass ica l case the Hamiltonian could be obtained without derivat ion in A.
• •J o •
35.2."Spin 1/2" XXZ model
The same field variables are used in the L.( ~\ ) matrix of the form
/ i.*\ +• cos A Sz 5+ \
t C - - ^ C • \ O. SinA - COS A O, y
The local Haailtonian is obtained from tr T„(A ) in the vicinitv of some A . This L - ( A ) is a product-average over the shifts of XXX L--
matrix with A = 0" .
35.3. N.S. Lattice model
Field variables are , ., i. wirh the commutation relations l i
H[ V] = S (127) The matrix L.(^i) coincides with that of the XXX model after using an infinite-dimensional representation for S. . Corresponding expressions
l ,a for the soin variables S. can be obtained from the classical ones given in.
i,a (156) using normal ordering. One must not forget also the multiplication by 0" from the left. The representation used is irreducible,self adjoint for g < 0 and corresponds to spin
[is 2) S(S-H) = -£_ ( A _[) or 5= - A.
To get the local Hamiltonian we must use a different realization of the auxiliary space. In general, a suitable choice for this auxiliary space V must coincide with the local quantum space VJ .
Without going into the details let us state that such a L.(^) can be constructed , local conservation laws are given by tr T ( )*) in the vicinity of A = i S and the Harailtonian has the form
H J £ f J S. 5 \ (l99> *.
where f (x) i s some funct ion with the smell argument behav ior s
f s ( x ) = * + 0 ( x i ) UocO
I t is clear that in the scal ing l i m i t (16.)
one is able to get the N.S. continuous Hamiltonian from these formulas.
The models of integer (or half-integer) spin j could be gotten from this iust by restricting S = j. The first non trivial f is given by
f ; % J. Î \ *} = X - -<
and it is a polynomial of order 2j for any j. (.201)
3 5 . 4 . S-G model
F ie ld ODerators a re T . , TT.
O; , *J = iS:w The mat r ix L. (X )
/ e - ; i K i Ç(-f:) e
kW - \ \ ~ 3. ft)
- i t r t " ^ \
^
(2*Z
f («,, . (i + J ^ - « f * ) ' 4 ; } t (»).- «- ( Pf * >)
can be considered as that of the trigonometric spin model for the infinite dimensional representation defined through
S + = S3 * l Y 2.
(2.03)
ram One must also remember the multiplication by d. from the left. The prog of finding the local Hamiltonian is not complete yet but it is quite plausible that H has the forra
» - I f f 'U,. S , H , 1 *S i ) 1 S i . l , 1 + a \ , S..,;J) Q where t(x) */ x and a is expressed in terms of ra, 3 and A .
r ' One sees that in the scaling limit this expression gives some
ZO
regularized version of the S-G Hamiltonian
( 1 7 P + ±%* + J^fN «**)) ix (205)
The R-matrices for all these models are Ax^r matrices of the abc-form
* ( » =
a 00 0 \
.(*) CÛ) o ! \ 0 c & ) L/ w / \ 0 ^ \ .'
<K*)/
i. 2 05
where a(^) can be normalized to be one and after that we have
k(a) =
in the rational case and
c(1) = -1 *+i
Uû7j
b ( ^ =
in the trigonometric case. Here "j" is defined as the period of the shift 2
and coincides with the coupling constant y = 3 /? in the case of the
S-G model.
(2-0 2)
36. Algebraic Bethe Ansatz
We shall show an algebraic procedure to get a family of eigenvectors
for the commuting operatorstr T (^), for the examples listed above. Let
us write T., in the form
T.fo) -- ÏÏ L-M =
so that
*t V>) ~- ANW + D,W
(ZW)
(2!o^
and B (^) is a quantum analogue of the angle variable b( A ) in the
classical infinite volume case. Now the angle variaoles being quantized
have to be spectrum generating operators. So it is natural to look for
eigenvectois in the form
+ (i>» . BO,) ... s(\o n (m)
where u is a particular eigenvector. We shall see that this program can be realized.
First we construct a suitable il . To do it, observe that for L. ( ) there exists a vector '-O. such that
Ac» * \ L- UÎ w. = I I w. (2iz:
when c< ( A ) and d(^k) are c-nurabers (local eigenvalues). The local vacuum 0). is the highest spin state for all models but S.G. In the latter case one can find CO only for the product L. ,L-.
It is clear now that the state
A s IT® "• ( 2 i ^ 4
is an eigenvector of A (^) and D„(à)
with a trivial modification in the S-G case.
To consider more general V ( ( 1 ) we need commutation relations between 3,.(") and A (^), D,,(^). We can read them from the relation \.i73)
N N N S(a-r) (TN(:,) ® T„i r)) = ( W , ® X ( ^ ) R0>-r) (zis
which until now was used only to show the commutativity of tr T ("X ).
In suitable form the relations look as follows
[B»W ,B„W] , 0
Using these formulae one can show that
11(7 .'
Î2I*)
A(*IW) +({*}) + 1 M*IW) B W - Bft)-3Mfl £*?!«*
where
and
A, , - Kv*> «M* TT _L_ - K»-»J sen" f i - i — %
Indeed the term proportional to can be obtained by using only the
first terms in the RHS of the commutation relation. To get A.( |{^})
one must use the second term in the RHS on the first step of commuting
A„(à), D,.(A) with the product B(^,)...B(/v ) and after that use only
the first term. The expression for A.(}{ [X\ ) follows from the symme
try in A ,,. .. A . 1 1 m
Now observe that
so that -dependence of A.( \ J {^\) factorizes. We can get rid of the
unwanted terms by requiring that the A . vanish. This gives m equations
<V
(2.21'.'
tor n*. unknowns r . -H
which ira nothing out the transcendental equations of the Bethe-Ansatz.
We shall list the local eigenvalues for concrete models :
i) General rational spin model
i
N.S. case is obtained for S = -1/g^ .
(ij) S-G model
'Mi 37. Comments on Bethe-Ansatz
3 7 . 1 . Continuous l i m i t in N.S. model
General equa t ions for sp in models have t he form
When S = a f t e r r e s c a l i n g g A -*~À we get
\\-i\ul i*l{ ^ - ^ -^ /
2L With N = — , the L.H.S. is exp 2ià L in the limit il-*0 and we obtain a the ordinary Bethe-Ansatz equations for N.S. model.
37.2. Nature ot the spectrum
There exists a sound reasoning showing that in any solution / .... I of the Bethe-Ansatz equations all 3 are different. Thus the spectrum of commuting operators is of fermionic type.
r 37.3. Additive observables
The form of the eigenvalue A is far from multiplicative. However the local eigenvalue 0 ( A ) has a zero ; in the spin case it is = is mentioned above. In the vicinity of *X , the second term in A is negli-;ible when N-»oo , so that -Cn tr T ( ) has an eigenvalue g N
N £.«00 -7,-2* c (*,-*) ^ A -X, (227-which is additive. It is nice to realize that this is nothing but the generating function for eigenvalues of the local observables.
37.4. Counting of Bethe states
In general there has been no proof up to now that Bethe states give a complete family of eigenvectors. For integer spin models one can show that Bethe states correspond to the highest weight in each SU(2) multiplet only. Other states are obtained by applying the S = T S operator. After that,the complete family of eigenstates is obtained. The proof is based on counting the number of states which happens to be equal to the dimension of full Hilbert space, namely (2S+I) .
It is exactly at this point that the real work begins. Given a particular model, one is to investigate and classify the solutions of the Bethe-Ansatz equations and describe possible eigenvectors in terms of particles or excitations. Some examples are presented in the lectures of J. Lowenstein. The reviews referred to below also contain much material and references to the original works. However my lectures have come essentially to a natural end. Still some comments will follow.
33. Generalization and questions unanswered yet
A lot about algebraic Bethe-Ansatz can be generalized to other Lie-algebras. In particular the R-raatrix can have the block abc - form, where a, b and c are themselves matrices. The eigenvalue A is a matrix itself and to diagonaii^e it we have to solve a new magnetic model with l/(c(^ - A.) playing the role of the matrix I,.. Thus a hierarchy of Bethe-Ansatze appears, many concrete examples of which are known. However there exists no systematic classification of all possible schemes.
The existence of the iocai vacuum U). is a very restrictive requirement. In particular it does not exist for the spin 1/2 XYZ model whereas one can write Bethe-Ansatz equations also in this case. The generalizations of the algebraic scheme to such cases are very desirable.
One possible route is to investigate the analytic properties of the eigenvalue A( *| { ^ V ) . Incidentally the Bethe-Ansatz equations are nothing but the statement that A is an entire function of A so that its apparent poles in A = A. vanish. General investigation of the possible form of /\ ( A j < X ] ) has not yet been done.
33. Infinite volume limit
The limit N-*«»(or L-»oo ) simplifies the investigation of the Bethe-Ansatz equations. More on this is contained in the lectures of J. Lowenstein. Tne outcome is as follows : the solutions /A\ in a typical case accumulate in the real axis with the density P (à )• The ground state^Dirac sea with all vacancies filled in") is characterized by the density f (A) ; the excitations are given by the density 'vac
?(*UM) = j ft) * ±1 <r^'\) ("*-where A. are holes in the Dime sea.
In these terms the ground state can be represented in the form
•J and excitations ar written as follows
nCPW • ? 3(\) i \ „ ^ &o) where
These formulas give a realization of the old program of Van Hove for the construction of the creation operators for the ground state and excitations.
It will be interesting to characterize the properties of the opera-tors B( A, ) without the recourse to B Q ) ; especially important is
establishing the connection between B(^) and the original field operators.
This connection is indispensable for the construction of Creen's functions, the problem discussed in more details in the lectures of H. Thacker.
40. Connection with statistical mechanics
The quantum L-operators in the case when auxiliary space V and quantum space Yj coincide can be used to construct the planar model in classical statistical mechanics. Introducing the matrix indices
>
L^ (r -*') for auxiliary (-v) and quantum (<x ) space we see that the matrix T (^) has still only two auxiliary indices but 2N quantum ones :
T £](T,TO . L L;;(-r > n ) L.*;(Tl,*) ... L%r/) and tr T (** ) is given by additional summation over "Jf = y'. This is nothing but the transfer matrix in planar models with the Boltzmann weight given by L ( 7, "j'). Thus the properties of tr T (}) investigated above can be used for solving these models. The role of FCR in this context was stressed by Baxter.
mz\
41. Connection with the factorizable S-matrix
Another interpretation of the four-index object L ( y, Y*) is in terms of two-body S-matrix, A being the relative rapidity of two particles. When V and Y> coincide we can essentially identify matrices R and L so that the FCR can be written in the form
R.,(»-r) V>-'> *„((—> • F»0"') *»<»-'> R » M ( 2 3 3 '
Here the following notations are used. Matrices R., are defined in the space V 9 V ® V, whereas each matrix acts non-trivially only in V $ V (it has four indices). The subscripts 12,13,23 indicate which spaces are involved.
We interpret the product of two-body S-matrices as a three-body S-matrix. The FCR is a statement that the definition of the latter is correct, namely it does not depend on the order of the consecutive
r two-body scattering!. It was shown by Zamol odchikov to be the only condition which allows the definition of a general N-body factorizable S-matrix.
42. Concluding remarks
We have seen that all exact results of low-dimensional mathematical physics, that is
i) Inverse scattering method ii) Bethe-Ansatz iii) Planar models in statistical mechanics iiii) Factorizable S-matrices
come together through the FCR or FPR. We also have given a fairly general classification of FPR with promising generalizations to FCR. This unification seems to be of fundamental value and throw a new light on all the subject of exactly soluble models.
A natural question is about the generalization to higher dimensions. The formalism presented here has used the one-dimensionality in a most essential way. Indeed the main object was a parallel transport over the line.
Connection with statistical mechanics shows a possible route to generalizations. The parallel transport along a surface must be introduced as a first step. It is not clear yet how useful this notion will be.
Two essential problems are still unsolved in 1+1 dimensional space-time :
i) construction of the Green's functions ii) solution of the non-linear C -model.
The first problem is under investigation in many places, see lectures of .H.Thacker. From the point of view of the present lectures the difficulty of the non-linear «r-model consists in the fact that it is associated with the non semi-simple Tip algebra E(3). Indeed beginning with the n-field
and the Lagrangian
L J i •• '
we introduce as the phase-space field variables n and Q.
o ) with the commutation relations
which for fixed x are the E(3) relations.
So we must think how to generalize the formalism to the non semi-simple case or find a suitable reduction of the 0(4) model. These questionsare under investigation now.
References
The goal of these lectures was to present a unifying view on the exactly soluble models. Real physical work consists in realizing this program in concrete examples and finding the true particle spectrum. We had no time to illustrate it in the lectures. Fortunately these exists a vast literature in this subject. The surveys wheremany references to the original works can be found are the following [I] L.D. Faddeev, Sov. Scient. Review CJ_, (1980) 107 [2] H.B. Thacker, Rev. Mod. Phys. y±, (1981) 253 [3] L.D. Faddeev, L.A. Takhtajan, Uspekhi Mat. Nauk 3£, (1979) 13 ;
translation by London Math. Soc. [A] P.P. Kulish and E.K.Sklyanin, in "Integrable Quantum Field Theories",
Proceedings of the Tvarraine Symposium, Finland, 1981 J. Hiotarinta and C. Montonen editors, Springer-Verlag, 1982
[5] A.G. Izergin and V.E. Korepin, Physics of Elementary particles and nuclei, (Proceedings Dubna Institute of Nuclear Research) 13 (1982) 501
[6] L.D. Faddeev, Proceedings of the 1981, Freiburg Summer Institute, Plenuir Press (to be published) We shall add several references to recent papers.
[7] V.E. Korepin, Commun. Math. Phys. _7£, (1980) 165 The detailed investigation of the mass spectrum in S.G. model
[8] L.D. Faddeev, V.A. Takhtajan, Proceedings of Scientific Seminars of the Stiklov Institute, 229 (1981) 134 Correct derivation of the excitation spectrum of XXX spin 1/2 models.
Contains also a survey of quantum inverse method in the application to this model.
Translation by the Consultants Bureau, as Soviet Journal of Mathematics [9] P.P. Kulish, N. Yu Reshetikhin, E.L. Sklyanin, Lett. Math. Phys. _5,
(1981) 393 Tensor product procedure for constructing higher spin XXX models
and much more.
A rather complete classification of the classical r-matrices was done recently by Belavin and Drinfeld. The results will be published in Doklady AUiderai: Nauk SSSR (Proceedings of Ac. of Sci.of USSR), a detailed paper is in preparation.
The results in the local quantum Hamiltonians and general classification of integrable models based on use of current algebras will be published by me in joint papers with Takhtajan and Reshetikhin, respectively.
The homogeneous quadratic, "commutation" relations which appeared in the lattice spin models, were introduced by Sklyanin. This paper is in the course of publication in " Functional Analysis and its Appli-cations.
Acknowledgements
I am grateful to the organizers of the School, professors R. Stor.i and J.B. Zuber for the invitation, which allowed me to think through the generalities of the classical and quantum inverse method. The lecture notes of Dr H. Braden and also of professor M. Peskin were indispensabi in course of preparation of this version.
