40 years in mathematical physics

Let Q = ft | M be the value of Q on M. The form Q is closed, but singular. We can show that, by (4), its rank is 2(n - m). It is not l e s s than this number, since the rank of a regular matrix cannot be lowered by more than 2m when it is r e s t r i c t e d to a submanifold of codimensionality m. On the other hand, we can select m linearly independent zero vectors from this form. We note that vector fields tangential to M are restrictions to M of vector fields Y on r satisfying

which, by (A. 1), we can rewrite as

By (4), we have

and, comparing this with (A.3), we see that the vectors X a are tangential to M and a r e zero vectors of the form S, which proves our assertion. Every singular closed 2-form Q, defined on some manifold, generates its foliation. Layers a r e maximal integral manifolds of an involutive distribution (cf., forexample, [25]), formedbythe zero vectorsof a f o r m . The set of such zero vectors is denoted by P and we shall show that since X E P a n d Y e p , we have [X, Y] e P . We have

and, by the condition on X and Y, the single t e r m which does not vanish by definition is the first t e r m on the right hand side. Thus, it also vanishes and this proves that P is involutive. We note that in the case under consideration, we have described explicitly the basis for the zero space of the form Q and the involutive nature of this distribution can be verified directly using (A.4). We now assume that foliation is indeed a fibration and let T* be its base. The form Q* of the inverse image of £2 on T* is closed and nondegenerate. Thus, r* is a phase space which is the object we look for The condition, formulated in this paragraph, for the existence of a fibration imposes very strong restrictions on the constraints
can be con­

which coincides with (8) and at the same time clarifies that condition. We now give an alternative algebraic definition of the algebra r . Let a be the algebra of observables on T with operations f + g, fg, and {f, g}. We denote by
63 that

imply [<$i, 92] e . Consider now the s e t a of functions f such that for all q> <e
and which can be called the normalizator of * in a with respect to the Lie operation. The existence of such a nontrivial (distinct from $) set is a strong condition on the constraints <pa, ccmparable with the abovee mentioned condition on fibration. The set a" is a subalgebra of the algebra 9 . Indeed, if / e a and J e= « , then, by (A.2), and the Jacobi identity

F u r t h e r , $ is, by definition, an ideal in 9 . The algebra a* = 8/d> is one further realization of the a l ­ gebra of observables of the system under consideration. In the actual description of the space T* in the second section we introduced the subsidiary condition (10). Equations (10) define a submanifold T* of the space M and condition (11) implies that r * is a t r a n s ­ versal integral manifold of the distribution P . We impose on cpa and Xb ^e condition that each integral manifold i n t e r s e c t s T* at only one point. In this case we have actually constructed a fibration and r * can be used as the realization of its base. r*.

Condition (12) has only technical value and simply makes it e a s i e r to choose canonical coordinates on Indeed, if it holds, then in some canonical coordinates on Tthe manifold F* is defined by the equation

The inverse image of the form in r * .

so that the variables q* and p* are canonical

In conclusion we wish to e x p r e s s our gratitude to V. I. Arnol'd, with whom discussions of the p r o b ­ lems of nonholonomic mechanics were of great value for this appendix. Responsibility for all noninvariant expressions not eliminated from the text r e s t s entirely with the author. LITERATURE

CITED

1. 2.

R. Feynman, Rev. Mod. P h y s . , 20, 367 (1948). P . A. M. Dirac, Can. J . Math., 2 , 129(1950).

3. 4. 5.

P. A. M. Dirac, Proc. Roy. Soc, A246, 326(1958). P. A. M. Dirac, Proc. Roy. Soc, A246, 333 (1958). P. A. M. Dirac, Phys. Rev., 114, 924(1959))

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

12

P . A. M. Dirac, Report at the symposium on Modern Physics at T r i e s t e (June, 1968). J. Klauder, T h e s i s at the Fifth International Conference on Gravitation and Relativity, Tbilisi (1968). A. Likhnerovich, The Theory of Cohesion and Holonomy Groups [in Russian], IL (1960). R. Feynman, P h y s . R e v . , 84, 108(1951). W. Tobocman, Nuovo C i m . , 3 , 1213(1956). C. Garrod, Rev. Mod. P h y s . , 3 8 , 483 (1966). L. D. Landau and E. M. Livshits, Mechanics [in Russian], Fizmatgiz (1958). P . Bergmann, Rev. Mod. P h y s . , 33, 510(1961). P . Bergmann and I. Goldberg, P h y s . R e v . , 98, 531 (1955). C. S. Yang and R. L. Mills, P h y s . R e v . , 96, 191 (1954). L. Faddeev and V. Popov, P h y s . L e t t . , 25B, 29 (1967). V. N. Popov and L. D. Faddeev, "Perturbation theory for gauge invariant f i e l d s , " P r e p r i n t of ITF, AN UkrSSR, Kiev (1967). R. Feynman, Acta P h y s . Polonica, 24, 697(1963). B . S. DeWitt, P h y s . R e v . , 162, 1195, 1239(1967). S. Mandelstam, P h y s . R e v . , 175, 1604(1968).

64 21. 22. 23. 24. 25.

M. Veltman, Nucl. P h y s . , B7, 637 (1968). S. L. Glashow and M. Gell-Mann, Ann. Phys. , 15, 437 (1961). J. Schwinger, P h y s . Rev., 125, 1043(1962). V. I. Arnol'd, Lectures 30-42, Lectures on Classical Mechanics [in Russian], Izd. MGU (1968). R. Jost, Rev. Mod. P h y s . , 36, 572(1964). R. S. P a l a i s , Mem. A m e r . Math. S o c , 22 (1957).

13

65

Covariant quantization of the gravitational field L D. Faddeev and V. N. Popov Leningrad Section of the V. A. Steklov Mathemaiical Usp. Fiz. Nauk 111, 4 2 7 ^ 5 0 (November 1973)

Institute.

USSR Academy of Sciences

A review dedicated to the contemporary methods of quantization of the gravitational field. In view of possible applications to elementary particle theory, the authors consider only asymptotically flat gravitational fields. The basis of the exposed method of quantization is the method of quantization of gauge fields in the functional integration formalism. The main result is the formulation of covariant rules for a diagrammatic perturbation theory. Its elements are the lines representing gravitons and the vertices of graviton-graviton interaction, as well as the lines and interaction vertices of fictitious vector particles ("Faddeev-Popov ghosts") characteristic for the theory of gauge fields. The expressions for the propagators and vertex functions are given explicitly. It is shown that the presence of fictitious particles in the covariant diagram technique guarantees the unitanty of the theory and the agreement between the covariant quantization with the canonical quantization. The bibliography contains 44 entries (54 names).

CONTENTS Introduction 1. The Feynman Rules for Covariant Perturbation Theory and the Feynman Functional Integral 2. Feynman Rules for the Gravitational Fields 3. A Derivation of the Modified Perturbation Theory Rules 4. The Hamiltonian Form of Gravitation Theory 5. The Hamiltonian Form of the Functional Integral References

777 778 779 782 784 786 788

INTRODUCTION The interest in a quantum theory of gravitation is maintamed to a large extent by hopes that inclusion of gravitation into the scheme of quantum field theory will allow us to construct a self-consistent closed theory of elementary particles. In this connection there are two directions. One is related to cosmology and to the use of cosmological considerations in the theory of elementary particles (cf. the papers of Wheeler- l ] M i s n e r ^ , Markov and collaborators « ) . The other direction, which does not make use of cosmological considerations, considers the field of the particles concentrated effectively in a finite volume and vanishing at spacelike infinity. In the latter approach the gravitational field is considered on the same footing as any other field ( c f . ^ ) and the theory is a variant of the theory of gauge fields. The role of gauge transformations is played by coordinate transformations which do not affect the spacelike infinity, and the role of the gauge group is played by the Poincare group (in more detail this problem is discussed in the report by one of the a u t h o r s c 0 ) . The role of Lorentz-invarince in the general theory of relativity was underlined by Fock C 6-. .... . „■1 The interest in elementary particle theory is mainly , ' , 1 * u uTTTI v • *«. - •* <■„ 1 related to the hope that it may be just the gravitational . . , , , . , , / , .. 1 . x , 11 u 1 field which could play the role of a natural "physical r e g u l a r i z e d which removes the singularities and infinit i e sfrom quantum field theory. The first results which supported this point of view were obtained by De W i t t " and K h r i p l o v i c h ^ . At the present time this direction is actively developed by Salam and collaborators 1 1 9 ] , making use of methods developed by Efimov, Fradkin, Volkov and Filippovt — ] .

In this review article we restrict our attention only to the problem of constructing covariant rules for a diagrammatic perturbation theory. The most convenient tool for m i s purpose is Feynman's functional integral. * this formalism the physical degrees of freedom, which are quantized, and the gauge-field degrees of freedom, w n ich remain c-numbers, are treated on the same footinS. This makes it possible to use a large class of different transformations. The first correct formalism for the quantization of the gravitational field was constructed by Dirac in 1958 C»3, within the framework of the Hamiltonian m e thod. Other methods, essentially equivalent to that de­ veloped by Dirac have been developed by the ArnowittDeser-Misner groupE* 3 , by S c h w i n g e r ^ and also some others C l T - l 9 ] . The absence of explicit covariance in the Hamiltonian formalism makes perturbation theory very cumbersome. It is appropriate to recall the analogous situation in quantum electrodynamics. The noncovariant perturbation methods of the thirties did not allow one to g 0 beyond the first nonvanishing approximation of perfurbation theory. The creation of a covariant perturbation theory toward the end of the forties and the development of the diagram technique associated to it have sim„. , , , ^ ' , ., . , ,, 1. ^ . pilied the calculations considerably, so that at present f " \ ". * the theory can be compared to experiment up to the c / . *"~ *" *: seventh f c ^ ^ce' to the the0^ of gravitation the tech i u cal A c u l t i e s a r e so great that a noncovariant quantization scheme becomes practically useless in concrete perturbation-theoretic computations. The first attempts of constructing a covariant quantization scheme for the gravitational field were contamed in the papers of G u p t a ^ . He followed the scheme

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66_ developed for quantum electrodynamics, and to overcome the difficulties related to the singularity of the free Lagrangian of the gravitational field, he made use of a trick analogous to the well-known Fermi method of quantization for the electromagnetic field (cf., e . g . / [ " ' 2 2 ] , .

Here are some details about the plan of our review article. Chapter 1 deals with the scheme of derivation of rules of covariant perturbation theory in the functional integration formalism, on the example of a scalar selfinteracting field theory. The Feynman rules for the gravitational field are formulated in Chap. 2. The deri-

It turned out, however, that an uncritical transfer of the Fermi quantization method (winch is justified in quantum electrodynamics to more complicated systems can lead to a violation of the unitarity condition of the theory. This was first discovered by Feynman in I M * * on the examples of the Yang-Mills field and gravitation theory. Feynman has mapped out the path to overcome the difficulties which he had pointed out. He has shown that the unitarity of a closed-loop diagram can be recovered if one subtracts from the appropriate matrix element the contribution of another diagram, also having the form of a loop and describing the propagation of a fietitious particle It was not possible to extend the Feynman method to more complicated diagrams. The solution of the problem for arbitrary diagrams was found in 1967 by De Witt [24 ] and by the authors of the present review-" 1 , using basically different m e thods. Both methods are unified by the use of the method of functional integration, which provides a scheme of covariant perturbation theory for gauge fields. In the case of a non-Abelian gauge group the method involves lines corresponding to fictitious particles and vertices describing their interactions with the quanta of the gauge

^X^lio^te^lorZ gravSnaT ££ poTJjustification for this is the possibility of f ^ ^ J u Hamiltcnianform. _ ... B . ,. „. . .. t. „ „, d l°r ^ S rea S f °n w e / * a I £ C ^ t a ^ o ' T T i the 1 ^ f ffh ^ n t " \ ^ h a t thafoni f nal ' t ^al e u n c 10 i n eg 7 ? . ' ' CJia Ptf r ^ s o w /^ ™ £ ™ * ^ useTin m e ^ r i v a t i o n of the " '"'""■**' "*»

duced by Feynman in order to recover the unitarity of the one-loop diagram for the real particles. The sign (—1) in front of the contribution of the closed IOOD shows ihat the fictitious particles (which are also known under the name of "Faddeev-Popov ghosts"-transl.) are subject to Fermi statistics. This makes a direct interpretation of the fictitious particles difficult, since they have integral spin (they are scalar for the Yang-Mills field and vectorjal for the gravitational field). The results obtained in*- ' s ] have been repeatedly rederived by v a r M a n d e l s W 2 9 } , Veltman^J Fradkin and T y u t i n ^ , K h r i p i o v i c h ^ , Boulware [30J , 't Hooft1 1 J , Altukhov and KH-^l«tf^ht3»3 ^"P10™11 ' The construction of a correct covariant quantization scheme for gauge fields has generated a series of papers where the (massless) gauge theory is compared to the corresponding massive (non-gauge) theory and the problems of the limit m - 0 are discussed (the papers of F a d d e e v a n d S l a v n o v ^ , Valnshteih and K h r i p l o v i i c ^ 1 , Z a k h a r o v ^ 3 , V e l t m a n ^ , Fradkin and T y u t i n ^ . The basic result of the majority of these papers reduces to the fact that the limit of a massive theory for m - 0 does not, in general, coincide with the corresponding massless theory As we already noted, methods of quantization of the gravitational field have been developed within the framework of the canonical HamUtonian formalism and in ex­ plicitly covariant form. The functional integration m e thod allows one to relate both formulations. The canonical Hamiltonian yields a unitary normalized theory and the covariant formulation allows one to construct a perturbation theory which is convenient for concrete calculations. 778

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+ T u c FFYMAN RUL ES FOR COVARIANT , „ „ „ . , . , - . , , - , , , - r t n v AMI-* - r u r CCVMIWIAM S E D ^ J ° » ™ ™ ,Y A N ° ™ E F E Y N M A N FUNCTIONAL INTEGRAL w « " c a U and discuss here the recipe for the derivation of the rules of covariant perturbation theory according to F e y n m a n ^ . We start with the simplest example <* a self-interacting scalar field

n'rnxl

The diagrams corresponding to the t e r m s of perturbation expansions is constructed out of two elements: the ^ G (propagator) and the vertex V: t

^

''

/

•*%. *

' "'

The scattering amplitude M n (ki, ..., k ) describing n (incoming or outgoing) particles with momenta kt, ..., k n j,s represented by a sum of contributions corresponding to diagrams with external (non-closed) lines. The contribution of a given diagram is obtained by associating to its elements the following expressions: to internal C(fc„ fcj- ' j f f i ' ,

(1.3)

:o the vertices one associates the vertex function (coupling constant) v

= u <*' + *« + * * P-*l ^ && integrating over all the momenta k of the internal elements of the diagram, the final result must be multiplied by ,p , ( L 5 ) H w ] ■ where I is the number of internal lines, v is the number of vertices and r is the order of the symmetry group of ^e diagram1'. To obtain the amplitude for a real process one has to go onto the energy shell k° = ± (k2 +■ m2)l/2; ^ sign depends on whether the particle is incoming or outgoing The described elements of the diagram technique are determined by the Lagrangian (1.1) in the following manner. We consider the action s y m f £ ( l } 9m (1 6) ' * as a functional of the Fourier transform 7j> (k) of fhe eield L. D. Faddeev and V. N. Popov

778

J>7 (p(x). This functional consists of a quadratic form in the field and a form of third degree: The function G(k t , k2) is the kernel of the integral operator inverse to the operator which defines the quad­ ratic functional form S 2 ; the function V is the coefficient function determining the form of third degree. This correspondence can also be seen in many other examples for which there exists a diagram technique, e.g., for the pseudoscalar theory of pions and nucleons. The only difference is the appearance of the factor ( - l ) s , in the weight (1.5) of the diagram, where s is the number of closed loops formed of fermion lines.

where the integrands have the form expfi(quadratic form)] x polynomial. Such integrals can be explicitly calculated and can be expressed in terms of the operator which is the inverse of the operator of the quadratic form as well as the co­ efficients of the polynomial. Indeed, let us consider an integral of this type for a finite number of variables:

It can be calculated by differentiating the generating function

The described recipe is not directly applicable to theories where the corresponding quadratic form is singular. The simplest example is provided by electro­ dynamics, where the action integral of the free photons

namely:

is degenerate owing to gauge invariance, and does not depend on the longitudinal component 9 AU of the poten­ tial A„. However, in this case it is known (cff C21 ' 22] )

The generating function can be computed by a shift of the variable, yielding

that one can u s e a«? photon p r n n a g a t o r a crpnera MzPH

inverse operator of the quadratic form (1 8V the momenturn-space representation of this propagator is the fol­ where the matrix B is the inverse of the matrix A. The first factor here plays the role of the normalizing con­ lowing stant which we have agreed to neglect. The differentia­ tion of the second factor leads to a result which in words can be formulated in the following manner: The integral where the constant is arbitrary. The scattering ampli­ J is represented by the sum of the coefficients c ^ . . . i n , tudes for real processes do not depend on this constant. contracted with products of the elements B^- of the ma­ However, as we have already pointed out in the intro­ trix B, carrying the same set of indices. duction, a direct adaptation of this recipe to theories with In our infinite-dimensional case the analogues of the nonabelian gauge groups leads to incorrect results. In coefficients c t >#>j are products of the coefficientorder to make clear the reason for these difficulties it is functions of the cubic form in the action. The role of useful to analyze the derivation of the rules of covariant the matrix A is played by the hyperbolic differential perturbation theory. In our opinion the most convenient operator (the Klein-Gordon operator) (D -m 2 2) or, in approach is the formalism of the Feynman path (func­ momentum space, the multiplication operator by tional) integral 2 '. (k2 - m22) A natural geneealization of our rinite-dimen­ We recall the main features of this method on the ex­ sional result leads then to the Feynman rules formulated ample of the scalar field with the Lagrangian (1.1). The above. scattering amplitude is obtained as an integral over all Two remarks are in place here. The first refers to possible fields with a given asymptotic behavior for the definition of the inverse of the operator A, i.e., of t — ± « of the functional exp(iS), where S is the action the Green's function of the operator ( □ - m22) There integral. We shall use the following notation for such are many such Green's functions: the retarded one, the integrals advanced one, the causal one, etc. The Feynman rules require the use of the causal Green's function. One can justify this within the framework of the functional integ­ where the symbol rxd

2. FEYNMAN RULES FOR THE GRAVITATIONAL FIELD

Expansion of the functional exp(iS) in powers of the coupling constant A leads to integrals of the form

The peculiarities of the gravitational field are rela­ ted mainly to its self-interaction. Therefore the major­ ity of this chapter will deal with the "free" self-interact­ ing gravitational field. The main result, namely the dia­ gram technique, is listed at the end of the chapter. We

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^ _ _ _

68 also indicate there the changes made necessary by the presence of a matter field or the electromagnetic field. Among the most frequently used parametrizations of the gravitational field the following two are the most im­ portant: the metric tensor and the moving-frame or tetrad (Vierbein) formalism. We summarize both of these. In the metric tensor formalism the gravitational field is described by the potentials gM„(x) and the Christoffel symbols r £ (x). The latter can be considered as inde­ pendent dynamical variables (the Paiatini formalism) or as functions of the g u „ :

(frame) e^ a (x) and the torsion coefficients u^ ^ ( x ) = _ u D a (x). The set of e^ a (x) form a mariix with positive determinant e(x). The action functional

The contravariant g ^ - m a t r i x is the inverse of g^, denotes the determinant of the matrix g .

A variation of S with respect to w leads to equations which allow us to express w in terms of e. The solution is conveniently written in the form

is invariant with respect to coordinate transformations

and with respect to local Lorentz transformations

g

In this review we shall only consider asymptotically flat gravitational fields. In this case the space-time manifold is topologicaily equivalent to four-dimensional Euclidean space and can be parametrized by global c o ordinates x^ ( - » <xM < +», ^ = 1, 2, 3). These coor­ dinates shall be compatible with the conditions at spacelike infinity such that

where If necessary, one may assume that this is already done ahead of time, so that S may be considered a s a functional only of the functions e ^ a .

where r = ((x1)' + (x2)2 + (x 3 ) 2 ) l/2 , and TI is the metric tensor of Minkowski space, with the signature (+ ). ~^ ^ ., ^. The action functional has the form

We shall talk about a formalism of the first order if the variables g ^ and r p (or e^a and w ^ ) are considered as indeDendent If the r are exore'ssed in terms o u c l p K ** ** . * , ,, " „ , of the g, and the e in terms of the w we shall talk about a formalism of the second order. The descriptions of the free gravitational field in terms of the g„,, or the e ^ a are equivalent. The differ*££ of components—10 m m e first ^ s e c o n d - i s compensated by the difference £ ^ g^Q ^ ^ which ^ ^ ^ ^ ^ ^ trized bv four functions and in the second case bv ten Thetetrad formalism Lconvenient for the description of interactions with spinor fields. The equivalence between first- and second-order formalisms may disappear when the interaction with other fields is switched on. Geometrically Eq. (2.11) defines a connection without torsion. The minimal inter­ action of the gravitational field with the spinor field in a first-order formalism leads to the appearance of torsion (cf C3al)

where * is the Newton constant is invariant under the group of coordinate transformations acting on the quantities et»>, r j „ according to the rules

We have written here the equations for infinitesimal transformations; r, fi are the infinitesimal components of a vector field which generates the coordinate transformations

Variation of the action (2.3) with respect to the r ^ leads to equations, the solutions of which are the func­ tions (2.1). In this sense one may consider the r ^ „ as The remainder of the exposition of this chapter will independent variables, which is sometimes convenient to be ^ ^ on *» ^ ^ ^ <* a tensor formalism of the do. ^ second order. We set Substituting into the expression (2.3) the explicit form (2.1) of the Christoffel symbols Tp „ in terms of the metric tensor, it becomes

where for convenience we have introduced the contravariant tensor d e n s i t v d e g r

and consider uM » a tensor field describing the gravitational field. The action functional (2.6) takes on the form

In the tetrad formalism (also known as the "movingframe formalism" or "Vierbein formalism") the gravitational field is described by the components of the tetrad

where S.j « " quadratic form and S n is a form of n-th e in the variables M^V and their first derivatives. The linearization (2.12) is in many respects not a natural one. It can violate the signature of the metric tensor if u ^ " is not sufficiently small. Recently exponential parametrizations have become popular, e.g., for the tetrad matrix e^a = expflcx"*). In principle the expansion (2.13) can be computed in this parametrization. We

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69 note that the quadratic form S2 does not depend on the parametrization. A Hir-^t a«r»n„„+,-„ «* «. • « ^u * i * A direct application of the recipe from Chapter 1 to the case of gravitation would lead to the following formulation of covariant perturbation theory rules: t t a q u a d ratic form S2 defines the propagator and the coefficientfunctions of the forms S give the expressions for the vertices, which in this case are S t e i n number. We note here that in a first-order formalism the linearization (2.12) and the substitution transforms the action into a sum of forms of second and third degrees, so that in this formalism the number of vertices is finite. As a consequence of the invariance of the action with respect to transformations (2.4) the quadratic form

In addition to these elements, in the internal parts of the diagrams one must make use of additional elements which can be interpreted in terms of vector particles i . „ .. ... * .. _ . . . £.. . . nte/ac'inS w l t * gravitons. Such an interpretation is int r ° f C,ed
is degenerate. It does not contain the longitudinal com­ ponents a uM". The example of electrodynamics suggests _ . , , . . the idea to use as propagator for the gravitons a generfh l ct l t l °us e,^ments p a r f c l Pate only m ,closed loops **» o alized inverse operator, e.g., the one-parameter family t e external hnes are always graviton lines. The contribution from a given diagram is obtained if the product of expressions of the form (2.16)-(2.18), (2.20) associated to its elements is integrated over the internal momenta, and the result is multiplied by with the elements of the S-matrix independent of the choice of the constant a. However, FeynmanC23] has shown that the matrix elements calculated according to the naive rules depend in an essential way on this constant, and the unitarity condition is violated. Feynman has also outlined the way out of this difficulty. As a result of the efforts of a large number of authors, as described in the introduction, correct rules for the perturbation theory have been obtained, rules which we describe here. Their derivation will be given in the following chapters. s * * The diagram technique contains, to be sure, all the elements of the "naive approach": graviton lines, for which the expressions have the form (2.16) and vertices generated by the forms S n + o of expansions of the type (2.13). Here is the explicit depression for the thirdorder vertex, corresponding to the linearization (2.12):

where s is the number of closed loops formed by fictitious particles. A comparison of this equation with (2.5) shows that the fictitious particles act like fermions, i.e., they violate the spin-statistics theorem. This shows that their role reduces to a subtraction of the contributions from unphysical degrees of freedom. * addition to * * described diagrams perturbation theory ^volves M i n i t e contributions of the renormalization type, contributions which a r e proportional to powers of the delta-function 6 (4,(0). The structure of these terms will be described below in Chap. 3. In the firstorder formalism the elements associated to ffctitious particles do not change. In addition to the tensor propagator (uu) the perturbation theory also involves the propagators (uy) and . In the momentum-space representation these propagators have the form:

where

The only graviton vertex is generated by the trilinear form

+ the sum over permutations of the pairs

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70

We do not write out the elements of the diagram technique in the tetrad formalism. The reader can obtain them by himself using the methods described below. The inclusion of matter fields does not lead to the appearance of new fictitious particles and the corresponding elements of the diagram technique, as long as the quadratic forms in the action functionals of the matter fields are nondegenerate. As an example we list the interaction Lagrangians of the gravitational field with a scalar field and with a spinor field. In the first case it is convenient to use the metric formalism and in the second case one must use the tetrad formalism:

point in M and a a group element, ? a denotes the action of the group element a on the point 4. Consider the quo­ tient manifold M/G = M*, formed by the classes of all points of the form £ a , where £ (the representative of the class) is fixed and a runs over the whole group ( | a is also called the orbit of the point { and M» *i she erbitspace of G—Transl.). measure on m e orbit M* can be actea. ded £ view of £ b consUnt on the classe S ) to a meisure M on M whichgis m v a n a n t under the group action. Conversely, given an invariant measure u on M it is not hard to construct a measure M* on M*, which extends to p in the sense indicated. One can make the selection of representatives from the classes concrete by defining in M a hypersurface which intersects each orbit (class) once. This means that if the hypersurface is defined by the equation then for given £ the system of equations

here we have utilized the standard notations for the components of the scalar and spinor fields; y a and crcd = V,(rcrd ~ r V ) are the usual Dirac matrices. We note that if there is no mass term it is easy to choose the parametrization of the gravitational field in such a manner that these functionals generate only a finite num­ ber of vertices.

must have a unique solution a (depending in general on O. « one uses such a parametrization the measure M* looks as follows: where the function Af(4) is defined by the relation

3. A DERIVATION OF THE MODIFIED PERTURBATION THEORY RULES Functional integration is a convenient heuristic means for the explanation and heuristic derivation of the rules of perturbation theory for the gravitational field, rules which have been enumerated in the preceding chapter. Within the framework of this approach the additional terms mentioned there are interpreted as a consequence of the nontriviality of the measure with respect to which the functional exp(iS) is being integrated. . . ... . . . ., ,. . Let us explain this in more detail, making use of natural geometric terms. The functional exp(iS) is in­ variant with respect to the infinite group of coordinate transformations in the metric formalism or with respect to the semidirect product of this transformation group and the group of local Lorentz transformations in the moving frame (tetrad) formalism Thus this functional is ^function on (equivalence) classes of fields where

and da is the invariant measure on the group G. The function Aj(£) is invariant, i.e., Aj(£ a ) = Af(?)- Equation (3.3) can be explained in the following way. We go over to the new variables where a is the group element defined by Eq. (3.2) and ? T = ? a . Let the invariant measure on M have the following expression in terms of the coordinates £: & ' , ... the new variables "

m

^

where

^ S r a ^ i n t o ^ ^ S r t o n S r m a l o n s from th dicated erouD ^ ^ 6 m S V ' We shall assume that it is a class rather than an individual field which describes a concrete physical situation. This is the content of the principle of general co„,„io„„ Q ir„ar.in,r =,„-h , f n , m l n ™ , nt th it ^^„^ir>i Q v a r i a n t . Keeping such ^formulation of tois principle in mind we can consider that m quantum theory the Feynman functional exp(iS) should be integrated with respect to such classes of fields, rather than individual fields. The nontriviality of the measure which we have mentioned is related just to this circumstance. Let us discuss how one can describe measures on the set of classes of fields. Following our method, we first consider the finite-dimensional case. Mathematically we are dealing with the following situation: we are given a manifold M (in the sequel this will be the set of all fields) and a group G (in the sequel this will be the group of gauge transformations) acting on M. Let E, be a

is * " measure Senerated by *e measure d ? on the hypersurface, Df is the Jacobian of the transformation to the variables ( | T , a). The measure on the orbit space M* is obtained from (3.7) by omitting the invariant measure da in the group coordinates. „, „ . . . . . . . We show that the Jacobian Df coincides with the in­ f J . I( ) ( ) consider the integral

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L. D. Faddeev and V. N. Popov

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of the arbitrary invariant function *(«), integrated with tresPect to " » measure M^d^ The V a r i a n c e condi-

allows one to assume that * depends only on £ T . Indeed, since ?T = 5 , 782

71 Making use of the expression (3.7) for the measure M(£)d£ in the integral (3.9), we transform the integral to the f o r m X

and thus is expressed in terms of the integral of the 5-functional n 5 ( 6 y ( h ^ ) a - |M) over the gauge group. , |x

) where M(G) = Jda is the "volume of the group." Another expression for the integral (3.9) can be obtained by introducing into the integrand the factor (3.4) which equals one, then carrying out the substitution | a - §, with respect to which the functions *, Af and the measure Md€ are invariant:

Let us discuss the calculation of this integral. The expression An[g] enters into the integral (3.15) only on the hypersurface determined by the equations (3.16). For such gV-v the total contribution to the integral from (3.16) comes from an infinitesimal neighborhood of the unit element of the group. In this nei°hborhood the action of group transformations on the 3 $ and the measure da can be parametrized by means of the infinitesimal func­ tions n E w S S e d a b S E t a a 5) W WUft thisparamelUl trization ^ ^ ^ ^ ^ * ^ ^ ^ ^

In view of the arbitrariness of the function # ( | ) the measures in the integrals (3.10) and (3.11) must coin­ cide:

At the unit element of the group, the measure da has the simple form

which yields the equality Df = Af. Let us return to the gravitational field. From what we just said it is clear that in order to define a measure in a class of fields it suffices to define the measure on a manifold of fields which is invariant with respect to co­ ordinate transformations (and local Lorentz transforma­ tions) and to specify the equations which parametrize the c l a s s e s . These equations should be Lorentz-invariant if we wish to obtain a covariant perturbation theory. In the metric formalism we choose as such equations the harmonicity conditions of de Donder-Fock:

where Z^(x) is a prescribed vector field. The a r b i t r a r i ness in the choice of l^(x) will be useful in the sequel for formal transformations. The condition (3.12) is not generally covariant and therefore can serve for the parametrization of classes. The analog of Eq. (3.2) is a complicated nonlinear equation for the parameters of the coordinate transformation which takes a given metric into a harmonic one. Within the framework of perturba­ tion theory this equation has a unique solution. In the following chapters it will be shown that one must select as the invariant measure the expression3' where

This will be done on the basis of an investigation of the Hamiltonian formulation, which we shall interpret as an alternative, non-Lorentz-invariant method of p a r a m e ­ trization of field classes. The advantage of the Hamiltonian formalism is the fact that in it the unitarity condition leads to the stan­ dard expression for the integration measure. Having the parametrization (3.12) of the classes and the measure (3.13) we obtain the following expression for the functional integral:

where according to (3.4) the functional A h [g] is defined by the e q u a t i o n w h e r

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So..Phys.-Usp., Vol. 16, No. 6, May-June 1974

Consequently, the integral in which we are interested has the form

Formally this integral equals (det A)"1 where A is the operator acting on the quartet of functions r?M: Thus we have found that

For the formulation of perturbation theory it is convenient to represent det A as an integral over auxiliary fields of a functional of exponential type. These fields must be anticommuting fields, since we need an integral which yields the first power of the determinant. These requirements are satisfied by the expression

where 0 and 9 are classical anticommuting fields satisfying ^e relations and similar relations for the pairs (9, 9), (0, 9). A definition and rules of operation with integrals over anti­ commuting variables can be found, e.g., in the mono­ graph of Berezin- 3 9 : i . Returning to the integral (3.15) we can not write it in the form

whi<=h can be used directly for the formulation of p e r ­ turbation theory. However, we still transform it, making use of the arbitrariness in the selection of u. The integral (3.24) does not depend on the choice of JH, by definition. We may therefore average it over IV- with an arbitrary weight. Let us use as a weight the exponential of the quadratic form in the fields e n is y ^ Minkowski metric tensor. The averag­ ing can be done explicitly and yields the expression L. D. Faddeev and V. N. Popov

783

72

where S e is the action of the free gravitational field and A[g] equals the product of the determinants

o f t e S 2 K U Mth t h f J b T t r a r v ^ J S S T ^ l S ' of the fields hM^ with the arbitrary coefficient a. It follows from our reasoning that the integral does not depend on a A method for seemg this directly has been proposed by DeWitt<> Here we have followed the simpler method of ' t H o o f t ^ . The diagram technique discussed in the preceding chapter follows from the expression (3.26) in the same manner as explained in Chap. 1 for the example of the scalar field. By introducing the "fictitious" fields 9» and 0M we have managed to make the quadratic form in the exponent nondegenerate Thus the inverse operators correspondmg to the operators of the quadratic forms in W, 9» and ?" become well defined, i.e., we obtain the propagators of the particles corresponding to the lines of the diagrams. The graviton propagator ( h ^ ^ ^ 0 ) contains the arbitrary constant a. The propagator (9^8U) of the fictitious vector particles in the k-representation is given by Eq. (2.18). The anticommutation of the fields 0V-, 9^ leads to the factor ( - l ) s for a diagram containing s closed fermion loops. The higher-degree forms in the expansion of the action (3.26) in powers of the fields h^", 8^, Bv give rise to the vertex functions of the diagram technique, as described in Chap. 2. Their concrete form depends on the choice of linearization used. . .. . ,.. . „ u -3/2, vWe note also the role of the local factor n h 5 2 fr in v x the measure. For the linearization (2.12) we have ... „.■f.* u i^u * i •* * • «. and thus this actor should be taken into account in the construc ion of perturbation theory Formally its role reduces to the appearance of a contribution of the form

where A is the operator (3.20>' The p r M e n c e of a ^ ^ thi duct shows that the in^ ^ 2 ^ £ ^ ^ J v b i c h could be introdescription of the electromagnetic field X ^ n t e r a c t s W l th the gravitational field Thus, in coS y ariant p e r ^ b a t i o n the ory for the electromagnetic and gravitational fields a fictitious neutral scalar particle participates in addition to the elements which have been described above. The reader who has understood me basic principles C 0 nstruct l 0 n of the diagram technique for gauge fields 4 approqpriate ^iculations J , '.r** *»f *• fortoemore comP il cated case. THE HAMILTONIAN FORMULATION OF RAVITATiniM THEORY r ^ A A justification of the correctness of the expression (3.13) for the invariant measure is based on the Hamiltonian formulation of gravitation theory. This formulation has been developed by D i r a c [ 1 4 ] . A series of variants for this formulation were obtained by diverse a u t h o r s ^ " 1 9 1 . The construction of an explicitly Hamiltonian form of the Einstein equations runs into the difficult problem of finding solutions to the constraint equations. For us it will be sufficient to consider a generalized Hamiltonian formulation of gravitation theory, . * . , \ . ' t where it is not necessary to solve the constraint equations, and one can restrict one's attention to a verifica­ tion of the commutation relations. We explain the generalized Hamiltonian formulation of the example of a system with a finite number of de^ * formulation the action funcf ^ ^ consideration ^ the form

in the action giving rise to vertices which are proportional to 6 (4) (0). The appearance of such renormalization terms is noted in many papers treating nonlinear theories (cf., e.g., C 4 0 ' 4 ^). We note that they are absent in the exponential parametrization. In this parametrization the measure (3.13) has, up to a constant factor, the simple form

Here p, q denote canonically conjugate coordinates and momenta which form a phase space of dimension 2n, Si is the Hamiltonian, <pa are the "constraints," A a are Lagrange multipliers (a = 1 m, m < n). The constraints <pa and the Hamiltonian & are in involution, i.e., satisfy the conditions

without any local additions. We have considered in detail the case of the gravitational field in vacuo. Introducing interactions with other fields does not change substantially the scheme of con­ struction of perturbation theory. For matter fields with

these ns usual Pois SO n brackets

SS^^i^S^^r^^sS^: s i 0 T i " a s fi\Tto,a

t i d e s and their correspondmg diagramfappear o n l / when a field with larger gauge group than the gravitational field is included, e.g., the electromagnetic field or fields of the Yang-Mills type. We shall not consider this case in detail here. We just list, as an example the expression of the functional integral corresponding to the electromagnetic and gravitational fields:

'

£°ed * » ~ £ £ * J ? " £ the m constraint equations

Tehductionof thebdimT £

£^t^rmine^t "

and the m supplementary conditions The functions x a are subject to the conditions

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L. D. Faddeev and V. N. Popov

784

73 In addition it is convenient to assume that the funcThe variables r P „ which differ from the r ^ are not tions x a commute with one a n o t h e r : d y n a m i c a l variables. They can be excluded with the help of the constraint equations In this case it is simple to introduce canonical varia­ bles on the submanifold r * . Indeed, in view of (4.7), a canonical transformation in r allows one to go over to new variables, where the >•a take the form

The tem (4 14) contains the equations (4.10) together with the equations

where p a are part of the canonical momenta of the new system of variables. Let q a denote the coordinates conjugate to them and p*, q* the other canonical variables. In the new variables the condition (4.6) has the form { '

The solution of the system (4.10), (4.15) expressing the 'nondynamical" quantities r ° n , rk r k in terms of the l0 l0 1J n ,,„ T^. and h ^ " is of the form

and can be interpreted as the condition for solvability of the constraints <pa = 0 with respect to the coordinates q a . As a result of this the surface r * is defined in r by the equations

, where r . k are the three-dimensional connection coeffii]

so that the p* and q* play the roles of independent variables on r * . By construction these variables are canonical. A more detailed discussion of the generalized Hamiltonian formulation for the finite-dimensional case with applications to the theory of the Yang-Mills field is contained in a paper by one of the a u t h o r s ^ . Let us return to the gravitational field. We shall show that its action can be reduced to a form which is a fieldtheoretic analog of (4.1), with the appropriate constraints and the Hamiltonian satisfying conditions of the type (4.2). We shall follow the general method proposed by one of the authors in a form especially adapted for the gravitational field C 1 9 ] . For our purposes it is convenient to make use of a formalism of the first order. We consider the expression of the action for the gravitational field in the form (2.3) and collect in the corresponding Lagrange function all the t e r m s which involve derivatives with respect to

This expression does not contain the variables r £ which occur in y(h, r ) linearly and play the roles of Lagrange multipliers. The factors (denoted by A°°) in front of r £ are the constraints. The constraint equations

allow us to express the variables rL , (r?« - r%) in 0 l0 n HU , ^ terms of the r £ and h^ ". Then the t e r m s containing time-denvatives take the form

if one omits the terms

cients, defined by the three-dimensional metric g i k (i, k = l, 2, 3). us substitute ^ expression (4.16) for the rf 0 , k r k into the ^ g ^ e function x(h, V). After l 0 6 v MO'Mj * ' omittinS several terms of * e ^ of a divergence, which vanish when integrated over three-space when the asymptotic conditions (2.2) are taken into account, the result of the substitution reduces to the form

where

here g3 = d e t g i k , R3 is the three-dimensional curvature scalar associated to the three-dimensional metric g l k M k, = 1, 2, 3). The symbol v k in the expressions for the constraints Ti denotes the covariant derivative with respect to the metric g ^ . As pointed out by Arnowitt, Deser and Misnercis :l , the canonical variables and the expressions for the con­ straints have an intuitive geometric meaning. The functions qik and "ik serve as the coefficients of the first an<* second quadratic forms associated to the surface x = const, submerged in the four-dimensional spacetime with the metric g and connection r £ „ . More p r e sicely, q1* are a covariant metric density of weight +2, and the n^ form a covariant density of weight - 1 . The constraints are then the well-known Codazzi-Gauss r e ­ lations in the theory of surfaces (cf. e.g., [ 1 3 -). .«,,,, _,, Equation 4.17) solves the problem of reducing the ac^n of the(gravitational field to the generalized Hamiltonian form, analogous to (4.1) for finite-dimensional systems with constraints The constraints T„, as is easily checked, commute with one another. In order to write explicit expressions it is convenient to introduce the quantities

whic , ,„..rat„.,bypart .■qua„„ 411) s u g g e s t X t the natural dynamical variables are the quantities

w h e r e TJ i s a v e c t o r field, pis* density of weight " U . The fo U owing relat l 0 ns hold

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L. D. Faddeev and V. N. Popov

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74

here [T]U rj2] is the Lie bracket of the vector fields, i.e., the vector field with the components r,l8,^-vh,n^, n

where M is the total mass which can be obtained by integrating<SC(X):

Thus, one may consider that St= [$(?xxd3x plays indeed the role of energy. The integrand

and is nondegenerate if the curvature of the metric g ^ is nonzero. , „ „ T l i r . ,_. , . , , « . , « . . A . 5. THE HAMILTONIAN FORM OF THE FUNCTIONAL INTEGRAL The functional integral for the quantization of a c l a s s ** system defined «» ^generalized Hamiltonian formuW i o n has the foU<wmB form ^ >:

The integration measure is here defined by the equation

To prove this we reduce the integral (5.1) with the measure (5.2) to an integral over paths in the physical phase space r*. For this purpose we go over to the coordinates p , p*, q a , q* described in Chap. 4. In these coordinates the measure has the following form:

which can be rewritten: [] 6 ( Pa )6( ? °-<,'(/>', q-))dp*dq" [[ lilJfL

which plays the role of energy density, has the form of a sum of two quadratic forms: one in the derivatives of q1* and another in the "momenta" n^, as required for the energy density of a wave field. In our case this is the energy of the gravitational field having two polarization states in agreement with the usual counting: 2 = 6(coordinates) - 4(constraints). We remark moreover, that in the weak field approximation the Hamiltonian is represented by a quadratic form in the densities of the transverse components of the linearized field. We now discuss the selection of supplementary condi­ tions. It is widely accepted to use the conditions which were first proposed by D i r a c [ 1 w h e r where q = det q ^ = (det g^) 2 . These conditions have a simple geometric sense: the surface x° = const is minimal and the coordinates x 1 , x2, x3 on it are "harmonic" coordinates. For us the following supplementary conditions will be more convenient:

The integration with respect to the p a and qa is reduced b the delta-functions. As a result the integral takes the form

which is standard for the usual Hamiltonian formula­ tion^ 4 3 . This proves the correctness of Eq. (5.1), which has ^ advantage over (5.3) of not requiring a solution of the constraint equations The integral (5.1) can be represented in the form

e the functional exp(iS) (S is the generalized action of the system (4.1)) is integrated over all independent variables p i ( q1, X a . Indeed, the Lagrange multipliers Aa enter linearly into this action and the integral with respect to them yields a delta function in the constraints <^a. Starting with this formula we shall not write out the factors of the type of it or the volumes of the integration lattices. mfc l BB "' « Let us return to the case of the gravitational field. We select the supplementary conditions in the form (4.22) and introduce the notations

where 4> is a function with the asymptotic behavior C/r at infinity. The commutation relations (4.7) are satisfied for these conditions. The Poisson bracket matrix of the conditions (4.21) with the constraints is determined by the equations

The analog of the integral looks as follows

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L. D. Faddeev and V. N. Popov

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.

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75

We intend to reduce this expression to a form where the integration is only over the field g^". This will allow us to identify the invariant measure we are looking for. For this purpose we must integrate with respect to the fields n^. This integration can be done explicitly, since the e x p r e s s i o n F o

depends linearly on ir^, and thus the integral turns out to be gaussian; here C is the operator defined by the r e ­ lations (4.22). Let us explain in more detail this feature of det C. The functions ffik enter only into the coefficients Cm, of the operator C, coefficients which do not contain derivatives; moreover the dependence on the rry, is linear. Thus, the operator C can be written in the form

We now show that this integral is an integral over classes of gravitational fields in the sense of Ch. 3, the classes being parametrized by the condition (4.22), and the invariant measure having the form

r this it is sufficient to verify that det B coincides with the factor A [h] obtained according to the rules of Chap. 3 The integral in this expression can be calculated in in the same way as the integral (3.16) in Chap 3 This yieldc: A [k] = det£' (5 13) where the operator B' is defined as follows:

where the operator Cz does not contain derivatives and for each n is determined by a matrix of rank 1. It is known from linear algebra that the determinant of a ma­ trix A + B, where B is one-dimensional, is linear in the matrix elements of B. The analog of this assertion in our case leads to the indicated linearity of det C with respect to the 77r.. The Gaussian integration over 7rik. reduces to the sub­ stitution

where 7rjk(h) is an expression which follows from formulas of ffie type (2.1), expressing the Christoffel symbols in terms of the metric. After such a substitution the a c ­ tion corresponding to the Lagrangian (4.17) turns into the initial covanant action (2.3). The determinant det C turns into the product

where B is the operator defined by the equations

Finally, the local factors in the products of differentials together with the local factor which appeared in the integration over ffik and the differentials themselves collect i to the express o

Here the factor in front of the differentials can be r e duced to the f

o

r

m

it is easy to see that

^deed, one can go over from one operator to the other by means of the triangular substitution Let us summarize. Starting from the obviously unitary Hamiltonian formulation of the functional integral, after formal changes of the integration variables we have transcribed it in the form of an integral over equivalence classes of fields with a concrete parametrization of the classes. The corresponding invariant measure has the form (5.12). This justifies the Lorentzinvariant expression for the functional integral in Chap. 3, which represents another writing of the same integral with another parametrization of the classes. With these considerations we conclude the derivation of the covariant rules of perturbation theory for the quantization of the gravitational field. The next problem which appears here is considerably more difficult. It consists in a consistent performance of the renormalization procedure based on an invariant r e g u l a t i o n . The difficulties are caused by the unwieldiness of the theory as well as by the fact that from a formal point of view the theory is non-renormalizable. We hope that symmetry and general covariance considerations will help in solving this problem.

'>For instance r = 2 for the diagram

">

.

.,, and the l a s t f a c t o r q i / 2 can be omitted owing to the c o n s t r a i n t q = exp $ . As a r e s u l t our functional i n t e g r a l t a k e s the f o r m

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L. D. Faddeev and V. N. Popov

Sov. Phys.Usp.. Vol. 16, No. 6, May-June 1974

)

^


^ Q{ funct l0 n a| (path) integrals in vanous problems of quantum mechanics and field theory is discussed in the recent review article o1" Blokhintsev and Barbashov in Uspekhi ( ;)7b ]. ])Any measure [\h*11dH^•„invariant with respect to the group of coordinate tranlformations. Indeed, under coordinate transformations this expression acquires a factor of the type [Jtdet (&'^'4^F- This factor should be considered equal to one, since for infinitesimal trans­ formation it is equal to exp [v Tr In (1 + S^igl = exp (Y8"> (0)J3^**) = t,

J

787

76

l

J. A. Wheeler, Geometrodynamics, Academic Press, N. Y. 1962 (see also C. W. Misner, K. S. Thome and J. A. Wheeler, Gravitation, W. H. Freeman and Co., San Francisco, 1973; added by Transl.). 2 C. W. Misner, Phys. Rev. 186, 1319, 1328 (1969). 3 M. A. Markov and V. P. Frolov, Teor. Matem. Fiz. 3, 3 (1970); V. A. Berezin and M. A. Markov, ibid., p. 161. 4 W. E. Thirring, Ann. Phys. (N.Y.) 16, 96 (1961). 5 L . D. Faddeev, Proc. Intern. Congress of Mathematicians, Nice, September, 1970, Vol. 3, p. 35. 6 V. A. Fock, Teoriya prostranstva, vremeni i tyagoteniya (The Theory of Space, Time and Gravitation), Fizmatgiz, Moscow, 1961. 7 B. S. DeWitt, Phys. Rev. Lett. 13, 114 (1964). 8 1 . B. Khriplovich, Yad. Fiz. 3, 575 (1966) [Sov. J. Nucl. Phys. 3, 415 (1966)]. 9 C. J. Isham, A. Salam and J. Strathdee, Phys. Rev. D3, 1805 (1971). 10 E. S. Fradkin, Nucl. Phys. 49, 624 (1963); 76, 588 (1966). 11 G. V. Efimov, Zh. Eksp. Teor. Fiz. 17, 1917 (1963) [sic!] 12 M. K Volkov, Ann. Phys. (N.Y.) 49, 202 (1968). 13 A. T. Filippov, Preprint JINR R-1439, Dubna, 1964. 14 P. A. M. Dirac, Proc. Roy. Soc. A246, 333 (1958). 15 R. Arnowitt, S. Deser and C. W. Misner, Phys. Rev. 117, 1595 (1960). 16 J. Schwinger, Phys. Rev. 130, 1253; 132, 1317 (1963). 17 P. G. Bergmann, Rev. Mod. Phys. 33, 510 (1961). 18 J. L. Anderson, Rev. Mod. Phys. 36, 929 (1964). 19 L. D. Faddeev, Gamil'tonova forma teorii tyagoteniya (The Hamiltonian Form of Gravitation Theory), Abstracts of the 5-th International Conference on Gravitation and Relativity, Tbilisi, 1968. 20 S. N. Gupta, Proc. Roy. Soc. A65, 1952. 21 N. N. Bogolyubov and D. V. Shirkov, Vvedenie v teoriyu kvantovannykh polef (Introduction to the Theory of Quantized Fields), Gostekhizdat, Moscow, 1957; Engl. Transl. Wiley-Interscience, 1959. 22 A. I. Akhiezer and V. B. Berestetskii Kvantovaya elektrodinamika (Quantum Electrodynamics), Nauka, Moscow, 1969; Engl. Transl., Wiley-Interscience, 1965.

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23

R. P. Feynman, Acta Phys. Polonica 24, 6 (1963). " B. S. DeWitt, Phys. Rev. 160, 1113; 162, 1195, 1239 (1967). M L. D. Faddeev and V. N. Popov, Phys. Lett. B25, 29 (1967); V. N. Popov and L. D. Faddeev, Teoriya vozmushchenii dlya kalibrovochno-invaraintnykh polef (Perturbation Theory for Gauge-Invariant Fields), Preprint, Inst. Theor. Fiz. Acad. Sci. Ukr. SSR, Kiev, 1967. 26S. Mandelstam, Phys. Rev. 175, 1580, 1604 (1968). 27 M. Veltman, Nucl. Phys. B7, 637 (1968); B21, 288 (1970). 28E. S. Fradkin and L V. Tyutin, Phys. Lett. B30, 562 (1969); Phys. Rev. D2, 2841 (1970). 29 I. B. Khriplovich, Yad. Fiz. 10, 409 (1969) [Sov. J. Nucl. Phys. 10, 235 (1970)]. 30 D. G. Boulware, Ann. Phys. (N.Y.) 56, 1 (1970). 31 G. 'tHooft, Nucl. Phys. B33, 173 (1971). 32 A. M. Altukhov and I. B. Khriplovich, Yad. Fiz. 11, 902 (1970) [Sov. J. Nucl. Phys. 11, 504 (1971)]. B A . A. Slavnov and L. D. Faddeev, Teor. Matem. Fiz. 3, 18 (1970). 34 A. I. Vainshtein and I. B. Khriplovich, Yad. Fiz. 13, 198 (1971) [Sov. J. Nucl. Phys. 13, 111 (1971)]. M V. I. Zakharov, ZhETF Pis. Red. 12, 447 (1970) [JETP Lett. 12, 312 (1970)]. 36 I. V. Tyutin and E. S. Fradkin, Yad. Fiz. 13, 433 (1971) [Sov. J. Nucl. Phys. 13, 244 (1971)]. 37 a) R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948); Phys. Rev. 84, 108 (1951); b) D. I. Blokhintsev and B. M. Barbashov, Usp. Fiz. Nauk 106, 593 (1972) [Sov. Phys.-Uspekhi 15, 193 (1972)]. 38 H. Weyl, Phys. Rev. 77, 699 (1950). 30 F . A. Berezin, Metod vtorichnogo kvantovaniya (The Method of Second Quantization), Nauka, Moscow, 1965; 40 H. Umezawa and Y. Takahashi, Prog. Theor. Phys. 9, 14, 501 (1953). 4l T . D. Lee and C. N. Yang, Phys. Rev. 129, 885 (1962). 42 L. D. Faddeev, Teor. Matem. Fiz. 1, 3 (1969). 43 G. Favard, Cours de geometrie differentielle locale (Russian transl.: IL, 1960). 44 G. Garrod, Rev. Mod. Phys. 38, 483 (1966). Translated by Meinhard E. Mayer

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Comments on Paper 5

This paper constitutes the lecture notes of a course given at the famous Les Houches summer school in August 1975.1 was to give an introductory course on the functional integral quantization. In these lectures, a point of special pride for me is the stress on the necessity of extra boundary terms in the action when the holomorphic quantization is used. Another technical novelty is the proper definition of the S matrix in terms of the asymptotics of the admissible trajectories in the functional integral. This approach was implicitly advocated by Feynman himself; however, I had not seen a selfcontained exposition of this idea in any written form before and was to devise it myself for the course. This text influenced the monograph on quantization of the gauge fields, which I wrote with A. A. Slavnov some time later.

79

COURSE 1 INTRODUCTION TO FUNCTIONAL METHODS

Ludwig D. FADDEEV V.A. Steklov Institute of Mathematics, Academy of Sciences, Fontanka 25, Leningrad 191011

§0. Contents

1. Functional integral and Feynman diagrams 2. Functional integral in phase space 3. Holomorphic form of the functional integral 4. Generating functional for the 5-matrix 5. The case of fermions 6. Some special features of the quantization of the Yang-Mills field References

R. Balian and J. ZinnJustin, eds., Les Houches, Session XXVII,. 1975 - Mithodes en theories des champs /Methods in field theory © North-Holland Publishing Company, 1976

3 5 11 19 28 32 39

81

Functional methods

3

1. Functional integral and Feynman diagrams We begin with the formulation of the main result of the functional ap­ proach. Consider for simplicity the example of the self-interacting scalar neu­ tral field (*) with tth Lagrangian

where V(
The functional Z(V) of the arbitrary function rj(X) (source)

is a generating functional for the Green functions of the system in considera­ tion,

There exists no rigorous definition of the integral "over all fields 0(JC)" which is symbolically written in (1.1). We will understand the expression (1.1) in terms of perturbation theory. Let us expand the expression for G(xlt...,x„) which follows from (1.2) in a power series in A:

The integrand in the last expression has the form exp{/ X quadrltic form} X polyntmial , if we understand that
82

4

L.D. Faddeev

Indeed let us consider the finite dimensional integral

where the matrix Aik of the quadratic form and the coefficients C^ _. of the polynomial are given. To evaluate / it is convenient to introduce the gen­ erating function

We are using the natural matrix notations. It is evident that

Now differentiating Z 0 (T?) we find

so that to get the non-vanishing result when r? = 0 one must differentiate also 77 in the second factor. It is clear from this observation, that / i s obtained by summing over all possible ways of saturating the indices of Ct by the { l products of the matrix elements of the matrix A ~ such thatthe index k comes in ik times. Let us return to our infinite dimensional expression and choose the nor­ malizing factor Was the integral

The operator

plays now the role of the matrix A, x being the analogue of the index k. The matrix elements of the inverse matrix are just the values of the free Green function

Jtt Functional methods

5

The polynomial we integrate has a particularly simple form

It is now easy to see that the finite dimensional rules described above lead iu the Feynman diagram prescription for the calculation of the Green functions, vacuum diagrams included, if we take as G(x - y) the Feynman propagator, obtained from the underdetermined expression (1.3) by the substitution m2 -> im2 - iie All the combinattons of the chronological lnd normal products and the Wick theorem are contained in the rules of the Gaussian integration. Two questions come to mind immediately after this observation: (i) Is it possible to explain the expression (1.1) or the corresponding gen­ erating functional for the S-matnx without any reference to the operator methods of the textbooks of quantum field theory? (ii) Why must the kernel of the operator, inverse to ( - D - m2) be taken in the form of the Feynman propagator? The following lectures are organized to give a systematic approach to an­ swer both these questions. We will begin with the derivation of the representa­ tion of the matrix elements of the evolution operator in terms of the function­ al integral. The 5-matrix will be obtained further by means of the appropriate limiting procedure.

2. Functional integral in phase space The general methods of quantum theory are more easily illustrated on the example of quantum systems with a finite number of degrees of freedom rath­ er than in the field theoretical case. That is why in these lectures we shall quite often describe the methods on the finite dimensional example before ex­ plaining their realization in field theory. Let us consider the mechanical system with one degree of freedom and the simplest phase-space two-dimensional plane. Canonical coordinates p and q, -°°
84 6

L.D. Faddeev

Let h(p,q) be a Hamiltonian function and h the corresponding energy opera­ tor. We are interested in the evaluation of the evolution operator

Let us suppose for definiteness that when quantizing h we have chosen the or­ der of factors in such a way that all operators p stand to the left of q. The for­ mal answer we shall obtain in this lecture does not depend on such a choice. With this convention we have a simple formula:

which follows immediately from (2.1). We shall now use it to evaluate the matrix elements of the evolution operator

The idea is to approximate the operator e~^e for small e in the form

and then consider the limit

To get the matrix elements of (2.4) we can use (2.3) and obtain

85 Functional methods

7

and to evaluate (2.5) we have to make convolutions of the products of such expressions. The strategy being clear, let us proceed with the calculations. For TV = 2 we get

where the factors ( l / 2 * ) W e * M " and (l/2ir)We*i«i convert the momen­ tum representation to the coordinate one to get the product of the operators involved. We rewrite the last formula in the form

where we understand that

It is clear from this example that the case of general N looks as follows

The limit N -> °°, e -> 0, TVe = t" - t\ corresponds to the integration over an infinite number of variables we can formally present as p{t\ q(t), t'
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L.D. Faddeev

i.e. that of the action functional for the phase-space trajectory p(t),q(t). This observation leads to the main result of this lecture: the matrix element (2.3) can be obtained by averaging the Feynman functional exp{/ X action} over all trajectories in phase space for a given time interval t < t < t" with the pre­ scribed initial and final values for coordinate variables

The last formula is attractive being expressed entirely in terms of the clas­ sical mechanical ingredients: action and Liouville measure on the phase space. The last objects are canonical invariants. This makes one suspect that with the proper definition of the functional integral one would be able to devise a canonically invariant procedure to obtain the quantum mechanical answers from the corresponding classical ones. This can not be true for all observables be­ cause it is well known that there exists no natural action of the group of gen­ eral (non-linear) canonical transformations in the space of states of the quan­ tum system. This shows that one must be very careful in dealing with such a formal object as the functional integral. I shall not make any attempt to devise an intrinsic definition of the r.h.s. in (2.6). Our understanding of this object will increase in the course of the for­ mal manipulations with it. The derivation above suggests the definition of (2.6) by means of a finite dimensional approximation of p(t) by piece-wise constant and of q(t) by piece-wise continuous functions and using the quadra­ ture formula to approximate the integral for the action. This procedure, how­ ever is by no means a natural one. Different possibilities appear if h(p,q) con­ tains a quadratic form - aa sn the field theoretical lxamplle One ccn then use a well-defined perturbation theory. The original formula of Feynman differs from (2.6) in the following way: the action is considered as a functional of the coordinate-space trajectory q(t) only. It is easy to get the Feynman formula in the case when the Hamiltonian is quadratic in the momentum variable

Indeed, the integration over p(t) in (2.6) is of a Gaussian type,

and after the shift of the integration variables,

87 Functional methods

9

we get instead of (2.8) the expression

Using it in (2.6) we obtain

which is the original Feynman formula. It is clear from the derivation that (2.9) is less general than (2.6); the former is valid only if the Lagrangian is quadratic in velocities. The case of any number of degrees of freedom can be treated analogously. With the appropriate choice of the vector notations

one can use formulas (2.6) and (2.9) in this case. The field theoretical applica­ tion is evident in the representation when the field operator 0(x) )i nome eiven time (say, x0 = 0) is diagonal,

Here His a Hamiltonian operator, corresponding to the Lagrangian £ of lec­ ture 1. One sees the relevance of (2.10) to understanding the role of the func­ tional integral in that lecture. For further use, let us note here, that one can rewrite (2.10) in the form

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L.D. Faddeev

where the first factor must be understood in terms of a power series expansion in X. We see that in the perturbation approach the dynamical problem of quan­ tum field theory reduces essentially to that for the non-self-interacting field influenced by the external source. We shall finish this lecture with a comment about the definition of the Smatrix by means of the functional integral. Let us return to the case of one degree of freedom and consider the non-relativistic particle interacting with the short range potential. The Hamiltonian of the system is given by (2.7) and we shall rewrite it in the form

The asymptotic behavior of the solutions of the Schrodinger equation is char­ acterized by the operator h0 and the S-matrix is defined in the usual way,

The matrix elements of the 5-matrix between two states \j/j and \J,2 described in the momentum representation by the functions Cj(p) and C2(p) can be written in the form

where
The momentum representation S(p",p) of the ^-matrix is defined by

To compute the limit (2.13) it is convenient to use the approximate form of the wave packet

which can be obtained by means of the stationary phase method. Substituting (2.15) in (2.13) and changing the variables x -+ pt/m ww ebtain

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Functional methods

11

Comparing (2.14) and (2.16) and remembering the functional integral expres­ sions (2.6) or (2.9) for (q",t"\q',t') we can say that to get the 5-matrix in the momentum representation one is to calculate the Feynman functional inte­ gral over the trajectories defined on the whole time axis -°° < t < °° having special asymptotic behavior for |r| -*•*•

This behavior is defined by the asymptotical dynamics which in our example is a free motion of the non-relativistic particle of mass m. Some factors are to be cancelled before the calculation of the limit. This observation shows that the S-matrix is defined by means of a func­ tional integral in a more elegant manner than the evolution operator. There is no reference to the fixed boundary conditions, they are substituted by the asymptotic behavior of the trajectories. This is especially attractive in the case of field theory because the expressions of the type (2.11) become manifestly Lorentz covariant, if the time JC0 varies on the whole axis. However the asymp­ totic behavior in this case is not easy to define in the (*) representation. In­ deed, the analogue of the operator H is not p2/2m, but rather an oscillator Hamiltonian h = p2/2m + |u2<7. This operator has a simple form in the socalled holomorphic representation, defined by Bargmann, Berezin and Segal. Having all this in mind it is natural to try to get the analogue of (2.6) in holo­ morphic representation before the discussion of the S-matrix for quantum field theory. This question will be discussed in the next lecture. Problem Choose some other ordering of factors for quantizing h(p,q) (i.e. such that all q are to the right of p) and repeat the derivation of (q"t"\q't'). Verify that the final formal expression for it coincides with (2.6). 3. Holomorphic form of the functional integral Let us consider once again a system with one degree of freedom but use the

90_

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L.D. Faddeev

complex coordinates a* and a instead of the real p and q in phase space. Hav­ ing in mind the application to oscillator type problems we shall choose the fol­ lowing form for a* and a:

co being an arbitrary rral larametee which will lecome e arequency in whha follows. In quantum mechanics the variables a* and a turn into the operators a* and a with the commutation relations

(the latter having sense only if we have more than one degree of freedom). The only difference between p and q is that i has disappeared from the com­ mutation relations. Operators a* and a are to be adjoint to.each other. The representation

when we do not specify the nature of the variable z, clearly satisfies the com­ mutation relations. We face the problem to find the inner product for the functions/(z) which the operators a* and a act on such that those operators are mutually adjoint. The authors mentioned above have found such an inner product. One is to consider polynomials/(z) in the complex variable z and define the scalar pro­ duct in the form

where it is understood that

This product is positive definite; indeed the monomials are orthonormal with respect to it as shown by the following computation

where we used polar coordinates z = pe'* and the r.h.s. is equal to 0 if n ± m and/*! ifn = m.

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Functional methods

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The mutual formal adjointness of a* and a in this representation follows from the identities

and the complex form of the Cauchy-Riemann condition

The complete space of state vectors on which a* and c act must be obtained by means of closure of the linear combinations of the monomials in the norm defined by the inner product. It can be shown that this space consists of the entire functions of the type \ and that a* and a when properly defined are truly adjoint to each other. The functions

are the eigenfunctions of the occupation number operator h or oscillator Hamiltonian h

We can use this to connect the representation described with the more famil­ iar ones, say the coordinate representation. This will be left as an exercise. Let us turn now to the problem of representing the operators. We shall use two ways to do it. The first is based on the matrix elements of a given opera­ tor in the
then we shall represent the operator^ by a kernel

It is clear that the action of A on an arbitrary vector/(z) is given by the for­ mula

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L.D. Faddeev

and the product of two operators looks as follows

One sees that to be able to realize these formulae it is necessary to consider A(z,z) as an analytic function of two independent complex variables z and z which are not to be taken as being complex conjugate one to another. The second representation is a well-known normal product, when ,4 is writ­ ten as a power series in a* and a such that all a* are to the left of a,

It is convenient at this point to make some change in the notation and use the variable a* instead of z

We see that one can associate with a given operator A two analytic functions of two variables: the kernel A(a*,a) and the normal symbol

Let us show that these two functions are connected by the relation

To prove it, it is sufficient to consider a monomial

so that

Now

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Functional methods

15

where 6{n>k) = 1 if n >k and 0 if n
QED. Formulae (3.1) and (3.2) are very similar to (2.1) and (2.2). The substitu­ tion

enables one to convert one set of formulae to another. This shows that we can repeat all the procedure of the previous lecture to get the expression for the kernel U(a*,a,t",t') of the evolution operator in terms of the functional inte­ gral over the trajectories a*(t), a{t) in the phase space. Indeed, let h(a*,a) be a Hamiltonian in a*-a variables which we quantize by choosing the normal product order

Then the kernel U(a*,a;e) of the evolution operator U(e) = exp{-/e/j} for small e is given by

For the finite-time difference t" -t'=Ne such kernels

where it is supposed that

we are to make a convolution of

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L.D. Faddeev

The formal limit N-+ - , e ^ 0 cac be bepressed in tht form

or after the symmetrization in a* and a,

It is especially evident in the last expression that we have got the result equiv­ alent to (2.6). Indeed, the integral

is nothing but the action of a trajectory in phase space given in terms of the complex coordinates. The extra term exp \(a*{t")a(t") + a*(t')a(t')) in the integrand takes care of the fact that we are using different boundary condi­ tions for the trajectory at t = t" and t = t'. Two temptations are to be avoided in these formal manipulations: (i) It is wrong to consider that a{t") is the complex conjugate of a*(t") or that a*(t') is the complex conjugate of a(t') and extract the factor exp \(a*(t")a(t") +a*(t')a(t)) from under the integration symbol. It must be understood that a(t") is the value at t = t" of the function a(t) for which only the value a(r') is fixed. (ii) It is wrong to say that the normal symbol of the evolution operator U(a*,a\t" - t'))t-a*a can be computed as a functional integral over the closed trajectories going through the point a*,a in phase space in spite of the fact that the factor

seems to provide the term we lack to convert the sum in (3.3) into the integral

_95 Functional methods

of the form

17

a dr. Indeed there ii no rrason to belleve that aQ muss be

close to aN in the limit e -* 0. We finish this lecture by applying formula (3.4) to describe the evolution operator for the harmonic oscillator in the interaction with the external source. The relevance of this example to our main subject is clear from the comment at the end of the previous lecture. The Hamiltonian of the system in question looks as follows where the complex function y(t) represents the time-dependent source. It is clear that all the derivations above can be used for time-dependent Hamiltonians. The only difference is that the evolution operator is not a function of the time difference and that the action is to be defined by

The action for our system is quadratic in a*(i) and a{t) and the functional in­ tegral (3.4) is a Gaussian one. We can evaluate it saying that it is equal to the value of the integrand at the critical point (i.e. at the stationary trajectory). The critical point is defined in terms of the classical equation of motion. Indeed for arbitrary h(a*,a) for fixed a*(t") = a* and a(t') = a,

because the boundary term we acquire when integrating by parts is cancelled by [This is one more confirmation that the strange looking term a I is quite natural.] In our example the equations of motion look as follows:

and have a unique solution subject to the boundary conditions

namely

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L.D. Faddeev

Let us note that this solution is not real in the sense that a*(t) is not a com­ plex conjugate of a(f). Nevertheless we are to use it to evaluate the Gaussian integral. Indeed, it is well known that the Gaussian integral is equal to the in­ tegrand at the critical point even if it does not belong to the space over which one integrates. We are now to substitute the trajectory found into the integrand in (3.4). The action part can be rewritten in the form

and the first term on the r.h.s. vanishes at the critical point. Collecting the terms both from the action and the boundary term we get

It is convenient to symmetrize the last term with respect to y(u) and y(u). After this is done we can write the answer for the kernel U(a*,a;t",t') we are looking for in the form

The system of several degrees of freedom, each having its own frequency u can be considered in the same way. Each degree of freedom gives its own con­ tribution to U(a*,a;t",t') which we are simply to multiply. This result will be used in the next lecture to get a final expression for the field theoretical 5-matrix.

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Functional methods

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Problems 1. Find the kernel K(x,z) such that the map

converts the holomorphic representation into the coordinate one. Describe the inverse map. 2. Show that

so that ta*a plays the role of the 5-function in the holomorphic representation. 3. Consider the integral

over the real variables z, when A is a complex matrix, Re A > 0 and b is a complex vector. Show that/= const. ew, W= -\{Az,z) + bz\z=A-\h.

4. Generating functional for the S-matrix We are now able to consider the main problem of these lectures, namely the construction of the ^-matrix by means of the functional integral. We begin with the case of a scalar field
and it leads to the Hamiltonian

which can be written in terms of a system of oscillators interacting with the time-dependent source. This well-known fact will enable us to use the result of the previous lecture to get the desired answer.

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L.D. Faddeev

Let us introduce the complex coordinates a*(k), a(k) instead of the real n(x), (x) by means of the familiar formulae

In terms of a*(k) and a(&) the Hamiltonian acquires the form

where

The analogy with the formulae of the previous lecture is apparent; the variable k plays the role of the label for the degrees of freedom. Generalizing the for­ mula (3.5) we get for the kernel of the evolution operator U(t",t') = Texp{-i / / H(t)dt} the following expression:

Let us use this to evaluate the S-matrix. We suppose that the source van­ ishes for | f | -*°° (note that the self-interacting terms we will be faced with in the general case effectively have such a property). Then the asymptotical dy­ namics are governed by the operator

so that the 5-matrix is defined by

99 Functional methods

21

It is easy to make the necessary convolutions to calculate the operator product in the r.h.s. Indeed, the kernel of the operator U0(t) = exp {-iH0r} has the form

and using the result of exercise 2 of lecture 3 we get for an arbitrary operator A with the kernel A(a*,a)

so that the multiplication by U0(f') from the right and by U~l{t") from the left reduces to the change of variables

in the kernel. The new variables are in fact the solutions of the classical equa­ tion of motion for the Hamiltonian HQ. As a result the kernel of the operator on the r.h.s. of (4.3) has the form

and the limit t" -> » and r'-> ^ is obtained simply by extendind the time in­ tegration to the whole axis. Let us rewrite the expression we should get in the limit using the source n(x,r) rather than its Fourier transform y(k,t),

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L.D. Faddeev

where Sn(a*,a) stands for the kernel of the S-matrix for scattering on the source T?(JC). It is clear now that it will be more elegant to use the normal sym­ bol. Indeed, to do this we are to drop the first term in the exponent. The re­ maining terms are manifestly Lorentz covariant. To see this let us introduce the solution of the free Klein-Gordon equation

and the Green function of this equation

It is easy to see that
Now the S-matrix for the self-interacting field has the form

and the discussion analogous to that at the end of lecture 1 shows that the last expression readily leads to the Feynman rules of perturbation theory. We have finished answering the two questions posed at the end of lecture 1. We have given the full derivation of the Feynman rules using the functional in­ tegral description of quantum dynamics. In particular we have shown how the correct account of the asymptotical conditions leads uniquely to the use of the Feynman propagator. We have used the scalar field as a simple example. The generalization for the spinor and vector fields will be discussed in the fol­ lowing two lectures. We shall finish this lecture with a brief discussion of the possible modifica­ tions of perturbation theory which follow from the general expression for the S-matrix we have found. Let us comment that it is by no means necessary to use the trick of introducing the source T?(X) and differentiating it to define the S-matrix. An alternative expression can be given directly in terms of the func­ tional integral. Let S(a*,a) be a kernel of the S-matrix. Then

101 Functional methods

23

where the trajectories over which we integrate have the prescribed values

We use here the notation Hint(a*,a) for the functional of two functions a*(k) and a(k) which we get by substitution of (4.1) into the interaction functional,

All the ideas necessary to derive this result have been illustrated earlier and we shall not repeat them. If we forget for the moment about the boundary terms in (4.5) we can say that the kernel of the •S-matrix is obtained by integrating the functional exp{/ X action} over the trajectories a*(k,t), a{k,t) (or 0(x), n(x)) defined on the whole time axis with a prescribed free-motion for the incoming wave a{k,t)~a(k)t-iu>t, f->-°°and the outgoing wavea*(fc,f) ~a*(k)ei<jJt, t -*■ °°, The boundary terms take care of the subtractions necessary to make the action functional on such trajectories converge. The most natural perturbation scheme based on this definition is given by the stationary phase approximation to the functional integral. To realize it, it is convenient to restore the Planck constant h, which until now we considered to be equal to 1. The Feynman functional will acquire the form exp{(z'/fc X ac­ tion}. Of course ti may enter the action itself through the mass term, for ex­ ample. This "internal" fi will be held fixed in the following and we will pro­ ceed with the construction of the power series expansion only in the "exter­ nal"/!. The main contribution to the Feynman integral will be given by the value of the integrand in (4.5) on the critical trajectory. We already know that the boundary terms in (4.5) have such a form that the critical trajectory is given by the classical equations of motion:

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L.D. Faddeev

The solutions subject to the boundary conditions (4.6) have a limit for t" -*■ °° and t' -* —°°; the asymptotic form can be found by the iteration of the follow­ ing integral equation:

Introducing the field 0c[(x), where

we see that it satisfies the classical field equation

and that the integral equations (4.7) are equivalent to the integral equation

where
103

Functional methods

25

The first term is to be dropped if we are interested in the normal symbol of the S-matrix. The other two can be rewritten in terms of the cl. Performing the trivial substitutions we find the main contribution to the normal symbol of the S-matrix has the form

The notations are explained as follows. We use the argument (J)Q(X) instead of a*(k), a(k) for the functional Stiee to emphasize the-Lorentz invariance of the answer; 4>Q(X) is given by (4.4) and is uniquely connected with a*(k) and a(k). The argument 0 C | on the r.h.s. is a solution Q through the integral equation (4.8). The tag "tree" on S means that the com­ puted approximation to the S-matrix contains the contribution to the normal coefficient functions of the S-matrix from the tree diagrams of ordinary per­ turbation theory. The proof of this is left as an exercise. It is instructive to observe that the exponent in (4.9) can be considered as an action functional

computed on the classical solution = 0C, in a specially prescribed regularized sense. Indeed we have formally

which is equal to the exponent of (4.9) if we agree to integrate by parts in the first two terms, use the free-motion equation to eliminate them and not care about the boundary terms. The next approximation is given by a Gaussian integration over the small deviations from the critical trajectory. Let us make the change of variables in the functional integral (4.5),

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L.D. Faddeev

We can now use as well the real variables

instead of the a and a* but the boundary conditions are more easily described in the former. The quadratic form in the deviations (*), 7r(x) is given by

where the particular choice of the first term on the r.h.s. is irrelevant due to the boundary conditions on 0 and if. The Gaussian functional integral

can be evaluated to give

where the operator ( - □ - m2) l is to be represented by the Feynman propa­ gator. The functional S0(0) generates the contribution to the normal symbol

105

Functional methods

27

of the S-matrix from the diagrams with one closed loop. To check it is a sec­ ond exercise. It is clear now how to proceed to the higher orders in (ft) 1 / 2 . One decom­ poses the action written in the variables $ and if,

and performs the Gaussian integration. There appears the natural generaliza­ tion of the Feynman rules. The quadratic form (1/ft) ^? 2 supplies the propa­ gator which is a Green function <^{x,y) of the differential operator

uniquely defined by means of the integral equation

Higher orders in (4.10) give the vertices, the total number of which is equal to the order of the polynomial V minus 2. The normal symbol of the S-matrix is given in terms of exp {i 2 vacuum diagrams}. The scheme described seems to be more complicated compared to usual perturbation theory. However it turns into the latter if one uses perturbation theory to solve eqs. (4.8) and (4.11). On the other hand it can be convenient for general theoretical investigations, because the combinations of the ordinary diagrams contained in one vacuum diagram in this generalized scheme can have nice invariance properties. This will be important for the discussion of the renormalization of gauge field theories and quantum gravity. Problems 1. Check that the functional Stiee(
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L.D. Faddeev

5. The case of fermions The functional methods described above explicitly take into account the Bose statistics of the fields involved. To treat the fermions one must use a dif­ ferent procedure. It is very desirable to have this procedure as close as possible to that used for bosons because the description of the interaction of bosons and fermions requires a unified scheme to display their dynamics. It was shown by Berezin that it is possible to devise a system of notations for fermions such that one can define the functional integral for fermions to look the same way as that for bosons. Let us begin to describe this system of notation on the example of the sys­ tem of fermions with one degree of freedom. The space of states for such a system is two-dimensional and two operators a* and a generate all algebra of the observables. These operators satisfy the anticommutation relations

and can be simply represented by means of 2 X 2 matrices. We shall devise an­ other representation for them. Let us consider two anticommuting variables a* and a,

considered as abstract algebraic symbols. In mathematics they are called the generators of the Grassman algebra. The general function f{a*,a) of these vari­ ables (an element of the Grassman algebra) can be written in the form

where f$,f\,f\ and/ 2 are complex numbers. We see that the linear space of such functions is four-dimensional. The "analytic" functions/(a*), i.e. the functions of the variable a* consti­ tute a two-dimensional space. We shall use it to represent the state space of our system. The operators a* and a are realized in the form

where the differentiation operation is defined as usual,

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Functional methods

29

It is easy to see that the anticommutation relations (5.1) are satisfied. Indeed a*2 = 0 because a*2 = 0, and a 2 = 0 because nothing remains after differentiat­ ing twice. Now

which gives the first relation in (5.1). Let us now introduce the integration of the functions f(a*,a). By definition

We will get this answer if we consider da* and da as anticommuting variables between themselves as well as with a* and a and introduce the rules of integra­ tion, devised by Berezin,

Note that the last two formulae show that the integral of a derivative is equal to zero. The notion of integration enables us to introduce the inner product into the space of functions/(a*). Define

Then

gives a positive definite inner product. Indeed

This shows that the monomials \ =a* constitute the orthonormal basis in the space of the functions/(a*). The operators a* and a are ad­ joint to each other in the inner product we have introduced. One can devise two ways of realizing the operators, either by means of a kernel or a normal symbol. The latter is defined as follows: with the normal form of a given operator A,

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L.D. Faddeev

we associate a function of the variables a* and a,

The kernel ,4 (a*,a) is given by

where Amn, m,n = 0, 1 are the matrix elements of the operator A in the O 0 , i^j) basis,

It is clear that

We see that to define these formulae we are to enlarge the set of the anticommuting variables by introducing the variables of integration a*,a. By defini­ tion they anticommute with a* and a. The normal symbol K(a*,a) and the kernel A (a*,a) of a given operator A are connected by means of the formula

We leave the proof as an exercise. All the formulae written above can be readily generalized to the case of a system with several degrees of freedom. One must use the set of 2n anticommuting variables

and pay attention to the order of factors. Arbitrary analytic functions/(a*) constitute the space of the state vectors. Creation and annihilation operators a* and a. act as follows

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Functional methods

31

The subscript L on the differentiation symbol means that the variable ai (if present in f{a*)) is to be pushed to the extreme left using anticommutativity before being dropped. These operators are adjoint with respect to the inner product

Here the * operation is defined by

and the integration is defined as above in the understanding that all the sym­ bols a*, a{, da*, dat are anticommuting. The monomials a,- ... aik are orthonormal with respect to this inner product. The kernels (a*, a) ana the normal symbol K(a*,a) of a given operator A are defined as in the case of one degree of freedom. The relations

are also satisfied. Now if we look at all the calculations in lecture 3, we realize that all the algebra is based on relations analogous to (5.3) and (5.4). This shows that all the results obtained there will be valid in the developed symbolism, including the fermion particles and fields. In particular, the kernel for the evolution op­ erator, generated by the Hamiltonian with the normal symbol h{a*,a,t) looks as follows

where it is supposed that

no 32

L.D. Faddeev

We shall not write the explicit formulae for the field theoretical case including the fermion fields. The education from the previous lectures suffices for one to do it him (or her) self. Problems 1. Show that

2. Obtain the rule for changing variables in the integral for an arbitrary nonhomogeneous linear substitution of the integration variables. 3. Prove (5.3). 4. Describe the generating functional for the Green function for the spinor charged field.

6. Some special features of the quantization of the Yang-Mills field This lecture is a transitional one to the course of Professor B. Lee. We shall give an introduction to the problem of quantization of gauge fields which will be discussed in his lectures in much more detail. First let me say several words about the definition and general meaning of the gauge field. We can associate it with an arbitrary compact Lie group G. Let ta,a- 1,... , n, be generators of the corresponding Lie algebra in the ad­ joint representation, n being the dimension of the group. We shall use the nor­ malization

then the structure constants fabc introduced in

are real and totally antisymmetric. Gauge field/I describes the parallel trans­ port of the arbitrary fields \p belonging to one or another of the representa­ tions of the group G. The space of this representation is called the internal space or space of charges to distinguish it from the ordinary space-time. Using

Ill

Functional methods

33

the words of H. Weyl, applied by himself to electrodynamics, one can say that the theory of gauge fields is a general relativity in the internal space. More specifically the gauge field is defined as a vector field A with the values in the adjoint representation of the Lie algebra. We shall use also the components^ of>l with respect to the basis ta,

By means of A^ one defines the covariant derivative of the field \p,

where T(ta) = Ta is a representation of the generators ta to which the field \p belongs. If the covariant derivative vanishes for some direction /,

we say that the field 1// at the point x^ + e/M is a parallel transport of the field i// at the point x^ for infinitesimal e. The parallel transport around the infinitesimal loop in general changes the value of \p. The measure of this change is called the curvature and is given by

where aM" is an infinitesimal surface bounded by the loop and

The change of the frame, with respect to which we define the components of the i// fields, can be considered as the following change in the field A^,

where e is a function e(x) belonging to the Lie algebra and describing the in­ finitesimal change of the frame. The transformation (6.2) is called the infinite­ simal gauge transformation. It can be integrated to define the gauge group

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L.D. Faddeev

which is formally written as a product of the group G taken once for each point in space-time. The group Gg auge plays the role of the group of the gen­ eral coordinate transformations. The general relativity means that the fields A^ and A'^ which can be obtained one from another by means of the gauge trans­ formation are physically indistinguishable. In other words it means that there exists no physical way to define a preferred frame to label the components of the charged field \p. This principle implies important conditions for the dynamics of the field A . Not all the components of this field are dynamical variables. Some of them are simply the group parameters. More specifically, the Lagrangian of this field is to be invariant with respect to the gauge transformations. This means that the time derivatives of some components of the fields will not enter into the Lagrangian, because the gauge transformations contain arbitrary functions of time. This in turn will lead to the singularity of the Lagrangian in the sense of Hamiltonian dynamics, i.e. to the fact that the canonical coordinates and momenta for the field A are not independent. The implications for the quan­ tization problem are evident. After these general considerations let us face the concrete problem. We con­ sider for simplicity the case of the pure gauge field. The Lagrangian introduced by Yang and Mills can be written in the second order form

or in the first order form

where in the former expression it is understood that F is to be written in terms of A according to (6.1) whereas in the latter expression .4 and F are independent variables. Using the 0 + 3 notations^ = (0,k), v = (Q,l), etc., k,l= 1,2,3 we can re­ write the last expression in the form

where

and we have made use of the time-independent equations of motion

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to express the Fik variables in terms of Ak. We see from (6.5) that

can be considered as canonically conjugate pairs whereas Ag has no conjugate momentum and can not be expressed in terms of other variables because it enters the Lagrangian linearly. Introducing the Poisson brackets

it is easy to check that

and

All this means that the Yang-Mills Lagrangian exemplifies the so-called gen­ eralized Hamiltonian form of dynamics. I shall explain this notion in the ex­ ample of a system with a finite number of degrees of freedom. Let us be given a phase space r 2 „ with the canonical coordinates Pj, q', i = I,..., n and consider the Lagrangian of the form

where Xa, a - 1,... , m are the dynamical variables additional to p and q and the functions <pa are called constraints for an obvious reason. We shall say that the Lagrangian (6.8) defines a generalized Hamiltonian dynamics if the follow­ ing conditions are satisfied:

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where Ca|3 and Caf3y can be functions of p and q. The condition m < n is nec­ essary for (6.10) to be true for independent <pa. The geometrical meaning of these conditions is as follows. Consider an arbitrary linear combination of constraints

as a Hamiltonian. It will define a trajectory — a line going through each point of the phase space. Varying the coefficients Ca in (6.11) we shall obtain an ra-dimensional submanifold intersecting each point of the phase space and spanned by those trajectories. We can say that the constraints a define a fibering of the phase space with an m-dimensional fiber. (We loosely.use this term from differential topology, not paying attention to the truly difficult global aspects of the dynamics.) The condition (6.9) shows that this fibering is invariant with respect to the dynamics, namely if two initial points belong to the same fiber, they will belong to one fiber for any time. The condition (6.10) shows that the procedure described above produces as well the fibering of the subspace f 2 „_ m of the space T2„ which is defined by the equations

It follows also from (6.9) that the function

has the same value for all points of a given fiber. As a result we get a 2(n - m) dimensional space T*, the points of which are the m-dimensional fibers of the subspace r 2 „ _ m . It is this space which plays the role of the physical phase space for the mechanical system defined by the Lagrangian (6.8) and the conditions (6.9) and (6.10), where the func­ tion H* plays the role of the Hamiltonian. To parametrize T* it is appropriate to use the subspace r 2 („_ m ) of T 2w _ m defined by the conditions

such that r 2 („_ m ) intersects each fiber of T 2 n _ m only once. The condition

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37

is sufficient for this. It is also convenient to suppose that

We shall call x a the subsidiary conditions. Indeed, if (6.15) is true we can make a canonical transformation in r 2 „ such that x a can be taken as the first m momentum variables

The condition (6.14) in the new variables looks as follows

so that the equations (6.12) can be solved for the qa. As a result the manifold T is given by the equations

and p* and q* play the role of the canonical coordinates in the physical phase space r*. One can quantize the generalized Hamiltonian system using non-constrained variables p*,q*. In the functional integral formulation the matrix elements of the evolution operator e\p{-iH*t} are given by

However it is difficult in practice to solve eqs. (6.12) and (6.13). For example in the case of the Yang-Mills field, the condition C(x) = 0 is a non-linear dif­ ferential equation without an explicit solution. This shows that it is desirable to get the generalization of (6.17) in terms of the constrained variables p,q,X. The answer looks as follows: formula (6.17) is equivalent to

Indeed integrating over X we can rewrite this as

which readily reduces to (6.17) if one uses the variables (6.16) described above.

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L.D. Faddeev

We can now apply this general scheme to the case of a Yang-Mills field. We know already the analogue of the variables p, q and A, Hamiltonian H and con­ straints 0. It remains to choose the subsidiary condition. The equation

is especially convenient because the Poisson bracket

does not depend on the dynamical variables, so that the determinant of the "matrix" (x,y) is a constant. We see that the functional integral for the Smatrix of the Yang-Mills field acquires the form

with the special asymptotical prescription for the fields A or after the Gaus­ sian integration over Ek,

where £(x) is given in (6.3). It is this form of the functional integral which will be a starting point for the lectures of Professor B. Lee. Note that he will use the re scaled fields

Problems 1. Let {<(>a, 4>n} = 0, {<pa, H] = 0. Find the explicit description of the physi­ cal phase space T* as an illustration of the general scheme described in the lec­ ture. 2. Show that (6.18) does not depend on the choice of x a 3. Find the expression for the functional integral in the so-called Coulomb gauge dkAk = 0.

117 Functional methods

39

References The formulation of quantum dynamics in terms of the functional integral was given by R. Feynman. He discusses the field theoretical application in [ 1 ] R. Feynman, Phys. Rev. 80 (1950) 440.

The equivalence of operator product combinatorics and Gaussian integra­ tion is shown in [2] N.N. Bogoliubov and D.V. Shiikov, Introduction to the theory of quantized fields (Interscience, New York-London, 1959) ch. VII.

One can find in the current literature the implication that the functional integral can be considered only as a heuristic tool, less rigorous (?) than the diagrammatic presentation of perturbation theory. The axiomatization of the notion of Gaussian integration which makes the functional integral at least as rigorous as other methods of quantum field theory is given in [3] A.A. Slavnov, Theor. Math. Phys. 22 (1975) 177. [4] J. Zinn-Justin, Bonn Lectures, Saclay preprint DPh-T/74-88.

The derivation of the main result of lecture 2 is taken from [5] W. Tobocman, Nuovo Cimento 3 (1956) 1213.

The trick with derivatives reducing the general dynamical problem to one in the external source belongs to J. Schwinger. See for example [6] J. Schwinger, Proc. Nat. Acad. Sci. 37 (1951) 452.

The holomorphic representation has a long history beginning with the works of V.A. Fock. The formulation used in the lectures was given by I. Segal, F.A. Berezin and V. Bargmann. A detailed description can be found in [7] F.A. Berezin, Method of the second quantization (Academic Press, NY-London, 1966).

The functional integral ever the complex variables a*(t), a{t) is discussed by S.S. Schweber and F.A. Berezin [8] S.S. Schweber, J. Math. Phys. 3 (1962) 831. [9] F.A. Berezin, Theor. Math. Phys. 6 (1971) 194.

Each of them has committed one of the sins mentioned at the end of lecture 3. Integration over anticommuting variables was introduced by F.A. Berezin in ref. [7]. The definition of the S-matrix in terms of the functional integral with the prescribed asymptotical behavior of the fields was pointed at by R. Feynman in [10] R. Feynman, Acta Phys. Polon. 24 (1963) 697. [11] R. Feynman, in Magic without magic: J.A. Wheeler, ed. J. Klauder (Freeman SF, 1972).

The use of the classical field as the argument of the generating functional in quantum field theory was suggested by B. De Witt, see for example

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[ 12] B.S. De Witt, in Relativity, groups and topology, Lectures at Les Houches 1963 (Gordon and Breach, New York-London, 1964).

The variant presented in lecture 4 describing the loop expansion is essential­ ly that of [13) 1. Arefieva, A.A. Slavnov and L.D. Faddeev, Theor. Math. Phys. 21 (1974) 311.

The formulation of the Feynman integral for generalized Hamiltonian sys­ tems is proposed in [ 14 | L.D. Faddeev, Theor. Math. Phys. 1 (1969) 3,

where one can also find the definition of the physical phase space in invariant terms of the observables rather than the coordinates.

119

Comments on Paper 6 Paper 6 has a very special value for me. As I already stated in Paper 1, the solution of the inverse problem of quantum scattering in the multidimensional case was one of the main goals I set for myself at graduation. Thus, when in 19701 finally got all the necessary tools for dealing with this problem, I felt very satisfied. The exposition of my approach is given in Part 3 of the paper. My scheme was developed further in coming years by R. Newton and Henkin-Novikov. The paper contains also a survey of the one-dimensional inverse problem. Though it plays mostly the role of an exercise, in which the appropriate technique is illustrated before the generalization to the real three-dimensional problem, it has value in its own right. Indeed, it plays a decisive role in the formulation of the inverse scattering method of solution of the KdV equation. This application was already known in 1973, and Part 2 is devoted to it. Unfortunately, Part 1, being rigorous from the mathematical point of view, contains several inaccuracies in estimates and statements. Paper 1 has some history of this. A completely correct proof of the main theorem of Part 1 can be found in the monograph of V. A. Marchenko (see Ref. 17 in the paper). The main theorem is true without corrections in the generic case, where the reflection coefficient r(k) vanishes at k = 0. In the exceptional case, where it is not so, the proper formulation can be found in Marchenko's monograph. I decided not to introduce the relevant corrections into this re-edition, bearing in mind that it is Part 3 which is most important.

121 INVERSE P R O B L E M OF Q U A N T U M

SCATTERING

T H E O R Y . II. L. D. F a d d e e v

UDC 517.9 .-530.1

INTRODUCTION The current survey summarizes the comparatively quiescent development of what is called the inverse problem of quantum scattering theory over the past 15 y e a r s . The preceding decade, during which this problem was formulated and subsequently intensively developed, was dealt with in my survey [25]. Its entire 25-yr history demonstrates that it is one of the most intriguing and instructive branches of mathema­ tical physics and reveals in its development new and unexpected aspects, and that it is far from being ex­ hausted. There exists somewhat self-consistent methods for presenting the formalism of the inverse problem. An elementary approach is based on the study of the properties of solutions of differential and integral equations c h a r a c t e r i s t i c for it by methods of classical analysis. The monograph of Z. S. Agranovich and V. A. Marchenko [1] is an example of this presentation. In the current survey, as in [25], we follow a dif­ ferent approach, using wherever possible an o p e r a t o r - t h e o r e t i c approach. The origin of this method of describing the inverse problem was set forth by Kay and Moses [38, 39], In this approach the inverse problem of scattering theory does not appear isolated, but finds a natural place within the framework of general scattering theory. Let us recall the general statements of scattering theory for the Schrbedinger operator, with which we will deal henceforth. It is a matter of comparing the spectral properties of two operators H and H0 defined in the Hilbert space £ = Z.2(R") by the formal differential equations

Here A is the Laplace operator

and v(x) is a real H0 defined in ft on these l e t t e r s . The alized by means of

function sufficiently smooth and rapidly decreasing as |x|— » . The operators H and the dense domain S> = Wl(R.n), define self-adjoint o p e r a t o r s , which we will denote by operator H0 has absolutely continuous s p e c t r u m . Its diagonal representation is r e ­ the Fourier transformation

Here

The assertion that the operator H has the same absolutely continuous spectrum as H0 is the fundamental r e s u l t of scattering theory. More precisely, there exists an invariant decomposition relative to H of the space J? in the direct orthogonal sum Translated from Itogi Naukii Tekhniki. Sovremennye Problemy Matematiki, Vol. 3, pp. 93-180, 1974. ©1976 Plenum Publishing Corporation. 227 West 17 th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

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122

of natural subspaces corresponding to the discrete and absolutely continuous spectrum of this opera­ t o r . Here the restriction of H to £ a c . is unitarily equivalent to H 0 . There exist among the o p e r a t o r s i s o m e t r i c in £ that realize this equivalence two o p e r a t o r s IP*' d i s ­ tinguished in t e r m s of their physical origin. They are called wave o p e r a t o r s and a r e defined by

where the limit is understood in the sense of a strong operator topology. There exists a broad l i t e r a t u r e that deals with the conditions on v(x) under which these limits exist. Our problem does not include the presentation of this "direct" problem of scattering theory, although s e v e r a l r e s u l t s relative to the e x i s ­ tence of operators t r * ' will also be mentioned in the text. Details on this question may be found, for ex­ ample, in the monograph of Kato [36], We note that the fact that the operator V = H - H0 is a function-mul­ tiplication operator plays no role in general discussions in scattering theory. The operators TjW a r e i s o m e t r i c :

Here P is a projector on the subspace £<j. which, as a r u l e , is finite-dimensional. The second relation is said to be a completeness condition. The unitary equivalence spoken of above is realized by the equation

The physical meaning of wave operators is based on the following concepts. In quantum mechanics the operator

d e s c r i b e s the evolution of a system, which in our c a s e , consists of a particle in the field of a potential center. Over a long period of time a particle with positive energy exits far from the center, becoming non-sensitive to its influence, and as a result its development over the course of time as |r|-*oo is actually described by the operator

corresponding to free motion. More precisely we may associate with every single-parameter family of vectors l^-(0> describing free motion (wave packet)

a solution of the Schrbedinger equation

such that

as t->- — oo. The precise equation defining such a solution and following from the existence of wave o p e r a ­ t o r s , has the form

335

123 Every solution of the Schrdedinger equation tf(t) from the given class as t — °° again reduces to a wave packet, in general differing from ^_(t):

as t — °°, where

The l a s t equation is justified since the operators U<+> and u(~) have a common range of values. The passage from the wave packet i^_(t) describing the initial state of a particle, to the wave packet 4+(t) describing its final state is also the p r o c e s s by which a particle is scattered by the center. All in­ formation on this p r o c e s s is contained in the operator S, which r e l a t e s both wave packets by the formula

Comparing the equations expressing 4 in t e r m s of 4- and 4 + in t e r m s of 4, we see that

where it follows from the properties of Tj(+) and u( ' that S is unitary and commutes with H0.

These relations reflect conservation of particle and energy flux in the course of scattering. The operator S is said to be the scattering operator. Its representation

in a diagonal realization of H0 is defined by the equation

where the 5-function in the integrand explicitly takes into account the fact that S and H 0 commute. The function f(k, l) defined for | k | = 11\ is called the scattering amplitude. T h e r e exists an alternative approach to the scattering theory, called the stationary approach, based on the study of the asymptotics of the eigenfunctions of H as |x| — » . The scattering amplitude f(k, I) h e r e explicitly occurs in the description of these asymptotics. Such an approach and its relation to the nonstationary approach will be illustrated in the text. The problem of reconstructing the potential v(x) corresponding to the scattering amplitude f(k, /) is said to be the i n v e r s e problem of scattering theory. This problem is not defined if the perturbation V = H - H0 is an a r b i t r a r y operator, since an entire set of operators V can easily be selected with respect to an a r ­ b i t r a r y unitary operator of the form S, such that the corresponding operator S is a scattering operator for the pair H0 and Ho + V. It becomes meaningful only under a further condition, that V is an operator of multiplication by a function. Henceforth, this condition will be said to be the locality of the potential. Over the past 25 years the inverse problem has been solved for the case most interesting for physical applications of a spherically symmetric potential, viz., when n = 3, and

In this case the scattering amplitude f(k, I) depends only on the lengths of the vector k and I, which a r e equal by the condition, and on the angle between them so that it is actually a function f(|k|, cosfl) of two v a r i a b l e s . The partial scattering amplitudes fj(|k|) a r i s e in decomposing f(|k|, cos6) in Legendre poly­ nomials

336

124 so c h a r a c t e r i s t i c for spherically s y m m e t r i c problems. The unitarity condition on an operator S can be explicitly borne in mind by setting

where i)j(|k|) is a r e a l function, called the asymptotic phase because of its role in the stationary formula­ tion of the scattering problem. The fundamental result obtained by V. A. Marchenko [16] and M. G. Krein [12] states that the potential v(x) is reconstructed by one of the asymptotic phases TJ,( | k | ) , one such phase being an arbitrary real function satisfying the integrality condition

Here m positive numbers associated with the characteristics of a discrete spectrum of the radial Schrodinger operator

whose reconstruction constitutes our problem, must be specified for a unique determination of the potential by a given phase rj,. It is precisely this result that was dealt with in the survey [25]. The subsequent development of the inverse problem, about which we shall speak in the current article, is associated with the Schrbedinger operator in the general case without spherical symmetry type assumptions. Here, two cases a r e distin­ guished, theoretically differing in t e r m s of technical difficulties: n = 1 and n S: 2. In the first c a s e , tools developed for the radial Schrbedinger operator with 1 = 0 turned out to be applicable. The chief role is played by the existence of a fundamental system of solutions for the corresponding one-dimensional dif­ ferential equation. In the multidimensional case, when we must deal with a partial differential equation contrary to ordinary differential equations, the concepts of a fundamental system vanishes. It may be that it is precisely this circumstance that constituted the hindrance that has extended the study of the multidimensional inverse problem over such a long period of time. In spite of this circumstance, which constitutes a technical distinction, it turned out that the one-di­ mensional and multidimensional inverse problems are to some degree analogous in many ways. This analogy can be particularly seen in an operator-theoretic language, which we have chosen for our p r e s e n t a ­ tion for precisely this reason. It is of interest that all these assertions on analogy refer to the one-di­ mensional, but not radial Schrbedinger operator. In this sense the one-dimensional case plays a fortunate role as an intermediate link between the radial Schrbedinger operator and the multidimensional operator, being technically close to the former and conceptually anticipatory of the fundamental outlines of the l a t t e r . We will now indicate the principal distinction between the inverse problem considered in this survey and the case of the radial Schrodinger operator. It consists in the overdeterminacy of this problem. In the radial case we must construct a function v(r) decreasing at infinity of a variable r that varies on the half-axis, in t e r m s of a function r\i(\k\) of a variable |k| also on the half-axis and also satisfying an a s y m p ­ totic condition as |^| —-co. it is therefore not remarkable that the function r\t(\k\) can be chosen a r b i t r a r i l y . A similar simple calculation of p a r a m e t e r s demonstrates that the scattering amplitude for m o r e complex problems cannot be arbitrarily selected. Let us first consider the Schrbedinger operator for n = 1. An a r b i t r a r y unitary scattering amplitude f(k, 1), where k, I € R1 can be parametricized by four r e a l functions of the variable |k|, running through the half-axis. A symmetry condition, which follows from the r e a l n e s s of the potential and which will be p r e ­ sented in the text, d e c r e a s e s this number down to three. At the same time the potential v(x) can be con­ sidered only as two real functions defined on the half-axis. Indeterminacy is present since it is difficult to imagine the problem of the physical origin in which the nondegenerate correspondence of sets of two and three arbitrary functions would be established. In other words these heuristic arguments demonstrate that the scattering amplitude f(k, /) will satisfy the additional necessary condition and so can be expressed in terms of two real functions of the half-axis. Such a condition in fact arises and is derived in the text.

337

125 When n a 2 this indeterminacy is significantly aggravated. The scattering amplitude here is a func­ tion of 2n— 1 variables, while the property of being unitary reduces to this function being r e a l . At the s a m e time the potential is a r e a l function of n variables. As a r e s u l t of such indeterminacy the problem of determining necessary conditions on the scattering amplitude that follow from the locality condition on the potential, arises and becomes of great importance. It was previously unclear that such conditions can in general be expressed in terms of the scattering amplitude in sufficiently explicit form. Nevertheless, as will be explained in the text, such conditions can be described. We will now formulate the fundamental statements of the formalism of the inverse problem. We will a s s u m e for the sake of definiteness that the operator H has one simple eigenvalue, so that the projector P under an isometricity condition is a projector on the one-dimensional subspace the corresponding eigen­ vector u spans. The choice of the transformation operator U, i.e., the choice of the solution of the equa­ tion

which differs from wave o p e r a t o r s , is the basis of this approach. Every transformation operator is ob­ tained from the wave o p e r a t o r s u W by multiplication on the right by a normalizing operator factor N^"' that commutes with H0,

In p a r t i c u l a r the scattering operator S is a normalizing factor for U^"^ with respect

Comparing these two formulas we can see that a factorization of the scattering operator

c o r r r e s p o n d s to every choice of the transformation operator U. We now assume that U is invertible in the sense that there exists a vector \ not belonging to the space j? , such that

Then the completeness condition expressed in t e r m s of U will be

where

The operator W will be called a weight operator. Equations (1) and (2) constitute the basis for solving the inverse problem. We must find a s u c c e s s ­ ful determination of U, such that Eq. (2) uniquely determines it in t e r m s of the weight operator W and that the corresponding factors N W is uniquely determined by the factorization condition of Eq. (1) in t e r m s of a given operator S. It turns out that such transformation operators exist and are distinguished by a Volterra property. Let us clarify in detail what we understand by a Volterra property. In the one-dimensional case this concept is formulated in the most classical fashion. The kernel A(x, y), where x, y € R ' is said to be t r i a n ­ gular if A(x, y) = 0 when x < y or A(x, y) = 0 when x >y. An integral operator with triangular kernel is said to be a triangular operator. Finally an operator of the form "identity element plus triangular operator" is said to be a Volterra operator. We have two possibilities for a Volterra operator in the one-dimensional case:

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Both formulas may be described uniquely if a direction y is introduced, i.e., actually a variable taking two values y = ± 1

In this form the calculation of the Volterra operator is naturally carried over to the case n S 2 . The variable y in this case is a unit vector and runs through, unlike the one-dimensional case, a connected set, namely the sphere S n _ 1 . An operator of the form

is said to be a Volterra operator with direction y of Volterra property. We will now prove that the Volterra property of a transformation operator Uy reduces the complete­ ness equation (2) to a linear integral equation for the kernel Ay(x, y) occurring in its definition. Suppose y is some direction and let Uy be a Volterra transformation operator with direction y of Volterra p r o p e r ­ ty. We consider the operator

This operator has direction of Volterra property opposite to that of the operator y , so that

Suppose W

is a weight operator for Uy, and we set

Equation (2) with the notation introduced, can be rewritten in the form

or, in m o r e detail, in t e r m s of the kernels A (x, y), fty(x, y), and Ay(x, y) in the form

The right side h e r e vanishes when (y - x, y)> 0. If this condition holds, we obtain the linear integral equa­ tion

which can be used to find the kernel A (x, y) in t e r m s of the known kernel S7y(x, y). This equation consti­ tutes a general formulation of the Gel'fand-Levitan equation introduced in [8] for the actual example of a S t u r m - L i o u v i l l e operator on the half-axis. Thus we will see how to reconstruct the Volterra transformation operator U y if the corresponding weight operator W y is known. To construct the operator W y in t e r m s of a known operator S it is n e c e s ­ sary to solve one m o r e problem in the factorization of Eq. (1) to determine the normalizing factors of U y .

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It is still not evident whether this problem for Volterra U r a l s o reduces to a linear equation nor whether it is even solved explicitly in the one-dimensional case. However it turns out that normalizing factors for Volterra transformation operators themselves turn out in some sense to be Volterra. We will not clarify this circumstance in more detail here and refer the reader to precise formulations in the text. Thus, the procedure for solving the inverse problem for given scattering amplitude is reduced to a set of factorization problems (1) to determine the normalizing factors N ^ . A weight operator W y is constructed in terms of these data and characteristics of the discrete spectrum, if such exists, and then using the Gel'fand-Levitan equation, the transformation operators U y a r e reconstructed. All the stages, in general, can be realized for an arbitrary initial operator S. Moreover if S is unitary, each of the operators

will be self-adjoint. Additional necessary conditions, about which we spoke above, begin to play a role at the next stage, when it is clarified that the operators Hy in fact are independent of y . This important s t a t e ­ ment simultaneously s e r v e s for investigating the properties for the reconstructed operator H and for prov­ ing that the initial operator S is in fact the scattering operator for the pair H and H0. The tools for proving the independence of Hy from y differ for n = 1 and n a 2, because of the difference between the range of values of the variable y . When n ^ 2 , i.e., when this set is connected, we can use differentiation with r e ­ spect to the p a r a m e t e r y . In the one-dimensional case it is necessary to use more artificial m e a n s . We conclude this description of the tools for solving the inverse problem, since further detail r e ­ quires a m o r e formal presentation, which will be presented in the text. We note only that, in our opinion, abstract scattering theory can be further developed so that Volterra transformation operators and the existence of separated factorizations of the form (1) of the scattering operator find a natural place within its framework. Apparently the formulation of scattering theory due to Lax and Phillips [45] is the most successful starting point for such a generalization, and a suitable causality condition will in a reasonable way be the appropriate language. Let us now indicate on the structure of the survey. Differences in the technique and elaboration of the cases n = 1 and n > 2 forced us to treat them separately. We will emphasize the analogies between the corresponding discussions and equations wherever possible. The one-dimensional case is discussed in Chap. 1. A significant technical simplification for study­ ing the one-dimensional Schrodinger operator lies in the existence of special fundamental systems of solutions of the correspondinding differential equation. All operator equations a r e suitably introduced and justified proceeding on the b a s i s of the well-known properties of these solutions. The description of these p r o p e r t i e s is discussed in Sec. 1, which plays an auxiliary role. In Sec. 2 the fundamental statements of scattering theory for a given concrete example are formulated and proved. Volterra transformation opera­ t o r s a r e introduced in Sec. 3 and the normalizing factors corresponding to them are obtained in Sec. 4. G e l ' f a n d - L e v i t a n - t y p e equations are formulated in the latter section. Section 5 t r e a t s the solvability of these equation. A relation is analyzed there between transformation o p e r a t o r s for y = 1 and y = — 1 . The general investigation of the inverse problem concludes h e r e . The last Sec. 6 contains a description of an explicit solution of the Gel'fand-Levitan equation for the particular case when the scattering amplitude is a rational function of a p a r a m e t e r k. Chapter 2 also t r e a t s one-dimensional problems. Here a generalization of the formalism developed in Chap. 1 to the case of potentials v(x) having nonzero asymptotic a s x » (Sec. 1) or to the case of an o p e r a t o r of the form

which is a d i r e c t generalization of the Schroedinger operator (Sec. 2), is analyzed at an elementary level. In the l a s t section we will describe so-called t r a c e identities, which relate certain functionals of the poten­ tial and scattering amplitude. These identities, through not a means for the direct solution of the inverse problem, can indirectly lead to information on the potential according to known properties of the s c a t t e r ­ ing amplitude, and conversely. Section 4 d e s c r i b e s an application of the inverse problem of scattering theory to the solution of one-dimensional nonlinear evolutionary equations. The starting point of this application was set forth in the important work of Kruskal et al. [42], Then P. Lax [44], V. E. Zakharov and A. B. Shabat [11], V. E. Zakharov and the author [10], and others further developed this subject. The inverse

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128

problem method of scattering theory for solving nonlinear equations currently draws ever g r e a t e r attention and is rapidly developing. The region of its applicability is still far from clarified and we will consider in the present survey only two c h a r a c t e r i s t i c examples. In Chap. 3 we r e t u r n to our fundamental theme and consider the inverse problem for the multidimen­ sional Schrbedinger operator. For the sake of definiteness we will deal with the physically interesting case n = 3, although all the discussions are trivially carried over to a r b i t r a r y n s 2. In the multidimen­ sional case no fundamental system of solutions nor proof procedure is available due to the far g r e a t e r c u m b e r s o m e n e s s . The scope of the present survey does not allow us to present somewhat instructive estimates needed to make all the constructions of Chap. 3 r i g o r o u s . We will therefore limit ourselves to a presentation of only the formal scheme of these constructions. We leave it to the r e a d e r to complete the algebraic framework of this scheme by appropriate analytic arguments. In Sec. 1 we set forth the fundamentals of scattering theory for the three-dimensional Schrodinger operator. General concepts that are of assistance in research on Volterra transformation operators U r a r e described in Sec. 2. The construction of normalizing factors and the weight operator W y a r e discussed in Sees. 3 and 4. The description of the Gel'fand-Levitan operator and a scheme for studying the inverse problem are presented in Sec. 5, with which Chap. 3 concludes. No special knowledge is required to read this survey. In particular, it can be read independently of the preceding survey [25], since all the necessary information on the Schrodinger operator a r e enumerated h e r e once again. We hope that for some mathematicians this survey can s e r v e as an introduction to s c a t ­ tering theory, a branch of functional analysis and mathematical physics which is constantly expanding the domain of its applications. In concluding this introduction we note trends associated with the inverse problem of scattering the­ ory that a r e not indicated in this survey. These include: 1. works of B. M. Levitan and M. G. Gasymov, M. G. Krein, and his students on canonical systems and Dirac-type s y s t e m s on the s e m i - a x i s . These works in t e r m s of the formulation of the problem and methods relate to the problems associated with the radial Schrbedinger operator. We refer the r e a d e r to new studies [7, 13] for references to the l i t e r a t u r e given there. 2. Works on inverse problems in t e r m s of scattering data for fixed energy. By this problem is under­ stood the reconstruction of the potential v(r) in t e r m s of a known set of asymptotic phases r?;(|k|) for all I = 0, 1, 2 , . . . and fixed |k|. An operator-theoretic formulation of this problem is not at all evident and the r e s u l t s obtained have yet to reach, in t e r m s of elegance and completeness, the level attained in the s p e c ­ t r a l formulation of the inverse problem. The most detailed presentation of well-known facts on this p r o b ­ lem can be found in Loeffel [47]. 3. Works of V. A. Marchenko and his students on the stability of the inverse problem, p r i m a r i l y for the example of the radial Schrbedinger equation. The recent monograph of V. A. Marchenko [17] d i s c u s s e s this subject. We will use no unusual notation. Constants appearing in the limits a r e denoted by C. An explicit d e ­ pendence of these constants on p a r a m e t e r s is indicated only if this is important. In numbered equations the first digit indicates the number of the section and the second, the number of the equation. The number­ ing of the sections begins anew within each chapter. A reference to an equation of a different chapter will use a number made of three digits of the type (II.3.14), whose meaning is self-evident. CHAPTER ONE-DIMENSIONAL

1

SCHROEDINGER

OPERATOR

In this chapter we will consider the Schrbedinger operator

where the potential v(x) is assumed to be a r e a l measureable function satisfying the condition

(P)

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129 Here the o p e r a t o r H defined on the dense set £> = U^(R) in the Hilbert space i} = L2{R) is a self-adjoint o p e r a t o r . We will introduce and c h a r a c t e r i z e the scattering data corresponding to this operator and d e ­ s c r i b e the procedure for solving the inverse problem for reconstructing the potential v(x) in t e r m s of these data. 1. F u n d a m e n t a l

System

of S o l u t i o n s

of S c h r d e d i n g e r

Equation

This section is auxiliary. Here we will describe two fundamental systems of solutions of the Schrde­ dinger equation

Henceforth k, as a r u l e , will be a real number, but sometimes we will a s s u m e it to be a complex number, particularly when specified. Condition (P) means that v(x) effectively vanishes as |JC|-> oo, so that we may naturally assume that every solution of Eq. (1.1) coincides at infinity with some solution of the equation

i.e., a linear combination of exponents

More rigorously, we may prove that there exist solutions f ( (x, k) and f2(x, k) of Eq. (1.1) which have the asymptotic

The proof is based on the fact that the differential equation (1.1) with the boundary conditions (1.2) and (1.3) is equivalent to the equations

where

and 0(x) is the Heaviside function

These equations a r e Volterra-type integral equations, so that the method of successive approxima­ tions always converges for them. Here the p a r a m e t e r k can have complex values from the upper halfplane. As a r e s u l t of analyzing the successive approximations we will prove that the solutions f,(x, k) and f2(x, k) exist and for fixed x are analytic functions of k when 1m k> 0 and are continuous when Im k = 0. Here we have the bounds for them

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130

Such assertions were first obtained by Levinson [46], It follows from the bound of Eq. (1.6) on the basis of the Jordan lemma that we have for the solution f^x, k) the integral representation

where the kernel A,(x, y) is quadratically integrable with respect to y for any fixed x. Similarly f2(x, k) can be represented in the form

where the kernel A2(x, y) also is quadratically integrable with respect to y. Such integral representations for solving the Schroedinger equation were introduced by B. Ya. Levin [15]. The detailed properties of the kernels Aj(x, y) can be obtained on the b a s i s of an investigation into integral equations equivalent to the corresponding equations (1.4) and (1.5) for the solutions f,(x, k) and f2(x, k)

Such equations were first derived by Z. S. Agranovich and V. A. Marchenko [1]. Agranovich and Marchenko proved the convergence of the method of successive approximations for these equations and obtained bounds on the solutions. To write the bounds it is suitable to introduce the monotone functions

Bounds on the kernels A^x, y) and A2(x, y) have the form

Further it is possible to prove using these equations the existence of the first derivatives of A^x, y) and A 2 (x, y) and to obtain bounds on them. For example,

and similar bounds hold for ^- A, (x, y) and ^-A2(x, y). Finally, it is evident from the equations that

343

131 so that

The p a i r s Ji(x,k), ft(x, — k)=fx(x,k) and / 2 (x, k), f2(x, —k) = /2(x, k) for real k * 0 are fundamen­ tal s y s t e m s of solutions of the fundamental equation (1.1). In fact, since the Wronskian {/,, / , } = / ! / , _ / , / , ' is independent of x, it coincides with its values as x—- °°, which may be calculated using the asymptotic for the solution fj(x, k) and its derivative. It can be proved that as x — °°

so that

We will see that when k * 0, the Wronskian is nonzero and the solutions fj(x, k) and f^x, - k ) are linearly independent. Similarly

so that f2(x, k) and f2(x,—k) a r e also linearly independent when k * 0. Any solution of Eq. (1.1) can be represented in the form of a linear combination of the solutions fj(x, k) and fj(x, - k ) or f2(x, k) and f2(x, —k). In particular, we have

Substituting Eq. (1.15) for f2(x, k) in Eq. (1.16) and performing the s a m e operation with fj(x, k) we find that the following equations must h o l d i n E q s . (1.15) and (1.16) a r e to be consistent:

We may e x p r e s s the coefficients Cjj(k), i, j = 1, 2 in t e r m s of the Wronskians of the solutions f,(x, k) and f2(x, k). In view of Eqs. (1.13) and (1.14) and also in view of the self-evident equations

we find

Comparing Eqs. (1.19) and (1.20) we find that

which, incidentally also follows from Eq. (1.17) since Cj2(k) -- (^(k). These equations imply also that

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132 We will see that the four coefficients cij(k) a r e in fact expressed in t e r m s of two complex-valued functions

satisfying the condition

Here

We will henceforth refer to these functions as transition coefficients. To derive further properties of the functions c(k) and b(k) we e x p r e s s them in t e r m s of the k e r n e l A 2 (x, y). F o r this purpose we note that Eq. (1.5) implies that f2(x, k) a s x - « has the asymptotic

Comparing this equation with the equation

which follows from Eq. (1.15) if we take into account definition (1.2) of the solutions fj(x, k) we obtain for a(k) and b(k) the equations

We now replace f3(x, k) by the kernel A2(x, y) using Eq. (1.9). We find

where

and

where

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133 Bounds on fl^x) and II2(x) follow from the bounds on A 2 (x, y):

which imply that the function n 2 ( x ) is absolutely integrable on the semi-axis 0 < x < °°, while fl|(x) is absolutely integrable on the entire axis. Thus, we have found for the transition coefficients a(k) and b(k) an expression in the form of a Fourier transform of absolutely integrable functions. In particular, it follows from the obtained representations that for large k we have for these coefficients the asymptotic

Moreover, we see that a(k) is the limiting value on the r e a l axis of a function analytic and bounded in the half-plane Im k> 0 and that the asymptotic of Eqs. (1.24) holds for all k with Im k ^ 0. We will consider the distribution of the zeroes of a(k) on the complex plane. Because of Eqs. (1.21), a(k) does not vanish on the real axis. F u r t h e r , it follows from the asymptotic of Eq. (1.24) that a(k) is also nonzero for sufficiently large |k|. It therefore follows that a(k) can have only a finite number of z e r o e s . It follows from the representation of Eq. (1.18) for a(k) by means of the Wronskian of the solu­ tions fj(x, k) and f2(x, k) that if a(k 0 ) = 0, these solutions are linearly dependent for k = k 0 , i.e.,

We note that when Imfc > 0 the solution fj(x, k) exponentially d e c r e a s e s a s x - » while the solution f2(x, k) behaves likewise as x ~ °°. When k = k 6 we may conclude on the basis of Eq. (1.25) that Eq. (1.1) has a solution that is quadratically integrable on the entire axis. The formal self-conjugacy of the equation im­ plies that this is possible only for real kjj, i.e., for purely imaginary k 0 . We have thus found that a(k) can have only a finite number of purely imaginary z e r o e s . We will prove that these z e r o e s are simple. For this purpose we obtain an expression for a(k0) = (d/dk)a(k)| k=k0We will proceed on the b a s i s of Eq. (1.1) for f t (x, k) and f2(x, k) and the equation

for fj(x, k) and f2(x, k). We obtain by the standard method the identities

On the other hand, using Eq. (1.18) we find

Suppose now that k coincides with one of the zeroes of a(k), which we again denote by k 0 . When k = k0 the Wronskians in Eqs. (1.26) taken for x = ± A vanish and the integrals in the right sides of these equations converge in limit as A —■«>. Comparing Eqs. (1.26) and (1.27) and recalling that o(k0) = 0, we find

346

134 The solutions f^x, k) and f2(x, k) for imaginary k are r e a l . The integral in the right side of Eq. (1.28) does not vanish because of Eq. (1.25), so that o(k0) * 0 and, consequently the zeroes of a(k) are simple. These z e r o e s will henceforth be denoted by in? .where 1 = 1,..., N. With this we conclude the study of the p r o p ­ erties of the transition coefficients a(k) and b(k). In concluding this section we present an expression for G r e e n ' s function of Eq. (1.1). Suppose X is a complex p a r a m e t e r and let us select a branch for y T , such that I m ] / ) 7 > 0 . The kernel

for fixed x and y is an analytic function of X on the plane with section on the positive part of the r e a l axis and with simple poles at the points X = _ x - ; - If \ does not coincide with these points and if X * 0, we have this kernel the bounds

Here R(x, y; X), as follows from Eq. (1.18), is a solution of the equation

We may prove using these facts that the integral operator R(X) with kernel R(x, y; X) is the resolvent ( H ~ XI)" 1 of the self-adjoint operator H. Moreover, we may use the properties of this kernel to define the operator H itself, which incidentally we have not done. Let us now turn our attention to the fact that the function a (V^) is in the denominator of the resolvent R(x, y; X) and has zeroes at the eigenvalues of H. In this it reminds us of the c h a r a c t e r i s t i c determinant det(H - XI). We may verify that such an interpretation of it is in fact justified. F o r example, we have the equation

where R0(X) is the resolvent of the operator H0. Subtraction by R0(X) plays the role of a r e q u i r e d regularization for the definition of det(H <- XI). 2. S c a t t e r i n g

Theory

Knowledge of the fundamental system of solutions for the Schroedinger equation (1.1) allows us to il­ l u s t r a t e in a simple way, using the operator H as an example, the general statements of scattering theory described in the introduction. We will prove how the wave operators TjW for the pair of o p e r a t o r s H and H0 defined in £ = £2(R) by the equation

can be expressed in t e r m s of appropriate solutions of the stationary Schroedinger equation (1.1). All the properties of the wave operators are subsequently obtained as simple corollaries of this relation. We begin with a description of the diagonal representation for H 0 . We consider the space £ 0 con­ sisting of p a i r s of functions

quadratically integrable on the semi-axis 0 < * . < oo and having the s c a l a r product

A diagonal representation for H0 can be realized in fc0. The corresponding isomorphism $-*.$ is p r o ­ vided by the Fourier transformation

347

135 where

The operator T0 is unitary:

The operator H0 under the isomorphism T0 is carried over into an operator for multiplication by the in­ dependent variable \ ,

We now consider two sets of solutions of the Schrbedinger equation:

For the sake of definiteness, k> 0 everywhere. The table of the asymptotics of these solutions as |x| — » has the form

The left column here being referred to x-+ — oo, and the right column to x-*- oo. The coefficients sjj(k) and iij(k) occurring in this table are expressed in the following way in terms of the transition coefficients a(k) and b(k):

These properties follow from relationships of the form of Eqs. (2.15) and (1.16) and the asymptotics of Eqs. (1.2) and (1.3) for the solutions fj(x, k) and f2(x, k). The functions Sjjfk) and s12(k) have meaning for all k * 0, since a(k) does not vanish on the real axis. We will prove that if |a(0)|= oo, the coefficients of s,, and s,2 equally have meaning up to k = 0. In this case, evidently, Su(0) = 0 and only s12(k) is to be considered. It is evident from Eq. (1.22) that |a(A)|-> oo, if

Here, as is evident from Eq. (1.23),

so that s12(k) is defined by continuity up to k = 0 and s12(0) = - 1 . We may similarly prove that in this case s21(0) = - 1 and s22(0) = 0. For larger |k|,

348

136 We can naturally continue Sjj(k) to the s e m i - a x i s k < 0 by the equation

We may easily verify based on the property of Eq. (1.21) that the m a t r i c e s

are inverses of each other, as is indicated by the notation, and are unitary matrices, so that for example

or, in more detail,

We will s e e , in particular, that for all k * 0,

and we recall that if |s 12 (0)| = 1, then

Finally, a comparison of the asymptotics of the set of solutions u> '(x, k) and uj '(x, k) implies the linear relation

where natural vector notation is used.

•

We note that the set of solutions uj (x. k) are naturally interpreted in their asymptotic in t e r m s of a radiation condition. However, we will not use this fact anywhere below. The solutions Uj (x, k) constitute a complete orthonormalized set of eigenfunctions of the continuous spectrum of the operator H. We may verify this fact by calculating the jump in the resolvent R(x, y; X) through a section in the positive part of the real axis, which corresponds to the continuous spectrum of H. We have the equation

which quite simply verifies the direct substitution of Eqs. (2.1) and (2.2) for the solutions UJ±»(JC, k) and u^±}(x, k) in t e r m s of f,(x, k) and f2(x, k) in the right side. The completeness equation which thereby fol­ lows has the form

Here the u/(x), I = 1 , . . . , N are orthonormalized eigenfunctions of the discontinuous spectrum of H. The orthogonality relation

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137

can be derived using the identity

which is t r u e for a r b i t r a r y solutions of the Schrbedinger Eq. (1.1), the asymptotic as jc[—*■ oo of the solu­ tions U J ^ J C , k), and the unitary condition on S(k). We construct using the solutions u^ix,

k) two maps T±:S}-+fy0 using the equations

The completeness and orthogonality relations are written in t e r m s of these operators as follows:

Here P is a projector into £ on the proper subspace of H spanned by its eigenvectors u;, I = 1

N.

We will now prove that the wave vectors can be introduced by the equations

F o r the proof it is sufficient to demonstrate that for any vector
vanishes in norm in ft as t->- ± oo. This is in turn easy to verify. In fact, recalling the definition of the operators T ± and T 0 , we can write the functions x^'C*. t) representing the vectors x ( ± ) ( 0 . i n t n e form

The functions 2(k2) can be considered as continuous and finite. By the R i e m a n n - L e b e s g u e l e m ­ ma the contribution to the integral

from an a r b i t r a r y finite interval |jc|
and

where [a, (i) is a finite interval on the s e m i - a x i s 0 < k < » and G(k) is a continuous function that vanishes at its endpoints. The a s s e r t i o n according to which A +\

350

A+)-+0,

(-voo; / i ^ . / J - U t ) ,

t-f-eo

138 whose proof we leave to the reader, concludes the proof with

In fact, we have not only proved the coinciding of Eq. (2.7), but have also given an independent proof for the existence of wave operators U w . Asymptotic completeness, i.e., the relation

for which there exists in the abstract theory a complicated proof, in our case immediately follows from the completeness condition on the function u(±)(x, k). The isometricity condition

which is trivially proved in the abstract theory is equivalent to the orthogonality of the functions u^'fx, k) and can be used for deriving it. Equation (2.6) can now be written in the form

where the operator S is defined by the equation

and the operator S is defined in f)0 by the matrix S(k),

Evidently, S commutes with H0,

We have thus obtained an expression for the scattering operator S in the given case. The matrix S(k) de­ fining it, which yields a representation of it in a diagonal realization of H0, is called the S-matrix. We say that s2i(k) and s12(k) are said to be the left and right reflection coefficients, respectively, and the coef­ ficient s n = s22, the transmission coefficient in accordance with the interpretation of its matrix elements sn(k) in the spirit of the radiation condition. These properties of the S-matrix allow us to reconstruct it if only one of the reflection coefficients is given. In fact, suppose Si2(k) is given. We may determine from the unitarity condition of Eq. (2.3) the modulus of the transmission coefficient

The argument of this coefficient (and thus the entire coefficient) is reconstructed in terms of its modulus based on the analyticity of the coefficient in the upper half-plane. We have the explicit formulas

The coefficient s21(k) can now be constructed on the basis of the unitarity conditions:

351

139 This procedure remains meaningful for any functions s12(k) that satisfies the conditions of Eqs. (2.4) and (2.5) and which possesses the asymptotic

The conditions of Eqs. (2.4) and (2.5) for the resulting s2i(k) also hold and for large |k| we have the asymp­ totic

We note that the analyticity of the transmission coefficient is just the additional necessary condition we spoke of in the introduction in discussing the overdeterminacy of the inverse problem. Further properties of the Fourier transform of the coefficients s n (k), s21(k), and s12(k) will be found in the next section. We stop here the description of the fundamental objects of scattering theory for the pair of operators H and H0 and pass to the inverse problem, the problem of reconstructing an operator H, i.e., the potential v(x), in terms of the matrix S(k), i.e., in fact in terms of one of the reflection coefficients. 3. V o l t e r r a Transformation

Operators

As already noted in the introduction, transformation operators which constitute the basis of the technique for solving the inverse problem are solutions of the equation

which have the structure of Volterra operators

These operators we have in fact already introduced. Indeed we will define operators Vj, i = 1, 2, operat­ ing from S) into $ 0 by the equations

Then the operators

are defined by Eqs. (3.1) and (3.2), where the kernels Aj(x, y) are defined in Sec. 1 by Eqs. (1.8) and (1.9). Let us discuss how the completeness condition of Eq. (2.8) appears in terms of the operators U] and U2. For this purpose we first calculate the normalizing factors, i.e., the operators N: \ i = 1, 2, realizing the equation

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140

As a consequence of the commutivity condition

which these operators must satisfy, they can be defined by the m a t r i c e s NJ (k), similar to how the o p e r a ­ tor S is defined by the matrix S(k). The definition of Eqs. (2.1) and (2.2) imply that

where ft, f2, i r * ' a r e the columns of the solutions f,(x, - k ) , fj(x, k), f2(x, k), f2(x, - k ) , u t tively, while the matrices M- (k) have the form

, \i\ ', r e s p e c ­

It therefore follows from the definitions of Eqs. (2.7), (3.3), and (3.4) that the normalizing factors expressed by the equations

where the operators N>

are

operate in j? 0 as matrix-multiplication o p e r a t o r s :

Comparing Eqs. (2.6) and (3.5), we also see that the m a t r i c e s Mj '(k) and M^'fk) factor the matrix S(k):

The factorization condition and the triagonal structure of the m a t r i c e s MP'flc), apparent in the explicit equations (3.6),yields a unique determination of them in t e r m s of a given matrix S(k). In fact, if the first equation of Eqs. (3.7) is rewritten in the form

we obtain a linear system of equations for determining the coefficients m u , m 1 2 , m 21 , and m 22 of the m a ­ t r i c e s M{ ' and M / ~ ' , which yields a unique solution. The second equation of Eqs. (3.7) may be similarly treated. Equations (3.6) yield the desired result. In other words a priori conditions on the structure of the normalizing factors corresponding to the transformation operators U| and U 2 uniquely determine them in terms of a given scattering operator S. The operators U i( 1 = 1,2, like the wave operators TjW, have the proper subspace H corresponding to its absolutely continuous spectrum as range of values. We will show how to expand the domain of defini­ tion of the operators by leaving the space !), so that their range of values subsequently coincides with j j . The discussion is based on the fact that the eigenfunctions of the discrete spectrum u ( (x) of H generat­ ing the defect subspace for the operators U are proportional to the solution fj(x, k) when k = i x ; . Thus,

where

141 nnd then m i 1 ' a r e normalizing

factors,

We note that Eq. (1.28) implies that

We now consider the spaces

where the finite-dimensional spaces 95, are spanned by the function ~/\ . duced by the equation

The s c a l a r product in 53, is in­

Equations (3.8) and (3.9) extend the o p e r a t o r s U] and U2 to the spaces $, range of values then coinciding with t).

and jj 2 , respectively, their

The completeness equation (2.8) in t e r m s of these extended operators then is

The weight o p e r a t o r s Wj operate in the spaces i)t and the decomposition of Eq. (3.12) reduces them. The operators Wj in the subspaces 95, are given in the bases {e ' } by the diagonal m a t r i c e s

The operators Wj in the subspace f} are given by the equation

We use Eq. (3.6) to e x p r e s s these operators in t e r m s of the matrix elements of S(k). We have on the b a s i s of the unitarity condition,

and similarly

Thus, the m a t r i c e s Wj(k) a r e expressed in t e r m s of only one of the reflection coefficients. Carrying out the F o u r i e r transformation necessary for the final calculation of the Wj, we find that they a r e expressed in the form

where the flj a r e integral operators with kernels depending on the sum of the arguments

where

354

142

where

The properties of Eqs. (2.11) and (2.12) imply that the functions n , ( x ) and £i 2 ( x ) a r e square integrable on the intervals a < x < » and -°° < x < b, respectively, for finite a and b. More detailed information on these kernels will be found in the next section. 4. G e l ' f a n d - L e v i t a n

Equations

The heuristic considerations presented in the introduction demonstrate that completeness conditions reduce to linear equations for the kernels Aj(x, y) and A2(x, y) for the transformation o p e r a t o r s Uj and U2,

These equations were first derived by Kay and Moses [40]. They superficially coincide in appearance with the equation of V. A. Marchenko [16] from the theory of the radial Schrodinger operator for I = 0. One change consists in the range of the variables becoming the entire axis. We will, however, call them the Gel'fand-Levitan equations, since their operator-theoretic content is s i m i l a r to that for the equations introduced by I. M. Gel'fand and B. M. Levitan in the theory of the inverse S t u r m - L i o u v i l l e inverse s p e c ­ tral problem. To strictly derive these equations we must investigate the operators Vf~l operating from ft; into ft or replace operator-theoretic concepts by more elementary concepts. One variant of these d i s ­ cussions conceptually s i m i l a r to [1] and carried out explicitly in [23, 26] will be set forth below. We will use the equations

which constitute a variant of Eqs. (1.15) and (1.16). We omit here the index (+) in uj + '(x, k), since the func­ tions u'~'(x, k) will no longer be used. We know that the functions f,(x, k) and f2(x, k) are analytic and bounded in the upper half-plane and are bounded there

where o(.-r-)

in general depends on x. The functions u,(x, k) and u 2 (x, k) a r e also analytic in the upper

half-plane except at the points k = ix.j, I = 1 N, where they have together with l/o(k) simple poles. The corresponding residues are simply associated with the values of the functions f,(x, k) and f2(x, k) at these points. For example

where ml 1 ' is defined in Eq. (3.10). Similarly,

At large |k| the functions Uj(x, k) have the asymptotic

355

143

where 0(j-.--.) may also depend on x nonuniformly. Functions with only the same number occur in each row in Eqs. (4.3) and (4.4) and both equations are completely identical to within the substitutions W 2 and elkx
Based on the analyticity properties of u,(x, k) described above,

By the convolution theorem, Eq. (4.3) takes the form, following a F o u r i e r transformation,

and when x < y we a r r i v e at Eq. (4.1) on the basis of Eq. (4.9). Equation (4.2) is similarly derived. The discussions we have presented on the b a s i s of Eqs. (4.3) and (4.4) together with analyticity-type conditions occurring in them for functions of the form of E q s . (4.1) and (4.2) can be r e v e r s e d . More p r e ­ cisely, suppose A,(x, y) is a solution of Eq. (4.1), such that the function f,(x, y) analytic for Im k > 0 con­ structed in t e r m s of it by Eq. (1.8) satisfies the condition of Eq. (4.5). We consider the kernel B](x, y) defined by Eq. (4.10) and construct using it a function u ^ x , k) by Eq. (4.8). Carrying out a Fourier t r a n s ­ formation, we find that f((x, k) and u,(x, k) a r e related by an equation of the form Eq. (4.3), so that, in p a r ­ ticular Uj(x, k) when Imk = 0 satisfies the condition of Eq. (4.7), Equation (4.1) implies that Eq. (4.9) is true for B](x, y) when x < y , so that Uj(x, k) has an analytic continuation into the upper half-plane Imk> 0 with poles at the points i x j , while Eq. (4.6) holds for the corresponding r e s i d u e s . The proof of the equiv­ alence concludes with this fact. In the next section we will study the solvability of the Gel'fand—Levitan equation and will formulate more precisely the corresponding assertion for the existence and uniqueness of a pair of functions u(x, k) and f(x, k) that satisfy such analyticity conditions and a r e related by an equa­ tion of the type of Eq. (4.3). In concluding this section we will use the Gel'fand—Levitan equation to refine the properties of the reflection coefficients, that i s , we will study m o r e precisely the behavior of the functions F^t) and F 2 (t), about which we so far only know that they are square integrable. We consider for the sake of definiteness the function F](t). We rewrite Eq. (4.1), setting x = y:

and consider this equation as that for Q](2y). This is a Volterra-type equation and the method of successive approximations always converges for it. We obtain based on the bound of Eq. (1.12) for the kernel A,(x, y) a bound for Q ^ x ) ,

356

144

Here and below we denote by C(x) a monotonically nondecreasing function bounded as x — « and, in general, increasing as x ——«. We conclude as a consequence of the differentiability of Aj(x, y) that ft,(x) is also differentiable and using Eq. (4.11), we find the bound

We can similarly prove using Eq. (4.2) the differentiability of fi2(x) and find the bounds

Here and below the function D(x) is a monotonically nondecreasing function bounded as x — - » and i n c r e a s ­ ing, in general, as x — <*>, The resulting estimates and the properties (P) of the potential imply that

The functions F^x) and F2(x) differ from ft,(x) and £2,(x) by a continuous t e r m with d e c r e a s e s as x — °° and x — - « , respectively. Consequently, inequalities of the type of Eqs. (4.12) and (4.13) are true also for F^x) and F 2 (x). We will see that the functions ^^M-*) and jj^aC*) behave s i m i l a r to v(x) as x->oo and x-+— oo, respectively. If v(x) is differentiable, it can be proved using Eqs. (1.10), (1.11), and (4.11) that this analogy extends also to the succeeding derivatives of fi,(x) and fi2(x). 5. I n v e s t i g a t i o n

of I n v e r s e

Problem

In the preceding sections we explained how the scattering matrix S(k) corresponding to a potential v(x) satisfying property (P) possesses the properties: 1. Unitarity:

2. Realness:

3. Symmetry:

4. Asymptotic behavior:

and the Fourier transforms F,(x) and F2(x) of the coefficients s12(k) and s 21 (k) satisfy the condition

357

145 5. Analyticity: the function Sn(k) is the limiting value of a function analytic in the half-plane Imk> 0, having there the asymptotic 1

+O(.TT)

and a finite number of poles on the imaginary axis.

It is possible that some of these p r o p e r t i e s are consequences of o t h e r s , but we will not bother our­ selves about this m a t t e r . In the current section we will prove that these necessary properties are also sufficient conditions under which a potential v(x) satisfying (P) corresponds to such a matrix S(k). Here we must specify N m o r e positive numbers, where N is the number of poles of s n ( k ) , for a unique defini­ tion of v(x), in addition to S(k). This result was formulated in [2.3] and proved in detail in [26]. The proof of this assertion will be found using the investigation of Gel'fand— Levitan-type equations, which we now begin to d e s c r i b e . We begin with an investigation of the solvability of Eqs. (4.1) and (4.2). Suppose we are given 1) The function

such that

and

2) distinct a r b i t r a r y positive numbers >tj, 7 = 1

N.

3) the same number of positive numbers m y ' , I = 1 , . . . , N. We construct using these data the func­ tion J2](x) in t e r m s of Eq. (3.14) and consider the equation

as the equation for A](x, y). This is an equation in t e r m s of the second independent variable of this func­ tion, where x occurs only as a p a r a m e t e r . Setting

we r e w r i t e Eq. (4.1) in the form of the operator equation

The free term w x (y) is absolutely integrable and bounded, and, consequently, is also a quadratically integrable function on the interval x<.y< ° ° , i.e., >or(y)g£,>(jc, oo). We will find a solution also from L 12 (x, <*>) and prove that it exists and is unique for any x, — oo < x < oo. For this purpose we first verify that we are dealing with an equation possessing a completely con­ tinuous operator in Lj(x, °°) and L 2 (x, °°). Suppose

358

146 The function TJ^X) on the b a s i s of Eq. (4.12) is absolutely integrable on the intervals [a, «=) for any a > — oo and we have the inequalities

Using the bound

we find that

i.e., the operator fix is a Hilbert-Schmidt type operator and its norm approaches zero a s x - « . The com­ plete continuousness of Qx in Li(x, °°) is also a well-known corollary of the absolute integrability of T)I(X). We now note that the operator I + Qx is positively defined for any x. In fact it is obtained by the restriction of the positive operator W from Sec. 3 to L 2 (x, °°). In particular, this implies that the homogeneous equation

has no nontrivial solutions in L 2 (x, ■»). We now prove that every solution of Eq. (5.2) in L,(x, °°) also b e ­ longs to L 2 (x, °°). We have

so that h[(y) is quadratically integrable on the interval [x, °°). We conclude that Eq. (5.2) has no nontrivial solutions in L t (x, « ) , so that Eq. (5.1) is uniquely solvable in Lj(x, °°) for any x. We will now consider how the bounds for the solution A t (x, y) follow from this fact. The operator (I + fix)-1 is uniformly bounded for all x from the interval [a, <*>), since the norm of fix approaches zero as x — •*>

so that

Substituting this bound in the integral occurring in Eq. (4.1), we find that

Using Eq. (4.1) we may also verify that the solution A,(x, y) is once differentiable, and we may find bounds on the derivatives. Let us estimate the function -^Ax(x,y). Differentiating Eq. (4.1) with respect to x, we a r r i v e at the equation for bx(y) = ^— A\(x, y) + ^— 2,(x + y) of

Here the free t e r m

359

147 has the bound

We therefore find for the solution b^y) that

We may similarly estimate the derivative j - At(x, y) and the result is given by

The integral equation (4.2) is similarly investigated. If

This equation is uniquely solvable and we have for the solution the bounds

where

If the functions F,(x) and F2(x) have more than one derivative, the kernels A,(x, y) and A2(x, y) are also multiply differentiable. Bounds on the corresponding derivatives are found in a way similar to what was done for b ^ y ) . Returning to the pair of analytic functions u(x, k) and f(x, k), we verify that if we are given a func­ tion s(k) satisfying the conditions repeatedly formulated for s12(k) and N unequal positive numbers x/, I = 1 , . . . , N, and further N positive numbers m^, there exists a unique pair of functions u(x, k) and f(x, k), such that 1) the function f(x, k) and u(x, k) are analytically continued in the upper half-plane l i n k s 0, where f(x, k)e~il°c is bounded for all k, Im k s 0, and u(x, k) has simple poles at the given points k = ix/, 1 * 1 , . . . . N; 2) the residues u(x, k) are connected to the values of f(x, k) when k = ixj by the equation

3) on the real axis by

4) for large |k| by

5) for real k we have the equation

360

148 We now construct using the solutions for the Gel'fand-Levitan equations A,(x, y) and A 2 (x, y) found, operators Uj and U2 using Eqs. (3.1) and (3.2) and consider the operators

We c a r r y out the investigation of these operators at the formal level, without going too deeply into justifica­ tions. A rigorous justification of the results obtained here can be derived m o r e simply by following the quite elementary, but laborious discussions of [1], We first prove that the operators U\ are self-adjoint. For the proof we note that the G e l ' f a n d - L e v i ­ tan equations derived from the completeness equation (3.13) a r e in fact equivalent to it, i.e., in other words the operators U( and U2 obtained by us satisfy this equation. We consider for the sake of definiteness the case i = 1. Suppose an operator A, has kernel A,(x, y), where

We construct the Volterra operator

The Gel'fand-Levitan equation can now be written in the form

The operator

is also self-adjoint since Wj is self-adjoint. But it is simultaneously a Volterra operator, since the opera­ t o r s Ui and Uj are Volterra with identical Volterra direction. These two properties a r e consistent only

which implies that U] satisfies the completeness equation

Using this equation we can rewrite the definition of the operator Hj in the form

which implies that Hj is self-adjoint since H0 and Wt commute. The case i = 2 is similarly considered. We now prove that the Hj a r e represented in the form

where the Vj a r e for multiplication by the functions

For this purpose, assuming again for the sake of definiteness that i = 1, we r e w r i t e Eq. (5.9) in the form

361

149 and take into account that the operator l', = I + A| is Voltcrra, so that

where 0(x) is the Heaviside function.

Equation (5.11) is rewritten in the form

where we set Vt = Ht — H0 and assume Vj to be an integral operator whose kernel can be a generalized func­ tion. The resulting equation is consistent with Vt being self-adjoint and U( being a Volterra only if

Moreover, we will see from this fact that A,(x, y) satisfies the partial differential equation

which, incidentally, we will not use. The operator H2 is analogously investigated. Our result is that the Hj a r e differential operators of the form

where the functions VJ(X) are given by Eqs. (5.10). Equation (5.11) also implies that the functions f,(x, k) and f2(x, k) constructed in t e r m s of the kernels of Aj(x, y) and A 2 (x, y) by means of Eqs. (1.8) and (1.9) are solutions of the differential equations

and have the asymptotics of Eqs. (1.2) and (1.3). Finally, the bounds of Eqs. (5.4), (5.5), (5.7), and (5.8) demonstrate that v t (x) and v2(x) satisfy bounds of the form

Moreover, if Fj(x) and F2(x) a r e n times differentiable, the potentials V](x) and v2(x) have n - 1 derivatives. Here v[ m J and v | m ' as x — » a n d as x « , respectively, behave as F [ m + 1 I and F J m + 1 ] . We conclude the study of the inverse problem by proving if fij(x) and fi2(x) are consistent, i.e., F t (x) and F 2 (x) a r e constructed in t e r m s of the given matrix S(k) satisfying the necessary properties given at the beginning of the section and m j 1 ' and mi 2 ' a r e related by Eq. (3.11), f,(x, k) and f,(x, k) satisfy the equations

It therefore follows that v t (x) and v2(x) coincide, and we obtain from the bounds of Eqs. (5.12) that

together with the corresponding refinements on differentiability in the case of the differentiability of F((x) and F 2 (x).

362

150 Wc have thereby found that the operator || = lln + V constructed by us belongs to the initial class of the Schrbedinger operators. Further, the equations (5.13) imply that H has the set of eigenfunctions u[ + ^(x, k) with asymptotic formulated in the table in Sec. 2, this asymptotic containing as the coefficient S]j(k) the matrix elements of the initial scattering matrix S(k). The construction of the operator H and thereby the proof of the sufficiency of the properties of the scattering matrix stated at the beginning of this section concludes with this fact. To prove Eqs. (5.13) we use the uniqueness theorem obtained above for the pair of functions u(x, k) and f(x, k), proving that the functions Uj(x, k) and fi(x, k) constructed using the Gel'fand-Levitan equations satisfy the equations

Equations (5.13) a r e thereby derived by using Eqs. (4.3) and (4.4). Suppose u 2 (x, k) and f2(x, k) a r e obtained using the equation

and the analyticity properties formulated above. We multiply Eq. (5.14) by s , , ( - k ) .

We obtain

The second t e r m in the left side is again replaced on the basis of Eq. (5.14),

In view of the unitarity condition of Eq. (2.3), the latter equation is rewritten in the form

We introduce the functions

The function u(x, k) is analytic everywhere in the upper half-plane except for the points k = i x j , where together with s,,(k) it has simple poles. The function f(x, k) has no singularities at k = ix?, since the singularities of u 2 (x, k) and s n ( k ) compensate each other. If s u ( 0 ) = 0, f(x, k) thereby lacks a singulari­ ty at k = 0. In fact if s,,(0) = 0, s12(0) = - 1 , and using Eq. (5.14) we find that u 2 (x, 0) = 0, so that the ratio u 2 / s n lacks a singularity at k = 0. Evidently, for real k,

We easily verify that the residues u(x, k) are related to f(x, k) by the equation

We need only use the condition of Eq. (3.11). Finally, Eq. (5.15) has the form in t e r m s of f(x, k) and u(x, k)

Based on the uniqueness of this pair of functions as formulated above, we conclude that

which implies, by Eq. (5.16), Eq. (5.13).

363

We conclude with this fact the general study of the inverse problem for the one-dimensional Schroedinger operator. 6.

Particular

Cases

of t h e

Solution

of t h e

Inverse

Problem

Here we will consider two examples where the inverse problem has an explicit solution: 1. Absence of reflection, i.e., the coefficients s 12 (k) and s 2 i(k) identically vanish and the entire nontrivial contribution to the G e l ' f a n d - L e v i t a n equation is provided by the discrete spectrum of H. 2. Rational reflection coefficient. Both examples a r e combined through the common property of the kernel Q(x + y) in the Gel'fandLevitan equation; it becomes degenerate and the solution reduces to q u a d r a t u r e s . However, we will con­ sider these examples separately. We will assume in the second example, in order to simplify the equations, that the discrete spectrum of H is absent. This will be sufficient for the reader himself to analyze, by combining methods for the first and second examples, which alternations to the equations arise in the general case. These examples make it possible to solve the inverse problem for a dense set of scattering data. Thus, let us consider the Gel'fand-Levitan equation (4.1) and assume that s12(k) - 0, so that the kernel ftj(x + y) has the form

The solution Aj(x, y) in this case is naturally found in the form

an algebraic system of equations naturally written in vector notation

arising for the function g/(x). Here g(x) is the desired column vector with components gj(x), I = 1 g0(x) is a column of the functions mj 1 '* - *'*, I = 1 , . . . , N and W,(x) is a matrix with elements

N,

The solvability of the resulting system is guaranteed by the general r e s u l t s of the preceding section and, solving it, it is possible to find gj(x) and, together with them, the k e r n e l A,(x, y). In particular, it can be easily verified that an expression is obtained for A](x, x) which in our notation can be written as follows:

We thereby find an expression for the desired potential,

We can s i m i l a r l y consider Eq. (4.2). We obtain for the potential v(x) the equation

where W2(x) is a matrix with elements

364

152 We note that the transmission coefficient in our case has the form

so that

and we recall that the constants m} 1 and m j 2 ' a r e related by the equation

It follows from the general considerations of Sec. 5 that Eqs. (6.1) and (6.2) for v(x) coincide under these conditions. Direct verification of this identity constitutes a nontrivial combinatorial problem. We now pass to the second example. Suppose s12(k) is a rational function of the variable k,

where P m (k) and Q n (k) are polynomials of degree m and n, m < n, having the identity element as coefficient for the highest degree k, and r is a constant. The r e a l n e s s condition

will hold if

where r 0 is a real number and the zeroes of the polynomials P m (k) and Q n (k) are located symmetrically about the imaginary axis. The constant r 0 must be sufficiently small in order that

For this purpose it is necessary that all the zeroes of Q n (k) have nonvanishing imaginary part. F o r s i m ­ plicity we will assume that all these z e r o e s are simple. The case of multiple zeroes can be considered by the corresponding passage to the limit. The procedure described in Sec. 3 for reconstructing the transmission coefficient s n ( k ) in terms of S]2(k) can be explicitly performed. We recall that we have assumed that the discrete spectrum of H is absent, so that s n ( k ) has no poles in the upper half-plane. The explicit equation for Sn(k) has the form

where a<+> and a}-> are the zeroes of Q n (k) in the upper and lower half-plane, respectively, and these a r e roots of the equation

in the upper half-plane. There are precisely n such r o o t s , since the equation is invariant relative to the substitution k k. Using Eq. (2.10), we can see that the second reflection coefficient s 2I (k) is also a r a ­ tional function and is represented in the form

365

153 where the polynomials P m ((k) and Qn((k) and tne constant r< possess properties similar to that for P m , Q n , and r . We will also assume that all the zeroes of Qn, are simple. The Fourier transforms Ft(x) and Tz(x) of the reflection coefficients s,2(k) and sa,(k) are calculated using the Jordan lemma. In particular, Fiix)-j}p,e'a}+)x,

*>0;

»_ (-1

where the sums are taken over the zeroes of Qn(k) and Qn-(k) in the upper half-plane. The coefficients p, and p] coincide to within a factor i with the residues Sj2(k) and su(k) at the poles located at these zeroes. We will see that the kernels Ft(x + y) and F2(x + y) of the Gel'fand-Levitan equations (4.1) and (4.2) are degenerate when x > 0 and x < 0, respectively. We can use a method in these regions for solving them already mentioned in the investigation of the first example. As a result we find the expression for the desired potential *(jc)=-2^1ndet(/+Z1(jc)), *>0;

{$3)

^W=-2^1ndeH/+Z s W), x<0,

<6>4)

where the matrices Zj and Z2 have as matrix elements the expressions ipi

7<"._,

-'fa<+>+ai+>R

The potential must be a continuous function if s12(k) is to sufficiently decrease as | * | - «.. In particular, when m
ln

M*)=-2^111 M*)

<6-5)

in terms of the Fredholm determinants At(x) and A^x) of the Gel'fand-Levitan equations (4.1) and (4.2). The appearance of the finite-dimensional matrices W(x) and Z(x) in the examples we are considering is explained by the degeneracy of the corresponding kernels in these equations. We present a brief and formal derivation of Eq. (6.5). The rigorously justified method is too long to be presented here. We will prove that A ( * . * ) - £ l n A, (*>; A>(x. *) = - £ l n A , ( * k after which Eq. (6.3) is implied by Eq. (5.10). Thus, suppose A,(,v) = det(/ + 2 J ).

where / + Q, was introduced in Sec. 5 and has the form PXW,PX, where Px is a projector into L2(x, - ) . PAW-Hy-*)$(?)■

366

154 We now note that

where T r is defined on the left in the space L 2 (x, » ) , and on the right, in L 2 (R). The operator P x t2, = Q x is an integral operator in L2(R) with kernel

The last equation implies that

Here we have introduced the resolvent r x of the operator Q x , i.e., an integral operator whose kernel satisfies the equation

Comparing this equation to the Gel'fand-Levitan equation (4.1) we may verify that

The t r a c e desired by us in this notation is expressed by

which also proves the first equation in Eqs. (6.5). The second equation is proved entirely analogously. CHAPTER SIMPLE GENERALIZATIONS

2 AND

APPLICATIONS

The technical apparatus described in the first chapter is carried over without significant variations to a number of one-dimensional problems in scattering theory. In this section we will consider s e v e r a l examples, limiting ourselves basically to only the formulation of the r e s u l t s , generalizing or modifying the assertions stated in Chap. 1. Moreover, we will consider in Sec. 4 an application of developed methods from scattering theory for solving nonlinear equations in the theory of one-dimensional continuous media. 1. P o t e n t i a l s

wi t h

Distinct

Asymptotics

at

Infinity

Here we will consider two examples of the Schroedinger operator

where the potential v(x), xGR, behaves differently as x-+ — oo

versus

JC->-OO:

Example 1.

was considered by V. S. Buslaev and V. L. Fomin [6], Example 2.

was studied by P. P. Kulish [14].

367

155 We will not exceed the limits of elementary stationary scattering theory. All the treatments can be embedded in an abstract scheme of scattering theory, but not very instructively. We will also not present any proofs, referring instead to the original works. Let us pass to a description of the first example. We assume that

Suppose k] = yk2 — c2 is defined so that Im k}>0 when lm£>0. The solutions f,(x, k) and f,(x, k) of the Schrdedinger equation

are distinguished by the asymptotic conditions

and are analytic functions for fixed x of the parameters k and k, in the upper half-plane. We have the in­ tegral representations

bounds similar to that presented in Sec. 1 Chap. 1 holding for the kernels A|(x, y) and A2(x, y). The solutions U](x, k) and u2(x, k) of Eq. (1.2) are uniquely determined by the radiation condition and have the form

The coefficients Sjj(k) determining the scattering matrix possess the following properties: 1. Unitarity:

2. Symmetry:

3. Analyticity: the coefficient s,,(k) is the limiting value of a function analytic in the upper half-plane and having there simple poles on the imaginary axis at the points k = ixj with residues

368

156

4. Behavior as

\k\->-oo: there exist Fourier transformations

such that

We shall b e a r in mind the modification of the unitarity and symmetry conditions. These properties reconstruct the entire matrix S(k) in t e r m s of the reflection coefficient s 12 (k). At the same time knowledge of s 21 at k> c is not sufficient for this purpose. The Gel'fand-Levitanequation for the kernel A,(x, y) is unchanged:

Here

where the function F)(x) has already been introduced, while

The Gel'fand-Levitan equation for the kernel A 2 (x, y),

contains a new term

Here the m j " are related to the m j 1 ' by the equation

An investigation of these equations and a proof for the necessity and sufficiency of these properties of S(k) corresponding to the potential v(x) satisfying the condition of Eq. (1.1) can be presented analogous to what was done in Chap. 1. We refer the reader to [6] for details. Let us now pass to a description of the second example. In this case the Schrbedinger equation (1.2) has a solution u(x, k) which p o s s e s s e s the asymptotic

369

157

where the reflection coefficient s(k) satisfies the unitarity condition |s(k)| = 1. At large k, s(k) rapidly o s c i l l a t e s . The inverse problem consists in reconstructing v(x) in t e r m s of given s(k). Let us refine the conditions on v(x) for which the r e s u l t s formulated below are t r u e . The nature of the i n c r e a s e in v(x) as x — —<*> is difficult to e x p r e s s explicitly in t e r m s of the asymptotic behavior of lns(k) as k —°°. It i s , however, possible to expand the c l a s s of potentials v(x) so that the necessary and sufficient p r o p e r t i e s of the corresponding reflection coefficients can be described. We will assume that for finite a,

and the spectrum of the Sturm— Liouville problem

for some x0 is semi-bounded from below and discrete. We will also assume that operator H does not have a discrete spectrum. The existence of the solution f(x, k), such that

its analytic p r o p e r t i e s in t e r m s of a p a r a m e t e r k, and, in p a r t i c u l a r , the integral representation

have already been proved by methods known to u s . The coefficient s(k) is uniquely determined by the fact that

is square integrable in a neighborhood of x = - °°. The properties of s(k) are as follows. 1. The F o u r i e r transformation

defined as a generalized function, coincides with an absolutely continuous function, such that

2. s(k) can be represented in the form

where |s 0 (A)| = l when ImA:=0 and s 0 (k) has meromorphic continuation onto the entire complex plane of the variable k, while when l m * > 0 ,

370

158 ?>. The function u(x, k) = f ( x , - k ) + f(x, k)a(k) will have an analytic continuation in the upper haltplane, such that the function p(X) defined by the equation

is the spectral function of the S t u r m - L i o u v i l l e problem on the s e m i - a x i s ( - » , x„) with boundary condi­ tion j - ty(x)=0 when x = x 0 . We note that the statement of the latter condition includes, in addition to s(k), the solution f(x, k) of the Schroedinger equation (1.2), so that this property at first glance is not formulated solely in t e r m s of s(k). In fact, the situation is not that bad. It is possible to solve a G e r f a n d - L e v i t a n - M a r c h e n k o equation for every function s(k) satisfying the first and second conditions and also the unitary condition

at x = x 0 and to find the solution f(x, k) and its derivative at x = x 0 . We next verify the third condition. Re­ construction of the potential v(x) at x < x0 must be carried out using the Gel'fand-Levitan procedure for solving the inverse Sturm-Liouville problem with discrete spectrum. Here we conclude the description of the second example and refer the reader for further details to [14]. 2. C a n o n i c a l

System

Many methods developed for the one-dimensional Schroedinger operator are naturally c a r r i e d over (and sometimes even simplified) to the differential operator of a canonical linear system, which we write in m a t r i x notation

Here J is a simplectic matrix and Q is a real symmetric matrix funciton with zero trace

The technical tools for studying the spectral properties of such an operator were developed by M. G. Krein and his students and also by M. G. Gasymov and B. M. Levitan. Corresponding r e f e r e n c e s can be found in [13, 7]. In both these works H is considered on the s e m i - a x i s 0 < x < ° ° . in this section we will present the fundamental equations and r e s u l t s of the stationary scattering theory and inverse problem for an operator H on the entire axis, following the thesis of Takhtadzhyan [20], We consider the system of differential equations

which plays in our case the role of the Schroedinger equation. The system (2.1) has solutions f,(x, k) and f2(x, k) which a r e column vectors and have the asymptotic

To prove the existence of these solutions it is sufficient to assume that the coefficients p(x) and q(x) of Q(x) a r e absolutely integrable functions,

371

159 The components fj''(x, k) and f>2'(x, k) of fj(x, k) and f,(x, k) for fixed x hnve an analytic continuation in the upper half-plane of the p a r a m e t e r k. For large k

For r e a l k the p a i r s of solutions f^x, k), f,(x, k) and f,(x, k), f2(x, k) form two fundamental s y s t e m s of solutions of Eq. (2.1). The transition coefficients a(k) and b(k) are introduced by the equations

These coefficients satisfy the identity

and can be expressed in t e r m s of fj(x, k) and f2(x, k) by means of the equation

Here {f, g}, an analog of the Wronskian, is defined as a bilinear form of the matrix

We will see that o(k) has an analytic continuation on the upper half-plane and is bounded there as k — °°,

The function a(k) does not have z e r o e s in the upper half-plane since such zeroes would correspond to com­ plex eigenvalues of the formally self-adjoint system (2.1). Knowledge of the fundamental system of solutions of the system of equations allows us to introduce solutions u(x, k) satisfying the radiation condition and to develop scattering theory similar to what was done in Sec. 2 Chap. 1 for the example of the Schrdedinger equation with decreasing potential. We will not carry this out here, limiting ourselves to stating that the matrix S(k) is determined by the transition coefficients a(k) and b(k) by means of equations that coincide letter for letter with those presented here. The sole difference is that the parameter k now runs through the entire real axis. In particular, the entire matrix S(k) can be reconstructed in terms of one of the reflection coefficients

We have for the solutions f^x, k) and f2(x, k) the integral representations

Here the column vector f0(x, k) has the form

and the kernels A,(x, y) and A 2 (x, y) a r e real matrix functions absolutely integrable with respect to y for fixed x. The kernels A,(x, y) and A 2 (x, y) can be used to construct the Volterra transformation o p e r a t o r s and, in p a r t i c u l a r , to express in t e r m s of them the completeness condition. We will not carry this out h e r e and p r e s e n t only equations expressing Q(x) in t e r m s of these kernels

372

160

and a formulation of one of the Gel'fand-Levitan equations

Here the matrix function F^x) is introduced by the equation

It is possible to prove, based on a study of this equation and its analog for the kernel A 2 (x, y), that the properties of S(k), viz., its unitarity and symmetry, the analyticity of the t r a n s m i s s i o n coefficient, and the absolute integrability of the Fourier transformations F,(x) and F,(x) of the reflection coefficients s 12 (k) and s21(k)

a r e necessary and sufficient properties of the scattering matrix for a canonical system with matrix Q(x) satisfying the condition of Eq. (2.2). We will now show how the canonical system is a generalization of the Schroedinger equation. If the matrix Q(x) has the form

the system of Eqs. (2.1) is equivalent to the Schroedinger equation with operator H of the form

which is evidently positively defined. That the equation

which also in fact reduces the range of variation of the p a r a m e t e r k to the s e m i - a x i s , is r e a l constitutes a necessary and sufficient condition on the scattering data corresponding to the matrix Q(x) of the form of Eq. (2.3). We note in conclusion that the case of the system (2.1), when the matrix Q(x) has the form

where m is a positive constant (system of Dirac equations with m a s s ) , was recently considered by I. S. Frolov [30]. 3. T r a c e

Formula

In this section we will derive identities relating the momenta of the function ln| a(k)|, where a(k) is one of the transition coefficients, with the integral of polynomials formed by derivatives of the potentials in the one-dimensional Schroedinger operator or in the operator of the canonical system. These equations first appeared apparently in [5], which in turn developed a paper by I. M. Gel'fand and V. M. Levitan [9]. In these works it was shown why such identities can naturally be called t r a c e formulas. The derivation which we present will not use general o p e r a t o r - t h e o r e t i c concepts and is taken from [10], We first consider the case of the Schroedinger operator. We will assume that the potential v(x) is a Schwartz-type function. Then the reflection coefficients s,,(k) and s 21 (k) d e c r e a s e as |*|--<» m o r e rapidly than \k\~n for any n > 0. The equation 373

161

which constitutes one variant of Eq. (1.2.9) implies that In a(k) has a decomposition in negative degrees of k,

where

Even d e g r e e s vanish as a consequence of the evenness of the integrand ln|a(k)|. Bearing in mind the inter­ pretation of a(S\) as a regularized c h a r a c t e r i s t i c determinant of H, we can say that the C 2 J + 1 are propor2;+'

J

tional to regularized t r a c e s of the half-integer d e g r e e s H 2 of H, while such t r a c e s vanish for inte­ gral degrees. We now calculate the coefficients c n directly in t e r m s of v(x). This calculation can be interpreted as the definition of the matrix t r a c e of the o p e r a t o r s H 2 . Identities which are obtained by the setting equal of the thus obtained expressions for c n are called the trace formulas. This interpretation will not be evident in the elementary calculations which follow. We consider the function

It can be proved that this function is analytic in k when I m k > 0 and sufficiently large |k| for any fixed x. F o r l a r g e |x| it has the asymptotics

so that

where

The Schroedinger equation implies thatCT(X,k) satisfies the Riccati equation

which can be used to determine the asymptotic decomposition

We find for the coefficients c n (x) the recursion relations

374

162 The first few coefficients have the form

We will see that a 2 (x) and CT4(X) a r e total derivatives. This property is also preserved for all CTR(X) for even n. Returning to In a(k) we will verify that the coefficients c n in the decomposition of Eq. (3.1) are written in the form

so that, for example

We have thus found the set of identites

called the trace formulas. Their interpretation in terms of the traces of the half-integer degrees of H is particularly evident due to the presence in the right side of the sum of half-integer degrees of its discrete eigenvalues. The differential operator of the canonical system is considered analogously. We will use as the function o(x, k) following [11], the expression

It can be proved that

The canonical system (2.1) for CT(X, k) implies the equation

which can be used for the asymptotic decomposition of the type of Eq. (3.2). Here the first coefficients <7 n (x) have the form

As a result an asymptotic decomposition of the form of Eq. (3.1) is obtained for In a(k), where all the co­ efficients c n are in general nonzero, the two different methods for calculating them leading to the identi­ ties

375

163

These formulas can be interpreted in t e r m s of regularized t r a c e s of the degrees of H of the canonical s y s ­ tem. If the condition for infinite differentiability of v(x) or of p(x) and q(x), which we have used in deriving the identities, does not hold, the number of true t r a c e formulas is determined by the number of the contiuous d e r i v a t i v e s . Here the absolute convergence of one integral in these identities guarantees the con­ vergence of the second integral. Thus, the t r a c e formulas a r e an indirect means for obtaining information on the bahavior of scattering data in t e r m s of properties of the potential. Identities s i m i l a r to those presented here for the radial Schroedinger operators and their relativistic analogs were also used in a number of works [31-33] by Italian physicists for m o r e meaningful assertions on the inverse problem. 4. Nonlinear

Evolutionary

Equations

The work of Kruskal et al. [42] and the succeeding work of Lax [44] opened a new range of applica­ tion of scattering theory. That is, it turned out that it is possible to describe using the scattering p r o b ­ lem the general solution of certain nonlinear evolutionary equations with a single spatial variable. Here we will d e s c r i b e two c h a r a c t e r i s t i c examples of such applications.

1. Korteweg-de Vries equation

2. Nonlinear Schroedinger equation

In the first example v = v(x, t) is a real function while w(x, t) in the second example is complex, w(x, t) = p(x, t) + iq(x, t). We will assume that they rapidly d e c r e a s e for large |x| and fixed t. We will prove that both equations constitute an equation of motion for an infinite-dimensional Hamiltonian system. We will recall that the definition of such systems includes the simplectic manifold (M, Q), where M is a differentiate even-dimensional manifold and Q is a closed nondegenerate 2-form on it, and the distinguished function (Hamiltonian) is h:M -> R. The trajectories of the Hamiltonian system are given by the differential equations

where $ i s a point of the manifold, | is a tangential vector to the trajectory at the point £, and J is the map of the 1-form in the vector fields defined by the form Q. (cf., for example, [2]). We now note that Eq. (4.1) can be written in the form

where the functional h[v] has the form

Comparing Eqs. (4.3) and (4.4) we see that the latter equation in fact is of Hamiltonian form, where h[v] plays the role of the Hamiltonian, while the antisymmetric operator J = 3/9x generates a simplectic struc­ ture in the set of functions v(x). The corresponding 2-form, which we write as a bilinear form of the vari­ ations 6{v and 62v of v(x) is determined by an operator inverse to J and having the form

376

164

The similar objects for Eq. (4.2) have the form

and

correspondingly. We see that the simplectic structure in the space of complex functions w(x) is induced by the natural complex structure of the real space of functions p(x) and q(x), where w = p + iq. The scattering problem is used to describe substitution of variables in Eqs. (4.1) and (4.2), under which they become explicitly solvable. We describe the corresponding scheme first for the example of Korteweg-de Vries equation. We consider the function v(x) as a potential in the Schrbedinger operator H studied in the preceding chapter. Suppose (S[2(k), KI, m/1*) a r e the corresponding scattering data which in turn uniquely define v(x). It turns out that the functional h[v] and the form Q cannot be expressed explicitly in t e r m s of the scattering data and their variations. The corresponding formulas have the form

and

where

the conversion factors a(k) and b(k) being constructed in t e r m s of the initial scattering data. We will see that the new variables are explicitly canonical, where P(k), p;, [ = 1 N play the role of canonical momenta, while Q(k), q/, Z = 1 , . . . , N play the role of canonical coordinates. Moreover, the Hamiltonian h[v] turns out to be a function only of the momenta, so that the Hamiltonians of the equa­ tion in new variables is trivially solved. Turning to the scattering data, the corresponding solution can be written in the form

These facts also make it possible to solve the Korteweg-de Vries equation using an auxiliary scat­ tering problem. Suppose v(x, 0) = v(x) is a Schwartz-type function defining the Cauchy data for this equati~" The sequence of maps

defines the corresponding solution. Here the first left a r r o w denotes the solution of the direct scattering problem for the Schroedinger equation with potential v(x) = v(x, 0), the next a r r o w is defined in Eq. (4.9), and finally, the last a r r o w constitutes the solution of the inverse problem.

377

165 It is precisely this proposition which is presented in the first work of Kruskal et al. [42]. The Hamiltonian interpretation presented here was obtained in [10]. Analogous formulas

and

where

can be found for the nonlinear equation of the second example. Here o(k) and b(k) are the conversion fac­ t o r s for the canonical system (2.1) constructed in t e r m s of the functions p(x) = Rew(x) and q(x) = Im w(x). We note that no factor k was present in the definition of the momentum P(k) and that the range of variation of the variable k is now the entire axis. These formulas were found in the thesis of L. A. Takhtadzhyan [20]. They imply that the general solution of the boundary-value problem for the equation is provided for by the sequence of maps

where

The s i m i l a r equation

(bearing in mind the opposite sign in front of the nonlinearity) was previously considered by V. E. Zakharov and A. B. Shabat [11]. The corresponding scattering problem is non-self-adjoint and has yet to be mathe­ matically investigated. Let us now dwell briefly on the derivation of Eq. (4.6), (4.7), (4.10), and (4.11). We consider only the Korteweg-de Vries equation, since the second example is considered analogously. Equation (4.6) has already been derived by us in the preceding section and constitutes the identity for the third t r a c e . In fact the coefficient c 5 is proportional to the functional h[v]. The right side of Eq. (4.6) a r i s e s if we bear in mind the definition of Eq. (4.8). We note that the preceding r e s u l t s imply that all the functional C2J+1 of the function v(x) p r e s e r v e their values for the Korteweg-de Vries equation. In fact the trace identities d e m o n s t r a t e that all these functionals depend only on momentum-type variables which do not vary with t i m e . This observation provides a simple and exhaustive approach to the description and completeness problem of the motion integrals for the Korteweg-de Vries equation, which has been dealt with in a broad l i t e r a t u r e . References can be found for example in [43]. To derive Eq. (4.7) for the form £2 we note that it is possible to obtain using the Gel'fand-Levitan equation an expression for the variation of the potential v(x) in t e r m s of the variations of the scattering data,

378

166 Here s ! 2 (k), f,(x, k), K J , and mi 1 ' are objects corresponding to the potential v(x) and f,(x, k) is the d e r i v a ­ tive of fj(x, k) with respect to k. Calculation of the form Q is subsequently found by substituting this ex­ pression into Eq. (4.5) and calculating the integral with respect to x and y. For this purpose the following equation turns out to be useful

which follows simply from the Schroedinger equation. Details on the calculations can be found in [10], to which we refer the r e a d e r . By carrying out the calculations presented t h e r e , they can be easily carried over to the second example considered by us and Eq. (4.11) can be obtained. Equation (4.10) constitutes an identity for the t r a c e s No. 3 for the canonical system. With this we conclude the description of an unexpected application of the formalism of scattering theory for solving one-dimensional nonlinear equations. The inherent r e a s o n s why this scheme works as well as its range of application has yet to be clarified. One "experimental'' approach towards its use is to consider a one-dimensional differential operator for which the direct and inverse scattering problems are investigated and to describe the trace identities. Next one is to find a simplectic structure on the set of coefficients of this operator which can be explicitly expressed in t e r m s of the scattering data. Then, the equations generated by this structure and by some functional of the trace formulas as a Hamiltonian can turn out to be exactly solvable. Finally, this scheme has not been studied to any great extent, but it must be used for lack of a better scheme. Equation (4.2) was found precisely by this method. Searches for new scattering problems that can be studied are of particular interest. Generalizations of the Schroedinger equation or of the canonical system to the case of vector functions are obvious candi­ dates. The monograph [1] demonstrated that most results known for s c a l a r equations can be c a r r i e d over without any difficulties in the case of vector equations. Equations of higher o r d e r s however a r e m o r e p r o m ­ ising. For example, V. E. Zakharov has recently proved that the t h i r d - o r d e r differential operator

generates the nonlinear equation

which plays the role of a continuum analog of the nonlinear Fermi—Pasta—Ulam problem. The inverse problem of scattering theory for the operator of Eq. (4.12) has yet to be solved. CHAPTER THREE-DIMENSIONAL

3

SCHROEDINGER

OPERATOR

In this chapter we will consider the Schroedinger operator in a space of functions depending on the three variables H=-/± +

v(x)=ff0+V.

Here *€R3, A is the Laplace operator, and v(x) is a function which we will assume to be r e a l , bounded, and sufficiently rapidly decreasing at infinity. Spectral analysis of this operator is significantly more complicated than the one-dimensional Schroedinger operator considered in Chap. 1. This particularly relates to the detailed investigation of such properties as continuity and the asymptotic behavior of the scattering amplitudes, which must be carried out in order to discuss sufficient or necessary conditions on this function corresponding to a potential v(x) of a given class. We will, therefore, in this chapter present only a formal scheme for solving the inverse problem, not making explicit each time under which conditions on v(x) do the discussions hold. A traditional condition on v(x) under which most of the bounds presented below hold asserts that

379

167

Under this condition the operator H defined in % = L7(R3) on the dense domain D = w'S(R3) is self-adjoint. We consider a three-dimensional case only for the sake of definiteness. All equations generalize without difficulty to the case of a r b i t r a r y n s 2. In appropriate places we will note analogies or divergences from the one-dimensional case treated in Chap. 1. The m a t e r i a l set forth first appeared in [29]. 1. S c a t t e r i n g

Theory

The construction of scattering theory for the pair of o p e r a t o r s H and H0 in a stationary variant is based on the existence of a set of solutions u'*'(x, k) of the Schroedinger equation

with the following asymptotic behavior at infinity, i.e., as |x| — °°,

(radiation condition). Here

The solutions uWfx, k) a r e s i m i l a r to the sets of solutions u j ^ x , k) and u ^ ( x , k) of Sec. 2 Chap. 1. The role of the index i = 1, 2 is played by the direction a = k/|k| of the vector k, which runs through the unit s p h e r e S2. An existence proof and an investigation of the solutions u ^ ' f x , k) was carried out by A. Ya. Povzner [18, 19]. The discussions presented in Chap. 1 and based on the existence of a fundamental system of solutions of the Schroedinger equation are not applicable to our case. Therefore, it is necessary to study directly the integral equations of scattering theory

Here G(±> (x, \k\) is G r e e n ' s function for the Helmholtz equation

which can be uniquely defined by the radiation condition. The explicit expression

well known in the three-dimensional case is obtained from the general equation

after calculation of the integral. In the last equation the well-known generalized function (x ± iO) * o c c u r s . The investigation of A. Ya. Povzner is based on Fredholm theory for Eqs. (1.3). An important role is played by the Kato theorem [35], which implies that homogeneous equations corresponding to Eq. (1.3) for real k do not have nontrivial bounded solutions. A. Ya. Povzner proved that the solutions u (±) (x, k) form a complete orthonormalized system of eigenfunctions of the continuous spectrum of H, which fills the e n t i r e positive s e m i - a x i s . This spectrum has uniform infinite multiplicity, so that the eigenfunctions a r e numbered in addition to the eigenvalue k2 by the point a 6S 2 . Besides a continuous spectrum, H can have a finite number of nonpositive eigenvalues of finite multiplicity. To simplify the equations we will assume that the entire discrete spectrum of H consists of a single negative eigenvalue, which we denote by - x 2 . The corresponding normalized eigenfunction, which can be assumed to be r e a l , is denoted by u(x).

380

168 The completeness and orthogonality conditions on the function u<±)(x, k) are written in the form

The scattering amplitude f(k, I) is simply related to the function f* + '(k, n) describing the asymptotic of the solutions u^ + '(x, k)

and can be expressed in t e r m s of the solution u^ '(x, k) by the equation

which is an analog of Eq. (1.1.22) and (1.1.23). We now present a relation between the functions u ^ x , k) and the wave o p e r a t o r s . F o r this purpose we introduce a diagonal representation for H a . Suppose the space $ 0 is formed by the functions
where d a is an element of surface of the sphere S2. We define the isomorphism T0:S)-*%0 by the formula

The operator T 0 is unitary,

and c a r r i e s H0 into a X-multiplication operator,

We introduce now two m o r e maps, T+:fy-+f)0

and

The completeness and orthogonality equations of Eqs. (1.5) and (1.6) a r e written in t e r m s of them as fol­ lows:

Here P is a projector into £ on a one-dimensional subspace spanned by the function u(x). The wave operators U ^ ' a r e given by

This fact can be proved using the scheme presented in Sec. 2 of Chap. 1. Detailed discussions as r e g a r d s this proof can be found in the article of Ikebe [34], which also contains s e v e r a l refinements of works of A. Ya. Povzner. Let us now relate the scattering operator S defined by the formula

381

169

to the s c a t t e r i n g amplitude f(k, I). We note for this purpose that the solutions ui+) (x, k) and u' '(.*, k) a r e linearly independent

and by comparing their asymptotics we note that the k e r n e l g(k, I) coincides with the scattering amplitude,

which in turn must satisfy a unitarity-type relationship. To write the latter it is convenient to introduce on £ 0 an o p e r a t o r s S by the equation

This operator is also unitary. A comparison of these definitions demonstrates that the scattering operator can be written in the form

which also yields the desired relation. We note the analogy of the equations obtained and those in Sec. 2 of Chap. 1. Henceforth, we will find it convenient to refer to the space $ 0 simply as Z-2(R3)

using the identity

In this case T 0 is a F o u r i e r transform,

The operator S in this notation is given by

Let us now r e t u r n to the solutions UP '(X, k) and say a few words about their p r o p e r t i e s . It can be proved that the functions u( + )(x, | k | a ) for fixed x and a have an analytic continuation into the upper halfplane of the p a r a m e t e r s = |k| and have a simple pole at the point s = i x . The principal part in the neigh­ borhood of this pole has the form

where

and u(x) is the already-mentioned eigenfunction. It is necessary to study Eq. (1.3) in the complex domain s = |k| for the proof. One variant of these discussions can be found in [24]. Further, for large |k|, we have the asymptotic

while if v(x) is differentiate, we can replace here o(l) by 0 ( l / | k | ) . A derivation can be found in [22]. In particular, these results imply that the forward scattering amplitude, i.e., the function 382

170

for fixed a has an analytic continuation into the upper half-plane of the p a r a m e t e r s = |k|, has a pole when s = in with principle part

and at infinity has the asymptotic

The proof requires the use of the representation of Eq. (1.7) for the scattering amplitude. The last property first found by the physicists Wong [48] and Khuri [41] is an analog of the analyticity condition on the transmission coefficient a(k) of Chap. 1. However, unlike the one-dimensional c a s e , this condition far from exhausts all the necessary conditions on the scattering amplitude which follows from the locality of the potential. In the next section we will find a profound generalization of this property, formulated in [28]. We will refer to the problem of reconstructing a potential v(x) in t e r m s of given scattering amplitude f(k, I) as the inverse problem. The results presented imply one important distinction of the multidimen­ sional case from the one-dimensional case considered above; when n - 2 there exists at most one poten­ tial that solves the inverse problem. In fact Eq. (1.14) implies that for large |k| = \l\,

so that, setting k ~ I = m and letting |k| tend to infinity, we can reconstruct the Fourier transformation of the potential in this limit. This long established and simple assertion found in [3, 21] was for a long time the only rigorous result on the multidimensional inverse problem. It finally is not of particular interest, though in every case it implies that, unlike the one-dimensional case, all the characteristics of the dis­ crete spectrum necessary for solving the inverse problem are to be calculated in terms of the scattering amplitude itself, 2. R e s e a r c h e s

on V o l t e r r a

Transformation

Operators

In the one-dimensional case the transformation operators U] and U 2 distinguished by the Volterra property, play an important role. As already noted in the introduction, the set of operators U y with Volterra direction ye S2,

is their natural multidimensional analog. In this section we will demonstrate how to prove the existence of such operators. Chapter 1 demonstrates that Volterra transformation operators are generated by a set of solutions of the Schroedinger equation possessing special analyticity p r o p e r t i e s . In our case it is necessary to find solutions f y (x, k) of Eq. (1.1) that have an analytic continuation into the upper half-plane of the variable s = (k, y) for fixed x and k± = k — {kri)-\ , such that as | JC| —>- oo ,

when l m s > 0 and has an asymptotic for large s,

383

171

In fact if such s e t s of solutions fy(x, k) exist, we may verify by introducing maps Ty from fy into # 0 by the equation

that the o p e r a t o r s

have the form of Eq. (2.1), where the kernel Ay(x, y), being perhaps a generalized function of x x and y x , will be a classical function of the variables (x, y) and (y, y ) , so that the condition according to which it vanishes when (X, y) > (y, y ) , is justified. We may attempt to find solutions of the type of f y (x, k) using integral equations of the form

for an appropriate choice of Green's function Gy(x, k) of the Helmholtz equation. In the one-dimensional case the solutions fj(x, k) and f2(x, k) generating Volterra transformation operators have been defined in precisely this way [cf. (1.1.4)]. To satisfy an analyticity condition on G r e e n ' s function G y (x, k) they must satisfy the requirement that for fixed y , x, and k x they must have an analytic continuation into the upper half-plane of the p a r a m e t e r s = ( k , y ) , such that

Such functions in fact exist. Guiding lines for searching for them and an analogy to the functions G](x, k) and G 2 (x, k) of Sec. 1 Chap. 1 were presented in [27]. Here we will limit ourselves to presenting an expression for G y in the form of an integral which, unfortunately, cannot be explicitly calculated:

where (x -\- iOa)-1 is understood as (JC + iO)"1 for a > 0 and as (x — i o ) - 1 when a < 0. It can be easily v e r i ­ fied that G (x, k) depends oh k only in t e r m s of the combinations s = (k, y) and n2 = k2 - (k, y ) 2 . When Im s> 0, Eq. (2.5) can be rewritten in the form

which imply the analyticity and boundedness-type properties formulated above. We further note that when Im s = 0, we have the r e a l n e s s condition

One important distinction between Gy(x, k) and one-dimensional G t (x, k) and G 2 (x, k) is that it is not Volterra. Therefore, to study Eq. (2.4) we cannot use the method of successive approximations and will need Fredholm theory. Here we find that the solution Uy(x, k) of this equation exists, is an analytic func­ tion of s = (k, y) when Im s> 0, and satisfies there a boundedness condition for all s, such that the homo­ geneous equation

has no nontrivial solutions satisfying the condition

Singular s such that these solutions exist are located discontinuously, have no accumulation points when lm s > 0, and are poles of finite order for u y (x, k).

384

172

Such singular s, in general, exist. In fact a comparison of Eqs. (2.5) and (1.4) makes it clear that whenk||y Green's functions G(+> (x, \k\) and Gy(x, k) coincide. Thereby the solutions u( + '(x, k) and u y ( x , k) for k|| y also coincide. The variable s = (k, y) under this condition is simply |k|. At the end of Sec. 1 we mentioned that u( + )(x, k) for fixed x has an analytic continuation into the upper half-plane of the v a r i ­ able s = |k| and has there a simple pole at s = m . Thus u y ( x , k) for k ± = 0 has a pole when s = i x . As y varies or, what is the same thing, a nonvanishing kj. appears, this pole will move without vanishing. Thus, we have verified that if an operator H has discrete spectrum, the singular values s exist. We cannot prove an analog of the Kato theorem for Eq. (2.4), i.e., we cannot guarantee that this equa­ tion is solvable for all real k nor that the singular values s do not leave the r e a l axis. We, therefore, must require that the potential v(x) be given such that these solutions do not exist. We will r e f e r to this r e q u i r e ­ ment as condition C. Henceforth, we will be able to formulate an equivalent condition in t e r m s of the scat­ tering amplitude. When this condition holds, Eq. (2.6) implies that

Thus, the solutions u Y (u, k) cannot be used to determine Volterra transformation o p e r a t o r s because of these singularities. It is, however, easy to refine them. For this purpose we consider the regularized Fredholm determinant Ay(k) of Eq. (2.4). The formal definition is provided by the equation

where we use obvious notation Gy(k) for an integral operator with kernel Gy (x - y, k). The t r a c e in the right side can be understood in an operator-theoretic sense if the operators under the sign of the t r a c e are symmetrized by the scheme

We will bear in mind that this method can be carried out and henceforth will not refer to it. The function A_,(k) depends on k only in t e r m s of the variables p and s and Ay(ji, s) for fixed y and n is analytic with respect to s in the upper half-plane, there having the asymptotic

and vanishes at singular s. Here the multiplicity of the corresponding z e r o e s is sufficient for all the poles in the product

to be annihilated, so that the set of solutions f y (x, k) satisfies all the requirements described at the begin­ ning of this section and can therefore be used to define Volterra transformation o p e r a t o r s . With this we conclude the description of r e s e a r c h into multidimensional Volterra transformation operators, which can in fact be far more exciting than can be seen from this presentation. In the next s e c ­ tion we will begin a calculation of the normalizing factor corresponding to these o p e r a t o r s . This section we conclude with a few more r e m a r k s on the determinant A y (k). The relation we have noted between the functions G<+> (jf, |A|) and Gy(x, k) at % imply that Av (k) when k\\j is the Fredholm determinant A ,+) (\k\) of the integral equation of scattering theory

We easily obtain from this expression that

so that A ,+) (/A.) is a regularized characteristic determinant of H. In this sense it is analogous to the transition coefficient a(VT) of Chap. 1. Henceforth, we have the formula

385

173 where S(k2) is a set of o p e r a t o r s in L2(S2) naturally generated by the operator S. The derivation of this r e l a t i o n , which is c h a r a c t e r i s t i c for t r a c e formulas, can be found, for example, in [4]. 3.

Normalizing

Factors

for

the

Solution

uy(x,

k)

Equations (2.10) and (2.3) demonstrate that we can find normalizing factors for the transformation o p e r a t o r s Uy if we know the determinant Ay(k) and the operators Qy*1. by means of which the solutions Uy(x, k) can be expressed as a linear combination of the solutions u( ± )(x, k). If we assume this to be an integral o p e r a t o r whose kernel Q ^ ( k , I) is a generalized function, the corresponding formula will have the form

In this section we will d e s c r i b e the set of such operators Q v (±) . which are in some sense an analog of the m a t r i c e s M>*'(k) described in Sec. 3 Chap. I . An appropriate expression for the Fredholm determinant Ay(k) will be found in the next section. Let us compare G r e e n ' s functions G(*)(X, |k|) and Gy(x, k) occurring in the integral equations (1.3) and (2.4). Equations (1.4) and (2.5) imply that

Here 0(t) is the Heaviside function. in the form

Using the first of these equations we can rewrite Eq. (2.4) for u y ( x , k)

We consider the first two t e r m s in the right side as a new free t e r m . Setting

where

we can rewrite them in the form

i.e., as a linear combination of free t e r m s in Eq. (1.3) for u( + )(x, k). The integral operator in the resulting equation also coincides with the operator of Eq. (1.3). We may a s s e r t based on the uniqueness theorem for this equation that Eq. (3.1) holds if the kernel Q( + )(k, I) is given by Eq. (3.3). Analogously we can find that

Equations (3.3) and (3.5) also constitute the desired equations determining the operators Qi*', which op­ e r a t e in i)0 according to the formula

Concepts by nowquite standard h e r e demonstrate that the operators Qy*' define a factorization of the s c a t ­ t e r i n g o p e r a t o r . Comparing Eqs. (1.9), (1.11), and (3.1) we find that

386

174

We now note that the kernel hy(k, Z) that occurs is Eqs. (3.3) and (3.4) is the same in these equations. We thus use Eq. (3.6) to uniquely determine hy(k, Z) in t e r m s of the scattering amplitude. In fact, we rewrite it in the form

and then substitute Eq. (1.11), (3.3), and (3.5) for § and

Q^'.

We obtain the equation

which can be considered as a linear integral equation for determining h v ( k , Z) in t e r m s of given f(k, I). This equation involves only the angular variables of the kernels occurring in it. The length of all equal vectors occur in it only as p a r a m e t e r s . In the next section we will verify that condition C is equivalent to a unique solvability condition on this equation. We now note an important property of the functions hy(k, Z) which is implied by analyticity p r o p e r ­ ties of the solutions u y (x, k). We consider the integral representation of Eq. (3.4) for hy(k, Z) and set (k, y) = (Z. y) = s in it. As a consequence of a bound of the type of Eq. (2.2), the integrand is absolutely inte­ g r a t e for all nonsingular s in the upper half-plane. It therefore follows that the function hy(k,Z) when (k, y) = (Z, y) = s and for fixed k ± , lx , and y has an analytic continuation into the upper half-plane of the variable s and poles of finite order at singular s. We emphasize that the locality of v(x) is highly important for deriving this analyticity property. In fact the growth oi the solution Uy(x, k) with respect to x at I m s > 0 is compensated by decreasing e"M*i x ) only because these functions are multiplied within the integral in Eq. (3.4). For nonlocal V, the independent variables x and y on which these functions will depend in an equation of the type of Eq. (3.4) will differ and no such compensation will occur. Henceforth we will verify in studying the inverse problem that the neces­ sary analyticity condition we have obtained is essentially also a sufficient condition on the scattering am­ plitude corresponding to a local potential. We should now state that this condition is a generalization of the analyticity of the foward scattering amplitude noted in Sec. 2. In fact it is evident from Eq. (3.7) that the amplitudes f(k, Z) andh (k, Z) coincide when k || y . Under this condition k = Z also if (k, y) = (Z, y ) . Thus,

and the analyticity we have indicated for the function on the left side implies the analyticity of the right side, which was noted above. We present one more useful equation relating the kernels h_,(h, Z) for different y . For this purpose we use the factorization of Eq. (3.6) and a unitarity condition on §. Rewriting the equation

in t e r m s of Eq. (3.6), using different y in the left and right s i d e s , we find

We now set y ' = — y and rewrite the resulting equation in the form

The operators Q^ 1 are Volterra, which is explicitly evident from the presence of the Heaviside function in their definition. The Volterra property of the operators in the right and left sides of the last equation are in opposite directions. It is thus consistent only if each side is separately a unit operator. We have arrived at the important equation

387

175

We also r e w r i t e the general equation (3.8) in more detail in t e r m s of the kernel of the o p e r a t o r s occurring in it,

Here we have used also the equation

which follows from the property of Eq. (2.8) and from the integral representation of Eq. (3.4). Equation (3.10) constitutes a generalization of Eq. (3.7). Because of Eq. (3.9) we must not solve integral equations to determine operators inverse to the normalizing factors Q(*\ which occur in the construction of the weight operator Wy. 4. D i f f e r e n t i a l

Equations

With R e s p e c t

to the

Parameter

y

Let us now turn to the transformation operator Uy. The normalizing factors corresponding to it we nearly already calculated; they a r e constructed by means of the operators Q$*' and the Fredholm d e t e r ­ minant Ay(k). Explicit equations can be written in the form

and the o p e r a t o r s Ny

acting in £ a r e described by the equation

In this section we will prove that the determinant Ay (k) can be explicitly expressed in t e r m s of the kernel hy(k, I) and thereby in t e r m s ofthe scattering amplitude. In o r d e r to state the Gel'fand—Levitan-type equation we must obtain, in addition to normalizing fac­ t o r s , an expression for the generalized element Xy, which is the p r e - i m a g e of the eigenfunction of the d i s ­ continuous spectrum u(x) under the map Uy. This can easily be carried out on the basis of the already noted coinciding of the function Uy(x, k) and u( + )(x, k) for k|| y and Eq. (2.10). We will proceed on the b a s i s of the equation

which constitutes a concrete variant of the more abstract definition of Eqs. (2.1) and (2.3). We note its analogy to Eqs. (1.1.8) and (1.1.9). Setting h e r e k = sy and expressing f y (x, k) in t e r m s of u( + )(x, k) and Ay(k), we find

Here we will use the already noted coinciding of Ay(sy) and A^ + '(s). Under our assumption on the simplicity of the discontinuous eigenvalue, the determinant A ( + ) ( S ) has a simple zero at s = in, so that setting s = ix in the last equation and using Eq. (1.12), we find

where

388

176

and c(y) was introduced in Eq. (1.13). This equation is also the result we require; we will see that

Thus to express all the variables occurring in the Gel'fand—Levitan equation in t e r m s of the s c a t ­ tering amplitude, it remains for us to find an expression for A y (k) and c(y). For this purpose differential equations for the functions u y ( x , k), h y ( h , I), A y (k), and c(y) with respect to the p a r a m e t e r y will turn out to be convenient. This variable runs through the unit sphere and it is therefore convenient for differentia­ tion to use the Lie operator corresponding to the operation of a group of rotations. We will not have to write explicit equations for these o p e r a t o r s , the following single equation being sufficient:

Here (y, a) is the scalar product of y and an arbitrary vector a, f(t) is an a r b i t r a r y function, Mt is the Lie operator corresponding to differentiation in the direction of £, and y x£ is the vector product of y and £. Differentiating Eq. (3.2) we find

where

This equation leads to a differential equation for all the variables mentioned at the beginning of the section. We begin with the function u y ( x , k). Differentiating Eq. (2.4) we find the equation

This equation differs from Eq. (2.4) only in a free t e r m , which can be written in the form of a linear com­ bination of plane waves of the form

where

The expression for o j y t ( k , I) includes 6-functions, so that I2 = k2 and (I, y) = (k, y) in the integral of Eq. (4.6). We recall that Green's function G y (x, k) depends on k only in t e r m s of k2 and (k, y ) . This together with Eq. (2.4) implies that

Now using the definition of Eq. (3.4) for the kernel h y ( k , I) we have

which is an integrodifferential equation for h (k, I). We now pass to the differential equations for the Fredholm determinant A y (k). A determination of this expression is given by Eq. (2.9). Differentiating it with respect to y and using Eq. (4.5), we find

389

177 where the constant

i s the asymptotic of the k e r n e l hy (I, I) for l a r g e \l\. solved. We consider the function

The resulting differential equation can be explicitly

Differentiating it with respect to y , we find

The first t e r m h e r e vanishes. In fact the function u) t(l, m) is antisymmetric with respect to I and m, while the remaining part of the integrand in this t e r m is s y m m e t r i c , so that we may set (m, y) = (I, y ) . Thus, the differential equations for In Ay(k) and gy(k) coincide so that these functions coincide to within a t e r m independent of y . We however know that when k|| y the determinant Ay(k) coincides with the F r e d holm determinant A^ + '(|k|) of the integral equation of scattering theory [Eq. (1.3)]. At the same time g (k) = 0 when y || k. We a r r i v e at the equation

We may analogously study the Fredholm determinant Ay(k) of the integral equation (3.7) in which an integral operator P y (k) with kernel

o c c u r s . In this case we may prove that

so that

On the basis of the Kato theorem we know that A (+) (| k 1)^0 for real |k|. Then Eq. (4.9) also demonstrates the equivalence between the unique solvability problem for Eqs. (3.7) and (2-.4). Let us now express the determinant A<+)(| k |) in terms of hy(k, I). For this purpose we recall that A (|fc|) has an analytic continuation in the upper half-plane of the variable |k| = s, there has a unique zero at s = i n , and when Im s = 0, Eq. (2.11) holds. Suppose Q Y + , ( | A | ) and Q} -, (|£|) are operators in L,(S2) d e ­ fined in t e r m s of Q' + ) and Q^ -) in the same way as S(|k|) was defined in t e r m s of S. (+)

The factorization of Eq. (3.6) leads to the equation

which leads to the equation

which implies that

390

since all the degrees of the operators Q ( * ) - I in a logarithmic decomposition, other than the first d e g r e e , yield a null contribution to the t r a c e as a consequence of the Volterra property of Q<+) and Qv-) . If we know argA l + , (|£|). we can reconstruct this function using the equation

where

Combining Eqs. (4.10), (4.11), and (4.12), we find a explicit expression for

A(+)(l*l)

in t e r m s of hy(k, I).

We will now demonstrate how to e x p r e s s the function c(y) occurring in the construction of the vector X y in t e r m s of the scattering amplitude f(k, I) [cf. Eq. (4.4)]. We consider the scattering amplitude f(k, I) as a function of the variable y = l/\ l\ and apply to it the operator M^ in t e r m s of this variable. Using the integral representation of Eq. (1.7) and the analyticity properties of the solution u( + '(x, k), we find that

like f(k, k) has an analytic continuation into the upper half-plane of the variable s = |k| with pole at s - i x . It can be easily verified using Eq. (1.12) that the corresponding residue has the form

Comparing this equation to Eq. (1.15) we find that

Solving this equation, assuming that the right side is known, we are able to obtain the equation

where b(y) is expressed in t e r m s of integral of this right side. Here multiplying both side by c(y) we find finally

so that the square of c(y) is explicitly expressed in t e r m s of the scattering amplitude. It is precisely this square that occurs in the Gel'fand-Levitan equation. We can now proceed to a direct formulation of this equation. 5. I n v e s t i g a t i o n

of I n v e r s e

Problem

By assuming that the potential v(x) satisfies the conditions a) v(x) is a continuous, rapidly decreasing function, and b) Eq. (2.7) does not have nontrivial bounded solutions for all real k we have proved that the scattering amplitude f(k, /) possesses the properties: 1. The integral equation (3.7) is uniquely solvable for any

(f

S 2 defining a family of kernels h,,(k, I).

2. The function Ay(k), constructed in terms of h y (k, /) by Eqs. (4.9), (4.10), and (4.11), has a bounded analytic continuation with respect to the variable s = (k, y) into the upper half-plane. 3. The function hy(k, /)AY(£) when (k, y) = (I, y) also has such a continuation with respect to s = (k, y) for arbitrary fixed lL and k j . . The last property presents a rich collection of necessary conditions that somewhat explicitly decrease the number of parameters in the scattering amplitude. It turns out that it essentially exhausts the necessary

391

179 conditions on the scattering amplitude, about which we spoke in the introduction. That is, we will prove that if it holds, there exists a local potential v(x), such that f(k, I) is the scattering amplitude. This will be carried out using the formalism of the inverse problem. We begin by writing an integral equation for determining the transformation operator Uy. The idea of the proof of this equation was already set forth in the introduction. The corresponding weight operator Wy has the kernel

where

In writing these equations we will use an abstract definition of the weight operator given in the introduc­ tion, a concrete form of the normalizing factors N ^ from Eq. (4.1), and Eq. (3.9). The Gel'fand-Levitan equation has the form

where

The kernel fty(x, y) is completely reconstructed in terms of the scattering amplitude f(k, I), as was dem­ onstrated in the preceding section. Equation (5.2) is an equation for A^(x, y) as a function of the variable y, while x and y play the role of parameters. The kernel of this equation is evidently positive, which ensures its unique solvability. Thus, we will find it possible to reconstruct the transformation operator Uy in terms of the scattering amplitude. We construct using the transformation operators Uy a family of operators

We intend to prove that if the properties formulated at the beginning of this section hold, the operator H is independent of y and the corresponding operator Vy = Hy - H0 is an operator for multiplication by the real function v(x), where the initial function f(k, I) is the scattering amplitude for this potential. The formal scheme of proof is significantly simplified if we assume that no discrete spectrum exists. We will limit ourselves here to a presentation of only this case, so that we will assume that the second term in Eq. (5.1) is absent. The investigation begins according to a scheme entirely analogous to the one-dimensional case of Sec. 5 Chap. 1. It is possible to prove that the Volterra operator

constructed in terms of the solution Ay(x, y) of Eq. (5.2) satisfies

which implies that H y is self-adjoint. We now prove that the operator Vy = Hy - H0 is local in the y d i r e c t i o n , i.e., that its kernel, a gen­ eralized function, is expressed in the form

392

180

We note for this purpose that in the identity

which follows from Eq. (5.3), t h e r e exist t e r m s with singularities of various o r d e r s on the plane (x, y) = (y, y ) . The kernel A y (x, y) itself has a discontinuity-type singularity,

After commuting with H0 there arises the more singular term of the form

which can be compensated for in the right side of Eq. (5.5) only by the potential V y (x, y), so that we will find an explicit expression for the potential V y (x, y) in t e r m s of the kernel A y ( x , y),

Until now we have not used the analyticity conditions on the kernel h y ( k , I). This property makes it possible for us to a s s e r t that the operator H y is independent of y , so that the potential V y is local in every direction and is therefore a function-multiplication operator. For this purpose we naturally use differen­ tial equations with respect to the p a r a m e t e r y . We note that the differential equation (4.8) can be derived by proceeding on the basis of the definition of h y (k, /) by means of the integral equation (3.7). In fact differentiating Eq. (3.7) with respect to y, we find

The free t e r m h e r e can be written in the form

[cf. definition of Eq. (4.7) of the kernel of R y | J . On the other hand, multiplying Eq. (3.7) by R-, t(q, k) and integrating over k, we obtain for

an equation with integral term of the form

Because of the presence of i\(q — k,t)] in the kernel Ry.i(q,k) we can h e r e replace 6 |(m— *,-f)l by 8 \(m — q, ■()]. The equations for Mihy{q, I) and r y ( q , /) then coincide, so that Eq. (4.8) can now be said to hold for the kernel hy(k, I) reconstructed from f(k, I). We now recall that the very procedure for constructing A^(k) in t e r m s of liy(k, I) is based on Eq. (4.8) for this function. The kernel L y (k, I) of the operator ZY = A/<+,_1 can then be said [cf. Eqs. (4.1) and (4.2)] to satisfy the differential equation

393

181 where

We introduce with this equation the operator riY,c in the space £ 0 . The analyticity of h y ( k , I) when (k, y) = (I, y) reduces to the operator DY,^ = rjnY,{7,0 being triangular,

We also note that

which is a simple corollary of Eq. (3.9), Recalling the definition of W y we find that

We now prove, proceeding on the basis of Eq. (5.4), that

F o r this purpose we note that the operator IVb_Uy is triangular and therefore uniquely determined by the equation

which is obtained by differentiating Eq. (5.4). In fact if we explicitly write out this equation in t e r m s of the k e r n e l s of the o p e r a t o r s occurring in it, the right side will vanish when (x, T)<(y. f) and we will obtain a linear integral equation for the kernel of Afjt/Y, which differs from the Gel'fand-Levitan equation only in the free t e r m . Using Eqs. (5.4) and (5.6) we easily find that — 6' v n v t satisfies this equation. TheVolterr a property of U y and the triangularity of ny,c implies that L/vIIY.t is also triangular. Equation (5.7) may therefore be said, because of the uniqueness noted above, to be proved. Let us now turn to the operator H y introduced in Eq. (5.3). We have

since the operator n Y ? commutes with H 0 . We have found the promised constancy of H y as a function y and together with it the locality of the potential V. In transferring this scheme to the case when a discrete spectrum is present, it is necessary to observe care in differentiating the contribution to Eq. (5.2) from the improper vector x y . It is convenient to first rotate all the variables so that a variation in y will not vary the Volterra direction of the desired o p e r a t o r s . Differentiation with respect to y then no longer causes complications. In proving an equation of the type of Eq. (5.6) it will be necessary to use a differential equation of the form of Eq. (4.13). It r e m a i n s for us to prove that the initial kernel f(k, I) is the scattering amplitude for the given Schrbedinger operator H. We need only prove here that the solutions u y ( x , k) constructed with respect to the transformation operator U y and the determinant A y (k) using Eqs. (4.3) and (2.10) have for large |x| the asymptotic

In fact once this equation has been proved, we easily verify that the set of solutions

394

182 which is independent of y because of Eq. (5.7),has the asymptotic of Eq. (1.2) in which the initial kernel f(k, I) o c c u r s . We will not p r e s e n t the arguments proving Eq. (5.8), since they require a significantly more detailed study of the kernel Ay(x, y) than we have so far limited ourselves to. This investigation can be carried out as soon as the formal discussions of this chapter have been made rigorous. The variants of it available to us are too cumbersome to fit within the present survey. We hope that the formal scheme for solving the multidimensional inverse scattering problem presented here will be a stimulus for some readers to develop better founded analytic tools for its justification. With this we conclude the description of the state of the inverse problem of quantum scattering theory through the beginning of 1973. LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

CITED

Z. S. Agranovich and V. A. Marchenko, Inverse Problem of Scattering Theory [in Russian], KGU, Kharkov (1960). V. I. Arnol'd, Lectures on Classical Mechanics [in Russian], MGU, Moscow (1968). Yu. M. Berezanskii, "Uniqueness theorem in the inverse problem of spectral analysis for the Schroedinger equation," T r . Moscow Matem. O-va, 7, 3-62 (1958). V. S. Buslaev, "Trace formulas for Schroedinger operator in three-dimensional s p a c e , " Dokl. Akad. Nauk SSSR, 143, No. 5, 1067-1070 (1962). V. S. Buslaev and L. D. Faddeev, "Trace formulas for the S t u r m - L i o u v i l l e differential singular operator," Dokl. Akad. Nauk SSSR, 132, No. 1, 13-16 (1960). V. S. Buslaev and V. L. Fomin, Inverse Scattering Problem for One-Dimensional Schroedinger Equa­ tion on the Entire Axis, Vestn. Leningrad Un-ta, No. 1, (1962), pp. 56-64. M. G. Gasymov, "Inverse problem of scattering theory for a system of Dirac equations of order 2n," T r . Moscow Matem. 0-va,"_19, 41-112 (1968). I. M. Gel'fand and B. M. Levitan, "Determination of a differential equation in terms of its spectral function," Izv. Akad. Nauk SSSR. Ser. Matem., 15, No. 2, 309-360 (1951). I. M. Gel'fand and B. M. Levitan, "Simple identity for eigenvalues of a second-order differential operator," Dokl. Akad. Nauk SSSR, 88, No. 4, 593-596 (1953). V. E. Zakharov and L. D. Faddeev, "Korteweg-de Vries equation— a completely integrable Hamiltonian system," Funktional'. Analiz.i Ego Prilozhen., 5, No. 4, 18-27 (1971). V. E. Zakharov and A. B. Shabat, "Rigorous theory of two-dimensional self-focusing and one-dimen­ sional self-modulation of waves in nonlinear media," Zh. Eksperim. i T e o r . Fiz., 6 1 , No. 1, 118-134 (1971). M. G. Krein, "Determining particle potential by its S-function," Dokl. Akad. Nauk SSSR, 105, No. 3, 433-436 (1955). M. G. Krein and F. E. Melik-Adamyan, "Theory of S-matrices of canonical differential equations with summable potential," Dokl. Akad. Nauk ArmSSR, 46, No. 4, 150-155 (1968). P . P . Kulish, "Inverse scattering problem for Schroedinger equation on the axis," Matem. Zametki, 4, No. 6, 677-684 (1968). B. Ya. Levin, "Fourier-type and Laplace-type transformations by means of solutions of a secondorder differential equation," Dokl. Akad. Nauk, H)6, No. 2, 187-190 (1956). V. A. Marchenko, "Reconstruction of potential energy in t e r m s of stray wave p h a s e s , " Dokl. Akad. Nauk SSSR, 104, No. 5, 695-698 (1955). V. A. Marchenko, Spectral Theory of S t u r m - L i o u v i l l e Operators [in Russian], Nauka Dumka, Kiev (1972). A. Ya. Povzner, "Decomposition of a r b i t r a r y functions in eigenfunctions of the operator Au + cu," Matem. Sb., 32, No. 1, 109-156 (1953). A. Ya. Povzner, "Decomposition in functions that are solutions of the scattering problem," Dokl. Akad. Nauk SSSR, 104, No. 3, 360-363 (1955). L. A. Takhtadzhyan, Method of the Inverse Problem for Solving the One-Dimensional Nonlinear Schroedinger Equation (Thesis), Matem. Mekhan. Dept., LGU (1972). L. D. Faddeev, Uniqueness of the Solution of Inverse Scattering Problem, Vestn. LGU, No. 7 (1956), pp. 126-130. L. D. Faddeev, Decomposition of Arbitrary Functions in Eigenfunctions of the Schroedinger Operator, Vestn. LGU, No. 7 (1957), pp. 164-172. L. D. Faddeev, "Relation of S-matrix and the potential for the one-dimensional Schroedinger o p e r a ­ tor," Dokl. Akad. Nauk SSSR, 121, No. 1, 63-66 (1958). 395

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44. 45. 46. 47. 48.

396

L. D. Faddeev, "Dispersion relations in nonrelativistic scattering theory," Zh. Eksperim. i Teor. F i z . , 35, No. 2, 433-439 (1958). L. D. Faddeev, "Inverse problem of quantum scattering theory," Usp. Matem. Nauk, 14, No. 4, 57119 (1959). L. D. Faddeev, " P r o p e r t i e s of the S-matrix of the one-dimensional Schroedinger equation," T r . Matem. In-ta Akad. Nauk SSSR, 73, 314-336 (1964). L. D. Faddeev, "Growing solutions of the Schroedinger equation," Dokl. Akad. Nauk SSSR, 165, No. 3 514-517 (1965). L. D. Faddeev, "Factorization of an S-matrix of a multidimensional Schroedinger operator," Dokl. Akad. Nauk SSSR, 167, No. 1, 69-72 (1966). L. D. Faddeev, "Three-dimensional inverse problem of quantum scattering theory," Sb. T r . AllUnion Symposium on Inverse Problems for Differential Equations [in Russian], Novosibirsk (1972). I. S. Frolov, "Inverse scattering problem for Dirac system on the entire axis," Dokl. Akad. Nauk SSSR, 207, No. 1, 44-47 (1972). F . Calogero and A. Degasperis, "Values of the potential and its derivatives at the origin in t e r m s of the s-wave phase shift and bound-state p a r a m e t e r s , " J. Math. Phys., 9, No. 1, 90-116 (1968). O. D. Corbella, "Inverse scattering problem for Dirac p a r t i c l e s , " J. Math. Phys., 1 1 , No. 5, 16951713 (1970). A. Degasperis, "On the inverse problem for the Klein-Gordon s-wave equation," J. Math. Phys., 11, No. 2, 551-567 (1970). T. Ikebe, "Eigenfunction expansions associated with Schroedinger o p e r a t o r s and their applications to scattering theory," Arch.Ration. Mech. and Anal., 5, No. 1, 1-34 (1960). T. Kato, "Growth properties of solutions of the reduced wave equation with a variable coefficient," Communs. P u r e and Appl. Math., 12, No. 3, 402-425 (1959). T. Kato, Perturbation Theory for Linear O p e r a t o r s , Vol. 19, Springer, Berlin (1966). I. Kay, "The inverse scattering problem when the reflection coefficient is a rational function," Communs. P u r e and Appl. Math., 13, No. 3, 371-393 (1960). I. Kay and H. E. Moses, "The determination of the scattering potential from the spectral m e a s u r e function. I," Nuovo Cimento, 2, No. 5, 917-961 (1955). I. Kay and H. E. Moses, "The determination of the scattering potential from the spectral m e a s u r e function. II," Nuovo Cimento, 3, No. 1, 66-84 (1956). I. Kay and H. E. Moses, "The determination of the scattering potential from the spectral m e a s u r e function. HI," Nuovo Cimento,_3, No. 2, 276-304 (1956). N. N. Khuri, "Analicity of the Schroedinger scattering amplitude and nonrelativistic dispersion r e l a ­ tions," Phys. Rev., 107, No. 4, 1148-1156 (1957). M. D. Kruskal, C. S. Gardner, J. M. Greene, and R. M. Miura, "Method for solving the Korteweg-de Vries equation," Phys. Rev. Lett., 19, No. 19, 1095-1097 (1967). M. D. Kruskal, R. M. Miura, C. S. Gardner, and N. J. Zabusky, "Korteweg-de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws," J. Math. Phys., 11, No. 3, 952-960 (1970). P . D. Lax, "Integrals of nonlinear equations of evolution and solitary waves," Communs. Pure and Appl. Math., 21, No. 5, 467-490 (1968). P. D. Lax and R. Phillips, Scattering Theory, Vol. 12, Academic P r e s s , New York-London (1967). N. Levinson, "On the uniqueness of the potential in a Schroedinger equation for a given asymptotic p h a s e , " Kgl. Danske Videnskab. Selskab. mat. F y s . medd., 25, No. 9 (1949). J. J. Loeffel, "On an inverse problem in potential scattering theory," Ann. Inst. H. Poincare, 8A, No. 4, 339-447 (1968). D. Wong, "Dispersion relation for nonrelativistic p a r t i c l e s , " Phys. Rev., 107, No. 1, 302-306 (1957).

185

Comments on Papers 7-10 Paper 7 was the first presentation of a new approach to exact quantization of the solitonic equations. I use the term "exact" to contrast with quasiclassical quantization, which I was developing in collaboration with Korepin, Takhtajan and Kulish in 1973-78. Here the role of Yang-Baxter type relations in proper quantization of the formalism of the inverse scattering method was duly emphasized. The accompanying Paper 8 was meant to uncover the algebraic structure inherent in the original papers of Baxter on the two-dimensional models of classical statistical mechanics. The term "Yang-Baxter relations" was coined there. I have chosen Papers 9 and 10 to represent my numerous reviews and/or lecture notes, describing the different aspects of the theory of mtegrable models in the classical and the quantum field theory in (l+l)-dimensional space-time. Paper 9 in its classical aspects is a short version of the monograph on the Hamiltonian methods in the theory of solitons, written by L. A. Takhtajan and me in 1986. The second monograph, devoted to the quantum theory of solitons, is still to be written. The beautiful object — lattice deformation of Kac-Moody algebra — described at the end of Paper 10 is still waiting to be properly used.

187

Quantum Completely Integrable Models in Field Theory L D. FADDEEV Institute of Mathematical Sciences, Leningrad

Contents

1.

Basic Features of the Classical Inverse Problem Method

2.

General Program of the Quantum Approach

3.

The N.S. Model

4. Other Models 5. Conclusion

The inverse problem method discovered in 1967 by M. Kruskal et al. [1] made it possible to find an extensive class of two-dimensional evolutionary equations admitting an explicit solution. This class in­ cludes equations, whose quantization leads to important models in quantum field theory, such as, for instance, the so-called Schrodinger nonlinear equation (the N.S. model)

analyzed by the inverse problem method in [2], [4], and the "SineGordon" equation (the S.G. model)

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L. D. FADDEYEV

to which [5]-[9] are devoted. The Hamiltonian interpretation of the inverse problem method [3] showed that its transformations may be considered as a transition to "action-angle" variables. This consider­ ation has led to a scheme of quantization of these equations within the framework of the quasi-classical approximation [ 10]—[ 13]. While working in this direction, my assoicates and I at the Leningrad section of the Steklov Mathematics Institute became in­ creasingly convinced that a direct approach to the quantum problem for models that are solvable by the inverse problem method may prove to be easier than the indirect treatment based on solving the classical problem and then quantizing it. This conviction was further reinforced by the fact that as far as the N.S. model is concerned the solution to the quantum problem—the determination of the eigenfunctions of the N particle problem with pair interaction via a 5-function potential—is in fact simpler than the classical solution (see [14]-[16]). A new interpretation of the Lax procedure [21] appeared in the spring of 1978. This procedure played an important part in formulat­ ing the inverse problem method in terms of the geometry of Lie groups (see [17], [18], and further development of these concepts in [19], [20]). The phase space of integrable models is the orbit of the coadjoint action of an infinite dimensional Lie group. An analysis of this action in the finite-dimension case proved to be very fruitful in due course and led to the development of the so-called orbit method in the representation theory of Lie groups [22]. In this method the represen­ tation is interpreted as a quantum mechanical system derived in the course of quantizing the classical action of the group. Naturally the relation between the inverse problem method and the orbit method also indicates the possibility of exact quantization of models that are completely integrable by the first method. This circumstance prompted me and my associates—V. Ye. Korepin, P. P. Kulish, and especially Ye. K. Sklyanin and L. A. Takhtadzhyan—to undertake the development of the quantum inverse problem method. The results obtained in the course of a year showed that such a method exists and is applicable to most of the problems previously solved classically. As the work progressed it became clear that there is a close connection between the inverse problem method and the following two methods in one-dimension mathematical phys­ ics (the term is due to Mattis and Lieb [24]):

189 QUANTUM COMPLETELY INTEGRA BLE MODELS IN FIELD THEORY

109

1. The ideas of Onsager, developed later, mostly by Lieb and Baxter, for solving two-dimensional lattice models in classic statistical physics (cf. the reviews in [23] and the original papers [25], [26]) 2. Explicit formulas for eigenfunctions of some quantum mechan­ ical problems, introduced by Bethe in 1932 [27], further developed by Hulthen [28], Yang and Yang [29], and many other authors, as can be seen from the reviews [23], [24]. As a result, both of these outstanding achievements in onedimension mathematical physics merge naturally with a third—the inverse problem method—further emphasizing its universality. Typical examples that led to the formulation of the quantum inverse problem method in [30]-[34] are discussed in this review. Incidentally, as this method developed, the role of the inverse prob­ lem method itself (i.e., the problem of reconstructing a differential operator from its spectral characteristics) diminished, so that it would probably be more natural to call this method "the auxiliary spectral problem method." The first title, however, has taken hold historically and it is already difficult to change it. In Section 1 we formulate the basic features of the classical inverse problem method in a form convenient for its subsequent quantum reformulation. General considerations for such a reformulation are discussed in Section 2. These general considerations are illustrated in detail in Section 3 using the N.S. model as an example for both attractive and repulsive interactions at finite density. Two other examples are discussed more briefly in Section 4: the S.G. model and the ATZ-model of the quantum theory of magnetism. Technical details about the behavior of these interesting models are being published separately [33], [34]. In the brief concluding section we summarize the results and indicate some unsolved problems. I would like to express my gratitude to V. Ye. Korepin, P. P. Kulish, A. G. Reiman, M. A. Semenov Tian-Shanskii, and especially to Ye. K. Sklyanin and L. A. Takhtadzhyan, for discussion and collaboration that led to the results described here. 1.

Basic Features of the Classical Inverse Problem Method

The basis of the inverse problem method is the investigation of an auxiliary spectral problem and the establishment of the relation

190

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L. D. FADDEYEV

between its coefficients and spectral data. For concreteness let us consider the N.S. model with the equation:

Here \p(x,t) is a complex function which at first we shall assume to be decreasing for |x|-> oo, the parameter K is the coupling constant. Eq. (1.1) is Hamiltonian in form, and is produced by the Hamiltonian

and the Poisson bracket The auxiliary spectral problem is given by a differential operator X, acting on the column vector

Here X is the spectral parameter, a = | K | 1 / 2 , e = sign K. We introduce the transition matrix TL(X) for problem (1.5) on the interval (—L,L). Let GL(x,X) be the matrix solution of the system (1.5), satisfying the boundary condition where / is the unit matrix. The transition matrix is the solution of (1.6) for JC - L: It can be shown that because of the particular form of the matrix elements in (1.5), the matrix TL(X) has the form

where the functions aL(X),bL(X)—the so-called transition coefficients —are entire functions of the parameter X, and for real X satisfy the

191

QUANTUM COMPLETELY INTEGRABLE MODELS IN FIELD THEORY

HI

"unitarity" condition

For decreasing \p(x), in the case of real A, the limit

exists, where V(X) is the numerical matrix

For the matrix elements aL(X) and bL(X) this means that there the limits

exist. The limiting coefficient a(X) remains a function analytic in the upper half-plane, whereas the coefficient b(X) is, generally speaking, defined only for real A. The relationship (1.9), of course, remains valid for both a(X) and b(X). For K < 0 Eq. (1.1) has a family of soliton solutions

The corresponding transition coefficients have the form

The transition coefficient

corresponds to a multisoliton solution, whose explicit form we shall not give. A systematic computation of the coefficients aL(X), bL(X) in terms

192

Ill

L. D. FADDEYEV

of «//*(*), \p(x) can be based on the integral equation

which can be solved by iteration. In particular, for b{\) we obtain the expression

which shows that the relation between \p*(x) and b(\) is a nonlinear generalization of the Fourier transform. The achievement of the transformation from v^*,^ to the transition coefficients a(\), b(X) is that ([3], [4]): 1. The Hamiltonian H is expressed explicitly in terms of a(k). Let

Then

2. The Poisson brackets of the transition coefficients are given explicitly. We list some of them:

In (1.21) we assume that ImX > 0. The relationships cited show that a(X), considered as a functional of \p* and \p, is the generating function for the commuting (in the sense of Poisson bracket) integrals of motion of the dynamical system under consideration. Furthermore, the equation of motion for b(n) has a very simple form. Let us rewrite (1.21) in the form

and expand both sides in series in powers of A '. Comparing coeffi-

193

Q UA NTUM COMPLETEL Y IN TEGRA BLE MODELS IN FIELD THEOR Y

113

cients of A 3, we obtain

from which we have the famous answer

Thus, in the variables b((i),b*(ji) the equation of motion becomes linear and is solved explicitly. It is in just this sense that they are the "action-angle" variables for the dynamical system under consider­ ation. The inverse transform from the variables b(\),b*(k) to the initial variables \p(x),\p*(x) essentially represents the solution of the inverse problem for the auxiliary linear system (1.5), which has given the name to the entire method. We shall not give the corresponding formalism, based on the Gel'fand-Levitan-Marchenko linear equa­ tion, since in this review our aim is to describe the quantum version of the formulas that have already been cited, in particular, relations of the type (1.19), (1.20), and (1.21). The formulation of a quantum analog of the formalism for the solution to the inverse problem is of undeniable interest and is now in the course of development. Together with solutions \p*(x,(),\p(x,t) for the problem (1.1) that decrease rapidly as |JC| —> oo, we also consider the solutions periodic in x: In this case the transition matrix TL(X) provides the spectral data, and its trace

is the generating function for the commuting integrals of motion. The Poisson brackets between the transition coefficients are computed explicitly, but there is no linear combination b(X) of transition coeffi­ cients for which relations of the type (1.21) would be satisfied

with the C-number function C(X, //.)■ For this reason the simple variables, such as the "action-angle" variable, have not yet been obtained for the periodic problem. To describe the dynamics we have to resort to more refined means, concerning which we refer the reader

194

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L. D. FADDEYEV

to the original literature (cf. reviews [35], [36]). We shall convince ourselves that, in the quantum version, the periodic problem is quite simple and that it can be dealt with practically within the framework of the objects and formulas already cited. Now let us present these formulas for a number of other interesting models: 1. The Heisenberg ferromagnet (H.F. model) [37]. The dynamical variables, the spin vector S(x), are a set of three functions S(x) = (S\x),S2(x),S3(x)), satisfying the supplementary condition

The equation of motion,

where we use the vector product

is Hamiltonian and is generated by the Hamiltonian function

and the Poisson brackets

The auxiliary linear problem has the form

where

and a', / = 1,2,3 are the Pauli matrices. The natural boundary conditions

correspond to the spin vector having a fixed value at infinity.

195 Q UA NTUM CO MP LETEL Y INTEGRA BLE MODELS IN FIELD THEOR Y

115

If this condition is fulfilled Eq. (1.33) has a matrix solution such that

where

and the transition matrix T{\) has the form

The transition coefficients a(k),b(k) have properties analogous to those of the coefficients for the N.S. model. In particular, the Poisson brackets, analogous to (1.20), (1.21), are:

Thus, the coefficient a(K) preserves its role as the generating function for the commuting integrals of the motion and the coefficients b(\), b*(K) are "action-angle" variables. 2. S.G. model ([5]-[9]). The equation of motion

contains two parameters, m and /?, which act as the mass and the coupling constant, respectively. This equation is Hamiltonian and is generated by the Hamiltonian function

and the Poisson brackets are Here it is understood that

196

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L. D. FADDEYEV

The auxiliary linear problem has the form

Under the conditions

eq. (1.45) has a matrix solution G(x,X), analogous to that introduced in the preceding models, by means of which the following transition matrix is determined:

The transition coefficients a(X),b(X) have an additional property for real X: so that they are determined by their values on the semiaxis 0 < X < oo. The basic Poisson brackets have the form:

The commuting integrals of motion are given by the coefficients in the expansion of lna(X), for X-> oo, and for A-^0:

In particular, the Hamiltonian H has the following expression:

In all the examples cited so far the auxiliary spectral problem is formulated for a two-dimensional vector function <E>, and the transi­ tion matrix TL(X) is a 2 X 2 matrix. The literature discusses examples

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QUANTUM COMPLETELY INTEGRABLE MODELS IN FIELD THEORY

117

of dynamical systems with an auxiliary spectral problem whose transition matrix has higher dimensionality (see, e.g., [38]-[44]). We shall not discuss these systems in this review. In addition to the continuous models in field theory, one can investigate lattice models using the inverse problem method. The most outstanding example is the Toda chain (T.Ch.) [45], investigated by the inverse problem method in [46], [47]. The equation of motion qn = £ « . ♦ . - * , _ £ « . - * , - ,

(1.54)

are Hamiltonian and are generated by the Hamiltonian function

// = S(-y+**'*'-*)

(1.55)

and the usual Poisson brackets [A,4L)-*LThere are three different types of boundary conditions:

(1-56)

a. the infinite chain - oo < n < oo f„-»0,

ft,-*0,

|«l->oo.

(1.57)

A simple subtraction must be made in the Hamiltonian in order for the sums in (1.55) to converge. b. a periodic chain of the length TV PN + \=PI*

?JV+I = ?I-

(138)

c. a free chain of the length N, where the Hamiltonian is given by (1.55), and the term «Kp{qN+l-qN) is omitted. The auxiliary linear problem is given by the system

where

HE)

(160)

*-(*- T)-

(161)

is again a two-dimensional vector, and Ln is the 2 x 2 matrix

-

198

118

L. D. FADDEYEV

The transition matrix TN(X) is given by a product of local matrices:

and its matrix elements are functions of the canonical variables pn and qn. It is quite easy to verify that the Hamiltonians of the free and the periodic chains are simply related to the expressions

respectively. A more symmetric, equivalent, expression for the matrix Ln may be more useful for investigating the periodic chain:

This expression is compatible with the general expression of the type (1.8). The matrices Ln and L'n are similar:

so that the corresponding transition matrices are also similar, while the expression (1.64) for the Hamiltonian remains unchanged. For the infinite chain the transition matrix is defined as the limit

where

It can be shown that a(X) and b(X) coincide with the transition coefficients introduced in [46], [47], [48] by another means, which clarifies their role as the generating function for the integrals of motion and "action-angle" variables. The basic Poisson brackets are

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QUANTUM COMPLETELY INTEGRABLE MODELS IN FIELD THEORY

119

so that the role of the transition coefficients as "action-angle" vari­ ables is preserved. The models listed represent a set of typical examples that, together with the Korteweg-de Vries equation, served as the basic training material for the mathematical development of the inverse problem method. Fortunately, they are all also of undeniable interest in physics as well. In this sense they represent an ideal field of activity for mathematical physics; this apparently explains the interest in them on the part of modern specialists in mathematical physics, which all of us are witnessing. 2.

General Program of the Quantum Approach

In the preceding section we clarified the dynamical role of the transition coefficients, introduced by means of the auxiliary spectral problem, for the solution of nonlinear evolution equations in classical field theory. Here we consider quantum models corresponding to these equations. The Cauchy data, \p*(x),\l/(x) for (N.S.), S(x) for (H.F.), U(X),IT(X) for (S.G.), and pn,qn for (T.Ch.), become operators whose commutation relations are obtained by the rule

where h is Planck's constant which we shall set equal to unity. We call these operators the Schrodinger canonical operators. The commutation relations (2.1) are realized in a Hilbert space 3B, characteristic for each model. In this section we will see to what extent the classical transformations using the auxiliary linear problem carry over to the quantum case. The linear operator

200

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L. D. FADDEYEV

for the continuous models (see (1.5), (1.33), (1.45)) now has as its coefficients operators in 2B. Therefore, if we consider the auxiliary spectral problem for the matrix GL(x,X) with operator coefficients

where I is the unit operator in 23, we should bear in mind that the matrix coefficients of Q and GL are non-commutative and we must introduce a definite operator ordering. We shall assume that this ordering is given by the parameter x itself, so that the formal solution to Eq. (2.3) is given by the expression

where «~s symbolizes ordering in x: the operators Q(x,X) are arranged from right to left with increasing x. A more correct definition of this ordering will be obtained after transition to the lattice. We divide the interval (— L,L) into 2N equal segments of length

and set

Here x_N, . . . , xN are coordinates of lattice points (the lattice in our one-dimensional case would be more naturally called a chain). The matrix elements of the 2 X 2 matrix Ln act in the Hilbert space 2B„, in which are represented the local Schrodinger canonical operators averaged over the interval (xn — A, xn). The Hilbert space 3RL, on which the Schrodinger operators prescribed in the interval ( - L,L) act, is expressed as a tensor product:

/„ in (2.6) denotes the unit operator in 9Bn. The most important property of the matrices Q(x,X) in auxiliary spectral problems is their "ultralocality": the matrix elements of Q(x,\) do not contain derivatives of the Schrodinger canonical

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QUANTUM COMPLETELY INTEGRABLE MODELS IN FIELD THEORY

121

operators. Hence the commutators of the matrix elements of the matrices Q(x,X) and Q(y, /A) are proportional to S(x — y), and the coefficients of the matrices Ln(X) and Lm( ji) commute if m ¥= n. The approximation (2.6) of the transition matrix in an infinitesimal segment (xn — A, xn) is sometimes too naive. Below we shall choose a more refined formula for Ln for the S.G. model. The transition matrix operator TL(X) is determined as follows:

where <-\ symbolizes ordering in which the operators Ln(\) are ar­ ranged from right to left with increasing n. Let us illustrate the objects we have introduced for specific models: 1. The N.S. model. The Schrodinger local operators \p*(x),\(/(x) satisfy the commuta­ tion relations The averaged operators

satisfy the commutation relations

so we may assume that x, anc< x£ a r e OI order A as A-^0. The space 2B„ is simply the Hilbert space £ for a quantum-mechanical system with one degree of freedom, which can be realized as £ = L2(R'). In the notation introduced, the matrix Ln is written in the form

2. The H.F. model. The Schrodinger canonical operators S'(x) satisfy the commutation relations

202

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L. D. FADDEYEV

and the condition

consistent with (2.13). The averaged operators

satisfy the relations and the condition

(to lowest terms as A-»0). The matrix Ln has the following form:

3. The S.G. model. The Schrodinger canonical operators tation relations

U{X),TT{X)

satisfy the commu­

We introduce the averaged operators

satisfying the relations

The naive matrix L„ of (2.6), as can be seen from (1.45), contains the operators

203 QUANTUM COMPLETELY

INTEGRABLE

MODELS IN FIELD THEORY

123

Instead of them we shall use the Weyl operators IV = Aexp{ ±ifaq„y

u* = exp{ ± ifjl

p\,

(2.23)

where we have introduced the convenient parameter y=|/?2-

(2.24)

The operators u,f,u„7 commute for n ¥= m; for n = m the nonttivial commutation relations have the following form: "X'=^'<exp{i«'y},

£,£'= ± .

(2.25)

and the space 9Bn is again realized as the space £ of a quantummechanical one-dimensional particle. It is also obvious that fn+ = ( t o * ;

u; =(un-)\

(2.26)

and we shall use the notation »„* = o„+; wn=w«";

%~%i

< = K-

(2-27)

In this notation the local transition matrix is written as

L

"(X)=

ml

1

\

■

( 2 2 8 )

For discrete models, as is obvious from the classical treatment of the Toda chain, the local transition matrix Ln is given from the very beginning. In this model there are no problems in replacing the classical variables p„ and qn by the corresponding quantummechanical operators and substituting them in (1.61) or (1.65) for Ln. Thus, the quantum T.Ch. model is characterized by a set of spaces 2B = £, in which the operators p and <J, with the commutation relations [^

pm] = iSnmIn,

act, and the local transition matrix "

\i pn-X-2ichqn 2\-i{Pn-\}-2shqn

i(pn-\)-2shqA pn-\ + 2ichqn]'

(2.29)

204 124

L. D. FADDEYEV

We see that in the quantum approach continuous models and models on a lattice are considered similarly. The sole difference between the chain models and continuous ones is that the lattice constant in the first case remains finite, so that there is no limiting transition in (2.8) for the transition matrix TL(X) as A->0, and the formula itself acquires the form of

where length The of the AL(X),

<-\ again symbolizes ordering with respect to n, and N is the of the chain. matrix elements of the transition matrix, the operator analogs transition coefficients aL(X) and bL(X), will be denoted by BL(\), Q(A), DL(\),

For real X the following relations are satsified for most of the models considered:

In fact, they are satisfied for the matrix elements of the infinitesimal transition matrices Ln and are preserved under multiplication of such matrices. It is interesting that specific objects of the type (2.31) have recently appeared in the quantum theory of lattice-dynamical systems inde­ pendent of the inverse problem method. Thus, in studying a spin chain with the Hamiltonian

(the XYZ model), where o„x = o„', a-,' = o*, o„z = a] are the Pauli spin matrices acting in the two-dimensional space 9Bn = C 2 , Baxter in [25], [26] introduced the set of matrices

where the coefficients wp(X), p = 0,1,2,3 are expressed in terms of the Jacobi elliptic function sn(u,k) = k~i/2H(u)/9(u) (see, e.g., [50])

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QUANTUM COMPLETELY INTEGRABLE MODELS IN FIELD THEORY

125

with modulus k and real period K. If we introduce the combinations then

The parameter p is inessential, the parameters k,jj are uniquely related to the ratios Jk :J :JZ,

Baxter showed that the Hamiltonian for the periodic problem for a chain of length N (o£ + , = of, p = 1,2,3) is given by the formula

analogous to (1.64), with Jz — cn(2rj,A:) dn(2ri,k). Baxter based his work on the relation of the one-dimensional-chain quantum mechanical problems to two-dimensional-lattice classical statistical physics problems. This relation, established by Onsager on the famous example of the Ising model, helped him find an exact solution to that model. Thus we see the intriguing relation between Onsager's ideas and the inverse problem method in its quantum version. Let us return to the investigation of the transition matrix TL(X). Its matrix elements are operators in %&L and have a rather complex dependence on the Schrodinger canonical operators. Nevertheless, it is possible to derive explicit commutation relations for these matrix elements. Furthermore, the computations are simpler than the corre­ sponding computations of Poisson brackets for transition coefficients aL(\) and bL(X) in the classical case. In fact, it is sufficient to calculate the commutation relations for the infinitesimal transition matrices Ln(K) in order to obtain the general commutation relations. Let us explain this in more detail. For a universal representation of commutation relations it is convenient to employ the tensor product; for two 2 x 2 matrices M and N with matrix elements Mn and TV-, i,j = 1,2 the matrix M <S> N is realized as a 4 X 4 matrix with 2 x 2

206

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L. D. FA DDE YE V

blocks:

In this notation the commutation relations of the matrix elements of the matrices Ln(X) and Ln( LL) have the form: R(X,

M )(L„(\)

® Ln( ix)) = (L„( /x) 8 Ln(X))R(X, LL).

(2.41)

Here R(X, LL) is a onumber 4 x 4 matrix independent of n. The validity of (2.41) can be checked directly, and in the computations we do not go outside the framework of the quantum mechanical problem for systems with one degree of freedom. It is evident that the very fact of the validity of (2.41) is a reflection of the rather special structure of the matrices Ln(X). For lattice problems (the T.Ch. and XYZ models) the relation (2.41) is exact. The matrix R(X, LL) for the XYZ model is

where

The quantity R(X, ju) has this same general form for the remaining models considered; they differ only in the explicit form of the coefficients a(X, LL), b(X, LL), C(X, LL), and d(X, /x). For the T.Ch. model these coefficients are given by the formulas:

For continuous models the relation (2.41) is satisfied approxi­ mately, omitting terms that vanish for A -> 0. The coefficients a, b, c, and d of the matrices R(X, fx) are given by the formulas: 1. The N.S. model

207

QUANTUM COMPLETELY

2.

The H.F. model

3.

The S.G. model

INTEGRABLE

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The local commutation relation (2.41) immediately leads to an analogous relation for the total transition matrix TL(X). In fact, the following relation is fulfilled for the matrix Ln with c-number (com­ mutative) matrix elements

But it is satisfied in our case as well for Ln = Ln(X) and L'n = Ln{ /x), since the matrix elements of the matrices Ln(\) and Lm(ii) commute for n T^ m. This property of ultralocality of the matrices Ln has already been mentioned. It follows from (2.8), (2.41), and (2.48) that for the transition matrices TL(X) the following relation is valid: which leads to a set of commutation relations for the matrix elements AL(X), BL(X), CL(X), DL(X). The explicit form of these relations will be presented in the follow­ ing section when we consider specific models. We shall see that (2.49) is actually the quantum analog of the Poisson brackets for the transition coefficients aL(X) and bL(X). The surprisingly simple deriva­ tion of the relation (2.49), actually based on the calculation of a few simple commutators, indicates that quantum dynamics is, as a rule, described more simply than the corresponding classical dynamics. In concluding this section we make several comments regarding the relations (2.41) and (2.49). The relation (2.49) was obtained for the XYZ model by Baxter [25], where, however, it was used only to prove the commutativity of the traces of the transition matrices AN(X) + DN(X)—the generating func­ tion for the commuting integrals of motion. In our approach we shall make full use of (2.49).

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A simple examination of the expressions for the matrix elements of the matrix R(\, /x) for the models considered shows that actually we are dealing with a universal matrix R. In fact, the most general expression (2.43) goes over into (2.47) as fc-»0 where k is the modulus of the elliptical functions, since sn(u, &)-»sinu;

(2.50)

In turn, (2.47) degenerates into (2.45) as K^> oo. The matrix (2.46) differs from (2.45) in the substitution \ - » 1/A (K = 1), and (2.44) is the particular case of (2.45) for K = — 1. Also note that the matrix (2.43) practically coincides with the operator-matrix L„ for the XYZ model. In fact, in this case the space SB„ is two-dimensional, and Ln can be written as a 4 X 4 matrix, if each matrix element in this matrix is expressed as a 2 X 2 matrix. It is evident from (2.35) that, on substituting explicit formulas for the Pauli matrices into this expression, Ln is written as

following which our assertion follows from a comparison of (2.42), (2.43) and (2.37). A matrix-operator like L (we omit the inessential label n of the spin space 2B„) for the XYZ model is naturally considered as a matrix with two sets of indices: two "auxiliary" and two "quantum" indices. We introduce the matrix

Here e 0 , o = 1,2 is a basis in spin space, the operator L- is a matrix element of L. The relation (2.41) for the XYZ model is rewritten in this notation as follows:

This relation has already appeared in the literature [51] during the discussion of the many-body factorizing 5-matrices. Here again we observe one more intriguing relationship of the quantum version of the inverse problem method with other approaches in onedimensional mathematical physics. A. Zamolodchikov [52] has shown that the relation (2.53) is inter-

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preted as the condition for associativity of the algebra with formal generators A,(\) that satisfy the relations

The generators At(A) in his work served as the creation operators of the "in" and "out" states of the many-body system. The relations (2.53) were later investigated both by A. Zamolodchikov as well as by other authors [52], [53]. They found examples where the indices i,k,a, /S run through not two, but an arbitrary number of values. These generalizations may, possibly, lead to new completely integra­ t e models describing particles with internal degrees of freedom. The relations (2.54) resemble the commutative relation due to H. Weyl from the theory of Schrddinger canonical operators. It is there­ fore natural to look for a realization by means of shift operators and multiplication by a function. Such a realization was recently con­ structed by I. Cherednik. We shall not cite his explicit formulas here, since we have already digressed far from our main theme. Our statement is sufficient to show that quantum completely integrable systems are associated with interesting mathematical objects requiring further study. In the next section we return to the main theme and see how the properties of transition matrix coefficients already obtained enable us to find the spectra of the quantum Hamiltonians for the models considered.

3.

The N. S. Model

In this section we shall make a detailed investigation of the N.S. model in its quantum version. First we examine the case of attraction K < 0. The equation is equivalent to a set of Schrodinger equations in the configuration representation:

studied in [14]—[15]. We shall show that the inverse problem method quickly brings us to the known answers for their spectrum.

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In the case of repulsion the spectral problem in a subspace with a finite number of particles is simpler than the preceding problem, since there are no bound states. We shall therefore investigate the problem of a Bose gas with a positive density. For this purpose, we introduce the chemical potential A, changing the Hamiltonian in the usual way:

where N is the particle number operator, and find a realization of the Hilbert space in which the operator HA is nonnegative. As a result, we naturally arrive at the formulas obtained by Lieb and Liniger [54], and Yang and Yang [56]. Let us begin with the first problem. We can immediately work in an infinite volume, using the Fock space generated by the operators \p*(x) from the vacuum ft, as the state space 2B; the vacuum state satisfies the condition The formal condition

symbolizes the fact that the particles in this case are retained in a finite volume. Under this condition the matrix Ln as |n| —> oo ceases to depend on n and becomes a numerical matrix

and we can pass to the limit L -> oo in the transition matrix as was done in the classical case (see (1.10)). We set

The matrix T(X) is the quantum analog of the classical transition matrix, composed of the transition coefficients a(\) and b(\). Our

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task consists in determining the commutation relations of its matrix elements A(\), B(\), B*(\), and A*(\), that generalize the Poisson brackets (1.20) and (1.21). For this purpose we have to pass correctly to the limit L -> oo in (2.49). Before doing this, let us present an alternative definition of the matrix T(\), based on the quantum generalization of the integral equation (1.16). We must consider the solution Gn(\) to the system

satisfying the boundary condition

Eq. (3.8) with the condition (3.9) can be written in the form of a single relation

which can be solved by successive approximations. As n -> oo the solution G„(\) has the asymptotic form

where the matrix T(\) is given by the expression

which provides a constructive definition of the transition matrix. The limiting transition as A -> 0 should be made in this expression. The boundary condition (3.5) is fulfilled only in the weak sense, so it cannot be used naively in determining the limit in the tensor product L n (X)® L„(JLI). In multiplying these matrices according to the rule (2.40) we encounter operator expressions xZx* and XiXj • T h e first may be assumed to vanish as |n|-»oo. The second expression, however, is rewritten in the form of

and tends to A as |«|-» oo. As a result, we may say that as |/i|->oo

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L. D. FADDEYEV

where the matrix W(X, /x) is given by the formula

and satisfies the relation

It follows from (3.14) that the limit as L-> oo of the expression

exists. Now let us approximate the basic commutation relation (2.49) as

using (3.7), and multiply both sides by W(fi, \)~N from the left and by W(X, n)~N from the right. Using (3.16), we move W(n,\yN to the left side through R(\, n), replacing it by W(K, \i)~N, and perform the analogous operation on the right side. Following this, combina­ tions of the type (3.17) will appear on the left and right sides so that we can pass to the limit as JV-»oo. Evidently, for this purpose we need to compute the limits

In this notation the relation (3.18) leads to the formula

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where

Let us calculate these matrices. Because of the block structure of the matrices R(X, n), W(\, n) and V(\) <S> V( ju), it is sufficient to multiply their middle blocks. From (3.15) we have

so that the limits (3.19) and (3.20) give

where

From (3.25) we see that

so it is clear that Then

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L. D. FADDEYEV

where

We have obtained a rich set of commutation relations:

for the matrix elements A(X), A*(\), B(X) and B*(\) of the transition matrix T(\) introduced in (3.7). We give some of them:

All these relations are obtained for A T^ /A; the last formulas become singular for A = /x. A more thorough examination of the limiting transition just made shows that (3.32) holds for Im A > 0, while for real A it is understood in the sense of a limit from the upper half-plane. It is difficult to give a meaning directly to the relation (3.33) for A = JLI. However, if we introduce the operator

this relation yields

The relations (3.32) are the quantum analogs of (1.20) and (1.21). They show that A(\) is the generating operator-function for the commuting integrals of motion and that the operator B(X) simply varies with time if the Hamiltonian H is computed from A(k) by means of formulas analogous to the classical (1.18) and (1.19). Using the integral equation (3.10) we can examine the behavior of the operator In A (A) as A —> oo and show that it has the asymptotic form

215

QUANTUM COMPLETELY INTEGRABLE MODELS IN FIELD THEORY

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where, in particular,

We shall not give these computations here because of lack of space. The relation (3.32) makes it possible to give the complete spectral characteristic of the operator A (X), and, with it, of all the commuting operators C„ and, in particular, H. In view of (3.4) the local transition matrix Ln, when applied to the vacuum Q, becomes a triangular matrix:

As a result, the action of the diagonal elements AL(X) and Al(X) of the matrix TL(X) on ft reduces to multiplication of the diagonal elements of the matrix L n , so that

and, as follows from the definition (3.7),

We shall now show that the operator B(X) serves as the creation operator. Let

From relations (3.32) and (3.40) we have

where

Thus, the \p(Xv . . . ,Xn) are eigenstates for all the operators A(X). The eigenvalues of the operators Cn are obtained if In a (A, /i) is represented in the form

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From a comparison of (3.36), (3.42) and (3.44) it is evident that

In particular,

These formulas show that the state »//(X,,..., X„) describes n free particles with momenta X,, . . . , Xn. Note that the eigenvalue a(X, /i) of the operator A(X) in the one-particle state I//(M) is an analytical function of X in the upper half-plane and has one zero there. It is just this property that is possessed by the classical transition coefficient a(X) for the soli ton potential (1.13). In order to have greater similarity of the classical and quantum formulas it is convenient to modify the operator A(X): We shall henceforth use the operator A (X). Its one-particle eigenvalue has the form

and coincides with (1.14) for TJ = /x -I- iy/2. For real X which is the quantum analog of the fact that a soliton potential is reflectionless. The many-particle eigenvalue

also has these two properties (1. analyticity for ImX > 0 and vanish­ ing once; 2. \a\ - 1, ImX = 0) for an appropriate choice of the complex parameters X,, . . . , X„:

These Xj are selected using the argument that the poles of the individual factors in (3.50) are cancelled by the zeros of neighboring

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QUANTUM COMPLETELY INTEGRABLE MODELS IN FIELD THEORY

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factors, as a result of which (3.50) becomes

and again coincides with (1.14) for TJ = fi/n + iy/2n. The eigenvalues of the operators N and H for the state

where \,j

<= I, . . . , n given in (3.51), have the form:

and correspond to bound states of n particles with total momentum ju,. The value of the binding energy

that we calculated from (3.52), (3.44), and (3.37), coincides with the exact value obtained in [14] by solving the Schrddinger equation. The eigenstate of the operators A (X) is given in general form by formula (3.42), where the X are combined into several chains of the type (3.51) of different lengths « , , . . . , np. Such a vector describes the state of p bound complexes, each of which consists of n particles. The eigenvalue of the energy operator H is obtained as the sum of the expressions E_(/0, given in (3.54), /i plays the role of the momentum of the bound complex. The quantization of the imaginary part of the parameter TJ, characterizing the one-soliton solution,

was noted in [10] in the quasiclassical approximation. We have verified here that this quantization is exact in a definite sense. We have also cleared up the mystery of the coincidence of the oneparticle states, obtained by quantization of the "continuous spec­ trum," and the lowest state of the quantized soliton. The states (3.41) are not orthonormal, which is explicitly manifest in (3.33). The relation (3.35) suggests that orthonormal states can be obtained in the form

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L. D. FADDEYEV

However, the operators R(X) for different A do not commute, and the formula (3.57) needs to be refined. The perturbation theory for Eq. (3.10) makes it possible to find exactly the first terms in the expan­ sion of the operators B(\) and R(X) in series in terms of normal products of the operators t//*(x) and \p(x). As a result we get the opportunity to compare vectors of the type (3.41) or (3.57) with the solutions to the Schrodinger equation ^(JC,, . . . , xn |A,, . . . , A„) from [14], using the formula

It then becomes evident that for Re A, > ReA2 • • • > ReA„ the vec­ tors ^(A,, . . . , A„) coincide with in-states, and for Re A, < ReA 2 < • • • < ReA„, with the out-states. As a result we are able to compute the S-matrix using the relation

which follows from (3.31) and (3.32). We shall not give details, because we are still unable to treat this problem without reference to perturbation theory. We shall only note that (3.59) is the simplest realization of the Zamolodchikov relations (2.54). We are convinced that the quantum inverse problem method actually provides an exact solution for the quantum problem for the N.S. model, leading to exact expressions for the eigenvectors, eigen­ values, and S-matrices. The formulas of the classical inverse problem method in their quantum interpretation become more elegant and visualizable. Now let us turn to the case of the Bose gas and consider the Hamiltonian (3.3). The basic difference from the case of attraction is that in Hilbert space, where the operator HA is given as where EQ is a constant (ground state energy), and Hex is positive, the relations (3.5) are not valid. Furthermore, the vacuum ft has no relation to the ground state ftphys, for which As a result, there is no sense to the procedure of passage to the limit for L -> oo, which led us to "action-angle" variables in the case of attraction.

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Under these conditions we are forced to make a more thorough investigation of the problem in the finite volume L < oo. Let us see what we already have. First, we have a simultaneous eigenstate for the operators AL(k) and DL(\), i.e., the vector £2:

(cf. (3.39) and the preceding discussion). Second, the commutation relations (2.49) are satisfied, and, in particular,

where

These data are sufficient to construct a large set of eigenvectors of the operator which is the quantum analog of the function fL(A) in (1.26), which appeared in the formulation of the N.S. model with periodic Cauchy data. Both the operators FL(X) and BL(\) commute for different A: and with no further comments we shall assume that FL(\) is the generating operator-function for the commuting integrals of motion. Let us again examine the vector and show that it is an eigenvector of the operator FL(\), if A,, . . . , A„ satisfy a certain system of transcendental equations. Let us note that, in view of (3.62) and (3.63) we have, carrying AL(\) through all the BL(\f) in (3.68),

220 140

L. D. FADDEYEV

where

and

The first summand on the right side of (3.69) is obtained when we carry AL(\) through BL(\j), using only the first term on the right side of (3.63). The second summand is obtained from all the remaining commutations. The operator expression

appears if we commute AL(\) with 2?L(A,), using the second term in (3.63), while for further commutations we use the first term. This observation brings us to the expression (3.71) for A,(A; A,, . . . , A„). The validity of the remaining formulas follows from the symmetry of all expressions with respect to the permutation of A,, . . . , A„. A similar computation shows that DL(\)\f/(\}, . . . , A„) is expressed in a form analogous to (3.69) with the replacement of A(A; A,, ...,A„)by

As a result we see that ^(A,, . . . , A„) is an eigenvector of the operator FL(\) with eigenvalue

if A , , . . . , \

satisfy the system of equations

which is rewritten more explicitly as

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QUANTUM COMPLETELY INTEGRABLE MODELS IN FIELD THEORY

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or

and does not contain A, serving only as a condition on A,, . . . , An. Equation (3.78) is the fundamental tool of investigation in the theory of the one-dimensional Bose-gas by Lieb and Liniger [54], [55]. It arose there when periodicity conditions were imposed on the eigenfunctions of the Schrodinger Eq. (3.2), obtained by means of the famous Bethe ansatz [27]. We may say that the formula (3.68) gives the algebraic form of this ansatz. Thus, we observe one more remark­ able connection of the quantum inverse problem method with other achievements of one-dimensional mathematical physics: it gives an algebraic interpretation of the Bethe ansatz and, thus, indicates its place in the theory of completely integrable quantum system's. Our further exposition does not contain the refinements of the discussions in [54], [55], or [56], and we shall cite only the basic formulas. The energy of the state \p(Xv . . . , \ , ) has the form:

It follows from an analysis of the normalization of the vector «|>(A,, . . . , \ ) that it vanishes if any two values A coincide. In other words, we may say that all the A in the expressions (3.68) and (3.79) are different, and, using the terminology of the free Fermi-gas, we may say that in the state ^(A,, . . . , A„) the levels with momenta A,, . . . , A„ are occupied. It is evident from (3.79) that in the ground state levels with small A, should be occupied. As L -> oo these A coalesce and fill the interval — q < A < q, where q is determined from the condition that excitations of the ground state have positive energy. Thus, we seek the ground state S2phys in the form

and assume that the A. satisfy the system (3.78), remaining in the interval - ^ < A < ^ as L -» oo. It is quite easy to verify that \j+, - Ay is of order 1/L and that the asymptotic density

222

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L. D. FADDEYEV

satisfies the following equation:

where

Elementary excitations of the ground state may be sought in the form (particle)

or (hole)

where in the last formula the prime indicates that the operator B(\j) for X = £ has been omitted in the product. The occupation numbers of the ground state X- and X" in these states differ from the popula­ tions in (3.80). The differences (vacuum polarization) arise in solving the system (3.78). Let us introduce the functions

These relations have a finite limit as L-» oo. The function Fh(p) has a discontinuity for X — £, and the jump at £ is equal to 1. Let us introduce the derivatives of these functions i*"p(X) and F^(X); we assume these functions to be continuous, subtracting the contribution of the jump in the second case. It follows from Lieb's arguments [55] that as a consequence of (3.78), in the limit as L^> oo, F'v and F'^ satisfy the integral equations

with the same kernel K that appears in Eq. (3.82). Let us consider the eigenvalues £0(X), Ep(\, £) and £h(X, £) of the operator FL(K) in the states &phys, * p (£), * h ( 0 - w e will assume that ImX > 0; in this case as L-> oo we may neglect the second term in the expression (3.75) for the eigenvalues. As a result we obtain for the

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eigenvalues E0(X):

for the ratio of Ep(K, £) to EQ(X) we have the expression

the ratio Eh(\,£)/E0(\) has a similar form with F'p replaced by F^. From these formulas and (3.37) we obtain the eigenvalues of the energy operator

Now we can determine the parameter q, by excitation energies cp(£) and eh(£) be positive for and ||| < q, respectively. From the results of Lieb Yang [56] it follows that for this it is sufficient to

requiring that the all £, when |£| > q [55] and Yang and set

or Equations (3.94) and (3.95) lead to the same relation, making it possible to express the maximum momentum q in the ground state in terms of the chemical potential A. With this we conclude the analysis of the Bose-gas theory. The reader will find the technical details in the original literature cited. Our purpose was to illustrate the quantum inverse problem method using the example of a system in a finite volume. There remains unsolved the attractive problem of obtaining the basic formulas for passage to the limit L -> oo more directly, as was done in the first half of this section for the attractive model. We hope that such an approach exists.

4.

Other Models

In this section we shall briefly describe the basic features of the quantum inverse problem method in its application to the most

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interesting models: the S.G. and the XYZ models. We have already seen above that the latter model is the most universal model—the other models considered are obtained from this XYZ model by appropriate limiting transitions. At the same time, as we shall see, the XYZ model requires the most refined investigating procedure, so that the N.S. and S.G. models, which are interesting in themselves, should be studied sepa­ rately, as we intend doing. Simultaneous with their investigation we more naturally and gradually approach the complexity of the basic method. Thus, let us consider the S.G. model with the local transition matrix L„, given in (2.28). We do not a priori know the asymptotic behavior of Ln as \n\ -» oo and we shall have to use the procedure that was already used on the example of the Bose gas, investigating the system in a finite volume and passing to the limit as L-> oo. Let us introduce the transition matrix TL(X) and try to find the set of eigenvectors for the operator FL = tr TL(X) by the algebraic Bethe ansatz. The commutation relations of the type (3.63), (3.64) are valid for our example also, as follows from (2.47). In this case

It remains to see if there exists an analog for the state £2. In the case of the N.S. model the state J2 had the form of the product

of the local vacua fin e 2B„, satisfying the condition For the S.G. model there is no state in SB„ that would be annihilated by the operator in the lower left corner of the matrix Ln. Such a state, however, exists for the two-step transition matrix Ln+lLn. We realize the space SB n+ , <8> 2B„ as the space of functions f(qi,q2) of two variables qx and q2(- oo < qt < oo), where the operators vn and un act on qx, and vn + v un+l act on q2. The lower left corner of the matrix Ln+lLn has the form

225

QUANTUM COMPLETEL Y INTEGRA BLE MODELS IN FIELD THEOR Y

145

and is realized in this space as follows:

The equation has the solution

We have selected the terms of order A2 in the last formula so that fn+i,n IS a n eigenvector of the diagonal matrix elements an+ln and 4,+ i.n o f ^ e matrix Ln + x,Ln:

Multiplying the local vacua f„+Xt„, we obtain the state

which is an eigenstate of AL(X) and DL(X):

where Further analysis differs only in some technical details from the Bose-gas case. We shall not present it here, referring the reader to the

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L. D. FADDEYEV

original work [33]. The role of Lieb's contribution [55] is served here by [57], where an investigation is made of the coordinate representa­ tion of the Bethe ansatz for the massive Thirring model, equivalent to the S.G. model [58]. The results obtained for the spectrum refine the quasiclassical answers in [11], [12]. The S-matrices predicted by A. Zamolodchikov [59], were computed from formulas equivalent to (3.57) and (3.59) by V. Korepin [60]. Now let us turn to the XYZ model and, for this purpose, let us consider the local transition matrix Ln (2.35). We assume that because, in the case of equality H>, = w2, the coefficient d in the matrix R of (2.42) vanishes and the local vacuum Qn exists, so that this case can be investigated by the procedure already described. Furthermore in this case, as shown by Ye. K. Sklyanin and P. P. Kulish [32], one can develop a version of the approach analogous to that used by us for the N.S. model with attraction, based on simple asymptotic conditions for the local Schrodinger operators and the existence of the matrices V(\) and W(\, fi). In satisfying the condition (4.14) there is no hope of obtaining the local vacuum either for the matrix Ln, or for a finite product of such matrices. We shall therefore use the following consideration: the transition matrix TN(\) is calculated if instead of the local transition matrices Ln we use matrices L'n, gauge-equivalent to Ln:

Here the matrices Mn are arbitrary nondegenerate c-number 2 x 2 matrices. Actually, the new transition matrix

differs from the matrix TN only by a simple linear transformation

It happens that the gauge transformations of Mn can be chosen so that each matrix L'„ has a local vacuum that is annihilated by its lower left element. The appropriate formulas can be found in Bax­ ter's papers [25], [61]. It turns out that there are many such matrices; henceforth we shall use the set

227 QUANTUM COMPLETELY INTEGRABLE MODELS IN FIELD THEORY

147

where

and g(u) = H(u)@(u), where H(u) and @(w) are the Jacobi thetafunctions already mentioned in §2. The normalizing factor 1/g in (4.20) has been introduced for convenience and is not important. The corresponding local transition matrices L'n will be denoted by

and the total transition matrix by T^(X):

and its matrix elements by A^(X), B^(X), CN(X), and D^X). The local vacuum Q'n G C 2 , satisfying the equation

exists and is given by the formula

where e* are the eigenvectors of the spin operator o*, but in constrast to the N.S. model case Q'n is not an eigenvector of the operators aln and 6„'. However, it transforms simply under the action of these operators:

These last formulas are checked on the basis of the following formulas:

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L. D. FADDEYEV

that are a convenient form of the addition formulas for the @functions introduced by Baxter. We see on the basis of (4.25) and (4.26) that the vectors

satisfy the relations

and, unfortunately, are not eigenfunctions for either A'N or DlN. However, by using them we can still construct a set of eigenvectors for the commutative operators The commutativity of the FN(X) follows from the similarity of the operator matrices TN(X)®TN(ji) and 7\,(/i)<8> TN(X), noted in (2.49). The remaining commutation relations following from (2.49) are more complex than, say, (3.63) and (3.64), and later we shall have to switch from the operators BN(X) to other linear combinations of the elements of the transition matrix TN(X). For this purpose we introduce the matrix set

and denote their matrix elements by Aki(X), Dk ,(X). We have, obviously,

Bkl(\),

Ckl(X),

and

for any /. Furthermore, the operators A^X) and D^(X) introduced above are written in the new notation as follows:

The commutation relations (2.49) lead to the following formulas for the operators just introduced:

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QUANTUM COMPLETELY INTEGRABLE MODELS IN FIELD THEORY

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where

For the proof one must use the explicit expressions for the elements of the 7?-matrix in terms of the ©-function and the addition theorem (4.28). These formulas generalize to this case the formulas (3.63), (3.64), and (3.67) which were used in §3 to construct the algebraic Bethe ansatz. We shall use them now to construct the generalized Bethe ansatz. For this purpose we consider the vector

where n = N/2 (we assume that N is even). This vector is symmetric with respect to permutations of the parameters X,, . . . , A„ by virtue of (4.36). We emphasize that, in contrast to (3.68), the number of operators B(K) in (4.40) is not arbitrary, but precisely equal to N/2. Using the commutation relations (4.37) and (4.38), as well as (4.34), we obtain, by analogy with the considerations in §3, that

where

230 150

L. D. FADDEYEV

and

We multiply the relation (4.41) by exp(/0/), where 9 is a real parame­ ter in the interval (0, 2TT), and sum over all integers /. Then we see that the vector

is an eigenvector of the operator FL(X), with the eigenvalue

provided the parameters X,, . . . , \, satisfy the system of transcen­ dental equations

Equations (4.48) were obtained by Baxter [25] who used a method very different from the Bethe ansatz. Later in [61] he proposed a coordinate form of the Bethe ansatz that again leads to Eq. (4.48). It seems to us that the considerations in these publications are very complicated. Their simple interpretation, obtained here in the course of developing the quantum inverse problem method, once more demonstrates its fruitfulness. The convergence of the series (4.46) must, of course, be investi­ gated. It may seem that this series sums to zero for almost all 9. Baxter's results show that the ground state is contained in (4.46) for 9 = 0. The situation is simplified if we assume, as Baxter did, that the coupling constant TJ is expressed rationally in terms of the periods K and K' of the ^-function, i.e., it satisfies a relationship where Q, p and q are integers. For sufficiently large Q any 17 can be approximated by a number satisfying (4.49). For such TJ all the objects Mk(\), Tkl(\) and \p,(X{, . . . , X„) introduced by us depend on the number / periodically with period Q. Because of this, in formula (4.40) for the Bethe ansatz it is sufficient to assume that 2n is

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congruent to N modulo Q

Furthermore, in sums of the type (4.46) it is sufficient to sum over a period, but one can choose for 8 only numbers of the type

Equation (4.48) for the occupation of the ground state and the elementary excitations in the XYZ model has been thoroughly inves­ tigated by the integral equation method in [62], to which we refer the reader for technical details. Our aim in this section was only to demonstrate that the inverse problem method leads naturally to a generalized Bethe ansatz for the XYZ model, which is the most universal quantum completely integrable lattice system. 5.

Conclusion

The material collected in this review shows that the inverse problem method for solving nonlinear evolution equations of field theory in one-dimensional space has a natural quantum generalization. Fur­ thermore, the quantum version makes many of the procedures of this method simpler and more visualizable and establishes interesting connections with other one-dimensional mathematical physics meth­ ods: the Onsager-Baxter methods for investigating transfer matrices in problems of classical statistical physics on plane lattices, the Bethe-Hulthen et al. methods for solving problems for quantum one-dimensional magnetics, etc. It may be asserted that all the achievements of one-dimensional mathematical physics in the investi­ gation of exactly solvable quantum models are to some extent in­ cluded within the framework of the quantum inverse problem method. This, undeniably, sheds new light on the extensive activity involv­ ing the inverse problem method that we observe in the literature of mathematical physics during the last decade. Let us make some comments about the prospects of the method and about unsolved problems. 1. The connection between the inverse problem method and the oribit method in the representation theory of Lie groups has already been noted in the introduction. A further development and refine-

232

152 L. D. FADDEYEV

ment of this connection will be fruitful for both theories. The quantum inverse problem method will lead to new results concerning the representations of infinite Lie groups. In its turn, the orbit method may provide an invariant interpretation of specific ap­ proaches in the inverse problem method that will be useful for its further generalization. The most convenient model for developing this connection is the Toda chain. It has a simple group interpretation [17]—[ 19], and a simple local matrix Ln (cf. (1.61)). However, so far, the techniques developed have not led to a suitable formulation of the Bethe ansatz for the periodic problem. 2. In this survey we have not considered matrix evolution prob­ lems corresponding to systems with several kinds of fundamental particles. Quantum models corresponding to such systems undoubt­ edly are of interest. In particular, in its classical formulation, it admits decaying solitons (cf. [39]-[42]), which after field quantiza­ tion, should lead to unstable particles. But the most urgent problem is the quantization of the nonlinear S2 o-model, or n-field theory. In this model the dynamical variable is the field n(x) with values on the two-dimensional sphere S2:

The equation of motion

represents the condition for zero curvature where the matrix operators

were introduced by Pohlmeyer [43] and Zakharov and Mikhailov [42] in a more general situation (see also [63]). The operator X contains derivatives of the /i-field with respect to x and is not ultralocal in the sense described in the text. Therefore we should either find a new ultralocal operator X for this model, or learn to work with weakly noncommuting local transition matrices. It is possible that along this line the relationship of the /i-field to the matrix generalization of the

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Toda chain, which has been repeatedly stressed by Polyakov [64], will be helpful. 3. In models with vacuum polarization, considered in the main part of the text, we had to work in a finite volume, solving the quantum periodic problem, and only then making the limiting transi­ tion to infinite volume. In spite of the successes along this line and, in particular, the determination of the general formulation of the Bethe ansatz, I do not find this situation satisfactory. It would be interest­ ing to trace which asymptotic conditions for the local Schrodinger operators appear in a Hilbert space built on the polarized vacuum (ground state), to find the correct analogs for the matrices V(X) and W{\, ft) from the theory of the N.S. model and to introduce the correct transition matrix T(X) for the auxiliary spectral problem over the entire axis — oo < x < oo. 4. I have already noted in the body of the text that the quantum generalization to date has been extended only to the "direct" part of the inverse problem method, i.e., the transformation from local Schrodinger operators to spectral data, the operator matrix elements of the transition matrix T(\) or TL(X). The "inverse" part of the method, i.e., the reconstruction of the initial Schrodinger and the time-dependent Heisenberg operators from the spectral data, is also very interesting. First, it may serve for the proof of the completeness of the eigenvectors found by means of the Bethe ansatz and its generalizations. Second, and this is the most important, it may lead to an expression for the Heisenberg operators in terms of the scatter­ ing data, which will make it possible to compute such interesting quantum system characteristics as Green's functions. 5. In the investigation of the S.G. model, as follows from [33], [57], to date only the eigenvectors of the Hamiltonian in the sector with zero topological charge have been found. Although this is sufficient for the answer to physical questions (for instance, to compute the whole 5-matrix), there still remains the important me­ thodical problem of constructing the charged sectors in Hilbert space and, in particular, the one-soli ton state. We can list more problems arising in connection with the quantum formulation of the inverse problem method. Of course, the main one involves finding methods for a multi-dimensional generalization of the whole formalism. It is not excluded that the quantum inverse

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L. D. FADDEYEV

problem method in this respect may prove to be more fortunate than the classical method. There are indications, unfortunately still very vague, that the role of the parameter x in the auxiliary spectral problem will be served by contours or surfaces in multi-dimensional space (see [64], [65]). My associates in Leningrad and I are actively engaged in further development of the quantum inverse problem method and, in partic­ ular, in solving the problems just formulated. We will appreciate the involvement of anyone who, with us, tackles this intriguing and highly promising area of modern mathematical physics. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

Gardner C , Green J., Kruskal M., Muira R., Phys. Rev. Lett. 1967, 19, 1095. Zakharov V. E., Shabat A. B., JETP, 1971, 61, 118; 1973, 64. Zakharov V. E., Faddeyev L. D., Functional analysis and appl. 1971, 5, 18. Zakharov V. E., Manakov S. V., TMP, 1974, 19, 332. Ablowitz M. J., Kaup D. J., Newell A. C , Segur H. Phys. Rev. Lett., 1973, 30, 1262. Takhtadzhyan L. A., JETP, 1974, 66, 476. Zakharov V. E., Takhtadzhyan L. A., Faddeyev L. D., DAN SSSR, 1974, 219, 1334. Takhtadzhyan L. A., Faddeyev L. D., TMP, 1974, 21, 160. Kaup D. J., Stud. Appl. Math., 1975, 54, 165. Kulish P. P., Manakov S. V., Faddeyev L. D., TMP, 1976, 28, 38. Dashen R. J., Hasslacher B., Neveu A., Phys. Rev. 1975, D l l , 3424. Korepin V. E., Faddeyev L. D., TMP, 1974, 25, 143. Faddeyev L. D., Korepin V. E., Phys. Reps., 1978, 28C, 3. Berezin F. A., PokhiJ G. P., Finkel'berg V. M., Vestnik MGU, 1964, Ser. 1, No. 1, 21. McGuire J. B., J. Math. Phys., 1964, 5, 622. Yang C. N., Yang C. P., Phys. Rev., 1966, 150, 321. Adler M., Inventiones Math., 1979, 59, 219. Kostant B., Inventiones Math., 1978, 48, 101. Reiman A. G., Semenov Tyan-Shanskii, M. A., Inventiones Math. 1979, 51. Manin Yu. I., Lebedev D. R., 1978. Preprint ITEP-155. Lax P. D., Comm. Pure Appl. Math. 1968, 21, 467. Kirillov A. A., Elements of the Representation Theory of Lie Groups. Nauka, 1972. Phase Transitions and Critical Phenomena. Ed. by S. Domb and M. S. Green, Academic Press, London, New York, 1972. Mathematical Physics in One Dimension, ed. E. H. Lieb, D. C. Mattis, Academic Press, New York and London, 1966. Baxter R. J., Annals of Physics, 1972, 70, 193. Baxter R. J., Annals of Physics, 1972, 70, 323. Bethe H., Z. Physik, 1931, 71, 205. Hulthen L., Arkiv Mat. Astron. Fysik, 1938, 26A, 1. Yang C. N., Yang C. P., Phys. Rev., 1966, 150, 321. Sklyanin E. K., Faddeyev L. D., DAN SSSR, 1978, 243, 1430. Sklyanin E. K., DAN SSSR, 1979, 244, 1337.

235 QUANTUM COMPLETELY 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

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Kulish P. P., Sklyanin E. K., Phys. Lett. A (1979). Sklyanin E. K., Takhtadzhyan L. A., Faddeyev L. D., preprint LOMI, E-1-79, 1979. Takhtadzhyan L. A., Faddeyev L. D., UMN, 1979, 34, 13. Dubrovin B. A., Matveev B. V., Novikov S. P., UMN, 1976, 37, 55. Krichever I. M., UMN, 1977, 32, 183. Takhtadzhyan L. A., Phys. Lett., 1977, 64A, 235. Manakov S. V., JETP, 1973, 65, 505. Zakharov V. E., Manakov S. V., JETP, 1975, 69, 1654. Budagov A. S., Takhtadzhyan L. A., DAN SSSR, 1977, 235, 805. Budagov A. S., Proceedings of LOMI scientific seminars, 1978, 77, 24. Zakharov V. E., Mikhailov A. V., JETP, 1978, 74, 1953. Pohlmeyer K., Comm. Math. Phys., 1976, 46, 207. Lund F., Phys. Rev., 1977, D15, 1540. Toda M., Progr. Theor. Phys. Suppl. 1970, 45, 174. Manakov S. V., JETP, 1973, 65, 1392. Flaschka H., Progr. Theor. Phys., 1974, 51, 703. Manakov S. V., Thesis, L. D. Landau ITF, 1975. McLaughlin, J. Math. Phys., 1975, 16, 96. Whittaker E. T., Watson G. N., Course of Modern Analysis pt. II, Moscow, Fizmatgiz, 1963. Yang C. N., Phys. Rev. Lett., 1967, 19, 1312. Zamolodchikov A. B., Zamolodchikov Al. B., Nucl. Phys., 1978, B133, 525. Karowski M., Thun H. J., Truong T. T., Weisz P. H., Phys. Lett., 1977, 67B, 321. Lieb E. H., Liniger W., Phys. Rev., 1963, 130, 1605. Lieb E. H., Phys. Rev., 1963, 130, 1616. Yang C. N., Yang C. P., J. Math. Phys., 1969, 10, 1115. Bergkhoff H., Thacker B., Phys. Rev. Lett., 1979, 42, 135. Coleman S., Phys. Rev., 1975, Dll, 2088. Zamolodchikov A. B., Comm. Math. Phys., 1977, 55, 183. Korepin V. Ye., TMP, 1979, 42, 211. Baxter R. J., Annals of Physics, 1973, 76, 2; 25; 48. Johnson J. D., Krinsky S., McCoy B. M., Phys. Rev., 1973, A8, 2556. Semenov Tyan-Shanskii H. A., Faddeyev L. D., Vestnik LGU, 1977, No. 13, 81-88. Polyakov A. M., Phys. Lett. 1979, B82, 274. Arefieva I. Ya., Lett. Math. Phys., 1979, 3, 270.

236 Uspekhi Mat. Nauk 34:5 (1979), 13-63

Russian Math. Surveys 34:5 (1979), 11-68

THE QUANTUM METHOD OF THE INVERSE PROBLEM AND THE HEISENBERG XYZ MODEL L. A. Takhtadzhan and L. D. Faddeev Contents

Introduction § 1. Classical statistical physics on a two-dimensional lattice and quantum mechanics on a chain §2.Connection with the inverse problem method §3.The six-vertex model §4. Generating vectors and permutation relations §5.The general Bethe Ansatz §6.Integral equations Conclusion Appendix 1 Appendix 2 References Introduction

In this survey we look at the problem of the diagonalization of the Heisenberg Hamiltonian H of a system of N interacting particles with spin 1 /2. This operator acts in a Hilbert state space Jg N

and has the following form:

Here Jx, Jy, and Jz are real constants; the spin operators o;n have the form li

11 15 25 29 37 43 50 57 58 63 65

237 12

L. A. Takhtadzhan andL. D. Faddeev

where the a7 are Pauli operators, which in the orthonormal basis of C2 have the following form:

and a4 = I is the identity operator in C 2 . In (3) it is assumed that the n-th factor in the tensor product is oJ and the remaining factors are identity operators, so that o'n acts non-trivially only in I)n. We also assume that

that is, periodic boundary conditions hold. The energy operator thus introduced is a matrix of order 2N x 2N. The quantum-mechanical model of magnetism thus described was proposed by Werner Heisenberg in 1928 [ 1 ] and was actively investigated by many scholars beginning with Hans Bethe. In the present-day literature on mathe­ matical physics it is known by the jargon name "the XYZ model", which is used in the situation of general position Jx =£Jy =tJz. The particular cases Jx = Jy =£ Jz and Jx = Jy = Jz = / are called the XXZ and the XXX models, respectively. Heisenberg's model turned out to be very fruitful in the theory of magnetism and there is an extensive physical literature devoted to it. In recent years it has attracted the attention of mathematical physicists, since it turned out that the problem of finding the eigenvalues and eigenvectors for H can be solved exactly, in a certain sense, and requires beautiful mathematical constructions. The first step on the way to a solution of this problem was taken by Bethe in 1931 [2], when he examined the completely isotropic case of the XXX model and found the eigenvalues and eigenvectors of its Hamiltonian. Bethe's exact solution counts as one of the fundamental results in the theory of spin models, and the method proposed by him, the famous "Ansatz" (substitution) of Bethe, has been applied successfully to other many-particle models in onedimensional mathematical physics (the term is due to Lieb and Mattis, see [3]). There is physical interest in the study of asymptotic properties of the model as N~+ °°, and particularly, of the ground state of excitation — the state with the least energy and the states with almost the least energy. The case / > 0 in the XXX model corresponds to ferromagnetism. In this case the ground state is very simply constructed and the excitations can be found with the help of Bethe's ansatz. When J < 0, we are concerned with antiferromagnetism. The ground state for this case was constructed in 1938 by Hulthen [4], who started

238 The quantum method of the inverse problem and the Heisenberg XYZ model

13

from Bethe's formulae and calculated the asymptotic expression for the energy of the ground state as N -* °°. Twenty-four years later des Cloiseaux and Pearson in [5] constructed excitations upon the antiferrOmagnetic ground state and found the asymptotic expression for their energy. The generalization of Bethe's method to the XXZ model presents no difficulties of principle. Yang and Yang ([6] - [ 8 ] ) showed in 1968 that Bethe's Ansatz in its classical form is applicable to this model too. The papers [6] - [ 8 ] contain a detailed investigation of the problems arising in the justification of the limit passages as N -*■ °° in the Bethe-Hulthen method, and raised this group of problems to a higher mathematical level. At the same time it became clear that the solution of the completely anisotropic XYZ model requires to all appearances new technical ideas, and the question of its solubility remained open right up to 1972. In 1972 Rodney Baxter in his remarkable papers [9] - [ 1 0 ] (the results were announced by him in 1971 in [ 11 ] —[ 12]) gave a solution for the XYZ model. He discovered a link between the quantum XYZ model and a problem of twodimensional classical physics, the so-called eight-vertex model (its exact definition will be given in the main text), — in fact, it had been his principal aim in his paper to investigate this. Baxter made use of the ideas of Kramers— Wannier [13] and, in particular, of Onsager [14] on the transfer matrix, and of Lieb's solution [ 15] —[ 18] of the special case of the eight-vertex model, the so-called six-vertex model connected with the quantum XXZ model. In [9] [10] he obtained a system of transcendental equations generalising the system derived from Bethe's method, and with its help he calculated the energy of the ground state of the XYZ model. In the subsequent series of papers [ 19] — [21 ] Baxter, by means of a very complicated and non-trivial generalization of Bethe's Ansatz, was able to construct the eigenvectors and to find the eigen­ values of the transfer matrix, and so to solve completely Heisenberg's XYZ model. In 1973 Johnson, Krinsky, and McCoy [22], using Baxter's results, calculated the energy of the excitations of the XYZ model. Although Baxter's work is rightly considered one of the most important achievements in statistical physics since the time of the famous paper of Onsager [ 14], only a few specialists understand his method. Many have made use of his results only, whereas the method itself remained totally without attention. This can be explained partly by the unusual difficulty of his work, partly by the abundance of cleverly-constructed devices based on deep technical intuition. We became acquainted with Baxter's work in the following way. In reading the paper of Luther [ 2 3 ] , in which Baxter's results and [22] are applied to the investigation of the spectrum of the Sine—Gordon quantum model we noticed that a number of Baxter's formulae resembled the formulae, already familiar to us, from the inverse problem method. This and other considerations promoted us, together with E. K. Sklyanin, to create a quantum version of the inverse problem method (see the survey [24]), which we applied to a complete

239 14

L. A. Takhtadzhan and L. D. Faddeev

solution of the quantum-field theoretical Sine-Gordon model [25]. As will become clear later, one of Baxter's formulae played an important part in this, and so it was only natural to turn to the XYZ model and to investigate it with the help of the inverse problem method. The results thus achieved form the content of this survey. It seems to us that the inverse problem method allows a simplification and algebraization of Baxter's formulae and to give a complete proof of them. We came to the conclusion that the inverse problem method is a natural unification of the basic achievements of one-dimensional classical physics, the ideas of Kramers—Wannier, Onsager, and Baxter, with Bethe's Ansatz and leads to a natural understanding of completely-integrable quantum systems. The inverse problem method for the solution of classical non-linear equations, which began in 1967 with the pioneering work of Kruskal and others [26], was further developed in [27] —[37] and now attracts the attention of a large number of experts in mathematical physics (see the sur­ veys [38] —[41 ] and [42], with the detailed bibliography there). Put very briefly, its basic propositions are as follows. Related to a non-linear evolution equation there is an auxiliary spectral pro­ blem and the Cauchy data occur in it as coefficients. The spectral characteristics of this problem (the elements of the monodromy matrix) are complicated functions of the Cauchy data. However, and herein lies the success of the method, the Hamiltonian of the system and the Poisson brackets can be expressed explicitly in terms of the spectral characteristics. The equations of motion in terms of the spectral characteristics become linear. Thus, the transition from the Cauchy data to new variables — the spectral characteristics — is a transformation to action — angle variables [29]. In the quantum version of the inverse problem method ([24] — [25], [43] — [44]) the coefficients of the associated spectral problem are constructed by the correspondence principle and can be expressed in terms of the Schrodinger canonical operators. The elements of the monodromy matrix are now operators in the Hilbert state space of the quantum system, and the quantum Hamiltonian can be expressed in terms of them. In this state space there is a particular vector such that all eigenvectors of the Hamiltonian are generated from it by means of the elements of the monodromy matrix. The existence of the generating vector and the simple commutativity relations for the elements of the monodromy matrix show that the latter are quantum analogues of variables of action-angle type. These general propositions will be illustrated in detail in the text by the examples of the XXZ and the general XYZ model. Now a few words about the contents of the paper. In the first section we give the classical results on the transfer matrix of two-dimensional lattice models, going back to Kramers-Wannier-Onsager. Here also we obtain Baxter's para­ meterization for Boltzmann weights and we comment on the formulae that arise there. In § 2 we show how the constructions of the preceding section fit naturally into the quantum method of the inverse problem, and we deduce a

240

The quantum method of the inverse problem and the Heisenberg XYZ model

formula connecting the Hamiltonian of the XYZ model with the transfer matrix of the eight-vertex model. The third section is devoted to the solution of the special case of the six-vertex model and the XYZ quantum model con­ nected with it. On this comparatively simple model we explain the appearance of the generating vector and the algebrization of Bethe's Ansatz. In §4 we construct a family of states generalizing the generating vector in § 3 to the case of the eight-vertex model. Here too we obtain a series of commutation relations for the operator elements of the monodromy matrix. In §5, using this family of generating vectors and the commutation relations, we construct an algebraic generalization of Bethe's Ansatz for finding the eigenvalues and eigenvectors of the transfer matrix for the eight-vertex model. In §6 we calculate the energy of the ground state of the XYZ model. This section may also serve as an intro­ duction to the method of integral equations in the theory of spin systems, which was first proposed by Hulthen in [4]. In the conclusion we sum up briefly and formulate some interesting mathematical problems arising in con­ nection with our account. In Appendix 1 we give a summary of definitions and formulae from the theory of Jacobi elliptic functions and theta-functions. Appendix 2 is devoted to a geometrical interpretation of the Baxter—Yang relations from § 1, which was recently given by Cheredrick [45]. In our exposition we follow the tradition of contemporary mathematical physics and draw attention mainly to the algebraic aspects of the problem at hand; therefore, our arguments often take on a formal character, especially in questions of convergence of series, of asymptotic estimates, and the like. A thorough airing of these questions would take us too far from the basic problem — to convey to a reader who is a mathematician the beautiful structures that arise in theoretical physics. We express our thanks to P. P. Kulish and E. K. Sklyanin for useful comments and remarks and to A. G. Reiman for a discussion of the results of [45]. We should also like to express our respect and our gratitude to Rodney J. Baxter, whose papers we have read with so much pleasure. The authors dedicate this paper to Academician N. N. Bogolyubov on the occasion of his seventieth birthday. §1. Classical statistical physics on a two-dimensional lattice and quantum mechanics on a chain

We take a rectangular lattice of order MxN on a two-dimensional torus, that is, a plane lattice with M + 1 rows and N + 1 columns in which the extreme rows and columns are identified. We give a definite direction, an arrow, to each edge of the lattice, that is, to each vertical or horizontal section joining adjacent nodes. Four edges come together at each node of the lattice, and so there are 16 distinct types of combinations of arrows at a node. We ascribe to each possible combination a positive number et (/ = 1, . . ., 16) that does not depend on

15

241 16

L A. Takhtadzhan and L. D. Faddeev

the number of the node (the energy of the combination). With each configuration of arrows on the lattice-configuration we associate a total energy, which is defined as the sum of the energies of the nodes, (1.1)

E=v

NJEJ,

where Nf is the number of nodes with a combination of arrows of type / in the given configuration. As a result we obtain a model of interacting arrows situated along the edges of the lattice. As an alternative definition, instead of arrows one may use a spin variable o with the two values ± 1, where + 1 corresponds to the arrows going to the right or upwards, and - 1 to the arrows going left or downwards. In this way there correspond four quantities a, a', 7, and 7' taking the values ± 1 to each combination of arrows at a node (Fig.l). The variables

Fig. 1.

oc and a' correspond to the vertical, 7 and 7' to the horizontal edges. The model described is called the general 16-vertex model and plays an important part in classical statistical physics. Interesting combinatorial problems are also connected with it (see [46]). The statistical sum is defined as the quantity w

(1-2) Z=2exp{-6£}, where the summation is over all configurations of arrows on the lattice, and E is the total energy of a configuration defined by (1.1). The statistical sum, as a function of the parameters p\ elt . . ., sM determines all the thermodynamic properties of the model. The parameter 0 is inversely proportional to the temperature, and the v} = exp{-pe/} (7 = 1 16) are called the Boltzmann weights. The problem of finding an exact expression for the statistical sum of a model on a finite lattice is very complicated. Fortunately, of greater physical interest is the simpler problem of finding an asymptotic value of 2 for large M and AT, more accurately, of calculating the so-called thermodynamic limit (1.3)

-P/=

Urn

-^ff\ogZ.

The quantity/is called the specific free energy. For a further investigation of the statistical sum Z it is useful to rewrite (1.2) in a more appropriate form. We denote by R^'(yt 7') the Boltzmann weight for

242

The quantum method of the inverse problem and the Heisenberg XYZ model

17

the combination of arrows in Fig. 1, and by {a-7} ={a1, . . ., a^Jan arbitrary configuration of vertical arrows between the neighbouring rows of the lattice numbered / and /' + 1. We note that {ctM+1} = {a1} by virtue of the periodicity condition. We now transform (1.2) with the outer summation over the vertical, and the inner over the horizontal configurations. As a result we get the expression

where 7" is an operator in the space !QN = C2 .In the orthonormal basis

its matrix elements are numbered by multi-indices {a}, {a'} and have the form

where it is also assumed that y^+l - y\; Sp in (1.4) denotes the trace of !QN. The operator T is called the transfer matrix of our model. It is an operator on $Qjf, the state space of the quantum system of N spins. In the expression R% (y, y') the indices a, a' and y, y' play different parts. The first are naturally called

1

2

3

^

5

6

7

6

Fig. 2.

"quantum" indices, and the second "auxiliary". Regarding the latter as the matrix indices of the 2 X 2 matrix R% , which number its rows and columns, we can rewrite (1.6) in the form

where tr signifies the trace in the auxiliary space C 2 . The relations (1.4)—(1.7) also establish a link between problems of quantum mechanics on one-dimensional chains and problems of classical statistical physics on two-dimensional lattices. For with their help we have reduced the problem of calculating the statistical sum (1.2) to a problem in quantum mechanics, the calculation of the eigenvalues of the transfer matrix T, an operator in !QN. For the simpler problem of calculating the free energy it is sufficient to know the asymptotic value as N~> °° of the eigenvalue A max (A0, of greatest modulus, of T. It is obvious that if the limit (1.3) exists and does not depend on the order of the limit operations over M and N, then

243

18

L. A. Takhtadzhan and L. D. Faddeev

Baxter's main result [9], which we reproduce in §6, consists in the calculation of/for the important special case of the system in question, the so-called eightvertex model, which we shall now describe. We consider only those configurations of arrows for which the number of arrows of a given type at each node of the lattice is even. The eight admissible combinations are drawn in Fig. 2. The model of interacting arrows that arises is called the eight-vertex model. In the expression (1.2) for the statistical sum the summation is now only over such configurations of arrows. In other words, for the eight-vertex model the Boltzmann weights uy, / = 9 , . . . , 16 are zero, that is, ej = oo (/ = 9, . . ., 16) (we assume that 0>O). We assume also that the interaction is invariant under simultaneous inversion of the directions of all the arrows on the lattice. This means that that is

For just such a model Baxter calculated the free energy and constructed a generahzation of Bethe's ansatz for finding the eigenvalues and eigenvectors of its transfer matrix. In the present survey we consider only this model. We introduce coefficients

in terms of which the Boltzmann weights R% (7, 7 ) corresponding to the com­ bination in Fig. 1 can be written in the following form:

Here we use the Pauli operators a1(j=l, 2, 3, 4), which were defined in the Introduction, where ayy> and aJaa- denote their matrix elements in the basis (4). To write out (1.7) in invariant form it is convenient to introduce operator matrices Xn of order 2 X 2 , defined by

The matrices Xn are of order 2 X 2 on the ring 23^ of operators in !QN. The product of the Xn and the trace tr are understood as operations in the algebra

244

The quantum method of the inverse problem and the Heisenberg XYZ model

19

of matrices over 23^. Comparing (1.7), (1.10), and (1.11) we see that the transfer matrix T can be written in the form Side-by-side with transfer matrix TN it is natural to look at the monodromy matrix STN

which is of order 2 X 2 over 93^- Then (1.12) takes the form (1.14) 7V = trjT„. We often omit the index TV, which indicate that the relevant objects operate i n £ * = C 2 ". The Boltzmann weights in the definition of the statistical sum (1.2) are positive real numbers. They can be regarded as defined up to a common factor since such a factor changes the statistical sum trivially. Since in what follows our arguments are of an algebraic character, we regard the weights u.- as com­ plex numbers. Thus, we consider the collection of coefficients wf- as a point in three-dimensional projective space over the field of complex numbers. The experience of exactly-solvable two-dimensional lattice models, such as Ising's model solved by Onsager, and the six-vertex model solved by Lieb sug­ gests that it is very fruitful to find conditions under which the transfer matrices with different values of the coefficients Wj commute. We investigate under what conditions on the w.- this is possible. Let us look at two sets of coefficients w}- and wj, and let JT and &"' be their monodromy matrices. The corresponding transfer matrices T and T' commute if there is a non-singular 4 X 4 numerical matrix M such that The Kronecker product in (1.15) is to be understood as a tensor product in the matrix algebra over 93^. For two 2 X 2 matrices Jh and 98 over an arbitrary ring, their tensor product Jb <8> 38 is the 4 X 4 matrix with the blocks

where the Aik are the matrix elements of A (i, k = 1 or 2). This definition was used implicitly in (1.11). From (1.15) it follows that To see this we have to multiply both sides of (1.15) by J? _ 1 on the left and to take the trace tr of the resulting 4 X 4 matrices over %$N.

245 20

L. A. Takhtadzhan and L. D. Faddeev

Now (1.1 5) is true if the analogous relation is satisfied for the matrices

Xn;

Indeed, for two matrices ^ n a n d <%n over a commutative ring there is the formula

It holds true also for matrices Xn and %'n, since it follows from (1.11) matrix elements of Xn commute with each other for distinct values of index n. We now find conditions under which the local formula (1.17) holds Wkhout loss of generality we look for a numerical 4 X 4 matrix M of

that the the good. the form

We substitute (1.19) in (1.17) and use the formula for the multiplication of Pauli operators

where ejkl is a completely antisymmetric tensor and e123 = 1. We find that for (1.17) to be true it is necessary and sufficient that for all permutations (/, k, I, n) of ( 1 , 2, 3 , 4 ) the following relation holds: In (1.21) there are six independent equations. If we regard them as a system of linear homogeneous equations for the four unknowns w'[, u/2', u/3', and 1U4, we find that the system is soluble if for all admissible values of/, k, I and n we have the relations

A natural parameterization of the conditions (1.22) has the form

Under the conditions (1.23) it follows from (1.21) that

hence, Thus, we have shown that (1.17) is valid if the coefficients Wj and wj have the form (1.23). Now the coefficients of the .52-matrix are defined by (1.25),

246 The quantum method of the inverse problem and the Heisenberg XYZ model

21

where for fixed ult u2, u3, u4, the parameter u" is a function of u and u . The choice of normalizing factors p, p', p" is immaterial by virtue of the homogeneity of the equations (1.21). To determine u" as a function of u and u we substitute (1.23) and (1.25) in (1.21) and differentiate with respects and «'. As a result we find that

where

We now introduce, instead of u and u', new variables v and v by

Then it follows from (1.26) that u" is a function only of v and v'. The differ­ ential equations (1.28) can be integrated explicitly in terms of elliptic functions (see Appendix I). With a suitable choice of the constants in (1.27) we obtain

where sn (v, /) is the Jacobi elliptic sine function with modulus /,

We define a parameter f by

Substituting the expression (1.29) for u in (1.23) and using the formulae for the Jacobi elliptic functions (see Appendix I), we obtain the following para­ meterization for the coefficients Wj:

The convenience of (1.32) consists in the fact that for two matrices <£nand X'n constructed from w;- and wj with identical values for the parameters f and / and differing values of v and u' there is a matrix M satisfying (1.17) whose elements depend only on u - 1 / . And so two transfer matrices with different values of the parameter v commute for fixed f and /. The parametrization (1.32) was found by Baxter in his famous paper [9], from which the arguments above are taken. We draw the reader's attention to the fact that (1.32), in fact, coincides with the expressions for the moments of a completely asymmetrical gyroscope (see, for example, [47]).

247

L. A. Takhtadzhan and L. D. Faddeev

22

We emphasize that the relations (1.32) do not constitute a restriction on the coefficients Wj. For given Wj the parameters /, f, and i> are defined by the formulae

To find the coefficients w" we substitute the expressions (1.32) for w}- and wj in (1.21). Using the addition theorems for the Jacobi elliptic functions (see Appendix I), we obtain the following expression for the solution of the system (1.21), the coefficients w":

More convenient in what follows is another parameterization of the Wj, which is obtained from (1.32) by Landen's transformation (see Appendix I). We set

Then we find (see Appendix I) that

For the coefficients w" we obtain, respectively,

where n = iv'/(l + k). Henceforth, when it is not necessary to indicate the modulus k, we write simply sn (X) instead of sn (A, k). Thus, the Wj in (1.11) are defined by (1.36) for any value of the common normalizing factor. For fixed k and rj we denote by Xn (X) the corresponding matrix Xn. Then (1.17) takes the form where the $} -matrix is

248 The quantum method of the inverse problem and the Heisenberg XYZ model

23

and the coefficients a, b, c, and d are given by

Since the relation (1.38) is homogeneous, it holds for any choice of common normalizing factors in (1.36) and (1.37). It follows from (1.39)—(1.40) that the J?-matrix is singular only for a discrete set of values of X - //, therefore, the validity of the equation for all values of X and /i follows from the principle of analytic continuation. Having at our disposal the formulae (1.11), (1.36), (1.39), and (1.40), we can now verify (1.28) directly with the aid of the addition theorems for the Jacobi theta-functions (see Appendix I). It is clear, however, that the task of finding a suitable parameterization (1.36) for the coefficients Wj has the nontrivial place in the reasoning advanced above. Therefore, instead of postulating (1.36), (1.37) and proving (1.38) directly we have first reproduced Baxter's arguments leading to the parameterization (1.36). Our relation

plays an essential part in what follows. We mention that Baxter himself in [9] used it only to prove (1.41). There he also gave another proof of this formula, not depending on (1.42). Baxter's formula (1.42) remained in the shade and did not attract the attention of other researchers. We have put (1.42) at the basis of our discussion. It is for us an important constituent part of the quantum version of the inverse problem method. We made an announcement of this in joint papers with Sklyanin [25], [48] and worked on field-theoretical examples of the non-linear Schrodinger equation [43] and the Sine—Gordon equation [25]. Here we show that this method works in the case of the XYZ model. We now look at the degenerate case when the modulus / of the elliptic functions tends to zero. In this case the Jacobi elliptic functions become ordinary trigonometric functions (see Appendix I) and the parameterization (1.36) has the form

From (1.9) we find with the parameterization (1.43) that u7 =vB= 0, that is, the cases 7, and 8, in Fig. 2 are forbidden. This model, which for obvious reasons

249 24

L. A. Takhtadzhan and L. D. Faddeev

is called the six-vertex lattice model, was solved by Lieb in 1967 by means of the classical Ansatz of Bethe. In §3 we explain the inverse problem method in the simpler example of this model. To simplify the formulae in § §4 and 5 it is helpful to choose a concrete realization of the projective parameterization (1.36). Taking y/k 0(2T7) @(X - rj) ©(A + T?) as the normalizing factor in (1.36) and using the formula (see Appendix I)

where H(w) and 0(u) are the Jacobi theta-functions (see Appendix I), we obtain the following expressions:

Similarly, for the elements of the M-matrix we obtain the representation

Using (1.45) and (1.46) it is easy to verify (1.38) by means of the addition theorem for the Jacobi theta-functions. To conclude this section we point out that (1.38) has already occurred, in a somewhat different form, in the literature of one-dimensional mathematical physics. For this way of writing (1.38) we introduce the following notation. Since the spin operators aJn act non-trivially only in the quantum space of spins at the n-th node, we may, allowing a certain liberty, forget the number n and identify a}n with the Pauli operators a]. Then the matrix Xn(%) changes to a 4 X 4 matrix, which in the basis has the matrix elements

where the summation is over the repeated indices. The basis vectors fa defined by (1.5). We put Then the matrix elements of the ^?-matrix take the form

a

are

250 The quantum method of the inverse problem and the Heisenberg XYZ model

25

In the notation (1.48)—(1.50) we can write (1.38) in the following form:

The relation (1.51) was first proposed by T. N. Yang in 1967 [49] in a dis­ cussion of many-particle factorizing ^-matrices, therefore, it is natural to call (1.38) the Baxter-Yang relation. Later it was used in the papers by Karovskii and others [50], A. B. and Al. B. Zamolodchikov [51]-[52] in which conditions were investigated for the factorizibility of ^-matrices in various models of the quantum field theory in two-dimensional space-time. Our attention to the cited form of (1.38) was drawn by A. B. Zamolodchikov after we had met him. He showed in [51] that (1.51) can be interpreted as a condition for the associativity of the algebra with the formal generators At{\) satisfying the relations The generators At(\) in [51] played the role of operators generating the 'in' and 'out' states of a many-particle system. In a recent paper [45] I. V. Cherednik proposed a realization of relations of the type (1.52) in terms of fibrations over Abelian varieties over the field of complex numbers. In Appendix II we give an account of Cherednik's results for our case, which corresponds to Abelian varieties of dimension 1, that is, elliptic curves. We turn now to the second section, in which, we hope, the apparent artificiality of our constructions will become more comprehensible, or at least more conventional. §2. Connection with the inverse problem method

The attentive reader will probably have guessed that the term "monodromy matrix" for JT(X) was not chosen accidentally. The equations (1.11)—(1.13) show that the linear problem

is connected with our model in a natural way. Here Xn(X) is given by the expression (1.11) and (1.36), and Hrn is a two-dimensional column with coefficients in 93^. We consider the matrix solution ^ ( X ) of (2.1) with the boundary condition

where IN is the identity operator in §N. As is known, the monodromy matrix S N(X) of the problem (2.1) on a chain of length N is defined as the value of the solution Vn (X) when n = N + 1, that is,

251 26

L. A. Takhtadzhan and L. D. Faddeev

The matrix Xn{X) itself is called the local transition matrix from the «-th node to the (n + 1 )-st. Systems of type (2.1) constitute the basis of the inverse problem method for the solution of non-linear classical equations on a chain. The coefficients of the local transition matrix Xn(k) in this case are, generally speaking, complexvalued functions of the characteristic canonical variables and the spectral para­ meter X. The system (2.1) is called the auxiliary spectral problem. As an example we look at a finite periodic Toda chain [53] whose equations of motion have the form

The local transition matrix Xn (k) in the case of a Toda chain has the follow­ ing form

where the pn are impulses, canonically conjugate in the coordinates xn. The monodromy matrix 3TN(k) is of order 2 X 2 and its elements are functions on the phase space QM of our system. The Poisson brackets on &fC have the form It turns out that the Poisson brackets for the elements of the monodromy matrix can be calculated explicitly. In particular, Thus, with the help of Xn(X) we have constructed o n i a family of commuting flows. The remarkable fact is that among them there is also the flow generated by the system (2.4). For the Hamiltonian h of the Toda chain then has the form

and can be expressed via the trace of the monodromy matrix by the method characteristic of the inverse problem:

The trace of the monodromy matrix for a Toda chain of length TV as a

252 The quantum method of the inverse problem and the Heisenberg XYZ model

27

function of X is a polynomial of degree TV with the highest coefficient 1. The relation (2.7) shows that the coefficients of the polynomial tr S~N(K) are involution integrals of the motion. The trace of the monodromy matrix is a generating function for commuting integrals of motion of the finite periodic Toda chain. One can prove that these coefficients are functionally independent as functions on the phase space aM. Their number is equal to half the dimension of <M. Hence, by Liouville's standard theorem (see, for example, [54]), there follows that the Hamiltonian system of the Toda chain is com­ pletely integrable, just as for any other system in which one of these integrals of motion can be taken as the Hamiltonian. Thus, (2.7) indicates that the finite periodic Toda chain is completely integrable. The inverse problem method for a Toda chain was applied by Manakov in 1973 [55] and independently by Flaschka in 1974 [56]. A matrix Xn(X) of type (2.5) can be abstracted from [55], in which for the proof of complete integrability a spectrum problem in a somewhat different form is used. In concluding this short digression into the classical method of the inverse problem we mention that variables of action-angle type for a finite periodic Toda chain, whose existence is guaranteed by Liouville's theorem, can be found by the methods of algebraic geometry (see the surveys [40] — [41 ]). Turning to our original theme we see that the study of the eight-vertex model leads quite naturally to the auxiliary spectral problem (2.1). However, in contrast to the example just studied, this is a quantum problem, since the ele­ ments of #„(X)are linear operators in
where H is the Hamiltonian of the XYZ model with the coefficients and A connection between Heisenberg's XYZ and the eight-vertex model was noted by Sutherland in 1970 [57]. He showed that under certain relations between the Boltzmann weights Vj and the parameters Jx,Jy, Jz the Hamiltonian of the XYZ model commutes with the transfer-matrix of the eight-vertex model. We emphasize here that the formulae (2.12) does not constitute a restriction

253 28

L. A. Takhtadzhan andL. D. Faddeev

on Jx, Jy, Jz, but give a parameterization of them up to an insignificant common factor. Let us prove (2.11)—(2.13). In the process of proof the special role of the value A, = Tj will become clear. In (2.11)—(2.13) it is naturally assumed that for the weights Wj the parameterization chosen is (1.36) with the common normal­ izing factor 1. For X = T| we obtain from (1.36) hence,

Using (1.7) we find that Thus, sn~N 2TJT(TJ) is an operator of cyclic displacement in fgN. We now differentiate (1.7) with respect to X and put X = TJ. With due regard for (2.14) we obtain the following expression:

Here we have used (1.10), that is,

Hence, the matrix elements of the logarithmic derivative of T(X) at X = n has the form

By transforming (1.10)

where

254 The quantum method of the inverse problem and the Heisenberg XYZ model

29

we can rewrite (2.17) as follows:

where the dash signifies the derivative with respect to X. The coefficients pj- are easily calculated with the help of the formulae for differentiation of Jacobi's elliptic functions (see Appendix I). As a result we get Other commuting quantum integrals of motion are given by the higher derivatives of log T(X) at X = 77. Liischer has calculated them explicitly in [58]. We do not need them in what follows; therefore we do not quote them. For k = 0, (2.11)—(2.13) set up a link between the transfer-matrix of the eight-vertex model and the Hamiltonian of Heisenberg's XXZ model. The para­ meters Jx, Jy, Jz have the form (2.21)

Jx = Jy = lf

/Z

= COS2T),

where 77 is an arbitrary complex number. We end with some preliminary conclusions. Following Baxter, for the operator H — the Hamiltonian of the Heisenberg model — we have found the auxiliary spectral problem (2.1). By its very statement it is a quantum problem. We have introduced the monodromy matrix $~{l) and by (2.11) have established a connection between H and the trace of the monodromy matrix, the transfer-matrix JT(X). The choice of the parameterization (1.36) for the coefficients Wj allowed us to write down the permutation relations between all the elements of Xn{K) and Xn{\x) in the compact form (1.38). As a consequence of the commutativity of the elements of %n(k) and %m{\i) for n =£ m an analogous relation (1.42) is valid for monodromy matrices. In the following section we shall show that the quantum method of the inverse problem works in the comparatively simple example of the XXZ model. We shall use throughout a significant simplification arising from the fact that wt = w2 and that the coefficient d in the M-matrix is zero. The general case wl =£w2 will be examined in § § 4 - 5 . §3. The six-vertex model

In the preceding section we have established, among other things, a con­ nection between the Hamiltonian of the Heisenberg XXZ model and the transfer-matrix of the six-vertex lattice model. Here we take up the problem of finding the eigenvalues and eigenvectors of the transfer-matrix of the six-vertex model. We solve it with the help of the quantum method of the inverse

255 30

L. A. Takhtadzhan and L. D. Faddeev

problem. At the end of the section we apply the results obtained to the analogous problem for the XXZ model. We recall that for the six-vertex model u7 = v& = 0, that is, wl =w2- In (1.36) this case corresponds to k = 0. Then the common parameterization of the coefficients Wj (1.36) becomes (1.43). We take the specific realization of (1.43), putting the common normalizing factor in (1.43) equal to 1:

The local transition matrix Xn{%)\id& the form

and satisfies (1.38), in which the M-matrix is given by the expression

where

The arbitrary common factor in (1.40) is chosen so that for X = fi the ^-matrix becomes the unit matrix. The monodromy matrix 3T N(k) is defined by (1.13), that is,

and as a (2 X 2)-matrix has the form

In this notation the transfer-matrix TN (X) is From this point on we again omit the index N in the relevant objects. The monodromy matrix 3T{X) satisfies (1.42), that is, We have already noted above that this equality contains commutation relations between all the operator matrix elements A(X), B(X), C(X), and D(X), of y(A,).We write down explicitly the ones that are basic.

256 The quantum method of the inverse problem and the Heisenberg XYZ model

31

We now proceed to the construction of a special condition, a generating vector in !QN. We rewrite Xn{%) in the form

(Note that from (3.1)—(3.2) it follows that in our case the operators /3„ and yn do not depend on X). The operators an,^n,yn, and 5„ act non-trivially only in the local space t)n, therefore, without changing the notation we can restrict them to f)n. Let e*n be the vector in f)n, (see (4))

From (3.1), (3.2), and (3.10) it follows that

where We call e*n a local vacuum. It annihilates the lower left element of ^n(X)and for all X is an eigenvector for its diagonal elements. From the local formulae (3.12) there follow analogous formulae for the elements of 3T{1) with respect to Q. Namely,

Indeed (3.12) shows tha.tXn(k) applied to en is triangular, and (3.15) follows from the rule for the multiplication of triangular matrices. The state Q thus con­ structed is called a generating vector. The resulting equations (3.9) and (3.15) are used to find the eigenvalues and eigenvectors of the operator ^4(X) + D(\), that is, the transfer-matrix T(k). We show that the eigenvectors of T(X) can be put in the form

where the X/ satisfy the system of transcendental equations

257 32

L. A. Takhtadzhan and L. D. Faddeev

and where the corresponding eigenvalues A(X; Xu .. ., Xn) are

The reader familiar with problems that can be solved by Bethe's Ansatz (see the surveys [3], [46] and the original papers [ 2 ], [ 6 ] - [ 8 ] ) will recognize here the basic formulae of this method. New are the compact formula (3.16) for the eigenvectors and the simple algebraic derivation of (3.16)—(3.18), which depends only on the commutation relations (3.9) and on the existence of a generating vector £2. It is natural to call the Z?(X/) generating operators of elementary excitations of the generating state £1. This algebraic version of Bethe's Ansatz was proposed by us together with Sklyanin in [25], where an exact solution of the Sine—Gordon quantum model was obtained with its help. We mention that in (3.16)—(3.18) the permissible values of the parameter n are 0, 1,. . .,N, because for« > N the expression (3.16) vanishes identically. For by using the commutativity of the operators ay, fy, y., and 5;- for various/' and the fact that $f = 0 we can easily verify that the product (3.16) vanishes for n> N. We also point out that since ^F(k) is periodic in X with period 27T, in (3.17) it must be understood that | Re X/ | < 7r, that is, the solutions X/ are considered only in the fundamental domain. We come now to the proof of (3.16)—(3.18). We rewrite the necessary commutation relations in (3.9) in the more convenient form

From the explicit form of the coefficients b and c in (3.4) it follows that

With the help of (3.19—(3.20) we can transform the expression

taking A(\) and Z)(X) through 5(X,) to £1 and using (3.15) and (3.21). Then there arise 2" terms, which naturally come together in n + 1 expressions of the type

and

258 The quantum method of the inverse problem and the Heisenberg XYZ model

33

where A and Ay- are numerical coefficients. The structure of the operator factors in (3.23) is obtained if on commutation we take account of only the first terms on the right-hand sides of (3.19) and (3.20). As a result we conclude immediately that that coefficient A(X; Xx, . . ., X„) is given by (3.18). The structure of the operator factors in (3.24) for/ = 1 is obtained if upon commutation of .4(A) and D(X) with £(Xj) we use the second terms in (3.19) and (3.20) and upon further commutation of the resulting ^(X^ ) and Z)(X/[ ) with ^(X,), / > 2, we again take into account only the first terms in (3.19) and (3.20). The coefficient A! (X; \ l t . .., X J then takes the form (using (3.21))

For calculating the coefficients A;-, (j = 2,. . ., n), we must combine a larger number of terms. However, it is not necessary to make these calculations, since by virtue of the fact that the B(X{) commute the expression (3.22) is a symmetric function of Xj, . . ., X„, consequently, the coefficients A;- are obtained from A t by a suitable permutation of the Xt. Thus, the coefficients Ay have the form

For the vector ^>(Xj, . . ., X„) defined by (3.16) to be an eigenvector of the transfer-matrix T(\) for all X the sum of the terms of form (3.24) must vanish. The system of equations (3.17) for the X7 results if we bear in mind that each term vanishes separately, that is,

Now (3.16)—(3.18) are proved. We note that a direct proof of (3.25) must depend on the addition formula for the coefficients b and c. We have already used these addition theorems in proving (1.38). It should also be mentioned that in the derivation of (3.25) we have used the condition that all the X, are distinct. We shall not consider in detail here the case of coinciding X/ in (3.16), which requires a special investi­ gation, since it would lead us too far from the basic theme of our survey. Comparison of (3.16), the operator representation for the eigenvectors of 7TX), with the coordinate representation in [6] - [ 8 ] is very useful. If we write

259

34

L. A. Takhtadzhan and L. D. Faddeev

out (3.16) in coordinate form, that is, in terms of the local operators c^j, j3w, 7 , and 5„ , then it differs only by a constant factor depending on A/ from the coordinate representations in [6] —[8] and, in contrast to the latter, it does not vanish when A,7 = A*. Hence, (3.16) can be used for coinciding A,,. It should also be pointed out that, as in the case of the classical Bethe Ansatz, not all solutions A, of the system of transcendental equations (3.17) for 0 < n < N correspond to eigenvectors of the form (3.16) with eigenvalues (3.18), because for certain solutions A/ the corresponding eigenvector $(\l, . . ., A„) may vanish. It can be shown that the eigenvector (3.16) con­ structed from a solution of (3.17) for n =N vanishes for almost all T?. However, in the situation of the general position with respect to the A, a vector of the form (3.16) does not vanish for n = N, but is proportional to

that is, it is an eigenvector of the operator T(\) and the same eigenvalue as the generating vector £2. To prove this latter assertion it is sufficient to note that the vector e~ annihilates /3„ and is an eigenvector of % (A) and 5„ (A) for all A, and then to repeat the derivation of (3.15). Thus, the investigation of this simple example shows that the vanishing of 4>(Aj,. . ., A„) for certain solutions A/ of (3.17) is connected with the degeneracy of the spectrum of T(\). We move now to the investigation of the Heisenberg XXZ model. The Hamiltonian H of this model takes the form

where it is assumed that the periodic boundary conditions (6) hold, and is con­ nected with the transfer-matrix T(X) of the six-vertex model by (2.11)—(2.13), that is,

Now (3.28) shows that the eigenvectors ^ ( A j , . . ., A„) of the commuting family of operators T(\) are eigenvectors also for the energy operator H. The eigenvalues of H can be found, from the eigenvalues (3.18) with the help of (3.28). We give the corresponding expressions. For consistency of the resulting formulae with the analogous formulae in [2] and [6] — [ 8 ] , we introduce, in place of A, a variable k (not to be confused with the modulus of the elliptic functions in § § 1 and 2)

and write

260

The quantum method of the inverse problem and the Heisenberg XYZ model

35

Then, after elementary transformations of (3.17)—(3.18), we obtain the follow­ ing results. The solution ^ of the eigenvalue problem of the Hamiltonian H can be put in the form (3.30), where the numbers k{ satisfy the system of trans­ cendental equations

and the eigenvalues E(kx, . . ., kn) have the form

The formulae (3.32)—(3.34) were obtained by Yang and Yang in [6] - [ 8 ] with the help of the classical Bethe Ansatz. We have shown how the quantum method of the inverse problem leads naturally to the classical results (3.32)— (3.34) for the XXZ model. In the study of the XXZ model it turns out to be helpful to use the variable X and the system (3.17), as well as the variable k just introduced and the system (3.32)—(3.33). In (3.17) the function c(X, n) depends only on X - / i and, as will be explained in §6, is transformed as iV -► °° into an integral equation with a kernel depending on a difference of arguments. This last circum­ stance is very convenient. On the other hand, in terms of the kt the expression for the energy E(k^,. . ., kn) takes the physically intuitive form (3.34), the sum of the energies of the elementary excitations in the state £2. It should also be noted that, in contrast to the system of equations (3.17), which in the natural range of the parameter 77, namely, | Re 77 | < IT, makes sense for 77 # 0, ± 7r/2, ± 7r, the system (3.33) has a meaning for all values of A. In particular, putting A = 1 in (3.27) and (3-32)-(3.34) we obtain the classical Bethe formulae for the XXX model [ 2 ] . So far we have been looking at the case of arbitrary complex values of the parameter A. Only real A are of physical interest, because only for such A is Ha Hermitian operator. Now (3.28) shows that it is natural to consider the following ranges of the real parameter A: I. - 1 < A < 1, II. A > 1, III. A < - 1. In I, the parameter 77 takes the real values 0 < 77 < -J-7T; in II, 77 = in', 0 < 77' < 00; in III, 77 = TT/2 + irj', 0 < 17' < °o. For 77 in I we put

for 77 in II

261 36

L. A. Takhtadzhan and L. D. Faddeev

and for 77 in III Then for the operator matrix elements of the local transition matrices Xn(X) thus defined we obtain from (3.1)

where the minus sign corresponds to the values of 77 in I and II, the plus sign in III, and the tilde and * denote complex and Hermitian conjugation, respectively. For a matrix X on the ring 9S^ we denote by X the matrix with the elements Xjh = Xfh- Then (3.38) can be rewritten in the form for 77 in I and II and for 77 in III. Let 3T(X) be the monodromy matrix constructed from the local transition matrices Xn(k) (3.35)—(3.37). Since the elements of Xn{X) commute for distinct n, From the local property (3.39)—(3.40) there follows an analogous property for J^(X), namely, and Thus, in the case of real A the monodromy matrix satisfies the supple­ mentary relations (3.42)—(3.43), that is,

where the minus sign corresponds to values of 77 in I and II, and the plus sign to III. Since Hin this case is Hermitian, the eigenvectors of the form (3.16) for distinct eigenvalues (3.18), are orthogonal. However, these states are nonnormalized. In the case of real A, for the calculation of scalar products of vectors of the form (3.16) it is convenient to use (3.44) and the commutation relation for B(\) and C(fi) in (3.9),

262

The quantum method of the inverse problem and the Heisenberg XYZ model

37

Setting out from (3.45), we can construct from the generating operator 5(A) for non-normalized elementary excitations on the state £1 a generating operator of normalized eigenstates for the Hamiltonian H. We do not go into further details here. In conclusion we summarize the contents of this section. Using (3.8) and the existence and properties (3.14)—(3.15) of the generating vector £1, we have obtained an algebrization of Bethe's Ansatz to find the eigenvalues and eigen­ vectors of the transfer matrix T{\) for the six-vertex model, the trace of the monodromy matrix JT(A,). The eigenvectors of T(X) are obtained by applying to 12 the generating operators B(\j), the matrix elements of S~(k) for values of A, satisfying the system of transcendental equations (3.17). The eigenvalues then are given by (3.18). We have shown how the results obtained for the sixvertex model lead to the classical results (3.32)-(3.34) for the Heisenberg XXZ model. We have looked at the physically interesting case of real A. In this case J~(k)allows the involution (3.42)-(3.43), and .5(A) turns out to be conjugate to C(A), the annihilation operator for elementary excitations on 12. We have pro­ posed in this case a scheme for constructing normalized eigenvectors of H, using (3.44) and (3.45). §4. Generating vectors and commutation relations

We devote this section to the analysis of the general case, the eight-vertex lattice model. The local transition matrix %n(k) corresponding to it has the form (1.11), that is,

where the w.- are defined by (1.45):

The matrices Xn{\) satisfy the Baxter-Yang relation in which the .#-matrix has the form (1.39), (1.46), namely

where

263 38

L. A. Takhtadzhan and L. D. Faddeev

The monodromy matrix jT(X) is defined by (1.3), and as a 2 X 2 matrix over 93^ has the form

It satisfies (1.42) which contains all commutation relations between the operators A(X), B(k), C(X), and D(X). (4.1)—(4.2) show that, in contrast to the case of the six-vertex model, the operator yn (X) is non-degenerate for almost all X. Hence, we cannot obtain a local vacuum, either for Xn(k) or for a finite product of such matrices. More­ over, in this case the coefficients d(X, ju) is non-zero, so that the commutation relations for the ^(X), B(k), C(X), and D(X) no longer have the simple form (3.9). Consequently, the method developed in §3 is not directly applicable to the case of the eight-vertex model. However, we can generalize the considerations in §3 appropriately, and this leads to a solution of the eightvertex model. Instead of the generating vector (1.14), we shall use a family of generating vectors and instead of the relations (3.9) a series of commutation relations for various linear combinations of A(X), B(X), C(X), and D(X). Let us proceed to the construction of the family of generating vectors. We use the following argument: the monodromy matrix JT(X) can be calculated if we replace the local transition matrices Xn(X) by matrices X'n(K) gaugeequivalent to Xn(X): where the Mn (X) are arbitrary non-singular 2 X 2 numerical matrices. Indeed, the new monodromy matrix

differs from y(k) only by the simple linear transformation It turns out that the gauge transformations Mn (X) can be chosen in such a way that each matrix X'n(X) has a local vacuum independent of X that is annihilated for all X by its lower left element. This condition defines the first column of Mn(X). The second column is defined, if necessary, so that the dia­ gonal matrix elements of X'n{%) act on the local vacuum in the simplest possible way. The corresponding formulae may be extracted from Baxter [ 19] —[20]. It becomes clear that there is a whole family of gauge trans­ formations Ml(\,s, t) depending on an integer / and arbitrary complex parameters s and t. We write

264

The quantum method of the inverse problem and the Heisenberg XYZ model

39

where

g(u) - H(u) 0(«), and

Here K is a half-period of @(u) (see

Appendix I). The corresponding local transition matrices X'n{X) are denoted by

In the next set of formulae we do not indicate explicitly when it is not necessary, the dependence of the relevant objects on s and r, since these para­ meters are assumed to be fixed. In terms of 3TlN{%) we write the monodromy matrix JT'(A.)

with the matrix elements AlN(\), BlN(X), Cy (A), and A local vacuum coln £ \)n with

DlN(\).

exists and is given by the formula In contrast to the case of the six-vertex model,
where (4.11)—(4.12) and (4.14)-(4.17) are easily checked on the basis of the addition theorems for the Jacobi theta-functions. From the local formulae (4.12)—(4.17) we find that the vectors

satisfy the relations

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L. A. Takhtadzhan and L. D. Faddeev

We call {^Miz!0^ a family of generating vectors for the transfer matrix of the eight-vertex model. The formulae (4.19) show that it is useful to examine apart from y{K) also matrices of the form (4.13). We introduce a collection of matrices with the operator matrix elements Ak ,(X), Bk t{\), Ck t(k), and Dk /(X). It is obvious that for all values of / The monodromy matrix .flN{V) above can be written in the new notation as It turns out that the commutation relations (4.6) for A(k), B(k), C(k), and D(k) lead to simpler relations for Ak }(\), Bk ,(X), Ck /(X), and Dk /(X), which in their structure somewhat resemble (3.9). We come now to the derivative of these commutation relations. We introduce covariant vectors X{(\) and F/(X) by

where the components xj, xf, y\, and yf are defined by (4.11); and contravariant vectors XZ(X) and Yt(X): From the addition theorems for the Jacobi theta-functions it follows that

that is det A//(X) is independent of/; we denote it by m(X). From (4.11), (4.20), (4.22), and (4.23) we obtain

266 The quantum method of the inverse problem and the Heisenberg XYZ model

The expressions (4.25) show that to obtain from (4.6) the commutation relations for A k , ( » , Bk ,(X), Ck {(K), and Dk t(\) we must apply the matrix J?(X, u) to the tensor products of the covariant vectors of the form (4.22) and of the contravariant vectors of the form (4.23). As a result we get vectors of the same form. We set

We then have the following relations:

(4.27)—(4.30) can be verified on the basis of the addition theorems for the Jacobi theta-functions. We note that (4.14)-(4.16) are contained in (4.27)(4.30). To see this we need only recall (1.48)—(1.50), which show that Xn(\), as a 4 X 4 matrix in the local space, has the same matrix structure as the M -matrix. Now we replace the variable u in (4.27) by s + 2(1 + 1 )rj - X and find that Replacing u in (4.28) by t + 2/77 + X, we obtain Finally, replacing u in (4.29) by t + 2(k + 1 )T? + X, v by s + 2lrj - X, and r by ?"(*+/+!)/2; and u in (4.30) by s + 2(k - l)i?~X, v by t + 2/tj + X, and r by T (k+i-1 )/ 2 w e come to the following formulae, respectively:

41

267 42

L. A. Takhtadzhan and L. D. Faddeev

and

Similar formulae hold for the contravariant vectors

With the help of (4.31)-(4.38) and using (4.6) we can obtain all the commutation relations between the operators (4.25). Relying on them and on (4.19) we shall construct in the following section an algebraic generalization of Bethe's Ansatz to find the eigenvalues and eigenvectors of the transfer-matrix for the eight-vertex model. We introduce here the commutation relations we need for this. We multiply (4.6) on the left by and on the right by

Using (4.25), (4.32), and (4.35) we find that for all values of k and /

268 The quantum method of the inverse problem and the Heisenberg XYZ model

43

Now we multiply (4.6) on the left by

and on the right by Using (4.25), (4.33), and (4.35) we find that for all values of k and /

Lastly, we multiply (4.6) on the left by and on the right by Now (4.25), (4.32), and (4.38) show that for all values of k

The remaining commutation relations for the operators Ak /(A), Bk ,(X), Ck ,(X), and Dk /(A) can be deduced similarly. We do not need them in what follows, and so we do not derive them. Let us sum up. We have established that in the case of the eight-vertex model it is useful to investigate apart from the 3T{k) also the matrices 3Fh,i(ty- We have constructed a family of generating vectors {Q^}/=-«>, on the elements of which the coefficients of the ^h,i(ty act in accordance with (4.19), and we have obtained a series of permutation relations for the operators Ak /(X), B k,i
In this section, using (4.19), and (4.39)-(4.41), we construct an algebraic generalization of Bethe's Ansatz to find the eigenvalues of eigenvectors of the transfer-matrix 7YX), the trace of &U(X). We rewrite (4.39)-(4.41) in a convenient form:

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L. A. Takhtadzhan andL. D. Faddeev

where

Let us examine the vector

where n = N/2 (we suppose that TV is even). By (5.1), ^/(Aj, . . ., A„) is a symmetric function of the variables A,, . . ., A„. We apply to ^liXy, . . ., A„) the operator Al /(A). Using the permutation relation (5.2) for k = I, we commute Aj /(A) with £ / + j i_ j (Aj) and, again using (5.2) with k replaced by / + / and / by / -/', we take the resulting operators Al+j l • through Bl+j+ lf /_/_ i (A / + ,) over to ft^". With the help of (4.19) we find'that

We emphasise that, in contrast to (3.16), the numbers of operators Bi+j i_j{\j) in (5.5) is not arbitrary but is exactly equal to N/2. This is due to the fact that after commuting Aj t with all the operators Bl+J-j_j in (5.5) we obtain Al+n t_n, which, as follows from (4.19), we can apply to Cll^n only on condition that n = N/2. The coefficients x A and j A'- (/= 1, • • •> " ) can easily be calculated by means of the device already used in § 3 , which is based on the symmetry of the lefthand side of (5.6) under permutations of \ x , . . ., A„. As a result we come to the expressions

and

Similarly we obtain

where

and

270 The quantum method of the inverse problem and the Heisenberg XYZ model

45

We multiply (5.6) and (5.9) by exp {2nilQ}, where 0 < 0 < 1, add them and sum the resulting equations over all integers / from - <» to °°. We obtain

where

Now (5.8), (5.11), and (5.12) show that ^ f l (X,, . . ., \n) is an eigenvector of T(X) with the eigenvalue provided that the X;- satisfy the system

The convergence of the series (5.13), of course, requires a special investigation. The analogy with the formula for Poisson summation and the theory of automorphic functions leads to the assumption that this series is summable to zero for all 8 except finitely many values 0;-. For these 8j, the numbers (5.14) are the eigenvalues of T(K). The system (5.15) for the special values of 8 was obtained by Baxter in [ 9 ] . His results show that among the 8j there is also the value 8 = 0. We shall verify this in §6. The situation is simplified if we impose restrictions on the range of r]. Let T be a lattice with the periods of sn {u, k), that is, with generators AK and 2iK' (see Appendix I). Let us suppose that 2r\ is a point of finite order on the elliptic curve % - C 1 / I \ that is, that there exists an integer Q such that 2Qr\ belongs to T. In other words

with integer mx and m2. Since the Jacobi theta-functions are quasi-doublyperiodic with quasi-periods 2K and 2iK' (see Appendix I), the Mkj(k), y fe j(X),and ^ / ( X j , . . ., X„) are quasi-periodic functions of A; and / with quasi-period Q. By the choice of a common normalizing factor in (1.36) it can be arranged that they become periodic functions of k and / with period Q.

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L. A. Takhtadzhan and L. D. Faddeev

We put A = mm2/4Qkrj and define From the quasi-periodic relations for the Jacobi theta-functions it follows that

We now choose a specific realization of the projective parameterization (1.36), having taken as a common normalizing factor We get

and similar formulae for the elements of the ^-matrix. The coefficients Wj in (5.19) differ from the corresponding coefficients (4.2) only by the common factor The matrices Mk /(A) and j h t t(k) which have been constructed from the coefficients (5.19) and the modified theta-functions H(u) and 0(u), and the vectors £2}y, are already periodic functions of k and / with period Q. A simple verification shows that the addition theorems for the Jacobi theta-functions in Appendix I remain valid for the modified theta-functions, provided that we replace g(u) by g (w) = 0(— u) H(u). Similarly, after replacing h(u) by h (u) = 0(O)0(- u) H(u), (4.19) and all subsequent^relations, in particular, (4.39)—(4.41), remain valid. Thus, the operators Ak [(k), Bk /(X), and Dk 7(X) and the vectors Q,lN satisfy relations similar to (4.19), (5.1)—(5.4), and we may, therefore, repeat the derivation of (5.6)-(5.8) and (5.9)—(5.11). By virtue of the periodicity in k and / in the expression the permissible values of n can now be found from the condition Next, in sums of the type (5.13) it is sufficient to sum over a period Q, and for 6 we can now take only values of the form In this way we have shown that in (5.16) the eigenvectors of T(X) can be represented in the form

272

The quantum method of the inverse problem and the Heisenberg XYZ model

47

where ^/(Xj,. . ., X„) is defined in (5.20), n in the congruence (5.21), and the X;- satisfy the system

here where n ~

We note that for Q = 2 the system of equations (5.24) breaks up into n independent equations for the unknowns \ x , . . ., \ n . This case corresponds to the dimer model, the Ising model and the model of free fermions (see [9], [46]). We emphasize that in the presence of the subsidiary condition (5.16) the infinite series (5.13) becomes the finite sum (5.23), and now the solutions of (5.21), are the admissible values of n in (5.20), and not only the value N/2 as in the general case. If we also suppose that Q divides N, then the admissible values of n are of the form, 0, Q, . . ., iV for odd Q and 0, Q/2, . . .,N for even Q. The system (5.24) and the formula (5.26) for m = 0 were obtained by Baxter in [ 21 ] by means of a very complicated and non-trivial generalization of Bethe's Ansatz. We have shown here how the quantum method of the inverse problem leads quite naturally to an algebraic generalization of Bethe's Ansatz. We note that in the derivation of (5.8) and (5.11), and hence of (5.14)—(5.15) and (5.24)—(5.26), we have used the fact that all the X;- are distinct. We do not discuss here the case of equal values of X/5 but we only point out that, in contrast to the coordinate representation for the eigenvectors in [21 ], the operator representation (5.23) does not vanish when some X;- are equal. As in the case of the six-vertex model, not all the solutions of (5.26), generally speaking, correspond to non-zero vectors of the form (5.23). This circumstance is connected with the possible degeneracy of the spectrum of T(X). We recall that, as (4.11)—(4.12) and (4.25) show, the vectors (5.23) depend on additional parameters s and / which do not occur in the expressions (5.24) and (5.26). It is to be hoped that the investigation of the dependence of vectors of the form (5.23) on the parameters s and t will turn out to be useful in the investigation of degeneracies in the spectrum of T(X). We now look at the special case when m2 = 0 in (5.16), that is,

273

48

L.A. Takhtadzhan and L. D. Faddeev

and A = 0, so that there is no need to modify the Jacobi theta-function. Where n = N/2 in (5.24) and by means of the given integer v we define an integer m from the congruence

The system of equations (5.24) and the formula (5.26) for the eigenvalues then take the following form:

By simple arguments founded on the continuous dependence of T(\) on 77 we can easily check that (5.29)-(5.30) remain valid for all 17 for which the ratio -q/K is real. We turn now to the case (5.16) and to the formulae (5.24)—(5.26). We write these in terms of ordinary (that is, not modified) Jacobi theta-functions. We find that the X;- satisfy the system of equations

and the corresponding eigenvalues have the form

By virtue of the fact that h{u + 2K) = -h(u) (see Appendix I) it is natural to examine the solutions X; of (5.31) only in the fundamental domain | Re X;- | < K. Let us apply our results to the Heisenberg XYZ model. We recall that the Hamiltonian of this model is connected with the transfer-matrix of the eightvertex model by (2.11)—(2.13). Here the coefficients Wj are defined by (1.36) with the common normalizing factor 1. In our case the Wj are given by (4.2), so that (2.11)—(2.13) can be rewritten as follows:

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The quantum method of the inverse problem and the Heisenberg XYZ model

49

where and Now (5.33)—(5.35) show that the eigenvectors of the Hamiltonian H are eigen­ vectors of the commuting family of operators T(X), that is,

and the X;- are defined by (5.31). The corresponding eigenvalues have the form

where and The admissible values of n here are subject to (5.21). We see from (5.36)—(5.39) that the eigenvalues of/fare degenerate. It can be shown (see [ 19]) that when N = 0 (mod Q), the eigenvalues (5.37) for n = 0 have the multiplicity IN. We must also note that then the eigenvalues of the transfer-matrix (5.26) for n = 0 are degenerate with the multiplicity 2JV/e(see[19]). In the case of arbitrary values of 77 the eigenvalues of H are, as before, given by (5.37) where now n = N/2 and the X;- are determined by (5.15). In conclusion we summarise the contents of this section. Using the family of generating vectors (4.18) and the series of commutation relations (5.1)—(5.3) we have constructed an algebraic generalization of Bethe's Ansatz to find the eigenvalues and eigenvectors of the transfer-matrix of the eight-vertex model. The eigenvectors form the infinite series (5.13) in which each term is a result of applying n operators B1+j /_/(X/) to the generating vector Sl1^". In contrast to the case of the six-vertex model, the number n is not arbitrary, but is equal to N/2. The X;- are defined by (5.15) and the corresponding eigenvalues are given by (5.14). We have also considered the interesting special case when 2r\ is a point of finite order on the elliptic curve %. Then the infinite series (5.13) becomes the finite sum (5.23). The admissible values of n are defined by the congruence 2n = A^'(mod Q), where Q is the order of 217 on %. Using these results we have solved the problem of finding the eigenvalues and eigenvectors of the Hamiltonian of the Heisenberg XYZ model. Thus, we have shown that

275 50

L. A. Takhtadzhan and L. D. Faddeev

the quantum method of the inverse problem works in the case of the XYZ model. In the following section, relying on our results, we shall calculate the free energy of the eight-vertex model in the physical domain, that is, for positive Boltzmann weights uy and we shall find the energy of the ground state of the XYZ model in the case of real coefficients Jx, Jy, and Jz. §6. Integral equations

Here we calculate the specific free energy of the eight-vertex model. We assume that the real coefficients w}- belong to the so-called basic domain, that is, In this case the parameters /, f, and v defined by (1.33) are real numbers and subject to Hence, the parameters 77 and X, which are connected with them by (1.35), are imaginary, and By K{ and K^ we denote respectively, as usual, the complete elliptic integrals of the first kind of modulus / and k' (supplementary to k), respectively. We must note that the case of the basic domain (6.1) for the Wj is nonphysical, since here the Boltzmann weight u3 is negative. Nonetheless, for the calculation of the statistical sum Z in the physical domain it is sufficient to consider only the case of the basic domain, because Z is an even symmetric function of the Wj, that is, for all permutations (j, K I, m) of (1, 2, 3, 4) and arbitrary signs The symmetry relations (6.4) for the eight-vertex model were obtained in 1970 by Fan and Wu [49]. They are a direct consequence of (1.11)—(1.14). For completeness we add here a proof of (6.4). We denote by Xn {wx, w2, w3, wk) the local transmission matrix Xn. From (1.11) we find that

where the aJ are Pauli matrices, that is, the matrices of the Pauli operators in the basis (4) of the auxiliary space C 2 . Moreover, for real w;where Xn is the matrix defined in §3. It is also obvious that

276 The quantum method of the inverse problem and the Heisenberg XYZ model

51

The local equations (6.3)-(6.9) imply analogous relations for the monodromy matrix 5~ (wv w2, w3, wk), from which we obtain the following formulae, which are valid for real w.- and even TV:

We denote by uJn the operators in $QN and by Uj the matrices in C2

From (1.11) it follows that

We put

From the local equations (6.17)—(6.19) we find that

From (1.11) we also obtain and We put, for even N,

Then from (6.24)-(6.25) it follows that

and

277 L. A. Takhtadzhan and L. D. Faddeev

52

From (6.10)-(6.14), (6.21 )-(6.23), (6.27)-(6.28), and from the definition of the statistical (see (1.2) and (1.4)), we now obtain (6.4). Thus, on the basis of the symmetry relations (6.4) and of (1.3) and (1.8) we conclude that to find the free energy/in the physical domain it is sufficient to calculate the asymptotic value as N -*■ °° of the eigenvalue Amax(A0 of greatest modulus of T(X) in the basic domain (6.1). Let us now look at the system (5.31) under the assumption that r\ satisfies (5.16). Using the theory of perturbations it can be shown (see [9]) that in the basic domain (6.3) of k, 77, and X the cases corresponding to the eigenvalue Amax(/V) in (5.31) are m = 0 and m = N/2. The corresponding solutions Xy- of (5.31) are here real numbers symmetrically situated on the interval [-Kk, Kk}, that is, and

Then (5.31) and (5.32) take the following form:

and

From the explicit form of (6.30)—(6.31) and the continuous dependence of T(X) on 17 it follows that (6.30)—(6.31) remain valid for all 17 in the basic domain (6.31). We come now to the direct investigation of the system (6.30). By taking the logarithm of (6.30) we obtain

where the /;- are integers and

278 The quantum method of the inverse problem and the Heisenberg XYZ model

53

The branches of the logarithm in (6.33) are uniquely determined by the expansions of these functions in Fourier series (see Appendix I). Using arguments based on the theory of perturbations ([9], [22], see also [6] - [ 7 ] ) , and taking into account the fact that if a is real then tp(a) is decreasing in the interval [~K, K] it can be shown that to the eigenvalue Amax(A0 there corresponds the solution of (6.32) for In what follows we use only Jacobi elliptic functions of modulus k, which allows us to omit the index k. The classical method of investigating (6.32) as N -> «>, which goes back to Hulthen [4], is based on the supposition that as N-* °°, the solutions X; are uniformly distributed on [—K, K] with positive density p(a). Thus, for each integrable function /

as N -* <», where n = N/2. In other words, as N-+ °°, the quantity \/N(XJ+1 - Xj) becomes a smooth function of p(a), that is, the difference X/+ j - \j is of order \/N for large N. We now subtract from (6.32) with /' = k + 1 the analogous equation with/ = k; then we divide the difference by N(\k+ x - Xfc); and in the resulting equation we pass to the limit N -*• °o. Using (6.35) we find that

where the dash signifies the derivative with respect to a. We note that 4>'(a:) is a Poisson kernel for the domain | Re a. | < K, \ Im a \ < K'/2 in the complex a-plane. Thus, (6.36) is connected with the problem of factorizing the function h(a + rj)/h(a

-17).

So we have shown that the density p(a) satisfies a linear integral equation with kernel depending on the difference of the arguments. Since the kernel and the free term of this equation are periodic functions with period 2K, and since the integration in (6.36) is over a period, (6.36) can be solved by means of the Fourier transform. We put

After going over to the Fourier coefficients, (6.36) takes the form

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L. A. Takhtadzhan and L. D. Faddeev

from which we see that

From the Fourier expansion of the logarithms of the Jacobi theta-functions (see Appendix I) it follows that tp'(a) and $'(oc), by (6.3), that is, when 0 < — vq < K'I2, have absolutely convergent Fourier series for real a, with the Fourier coefficients

and

Thus, from (6.38)-(6.40) we find that and, consequently, 1

The solution p(a) is compatible with the conditions (6.29) and n = N/2, on the basis of which we obtained the integral equation (6.36). For from (6.42) it follows that p(a) is an even function of a, so that

hence, by (6.35), we see that (6.29) is satisfied. Moreover, the equality

shows that n = N/2 also. Finally, if we apply Poisson's summation formula to

we obtain

1

Here the modulus k is defined in terms of q = exp (nirj/K) by (1.7) in Appendix I (see also (62)). We do not need this fact in what follows.

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The quantum method of the inverse problem and the Heisenberg XYZ model

55

that is, the density p(a) is indeed positive. We turn now to the calculation of Amax(7V) and with this in view we consider (6.31). From the Fourier expansion of the logarithms of the ratios of Jacobi theta-functions it follows that in the basic domain (6.3) the second term in (6.31), as N-> °°, is exponentially small in comparison with the first. Thus, as 7V-*-oo, we obtain

From which, on the basis of (6.35), we conclude:

Using (6.39), (6.41), and the Fourier series for the logarithms of the ratios of Jacobi theta-functions we finally see that in the basic domain of the coefficients w- the free energy of the eight-vertex model has the form

Baxter obtained (6.46) in [ 9 ] . The formula is valid for any choice of the common positive normalizing factor in the parameterization (1.36) for the wf-. We emphasize once more that we have derived (6.46) subject to condition (6.1). To calculate the free energy / i n the physical domain, we have to arrange the coefficients Wj in the series (6.1), using (6.4), to find the parameters k, T?, and X from (1.33) and (1.35), to calculate the complete elliptic integrals K and K', and to substitute the values obtained in (6.46). From (6.46) it follows t h a t / i s continuous also in the extended domain (6.47)

0 < . w 4 O 3 1 .

Moreover, it can be shown (see [9]) t h a t / , generally speaking, is singular as w2 ->u>3 in the basic domain (6.1). Thus, the case w2 = w3 corresponds to phase transition in the eight-vertex model. We also point out that when w i ~* w2, then the standard expressions for the free energy of the six-vertex model can be derived from (6.46) (see [ 15 ] - [ 18 ]). We now use our results to calculate the energy of the ground state of the XYZ model. We recall that the Hamiltonian H of the XYZ model is connected

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L. A. Takhtadzhan and L. D. Faddeev

with the transfer-matrix T(X) of the eight-vertex model by (2.11)—(2.13), which we rewrite in the following form:

where

In (6.48) it is assumed that we have chosen for the coefficients Wj the realization of the projective parameterization (1.36) with the common normalizing factor 1. Using the properties of the Jacobi elliptic functions (see Appendix I) we easily find that the basic range of the parameters k and x\ is subject to

Using perturbation theory it can be shown (see [ 10]) that if then the eigenvector of T(\) associated with A max (A0 corresponds to the minimal eigenvalue E0(N) of the Hamiltonian H. Thus, from (6.48)—(6.51) it follows that when we choose in the parameterization (6.49) a negative common normalizing factor in the basic domain (6.3), then we have the following expression for the energy of the ground state:

It can be proved (see [9]) that (6.46) is a differentiable function when A = n. Hence, if we define the energy density of the ground state

then in the basic range (6.51) of the coefficients Jx,Jy,

and Jz we find that

Using the properties of the Jacobi elliptic functions (see Appendix I) and (6.46) we obtain finally that in (6.51) the energy density of the ground state of the XYZ model has the form

(6.55) was obtained by Baxter in [ 1 0 ] .

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57

To calculate the energy of the ground state in the case of arbitrary real Jx,Jy, and Jz it is sufficient to note that E0(N) is a symmetric function of Jx,Jy, and Jz, which does not change under a simultaneous change of signs of any two of its arguments. Therefore, we can always arrange Jx, Jy, and Jz in such a way that they satisfy (6.51). By a limit passage we can derive from (6.55) (see [ 10] ) standard expressions for the energy density of the ground state of the XYZ model (see [ 6 ] - [ 7 ] ) . To conclude this section we say a few words about excitations of the XYZ model. To arbitrary values of 77 subject to (6.3) there correspond complex solutions \j of the system of equations

This results if in (5.31) we put m = 0, n = N/2 and note that for values of rj in (6.3) it follows from (5.16) that im2/Q = rj/K'. By a limit passage we find that (6.56) holds for all values of 7? in (6.3). The simplest excitations of the XYZ model are constructed in the following way. We remove from the interval [~K, K) one value X,- and put it at the point \j +\ iK' of the interval Im a = Jr K', | Re a | < K; here the values of Re X;- again fill out the interval [~K, K] symmetrically as Af-* °°. Similarly we can construct excitations in which certain values X^ are replaced in the interval Im a = K'/2. The remaining excitations are their bound states and are more difficult to construct. The energy of these excitations is calculated in [22] to which we refer the reader for technical details. Let us say only that the calculations in [22] become clearer if one uses simplifications in the method of integral equations, as proposed in the paper [60] of Korepin and in [25]. We note also that in contrast to the XXZ model, in the general XYZ model not only the ground state but also the excitations on it correspond to the case n = N/2, that is, occur in the sector with zero "charge" q, where q = 2« — N, so that the case (5.16), in which there are also sectors with nonnull "charge", is of special interest. We do not wish to speak in passing of these interesting questions, but hope to return to them elsewhere. Conclusions

In this survey we have shown that the recently-created quantum method of the inverse problem works in the example of Heisenberg's XYZ model. Following Baxter, we have found for the Hamiltonian of the XYZ model an auxiliary spectral problem, which by its nature is a quantum problem. The monodromy matrix corresponding to it turns out to be connected with the eight-vertex model of classical statistical physics. In the example of the XXZ model we have shown how simple permutation relations for the elements of

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the monodromy matrix and the existence of a generating vector lead to the algebraization of Bethe's Ansatz; here the trace of the monodromy matrix, the transfer-matrix, is a generating function for the quantum integrals of motion. Thus, with the help of the quantum method of the inverse problem for the Hamiltonian of the XXZ model, we have constructed explicitly both com­ muting integrals, the quantum analogues of "variables of action-type", and also its eigenvectors, the quantum analogues of "variables of angle type". Using a series of commutation relations for the elements of the monodromy matrix of the eight-vertex model and a family of generating vectors, we have constructed an algebraic generalization of Bethe's Ansatz to find the eigen­ values and eigenvectors of the Hamiltonian of the XYZ model. In this way, we have found quantum analogues of "variables of action-angle type" and so have shown that the general Heisenberg XYZ model is a completely integrable quantum system. Basically our presentation has a formal algebraic character, since in the first place we wished to present the reader with interesting algebraic structures that rise naturally in the quantum method of the inverse problem. We now state a number of mathematical problems closely connected with the contents of our survey. 1) To clarify conditions under which the eigenvectors of the transfer-matrices T(X) in § 3 , which were obtained with the help of Bethe's Ansatz, form a basis in $QN. To investigate the degeneracies in the spectrum of the transfer-matrix T(X) and the Hamiltonian H of the XXZ model. 2) Similar questions remain also for the XYZ model provided that parameter 77 satisfies (5.16). In the general case, to examine the convergence of the series (5.13) and the completeness of the appropriate eigenvectors. 3) to give a strict justification of the limit passages in §6, without recourse to the theory of perturbations. APPENDIX I Here we recall the definition of the Jacobi elliptic functions and thetafunctions and list of their properties those that are used in the main text. For proofs of these formulae, see [61 ] — [ 6 4 ] . Let r be a complex number, Im r > 0, and q = enlT. We put

The series (1.1) converges absolutely for all complex z and represents an entire function of z. The function # 4 (z, q) = #(z, q) satisfies the relations

The remaining three theta-functions are defined as follows:

284 The quantum method of the inverse problem and the Heisenberg XYZ model

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By (1.2) they are quasi-doubly-periodic functions of z with quasi-periods 1 and T, where §x (z) is an odd function, and #j (z), t? 3 (z), # 4 (z) are even. The functions $x (z), # 2 (z), # 3 (z) and # 4 (z) have zeros at m + nr. m + 1 /2 + «r, m + 1/2 + (« + l/2)r, and w + (« + 1/2)T. respectively, where w and n are integers. The general addition theorems for theta-functions were obtained by Jacobi in his classical treatise "Fundamenta Nova Theoriae Functionum Ellipticarum" [61] (see also [64]). Here we cite only those formulae that are used in the text,

where for brevity the parameter q is omitted. To prove (1.4) (and similarly (1.5)) it is sufficient to note that, by (1.2) and (1.3), the ratio of the left-hand side of (1.4) to the right-hand side is an entire doubly-periodic function, that is, a constant. It is equally simple to prove the relation

We put

and let

Then

The Jacobi elliptic functions of modulus k are defined in the following way:

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where u = u/2Kk. They are doubly-periodic functions of u. The following relations hold:

where, if the modulus k is not indicated, the expressions sn u, en u, dn u, K, and K' denote sn {u, k), en (u, k), dn (u, k), Kk, and K'k, respectively. Thus, the elementary periods of sn u are AK and 2iK'\ of en u and 4K and 2K + 2iK', and of dn u are 2K and 4iK'. The number k' is called supplementary to the modulus k, and the quantities K and A" are the complete elliptical integrals of the first kind of modulus k and k' respectively. They have the form

For a given modulus k to find the number r, Im r > 0, for which (1.7) holds is called the inversion problem. It is uniquely soluble when k2 =£ 0 or 1. The function y = sn (u, k) satisfies the differential equation

and therefore, the differential equation of the general form

also can be integrated in terms of elliptic functions. So we obtain (1.29) and (1.30) in § 1 . The following relations hold:

with the help of which we obtain (1.32) of § 1. The addition theorems for the Jacobi elliptic functions of modulus k have the form

286 The quantum method of the inverse problem and the Heisenberg XYZ model

61

The imaginary Jacobi transformation connects elliptic functions of argument u and modulus k with those of argument iu and supplementary modulus k' and looks as follows:

The connection of the Jacobi elliptic functions of argument (1 + k')u and of modulus fcj, with those of argument u and modulus k is given by the Landen transformation

where By using (1.16)—(1.18) we obtain from the parameterization (1.32) in § 1 the basic parameterization (1.36) for the coefficients Wj. If k -* 0, the supplementary modulus k' tends to 1, K -+ 7r/2, K' -*■ °°, and the elliptic functions sn u, en u, dn w become sin u cos w, and 1, respectively. The differentiation formulae for Jacobi elliptic functions have the form

From a. 11) and (1.19) we deduce (2.20) in §2. Let us also point out that when 0 < k < 1, as u ranges from 0 to iK', sn u takes on pure imaginary values from 0 to i°°, while sn u and dn u take real values from 1 to °°. Together with (1.15) this proves (6.50) in §6. In the main part of the text we have used the Jacobi theta-functions H(w) and 0(H), which were also introduced by Jacobi in his treatise [61 ] . They can be expressed in terms of the theta-functions ^1 (z) and # 4 (z) as follows:

In this notation

The Jacobi theta-functions H(u) and 0(H) have the following properties:

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where The addition theorems (1.4)—(1.5) for the Jacobi theta-functions H(u) and Q(u) allow us to prove (1.38) in §1 directly. To prove (4.14)—(4.17), (4.24), and (4.27)—(4.30) in §4 we have to use, apart from the addition theorems (1.4)—(1.5), also (1.6), which in terms of H(w) and 0(w) takes the following form:

where From the expansion of the theta-functions in infinite products we can obtain expansions of their logarithms in Fourier series. In this manner we find

In (1.25) the principal branch of the logarithm is chosen; the series (1.25) converges absolutely provided that

Using (1.22) we find that

The series (1.27) converges absolutely provided that

From (1.20), (1.25), and (1.26) we conclude that for the functions (a) (defined in §6) when 0 < k < 1 (that is, for pure imaginary r) the follow­ ing relations hold:

and

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63

The series (1.29) and (1.30) also determine the branches of the logarithm in (6.33) in §6. (1.29) converges absolutely provided that

and (1.30) provided that Therefore, (1.29)—(1.30) converge absolutely for real a and 77 in the domain (6.3) in §6. The coefficients (6.39)-(6.40) in §6 are obtained by term-byterm differentiation of these Fourier expansions. From (1.29) it also follows that when (6.3) in §6 holds, the real part of iip(\) is positive. This fact was used in §6 to obtain (6.44). APPENDIX II

We now consider the associative algebra with the generators At(x) satisfying (1.52) in §1, that is, where /, /, k, I = 1 or 2 and the summation is over repeated indices. Here we give a realization of this algebra in terms of fiberings on elliptic curves. This realization was proposed by Cherednik [45]. Let L be a lattice in the plane C1 with the generators 1 and T, Im r > 0; % the elliptic curve corresponding to it, that is, % = C1 /L. We consider a linear analytic fibration V on £. It is known (see [65] - [ 6 6 ] ) that all such fibrations are obtained in the following way. Let V be the trivial linear fibre of C 1 , that is, V = C1 X C 1 . We define an action of L on V in the following way: where where co = «j + « 2 r i s a n element of L, iz, u) a point in V, and N is an integer, N> 1, and |3 is an arbitrary complex number. Then V = V/L. The sections of a fibre VN $ can be identified with the analytic functions / on C1 for which

The space of these functions is of dimension Af and is denoted by $£{N, p). We fix a complex parameter 17 and examine the space <$?(2, p0),where 0o =m(l- IT). The basis of dtf(2, po) has the form

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The generators At{x) are realized as operators taking the space of sections

$S{N, p)into and look as follows: The operators Ai{x)Ak{y) space of sections

and Al(y)AJ(x) take 3{(N, |3) into one and the same

and so these operators can be equated. Thus, we find that if (II. 1) the At(x) satisfy (II. 1), then the Shl satisfy the Baxter-Yang relation (1.51), because it obviously follows from (II.6) that the algebra with the generators A{(x) is associative. For (II. 1) to hold it is sufficient that for all values of the indices We write w = z -y, v = x —y. Then (II.7) takes the form The functions form a basis in $6(4, 2$0 + Ani(v + n)), while (II.8) establishes a connection between them. Therefore, the SJj (v, rj) are uniquely determined by (II.8). Using the antiperiodicity off1 (z) under a shift of the argument by ^, and the periodicity of/ 2 (z), we conclude that And so from (1.50) in § 1 we again find that the ,52-matrix has the form (1.39). The remaining coefficients Sh (u, 77) are determined from (II.8) with the help of the addition theorems (1.4)—(1.5). Finally,

Comparison with (1.40) shows that after obvious renotations and multiplication of (11.10) by the common factor W)h(v

+ T\)U{V + r\)

we again obtain the elements of the J?-matrix in the form (1.40). Thus, we have shown that the geometric interpretation (II.6) of the operators Ai(x) again leads to the elliptic parameterization (1.36) in § 1.

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The construction described here can be generalized to the case when the indices take a larger number of values and also when x and y are vectors, but then instead of elliptic curves we have to analyse Abelian varieties. We also note that the formulae (II. 1) recall the famous Weyl relations from the theory of canonical Schrodinger operators. This is borne out by the con­ crete realization (II.6) of the At{x) as multiplication and shift operators. It would be very interesting to follow through this analogy. Possibly in this con­ text it would be useful to look at the projective representations of the square % x % of the elliptic curve £.

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Translated by D. W. Jordan

294

COURSE 8 INTEGRABLE MODELS IN (1 + l)-DIMENSIONAL QUANTUM FIELD THEORY

Ludvig FADDEEV Steklov Institute, Fontanka 27, Leningrad 191011, USSR.

J.-B. Zuber and R. Stora, eds. Les Houches, Session XXXIX, 1982—Developpements Recents en Theorie des Champs et Mecanique Stalistique Recent Advances in Field Theory and Statistical Mechanics © Elsevier Science Publishers B. V., 1984

295

296 Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

Arguing in favor of the (I + l)-dimensional models Possible approaches People involved What is a completely integrable model? A representative model Zero curvature representation Conserved quantities The fundamental Poisson brackets Poisson brackets for the monodromy matrix The Hamiltonian is a member of the commuting family FPR substitutes for the zero curvature condition The L - o o limit Simplifying the Poisson brackets Linearization of the equation of motion Elements of scattering theory The Riemann-Hilbert problem Inverse problem reduced to the Riemann-Hilbert problem Inverse problem investigated Angle-action variables Heisenberg ferromagnet Sine-Gordon model Non-isotropic magnet (Landau-Lifshitz model) Lie-algebraic Poisson brackets FPR as Lie-Poisson brackets Orbits for NS and HM models Trigonometric and elliptic /--matrices Lattice systems A representative example NS as a Jimit of a spin lattice model Trigonometric lattice model Quantum systems Rational example Comments on trigonometric and elliptic reductions Comments on higher spins Concrete models 35.1 Spin \XXX model 35.2. "Spin \" XXZ model 35.3. NS lattice model 35.4. SG model

563

565 565 565 566 567 567 568 569 569 571 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 589 591 592 594 595 596 597 597 598 598 598 599

297 36. Algebraic Bethe Ansatz 37. Comments on the Bethe Ansatz 37.1. Continuous limit in NS model 37.2. Nature of the spectrum 37.3. Additive observables 37.4. Counting of Bethe states 38. Generalization and questions unanswered as yet 39. Infinite volume limit 40. Connection with statistical mechanics 41. Connection with the factorizable S-matrix 42. Concluding remarks References

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600 603 603 603 603 603 604 604 605 605 606 607

298

1. Arguing in favor of the (1 +l)-dimensional models There exist several reasons why (some) people are interested in such models: (1) Every exact (non-perturbative) solution of a field-theoretical model can teach us about the ability of quantum field theory to describe particle spectra and scattering. (2) Some (l-(-l)-d models have physical applications i.e. in the solid state theory. (3) The mathematics of the subject is quite beautiful. 2. Possible approaches There are several ways to become acquainted with the methods of exactly soluble models: (i) Via classical statistical mechanics. This way is associated with the names of Onsager, Lieb, Baxter, . . . . (ii) Via the 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 co-workers and developed by Lax, Zakharov, . . . . I shall use the last way because it is nearer to my own interests. However in the course of the 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 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 of Paris VI (de Vega, ...), the Landau Institute (Belavin, Wiegman, ...) and Kyoto University 565

299 L. Fuddiw

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(Sato, Miwa, Jimbo, ...). In my lectures the point of view of the Leningrad group will be presented. This group consists of Kulish, Korepin, Sklyanin, Takhtajan, Izergm, Reshetikhin, Semyonov-TjanShansky, 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 [1-6] cover many parts of my lectures. However the method of exposition as well as some technical details are new. They reflect the attitude taken in the forthcoming monograph on classical and quantum theory of solitons which I am preparing in collaboration with Takhtajan. 4. What is a completely integrabk model? The term originates in the classical Hamiltonian mechanics. The system given by the phase space r2n with coordinates c = (r^/ 1 ), i = 1 n and Hamiltonian Hfaq) is called completely integrable if there exist n independent functions Ifaq) such that

lHJt) =0, {/„/,} =0. •

(1)

The /,- are called the (commuting) integrals of motion or conserved quantities. The Hamiltonian depends on p, q only through them, H

= mn

(2)

Moreover one can in principle find a change of variables (p, (/,
(3)

such that the following relations are true

{%/ t }=Wn

(4)

and, in particular {H,q>i} =o);{/).

(5)

In the new variables the equations of motion simplify / = {H,I} =0,

} = (o(I),

(6)

so that /=constant; is a solution.

t

(?(

300 Inteuruhlc moclt'l.s in (I + 11-dimensional quantum field theory

567

The (
and the Hamiltonian

lead to the equation of motion

which has quite a number of physical applications. In particular it is a Hartree—Fock equation for the system of non-relativistic particles with (5-function interactions, g playing the role of the coupling constant. We are going to show that this model is completely integrable. 6. Zero curvature representation Consider the pair of first order differential derivatives)

operators

(covariant

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L. Faddeev

where U and V are 2 x 2 matrices

parametrized by \p*, ift and a complex parameter /. The (zero curvature) condition

being fulfilled for all /. is equivalent to the NS equation. From the zero curvature condition it follows that for a closed contour 7 in the space-time plane the ordered integral

is a unit matrix. 7. Conserved quantities Suppose we have periodic boundary conditions

Take the boundary of the rectangle -L^x^L, the notations

O^t^T

as y. With

the following relation is true

so that tr TL(A,f) is independent of f; A being arbitrary we have an infinite family of conserved quantities.

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Integrable models in (I +1)-dimensional quantum field theory

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To show the complete integrability we have to answer several questions (i) do these quantities commute; (ii) do they commute with / / ; (iii) do we have enough of them. The answers will take some 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 \j/ the symbol {A ® B} denotes the 4 x 4 matrix of Poisson brackets of all their matrix elements. Explicitly it means that

In this notation the following relation can be easily checked

Here r(X~n) is a 4 x 4 matrix

where P is a permutation matrix. In terms of Pauli matrices we have

and in a natural basis P looks as follows:

In what follows the relation (18) will play a fundamental role so that we shall give it a special abbreviation: FPR (fundamental Poisson bracket relation). 9. Poisson brackets for the monodromy matrix The matric TL(X) will be called the monodromy matrix because it describes the transport along the "circle" -L ^ x ^ L. We shall

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I. Fadtkev

570

calculate the Poisson brackets of all its matrix elements using FPR. We divide the interval -L ^ v ^ L into infinitesimal sub-intervals of length A and introduce the corresponding transport matrices ■

L,U) = exp

U(x./.)dx • j ,

= /+

U(.v,;.)d.Y + 0(zl->.

(22)

Integrating FPR we get the relation ;L ; <;.)®L,(, ( )! = [r ) p , , + 0(d 2 ).

(23)

In terms of LJAI the matrix TL{X) is written in the form TLU) = f\LiU)

+ 0(AY

(24)

J

Now use the general formula {A ® BC\ = I ® B{A ®C\ + \A®B\l®C

(25)

to get

{Y[L,U)®Y\Lk(ti)} i

k

= n ( n umm n Lk(Li)[LPu)®Lqut)} n *,-u>® n MTOJ (26) where /7 p + 1 means product starting from i = p+l, 77'"' the same ending at i = p-1, etc. Using FPR we rewrite the rhs in the form X i l 7.,.(A)®L ( ^)[rU-/i),L p (;.)®L>)] n p

L,U)®f»,

p+i

which is nothing but the commutator of 77L,(/l) ® L,(A0 and r{A-n). Thus in the limit z l » 0 w e get the desired equation {TL(V® TL(ft)} = [r(A-vlTLU)®TL(M)l

(27)

In particular it follows that the family of conserved quantities combined in the generating function tr TL{X) commute among themselves. {tr7i.(/l),tr7 L (/i)}=0.

(28)

304 Inteyrable models in (I +1)-dimensional quantum livid theory

571

10. The Hamiltonian is a member of the commuting family We shall show that tr TL(X) can be represented in the form

where the function PL(k) (called quasi-momentum) can be written as a formal series

C„ being local functional of
It satisfies the equations

The solution can be found in terms of the gauge transform of a diagonal matrix

when Z is diagonal and W anti-diagonal, admitting the formal series expansion

where ^ ( x ) are local functions of ^*(x), ij/(x) and their derivatives. To check it, substitute this representation into the differential equation and separate the diagonal and anti-diagonal parts to get

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L. Faddeev

where we introduced the notation

so that U0 is anti-diagonal and U1 is diagonal. Eliminating Z from the first equation we get the Ricatti equation for W

This allows a formal \/X expansion, Wx being given by and subsequent Wn are recursively obtained as functions of (/>*, ij/ and their derivatives at the point x. In particular all W„{x) are periodic in x, WB( — L) = W„(L). Moreover one can see that in general W can be represented in the form

with some scalar function w. Integrating the equation for Z(x, y\X) we get

Observing that tr Z = 0 because of the unimodularity of T(x, y\X) we get finally the desired representation for tr TL(A) with PL(A) given by the expression

It is instructive to go through the first steps of calculating Wn to see that coefficients C,, C2 and C3 are proportional to the functionals N, P and H where N and P are the number of particles and the momentum

and H is the Hamiltonian. Thus questions (i) and (ii) are answered affirmatively.

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Iniegrahk' models in (I + I }-dimensional quantum field theory

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11. FPR substitutes for the zero curvature condition Before proceeding to answer 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

covering all possibilities in one formula. Given this equation, U{x, /.) satisfies the equation

The rhs can be read off the more general expression

where the definition of T(x, y\fi) in terms of U(x, n) was used. Using FPR the rhs here is rearranged as

which with the help of the differential equations for T(.x, y\X) can be written in the form

where Q(x\n, / ) is given by

Let tr, denote the trace over the first space in our tensor product. With this notation we have

where Qi(x\fx,X) = trx Q{x\y.,X). This is already a kind of zero curvature condition. Let us now simplify the expression for Qi(x\fi,X)) with the help of the formula for T(x, y\n) in terms of W{x,n) and Z(x, /i). Periodicity and cyclic symmetry of the trace allow to write Qi(x\n,X) in the form:

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L. Faddcev

and the first term gives an inessential contribution to Ql : it is xindependent and proportional to the unit matrix, so it does not contribute to the zero curvature condition. Omitting it we get finally:

Here the property

was used. Note that

is a local function of tA*(x), ij/(x). Remember now that tr TL(//) = 2 cos (PL(/v)L) so that

Dividing by sin PL(n) we get a local zero curvature representation

the matrix V(x\n, X) 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-KX) limit We are going to return to question (iii). It is difficult to count properly the number of integrals of motion when it is infinite. Things simplify in the limit L -» oo 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

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Inteqrahle

models in (I + I)-dimensional

quantum field theory

575

We shall consider the simplest case where ij/* and tp vanish when |.v| -» x . Then the asymptotic behaviour of solutions is governed by the matrix

In particular the following limit exists

called a transition matrix. It can be written in the form

( ± ) corresponding to the sign of g. Bearing in mind that W„[x) vanish when |x| -» oo we have the relation

so that In a(X) plays the role of the generating function for the conserved quantities. 13. Simplifying the Poisson brackets The Poisson brackets for the elements a{X), a*(A), b{/.) and />*(/) can be found explicitly. Begin with the already known formula

multiply it by £( — L, X) ® £ ( - L , ^) from both sides and go to the limit L -» oo. In the rhs it is convenient to consider the two terms in the commutator separately. Consider the function 1/A — /i to be v.p./X—fi for definiteness. Use the property

of the permutation matrix. Then the following relation holds

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L. Faddecv

where

These limits can be found explicitly with the help of the formula

thus leading to the expressions

14. Linearization of the equation of motion In particular we have the following relations

(as it must be) and

From the last one we find that the equations of motion, generated by In a(n) are linear for b(n)

In particular for the Hamiltonian, which coincides up to a factor with the third coefficient in the expansion

we get

310 Iniearuble models in (I +1)-dimensional quantum field theory

577

Thus the map

(«^)-(W),fc*tf)), linearizes the equation of motion for the NS model. To answer question (iii) it remains to investigate to what extent this map is one to one. 15. Elements of scattering theory We continue collecting information about the solutions of the auxiliary linear problem. It is useful to realize that it is nothing but the scattering problem for the stationary massless Dirac equation. Breaking TL{X) into the product

and taking the limit L -> oo we see that T(X) has a representation

where

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 /. It turns out that the columns of T±(x,X) allow an analytical continuation in one of the half-planes I m A ^ O . Introduce special notations T(1), T (2) for the columns, so that

Then T{+1) and Ti2) have a continuation onto Im/i > 0, whereas T(+2) and Ti1* are analytic in Im X < 0. The relation

for fixed x can be considered as a sort of boundary problem (Riemann-

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L. FaiMivv

Hilbert problem) in the theory of analytic matrix-functions and used for the reconstruction of T+(x, /.) in terms of T(/.). 16. The Kiemann-Hilbcrt problem

The simplest example is as follows: let two (matrix) functions G + (/) be analytic for + I m / > 0 and have there neither zeros nor essential singularities, for instance

For real / we are given the relation

Then if some conditions on the given C(/.) are fulfilled, G±(/.) can be found. One necessary condition is that

vanishes. In a scalar case this condition is also sufficient and the solution G + (A) can be found through dispersion relations. In a matrix case a convenient sufficient condition is the positivity of one of the two matrices

Representing G(A) and G±(A) in the forms

we reduce the Riemann-Hilbert problem to the Wiener-Hopf equation. Indeed, in the relation

the lhs vanishes for r > 0 leaving a linear integral equation for F+(t).

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(nteyrable models in (I + I)-dimensional quantum field theory

579

17. Inverse problem reduced to the Riemann-Hilbert problem The equation

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 /. assumes the form

where S(A) can be expressed through a(/.). h(/.)

Note now that ci(/.) is analytic in the Im /. > 0 half plane: indeed it can be expressed in the form and can vanish at points /.j whenever the solutions T | " and Ti 2) are linearly dependent

Both sides vanishing exponentially for |x| -» oo, this can happen when /., 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(/) can be essentially arbitrary. In what follows we require a(/.) to have a finite number of simple zeros. Let us first suppose that a(k) has no zeros. Then we can include l/a(A) into the analytic matrix S + (X) without spoiling it. With new notations

we have a linear relation

which is equivalent to the normal Riemann-Hilbert problem because for

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L. Faddeev

\X\ - » o o w e have the following asymptotic behavior

so that G±(x, X)E( — x . / ) have no essential singularities in their half planes. Now one can see that the matrix G(X) + G*(X) 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(/.)

shows that \b\ does not exceed 1 and so G is positive. Thus when a{X) has no zeros the matrices G + (\. /.) and G(X) are in one-to-one correspondence. If we allow now a(X) to vanish at points /. = Xh the matrix G + (A) has simple poles there, whereas G_(A) has simple zeros at /. = kf. We can absorb these singularities into a left multiple of the form

where each factor has the form

and Pj is a one-dimensional projector. We can uniquely find B(X) in such a way that G ± (x, A) will be represented in the form

where G + (x, X) have neither zeros nor poles. The factors y, come explicitly into Py 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 for large |A| it is easy to

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Inteqrahle models in (I + I)-dimensional quantum field theory

581

show that the solutions C + (.v,/) and G_(x,/.) satisfy the differential equation of the auxiliary linear problem. Indeed, differentiating both sides we find

Thus the rhs gives an analytic continuation of the Ihs into the upper half-plane. The essential singularity at infinity being cancelled the only possibility is that the Ihs is a polynomial in /. From the asymptotic behaviour we get

thus reconstructing the auxiliary linear problem with matrix U0 given by

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(A). It is less trivial to show that the matrix t/ 0 (x,/.) vanishes when |x| -► oo and that the solutions G + (x, A), in their linear combinations T+{x,X) have the prescribed asymptotics. However it can be done, thus showing that the system of scattering data (6(A), /?*(/), /_,-,}'_,■) are in oneto-one correspondence with the initial data (iA*(x), t/^x)). 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 scattering. 19. Angle-action variables We have found a change of variables

(discrete variables appearing only in the attractive case g < 0) which linearizes the equation of motion. The new variables can be interpreted as the angles and actions. Indeed the dispersion relation for lna(A) is given by

315

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L. Faddeev

Thus the generating function for the conserved quantities depends only on the variables

—half as many as the set (b,b*,A,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 arg/>(/), In \yj{ and argy,, the first and the last being of the phase type. The main conservation laws now can be written as follows

exhibiting the spectrum of modes. The continuous spectrum describes a particle of mass m = \ which is the only mode of perturbation theory. The discrete spectrum introduces a non-perturbative contribution in terms of particles of mass r]m, r\£, playing the role of momentum and (#2/12)r/3 that of internal energy. When quantized in the quasi-classical approximation rj 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 ifr, i//*, corresponding to b = 0 and just one zero A>, describe the classical soliton solution. This concludes the discussion of complete integrability of the NS 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

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Intearuble models in (I + I)-dimensional quantum field theory

583

subject to the commutation relations

The Hamiltonian is given by

Equations of motion look as follows

The auxiliary linear problem is defined through operator L

where a" are Pauli matrices. The fundamental Poisson brackets (18)

are satisfied with an /--matrix coinciding with that of NS model

The local conserved quantities (and the Hamiltonian among them) are given by expanding the transition matrix T(x, y\X) in the vicinity of A = 0. 21. Sine-Gordon model The relativistic equation

is a zero curvature condition for a connection given by

317

584

L. Faddcev

if the vector k = (k0>A,) is on the mass shell

Parametrizing it by the rapidity /.:

we get the matrix U[x, A) which satisfies FPR with the /--matrix

which we shall call trigonometric to distinguish it from the rational rmatrix of the two previous models. 22. Non-isotropic magnet (Landau-Lifshitz model) The same vector variable S"(x) can be used in yet another model with equation of motion

where J(S) is a vector with components (J lSl,J2S2-,J3S3), Ju J2, J3 being given constants. If we look for an appropriate connection of the form

where Ua, Va, Wa are some coefficients we find that the zero curvature condition leads to the following relations among them

The last condition gives two equations on three coefficients so that they can be parametrized by one variable X. Explicit expressions involve elliptic functions. The FPR are satisfied with the r-matrix given by

318

Intcgrable models in {I + I)-dimensional quantum field theory

585

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 cannot invent many natural Poisson brackets. One important family is generated by Lie algebras. Let G be a Lie algebra with the generators X0 and structure constants Ccab

Consider a linear space G* with variable ca and introduce the Poisson bracket

or for two arbitrary functions on G*

This Poisson bracket satisfies the Jacobi identity, however it is degenerate, namely there exist functions Z(£) which commute with arbitrary functions

This bracket is associated nowadays with the names of Kirillov, Kostant, Souriau, Berezin, ... but apparently it was known to Lie himself. To make the bracket non-degenerate one must restrict it on a submanifold. 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 non-degenerate. 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.

319

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L. Faddeec

24. FPR as Lie-Poisson brackets We shall show that FPR appear naturally in connection with current algebras. Let us omit the variable v for the time being and consider the Lie algebra C of functions »/(/) with values in a finite-dimensional Lie algebra G with generators Xa and structure constants Cah. A natural basis of generators in C is supplied by

with the commutators

We shall distinguish two "triangular" sub-algebras: C + is generated by X™ for m ^ 0; C_ is its complement. After changing m -» — 1 — m, the new index being non-negative, the commutation relations in C_ can be written as

Thus the dual spaces C* and C* are supplied with the Poisson brackets

respectively. Let us introduce now the generating functions for the variables

It is easy to calculate that these functions satisfy the following relations

The next trick is to saturate the index a. Let Kab be a Cartan-Killing matrix corresponding to the basis Xa and Kab its inverse. In what follows we require that such inversion is possible, reducing the admissible Lie-algebras to semi-simple. Introducing the matrices

320

Intern able models in (I + I)-dimensional quantum field theory

587

the following relation can be checked irrespective of the representation for X..

Using this for

we obtain the relation

Now we remember about x which we reintroduce as a second variable in the current algebras C + . The generators now acquire an additional index v and commutation relations look as follows

Repeating what we have done and keeping x untouched we end up with the general forms of rational FPR

25. Orbits for NS and HM 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 evidently invariant with respect to the action of C + defined through the equations

for all b and n. Moreover, only £$(y) acts non-trivially in this subspace and

is conserved.

321

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L. Faddeev

Choosing the orbit by the condition

we get the matrix

satisfying FPR and defining a non-degenerate Poisson bracket. It is exactly the matrix U{x, A) used for the HM model. The NS model is defined in an analogous way as the simplest nontrivial orbit in C*

when

with the commutation relations

It is clear that the matrices U0 and (7, used for NS model do satisfy 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

Then it is easy to check that U{X) satisfies the relation

322

Intearable

models in (I + I)-dimensional

quantum field

theory

589

where

In particular the r-matrix for in the case G — SU(2) and A In some cases (but not for only once) again. The elliptic

the SG model can be obtained in this way = a3. all G) one can repeat this trick once (and r-matrix is obtained thereby.

27. Lattice systems We have used above lattice approximations to continuous models. The role of covariant derivative

was played by the approximate FPR

infinitesimal

parallel

transport

L, with

the

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 TN(X)

are satisfied, so that the family of dynamical systems defined by

have an infinite number of conservation laws. 28. A representative example Let us fix the r-matrix to be rational

and suppose that Yl plays the role of permutation, namely for any X from G the following relation holds

323 590

L. Faddeev

Then it is easy to check that an L-matrix of the form

satisfies the lattice FPR if the field variables S" satisfy a discrete version of the Poisson brackets used in C* :

This procedure works in particular for G = SU(/V). For G = SU(2) the matrix L, is 2 x 2 and given explicitly by

where we multiplied the previous L,(/.) by /.. The model associated with this L,(A) is a discrete version of the Heisenbe'rg magnet. To get local conservation laws from tr TN(X) we observe that

If we fix the length Sf to be independent of i (as was done also in the continuous case)

then for X = ±S, the matrix L,(A) is degenerate

We can use it to expand tr TN(X) in the vicinity of X = ±\S\ in power series of (A±|S|). In particular we have

and explicit calculations give

Thus ln(tr IJ,(|S|)tr7J,(—|S|)) gives a nearest neighbour interaction Hamiltonian

324 Inlegrable models in (1 + I)-dimensional quantum field theory

591

which is a natural lattice approximation to the continuous Heisenberg magnet after rescaling

29. NS as a limit of a spin lattice model We can realize the Poisson brackets for 5, a using the ordinary canonical variables \(/h ij/*

as fields. The formulas

where give such a realization for S2 — 4/g2A2. We have intentionally split S2 into a product to indicate the scale of the continuous limit A -* 0, when ^,- and \p* are of order A112. In this limit L,(A) has an approximate form

which coincides with the expression

after multiplication by (gA/2)o3 from the left and changing gk - » \ L Additional matrices o3 in the product /7 L, can be combined to multiply each Lj with odd i from both sides. Now it is easy to see that

All this shows that the continuous NS model can be obtained from the

325

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L. Fciddeei

lattice spin model in the scaling limit if the following convention is used

In particular one gets the Hamiltonian of the NS 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 /-matrix. 30. Trigonometric lattice model We can repeat the trick of averaging over the group of shifts. However instead of summing shifted L,(/) we must use a product. Suppress the index i and define L(A,») as follows

where A is a representation of an automorphism of G as above. Define L{X) to be an ordered product

We show that L(X) satisfies the lattice FPR with the trigonometric rmatrix. Indeed, from FPR for L,(A) we get

where

Furthermore, differentiating the product we have

and after using the FPR this is rewritten as

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In ley rank' models in (I +■ I)•dimensional quantum field theory

$93

Making resummation by parts with the natural convention

we get the usual FPR for the matrices L(X) with the trigonometric rmatrix

It would be interesting to evaluate explicitly the product L(/) in the case of the spin system. This requires the calculation of products of the form

where B is an arbitrary 2 x 2 matrix and n is some number. I do not know any direct means of calculating such products. Some indirect considerations show that L(/) is given by

where

Such a matrix L(X) can be used to define a non-isotropic lattice spin system. It can serve also as a lattice version of the Sine-Gordon model. Indeed after parametrization

where

we get an L-matrix which after multiplication by al from the left coincides with the matrix

327

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L. Faddce v

in the limit A -* 0 (in some particular units for m and (1 which can be reintroduced). Thus once more we see that there exists essentially only one lattice model for the trigonometric /-matrix. The elliptic case can be considered in an analogous way. The lattice version of a completely nonisotropic magnet is obtained. Its realisation as an elliptic-sine-Gordon has not 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 f]{ the full Hilbert space is

Of course all f/, are the same. In terms of the field operators acting in JJ, we have to construct a matrix L,{X) in an auxiliary space V depending in X. The FPR relations in quantum version appear as follows

Here R(X — n) is a matrix in V ® V independent of the field variables. These relations could be guessed through the correspondence principles. Indeed, they turn into classical FPR if /?(/. — /i) allows an expansion

where P = I + fJ. However their main merit consists in the fact that they allow to get a simple form of the commutation relations for the quantum transport matrix

which is well defined because the field operators for different lattice sites are supposed to commute. In what follows we shall call them fundamental commutation relations—FCR. Multiplying FCR over all i we get these relations in the form

It follows, in particular, that the family of operators generated by tr TN(X) is commuting. Whenever we find an interesting Hamiltonian in

328

Intearahle models in (I + 11-dimensional quantum field theory

595

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 5" of Liealgebraic origin, satisfying the commutation relations

and let Xa be a representation of the algebra G (defined by Ccab) such that P = I + n is a permutation and is normalized as P2 = a21. For SU(/V) algebras this representation is the fundamental one. Then for

and

the FCR are satisfied. In the process of checking it, the non-trivial terms generated by the first term in /?(/ — n) are

whereas the second term produces the expression

It is exactly the permutation property which allows those contributions to cancel. Indeed in a more explicit form we have

two

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Now we have to use the field commutation relations and the following property of the matrices A" to reduce eq. (185) to

so that the two contributions above indeed cancel each other. In the particular case of G = SU(2), the L-operator

where ot a are Pauli matrices acting in rj( = C2, defines the spin quantum Heisenberg chain. 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 K-matrices as follows:

Unfortunately I do not know any general proof that such L,(A) and /?(A) satisfy FCR. In the particular example of the spin \ 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, operators has the form

where index i means that /, and oai acts in tfit the functions W0(X), Wa(X)

330 Intearahle models in (1+ I) -dimensional quantum field theory

597

are trigonometric or elliptic functions of X. I hope that some way could be found to justify the process of averaging introduced here. 34. Comments on higher spins The construction of L-operator in the rational case used above works only for the 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 an L, of the form

for any representation Sa of SU(2) with the same r-matrix as above. However if we want to have another representation for the auxiliary space (below we shall see when it is needed) the expression will be more difficult. Moreover even with a two-dimensional auxiliary space in the process of averaging we will obtain L,(A) of the form

where T0 and Ta, a = 1,2,3 are non-trivial functions of the spin operators Sa in contrast with the spin j case. It is interesting to observe that the commutation relations for T0 and Ta which can be read from FCR have the following general structure

where q>ab are some constants. These relations could be looked upon as those of deformations of the SU(2) Lie-algebra we begin with. For the spin \ representation they reduce to the representation used above (see eq. (188))

35. Concrete models Let us collect several quantum lattice models associated with the SU(2) Lie-algebra which have already been mentioned. At first the

331 598

L Faddeet

corresponding L-operators will be given in two-dimensional auxiliary space. 55.7. Spin | XXX model Field variables are Pauli matrices Sa, = \va,. The L,(/) matrix has the form £t-AI$/f+£2>,&
(193)

where we have made a convenient change / - \X. The Hamiltonian ff*+is?x<

W=-|Xffii0ffi+i,„.

(194)

■if

can be obtained from as follows d H = constant x —- In tr T{X)\x = il2.

(195)

The role of the point X = i/2 will be clearer later. Note that in the classical case the Hamiltonian could be obtained without derivation in X. 35.2. "Spin | " XXZ model The same field variables are used in the L,-(A) matrix of the form /sinA+cos/lS-,

S+

\

The local Hamiltonian is obtained from tr TN{X) in the viciniiy of some X. This L((X) is a product-average over the shifts of the XXX L.-matrix with A = (j3. 35.3. NS Lattice model Field variables are ^,-, # with the commutatton relations [#• « ] = **•

(197)

The matrix Lt(X) coincides with that of the XXX model after using an infinite-dimensional representation for S, a. Corresponding expressions for the spin variables S, a can be obtained from the classical ones given

332

Integrahk models in (I + I)-dimensioned quantum field theory

599

in eq. (156) using normal ordering. One must also not forget the multiplication byCT3from the left. The representation used is irreducible, selfadjoint for g < 0 and corresponds to spin

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 r\. Without going into the details let us state that such a Lt{X) can be constructed, local conservation laws are given by tr TN(A) in the vicinity of X — iS and the Hamiltonian has the form

where fs(x) is some function with the small argument behavior

It is clear that in the scaling limit (161)

one is able to obtain the NS continuous Hamiltonian from these formulas. The models of integer (or half-integer) spin j could be obtained from this just by restricting S = j . The first non-triviai / s is given by

and it is a polynomial of order 2/ for any / 35.4. SG model Field operators are
The matrix L,(A)

can be considered as that of the trigonometric spin model for the infinite

333 600

L. Faddeev

dimensional representation defined through S+ = e^'4,

s 3 = i/ty.

(203)

One must also remember the multiplication by <x, from the left. The program of finding the local Hamiltonian is not complete yet but it is quite plausible that H has the form « - ZMSi.iSi+ut+Si2Si+ia+aSit3Sl+lti)>

(204)

where jp(x) ~ x and a is expressed in terms of tn, 0 and A. One sees that in the scaling limit this expression gives some regularized version of the SG Hamiltonian Un2+±
H=

(205)

The /^-matrices for all these models are 4 x 4 matrices of the afec-form la{k) v

'

0

\ 0

0

0

c(X) b(X)

0

0

0

\

0

a{X) I

where a(X) can be normalized to be one and after that we have W = y^r, c(A)=7iT,

(207)

in the rational case and isiny

,.

sh X

,„„

in the trigonometric case. Here y is defined as the period of the shift and coincides with the coupling constant y = 02/8 in the case of the SG model. 36. Algebraic Bethe Ansatz We shall show an algebraic procedure to get a family of eigenvectors for the commuting operators tr TN(X\ ffo the eexmples sisted above. Let us write TN in the form

334

Iniegrable models in (I + I) -dimensional quantum Jield theory

601

so that

and BN(X) is a quantum analogue of the angle variable b(X) in the classical infinite volume case. Now the angle variables being quantized have to be spectrum generating operators. So it is natural to look for eigenvectors in the form

where Q is a particular eigenvector. We shall see that this program can be realized. First we construct a suitable Q. To do it, observe that for L{{X) there exists a vector co, such that

when a{X) and S(X) are c-numbers (local eigenvalues). The local vacuum co, is the highest spin state for all models but SG. In the latter case one can find w{ only for the product Li+lLj. It is clear now that the state

is an eigenvector of AN(X) and DN{X)

with a trivial modification in the SG case. To consider more general ip({X}) we need the commutation relations between BN(X) and AN{X), DN(X). We can read them from the relation (179)

which until now was used only to show the commutativity of tr TN{X). In suitable form the relations look as follows

335 602

L. Ftulileei

Using these formulae one can show that

where

and

Indeed the term proportional to i// can be obtained by using only the first terms in the rhs of the commutation relation. To get yl x (x[{A}) one must use the second term in the rhs on the first step of commuting AN(k), DN{X) with the product B(kx)...B(Xm) and after that use only the first term. The expression for /1{(A|{/}) follows from the symmetry in Ai,... Am. Now observe that so that the A-dependence of /t,(A|{A}) factorizes. We can get rid of the unwanted terms by requiring that the A, vanish. This gives m equations for m unknowns

which are nothing but the transcendental equations of the Bethe-ansatz. We shall list the local eigenvalues for concrete models: (i) General rational spin model

The NS case is obtained for S = — 1/gA. (ii) SG model

336

Inteyrable models in (I + 11 -dimensional quantum field theory

603

37. Comments on Bethe Ansatz 37.1. Continuous limit in NS model General equations for spin models have the form

When S = - \/gA after rescaling gX-*X we get

With N = ILIA, the lhs is exp2i/.L in the limit A -* 0 and we obtain the ordinary Bethe-Ansatz equations for NS model. 37.2. Nature of the spectrum There exists a sound reasoning showing that in any solution {A, ...Xm) of the Bethe-Ansatz equations all /., are different. Thus the spectrum of commuting operators is of fermionic type. 37.3. Additive observables The form of the eigenvalue A is far from multiplicative. However the local eigenvalue S{/.) has a zero; in the spin case it is X0 = \S as mentioned above. In the vicinity of A0, the second term in A is negligible when N -> oo, so that In tr TN(k) has an eigenvalue Anna(A)-£lnc(A-A),

X ~ AQ,

(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~ = £ a S a ~ 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 the full Hilbert space, namely (2S + 1)N. It is exactly at this point that the real work begins. Given a particular model, one investigates and classifies the solutions of the Bethe-Ansatz equations and describes possible eigenvectors in terms of particles or

337

604

L. Faddeev

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. 38. Generalization and questions unanswered as yet A lot about the algebraic Bethe-Ansatz can be generalized to other Liealgebras. In particular the i?-matrix can have the block abc form, where a, b and c are themselves matrices. The eigenvalue A is a matrix itself and to diagonalize it we have to solve a new magnetic model with l/c(A — A,) playing the role of the matrix L,. Thus a hierarchy of BetheAnsatze appears, many concrete examples of which are known. However there exists no systematic classification of all the possible schemes. The existence of the local vacuum a>i is a very restrictive requirement. In particular it does not exist for the spin \X YZ 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 /1(A|{A}). 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 /1(A|{A}) has not yet been done. 39. Infinite volume limit The limit N -» oo (or L -* oo) simplifies the investigation of the BetheAnsatz equations. More on this is contained in the lectures of J. Lowenstein. The outcome is as follows: the solutions {A} in a typical case accumulate in the real axis with the density p{X). The ground state (Dirac sea with all vacancies filled in) is characterized by the density pvac(A); the excitations are given by the density

where A, are holes in the Dirac sea. In these terms the ground state can be represented in the form

and excitations are written as follows

338

InWi/rahlc models in 11+ I)-dimensional quantum field theory

605

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 operators B(At) without the recourse to B(/.); especially important is establishing the connection between B(A) and the original field operators. This connection is indispensable for the construction of Green's functions, the problem is discussed in more detail 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 r\ coincide can be used to construct the planar model in classical statistical mechanics. Introducing the matrix indices L%(y,y') for auxiliary (y) and quantum (a) space we see that the matrix TN{X) has still only two auxiliary indices but 2/V quantum ones:

and tr TN{k) is given by additional summation over y = y'. This is nothing but the transfer matrix in planar models with the Boltzmann weight given by Ha(y, y'). Thus the properties of tr TN{A) investigated above can be used for solving these models. The role of FCR in this context was stressed by Baxter. 41. Connection with the factorizable S-matrix Another interpretation of the four-index object Z?a(y, y') is in terms of two-body S-matrix, X being the relative rapidity of two particles. When V and rj coincide we can essentially identify matrices R and L so that the FCR can be written in the form

339

606

L. Factdee i

Here the following notations are used. Matrices Rik are defined in the space V ® 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 Smatrix. The FCR is a statement that the definition of the latter is correct, namely it does not depend on the order of the consecutive twobody scatterings. It was shown by Zamolodchikov to be the only condition which allows the definition of a general JV-body factorizable Smatrix. 42. Concluding remarks We have seen that all the exact results of low-dimensional mathematical physics, that is: (i) inverse scattering method, (ii) Bethe-Ansatz, (iii) planar models in statistical mechanics, (iv) factorizable S-matrices, come together through the FCR or FPR. We have also given a fairly general classification of FPR with promising generalizations to FCR. This unification seems to be of fundamental value and throws a new light on the whole subject of exactly soluble models. A natural question is about the generalization to higher dimensions. The formalism presented here has used 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 + l)-dimensional spacetime: ^(i) construction of the Green's functions, /(ii) solution of the non-linear a-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 cr-model consists in the fact that it is associated with the non-semi-simple Lie algebra E(3). Indeed beginning with the n-field

340

Inteurakle models in (I + I j-dimensional quantum field theory

607

and the Lagrangian

we introduce as the phase-space field variables n and / / = cQn x n,

/■ n = 0,

with the commutation relations [/"(x ),/"()■)] =

k"bclc(x)S(x-y),

[/"(x), n*(v)] =

ieabcnc(x)6(x-y),

[ » % Y ) V ( V ) ] = 0,

which for fixed x are the E(3) relations. So we must think how to generalize the formalism to the non-semisimple case or find a suitable reduction of the 0(4) model. These questions are under investigation now. Acknowledgements

I am grateful to the organizers of the School, Professors R. Stora and J.B. Zuber for their 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 indispensable in the course of the preparation of this version. 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 there exists a vast amount of literature on this subject. Surveys where many references to the original works can be found are the following: [1] L.D. Faddeev, Soviet Scientific Reviews, Section C, Vol. 1 (Mathematical Physics) ed., S.P. Novikov (1980) p. 107. [2] H.B. Thacker, Rev. Mod. Phys. 53 (1981) 253. [3] L.D. Faddeev and L.A. Takhtajan, Usp. Mat. Nauk. 34 (1979) 13; translation by London Math. Soc. [4] P.P. Kulish and E.K. Sklyanin, in: Proc. Tvarmine Symp. on Integrable Quantum Field Theories (1981), eds., J. Hietarinta and C. Montonen (Springer-Verlag, New York, 1982). [5] A.G. Izergin and V.E. Korepin, Physics of Elementary Particles and Nuclei (Proc. Dubna Institute of Nuclear Research) 13 (1982) 501.

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L. FatUk'cv

[6] L.D. Faddeev, Proc. 1981, Freiburg Summer Institute, (Plenum Press, New York), lo be published. We shall add several references lo recent papers. [7] V.E. Korepin, Commun. Math. Phys. 76 (1980) 165. A detailed investigation of the mass spectrum in the S.G. model. [8] L.D. Faddeev and V.A. Takhtajan, Proc. Scientific Seminars of the Steklov Institute, 109 (1981) 134. Correct derivation of the excitation spectrum of XXX spin j models. Also contains a survey of the quantum inverse method in the application to this model. Translation by the Consultants Bureau, as Sov. J. Math. [9] P.P. Kulish, N. Yu Reshetikhin and 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 /-matrices was done recently by Belavin and Drinfeld. [10] A.A. Belavin and V.G. Drinfeld, On the solutions of the classical Yang-Baxter equation for simple Lee algebras, Funct. Anal. Appl. 16 (1982). The results on the local quantum Hamiltonians and general classification of integrable models based-on the use of current algebras will be published by me in joint papers with Takhtajan and Reshetikhin, respectively. [11] L.D. Faddeev and N.Yu. Reshetikhin, The Hamiltonian structures for integrable field models, Teor. Mat. Phys. 56 (1983) 323. The homogeneous quadratic, "communication" relations which appeared in the lattice spjn models, were introduced by Sklyanin. [12] E.K. Sklyanin, On some algebraic structures connected with the Yang-Baxter equation, Funct. Anal. Appl. 16 (1982) 27. [13] E.K. Sklyanin, On some algebraic structures connected with the Yang-Baxter equation. II. The representation of quantum algebra, Funct. Anal. Appl. 17 (1983) 47. See also papers on the solution of the nonlinear cr-model. [14] L.D. Faddeev and L.A. Takhtajan, Integrability of the quantum nonlinear
342 F R O M I N T E G R A B L E MODELS TO C O N F O R M A L FIELD THEORY VIA Q U A N T U M G R O U P S

L.D. FADDEEV

St. Petersburg Branch of Steklov Mathematical Institute St. Petersburg, Russia and Research Institute for Theoretical Physics University of Helsinki, Finland

Abstract In these lectures, which are a variant of ones given during the last year (see references), I present a historical development in the 1 + 1 dimensional integrable models, leading to the notion of quantum groups. I also give a modern exposition of this notion in the /2-matrix language and explain a new application to Conformal Field Theory.

The mathematical theory of solitons is about 25 years old. It started with the in­ vention of the so-called inverse scattering method. The inverse scattering method was based on the introduction of the Lax equation, a very important additional idea was the Hamiltonian interpretation of these concepts. The Hamiltonian interpretation of the Korteweg-de Vries equation was first given by Gardner, Zakharov and Faddeev, in 1971. In our approach we were mainly led by the idea of a future quantization of this subject, which was completely classical at that time. To quantize something one has to know first of all the Hamiltonian structure of the corresponding classical problem. Going step by step deeper into quantum mechanics an algebraic structure evolved, which was on one side very simple and on the other side quite universal. Our framework will be a 1 + 1 dimensional quantum field theory. We will consider mainly systems with discrete space variable and continuous time variable. Let x = n ■ A be the space, t the time variable, with n = 1 . . . N and N + 1 = 1, (one dimensional chain with periodic boundary conditions). With each site 72 we connect a Hilbert space 7{n . The total space of physical states is thereby given by l L. A. Ibort and M. A. Rodriguez (eds.), Integrable Systems, Quantum Groups, and Quantum Field Therapy 1-24. © 1993 Kluwer Academic Publishers. Printed in the Netherlands.

343

2

We are given some dynamical algebra, generated by the operators

where a is some additional index, e.g. with respect to a Lie algebra basis, and all dynamical variables are required to be functions of the X°. Next impose the condition of ultralocality:

The dynamical equations are given as the usual Heisenberg equations:

(a dot denoting derivation with respect to the time variable t). The idea of soliton theory is to associate to the Hamiltonian H a large series of commuting integrals of motion. To achieve this we introduce a new object, the so-called Lax operator

which is an m x m matrix in the auxiliary space V = C m and with matrix entries which are operators on the Hilbert space 7in. It depends also on the additional parameter A, called the spectral parameter. We look for a nice instrument to display the various commutation relations be­ tween the matrix elements of the Lax operator in a compact way. By ultralocality these commutation relations are to be given for a fixed value of the index n. Skipping threfore the n-dependence we may express everything by terms of the form

and the products with reversed order of factors. Therefore it seems convenient to define the following operators on V ® V constructed in terms of L:

The commutation relations among the matrix elements may now be written as:

with a matrix R(X) : V ® V —> V ® V. We will call this relation the fundamental commutation relations (FCR). Considering Ln as a local transport matrix, we put now:

344

3

This relation is called the auxiliary problem. It should be understood as a system of matrix equations with operator coordinates, i.e. as a system of equations in a noncommutative space. If Ln for small A is of the form:

the auxiliary problem becomes in the continuous limit

Furthermore we have As monodromy it is natural to define the operator

The matrix entries of M are global operator on 7i. The special feature of the FCR is, that a pure local relation Eq. (1) gives a global relation (global on H), for the monodromy M:

The proof is as follows: (In the following, we suppress the arguments A and fi). By the FCR, Eq. (1) and ultralocality we have for the products LkLk-\~-

and the rest follows by induction. The matrix entries of the monodromy are global operators. In order to get scalar commuting operator on the full Hilbert space 7i we take the trace of the monodromy:

Assuming R(X) to be invertible and using Eq. (2) we get

The F(K) are therefore generators of an algebra of commuting operators. Now we will illustrate the above framework by examples.

345

4

EXAMPLE 1. Let 7{n = C be the spin quantum Hilbert space and the spin operator a where cr ,a — 1,2, 3, are the Pauli matrices. The 5° satisfy the commutation relations of 5/(2, C): The Lax operator in our example is now:

So in this example V = 7Yn, which is not true in general. With the i?-matrix given by:

where 6 = jrr and c = y ^ , the Lax operator Ln satisfies the FCR as may be verified by direct computation. As a Hamiltonian we choose the following element of the abelian algebra generated by the F(X):

which turns out to be the Hamiltonian of the isotropic Heisenberg magnet. In this particular case the FCR holds for any spin representation, so that the example might be easily extended to higher spin j , with the corresponding Hilbert space Hn —
where 5 is any complex number. In the case where 25 is a positive integer one gets subrepresentations (just the Verma modules of 5/(2, C)), which are finite dimensional.

346

5 If S tends to infinity ("the quasiclassical limit"), we have the following behaviour of the spin operators in the vicinity of the lowest weight state (playing here the role of the ground state).

With this, one obtains the asymptotic behavior of the Lax operator Eq. (3).

if we put S =

and take into account that

The continuous quasiclassical limit of the Lax operator of the isotropic Heisenberg magnet is therefore nothing else than the well known Lax operator for the nonlinear Schrodinger equation. E X A M P L E 3. Another model might be obtained by substituting A by sinhA, so we can define the Lax operator

for the spin operator 5 ° . One realizes, that after the substitution one gets the same form of the Lax operator as in example 1 in the limit 7 = 0. The /^-matrix has the same form as in example 1, Eq. (4) but the coefficients 6, c are now substituted by:

T h e Hamiltonian may now be calculated to be

which is the Hamiltonian of the anisotropic Heisenberg model (XXZ-model). Lax operator may alternatively be written as

The

347 6

Now the fundamental commutation relations do not hold in the case of higher spin. But a slight modification leads again to fundamental commutation relations. Instead of the 57(2, C) commutation relations, we impose the following deformed 5/(2, C) relation:

If these relations are satisfied, then the Ln(X) of the form (5) satisfy the funda­ mental commutation relations but the operators S„ , S„ no longer form a Lie algebra. Instead they "generate" a new algebraic structure which gives in the limit 7 —> 0 a Lie algebra. A realization of the relations (6) by usual field operators:

is given by:

Here now one obtains the quantum Sine-Gordon model as a special case of the Heisenberg XXZ-chain. Indeed, after multiplication of Eq. (5) by 2a sin 7 which do not change the FCR Eq. (1), we get the L-operator:

If one puts 2a =

and

this leads to continuous auxiliary problem for the Sine-Gordon equation. These examples cover the case of "rank 1", namely group 5/(2, C). One can use now groups of higher rank, there are generalizations of Lax operators for them and corresponding integrable models, the parameters in this list are: group, repre­ sentation, anisotropy parameter (i.e. 7). It will be interesting to see if that is a classification. EXAMPLE 4. Let us return to the S-G L-operator (7) and consider the "Liouville limit":

348

7

We get a new L-operator

with a trivial dependence on A. The A independent L-operator:

where

satisfy the modified FCR

where (after changing the normalization)

The A-dependence in R(X) is also separated. We have

where Rq is a simple triangular matrix

349

8

P-permutation

matrix in V ® V

and we denoted

It is clear, that L„u

satisfy the A-independent FCR

Thus the development of Lax operators presents us with the generalization of Liealgebra relations (5), the relation Eq. (8) and their "stability" under the multipli­ cation of matrices in the auxiliary space with independent matrix elements. All this leads to the general definition of Quantum Groups and Quantum Lie Algebras, to which we turn now. Let us begin with the first one, and consider the relation.

where the matrix T in the auxiliary space V has as entries the generators T{} of the associative algebra A. Define a comultiplication in A

on the generators Tt] by:

The FCR Eq. (9) guaranties that this indeed defines an algebra homomorphism as already was shown on the example of Lax operators Ln+l and Ln. We have to require certain conditions on R for the algebra A to be rich enough. Consider the operator TXT2T3 acting as a matrix on V <8> V <8> V, where T' acts as T on the i-th component of V ® V ® V and as the identity on the others. There are two possibilities to interchange these three operators according to the paths:

350 9

Define and: The realization of (9) gives:

In this way one obtains higher and higher relations on the matrix entries of T. But it turns out, that it is enough to require R123 = R32x to get rid of all higher order relations simultaneously. Thus we impose the main condition

on the structure matrix R to define a deformed (or "quantized") matrix algebra A. Here the term "deformed" or "quantized" is used in the spirit of noncommutative geometry. T h e relation (10) is called the Yang-Baxter equation. One may look at the Yang-Baxter equation as a kind of Jacobi equation for the "structure constants" of the quantized matrix algebra. It appeared previously in statistical mechanics as well as in the theory of factorizable S-matrices. T h e Liouville example has given us a solution Rq of the Yang-Baxter equation for the two-dimensional space V. It will be convenient in the following to use Rq with a different normalization

and

T h e defining relation (9) is now: TXT2R±

=

R±T2rFx

(where both signs can be used). Let us look at them more closely and introduce the matrix elements of T:

351

10

The six nontrivial conditions resulting out of the FCR are:

The algebra generated by the a, b, c, d together with these relations will be called Glq(2, C). More exactly we quantize the algebra of functions over the Lie group. One may look at q as a deformation parameter, as one gets for q = 1 a commutative algebra, corresponding to usual matrices T. As we have seen these relations were found by looking at the quantized Liouville model on the lattice. A natural algebraic question which arises, is whether there exist central elements of the algebra A. We find

which we may fix to be 1 to get a subalgebra, which we call 57,(2, C). Another formal central element is -, which may be singular as c is not required to be invertible. T has an inverse with respect to matrix multiplication:

T —► S = T Ms therefore an antiautomorphism from S/ g (2,C) to S7i(2,C) i.e. S satisfies the following fundamental commutation relations:

We expect now, that there will exist such quantum deformations for all classical groups. Corresponding /^-matrices were found by Jimbo and Bazhanov. Let us turn now to another algebra B associated with the relations (5) that ap­ peared in the XXZ model. Once more we will consider first a two-dimensional exam­ ple, but this time we take two matrices L± as generator matrices. In order to get not too many generators we restrict ourselves to triangular matrices, which corresponds to the choice of Borel subalgebras in a matrix Lie algebra. The matrices

352 11

contain three generators X±,H

and satisfy the following F C R ' s :

only three of which are independent. By direct computation of the FCR we get

as denning relations on a new algebra B, which is generated by X± and / / . These relations coincide with (5). This example also naturally generalizes to higher rank classical Lie algebras. T h e components of the corresponding matrices L± constitute the analog of the CartanWeyl basis. In a different abstraction from relation (5) Drinfeld and Jimbo use generators, corresponding to the Chevalley basis. So they have additional Serre relations, which are absent in our formulation. T h e comultiplication is given by

in accordance with Eqs. (11)(12). The determinant of L± is equal to 1 by construc­ tion, so that Z/J1 are defined analogously to T~l. For example

An alternative way of description the quantum Lie algebra uses only one matrix

T h e commutation relations for L follow from (11)-(14) and look as follows

353

12

The comultiplication can not be written explicitely in terms of L only. However this formulation is useful in many respects. Let us show how the undeformed Lie-algebraic relations follow from (17) in the limit q —► 1 on the rank .one example. Posing as usual q = en we have for 7 —► 0

where the matrix r ± is given by:

and

Now we look for the expansion of L in the form:

in order 7 2 . Here

is a realization of the Casimir operator:

in terms of Pauli matrices aa. One recognizes in Eq. (19) the ordinary Lie-algebraic relations for the generators /Q, combined into the matrix:

The formula (18) shows that after deformation the matrix L is more "grouptype" than a "Lie-algebra type" matrix /. So the difference between the Lie group

354

13

and Lie-algebra is much less in the deformed case than in the classical undeformed situation. Now we shall introduce the important notion of g-trace: tr,>4 =

trVA

where P is a diagonal matrix, which is attributed to any classical group. 5/(2, C) case, T> can be chosen as:

In the

One can show now, that

where trj means trace over the first auxiliary space and I2 is a unit matrix in the second auxiliary space. Using this relation and Eq. (17) we see t h a t trqL commutes with all the elements of the matrix L and thus is a central element. It plays the role of the lowest Casimir operator. In 5/(2, C) example we have:

Another way to find elements in the center of the q u a n t u m Lie-algebra consists in diagonalizing the matrix L. One can show ( see i.e., my Cargese lectures) that /, can be represented in the form

with the matrix u satisfying the relations:

where

355

14

T h e operator matrix u has an interpretation as a combination of deformed ClebschG o r d a n coefficients and R(0) is the corresponding 6 — j symbol. T h e central elements 0 and trqL are related as follows

which formally coincides with the usual trace of the diagonal form of L. In the case when the deformation parameter 7 is real, so that \q\ — 1, we have:

Hence we can introduce the real unitary form of 5/(2, C) by the requirement:

or:

T h e Casimir tvqL is then selfadjoint. Its eigenvalues are real for 0 real 01^ 0 imaginary. T h e first case can be called elliptic, and the second, hyperbolic. For the elliptic case 0 acquires quantized values:

in t h e interval 0 < 0 < - . In particular, for:

where k is an integer, j varies between 0 and T h e corresponding represen­ tations Vj have dimensions 2j + 1. In the hyperbolic case the representations are infinite dimensional; however for 7 having the form (22) they are cyclic and have dimension k. T h a t finishes the description of the quantum groups and the q u a n t u m Lie algebras. We proceed to describe the action of the latter on the former. In the classical nondeformed case the generators / can be realized as vector fields acting on functions on the group variables TtJ. In particular the action V/ of / on the coordinate functions T can be described as

where C is a matrix Casimir (20 ). Alternatively, we can say, t h a t vector fields / and coordinates T have the commutation relation:

356 15

characteristic for the regular representation of the group. The corresponding relation in the deformed case can be written as:

or in terms of L,

It is clear that Eq. (24) reduces to Eq. (23) in the classical limit 7 —> 0 if we take into account (13), (18) and (20 ). The relations:

give us an algebra which can be called (TG)q. It represents the quantization and deformations of the classical contangent space, which constitutes a phase space of a top. Putting: when h is the Planck constant of quantum mechanics we can draw a diagram: classical top

regular representation of G

deformed classical top

(T*G)q

so that the regular representation of the group, classical deformed and nondeforrned top, could be obtained in the limit 7 —* 0 or k —* 0, or both. One can extend the involution (21) onto (TmG)q. It would be nice to be able to restrict this noncommutative manifold to the case of elliptic L. One could hope, that the corresponding Hilbert space would be finite dimensional and have the structure:

with the finite sum over j ' s corresponding to real 0 . The condition (22) on 7 must be essential for that. The work in this direction is in progress. Now we turn to the generalization of our object on the loops over the chain with sites n, n = 1,. . . TV, n-\-N = n. The ultralocal FCR for Ln independently prescribed to any site n reads

357 16

can be written in one line as

with:

We shall show that a simple twist changing these relations into:

accounts for a central extension of the loop algebra. Indeed in the continuous classical limit A -> 0, h -¥ 0 if we put:

and take into account that

then we get in order

which give us the classical realization of the Kac-Moody algebra with level

if we identify with the Poisson bracket. Thus the relation (26) could be interpreted as giving a lattice deformation of a Kac-Moody algebra. This opens to us the way in the direction of Conformal Field Theory. Unfortunately in this purely algebraic setting there is no simple way to describe a continuous limit A —» 0 without recourse to the classical limit h —> 0. However some indirect considerations, based on the operator expansion, lead to the identification of the level / and deformation parameter 7 as follows:

358

17

where c is the Coxeter number of the group G, c = 2 for 57(2, C). T h e nontrivial relations in (26) are:

It follows from these relations that Ln commutes with the product of its three neighbors.

This allows to compute the commutation relations between products of Ln. particular for the monodromy:

In

we obtain the relation

which is nothing but the defining relation of the quantum Lie algebra. So we see that the lattice Kac-Moody algebra (Latt K-M) contains a quantized Lie algebra.

One can calculate the relations between M and the generators Ln. nontrivial ones are for L\ and L/v- We have:

T h e only

This means that the center of B, namely t r q M , (or more generally, invariants of M) is also a center of Latt K-M. Thus any representation V of the q u a n t u m Lie algebra B labels a representation Tij of Latt K-M with the same values for the center:

We are ready now to turn to Conformal Field Theory. We shall consider the typical model of C F T , namely the WZNW model. First, I will describe the formulation of the W Z N W model as a classical field theory. T h e space time M is a cylinder Sl x IR1 with coordinates a:,/., 0 < x < 27T,

359

18

—oo < t < oo with Minkowskian metric. The field g(x,t) 2 x 2 matrix

is a unitary unimodular

The action functional looks as follow

where the pullback of the integrand of the closed, but not exact, 3-form tr(dgg~1)3, is used to define the WZ term in the action. This nonunivalent action ( in the terminology of Novikov) can be used in quantum theory only if the coupling constant 7 is quantized, namely

must be an integer. The phase space of the model is spanned by the Cauchy data g(x), Jo{x) for fixed time, where we use the notation JM for right invariant currents

To specify their Poisson brackets it is convenient to use the first order formalism when g and J 0 are considered as independent variables. The action, equivalent to (33) can be taken in the form.

where WZ is a WZ-term, i.e. the second term in the RHS of (33). It is considered as a functional of g alone as it contains dog only linearly. Furthermore,

The variation of the canonical 1-form in A leads to the symplectic form

In the derivation, one has to use the property of WZ

which follows from

360 19

The symplectic form Eq. (34) leads to the Poisson brackets:

and

Furthermore we have

or

If we define / = it follows

t

o

be the current with respect to light cone variables.

and in coordinates with respect to the Pauli matrices, / = laaa

These are the commutation relations for a Kac-Moody algebra. A part of the phase space constitutes therefore a Kac-Moody algebra. Let us define a right current by

for which we have

and

361

20

So we have a chiral decomposition of the phase space in left and right movers, / and r respectively. However, it will be shown later on, that these variables are not enough to characterize the phase space completely. We are to introduce another scalar cyclic coordinate. With this we have the coordinate transformation:

and the Hamiltonian H decouples into two currents:

T h e equations of motion are:

Substituting the expressions for Jo and J\ , into / and r and subtracting after multiplication by g from the right and left, respectively, we get for the old field g(x, t)

which is solved by:

where u and v satisfy dxu — hi and dxv = —vr with initial conditions u(0) = 1, v(0) = 1, where we have chosen a fixed point XQ € Sl. We define the monodromy for u and v by:

Mft

From the periodicity of g(r) it follows that A' = M^KMR. Therefore Mi are conjugated by A" and have the same spectrum. We set

where D is a diagonal matrix:

with 0 < 0 < ir.

and

362 21

Finally we have:

and

This implies that D and ZLlKZRl

commute and the only freedom we have for

K is:

where

and a is the looked for cyclic variable. The above equations bear a strong resemblance to the equations for the quantized top, so that we may suspect, that the quantum top is hidden inside this model of conformal field theory. Now we turn to the quantization. We introduce the lattice and use the algebra (26) as quantization of currents l(x) and r(x). Thus we come to lattice analogues of the chiral fields u(x) and v(x)

Using the rules (27) and property (30) we can easily get the commutation relations:

and Similarly we obtain for v.

and while un and vm commute for all m,n. Now let us turn to the local field

363

22

where K plays the role of the initial condition.

In the classical case we have constructed K classically through the matrices Z>L,ZR, diagonalizing ML and MR and an additional variable a. This process could be repeated here. However we shall propose instead some characterization of K in terms of simple commutation relations and check their consistency. We require that the following relations take place:

and K commutes with Ln, n = 2 , . . . , N — 1. This is the quantum counterpart of the Poisson commutation relation between JQ{X) and g(y) (at y = 0). Moreover, K itself constitues a quantum group

where R stands for R+ or /?_. It follows that K has similar relations with un,

and monodromy Mi = u^:

The corresponding relations with vn look as'follows:

(one reads the relations for u from right to left). Now by simple algebra one can check that the local fields gn satisfy the relations:

among itself and

364

23

between gn and Lm. These relations turn into Eqs. (35),(36) in the classical continuous limit. T h e consistency with the periodicity condition is now checked in the following manner, M^KMR has exactly the same relations with K, Ln and ML and itself (and their right counterparts) as K. It follows that

where a commutes with all dynamical variables, so it is a constant and must be equal to 1 by comparison of determinants. Now we observe that the relations (37), (38) together with the relation of type (32) for Mi define us an algebra (T*G)q with the identification:

Thus the local field at a fixed point and the monodromy of a chiral component around the circle from this point give us a representation of (T'G)q. These d a t a com­ prise zero modes of the W Z N W model and essentially define its full structure. This establishes the intimate connection of Q u a n t u m Groups and Kac-Moody algebras. In particular the Hilbert space of the top (25) generates the Hilbert space of the W Z N W model:

in accordance with (26). T h e oscillators, generating 7i 0 do not influence the nontrivial part of the dynamics. We believe, that the unravelled structure will be useful in the calculation of the correlation functions of the local field. However this truly dynamical problem is out of the scope of these lectures.

365

24

References: I add now some comments on the literature. Lattice Lax operators were introduced in the course of development of the method of quantum inverse problem, for a survey see: [I] L.D. Faddeev: Sov. Sci. Rev. Sec. C: Math. Phys.1,107 (1980), [2] A.G. Izergin, V.E. Korepin: Fisika Echaya, (JINR, Dubna) 13, 501 (1982). The deformed Lie algebra Slq(2) appeared first in [3] P.RKulish, N.Yu. Reshetikhin: Zap. nauch. seminarov LOMI 101, 101 (1981) (in Russian); J.Sov.Math. 23, 2435 (1983), and the Lie groups SLq(2) in [4] L.D.Faddeev, L.A. Takhtajan: Lectures Notes in Physics 246, 166 (1986), in connection with XXZ and Liouville models. The general notion of quantum groups was developed by Drinfeld and Jimbo: [5] V.G. Drinfeld: Dokl. AN SSSR 273, 531 (1883), [6] V.G. Drinfeld: Dokl. AN SSSR 283, 1060 (1985), [7] V.G. Drinfeld: in Proc. ICM Berkeley, 786 (1986), [8] M.Jimbo: Lett. Math. Phys. 11, 247 (1986). The description of the lecture follows [9] L.D.Faddeev, N. Yu. Reshetikhin, L.A. Takhtajan: Algebra and Analysis 1, 178 (1989). The notion of (T'G)q is introduced in a paper with A. Yu. Alekseev: [10] A.Yu. Alekseev, L. D. Faddeev: Commun. Math. Phys. 141, 413-422 (1991). Its role in WZNW was indicated in [II] A. Alekeseev, L. Faddeev, M. Semenov-Tjan-Shansky, A. Volkov: preprint CERNTH-5981/91 (1991). [12] A.Alekeseev, L.Faddeev, M. Semenov-Tjan-Shansky: (1991) and elucidated by me in Cargese and Berlin lectures in 1991. [13] L.Faddeev: preprint HU-ITF 92-5 and [14] L.Faddeev, preprint SFB 922 Nl Berlin.

LOMI preprint E-5-91

367

Comments on Paper 11 This is an English translation of my lectures at the Alushta summer school in 1976. I was in the middle of a work on quasiclassical quantization of solitons and naturally was desperately searching for the possible three-dimensional generalization of results which were known in the one-dimensional case. Several of my comments stayed in the literature (i.e. the Coleman theorem); some were independently introduced by others (i.e. the distinction of large and small gauge groups and the role of the ChernSimons functional). The idea of the soliton with Hopf invariant as the topological charge is discussed in more detail in Paper 15. Until now there has been no quantitative description of this object, which is rather interesting: the solution here is concentrated on the contour, rather than on the point, as it is in the case of the Skyrmion or the 't Hooft-Polyakov monopole. Especially interesting is the possibility of making the knots out of this contour, restoring the old dream of Lord Kelvin. The Lagrangian, including the gauge field in the way indicated at the very end of the paper and admitting the localized solutions, was introduced by me in Lett. Math. Phys. 1, 289 (1976).

369

T H E S E A R C H FOR M U L T I D I M E N S I O N A L SOLITONS L. D. Faddeev Leningrad Division, V. A. Steklov Mathematics

Institute

From the Proceedings of the IV International Symposium on Nonlocal Field Theories (Nonlocal, Nonlinear, and Nonrenormalizable Field Theories), 20-28 April 1976, Alushta, USSR (Preprint D2-9788, J I N R , D u b n a ) . Translated from the Russian.

In the past two years, major efforts of many physicists have been directed toward the development of a new theoretical mechanism for the emergence of the mass spectrum of elementary particles. It has been shown that localized solutions of classical field equations correspond to particles in quantum field theory. These particles are coherent excitations of a fundamental field, which contain an infinite number of particles corresponding to this field when it is quantized according to perturbation theory. The new particles, called solitons,* together with their bound states provide a rich mass spectrum which cannot be obtained by means of ordinary perturbation theory. The connection between localized solutions and elementary particles was already discussed in the literature many years ago. 2 - 4 The progress during the past two years has led to the following achievements: 1. A consistent scheme of quantization has been developed on the basis of a modified perturbation theory, which is essentially the same as the semiclassical method applied to quantum field theory. 2. Approximate expressions have been found for the masses of the solitons and their bound states, and for their scattering amplitudes. It has been found that in a theory of weakly interacting fundamental fields the solitons have large masses and interact strongly. A voluminous and rapidly growing literature is devoted to these questions. We cite here references to only a few original articles 5 - 7 and reviews. 8 - 1 0 Unfortunately, the successes mentioned above have been achieved, in the main, only for examples of field-theory models in two-dimensional space-time. To extend them to the realistic four-dimensional case, it is necessary, above all, to have a fairly rich set of examples of localized solutions of the classical equations of field theory. In the four-dimensional case, there exist a number of specific obstacles for the existence of such solutions. For this reason, the soliton mechanism for describing the mass spectrum of elementary particles for four-dimensional field-theory models is still in an embryonic state. In this lecture, I shall describe a number of methods which have already been found to be useful, or which may prove useful in the future, in seeking multidimensional solitons. These methods are scattered among many studies in mathematics and physics, often in connection with different problems. My contribution is perhaps only to systematize them "The term "soliton" arose in plasma theory, where it refers to a localized solution of the equations of motion, and is derived from the term "solitary wave"; this term was introduced in Ref. 1 because of the "obvious analogy with elementary particles."

370 and to concentrate on the subject of solitons. The content of this lecture has been influenced by conversations with A. Polyakov and S. Coleman.

1. Scale transformation and how to contend with it Here we shall give a simple explanation of why the existence of stationary localized solutions of nonlinear equations tends to be the rule for one-dimensional space and the exception for three-dimensional space. We shall consider boson fields and assume that the equations of motion are of the second order. Let us consider a system of scalar and vector fields possessing internal degrees of freedom. The complete set of these fields will be denoted by pa(x), so that the index a refers to both the spin and the internal degrees of freedom. Stationary solutions provide a minimum of the Hamiltonian, which we shall write, as usual, as a quadratic form in the first derivatives, plus a local function of the fields:

The distinction between spaces with different dimensionalities manifests itself in different behaviors of H\(ip) and H2{{f) under a scale transformation

To be specific, where D is the dimensionality of the space. For D = 1 the first term decreases, while the second increases as 1/A becomes larger. Thus, if (p provides a minimum, the stationarity condition gives the usual virial relation For D — 3 both terms increase monotonically as 1/A becomes larger, and this shows that the functional H{
371 and for D — 3 it decreases as 1/A becomes larger. We shall see later that it is possible to devise a Lagrangian that leads to a Hamiltonian of this type. 2. One can use slowly decreasing fields for which Hi(tp) and / ^ ( y ) a r e n ° t finite indi­ vidually. An example of this situation is provided by a Lagrangian that includes a matter field ip and a Yang-Mills field A^ which have a nontrivial asymptotic behavior for r —* oo. If d^if) and AM decrease slowly, so that J \ijj\2dx is divergent, but

decreases sufficiently rapidly, so that is convergent, the scale transformation (1) is inadmissible. The last integral will remain finite under the more correct scale transformation

Under this transformation, the Hamiltonians of the matter field and of the Yang-Mills field change according to the laws

so that the obstacle for the existence of solitons disappears. It is this possibility that is used in the construction of the 't Hooft-Polyakov monopole. Another example in which one requires a scale transformation of the form

for the Yang-Mills field is provided by a boundary condition of the type

where the integral of one of the components of the Yang-Mills field is taken over a contour that extends to infinity. The possible appearance of such a boundary condition will be discussed in Sec. 3. The role of scale transformations in considering the existence of solitons was apparently pointed out for the first time by Derrick in Ref. 11.

2. Substitutions and how to find them From the technical point of view, the basic method for finding solitons is to discover a substitution for the desired solution in the equation of motion. As a rule, such a substitution makes use of separation of variables into radial and angular parts. After the substitution, the equations of motion become a system of ordinary nonlinear differential equations in the radial variables. Because of the nonlinearity of the equations of motion, such a separation of variables is based on artificial procedures. The following theorem helps us to provide a scientific basis for these procedures. We shall formulate the theorem in terms of finitedimensional notation, and we shall not indicate all the necessary reservations. Suppose that we are given a manifold X, a group G acting on X , and an invariant function

372 We adopt the notation Xo for the set of all fixed points of the group G in X , i.e., the set of points xo such that xog = xo for all g E G. Theorem. An extremum of / on XQ is an extremum of / on X. In applications to field theory, the role of X is played by the set of all fields at fixed t, and the role of / by the Hamiltonian; for Xo we must take the set of fields of the special form used in the substitution, and G must be the subgroup of the complete invariance group of the Hamiltonian with respect to which the fields in Xo are invariant. The theorem means that we need not verify that the solution of the equations of motion can be sought in the form of a field in Xo- Such verification usually requires cumbersome and laborious calculations. S. Coleman has drawn attention to the role of the theorem in verifying the admissibility of substitutions in nonlinear equations. We shall analyze several typical examples of the application of the theorem. At the same time, we shall become acquainted with the list of systems of fields that are used in the search for multidimensional solitons. 1. The nonlinear chiral field. Use is made in the literature of such fields g(x) with values in a compact group G (below, principal chiral fields) or in a homogeneous space G/H. A typical example in the second case is a field x(x) w ^ t n values on the unit sphere SN in a space of dimensionality N + 1, i.e., Chiral fields have arisen many times in various aspects of field theory (Weinberg's soft-pion model, Sugawara's current model, etc.), but apparently they first appeared in connection with multidimensional solitons in the studies of Skyrme 3 (see also Refs. 12 and 13). We shall analyze two substitutions for the field x(a) D = 2, N = 2. I like the notation n(x) for this field, following the obvious association with a direction in three-dimensional space. We shall assume that the Hamiltonian for a stationary solution is invariant under isotopic rotations n —* Rn, with R E 0(3). We define the group h(x) -> R3{a)n(r-\a)x). Here x — (x\, x2) - (/>,
where 6{p) is an arbitrary function such that 0(0) = n7r. A substitution of the form (4) has been used many times in the literature; in particular, the Hamiltonian

373 which is admissible in the two-dimensional case, becomes, on n fields of the form (4),

and admits a family of solitons of the form

where p0 is an arbitrary dimensional parameter. (b) D = 3, N = 3. We introduce the notation x = (x> Xo)- We shall assume that the Hamiltonian is invariant with respect to the group 0(4) = 0(3) x 0(3). Consider a subgroup G of the form

The set of invariant fields is given by a formula analogous to (4):

It follows from Derrick's theorem that the Hamiltonian

does not admit solutions of the soliton type. However, such solutions exist for the Skyrme Hamiltonian

The substitution (4) leads to H of the form

and the corresponding equation of motion has a solution that decreases at infinity. 2. The Yang-Mills field. Let us consider a stationary Yang-Mills field A^x) associated with the group 0(3) in the gauge AQ = 0. This field is specified by a 3 X 3 matrix, whose first index is spatial and second index is isotopic. A spherically symmetric substitution is defined by the action of the group G = 0 ( 3 ) : The general form of the invariant fields is given by the formula

374 The first two terms are even, while the third is odd, so that it can be used individually as an admissible substitution. Such a substitution was first used by Yang and Wu. 14 Derrick's theorem does not admit solitons of this form. 3. The 't Hooft-Polyakov monopole. 15 ' 16 Let us consider an 0(3) Yang-Mills field interacting with an isovector field ip(x). It is clear from the foregoing discussion that the substitution

is consistent with the equations of motion that follow from the Higgs Hamiltonian

The Hamiltonian H is finite if V and A satisfy the following boundary conditions for r —► oo:

A scale transformation is admissible only in the form (2), and Derrick's theorem is consistent with the existence of solitons. 4. The model with an n field. Let us consider the model of an 0(3) Yang-Mills field interacting with an n field. We take the stationary Hamiltonian in the form

The first term gives mass to vector fields orthogonal to the field n, and if such fields exist, they decrease rapidly in the limit r —> oo for any stationary solution. At the same time, the component (A,n) parallel to n remains massless and may decrease slowly. In particular, a boundary condition of the type (3) can be imposed on it, thereby eliminating the obstacle of Derrick's theorem. Unfortunately, no successful substitution for this model has yet been found. To conclude this section, we note that in all the successful examples the group G identified the spin and isospin of the fields under consideration. The possible appearance of halfinteger spin in a theory containing a field with half-integer isospin was discussed recently in Ref. 17.

3. Topological charges and how to determine t h e m In considering the existence of multidimensional solitons, the fact that there exist cur­ rents that are conserved independently of the equations of motion is acquiring ever-increasing importance. These currents are not associated with any invariance of the Lagrangian. In­ stead, we can say that they are associated with specific features of the structure of the fields that enter into the Lagrangian. In this section, we shall illustrate these currents for specific examples and explain why it is natural to call them topological currents. In the following section, we consider their importance for solitons.

375

1. The scalar isovector field \j) — (ij)1, il>2, V>3)Let us consider the expression

It is obvious that The charge density, the zeroth component Jo of the current / M , is the total divergence

so that the charge

is nonzero only if the field V does not vanish at infinity. In particular, for the field if) in the 't Hooft-Polyakov model we have 2. The Higgs model. Let us consider a Yang-Mills field A* associated with an arbitrary compact group, and a scalar field iba with values in the adjoint representation. The current

is conserved. Here fabc are totally antisymmetric structure constants. In fact,

The first term on the right vanishes by virtue of the Bianchi identity. Furthermore, in the second term is antisymmetrized,

and, as a result, it also vanishes. The charge density JQ is again a total spatial divergence

(to verify this, one must again make use of the Bianchi identity). The charge Q is nonzero only if F and ip do not decrease too rapidly at infinity. For example, Q ^ 0 if the group is 0(3) and

as is obtained for the 't Hooft-Polyakov monopole. We note that in this case the charges in Subsecs. 1 and 2 are equivalent.

376 3. The principal chiral field. Let g(x) be a field with values in a group G. We define the left current

which specifies a matrix in the adjoint representation of the group G. The current

is conserved. Here tr denotes the trace in the adjoint representation. To verify this, we note that it follows from (7) that

and the conservation of JrM follows from the Jacobi identity for the commutators. The charge Q = J Jodx takes (after appropriate normalization) integer values for fields g(x) that are regular at infinity, i.e., fields such that

where go is a constant matrix, and the asymptotic value is approached sufficiently rapidly. We shall not prove this in a general form, but shall consider only the case of the group SU(2). In this case, the field g(x) can be parametrized by a chiral field x — (x^Xo) with x2 — 1The current (8) is equivalent to the current

which is conserved, since the vectors d^xa for o = 0, . . . , 3 are linearly dependent. The charge density JQ cannot be expressed as the divergence of a nonsingular vector field. If the field x is regular, i.e., if

and the limiting value is reached sufficiently rapidly, then it gives a regular mapping of the space R3 onto the sphere S3 for which the neighborhood of infinity in R3 is mapped into a fixed point x° o n the sphere S3. It is easy to see that J0d3x in the variables x g ° e s into the element of volume on the sphere, so that the integer

shows how many times x covers the sphere S3 when x varies in R3. Here 2n2 is the volume of the unit sphere S3. 4. The direction field n(x). For the n field, a current analogous to the one considered in the preceding subsection can be introduced only in three-dimensional space-time. However, it is also possible to

377

associate an integer with the n field in the four-dimensional case, i.e., for three-dimensional space. Consider the vector [the analog of the current (9)] with zero divergence. There exists a vector A% such that

For a regular n field, the integral

is convergent and takes integer values. This characteristic of an n field in topology is called the Hopf invariant. It can also be described in terms of integrals of H{ over two-dimensional surfaces; i.e., the Hopf invariant, like the charges in the preceding subsections, has a local density. With this, we conclude our examples of topological charges. These examples show clearly how important the dimensionality of space is for the very definition of this concept. A natural language for the general definition is the terminology of homotopy groups (see the elementary exposition in Ref. 18). We note that there is an important distinction between the topological charges in Subsecs. 1 and 2, on the one hand, and those in Subsecs. 3 and 4, on the other. The former are nonzero only for fields with a nontrivial asymptotic behavior at infinity (monopoles, vortices, etc.). The latter require, for their very existence, that the fields differ from a fixed asymptotic value only in a localized region.

4. Topological charges and how to use t h e m The great importance of topological charges for solitons consists in the fact that, as a rule, they take nonzero values for solitons. Moreover, in favorable cases the topological charge gives a lower bound on the energy of stationary solutions in which it has a nonzero value. This property can be used for the actual proof that solitons exist, and also to estimate their mass. Estimates for nonlinear functionals defined on function spaces that admit topological charges have been known for a long time in the mathematical literature (see, for exam­ ple, Ref. 19). The possibility of applying such estimates to the problem of solitons was mentioned by Skyrme 3 and was later revived by a number of authors. 2 0 ' 2 1 In particular, in Ref. 20 it was shown that such estimates make it possible to simplify the equations of motion for the soliton solutions, reducing their order. We shall now illustrate all these points by means of examples. 1. The n field in two-dimensional space. The current (10) is quadratic in the n field and can therefore be estimated in terms of a sum of squares, i.e., the Hamiltonian density. An accurate estimate can be obtained on the basis of the following observation. 20 We introduce the vector isovector field

378 By virtue of the condition n2 — 1, we have

On the other hand, for the current of the n field. The trivial inequality

leads to the estimate which becomes exact for or, more concisely, The last equation is of the first order, and in an appropriate parametrization it reduces to the Cauchy-Riemann condition in the theory of analytic functions (see also Ref. 19). In particular, the solution mentioned in Sec. 2 satisfies it trivially. 2. The Higgs model. Let us consider the Hamiltonian for the static solutions of the Higgs model in the limit in which the self-action of the scalar field vanishes:

It is obvious that the charge associated with the current (6) gives a lower bound for this Hamiltonian in the form and the inequality becomes exact if

This equation replaces the equations of motion. The 't Hooft-Polyakov substitution reduces it to a system of first-order equations for / and g. The solution

was first obtained in Ref. 22 and gives the explicit form of the coefficient functions of the 't Hooft-Polyakov monopole in the limit for X —> 0. If A ^ 0, this reduction to first-order equations is lost, but the inexact estimate of the type (11) naturally remains valid. Unfortunately, this trick leading to a reduction of the order of the equations of mo­ tion does not always work. For example, the Hamiltonian for the stationary solutions of

379 the Skyrme model for the principal chiral field g(x) admits an estimate in terms of the topological charge of the current (8) in the form

which becomes exact if However, the last equation does not have localized solutions. At the same time, the secondorder equations which follow from the Skyrme Hamiltonian, have a localized solution, at least for the group SU(2). For this solution, the topological charge is unity, and the estimate of the type (11) is satisfied with a margin. It appears that the last equation differs from the first-order equations in the preced­ ing examples in that its left- and right-hand sides are of different orders in the unknown functions. We conclude this section by discussing the role of topological charges of the second type (in the terminology of the preceding section) in models with gauge fields. The natural question of gauge invariance of the topological charge immediately raises the related question of how to understand gauge invariance. We shall consider this in more detail, confining ourselves to the case of stationary solutions. In this case, the gauge group is formed from functions Q,(x) defined on the space R3 and having values in a group G. Formally, the elements of the gauge group coincide with the principal chiral fields. In particular, it is in this case of three-dimensional space that the complete group of matrices is not connected. Elements belonging to different components are distinguished by different values of the topological charge. The component of connectiv­ ity of the unit element consists of the elements with zero charge and is itself a group, which we shall call the little group. The question now is what to take as the gauge transformations: the little group or the complete group. We recall that in the Hamiltonian approach gauge transformations are generated by constraints which are the elements of the Lie algebra of the group of gauge transformations. The constraints condition corresponds to factorization with respect to the little group. Thus, if we assume that the complete group is the gauge group, we are extending the principle of relativity somewhat further than is required by the dynamics. It is interesting to note that the Yang-Mills field makes it possible to construct a quantity that is invariant with respect to the little group but changes under transformations with nontrivial topological charge. This quantity is defined by the expression

and has the following property. Let AQ denote the image of the action of the element fi on the field A:

380 Then where The quantity B[A] is well known to mathematicians (see, for example, Ref. 23) and has been used recently in the physics literature (see, for example, Ref. 24). Unfortunately, it is not an integral of the motion. If the complete group is the gauge group, then (13) shows that B[A] has physical meaning only up to an additive integer. In this case, we can say that the configuration space of the Yang-Mills fields has the topology of a cylinder. The topological charge of the principal chiral field and the Hopf invariant of the n field [in the case of the group 0(3)] are invariant only with respect to the little group. Furthermore, we have the relations

This shows that if the complete group is the gauge group, the topological charge of nonlinear fields has no physical meaning. Irrespective of the answer to the question of the gauge group, it is clear from the foregoing discussion that the topological charge of chiral fields cannot provide a lower bound on a gauge-invariant Hamiltonian. This additional obstacle for the existence of solitons is typical of gauge theories in realistic four-dimensional space-time. A possible way of avoiding this obstacle is to use two chiral fields. For example, in the case of the group SU(2) or 0(3) one can make combined use of the principal chiral field and the n field. A model of the electromagnetic and weak interactions of leptons which generalizes the model of Ref. 25 in this way will be discussed elsewhere. I have tried to explain here all the general arguments that have appeared in the literature in connection with the search for multidimensional solitons. The result of this search is as yet not very comforting. So far, the most successful soliton is the 't Hooft-Polyakov monopole, which is not truly localized. A more attractive soliton for a gauge-invariant theory of chiral fields has not yet been constructed. Nevertheless, I hope that this will soon be done. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

M. Kruskal and N. Zabusky, J. Math. Phys. 5, 231 (1964). R. Finkelstein, R. Le Levier, and M. Ruderman, Phys. Rev. 83, 326 (1951). T. H. R. Skyrme, Proc. R. Soc. London Ser. A 247, 260 (1958); 260, 127 (1961); 262, 237 (1961). D. Finkelstein and C. W. Misner, Ann. Phys. (N.Y.) 6, 230 (1959). L. D. Faddeev and L. A. Takhtadzhyan, Usp. Mat. Nauk 29, 249 (1974); Teor. Mat. Fiz. 21, 160 (1974). R. F. Dashen, B. Hasslacher, and A. Neveu, Phys. Rev. D10, 4114, 4130 (1974); 11, 3724 (1975). J. Goldstone and R. Jackiw, Phys. Rev. D l l , 1486 (1975). L. D. Faddeev, IAS Preprint (1975). R. Rajaraman, Phys. Rep. 21C, 227 (1975). S. Coleman, Lecture Notes, Erice Summer School, 1975. G. H. Derrick, J. Math. Phys. 5, 1252 (1964). U. Enz, Phys. Rev. 131, 1392 (1963).

381 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

D. Finkelstein, J. M. Jauch, S. Schiminovich, and D. Speiser, J. Math. Phys. 4, 788 (1963). C. N. Yang and T. T. Wu, in Unusual Properties of Matter (1971). G. 't Hooft, Nucl. Phys. B79, 276 (1974). A. M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 20, 430 (1974) [JETP Lett. 20, 194 (1974)]. R. Jackiw and C. Rebbi, MIT Preprint (1976); P. Hasefratz and D. A. Ross, Utrecht Preprint (1976). D. Finkelstein, J. Math. Phys. 7, 1216 (1966). J. Eells and J. H. Sampson, Am. J. Phys. 86, 109 (1964). A. A. Belavin and A. M. Polyakov, JETP Lett. 22, 245 (1975). J. Honerkamp, A. Patani, M. Schlinwein, and Q. Shan, Univ. Freiburg Preprint (1975). M. K. Prasad and C. M. Sommerfield, Phys. Rev. Lett. 35, 760 (1975). S. S. Chern and J. Simons, Proc. Natl. Acad. Sci. USA 68, 791 (1971). A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin, Phys. Lett. 59B, 85 (1975). L. D. Faddeev, Dokl. Akad. Nauk SSSR 210, 807 (1973) [Sov. Phys. Dokl. 18, 382 (1973)].

Addendum 1. A. M. Polyakov has drawn my attention to the fact that a reduction of the order in the equations for the monopole in the case A = 0 was obtained by E. B. Bogomol'nyi (preprint, Institute of Theoretical Physics, Chernogolovka, 1975). 2. There is one more way to avoid the difficulty, mentioned at the end of this paper, with regard to the noninvariance of the topological charge under gauge transformations. One must make use of the principal chiral field g(x) and introduce a gauge transformation in the form The simplest invariant Lagrangian takes the form

where

383

Comments on Paper 12 These lecture notes essentially cover my point of view on quantization with anomalies. I interpret anomalies not as a breaking of symmetry, but as a change of the mode of action of the symmetry group; in the case of the first anomaly the pure shift is accompained by a factor, whereas in the case of the second anomaly the action acquires an extension. Thus we are dealing with the cocycles of the group, which are exactly the manifestations of anomalies. In this perspective anomalies seem rather attractive mathematically and one must learn how to live with them instead of discarding them from the start. A rather short exposition of the functional integral derivation of the Schwinger term for the model Yang-Mills + chiral fermions in 3+1 dimensions was given in a paper by A. Alekseev et al., Theor. Math. Phys. 73, 187 (1987). The practical applications of my proposal in 3+1 dimensions stumble across a problem of exact evaluation of the Wess-Zumino term. I hope to be able to return to this problem some time in the future.

385 137

Hamiltonian Approach to the Theory of Anomalies L. Faddeev LOMI, Pontanka 27, SU-191011 Leningrad, USSR

Lecture 1 Introduction It is used to speak of anomaly if a p a r t i c u l a r symmetry of t h e classical field theory is not satisfied

after

quantization.

In t h e s e l e c t u r e s I shall consider some

recent

developments in t h e case of nonabelian anomaly, namely the breaking of t h e gauge i n v a r i a n c e in t h e model of the Weyl fermions i n t e r a c t i n g with the Yang-Mills The

literature

on

this

subject

is

quite

numerous

and

I can

refer

to

field.

original

monograph [1] and review a r t i c l e s in [ 2 ] . The model is described by 2 - s p l n o r field Y1* (x), V(x) and vector field A^(x) being vectors

and

matricies

in

a given

representation

a = 1

n are corresponding g e n e r a t o r s

of the

compact

group G. If

ta.

we h a v e components A£(x) of t h e Yang-Mills field

It will be c o n v e n i e n t to use a 1-form matrix

The field s t r e n g t h ( c u r v a t u r e ) 2-form is given t h e n by

where A are multiplied as matricies and as differential The gauge group

consists of m a t r i x - v a l u e d

forms.

functions

g(x)

with

t h e group G. The action of t h e gauge group on t h e fields is given by

the

values

in

386 138

This action Is right, for example

The Dlrac operator

where

Paull matrlcies, transforms as follows

and

Thus the determinant

of Dk which can be defined

as a functional

Integral

over

1

grassman v a r i a b l e s V *, V

seems to be gauge I n v a r i a n t . However t h e definition of Z(A) requires

regularizatlon

and t h i s leads to noninvarlance

In

FUJIKAWA

[3]

Interpretation

this

is

attributed

to the

noninvarlance

of

the

measure of integration in t h e integral r e p r e s e n t a t i o n for Z(A). The measure

implicitly depends on A and after the gauge transformation a "Jacoblan" occurs

The Fujikawa jacoblan is a phase factor, depending both on A and g. It is defined independently of Z(A). We see now t h a t

387 139

or Introducing the effective action

we can rewrite this in the form

For the infinitesimal transformations

we have

so that

and

Identifying

the <5W/<5Ap with the induced current J^ we have finally

that

this

current is not conserved

which is currently considered as a breaking of the gauge symmetry. In the second part of this lecture I want to ameliorate this statement and show, that the gauge symmetry does not disappear completely, it is not broken but rather modified. To explain this I need a short mathematical interlude. Let us consider a generic situation of an action of a group G with elements g on a m a n i f o l d M which p o i n t s a

The q u a s i r e g u l a r r e p r e s e n t a t i o n U c (g) of G t h e n i s d e f i n e d i n t h e s p a c e of functions over M

The c h o i c e of t h e r i g h t a c t i o n i s e s s e n t i a l i n p r o v i n g t h a t U c (g) i s i n d e e d

388 140

a

representation

Next in simplicity is a r e p r e s e n t a t i o n with a factor

This formula defines a r e p r e s e n t a t i o n

if the function a(A,g) satisfies the

following

relation

for any a, gx and g 2 . This relation admits a trivial solution

where

a(a)

is

a

function

of

one

variable.

In

case

of

a

trivial

a(a,g)

the

r e p r e s e n t a t i o n U(g) easily reduces to U„(g)

Nontrivial

solutions

considered

modulo

addition

of

the

trivial

ones

are

called

1-cocycles of the group. Now let us r e t u r n to our example and consider "manifold" of all Yang-Mills fields as a manifold M with the gauge group action on it. The simple chain of transformations

shows

that

the

Fujikawa

Jacobian

satisfies

the

cocycle

condition.

From

the

expression for a(A,g) in terms of W(A) it seems t h a t <*(A,g) is a t r i v i a l solution of t h i s condition. However there e x i s t s an important o b s e r v a t i o n . The functional W(A) is highly n o n - l o c a l functional

of t h e Yang-Mills field. The explicite expression

for

«(A,g) which we shall readily obtain shows t h a t it is a local functional, namely it will be written

as an Integral

of a d e n s i t y depending

field. In terms of local functlonals

locally

a(A,g) is a n o n t r i v i a l

on the

Yang-Mills

1 cocycle of the

gauge

group acting on t h e Yang-Mills fields. The gauge transformation of Z(A) can now be rewritten In the form

389 141

so that Z(A) Is invariant with respect to the new representation of the gauge group. Thus the gauge lnvarlance is not broken, it rather modifies. The cocycle condition in its infinitesimal form in terms of A(x) was introduced by J. WESS and B. ZUMINO, who called it a consistency condition [4]. I shall present now a trick to calculate the particular cocycle a,(A,g) of the gauge group on V-dimensional space time which lately will be identified with a(A,g). C o n s i d e r t h e 6-form

(for time being we suppose that the number of x-varlables is ^sufficiently large): The form W_, is exact

where the 5-form W0 is given by

In checking one must remember to change sign when transmitting the differential d t h r o u g h an odd form o r commuting odd forms u s i n g t h e c y c l i c p r o p e r t y of trace.

In p a r t i c u l a r

t r A2n = 0 f o r a n y n ,

indeed

The form W0(A) is not gauge invariant and particular calculation shows that

D e f i n i n g t h e 4-form 0 = d 1

we have

t r ( d g g - 1 ) 5 by means of t h e

formula

390 142

The factor In front of W- t was chosen in such a way that IP Is defined modulo the Integers. Introducing the functional

we see that exp i<x Is defined nonambiguously. Concrete expression for a^A.g) shows that it depends only on A and g given on IR* and one can forget now about the additional auxiliary x-variables used in the derivation. Last expression for a^A.g) shows that «, satisfies a cocycle condition. Moreover one can not write It down in a trivial form without extention to higher dimensional manifolds.

So ^(A.g)

Is

a

nontrivlal

1-cocycle

of

a

gauge

group

on

four

dimensional space-time. The procedure of the calculation of a,(A,g) is a particular example of finding so called secondary characteristic classes in differential topology, explored by Chern, Simons, B o t t , Gelfand and o t h e r s . In m a t h e m a t i c a l p a p e r s one c o n s i d e r s a g e n e r a l p e r t u r b a t i o n of a c o n n e c t i o n A. The f a c t t h a t gauge p e r t u r b a t i o n l e a d s t o t h e c o c y c l e of t h e gauge group was e x p l a i n e d i n my p a p e r [ 5 ] . In [6] t h i s p r o c e d u r e was used f o r c o n s t r u c t i o n of h i g h e r c o c y c l e s . In t h e next

lecture

a simplified

infinitesimal

version will

be p r e s e n t e d

in

detail. We f i n i s h t h i s l e c t u r e w i t h t h e i d e n t i f i c a t i o n of ai(A,g) and cc(A,g) . We s e e t h a t f o r i n f i n i t e s i m a l g

so that if «! = <x we are to have-the following expression for the anomaly A(x)

This is exactly the answer of the original calculation [7]

Lecture 2. Lle-Algebralc Cocycles and Their Interpretation The cocycles of a Lie group, the first of which was described in the lecture 1, have corresponding Lie-algebraic analogs. I begin this lecture with one more mathematical diversion, where, in particular, the role of the 2-cocycle will become clear.

391 143

Let G once more be a group acting on a variable

a, running through

the

manifold M. Infinitesimal action Is given by the vector fields Tu, defined us follows

Here u is an element of the Lie algebra, corresponding to G; with [u.v]; Tu realizes the representation of this algebra

commutator

which coresponds to the representation U0(g) of the previous lecture. The presentation, analogous to U(g) is given by the lnhomogenous operator

where 0,(a,n) is an operator of multiplication by a function linearly.

and depends on u

The condition

is fulfilled if /?, satisfies the equation

Here we Introduced an operation <5, mapping functions of a and one Lie-algebraic variable u Into functions of two such variables, linear In both and antisymmetric. The last condition Is trivially satisfied by 0,(a,u) of the form

where an operation <5 now maps functions of a Into functions of a and u, linear in u. In this way a 1-cocycle of a Lie algebra appears. The second cocycle appears naturally if we consider operators of the form

acting

an

vector-functions

requirement on Su

f(a),

so

that

J(a,u)

Is

a

matrix,

and

relax

the

392 144 Here /? a (a;u,v)

Is

a

function,

antisymmetric

(and

linear)

In

u and

v.

The

last

relation Is compatible with the Jacobl Identity

If 0 2 (a;u,v) satisfies the equation

Introducing one more example of t h e operation <5. It Is easy to see t h a t p2 of t h e form

is a t r i v i a l solution. Nontrivlal solutions are called 2-cocycles of the Lie algebra. In general, one can define an operation <5

acting on the antisymmetric multilinear functions on t h e Lie algebra in such a way that

Three

lowest

terminology

of

p * 67-cocycle.

examples

were

cohomology We

have

already

theory: seen

presented.

0-cochain,

the

role

of

This

P = the

allows

to

<5-y-coboundary, first

two

introduce sp

cocycles

=

the 0

for

but the

r e p r e s e n t a t i o n s of the Lie algebra: t h e first cocycle defines an inhomogeneous shift, the second appears In the projective r e p r e s e n t a t i o n . The role of the higher cocycles is not so clear and we shall not discuss them in t h e s e lectures. The trick allowed us to find the 1-cocycle of the gauge group in t h e lecture 1 can

be

used

for

calculating

of

the

cocycles a

of 3

the

corresponding

Lie

algebra,

consisting of m a t r i x - f u n c t i o n s u(x), u(x) = u (x)t . The action of 7^ is defined by

We begin once more with the 6-form

393 145

omitting for some time the multiplier l/24rr 2 . The form W_, Is gauge invariant, so

We have seen, that

where 5-form W0 Is not gauge invariant. However <5W0 is exact

where W,(A,u) is 4-form which is relative easily calculated

The closeness of <5W0 is trivial

and <5W0 is exact because the term tr (dgg _ , ) s , which appeared in the lecture 1, Is negligeable being of order 0(u 5 ). We can go further and show that

where W2 = W2(A,u,v) is a 3-form. Indeed

because <5J = 0. Explicetely one finds

This "procedure of descent" can be continued by the ladder of relations

where Wk is (
uk. In the original approaches of ZUMINO [8] and STORA [9] they used multilinear

functions of a grassman variable u(x), which Is evidently equivalent to using the polylinear antisymmetric functions of u,...u n . The language of this lecture is nearer to the mathematical traditions.

394 146 Given 3-form W,(A;u,v) such that

we can construct the 2-cocycle of our Lie algebra by Integration

where we have restored the factor 1/24TT2. One can directly check the nontrivlality of p2 in the complex of local functionals. There exist only two local 1-cochalns on IP, given by the 3-forms

Combining <5e,, i = 1,2 , with the form W2 one can obtain equivalent expression for the 2-cocycles. The expression written above is distinguished being the first order in A. One can find an ultralocal expression where the derivatives of u and v are absent

Corresponding 2-cocycle defines antisymmetric tensors

Aab(x)

which will be referred to below. We can ask where such 2-cocycle can appear? It is meaningful In the 3-dimensional space and the general considerations in the beginning of this lecture show that it must be associated with the operator in the form of sum of translation along the gauge orbit and rotation. Such operator exists in the theory of the fully quantized model which we consisder and it is nothing but the Gauss Law constraint. Classically in the Hamiltonian gauge A0 = 0 we have canonical variables

with the fundamental Poisson brackets

395 147

where for odd grassman variables V-*. i> we are to use "symmetric" Poisson brackets. The functional

satisfies the commutation relations

and one believes that these relations survive after the quantization. Our proposal can be formulated

as follows: the anomaly manifests

Itself through making the

representation projective so that in quantum case we have the modified commutation relations

More explicetely these relations look as follows

where the antisymmetric kernel pab(x\y") Is given by

or

depending on the choice of the realization of 2-cocycle p2 ■ It is instructive to present the corresponding expression for P2 i n case

1-dimensional

396 148 We see, t h a t

we get In 3 dimensions

a natural

generalization

of the

Kac-Moody

algebra. The presence of the Yang-Mills field is e s s e n t i a l for such a generalization. The cocycle §2 Gauss

Law

commutation

Independently authors

and proposal on its possible role as a Schwinger

it

relations

was found

did not consider

were

by ZINGER

Its

connection

Introduced [11] with

and

by me

term in

in [ 5 ] , see

MICKELSON

the q u a n t i z a t i o n

the

also

[10].

[ 2 ] , however

these

of

the

anomalous

fermionic models. In the l a s t lecture I shall describe you the present s i t u a t i o n on t h e derivation of t h e modified commutation r e l a t i o n s . Firstly In t h e next l e c t u r e we shall consider the implications of t h e s e r e l a t i o n s on t h e q u a n t i z a t i o n .

Lecture 3. Quantization of the Anomalous Model. The appearance of cocycle In the commutation relations of t h e Gauss Law changes d r a s t i c a l l y t h e quantization of our model. Indeed the operators G(u) play the role of constraints

in

the

Hamiltonian

c o n s t r a i n t s would be of the presence

of cocycle

they

quantization.

If

the

cocycle

1 class (see lectures of professor

turn

Into

2 class

changing

the

was

absent,

A. Hurst). In

definition

of

there the

physical

s t a t e s . So It seems t h a t one can quantize consistently an anomalous model but only after proper understanding of t h e s e s t a t e s . In what follows I use ideas published in a Joint work with SHATASHVILI

[13] .

Let us consider the difference of t h e 1 class and 2 class c o n s t r a i n t s on a t r i v i a l example of a system with finite number degrees of freedom. The phase space has canonical coordinates

and t h e wave function in the coordinate r e p r e s e n t a t i o n is a function of q - s .

First example is a c o n s t r a i n t of 1 class

One can impose it as a strong condition on a s t a t e vector

with solution

r2n

397 149

Thus the constraint of the 1-class kills full degree of freedom. As example of 2 class constraints we consider

so that

The equations

are Incompatible and can not be used for the definition of the physical states. The constraints can be satisfied only in the weak sence

for the physical states with appropriate scalar product. One of the tricks is to choose an "analytic" half of constraints

with the solution

Thus two second class constraints kill only one degree of freedom. In more Imaginative language one can say that 2 class constraints comprize a Heisenberg type subalgebra In the algebra of observables will essentially

unique

irreducible representation. The full space of states factorlzes into tensor product

of the representation space K and physical space P. Observables, commuting with constraints, act only In P. These simple observation constraint does not kill full

show, that

In the presence

of cocycle Gauss Law

(longitudinal) polarization of the vector

Yang-Mills

field. Rather 1/2 of polarization remains physical and we are to see what does it practically mean.

398 150

There e x i s t s a simple way to change the 2 - c l a s s

c o n s t r a i n t s into the

1-class

ones by adding new auxiliary v a r i a b l e s . In our simple example it goes as follows. We introduce one more degree of freedom p0.Qo and modify t h e c o n s t r a i n t s 4>i and * 2

Now new c o n s t r a i n t s commute

and we can find the physical vectors by solving t h e e q u a t i o n s

with general solution

The modified Hamiltonian

commutes with ♦, and * 2

and

so

in

a

new

formulation

our model

reduces

to

the

form,

suitable

for

the

functional integral q u a n t i z a t i o n . The functional integral aquires the form

with an a p p r o p r i a t e measure dp

and the gauge fixing procedure not shown explicetely. To apply t h i s trick to our gauge model we are to find a n a t u r a l candidate for "(Po

Qo)" v a r i a b l e s .

Number

of

them

Is

to

be

equal

to

number

of

Gauss

Law

399

151

c o n s t r a i n t s G a (3). The chiral field h(x) with the v a l u e s in the gauge group is j u s t such a field. Furthermore t h e functional a,(A,h) can be i n t e r p r e t e d as action of the form

Indeed from the explicite expression

we see t h a t

In all we propose to add chiral field h(x) to Yang-Mills field A^ and Weyl field f" , 1> and modify an action S

The functional

integral

with a p p r o p r i a t e measure n and a gauge fixing prescription is our main proposal. It is gauge i n v a r i a n t if the gauge group a c t s on h as follows

Indeed

due

to

the

cocycle

condition

gauge

variances

of

a,(A,h)

and

Fujikawa

measure cancel each other. The measure dp is to be Liouville measure for t h e fields A j , E,

and h times

dA 0 . To control t h i s Liouville measure let us mention t h a t one can reconstruct

the

symplectic form n = dW knowing t h e corresponding action /W. Indeed if we take W in general form

v a r i a t i o n of /W gives

400

152

t h u s enabling one to find o. Appling t h i s to a, we obtain

so t h a t the symplectic form can be expressed schematically as follows

The matrix n is of course even dimensional because we have x v a r i a b l e besides the isotopic

indices

to

label

its

matrix

elements.

It

turns

out

that

this

degenerate at points h = 1 or A = 0. This precludes us from simple calculations

matrix

is

perturbative

with "vacuum" A = 0, h = 1 a s a point of d e p a r t u r e . So only

the

background field method is to be applied. This means t h a t propagators and vertices in

the

Feynman

diagramms

will

contain

coordinate space. The renormalization

arbitrary

functions

and

one

uses only singular p a r t s of the

which can be evaluated explicitely. Thus the renormalization

is

use

propagators

programm is

but quite cubersome. The naive considerations make one hope, t h a t

to

feasible

the model is

renormalizable. Indeed action <*,(A,h) is dimensionless and the numerical

coefficient

in front of it is fixed by condition of u n i v a l u e d n e s s . Recently a simplified derivation of our proposal appeared in the l i t e r a t u r e One begins with the functional

[14].

integral

without gauge fixing and then uses the F a d d e e v - P o p o v trick of introducing 1 in the guise

where dh is an i n v a r i a n t measure on t h e group. After change of v a r i a b l e s Ah -* A the

action

a, (A,h)

appears

as

a Fujikawa

jacobian.

This

derivation

can

not

be

accepted in s p i t e of t h e fact t h a t it gives almost t r u e answer (only local measure differs

from

unitarity.

ours). Indeed

only

Hamlltonian

derivation

can

give confidence

about

401 153

It is worthwhile to mention, t h a t our proposal is similar to t h a t of POLYAKOV [15]

In

his

approach

to

string

model

in

noncritical

dimension.

Two

classical

c o n s t r a i n t s T 0 0 and T 0 1 t u r n Into 2 class after q u a n t i z a t i o n . To cure t h i s one is to add one degree of freedom which is t h e Liouville field introduced by Polyakov. To

finish

this

lecture

I shall

polarizalton. In 1-dimensional

make

case t h i s

some

comments

on

the

nature

of

is easy. The t r i v i a l example is given

1/2 by

free massless scalar field ♦ with lagrangian

and e q u a t i o n of motion

which admits a general solution of t h e form

In t h e hamiltonian approach one introduces instead of the canonical v a r i a b l e s * and 7T = a 0 * new v a r i a b l e s

with Poisson brackets

One sees

that

the full

phase space s e p a r a t e s

Hamiltonian

which produces the e q u a t i o n s of motion

for each of two 1/2 polarizations f and g.

in two. The same is t r u e

for

the

402

154

The analogue of this trick is known in nonabelian lagrangian

with

Wess-Zumino

term

leads

to two

case also where the

independent

Kae-Moody

chlral algebras

realized by c u r r e n t s

where

and h is chiral field. In three explicite

dimensions no such

expression

for

simple description

2 cocycle

shows,

the

can be found.

Yang-Mills

field

Indeed,

must

as

enter

the in

a

picture of 1/2 polarization field.

Lecture 4 Derivation of the Anomalous Commutation. Relations In t h e

last

derivation

two years

quite dramatic development

has t a k e n

place in a way of

of Schwinger term in the commutator of Gauss Law. I can devide

the

existing derivations in 5 partly overlapping classes: a) Operator derivations [13, 16] are based on the fact t h a t Gauss Law is given as a quadratic form of canonical variables

A, E, 1>m and f.

However the mistake

in

[13] make this derivation incomplete. b) Diagrammatic derivations

[16,17]

are based

on t h e BJL prescription

(see

[1])

which uses the fact t h a t the fixed time commutator of fields A(x) and B(x) is given in terms of the l/p 0 ~qo

term in the expansion of the Fourier integral F(p,q) of any

matrix element of their T-product for large p,, -q> . This derivation gives cocycle in the

ultralocal

form

which

was not

realized

at

the

beginning,

so t h a t

the

first

r e s u l t s where terms linear in A were sought for were also inconclusive. c) Topological gauge

group

derivations orbit

such

[18, 19] introduce

the "vacuum" U(l)

that

a

2-cocycle

is

curvature

of

bundle over the

the

corresponding

connection. These derivations are r a t h e r impllcite. d) Holonomy derivations [20, 21] introduce t h i s connection explicitly but until now I was not able to check the consistency of the regularizatlons used thereby. e) Functional integral derivations

[22] and partly

[17] as always p r e s e n t a s h o r t

cut for the diagrammatic ones and I shall p r e s e n t an oversimplified

description in

the end of this lecture. But firstly I shall present several formulas which will enable me to comment on the derivations listed above.

403 155

Let A in more d e t a i l be a manifold of all Yang-Mills fields A over II? at

infinity).

corresponding

The

Fock

space

HA

creation-annihilation

for

fermions

operators.

is

One

introduced uses

the

by

(vanishing means

of

decomposition

of

V'-operators

where u A and v A are positive and n e g a t i v e energy solutions of Dlrac equation

and

We use the fact t h a t with our boundary conditions Dirac operator

h a s purely continuous spectrum. We can formally introduce an operator UA such t h a t

however UA is not generally speaking a well defined operator in a free Fock space H with vacuum I0> a n n h i l a t e d by a and b

Indeed

the

criterium

for

the

unitary

implementarity

of

the

linear

canonical

transformation introduced in the formula of expansion of V(x) looks as follows

where P\. the

PJ and P+, P- are projectors on the p o s i t i v e and n e g a t i v e subspaces of

corresponding

Dirac

operators.

This

condition

is

not

fulfilled

3 - d l m e n s i o n a l case which is most possibly a reason for a failure of the in [ 1 3 ] .

in

the

derivation

404 156

If one p e r s i s t s in using UA with some regularization in mind, then the

vacuum

IA>

can be introduced and it is a n n i h i l a t e d by aA and b A . The U(l) connection is then introduced by the analogy with the

Berry phase t r e a t m e n t (cf. lecture of professor

Seiler) as follows

Its c u r v a t u r e is to give 2-cocycle when one r e s t r i c t s A to run through t h e gauge group orbit A9. In one dimension the scalar product makes sence

without

any regularizations and one can recover 2 cocycle as a c u r v a t u r e of WA. However in 3-dimensions this scalar product requires a regularization. Apparently it is given in [20] and [21] but I did not check this myself. In the operator approach of [13] the idea was to regularize the c u r r e n t J(u) by going to the nonlocal expression

s u b s t r a c t i n g the divergent part of its expectation in vacuum IA> and taking to the local limit

afterwards. This introduces the dependence of J(u) on A. More explicetely

where

^jng(x-y)

is

a

contribution

to

the

kernel

P^lx.y)

Formally we are s u b s t r a c t i n g the expression like

In 1-dimensional case the kernel P ^ i n g ^ . y ) is A - i d e p e n d e n t

and one can easily get

of

the

projector

PA.

405 157

leading to t h e <5'(x-y) form of 2 cocycle. In 3-dlmensional case we h a v e

The

second

enforcement

term,

quadratic

we h a v e found

In

A.

in

was

missed

in

[13] the expression

[13].

Without

of 2-cocycle

it

with

some

linear in A. The

visitor from Poland A. Madajchyc h a s found a mistake in the last spring and attempts

to

cure

the

derivation

in

[13]

were

Inconclusive

until

now

(see

the also

[ 2 3 ] ) . One could probably t r a c e s t h e reason to t h e fact of nonimplementability of t h e transformation a •» a A . I finish t h i s lecture with a s h o r t comment on a functional the

anomalous

derivation

of

commutation the

relations.

Ward-Slavnov

I

still

identities.

use

an

integral d e r i v a t i o n of

old-time

More modern

and

approach rigorous

to

the

derivation

based on t h e BRS procedure is given by Fujikawa [ 2 2 ] . We begin with the functional integral in A 0 = 0 gauge

and change t h e -^-variables

This is t h e same as if A in the Dirac operator changes into

Moreover t h e Fujikawa jacobian a(A,g) will appear when we change t h e

transformed

local measure into t h e original one. Expanding the r e s u l t up to second order in u and

taking

into

Ward-Slavnov

account

the

explicite

expression

for

a(A,g) we get t h e

modified

identity

The terms in s q u a r e

brackets

define

a symmetric q u a d r a t i c

form

of an

arbitrary

406 158 function u(x)

In the BJL limit one can use the classical equation of motion

Thus to restore t h e commutator of two G one is to multiply the Fourier ab

of the first term in t h e coefficient

K (x,y)

by l / p 0 q 0 -

transform

Combined with t h e

factor

Po~Qo p e r t i n e n t to the BJL procedure t h i s leads to the prescription to c a l c u l a t e

In t h i s way the first term gives

the second leads to

the third term v a n i s h e s and t h e l a s t gives t h e Schwinger term in its ultralocal form

Acknowledgement These lectures were prepared in the nice atmosphere of Schladming during t h e first week of the 26th school. I would like to express my deep g r a t i t u d e to H. Mltter

for

his

hospitality.

Influence

of Professor

It

is

W. Thirring's

pleasant forthcoming

to

acknowledge

birthday

on the

the

Professor

overwhelming

program

of

this

session.

References 1.

S. Trelman, R. Jackiw, B. Zumino, E. Witten: Current Algebra and Anomalies, World Scientific, Singapore, 1985

2.

W. Bardeen, A. White: Anomalies, Geometry. Topology, World Scientific, Singapore, 1985

3.

K. Fujikawa: Phys. Rev. D21_. 2848 (1980)

407

159 4.

J. Wess, B. Zumino: Phys. Lett. B37, 95 (1971)

6.

L. Faddeev: Phys. Lett. 145B, 82 (1984)

6.

A. Reiman, M. S e m e n o v - T j a n - S h e n s k y , L. Faddeev: Func. Anal. Appl. 18, 64 (1984)

7.

W. Bardeen: Phys. Rev. 184, 1848 (1969)

8.

B. Zumino: Les Hauches Lectures, 1983

9.

R. Stora: Cargere Seminar 1983, preprint LAAP TH94 1983

10. L. Faddeev, S. S h a t a s h v l l i : PMPh. 60, 206 (1984) 11. I. Zinger: Asterisque 1985 (EMF, Lion) 12. J. Mickelsson: Comm. Math. Phys. 97, 361 (1985) 13. L. Faddeev, S. S h a t a s h v i l i : Phys. Lett. 167B, 225 (1986) 14. O. Babelon, F. Shaposnik, C. Vlallet: Phys. Lett. 177B, 385 (1986); A. Kulikov: Serpukbov p r e p r i n t IHEP 8 6 - 8 3 ; K. Harader, I. T s u t s u i : preprint TIP/HEP-94 15. A. Polyakov: Phys. Lett. 103B, 207 (1981) 16. I. Zinger, I. Frenkel: (in p r e p a r a t i o n ) S.-G. Jo: Phys. Lett. 163B, 353 (1985) 17. M. Kobajashi, K. Seo, A. Sugamoto: Nucl. Phys. B273, 607 (1986) 18. G. Segal: Oxford preprint, 1985 19. L. Alvarez-Gaume, Nelson: Comm. Math. Phys. 99, 103 (1985) 20. A.J. Niemi, G.W. Semenoff: Phys. Rev. Lett. 56, 1029 (1986) 21. H. Sonoda: Nucl. Phys. B266. 410 (1986) 22. K. Fujikawa: Phys. Lett. 171B, 424 (1986) 23. S.-G. Jo: MIT p r e p r i n t CTP W1419 (1986)

409

Comments on Paper 13

This is a survey of the Hamiltonian approach to the Einstein theory of gravitation in the asymptotically flat space-time. I advocated the point of view of V. A. Fock, that the Poincare group stays in this case as a dynamical group, in contrast to the localized coordinate transformations. This allows one to introduce the physical generators of the group, in particular the energy. I used the paper of Witten, where the positivity of this energy was ingeniously proved and which just appeared as a preprint at that time. The paper had also some political value, rebuffing the activity of a rather influential person in Soviet scientific life who denounced the Einstein theory of gravitation.

411

The energy problem in Einstein's theory of gravitation (Dedicated to the memory of V. A. Fock) L D. Faddeev Leningrad Branch, V. A. Sieklou Mathematics Institute, USSR Academy of Sciences Usp. Fiz. Nauk. 136, 435-457 (March 1982) The review is devoted to a discussion of the definition and properties of energy in Einstein's theory of gravitation. Asymptotically flat space-time is defined in terms of admissible asymptotically Cartesian coordinates and a corresponding group of coordinate transformations. A Lagrange function is introduced on such a space-time, and a generalized Hamiltonian formulation of the theory of gravitation is constructed in accordance with Dirac's method. The energy is defined as the generator of displacement with respect to the asymptotic time. It is shown that the total energy of the gravitational field and the matter fields with normal energy-momentum tensor is positive and vanishes only in the absence of matter fields and gravitational waves. The proof follows Witten's proof but contains a number of corrections and improvements. Various standard criticisms of the energy concept in general relativity are discussed and shown to be without substance. PACS numbers: 04.20.Fy, 04.50. + h

CONTENTS Introduction 1. Generalized Hamiltonian formulation 2. Asymptotically flat space-time 3. Generalized Hamiltonian formulation for the gravitational field 4. Proof of positivity of the energy Conclusions Appendix I Appendix II References

INTRODUCTION _, , . , The energy concept plays a central role in modern . , . . . , _, . , .. , theoretical physics. The Law of conservation of energy 1 . . , , . x. (and also momentum and angular momentum) is a con­ sequence of the homogeneity of time (respectively, the homogeneity and isotropy of space). In this sense, the energy concept is associated with the fundamental structure of space-time. A characteristic property of energy is its positivity, reflecting stability of a physical system. The traditional method for determining the energy and momentum in relativistic field theory is based on the introduction of the energy-momentum tensor. This tensor is defined as the variation of the action with r e ,. .. , ,. , , „ , ... spect to an external gravitational field. Such a method . .. . ,. , , .. is not valid in the case when the gravitational field it., . • 1 ■ ui n. self is regarded as a dynamical variable, since the r e ., . . . . . . , ., suiting tensor vanishes identically by virtue of the equa, . .. .. ., tions of motion. As a result, the energy concept in the ,. , . .. theory of gravitation requires further discussion. The problem of determining the fundamental integrals of the motion—the energy, momentum, and angular mo­ mentum—arose immediately after the final formulation of the theory of gravitation by Einstein and Hilbert at the end of 1915, and it was essentially solved by Einstein by 1918 (see Ref. 1). His proposal originally evoked many questions and objections from his contemporaries, who included Lorentz, Levi-Civita, Schrodinger, and others. This discussion is well reflected in Pauli's review article of Ref. 34. However, the situa130

Sov. Phys. Usp. 25(3), March 1982

130 13] 133 135 137 J30 130 140 142

tion was gradually clarified and a conception formulated that has found its way into the textbooks and mono, „ , „ ^ graphs (see, for example, Refs. 2-5); this can be fora r \ < *■ mulated as follows. 1. The energy (and also the other fundamental integrals of the motion) of the gravitational field interacting with a system of masses and matter fields can be introduced if space-time is asymptotically flat, i.e., becomes identical with Minkowski space asymptotically at spatial infinity. 2. The energy of the gravitational field is not localized, i.e., a uniquely defined energy density does not exist. The asymptotic condition 1) replaces the homogeneity _. , ,., , J , / . _ , . L , '. T of time in ordinary relativistic field theory. It makes it _,-._, • j. ... possible to define a dynamic displacement in time as a ,. , ,... . .. ,_ ... displacement with respect to an observer far from the r F gravitating matter, and to associate energy with the ° , , , displacement. In contrast, in cosmological models r , . . , . , , ,. , there is no natural time displacement and accordingly no energy concept. The nonlocalizability of the energy of the gravitational field is due to a specific property of the relativity principle in the theory of gravitation. It is not the metric of space-time that is a physical quantity but the class of equivalent metrics differing by an arbitrary coordinate transformation consistent with the asymptotic conditions. The value of the metric at a given point of space-time does not have absolute significance, and the

0038-5670/82/030130-13S01.80

© 1982 American Institute of Physics

130

412 cheory of gravitation itself is in this sense fundamentally nonlocal. _, , . . . . . . . These matters are discussed in detail and critically in the quoted monographs. The above two propositions must be augmented by a

3. The total energy of the gravitational field and gravitating matter is positive and vanishes only in the absence of matter and gravitational waves, when the metric becomes identical with the flat Minkowski metrie. However, this result has not yet found its place in the 3

v

monographs. The proof is a difficult problem of mathe3

p

r

v

matical physics, and it has been solved only very r e , ' ' cently. A particularly elegant solution had just been , . „ , . , found by Witten, and it is given in the main text. & ' ' During the 60 years that Einstein's theory of gravitation has existed, Einstein's solution to the energy problem has continued to be doubted. The criticism has crystallized in a number of fixed ideas, which have appeared periodically in the publications of various authors. In a recent series of papers by Logunov et al. (see Refs. 7 and 8), this criticism was the stimulus for the construction of a new theory of gravitation. In the main text, we mention some of the main arguments of this criticism, and we show where they are defective. In the present paper, we review the energy question in Einstein's theory from the point of view of Hamiltonian dynamics. In such an approach, the energy plays the part of a dynamical observable—it is the generator r & ' of displacement in time. Together with the energy Pn, v & °} °' the momentum P„, the angular momentum Mit, and the Lorentz moments M0k form the 10 generators of the Poincare group, which act on the phase space of the system consisting of the matter fields and the gravitational field. From this point of view, the dynamical group of the theory of gravitation in the case of asymp­ totically flat space-time does not differ from the dynam­ ical group of any other relativistic dynamical system. Fock : particularly insistently emphasized the distinguished role of the Poincare'group in the theory of gravitation. The history of our approach began with Dirac's studies 9 in 1958-1959, in which he applied to the gravita­ tional field the general theory of dynamical systems described by singular Lagrangians that he had created earlier in Ref. 10. In the sixties, this approach was adopted by many theoreticians, among whom we mention the Arnowitt-Deser-Misner team, 11 Schwinger, 12 DeWitt,13 and Regge and Teitelboim. 14 Although this method is fundamental, it has not yet found its place in the textbooks and is regarded by many as something exotic rather than a basic method of exposition of gravitational theory. Its importance has been widely recog­ nized only in the field of the quantum theory of gravita­ tion (see, for example, the review of Ref. 15). I hope that this review will serve to popularize the Hamiltonian approach to the theory of gravitation. The 131

Sov. Phys. Usp. 25(3), March 1982

satisfactory solution to the energy problem by means of this approach convincingly illustrates the-power of the method and demonstrates once more that the theory of gravitation itself is in need of neither revision nor .... modification. . . , . .. , T „ We give a brief summary of the review. In Sec. 1, we give the fundamentals of the generalized Hamiltonian dynamics of Dirac for systems defined by means of a singular Lagrangian. In Sec. 2, we introduce the concept of asymptotically flat space-time and discuss its group of transformations. In Sec. 3, the generalized Hamiltonian formulation will be given for the theory of gravitation, and the generators of the Poincare' group, ... .. ... _ . _,. ,-,. ,, with the energy among them, arise naturally. Finally, . „ . ,. , . ,, ... _ . . . .. c ... in Sec. 4 we discuss briefly the history of the problem , .. , ., . „ „ „ „ >„ „ „ r of the positivity of the energy and give Witten s proof. , .. , . _ .. . , Some lengthy calculations are given in Appendices I and IT II. We use the usual relativistic notation: n = (0,0 is coordinate index, a- (0, a) a local Lorentz index, and a ^ [s the metric tensor of Minkowski space with signature (-+ + + ). 1. GENERALIZED HAMILTONIAN FORMULATION The conce

P l o f a generalized Hamiltonian formulation dynamics of a mechanical system appeared in Dirac's paper Ref. 10 and lectures Ref. 16 devoted to singular Lagrangians. Dirac himself extended this formulation to field theory and applied it to the theory of gravitation in Refs. 9 and 17. „. _.__i#„ tha ; J „ „ *;„,.,. ,,„.:„„ n,- „„„ ■„ nf „ „,„ We clarify the idea first using the example of a me„u„„ ; . „„„i.„„ ,„;n, „ c;„^„ „ i,„ „r A „„„ „* chanical system with a finite number of degrees of f r e e d o m . Suppose the system is described by n pairs < - ! , . . . , „ , and by a further m Qf v a r i a b l e s / ) qif variables X«, a - 1 , . . . , « , the Lagrangian having the form of t h e

where cpa, a= 1, . . . ,m, and h are certain functions of p and q. We use the so-called "first-order formalism," in which the derivatives of the independent dynamical variables appear linearly in the Lagrangian. The La­ grangian (1.1) is singular, since the equations of motion d o n o t contsdn t h e derivatives of the variables X". II

is natural to call pt and qi canonical variables, V Lagrangian multipliers, <pa constraints, and h the Hamiltonian. By [f,g\ we denote the ordinary Poisson brackets:

Tne ai

equations of motion that follow from the variationprinciple are

We shall say that the Lagrangian (1.1) defines a generaiized Hamiltonian formulation if the constraints and the Hamiltonian satisfy the conditions L. D. Faddeev

131

413

where c\b and c\ are arbitrary functions of p and q. The condition (1.4) means that the Poisson brackets of the constraints with one another and with the Hamiltonian vanish on the constraint surface <pa= 0. It guarantees that this surface remains invariant during the motion for any choice of the time dependence of the Lagrangian multipliers Aa(/). The condition m < n is necessary for (1.4) to hold. The generalized Hamiltonian formulation reduces to the ordinary one if the constraint equations are solved and the solution substituted in the Lagrangian (1.1). We then obtain the new Lagrangian

where h= 1 , . . . ,n-m and the Hamiltonian h* is equal to h restricted to the constraint surface:

Indeed, on the constraint surface, the Hamiltonian h does not, by virtue of (1.4), depend on the m variables canonically conjugate to the constraints
Sov. Phys. Usp. 25(3), March 1982

wnere h

c i s t n e energy density of the charged field in the external electromagnetic field, which enters h through the spatial covariant derivatives Vk= 9„+ iA>; p(cp,T,) i s the charge density and G, = t{l„Fjk/2 is the magnetic field - W e h a v e s e t t n e coupling constant e equal to 1. The function he(tp,it,AM) is positive. lt

is

obvious that (1.7) has the form (1.1) and we must make the identifications

so that x also plays the role of a constraint label. The Lagrangian (1.7) is associated with the Poisson brackets

The constraint
. .. .. . . , u. . . and consider the canonical transformation which it gene r a t e s . We have transforma­ densi. ty T n u S j ^ A ) . g t h e g e n e r a t o r o f a g a u g e transformation for the electromagnetic field and the charged field. 1° electrodynamics, the group of such transformations is commutative, and the Hamiltonian is invariant with respect to it. This is expressed in the relations md

fi

,

Q} i g t h e i n f i n i t e s i m a l p h a s e

t i Q n o f t h e chJged

field g e n e r a t e d

by tne c n a r g e

which

can be readily verified directly. These relations realize the determining property (1.4) of the generalized Hamiltonian formulation. T h e e q u a t i 0 n s 0 f motion

... ,, „. .. ,, t ■■ , -n together with (1.9) give all the nontrivial Maxwell equa. . . .. _ „ _ , „ _ , . „ , , ~s Ana„ { „ J ^ J „ „ » « tions, so that the Lagrangian (1.7) does indeed c o r r e . , , , . .. . . ., .. . spond to electrodynamics. In fact, it is simply identical to ^ m a n i f e s t l y r e l a t i v i s t i c a l l y invariant Lagrangian ^ the f i r s t . o r d e r f o r m a i i s m

after the magnetic field F.b has been expressed in terms ^ tential

Qf t h e y e c t o r

In the considered simple example, the constraints can be solved explicitly. The role of the variables q* and p* is played by cpa, ir" and the three-dimensionally transverse components A% and El of the fields Ah and Ek. The longitudinal component A% of the field Ak is L. D. Faddeev

132

414 canonically conjugate to the constraint and does not occur in the Hamiltonian h. The longitudinal component E£ of the field E„ can be expressed in terms of/)* and q* by means of the constraint equation. If we set

then Eq. (1.9) takes the form of the Poisson equation

which can be solved explicitly. The contribution |/(£^) 2 d 3 x of the longitudinal field to the Hamiltonian gives the instantaneous Coulomb interaction of the charges:

considers the more general case when such coordinates can be introduced only in the asymptotic region. However, as follows from the results of Ref. 32, our r e striction is not fundamental. Let *"= {x°,x{) be one such system, guv be the pseudo-Euclidean metric with signature (-+ + + ), and _. . ,, . ... ... ~. r*„ be the components of the connection. These quanti­ ties define an asymptotically flat space-time if in the limit r — °° and for finite /

where

and 7j However, it is not necessary to solve the constraint. For example, the positivity of the total energy, which is made up of the energy of the electromagnetic waves, the energy of the waves of the charged field, and the Coulomb energy,

is the metric tensor of flat Minkowski space:

particular, that in ^ U m i t r _ „ t h e c o o r d i n a t e s xt a r e s p a c e - l i k e and Cartesian, and the coordinate x° is timelike. The condition on the masses and matter fields which ensures their effective localization in a compact region of space can be formulated as Thg conditions (2A)

indicate,

m

follows from the explicit positivity of the generalized energy (1.8). We have intentionally gone into such detail for the standard and noncontentious example of electrodynamics to have the possibility, when analyzing the theory of J * ' ' ° ' gravitation, to draw a parallel with this more simple r r " ' case. In the following sections, we shall see that the Hamiltonian formulation of the theory of gravitation dif­ fers little from the formulation of electrodynamics that * we have just given. The only important difference will ' ° . , . be the circumstance that in the case of gravitation the energy in which we are interested is itself the source of the graviational field, whereas in electrodynamics the charge is the source. 2. ASYMPTOTICALLY FLAT SPACE-TIME

where Tuv is the corresponding energy-momentum tensor. _. .... ,„ ,, . . „ . t .. .. „. The condition (2.1) does not r e s t r i c t the coordinate _ transformations

.... . . . .._ t ... in a finite region; however, at large r the functions „, . . . .. . . i)"u) must have the asymptotic behavior

Where A

* i s t h e m a t r b c o f L o r e n t Z transformations, and a is an arbitrary constant vector. We shall assume that these transformations act on the set of metrics and connections referred to a fixed coordinate system. The corresponding infinitesimally small transformations are given by u

There exist several definitions of asymptotically flat space-time, differing in the level of covariance and mathematical rigor. We shall use here the most naive but at the same time perspicuous definition based on the introduction of admissible coordinates. As is cus­ tomary in geometry, the invariance of this definition is ensured by the introduction of an admissible group of transformations. .. c, . Physically, asymptotically flat space-time corresponds to situations when the gravitating masses and matter fields at finite times are effectively concentrated in a finite region of space. Then far from such a r e gion, in the spatial directions, there exists only the Newtonian tail of the graviational field due to all the masses and the energy of all the wave fields, including the energy of gravitational waves. Clearly, to describe such a situation it is sensible to use coordinates that match this picture. ,„ ., .. , ,. We shall consider the case of a topologically simple space-time whose points can be uniquely parametrized by four coordinates x", -°° < xu < « . One sometimes 133

Sov.Phys. Usp. 25(3), March 1982

where s w is a vector field having in the limit r~°° the asymptotic behavior (2.6) with infinitesimal w" = A? " * M " We denote by G the infinite-dimensional group generated by these transformations. The group G has a normal subgroup G0 generated by the transformations (2.7) for which A" = 8" and au = 0, i.e., they are identify transformations as r ~ ° ° . The factor group . . _ ° is identical with the Poincare'group—the 10-parameter group of displacements and rotations in Minkowski

The group G0 is the gauge group of the theory of I . D. Faddeev

133

415 gravitation in asymptotically flat space-time. Two metrics that differ by transformations of this group describe the same physical situation, provided, of course, a corresponding transformation of the matter fields is made as well. At the same time, the symmetry group of the Lagrangian, the equations of motion, and the boundary conditions is the group G. This means that the Poincare group acts nontrivially on the space of interacting matter and gravitational fields. In particular, the time displacement defined by

which is invariant with respect to the group G. Indeed, it follows from (2.7) that

° n integration by parts, the integrated t e r m s for 5S disappear at spatial infinity on account of_(2.13) and (26) - B u t i n t n e analogous integral for 6S they do not in general vanish. All this shows that the correct variational principle for the gravitational field in asymptotically flat space-time must be based on the action S. T n e u s e of t n e Lagrange function density L is sharply criticized in Ref. 20 on account of its noncovariance. However, as we have just shown, the Lagrange function jLtfx i s invariant with respect to the allowed transformations of the group C. Thus, the objections are witho u t substance, since they do not take into account boundary effects in the noncompact asymptotically flat space.

makes it possible to define the energy up to transformations in G Thus, from the point of view of dynamics the theory of gravitation in asymptotically flat space-time does not differ from other relativistic field theories, since the Poincare' group plays the part of the dynamical group in it. In this sense, the expression "general relativity" does not apply to the dynamics but to the definition of the gauge group. This point of view was formulated in Ref. 19 in the t e r m s just introduced. It is close to many parallel formulations of other authors, in particular Fock's 2

to

the

definition of asymptotically flat space-time, essentially relax the condition of decrease °^ t n e r e m a - m < : i e r terms in (2.1) and (2.6). For example, if instead of (2.1) we use one cannot

The two last paragraphs might appear rather peremp­ tory. However, they summarize in a few words the r e ­ sults that will be given in the following two sections. We now discuss the Lagrange function of the gravitational field. As is already clear from Sec. 1, we shall find it convenient to use the first-order formalism, in which guv and r^„ are regarded as independent dynamical variables. The density of the Lagrange function is to usually taken to be the scalar density

then for a-s | the action (2.12) becomes meaningless, t n e s p a t i a l integral diverges. The interesting f c a s e 0 gravitational fields with Newtonian asymptotic behavior belongs to the class (2.1), so that there are n o physical reasons for relaxing the conditions (2.1). since

tneir recent

P r e P r i n t o f R e f - 35> Denisov and Logunov use a coordinate transforaiation with asymptotic behavior

or the function

which differs from ^TgR by a total divergence and therefore gives the same equations of motion. Adherents of the second variant usually adopt a defensive position, recognizing the fundamental role of J=gR and invoking only the formal convenience of L (for example, the absence in L of the second derivatives of guv which occur in the second-order formalism). However, in asymptotically flat space-time L is the only admissible density in the definition of the action

where the integration is over the whole of space and a finite time interval. Because of the conditions (2.1), we have in the limit r - » o f

where the first estimate is correct only if it is assumed in addition to (2.1) that 3XVUV= OU/r 3 ). Therefore, it is S and not

134

Sov. Phys. Usp. 25(3), March 1982

which c a r r i e s the metric from the class (2.1) into the class (2.16) with a = j . With respect to such transfor­ mations, the definition of the energy given below is not From our point of view, such coordinate invariant. transformations are inadmissible. For example, the a c t k ) n (2 12) i s n o t i n v a r i a n t w i t h r e s p e c t t o

them

be based on a detailed d t s c u s S i o n o f t h e compactification of the manifold corr e s p o n d i n g to the asymptotically flat space-time. The coordinate transformations (2.17) are singular on this However, a detailed discussion goes beyond mani{old. t n e f r a m e w o r k of the naive but perspicuous definition of admissible coordinates adopted in this review. More rigorous arguments must

Let us conclude with a few words on general covariQf c o u r s e > a g ^ M y t h e o r y f o r m u l a t e d in s p a c e „ t i m e > a r b i t r a r y c o o r d i n a t e s can be used in the theory gravitation. However, the concept of asymptotically flat space-time, which has an objective physical mean­ ing, can be formulated simply and clearly in the coordinates that we have used that satisfy the asymptotic condition (2.1). In particular, the important concept of displacement in time is given by the simple formula (2.9). In some papers, 2 1 ' 2 2 criticism has been advanced of the definition of energy in Einstein's theory of gravi^

L. D. Faddeev

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416 tation based on the use of homogeneous coordinate transformations which are not Lorentz transformations. It is clear that the paradoxes which then arise are associated with such a violation of the asymptotic conditions.

We note that the scalar curvature contains the second derivatives of the metric tensor linearly, so that Q = -yR3- S^n?'* is a quadratic form of the first derivatives of the metric with respect to the spatial variables:

3. G E N E R A L I Z E D H A M I L T O N IAN F O R M U L A T I O N FOR THE G R A V I T A T I O N A L FIELD

In particular, H{x) has the asymptotic behavior

Just as we treated electrodynamics in Sec. 1, we shall now describe the Hamiltonian dynamics for a gravitational field interacting with a matter field, using initial data on the surface JC° = 0, which we shall assume to be spacelike. As the variables of the gravitational field, we take gliu and r"„. These data induce on the initial surface a metric g(b (the first quadratic form), which is positive, and the second quadratic form JT° .

and the integral in (3.3) converges. T h e P o i s s o n brackets

induced natural . ti, density

It is convenient to use tensor densities instead of these tensors. L

e

t

t

i

o

n

s

by the Lagrangian (3.3) are consistent with the interpretation of the 5 function as a (bi)scalar . „■ k . , ,Q „„ >■,. -__,_„i_n__ „„■„ of weight 1. To express the commutation relao f t h e c o n s t r a i n t s QO and C„, it is convenient to introduce functionals of the vector Xk(x) and the scalar density/(x) of weight - 1 :

be the contravariant tensor density of the four-dimen­ sional metric tensor. We denote The matrix qik is the contravariant density of weight 2 of the metric gik,qltglk = &&, y=det\\gih\\. The matrix XI jk is a covariant density of weight - 1 . For details, see Appendix I, in which it is shown in detail that after elimination of the unimportant variables in the action (2.12) the Lagrange function of the gravitational field and the matter fields, which we denote by