On the quantization of continuous non-ultralocal integrable systems
aa r X i v : . [ h e p - t h ] N ov On the quantization of continuous non-ultralocalintegrable systems
A. Melikyan ∗ and G. Weber † 21 Instituto de F´ısica, Universidade de Bras´ılia, 70910-900, Bras´ılia, DF, Brasil Escola de Engenharia de Lorena, Universidade de S˜ao Paulo, 12602-810, Lorena, SP, Brasil
October 15, 2018
Abstract
We discuss the quantization of non-ultralocal integrable models directly in the con-tinuous case, using the example of the Alday-Arutyunov-Frolov model. We show thatby treating fields as distributions and regularizing the operator product, it is possibleto avoid all the singularities, and allow to obtain results consistent with perturbativecalculations. We illustrate these results by considering the reduction to the massivefree fermion model and extracting the quantum Hamiltonian as well as other conservedcharges directly from the regularized trace identities. Moreover, we show that our regu-larization recovers Maillet’s prescription in the classical limit.
Keywords:
Exactly Solvable Models, Bethe Ansatz; Continuum models; Integration ofCompletely integrable systems by inverse spectral and scattering methods; Quantum FieldTheory.
The Alday-Arutyunov-Frolov (
AAF ) model is a purely fermionic classical integrablemodel arising from the reduction of the
AdS × S superstring theory to the su ( | ) subsec-tor in the uniform gauge [1, 2]. Applying the inverse scattering method to the AAF modelhas proven to be a very non-trivial task as a result of its non-ultralocality, which manifestsin an even more complicated form than in the usual examples of non-ultralocal integrablesystems [3]. This prompts a more detailed investigation of non-ultralocal models and thedevelopment of new methods to regularize and therefore quantize such theories.As a matter of fact, the quantization of non-ultralocal integrable models is one of themost intriguing and challenging open problems in the context of integrability. To date onlyin very few examples this question has been properly addressed, i.e., the SU ( ) PrincipalChiral Model (PCM) [4], the Wess-Zumino-Novikov-Witten (WZNW) model [5, 6] and the ∗ [email protected] † [email protected] AdS × S string theoryis a classically integrable system of this type (for a review see [8] and references therein). Inspite of all the attention devoted recently to this area, because of its importance to quan-tizing the AdS × S superstring and thus improving our understanding of the AdS/CFTcorrespondence [9–14], there is still no satisfactory general method to resolve all the difficul-ties involved in the quantization process of non-ultralocal theories.There exist, though, some standard approaches to this problem. The method proposedby Maillet and collaborators [3,7,15] seems, however, to be the simplest and most systematicin order to construct the action-angle variables, and understand the classical integrability.It involves a generalization of the concept of the r -matrix to a pair of ( r , s ) -matrices andthe simultaneous regularization of the ill-defined Poisson brackets with the use of a sym-metrization procedure to introduce the so-called Maillet brackets. The most fundamentaldifficulty that prevents the full implementation of the quantum inverse scattering methodto non-ultralocal theories lies in finding the appropriate regularization/quantization of thecorresponding Maillet brackets. In particular, one that recovers the symmetrization pre-scription pertaining the definition of the Maillet brackets in the classical limit. Thus, forthe non-ultralocal systems there is no direct generalization of the Yang-Baxter type equationfrom which one can extract the quantum Hamiltonian and other quantum charges, and findthe spectrum.Although the lack of a general procedure to properly quantize the Maillet bracket hasprecluded the full implementation of the quantum inverse scattering method to many in-teresting models, one notable exception is the SU ( ) PCM which has been quantized byFaddeev and Reshetikhin ( FR ) in [4]. The FR quantization method is based on the ultralo-calization of the theory and its subsequent regularization in terms of a magnetic lattice al-gebra. The original non-ultralocality can be shown to be restored in the continuous theoryby taking the large spin limit. Thus, by replacing the original non-ultralocal Poisson al-gebra by a new ultralocal one, while preserving the equations of motion with respect toa new Hamiltonian, Faddeev and Reshetikhin avoided dealing directly with the problem-atic Maillet bracket. The recent identification of the algebraic mechanism underlying theabove ultralocalization procedure enabled its application in more general contexts such assigma models on symmetric and semi-symmetric spaces, including the AdS × S super-string [9, 10]. Notwithstanding all this effort the quantization of sigma models on symmetricand semi-symmetric spaces is still unknown, even though some candidate lattice Poissonalgebras have been proposed [9, 10, 16].From the few cases where the quantization of the Maillet bracket was successful [4–7]there emerges some general strategy to be followed. It reduces essentially to the followingfour steps: ( i ) ultralocalize the Kac-Moody type algebra satisfied by the classical continu-ous theory; ( ii ) regularize the ultralocalized current algebra to get rid of the singularities atcoinciding points by invoking a lattice discretization; ( iii ) quantize the lattice current alge-bra by means of the quantum inverse scattering method; ( iv ) check that in the scaling limitthe quantized discrete algebra reproduces the classical Kac-Moody algebra. However, thisrecipe breaks down for the AAF model already in step ( i ) , as all the ultralocalization proce-dures so far developed work only for models plagued by non-ultralocalities up to the firstderivative of the delta function, while the algebra of Lax operators for the AAF model iseven more non-ultralocal, including terms proportional to the second derivative of the deltafunction [17, 18]. Moreover, being a purely fermionic model, any naive lattice discretization2ecessarily incurs in fermion doubling.Thus, despite the absence of appropriate methods to directly quantize the Maillet bracketfor the
AAF model and the inherent difficulty in generalizing the available methods, one canstill try to find the quantum Hamiltonian via coordinate Bethe Ansatz. For example, in thecase of the
AAF model, it has been shown in [19] that this is indeed possible, and the quan-tum Hamiltonian, which, after a field redefinition to make its Poisson structure canonical,acquires a very complex form, containing terms up to the eighth order in the fermion fieldand its derivatives, can be diagonalized. The key point here, as was shown in [19], is that thewave-functions (and their derivatives) in the quantum mechanical picture are not continu-ous functions, and to avoid meaningless expressions in the calculation one has to: (i) treatthe quantum fields as operator valued distributions, and (ii) employ the principal value pre-scription in the resulting integrations where the discontinuities arise. It was shown that thisprescription indeed does the desired job, and the diagonalization process reproduces thecorrect S -matrix of the AAF model, found earlier via perturbative calculations [20, 21].In this paper we take another step towards the full implementation of the quantum in-verse scattering method to the
AAF model. This program was initiated in [17] where weidentified a surprisingly simple 2 × ( r , s ) -formalism to accommodate the second derivative of the delta function in the algebra of Laxoperators, was taken in [18]. Here we make the next step, and implement the principal valueprescription, which had to be manually performed in the previous calculations, directly onthe operator level. Namely, we show that by treating the quantum fields as operator valueddistributions, and regularizing the product of operators by means of Sklyanin’s product [22]in the quantum Hamiltonian, as well as any relevant operator quantities, such as the Laxoperator, the principal value prescription follows naturally without any manual input. Animmediate consequence of this implementation is the reproduction of Maillet’s symmetriza-tion prescription in the classical limit. We stress that differently from [4, 5, 7, 15, 23–26] wework directly in the continuous case, without appealing to any lattice regularization of thetheory, as the latter is not always an easy task to formulate.Sklyanin’s product (or the ◦ -product) is a type of split-point regularization, which wasoriginally introduced in [22] in order to regularize the product of two operators at the samepoint and therefore obtain the Yang-Baxter relation for the Landau-Lifshitz model. The latteris an ultralocal model, and so the difficulties of the quantization are associated only with thesingularities appearing in the product of operators at the same point. Thus, the regularizedquantum Hamiltonian can be naturally obtained from the fundamental regularized Yang-Baxter relations [27, 28]. In contrast, in the case of the AAF model one does not have, asexplained above, the Yang-Baxter type equations, and it is not clear from which fundamen-tal relations such regularization with Sklyanin’s product can appear. To address these pointswe consider the consistent reduction of the
AAF model to the free massive fermion model.Such a procedure allows to avoid all the unnecessary technical complications of the
AAF model, and automatically gives the Lax pair for the free fermion model which leads to analgebra with the same degree of non-ultralocality as the
AAF , and thus it is still sufficientlynon-trivial in order to test our approach. Then we show that if one regularizes the Lax oper-ator via Sklyanin’s product, the regularized quantum Hamiltonian can be obtained from theintegral equations defining the quantum transition matrix. Moreover, for the classical the-ory obtained from this regularized quantum theory, one reproduces Maillet’s symmetrized3oisson bracket prescription.Our paper is organized as follows. In section 2 we present the most essential aspects ofthe
AAF model and briefly discuss how to generalize the classical inverse scattering methodto accommodate its higher degree of non-ultralocality. Then, in section 3, we address the fun-damental problem of ill-defined operator products when formulating a continuous quantumalgebra and introduce Sklyanin’s product as our regularizing prescription. Next, in section4, we particularize the discussion of the previous section on the role of Sklyanin’s product tothe case of the
AAF model. In section 5, we consider the reduction of the
AAF model to thefree fermion model, which gives the explicit Lax operator and the associated non-ultralocalalgebra. We explicitly work out the regularized quantum monodromy matrix, showing howto extract the quantum conserved charges. The relation between the normal product andSklyanin’s product is also explained. In section 6, building upon the results of the previoussection, we conjecture the form of the quantum algebra of transition matrices for a non-ultralocal continuous theory and show that it consistently reduces to the Maillet algebra inthe classical limit. In section 7, we summarize our results and point out some interesting di-rections and open problems. Finally, in appendices we collect various computational detailsused in the text.
In this section we briefly overview the essential properties of the
AAF model, referringthe reader to the papers [1, 2, 17–19, 21] for all the technical details. As we mentioned in theintroduction the
AAF model arises from the reduction of the superstring on
AdS × S tothe su ( | ) subsector, where in the process of constraint analysis all the bosonic degrees offreedom are eliminated in favor of fermionic ones. The resulting theory is a two-dimensionalLorentz-invariant fermionic model which is described by the following action (see appendixA for notations): S = ˆ dy ˆ J dy h i ¯ ψ∂ / ψ − m ¯ ψψ + g m ǫ αβ (cid:0) ¯ ψ∂ α ψ ¯ ψ γ ∂ β ψ − ∂ α ¯ ψψ ∂ β ¯ ψ γ ψ (cid:1) −− g m ǫ αβ ( ¯ ψψ ) ∂ α ¯ ψ γ ∂ β ψ i . (2.1)The two-particle scattering S -matrix has been first found from perturbative calculations andhas the form [20]: S ( θ , θ ) = − img sinh ( θ − θ ) + img sinh ( θ − θ ) , (2.2)where θ and θ are the rapidities of the scattered particles with momenta p = m sinh θ and p = m sinh θ . The coupling constants g and g in (2.1) were introduced in [21], wherethe S -matrix factorization property, underlying the quantum integrability of the model, wasproved up to the first loop approximation, provided the relation g = g between the cou-pling constants is satisfied.The Lax pair for the AAF model found in [1, 17] leads to a non-ultralocal algebra for the L -operators of the form [18]: Here the symbol ⊗ stands for the supertensor product , which extends the concept of the tensor product forbosonic fields to the fermionic case. For detailed mathematical definitions and the relevant constructions werefer the reader to the monograph [29] and the original papers [30–34]. For a comprehensive review, see [35]. L ( σ ) ( x ; λ ) ⊗ , L ( σ ) ( y ; µ ) } = A ( x , y ; λ , µ ) δ ( x − y ) + B ( x , y ; λ , µ ) ∂ x δ ( x − y )+ C ( x , y ; λ , µ ) ∂ x δ ( x − y ) . (2.3)It has a more complicated form when compared to the standard case studied in [3], sinceit contains terms proportional not only to the first derivative of the delta-function, but alsoto its second derivative. To slightly simplify the discussion bellow, we shall classify thenon-ultralocal algebras by the highest order of the derivative of the delta-function present.For instance, the algebra (2.3) is a second order non-ultralocal algebra, while the standardcase [3], a first order. Thus, to properly take into account the contribution of the last term in(2.3) it is necessary to consider a generalization of Maillet’s ( r , s ) -matrix formalism, whichamounts to the introduction of a third matrix. In terms of the triple ( r , s , s ) , the algebra(2.3) becomes: n L ( σ ) ( z ; λ ) , L ( σ ) ( z ′ ; µ ) o = δ ( z − z ′ ) (cid:18) ∂ z r ( z ; λ , µ ) + h r ( z ; λ , µ ) , L ( σ ) ( z ; λ ) + L ( σ ) ( z ; µ ) i (2.4) + h s ( z ; λ , µ ) , L ( σ ) ( z ; µ ) − L ( σ ) ( z ; λ ) i + h ∂ z s ( z ; λ , µ ) , L ( σ ) ( z ; λ ) + L ( σ ) ( z ; µ ) i + hh s ( z ; λ , µ ) , L ( σ ) ( z ; λ ) i , L ( σ ) ( z ; µ ) i + hh s ( z ; λ , µ ) , L ( σ ) ( z ; µ ) i , L ( σ ) ( z ; λ ) i (cid:19) − ∂ z δ ( z − z ′ ) (cid:2) s ( z ; λ , µ ) + s ( z ′ ; λ , µ ) (cid:3) + ∂ z δ ( z − z ′ ) (cid:2) s ( z ; λ , µ ) + s ( z ′ ; λ , µ ) (cid:3) ,where we used the following standard notation for tensor products L ( z ; λ ) ≡ L ( z ; λ ) ⊗ and L ( z ; λ ) ≡ ⊗ L ( z ; λ ) . For the AAF model the exact form of the matrices ( r , s , s ) has avery complicated non-linear character [17].Nonetheless the resulting classical algebra of transition matrices corresponding to equaland adjacent intervals with x > y > z has exactly the same structure as the originally pro-posed by Maillet [3, 7, 15] for the simpler first order case, { T ( x , y ; λ ) , T ( x , y ; µ ) } M = r ( x ; λ , µ ) T ( x , y ; λ ) T ( x , y ; µ ) − T ( x , y ; λ ) T ( x , y ; µ ) r ( y ; λ , µ ) , { T ( x , y ; λ ) , T ( y , z ; µ ) } M = T ( x , y ; λ ) s ( y ; λ , µ ) T ( y , z ; µ ) . (2.5)The effect of the second derivative of the delta-function in (2.4) amounts to the followingshift of the pair of intertwining matrices ( r , s ) : r ( z ; λ , µ ) → u ( z ; λ , µ ) = r ( z ; λ , µ ) + ∂ z s ( z ; λ , µ ) + [ s ( z ; λ , µ ) , L ( z ; λ ) + L ( z ; µ )] , (2.6) s ( z ; λ , µ ) → v ( z ; λ , µ ) = s ( z ; λ , µ ) + [ s ( z ; λ , µ ) , L ( z ; λ ) − L ( z ; µ )] . (2.7)In (2.5) the subscript M indicates that the Poisson bracket has to be symmetrized according toMaillet’s prescription to avoid ambiguities arising from coinciding points. Starting from (2.5)one can then construct the angle-action variables following the standard procedure [35–38].This program has been realized for simpler models in [18].The fundamental construction underlying Maillet’s method is the symmetrization pre-scription for Poisson brackets and the corresponding generalization for nested Poisson brack-ets. To introduce Maillet’s symmetrization procedure one considers n -nested Poisson brack-ets for transition matrices T ( x i , y i ; λ i ) : ∆ n ( x i , y i ; λ i ) = (cid:8) T ( x , y ; λ ) ⊗ , (cid:8) . . . ⊗ , (cid:8) T ( x n , y n ; λ n ) ⊗ , T ( x n + , y n + ; λ n ) (cid:9) . . . (cid:9)(cid:9) , (2.8)5nd for any subset of l = p + q coinciding points x α = . . . = x α p = y β = . . . = y β q = z , onedefines the left-hand side of (2.8) by: ∆ n ( z ; λ i ) : = lim ǫ → l ! ∑ σ ∈ P ∆ n (cid:0) x α + ǫσ ( ) , . . . , y β q + ǫσ ( l ) ; λ i (cid:1) , (2.9)where for simplicity of notations we omitted in ∆ n ( x i , y i ; λ i ) the dependence on the coordi-nates different from z , and the symbol P indicates the sum over all possible permutations of (
1, . . . , l ) . For example, this symmetrization procedure yields: { T ( x , y ; λ ) ⊗ , T ( x , y ′ ; µ ) } M =
12 lim ǫ → (cid:0) { T ( x − ǫ , y ; λ ) ⊗ , T ( x + ǫ , y ′ ; µ ) } + { T ( x + ǫ , y ; λ ) ⊗ , T ( x − ǫ , y ′ ; µ ) } (cid:1) . (2.10)The quantization of classical algebras for transition matrices of the form (2.5) has notbeen successful except in few very specific cases. One of the principal difficulties is that thecommutators in the quantum theory cannot immediately reproduce the symmetrized Mailletbrackets on the left hand side of (2.5). Another important difficulty is the quantization ofintegrable models directly in the continuous theory. This is especially relevant for the AAF model as its lattice version is not known. Quantization of continuous models, as discussedin the introduction, presents a challenge due to the singularities arising from the product ofoperators. To correctly quantize the system, one should first remove such singularities bymeans of a proper regularization of the fields or products. Before addressing this problem forthe
AAF model, we briefly explain in the next section how to quantize a simpler continuousintegrable model - the Landau-Lifshitz ( LL ) model, which although ultralocal exhibits thesame type of interaction terms in the Lagrangian as the AAF model.
To formulate a well-defined algebra for quantum transition matrices for a continuous the-ory one must first deal with the singularities associated with operator products at the samepoint. The most natural way to solve this problem is to resort to the methods of quantumfield theory where such singularities are dealt with by means of renormalization techniques.The latter can be strictly formulated in the framework of axiomatic quantum field theory(see, for example, the monograph [39]), where the quantum fields are treated as operator-valued distributions: Φ F ( x ) = ˆ dy Φ ( y ) F ( x , y ) , (3.1)where F ( x , y ) ≡ F ( x − y ) is an element in the Schwartz space of test functions.The necessity to treat fields as distributions in the context of the integrable systems wasfirst realized on the example of the LL model in [27], following an early attempt by Sklyanin[22] to regularize the product of operators in order to satisfy the Yang-Baxter relation. Morerecently this approach was also applied to the
AAF model [19]. For both models it was In the original Sklyanin’s approach [22], instead of treating fields as distributions, a product between twooperators was introduced in order to to regularize arising singularities. The original definition of [22] is asfollows: A ( x ) ◦ B ( x ) ≡ lim ∆ → ∆ ˆ x + ∆ x d ξ ˆ x + ∆ x d ξ A ( ξ ) B ( ξ ) . (3.2) n -particle sector wave-functions, one has to regularize the fields as in (3.1). Thisallows to avoid meaningless singularities of the type ∂ x δ ( ) , and permits the construction ofthe correct spectrum and S -matrix. Furthermore, it was also shown that the correspondingquantum-mechanical Hamiltonian is a self-adjoint operator. These results would have beenimpossible to obtain without regularizing the fields as in (3.1), i.e., treating the fields asdistributions. Before turning to the more complex, non-ultralocal integrable AAF model,we first give a brief review of this construction for the simpler, ultralocal LL model usingthe methods previously elaborated in [22, 27, 28]. Then we present a new, more convenientformulation, which is more appropriate when dealing with non-ultralocal models.We recall, that the Hamiltonian for the isotropic LL model for the su (
1, 1 ) case has theform [40]: H = ˆ dx (cid:16) ∂ x ~ S · ∂ x ~ S (cid:17) , (3.3)where the fields S i ( i =
1, 2, 3 ) satisfy the following Poisson structure: (cid:8) S ( x ) , S ± ( y ) (cid:9) = ± iS ± ( x ) δ ( x − y ) , (3.4) (cid:8) S − ( x ) , S + ( y ) (cid:9) = iS ( x ) δ ( x − y ) .It is now possible to show (for full details see [28]) that by passing to regularized fields as in(3.1): S i F ( x ) = ˆ dy S i ( y ) F ( x , y ) , (3.5)the Lax operator: L F ( λ , x ) = i λ (cid:18) S F ( x ) − S + F ( x ) S − F ( x ) − S F ( x ) (cid:19) (3.6)satisfies the fundamental intertwining relation:lim n R ( λ − λ ) h L F ( λ , x ) + L F ( λ , x ) + L F ( λ , x ) · L F ( λ , x ) io == lim nh L F ( λ , x ) + L F ( λ , x ) + L F ( λ , x ) · L F ( λ , x ) i R ( λ − λ ) o . (3.7)The limit on both sides of the equation (3.7) corresponds to removing the regularization,i.e., when F ( x ) ∼ δ ( x ) . The quantum R ( λ ) -matrix in the above expression is given by thefollowing formula: R ( λ ) = ∑ a = w a ( λ ) σ a ⊗ σ a , (3.8)where w ( λ ) = λ − i / , w = − i / , and σ a = ( , σ i ) .Furthermore, it can be shown [22] that the intertwining relation (3.7) leads to the follow-ing Yang-Baxter relation: R ( λ − µ ) T ( λ ) T ( µ ) = T ( µ ) T ( λ ) R ( λ − µ ) , (3.9) Although this was enough to obtain the Yang-Baxter equation (3.9), Sklyanin’s product, as discussed in [28], wasnot enough to diagonalize the quantum Hamiltonian, or to obtain the higher order local conserved charges. Italso led to other problematic singular expressions. Here S ± = S ± iS . T ( λ ) is obtained from the corresponding quantum L F ( λ , x ) .The Yang-Baxter relation (3.9) allows the quantization of the LL model using the standardmethods. In particular, using the regularized fields as discussed above, one can diagonalizethe quantum Hamiltonian for any n -particle sector, as well as construct the higher orderconserved charges.In order to present our results, it is necessary to first explain how the diagonalizationprocedure for the LL model should be carried out when the fields are regularized accordingto (3.5). Afterwards, we introduce an alternative formulation which is more suitable fornon-ultralocal models such as the AAF model. The main result of [28] is that the quantumHamiltonian of the LL model written in terms of the F -regularized fields has the form: H F = ˆ dx (cid:2) − ∂ x S F ( x ) ∂ x S F ( x ) + ∂ x S + F ( x ) ∂ x S − F ( x ) + ∂ x S − F ( x ) ∂ x S + F ( x ) (cid:3) , (3.10)while the n -particle states are: | f n i = ˆ n ∏ i = dx i f n ( x , . . . , x n ) n ∏ j = S + ( x j ) | i , (3.11)and provide a representation space for the su (
1, 1 ) algebra for the operators in terms of S i F fields. Here, the wave functions f n ( x , . . . , x n ) can be shown to be continuous and suffi-ciently fast decreasing, symmetric functions of x , . . . , x n , which, however, have discontinu-ous first derivatives.The crucial comment is that due to the presence of the derivatives in the quantum Hamil-tonian (3.10), and the discontinuity of the first derivatives of the wave functions, during theprocess of the diagonalization, the resulting integrations should be understood in the princi-pal value sense. In other words, even though the field regularization (3.5) is enough to obtainthe Yang-Baxter relation (3.9), one still has to treat the integrals arising in the diagonalizationprocess in the principal value sense. As was shown in [28] this leads to the boundary con-ditions on the first derivatives of f n ( x , . . . , x n ) , and indeed reproduces the correct S -matrix.Thus, whenever integrals containing the derivatives ∂ x i f n ( x , . . . , x n ) occur, one has to un-derstand such integrals in the principal value sense. For example, for the case n =
2, thearising integrals are of the type: ¨ dx dy ∂ lx ∂ ky [ ∂ x f ( x , y ) . . . ] ; l , k =
0, 1, . . . ,and should, in general, be understood as follows: ˆ dx (cid:20) ˆ x − ε − ∞ dy + ˆ ∞ x + ε dy (cid:21) ∂ lx ∂ ky [ ∂ x f ( x , y ) . . . ] .For the higher n -particle sectors, the resulting integrations are over an n -dimensional space,and one has to divide the integration region into n ! subspaces, similar to the example above,in such a way as to remove a small strip around the singularity region.Thus, for a general model, when performing the direct diagonalization one has to man-ually take care of ill-defined integrals due to discontinuities of the wave functions or theirderivatives. The natural question is whether it is possible to have a more fundamental for-mulation which does not require any such manual input, and the principal value prescrip-tion or its n -dimensional generalization is taken into account from the beginning. We show8ext that this is indeed possible, and can be formulated in terms of an operator product. Fortwo operators A ( x ) and B ( x ) , the ◦ -product is defined as follows: A ( x ) ◦ B ( x ) ≡ lim ∆ / → ǫ ( ∆ − ǫ ) ∆ S ∪ ∆ S d ζ d ξ A ( ζ ) B ( ξ ) . (3.12)The notation ffl ∆ S ∪ ∆ S means that the integration is taken over a square of side ∆ , minus astrip of width ǫ around the diagonal ζ = ξ , and the areas ∆ S and ∆ S correspond to theregions above the line ζ = ξ + ǫ and below the line ζ = ξ − ǫ . This essentially means thatwe “smear” the product of two operators around an arbitrary small area of size ∆ , avoidingthe singularity at ζ = ξ . The parameter ǫ is the regularization parameter of the theory, andshould be taken to zero only at the end of all computations. It is clear from (3.12) that if theproduct of two operators is not singular, then, in the limit ǫ →
0, it reduces to the usualproduct. In other words, we explicitly exclude the entire singular region, parametrized bythe length scale ǫ .In general, for a product of k operators A ( x ) , . . . , A k ( x ) , the ◦ -product is defined sim-ilarly to (3.12). The integration should now be performed over the k -dimensional cube ofside ∆ , where all possible singular regions are taken out of the integration domain. Thus,there are k ! integrations over disconnected volume elements of size ∆ V i corresponding toall possible orderings of the variables ζ , . . . , ζ k separated by the length of the regularizationparameter ǫ . More precisely, we have: A ( x ) ◦ . . . ◦ A k ( x ) ≡ lim ∆ / → ǫ ∆ V k ! ∑ i = ˆ ∆ V i d ζ . . . d ζ k A ( ζ ) · · · · · A k ( ζ k ) , (3.13)where ∆ V is the sum of all disconnected volume elements ∆ V i . Principal value prescription from operator product
We now state one of our central results. In order to account for the principal value pre-scription from the beginning, which is needed, as discussed above, to treat ill-defined inte-grals due to the discontinuities of the wave functions f n ( x , . . . , x n ) or their derivatives, onehas to replace the usual operator product in the quantum Hamiltonian (or in other relevantoperators, e.g., higher-dimensional charges) by the ◦ -product defined in (3.13). We stressthat this is in addition to regularizing the fields as in (3.1). To prove this statement we ex-plicitly consider, for the the sake of concreteness, the action of the following term from theHamiltonian of the LL model (3.10): H F = ˆ dx ∂ x S F ( x ) ∂ x S F ( x ) (3.14) We also refer to it as
Sklyanin’s product , since it can be seen as a modification of Sklyanin’s original definition(3.2). We note, nonetheless, that a similar situation takes place in other integrable models, including the
AAF model, and, therefore, the proof that follows can be easily generalized for other types of interactions, with suit-able modifications.
9n the two-particle state (3.11): | f i = ˆ d ξ d ζ f ( ξ , ζ ) S + ( ξ ) S + ( ζ ) | i . (3.15)The result reads: H F | f i = ˆ dx du dv f ( u , v ) ∂ u F ( x , u ) ∂ v F ( x , v ) S + ( u ) S + ( v ) | i + . . . , (3.16)where the terms in ellipses are suppressed in order to avoid cluttering, and have a similar tothe first term structure. To evaluate such terms and to obtain meaningful expressions, onemust employ the principal value prescription, as discussed earlier. After some transforma-tions one arrives at the following result [28]: H F | f i = ˆ dx du (cid:20) ˆ u − ǫ − ∞ dv + ˆ ∞ u + ǫ dv (cid:21) (cid:8) ∂ v ∂ u f ( u , v ) F ( x , u ) F ( x , v ) S + ( u ) S + ( v )+ ∂ u f ( u , v ) F ( x , u ) F ( x , v ) S + ( u ) ∂ v S + ( v ) (cid:9) | i− δ α ( ) ˆ du ∂ u f ( u , v ) | v = u − ǫ v = u + ǫ S + ( u ) S + ( v ) | i + . . . . (3.18)Next, we show that by employing the operator product (3.12) one arrives at the same result(3.18), but without the necessity to use any principal value prescription. In other words, weshow that the principal value prescription can be implemented on the operator level. Theadvantage of this will be discussed further in the text.We start by writing the Hamiltonian (3.14) in terms of the F -regularized fields and theoperator product (3.12): H SP F = ˆ dx ∂ x S F ( x ) ◦ ∂ x S F ( x ) (3.19) = lim ∆ → ǫ ( ∆ − ǫ ) ˆ dx dy dz x + ∆ ˆ x − ∆ + ǫ du u − ǫ ˆ x − ∆ dv + x + ∆ − ǫ ˆ x − ∆ du x + ∆ ˆ u + ǫ dv · ∂ u F ( u , y ) ∂ v F ( v , y ) S ( y ) S ( z ) .Using the commutation relations following from the Poisson algebra (3.4) and that F ( x , y ) depends only on the difference of its arguments [28], it is a straightforward computation to Here δ α ( x ) denotes a regularization of the δ -function with respect to the regularizing length parameter L = / α [41]: δ L ( x ) = π ˆ L /2 − L /2 e ixu du . (3.17)The length parameter L is very large and is associated with the size of the box in which we consider our system,resulting, as a consequence, in asymptotic Bethe ansatz equations. H SP F | f i = lim ∆ → ǫ ( ∆ − ǫ ) ˆ dx dy dz ( x + ∆ ˆ x − ∆ + ǫ du u − ǫ ˆ x − ∆ dv + x + ∆ − ǫ ˆ x − ∆ du x + ∆ ˆ u + ǫ dv · f ( y , z ) ∂ y F ( u , y ) ∂ z F ( v , z ) S + ( y ) S + ( z ) ) | i + . . . . (3.20)Here we have explicitly written down the term corresponding to the first term in (3.16),while the other terms represented by ellipses are again suppressed. The ◦ -product splits theintegration into two disconnected regions which differ only on the domain of integration forthe variables u and v , as one can clearly see in (3.20). Thus, in the following we concentrateon the evaluation of I a = ˆ dx dy dz x + ∆ ˆ x − ∆ + ǫ du u − ǫ ˆ x − ∆ dv ∂ y F ( u , y ) ∂ z F ( v , z ) f ( y , z ) S + ( y ) S + ( z ) , (3.21)noting only that the computation of I b = ˆ dx dy dz x + ∆ − ǫ ˆ x − ∆ du x + ∆ ˆ u + ǫ dv ∂ y F ( u , y ) ∂ z F ( v , z ) f ( y , z ) S + ( y ) S + ( z ) (3.22)is completely analogous.Since for the LL model the wave function f ( y , z ) is itself continuous, all the functionsbeing integrated in I a (3.21) are continuous in the region above the line u = v , so that we caninvoke the mean value theorem to compute the integrals over u and v . This makes u = x + c and v = x − k , where c , k ∈ (cid:0) ǫ , ∆ (cid:1) , and (3.21) reduces to: I a = ( ∆ − ǫ ) ˆ dx dy dz f ( y , z ) ∂ y F ( x + c , y ) ∂ z F ( x − k , z ) S + ( y ) S + ( z ) . (3.23)Next, we integrate by parts with respect to y and use the fact that the fields S + ( x ) vanish as | x | → ∞ to obtain: I a = − ( ∆ − ǫ ) ˆ dx dy dz n ∂ y f ( y , z ) F ( x + c , y ) ∂ z F ( x − k , z ) S + ( y ) S + ( z ) (3.24) + f ( y , z ) F ( x + c , y ) ∂ z F ( x − k , z ) ∂ y S + ( y ) S + ( z ) o .The first term in (3.24) is proportional to the derivative of the wave function, ∂ y f ( y , z ) , whichis no longer continuous on the line y = z for the LL model. Thus, when evaluating (3.24) oneneeds to carefully consider the first term, while the second term can be trivially integratedby parts with respect to z . Hereafter, we will drop the contribution from the second termin (3.24). We note, nonetheless, that the contribution from the corresponding term has alsobeen neglected in the computation that led to (3.18).11he principal value prescription was introduced in (3.18) with the sole purpose of avoid-ing the discontinuity of the derivative of the wave function on the line y = z . Here, we willshow that the ◦ -product, by dividing the integration domain over u and v into two disjointregions, naturally avoids such discontinuity. In order to do that we will explicitly use thefact that the fields S + F ( x ) are smeared around the point x , i.e., they are localized in a small,but finite, neighborhood of the point x . The key point being that such neighborhoods can betaken sufficiently small to be completely separated by the width ǫ introduced by Sklyanin’sproduct. Since, in the end of the calculation, we are going to remove the F -regularization,it is sufficient to show that there is a representation of the F -functions in terms of delta se-quences that satisfies such separation property. For the sake of concreteness, we consider thefollowing explicit representation [42]: F α ( x , y ) = C α e − α α −| x − y | if | x − y | ≤ α | x − y | > α , (3.25)where C α are constants satisfying: ˆ dz F α ( z ) = ⇐⇒ C α α n ˆ | z | < dz e − −| z | =
1. (3.26)Here we denoted F α ( x − y ) ≡ F α ( x , y ) .It can be proved that provided the regulator α is kept finite the representation (3.25) of F α ( x , y ) and its derivatives are smooth functions with support in | x − y | ≤ α . Thus, theintegrations over y and z in (3.24) have only non-zero contribution from the intervals: x + c − α ≤ y ≤ x + c + α , (3.27) x − k − α ≤ z ≤ x − k + α . (3.28)Noting that c , k ∈ (cid:0) ǫ , ∆ (cid:1) , it is easy to see that whenever α , α < ǫ the following inequalitieshold α + α < c + k ⇔ z ≤ x − k + α < x + c − α ≤ y (3.29)and the line y = z is never reached in (3.24). In particular, if we choose α , α < ǫ , theinequalities (3.27) and (3.28) together with the fact that c , k > ǫ restrict the domain of inte-gration over z to ( − ∞ , y − ǫ ) . The choice of α , α to be smaller than ǫ can be justified asfollows. Recall, that ǫ is the characteristic length around the singularity region, which is re-moved in the definition of the ◦ -product (3.12). In other words, the inverse Λ = / ǫ can beinterpreted as the cut-off length, which is a fixed momentum and remains constant through-out the calculation. Any other parameters should be taken to zero before considering (ifnecessary) the limit ǫ → z , (3.24) becomes: I a = − ( ∆ − ǫ ) ˆ dx dy dz n ∂ f ( y , y − ǫ ) F α ( x + c , y ) F α ( x − k , y − ǫ ) S + ( y ) S + ( y − ǫ ) − ∂ z ∂ y f ( y , z ) F α ( x + c , y ) F α ( x − k , z ) S + ( z ) S + ( y ) − ∂ y f ( y , z ) F α ( x + c , y ) F α ( x − k , z ) S + ( y ) ∂ z S + ( z ) o + · · · . (3.30) Here, ∂ j f denotes the derivative with respect to the j-th argument of f . ( ∆ − ǫ ) and taking the limit ∆ → ǫ forces the open interval (cid:0) ǫ , ∆ (cid:1) toshrink to ǫ . Finally, in order to remove the F -regularization, we can use the relation between F -functions and delta sequences [28]. Hence, after considering the appropriate limit, we canperform the integration over x to obtain:lim ∆ → ǫ I a ( ∆ − ǫ ) = − δ α ( ) ˆ dy ∂ f ( y , y − ǫ ) S + ( y ) S + ( y − ǫ ) (3.31) + ( δ α ( )) ˆ dy (cid:2) ∂ ∂ f ( y + ǫ , y − ǫ ) S + ( y + ǫ ) S + ( y − ǫ )+ ∂ f ( y + ǫ , y − ǫ ) S + ( y + ǫ ) ∂ S + ( y − ǫ ) (cid:3) + · · · .Proceeding analogously with (3.22), we derive a very similar expression for the contributionfrom bellow the line u = v . Summing these two results, we finally obtain: H SP F | f i = lim ∆ → ǫ I a + I b ( ∆ − ǫ ) | i + · · · (3.32) = δ α ( ) ˆ dy (cid:2) ∂ f ( y , y + ǫ ) S + ( y + ǫ ) − ∂ f ( y , y − ǫ ) S + ( y − ǫ ) (cid:3) S + ( y ) | i + ( δ α ( )) ˆ dy n [ ∂ ∂ f ( y + ǫ , y − ǫ ) + ∂ ∂ f ( y − ǫ , y + ǫ )] S + ( y + ǫ ) S + ( y − ǫ )+ ∂ f ( y + ǫ , y − ǫ ) S + ( y + ǫ ) ∂ S + ( y − ǫ ) + ∂ f ( y − ǫ , y + ǫ ) S + ( y − ǫ ) ∂ S + ( y + ǫ ) o | i + · · · ,which reproduces (3.18) without the need to invoke any principal value prescription in themiddle of the calculation.Hence, we see that Sklyanin’s product (3.13) naturally avoids the discontinuities in thederivatives of the wave functions both in the boundary and bulk terms. Moreover, all the op-erator products in (3.32) are explicitly symmetrized, indicating that the ◦ -product (3.12) pro-vides a quantization which is compatible with Maillet’s symmetrization prescription (2.10)in the classical case. Later, in section 6, we will work out in details the relation between thequantum regularization provided by the ◦ -product and the classical regularization given byMaillet’s brackets.We conclude this section with remarks on the physical meaning of the ◦ -product (3.12).It follows directly from the definition that this product is a point splitting type of regulariza-tion, which is frequently used in quantum field theory in order to regularize singular opera-tor products, e.g., in relation to anomaly computations (for an overview and the relation ofpoint-splitting regularization to other methods, see [43, 44]). We recall the standard exampleof the axial fermionic electromagnetic current, regularized via point splitting method [45]: j reg µ ( x ) = ¯ ψ ( x + η / ) γ µ γ e − ie ´ x + η /2 x − η /2 dy ν A ν ψ ( x − η / ) . (3.33)To avoid the dependence on the fictitious point η µ , one has to take the limit η µ → symmet-rically , which means that it should be taken in such a way as to ensure that the final formulasdo not depend on the specifically chosen vector η µ . More explicitly, the symmetric limit isdefined as follows: (for details see [45]):symm lim η → (cid:20) η µ η (cid:21) = η → (cid:20) η µ η ν η (cid:21) = g µν . (3.35)13n contrast, the product in (3.12) is automatically smeared homogeneously over the entireregion around the singularity strip, and does not depend on any chosen point η µ .To summarize, in order to derive the standard intertwining relation (3.7) for the LL modeland perform diagonalization of the quantum charges one has to treat quantum fields asoperator valued distributions (3.1), and, in addition, employ Sklyanin’s product (3.12) toaccount in a natural way for principal value prescription when dealing with discontinuouswave-functions or their derivatives. One of the consequences of our regularization is that itreproduces Maillet’s symmetrization prescription for non-ultralocal systems in the classicallimit. Further implications of this new formulation will be considered in the next section forthe AAF model.
AAF model
We now turn our attention to the
AAF model, which unlike the LL model, is not an ul-tralocal integrable model. The immediate difficulty in this case is the following. For theultralocal models, such as the LL model, the quantum Hamiltonian as well as the other con-served charges can be found from the fundamental Yang-Baxter relation (3.9). In contrast, fornon-ultralocal models, such as the AAF model, there does not exist such quantum relation,from which one can, for example, extract the quantum Hamiltonian. We emphasize, thatsuch quantum Hamiltonian should be written, according to our main result in the previoussection, in terms of the regularized fields (3.1), where the operator products are written interms of Sklyanin’s product (3.12), in order to avoid the problems associated with singulari-ties and to implement the principal value prescription from the beginning.The
AAF model (2.1) presents further complications in comparison to simpler models.Namely, the Dirac brackets for the fermionic fields have a very complex structure, extendingup to the eighth order in the fields and their derivatives. This immediately creates a compu-tational difficulty when dealing with regularized fields (3.1). An alternative approach wastaken in [19] where the so-called equivalence theorem for field theories [46] was proven forthe
AAF model. The equivalence theorem states (see [46] and the references therein) thatthe n -particle S -matrix of the theory, which for integrable models plays a central role andcan be used to reconstruct the spectrum of the model, does not change under an appropriatetransformation of the quantum fields in the action (2.1).One such change of fields, which was studied in details in [19], results in the theory forwhich the complicated non-linear Dirac brackets of the original theory (2.1) are reduced tothe canonical relations. This in turn allows the possibility of directly diagonalizing the re-sulting quantum Hamiltonian. The computational details, although much more involved inthis case, are essentially similar to that of the LL model discussed in the previous section.Here one has also to employ the principal value prescription due to discontinuities in thewave-function and its derivatives. As explained in the previous section, we can implementthe p.v. prescription by using the operator product (3.13) instead. Therefore, we can refor-mulate one of the main results of [19] in terms of the operator product (3.13) as follows: afterthe aforementioned field transformation, the quantum Hamiltonian which can be explicitly14iagonalized has the form: H = − i (cid:16) ψ † i γ i i ∂ ψ i − ∂ ψ † i γ i i ψ i (cid:17) + m ψ † i γ i i ψ i + i g (cid:16) ψ † i ψ † j γ i i γ j j ψ i ∂ ψ j − ψ † i ∂ ψ † j γ i i γ j j ψ i ψ j (cid:17) + g m (cid:16) ψ † i ψ † j γ i i γ j j ∂ ψ i ∂ ψ j + ∂ ψ † i ∂ ψ † j γ i i γ j j ψ i ψ j (cid:17) − (cid:18) g + g m (cid:19) (cid:16) ψ † i ψ † j ∂ ψ † k γ i i γ j j γ k k ψ i ψ j ∂ ψ k (cid:17) + i g m (cid:16) ψ † i ψ † j ∂ ψ † k γ i i γ j j γ k k ψ i ψ j ∂ ψ k − ψ † i ψ † j ∂ ψ † k γ i i γ j j γ k k ψ i ψ j ∂ ψ k (cid:17) − i g m (cid:16) ψ † i ψ † j ∂ ψ † k γ i i γ j j γ k k ψ i ∂ ψ j ∂ ψ k − ψ † i ∂ ψ † j ∂ ψ † k γ i i γ j j γ k k ψ i ψ j ∂ ψ k (cid:17) + g m (cid:16) ψ † i ψ † j ∂ ψ † k ∂ ψ † l γ i i γ j j γ k k γ l l ψ i ψ j ∂ ψ k ∂ ψ l (cid:17) . (4.1)We stress that the fermionic fields in (4.1) are the regularized fields as in (3.1), and the oper-ator product is understood to be the ◦ -product. We omitted here for simplicity the index F in fermionic fields, as well as the explicit ◦ -product symbol in (4.1).Some comments are in order. Firstly, in the quantum mechanical picture, the wave-functions (c.f. (3.11)) as well as their derivatives are not, as we discussed, continuous func-tions and satisfy a rather involved relation. A consequence of this relation is the fact thatthe quantum mechanical Hamiltonian is a self-adjoint operator. Secondly, it was shownthat in the process one recovers the correct S -matrix (2.2), which was previously found onlyfrom the perturbative calculations from (2.1). The direct diagonalization provides a non-perturbative confirmation of the S -matrix.Although this initial step of constructing the quantum Hamiltonian reproduces all theknown results, it is still not obvious how to derive such quantum charges using the methodsof integrable systems. The AAF model is a non-ultralocal model, and no standard procedureexists to construct the quantum Hamiltonian and other quantum charges. The result aboveshows however, that whatever the method, while working with continuous quantum sys-tems, it should automatically produce the quantum Hamiltonian (4.1), i.e., it should alreadycontain the ◦ -product from the beginning on a more fundamental level. In other words, ifthere exists some generalizations of Yang-Baxter relations for non-ultralocal systems, suchoperator relations should already be written in terms of the ◦ -product (3.13), as well as interms of the F -regularized fields (distributions).As a demonstration of this point of view and its consequences, we consider in the nextsection the simpler model of a free massive fermionic field, for which the Lax pair and thecorresponding algebra can be readily obtained from the AAF model. Although a free model,it is still a non-ultralocal model of the same order as
AAF with a rather involved Lax pair,on the example of which we show how to carry out the quantum calculations, without thetechnical complications of the full
AAF model. We show below the relation of the ◦ -productwith normal ordering for this case, and explain how to obtain the quantum Hamiltonian (4.1)reduced to the free case. Another consequence will be explored in the subsequent section,where we make a connection of the ◦ -product of operators with Maillet’s symmetrizationprocedure in the classical case. 15 Free massive fermion model
The massive free fermion model can be obtained as a consistent reduction of the
AAF model by setting the coupling constants g = g = AAF model. The explicit formulas and computational details are given in [17]and [18]. Here we only list the necessary results.
We start from the Lax pair obtained as a result of this reduction: L ( τ ) ( x ; λ ) = ˆ ξ ( τ ) ( x ; λ ) + ˆ ξ ( τ ) ( x ; λ ) σ + ˆ Λ ( − ) τ ( x ; λ ) σ + + ˆ Λ (+) τ ( x ; λ ) σ − , (5.1) L ( σ ) ( x ; λ ) = ˆ ξ ( σ ) ( x ; λ ) + ˆ ξ ( σ ) ( x ; λ ) σ + ˆ Λ ( − ) σ ( x ; λ ) σ + + ˆ Λ (+) σ ( x ; λ ) σ − . (5.2)The explicit form of the functions ˆ ξ ( σ , τ ) j ( x ; λ ) , j =
0, 1, and ˆ Λ ( ± ) σ , τ ( x ; λ ) is given in appendix A.The classical algebra of transition matrices (2.5) has been given in [18] for the infinite linecase, and has the same structure as that of the full AAF model (2.3). We stress that evenin the free fermion case, the coefficients A ( x , y ; λ , µ ) , B ( x , y ; λ , µ ) and C ( x , y ; λ , µ ) are non-vanishing and nonlinear functions of the fermionic fields. Nonetheless, the algebra for thereduced monodromy matrix: T ( λ ) = lim x → + ∞ y →− ∞ (cid:20) e − (cid:16) ˆ ξ ( σ ) ( λ ) σ x (cid:17) T ( x , y ; λ ) e (cid:16) ˆ ξ ( σ ) ( λ ) σ y (cid:17) (cid:21) , (5.3)can be easily obtained: (cid:8) T ( λ ) ⊗ , T ( µ ) (cid:9) M = u + ( λ , µ ) T ( λ ) ⊗ T ( µ ) − T ( λ ) ⊗ T ( µ ) u − ( λ , µ ) . (5.4)The matrices u + ( λ , µ ) and u − ( λ , µ ) in (5.4) have the following form: u + ( λ , µ ) = − p.v. a ( λ , µ ) − b ( λ , µ ) i π c ( λ ) δ ( λ − µ ) − i π c ( λ ) δ ( λ − µ ) b ( λ , µ )
00 0 0 p.v. a ( λ , µ ) , (5.5)and u − ( λ , µ ) = − p.v. a ( λ , µ ) − b ( λ , µ ) − i π c ( λ ) δ ( λ − µ ) i π c ( λ ) δ ( λ − µ ) b ( λ , µ )
00 0 0 p.v. a ( λ , µ ) , (5.6)with a ( λ , µ ) : = coth ( λ − µ ) sinh ( λ + µ ) k , (5.7) b ( λ , µ ) : = sinh ( ( λ + µ )) k , (5.8) c ( λ ) : = − k sinh ( λ ) . (5.9)One can use these formulas and the standard methods of integrable models to obtain theaction-angle variables as well as the classical conserved (local and non-local) quantities [18].16 .2 Quantum integrability For the quantum case, in the absence of a Yang-Baxter-like relation, we have first to care-fully define the quantum transition matrix. Therefore, we start from the standard classicaldefinition of the transition matrix T ( x , y ; λ ) via the following differential equations: ∂ x T ( x , y ; λ ) = L ( σ ) ( x ; λ ) T ( x , y ; λ ) , (5.10) ∂ y T ( x , y ; λ ) = − T ( x , y ; λ ) L ( σ ) ( y ; λ ) , (5.11)lim x → y T ( x , y ; λ ) = , (5.12)where the classical Lax operator L ( σ ) ( x ; λ ) has the form: L ( σ ) ( x ; λ ) = (cid:18) ˆ ξ ( σ ) ( x ; λ ) + ˆ ξ ( σ ) ( λ ) ˆ Λ ( − ) σ ( x ; λ ) ˆ Λ (+) σ ( x ; λ ) ˆ ξ ( σ ) ( x ; λ ) − ˆ ξ ( σ ) ( λ ) (cid:19) . (5.13)Equivalently, one can define the transition matrix T ( x , y ; λ ) via the corresponding integralequations: T ( x , y ; λ ) = E ( x − y , λ ) + ˆ xy T ( x , z ; λ ) U ( ) ( z , λ ) E ( z − y , λ ) dz , (5.14) T ( x , y ; λ ) = E ( x − y , λ ) + ˆ xy E ( x − z , λ ) U ( ) ( z , λ ) T ( z , y ; λ ) dz , (5.15)where in our case: E ( x , λ ) = e ˆ ξ ( σ ) ( λ ) σ x , (5.16) L ( σ ) ( x ; λ ) = U ( ) ( λ ) + U ( ) ( x , λ ) , (5.17) U ( λ ) = ˆ ξ ( σ ) ( λ ) σ , (5.18) U ( ) ( z , λ ) = (cid:18) ˆ ξ ( σ ) ( x , λ ) ˆ Λ ( − ) σ ( x , λ ) ˆ Λ (+) σ ( x , λ ) ˆ ξ ( σ ) ( x , λ ) (cid:19) . (5.19)Now we turn our attention to the quantum case. By considering the Lax operator (5.13)with the fields replaced by the corresponding quantum operators treated as operator val-ued distributions as in (3.1), we use the integral equations above to define iteratively thequantum transition matrix in each order of iteration. According to our prescription (see dis-cussion in section 3), we regularize the operator products by employing ◦ -product (3.12).Thus, our quantum Lax operator will have the same form as in (5.13), but with ˆ ξ ( σ ) ( x ; λ ) regularized via Sklyanin’s product. Namely, we take (see appendix A for notations):ˆ ξ ( σ ) ( x ) = J (cid:2) − χ ( x ) ◦ χ ′ ( x ) + χ ( x ) ◦ χ ′ ( x ) − χ ( x ) ◦ χ ′ ( x ) + χ ( x ) ◦ χ ′ ( x ) (cid:3) . (5.20)Since we are dealing with the free fermion theory, we pause here to analyze Sklyanin’s ◦ -product in details. The standard formulas (for x = y ) read: ψ α ( x ) ¯ ψ β ( y ) = : ψ α ( x ) ¯ ψ β ( y ) : − iS + αβ ( x − y ) , (5.21)¯ ψ β ( y ) ψ α ( x ) = : ¯ ψ β ( y ) ψ α ( x ) : − iS − αβ ( x − y ) . (5.22) Here and below we omit for simplicity the index F in fermionic fields. x → y is entirely contained in the func-tions S ± αβ ( x − y ) . Their explicit form is: S + αβ ( x ) = i ˆ dp π (cid:18) / p + m ω ( p ) (cid:19) e − ipx , (5.23) S − αβ ( x ) = i ˆ dp π (cid:18) / p − m ω ( p ) (cid:19) e ipx , (5.24)where ω ( p ) : = p ( p ) + m . Since the normal ordering is free of singularities, we obtainfrom (5.21), (5.22) and (3.12): ψ α ( x ) ◦ ¯ ψ β ( x ) = : ψ α ( x ) ¯ ψ β ( x ) : + Γ + αβ ( x ) , (5.25)¯ ψ β ( x ) ◦ ψ α ( x ) = : ¯ ψ β ( x ) ψ α ( x ) : + Γ − αβ ( x ) , (5.26)where the functions Γ ± αβ ( x ) take the form: Γ ± αβ ( x ) : = lim ∆ → ǫ ( − i )( ∆ − ǫ ) x + ∆ ˆ x − ∆ + ǫ du u − ǫ ˆ x − ∆ dv S ± αβ ( u − v ) + x + ∆ − ǫ ˆ x − ∆ du x + ∆ ˆ u + ǫ dv S ± αβ ( u − v ) .(5.27)Thus, in the free fermion case the difference between Sklyanin’s product and the normalordering is merely a function. Moreover, we are interested in the equal time relations, andin this case the relation: S + αβ ( x ) (cid:12)(cid:12)(cid:12) x + S − αβ ( x ) (cid:12)(cid:12)(cid:12) x = i γ δ ( x ) (5.28)implies that (for a fixed regularization parameter ǫ ): ψ α ( x ) ◦ ¯ ψ β ( x ) = − ¯ ψ β ( x ) ◦ ψ α ( x ) , (5.29)which also follows directly from the definition (3.12). Hence, in the free fermion case the ◦ -product enjoys the same properties as the normal ordering, and it is, as commented in sec-tion 3, a point splitting type regularization that symmetrically takes into account all pointsaround the singular region.It is not enough to regularize only the Lax operator (5.13). The integral equations them-selves must be regularized, due to the product of the operators at the same point under theintegral in (5.14) and (5.15). Thus, together with the regularization of the Lax operator (5.13),where now the formula for ˆ ξ ( σ ) ( x ; λ ) must be regularized as in (5.20), one has to replace theintegral equations (5.14) and (5.15) by their non-singular versions:ˆ T ( x , y ; λ ) = E ( x − y , λ ) + ˆ xy h ˆ T ( x , z ; λ ) ◦ U ( ) ( z , λ ) i E ( z − y , λ ) dz , (5.30)ˆ T ( x , y ; λ ) = E ( x − y , λ ) + ˆ xy E ( x − z , λ ) h U ( ) ( z , λ ) ◦ ˆ T ( z , y ; λ ) i dz . (5.31)18ince the above expressions are well-defined, one can now find the differential equationsatisfied by the quantum transition matrix ˆ T ( x , y , λ ) : ∂ x ˆ T ( x , y ; λ ) = L ( x ; λ ) ◦ ˆ T ( x , y ; λ ) , (5.32) ∂ y ˆ T ( x , y ; λ ) = − ˆ T ( x , y ; λ ) ◦ L ( y ; λ ) . (5.33)In order to check these relations one has to verify the Leibniz rule for Sklyanin’s product.This is done in appendix B.Having defined the quantum transition matrix ˆ T ( x , y , λ ) , we can now solve the integralequations (5.30) and (5.31) iteratively. Denoting:ˆ T ( x , y ; λ ) : = (cid:18) ˆ t ( x , y ; λ ) ˆ t ( x , y ; λ ) ˆ t ( x , y ; λ ) ˆ t ( x , y ; λ ) (cid:19) , (5.34)we find from (5.30) and (5.31) that in the second iteration the components ˆ t i ; i =
1, . . . , 4 havethe form: e − ˆ ξ ( σ ) ( λ )( x − y ) ˆ t ( x , y ; λ ) = + ˆ xy dz ˆ ξ ( σ ) ( z ; λ ) + ˆ xy dz ˆ xz du ˆ ξ ( σ ) ( u ; λ ) ◦ ˆ ξ ( σ ) ( z ; λ )+ ˆ xy dz e ξ ( σ ) ( λ ) z ˆ xz du e − ξ ( σ ) ( λ ) u ˆ Λ ( − ) σ ( u ; λ ) ◦ ˆ Λ (+) σ ( z ; λ ) , (5.35) e − ˆ ξ ( σ ) ( λ )( x + y ) ˆ t ( x , y ; λ ) = ˆ xy dz e − ξ ( σ ) ( λ ) z ˆ Λ ( − ) σ ( z ; λ ) + ˆ xy dz e − ξ ( σ ) ( λ ) z ˆ xz du ˆ ξ ( σ ) ( u ; λ ) ◦ ˆ Λ ( − ) σ ( z ; λ )+ ˆ xy dz ˆ xz du e − ξ ( σ ) ( λ ) u ˆ Λ ( − ) σ ( u ; λ ) ◦ ˆ ξ ( σ ) ( z ; λ ) , (5.36) e ˆ ξ ( σ ) ( λ )( x + y ) ˆ t ( x , y ; λ ) = ˆ xy dz e ξ ( σ ) ( λ ) z ˆ Λ (+) σ ( z ; λ ) + ˆ xy dz e ξ ( σ ) ( λ ) z ˆ xz du ˆ ξ ( σ ) ( u ; λ ) ◦ ˆ Λ (+) σ ( z ; λ )+ ˆ xy dz ˆ xz du e ξ ( σ ) ( λ ) u ˆ Λ (+) σ ( u ; λ ) ◦ ˆ ξ ( σ ) ( z ; λ ) , (5.37) e ˆ ξ ( σ ) ( λ )( x − y ) ˆ t ( x , y ; λ ) = + ˆ xy dz ˆ ξ ( σ ) ( z ; λ ) + ˆ xy dz ˆ xz du ˆ ξ ( σ ) ( u ; λ ) ◦ ˆ ξ ( σ ) ( z ; λ )+ ˆ xy dz e − ξ ( σ ) ( λ ) z ˆ xz du e ξ ( σ ) ( λ ) u ˆ Λ (+) σ ( u ; λ ) ◦ ˆ Λ ( − ) σ ( z ; λ ) . (5.38)These are well-defined expressions as long as the regularization parameter ǫ is a fixed num-ber different from zero. Thus, one can easily continue this iterative process and obtain well-defined components of the quantum transition matrix for any iteration order.Using that the fields vanish at infinity, i.e., χ i ( z ) z →± ∞ −−−→ i =
1, . . . , 4, one can readilyverify that the component ˆ t ( λ ) is a conserved quantity, expansion of which has the form:ˆ t ( λ ) = − i ( λ ) ˆ + ∞ − ∞ dz P ( z ) + i k sinh ( λ ) cosh ( λ ) ˆ + ∞ − ∞ dz Q ( z ) − i k sinh ( λ ) ˆ + ∞ − ∞ dz H ( z ) + sinh ( λ ) N ( λ ) + O ( χ ) (5.39)19nd contains the quantum Hamiltonian H for free fermion model, H ( z ) = J − k J (cid:2) χ ( z ) ◦ χ ′ ( z ) − χ ′ ( z ) ◦ χ ( z ) − χ ( z ) ◦ χ ′ ( z ) + χ ′ ( z ) ◦ χ ( z ) (cid:3) + χ ( z ) ◦ χ ( z ) − χ ( z ) ◦ χ ( z ) . (5.40)The other terms in the expansion (5.39) correspond to the momentum and the charge opera-tors, P ( z ) = − iJ (cid:2) χ ( z ) ◦ χ ′ ( z ) + χ ( z ) ◦ χ ′ ( z ) (cid:3) , (5.41) Q ( z ) = − χ ( z ) ◦ χ ( z ) − χ ( z ) ◦ χ ( z ) , (5.42)as well as conserved non-local charges, N ( λ ) = k ˆ + ∞ − ∞ dz [ χ ( z ) ◦ χ ( z ) − χ ( z ) ◦ χ ( z )]+ Jk ˆ + ∞ − ∞ dz e ξ ( σ ) ( λ ) z ˆ + ∞ z du e − ξ ( σ ) ( λ ) u (cid:2) l ( λ ) χ ( u ) ◦ χ ( z ) + l ( λ ) χ ( u ) ◦ χ ( z )+ il ( λ ) l ( λ ) ( χ ( u ) ◦ χ ( z ) − χ ( u ) ◦ χ ( z ))] . (5.43)The expression (5.40) coincides with the free part of the quantum Hamiltonian for the AAF model (4.1), with operator product already regularized via Sklyanin’s product H = − i (cid:16) ψ † i γ i i ◦ ∂ ψ i − ∂ ψ † i γ i i ◦ ψ i (cid:17) + m ψ † i γ i i ◦ ψ i . (5.44)Thus, we have shown, on the example of the free fermion model how to obtain the quantumconserved charges, and in particular, the quantum Hamiltonian, from the quantum transi-tion matrix with products regularized via Sklyanin product.We conclude this section with a remark concerning the interacting case. One can, inprinciple, repeat the above calculations for the interacting theory, i.e., the full AAF model.Starting from the integral equations (5.30) and (5.31), the quantum transition matrix is a well-defined object from the beginning, and one does not have to worry about singular expres-sions. For interacting fields, however, we do not have a simple relation between Sklyanin’sproduct and the normal ordering, as it is the case for the free fields, cf. (5.25) and (5.26). Inthis case, the normal ordering does not solve the singularity problem, and one has to usethe normal product introduced by Zimmermann (for a review see [47]) instead. The pointsplitting regularization in the interacting theory can be expressed via the operator productexpansion [47–50] as: ψ ( x ) ¯ ψ ( y ) = ∞ ∑ n = C n ( x − y ) O ( x ) , (5.45)where O ( x ) correspond to composite local operators defined via normal product, and thesingularities are exhibited in the functions C n ( x − y ) when taking the limit x → y .Some comments are in order. First, the expansion in (5.45) can be strictly proven for(BPHZ) renormalizable theories in each order of the perturbation theory. Although there isno strict proof of the renormalizability of the AAF model, it has been proposed to be such a20heory in [1], and the recently obtained explicit diagonalization of the quantum Hamiltonian(4.1) makes this proposal more plausible. Then proceeding exactly as in the free case, bysmearing the product ψ ( x ) ¯ ψ ( y ) over the entire region around the singular strip, we canwrite: ψ ( x ) ◦ ¯ ψ ( x ) = ∞ ∑ n = ξ n ( ǫ ) O ( x ) , (5.46)where ǫ is the regularization parameter entering into the definition of ◦ -product (3.12), and ξ n ( ǫ ) are functions obtained from C n ( x − y ) , which diverge in the limit ǫ →
0. Thus, inperturbation theory, one can naturally relate Sklyanin’s product to renormalized compositeoperators O ( x ) , which can be written in terms of Zimmermann’s normal products and foundperturbatively in each order. It would be interesting to carry out such explicit analysis forintegrable models, as in this case the diagrammatic analysis significantly simplifies due topowerful non-renormalization theorems (see for details [20, 21, 27, 51, 52]). We are finally in a position to conjecture the form of the quantum algebra of transitionmatrices for non-ultralocal systems directly in the continuous case. As we discussed previ-ously, one of the main difficulties is to formulate a regularization which consistently reducesto the classical algebra (2.5) in the classical limit. Indeed, it is not obvious how a commu-tator between two operators, or in general n -nested commutators, restores Maillet’s sym-metrization prescription (2.8) and (2.9) upon taking the classical limit. Bellow, we considerthe classical limit of a quantum commutator regularized in terms of Sklyanin’s product andits relation to the Maillet bracket to propose the form of the quantum algebra of transitionmatrices to models such as the AAF model or the free fermion model.To obtain the aforementioned relation, we consider here, following [7, 15], the simplercase of the general classical algebra (2.5): { T ( x , y ; λ ) , T ( x , y ; µ ) } M = a ( λ , µ ) T ( x , y ; λ ) T ( x , y ; µ ) − T ( x , y ; λ ) T ( x , y ; µ ) d ( λ , µ ) , { T ( x , y ; λ ) , T ( y , z ; µ ) } M = T ( x , y ; λ ) b ( λ , µ ) T ( y , z ; µ ) , (6.1)and recall the lattice algebra for the corresponding quantum case proposed by Freidel andMaillet in [7, 15]. It has the following form:ˆ A T ( n ) T ( n ) = T ( n ) T ( n ) ˆ D , (6.2) T ( n ) T ( n + ) = T ( n + ) ˆ C T ( n ) , (6.3) h T ( n ) , T ( m ) i =
0, for | n − m | >
1. (6.4)Using the quasi-classical expansionsˆ A = + i ¯ ha + . . . ,21or all matrices in (6.2)-(6.4), one obtains the classical lattice algebra: n T ( n ) ( λ ) , T ( n ) ( µ ) o = a ( λ , µ ) T ( n ) ( λ ) T ( n ) ( µ ) − T ( n ) ( λ ) T ( n ) ( µ ) d ( λ , µ ) , (6.5) n T ( n ) ( λ ) , T ( n + ) ( µ ) o = − T ( n + ) ( µ ) c ( λ , µ ) T ( n ) ( λ ) (6.6) n T ( n ) ( λ ) , T ( m ) ( µ ) o =
0, for | n − m | >
1, (6.7)where T ( n ) ( λ ) ≡ T ( x n + , x n ; λ ) is defined so as to make a connection with the continuous al-gebra (6.1). In passing from the quantum algebra (6.2)-(6.4) to the classical one (6.5)-(6.7), oneclearly obtains the usual Poisson brackets, which are not symmetrized according to Maillet’sprescription (2.10), as the lattice spacing already regulates all products. Moreover, the Ja-cobi identity for the classical lattice algebra (6.5)-(6.7) can be understood as a consequenceof the consistency conditions (Yang-Baxter relations) satisfied by the quantum algebra (6.2)-(6.4) [7, 15]. Nonetheless, there remains the problem of restoring the symmetrization pre-scription upon removing the lattice regularization.To solve this problem, we use the results explained in the previous sections to bypass thelattice reformulation of the quantum theory in favor of regularizing the continuous theorywith Sklyanin’s product (3.12) and (3.13). The main idea is to formulate the quantum algebraof transition matrices T ( x , y ; λ ) so that the singular products are replaced by well-defined ◦ -products.First, we make a key observation and show that the commutator of two operator-valuedfunctions regularized with Sklyanin’s product goes to the symmetrized Poisson brackets(2.10) in the classical limit. In this way, Maillet’s symmetrization prescription appears natu-rally in the classical continuous theory from simply regularizing the singular operator prod-ucts in the quantum case. Indeed, writing explicitly the commutator between two operator-valued functions ˆ A ( x ) and ˆ B ( x ) in terms of the definition (3.12), we obtain: (cid:2) ˆ A ( x ) ◦ , ˆ B ( x ) (cid:3) = lim ∆ / → ǫ ( ∆ − ǫ ) ¨ ∆ S d ζ d ξ (cid:2) ˆ A ( ζ ) , ˆ B ( ξ ) (cid:3) + ¨ ∆ S d ζ d ξ (cid:2) ˆ A ( ζ ) , ˆ B ( ξ ) (cid:3) . (6.8)In the classical limit, denoted by CL bellow, equation (6.8) becomes: (cid:2) ˆ A ( x ) ◦ , ˆ B ( x ) (cid:3) CL = lim ∆ / → ǫ ( ∆ − ǫ ) ¨ ∆ S d ζ d ξ { A ( ζ ) , B ( ξ ) } + ¨ ∆ S d ζ d ξ { A ( ζ ) , B ( ξ ) } , (6.9)where A ( ζ ) and B ( ξ ) are already the corresponding classical functions, and { A ( ζ ) , B ( ξ ) } inthe right-hand side is the usual Poisson bracket. It is then clear, by invoking the mean valuetheorem in the regions ∆ S and ∆ S , where the integrands are smooth functions, that in thelimit ∆ → ǫ , followed by the limit ǫ →
0, the right-hand side of (6.9) reduces to: (cid:2) ˆ A ( x ) ◦ , ˆ B ( x ) (cid:3) CL =
12 lim ǫ → ( { A ( x + ǫ ) , B ( x − ǫ ) } + { A ( x − ǫ ) , B ( x + ǫ ) } ) . (6.10)Finally, comparing this formula with Maillet’s definition of the symmetrized Poisson bracket(2.10) we conclude that: (cid:2) ˆ A ( x ) ◦ , ˆ B ( x ) (cid:3) CL = { A ( x ) , B ( x ) } M . (6.11)22his simple observation shows the connection between Sklyanin’s product in the quan-tum case, and Maillet’s ad hoc construction of symmetrized Poisson brackets. Namely, theclassical limit of the commutator, regularized via Sklyanin’s product, reproduces preciselythe symmetrized Poisson bracket. Of course, if the integrable system is ultralocal, then bothterms in (6.11) coincide, and Maillet’s symmetrization procedure reduces to the usual Pois-son brackets. This consideration can also be easily generalized, and the n -nested Poissonbrackets (2.8) can be similarly obtained from the general Sklyanin product (3.13).To conclude this section, we speculate on the form of the quantum continuous algebra.Namely, we propose the following quantum algebra, naturally reproducing the classicalMaillet algebra (6.1):ˆ A T ( x , y ; λ ) ◦ T ( x , y ; µ ) = T ( x , y ; µ ) ◦ T ( x , y ; λ ) ˆ D , (6.12) T ( y , z ; λ ) ◦ T ( x , y ; µ ) = T ( x , y ; µ ) ◦ ˆ C T ( y , z ; λ ) . (6.13)These relations are well defined due to Sklyanin’s product, and upon using as before thequasi-classical expansion for the matrices ( A , B , C , D ) , one obtains: [ T ( x , y ; λ ) ◦ , T ( x , y ; µ )] = − ia ( λ , µ ) T ( x , y ; λ ) ◦ T ( x , y ; µ ) (6.14) + iT ( x , y ; µ ) ◦ T ( x , y ; λ ) d ( λ , µ ) , [ T ( y , z ; λ ) ◦ , T ( x , y ; µ )] = iT ( x , y ; µ ) ◦ c ( λ , µ ) T ( y , z ; λ ) . (6.15)Then, invoking the relation between Maillet’s symmetrized brackets and the quantum com-mutator regularized by means of Sklyanin’s product (6.11), we can accordingly concludethat, in the classical limit, the quantum algebra given by (6.14) and (6.15) reduces to theclassical algebra (6.1). Moreover, the consistency conditions for the classical algebra (6.1) de-rived by Freidel and Maillet in [7, 15] from the classical Jacobi identity follow similarly fromthe corresponding quantum Jacobi identity. We refer the reader to appendix C for a detaileddiscussion of the Jacobi identity for the quantum algebra (6.14) and (6.15). In this paper we have considered the quantization problem of continuous non-ultralocalintegrable models, such as the
AAF model, without resorting to any lattice discretization.To do so, we have shown that it is necessary to treat the quantum fields as distributions, andemploy a regularized product of operators - Sklyanin’s product, in order to avoid singulari-ties in diagonalization procedure, as well as to correctly reproduce the S -matrix known fromperturbative calculations. Sklyanin’s product corresponds essentially to a “smeared” prod-uct of operator-valued functions over a small neighborhood around the singular region, andit is naturally related, as discussed in the text, with renormalized local composite operators,defined via Zimmermann’s normal products. As an example of this procedure, we were ableto extract the quantum Hamiltonian as well as other conserved charges for the free fermionmodel directly from the quantum trace identities. In particular, the quantum Hamiltonianthus obtained coincides with previous results [19]. In addition, we demonstrated that thequantum algebra of transition matrices, written in terms of the ◦ -product consistently re-produces Maillet’s symmetrization prescription for non-ultralocal integrable systems in theclassical limit. 23e outline here the possible future directions. As it is well known, the key obstaclein formulating a well-defined quantum algebra for transition matrices, corresponding to theclassical expressions (6.1), is Schwartz’s theorem on the impossibility of defining a product oftwo distributions with natural properties. Although it is possible to define a product of dis-tributions in some exceptional cases, for instance when their singular supports are disjoint,in general one must look for extensions of Schwartz’s distribution theory. Out of several dif-ferent approaches to evade Schwartz’s impossibility theorem, two look specially promising:the microlocal analysis based on the concept of wavefront sets (for a review see [55] and ref-erences therein) and Colombeau algebras [56, 57]. The former has been used in the contextof quantum field theory, making it possible to define the product of distributions. However,even in this approach some interesting products of distributions, e.g., the powers of the Diracdelta distribution, remain ill-defined. As for the latter, the space of Schwartz distributionsis embedded into an associative algebra which satisfies the Leibniz’s rule, nevertheless theassociation between a distribution and an element of the Colombeau algebra is not alwaysunique. Thus, it is still an open and interesting question whether any of these approacheswill prove to be successful in the context of the quantization of continuous non-ultralocal in-tegrable systems. We also mention here that another compelling possibility to approach thisproblem lies within Sato’s theory of hyperfunctions [58, 59], which also contains Schwartz’sdistribution theory and relies on the boundary behavior of analytic functions.Furthermore, as we discussed earlier at the end of section 3, Sklyanin’s product is essen-tially a point splitting procedure taken uniformly over the entire region around the singular-ity strip. Thus, instead of taking the limit x → y symmetrically at the end of the calculation,as it is in the standard case of the point splitting procedure for the axial fermionic current(3.33), we simply take into account all the points around the singularity equally from thebeginning. In principle, for a renormalizable theory the quantum field equations can bewritten in terms of the Zimmermann’s normal product in all orders. It is, therefore, possi-ble to define the quantum Lax operator and the quantum transition matrix only in terms ofthese normal products. However, this is valid only for BPHZ renormalizable theories, andfor models such as the
AAF model, there is no proof of such renormalizability. Nevertheless,it is an interesting problem to formulate quantum integrable models (the Lax operator, thetransition matrices, etc.) in terms of only Zimmermann’s normal products. In general, in theabsence of renormalizability of the theory, we can only use the generic Sklyanin’s product,which simply homogeneously removes the singular region. Nonetheless, it is not immedi-ately clear if Sklyanin’s product is associative, presenting therefore an interesting problemto consider in the context of Schwarz’s impossibility theorem for defining an associativeproduct of distributions. Finally, a related issue that will be interesting to consider regards the ultralocalization ofsecond order non-ultralocal algebras such as the ones appearing in the
AAF model. Eventhough, a purely classical problem, it is the first step in the usual approach to non-ultralocalmodels, and it should, therefore, provide also a good insight in the quantization of second or-der non-ultralocal models. The natural starting point should be to consider a generalizationof either the generalized FR ultralocalization technique identified by [9, 10] in the context ofsigma models or the procedure proposed by [26] for the WZNW model to accommodate forthe second derivative of the delta function appearing in (2.3). Moreover, it may also help us For an overview, see, for example, [53, 54]. For a recent discussion of the problem of associativity in the context of operator product expansion, see [60].
Acknowledgments
G.W. would like to thank D. Guariento for useful discussions. The work of A.M. is par-tially supported by CAPES.
AppendicesA Notations
In this paper we consider results derived from two different but equivalent descriptionsof the
AAF model. The goal of this appendix is, therefore, to fix our notations and show howthese two descriptions are related. The action (2.1) S = ˆ dy ˆ J dy h i ¯ ψ∂ / ψ − m ¯ ψψ + g m ǫ αβ (cid:0) ¯ ψ∂ α ψ ¯ ψ γ ∂ β ψ − ∂ α ¯ ψψ ∂ β ¯ ψ γ ψ (cid:1) −− g m ǫ αβ ( ¯ ψψ ) ∂ α ¯ ψ γ ∂ β ψ i , (A.1)was originally used in the papers [20, 21] to study the AAF model from the perspective ofperturbative quantum field theory. In (A.1) g and g stand for the coupling constants of themodel and the mass m is related to the ’t Hooft coupling λ ′ via: m = π √ λ ′ . (A.2)Here, the following representation for the Dirac matrices is employed γ = (cid:18) (cid:19) , γ = (cid:18) −
11 0 (cid:19) , γ = γ γ . (A.3)The action (A.1) can be deduced from the original lagrangian proposed in [1] L = − J − iJ (cid:0) ¯ χρ ∂ χ − ∂ ¯ χρ χ (cid:1) + i κ (cid:16) ¯ χρ ∂ χ − ∂ ¯ χρ χ (cid:17) + J ¯ χχ + κ g ǫ αβ (cid:0) ¯ χ∂ α χ ¯ χρ ∂ β χ − ∂ α ¯ χχ ∂ β ¯ χρ χ (cid:1) − κ g ǫ αβ ( ¯ χχ ) ∂ α ¯ χρ ∂ β χ , (A.4)where χ = ψ − ψ √ χ = i ψ + ψ √ ρ = (cid:18) − (cid:19) , ρ = (cid:18) ii (cid:19) , ρ = ρ ρ (A.6)with κ = √ λ ′ . 25he constant J in (A.4) corresponds to the total angular momentum of the string in S .More importantly, the two lagrangians (A.1) and (A.4) differ by some overall minus signthat was introduced in [21] in order to ensure a positive definite mode expansion. For allthe details and a thorough derivation of (A.1) from (A.4), we refer the reader to the originalpapers [1, 21].The equations of motion for the free massive fermion following from (A.4) with g = g = L ( τ ) ( x ; µ ) = ˆ ξ ( τ ) ( x ; µ ) + ˆ ξ ( τ ) ( x ; µ ) σ + ˆ Λ ( − ) τ ( x ; µ ) σ + + ˆ Λ (+) τ ( x ; µ ) σ − , (A.7) L ( σ ) ( x ; µ ) = ˆ ξ ( σ ) ( x ; µ ) + ˆ ξ ( σ ) ( x ; µ ) σ + ˆ Λ ( − ) σ ( x ; µ ) σ + + ˆ Λ (+) σ ( x ; µ ) σ − , (A.8)where the functions ˆ ξ ( σ , τ ) j ( x ; µ ) , j =
0, 1, and ˆ Λ ( ± ) σ , τ ( x ; µ ) have the following explicit form:ˆ ξ ( σ ) = J (cid:2) − χ χ ′ + χ χ ′ − χ χ ′ + χ χ ′ (cid:3) , (A.9)ˆ ξ ( σ ) = il J k , (A.10)ˆ Λ ( − ) σ = √ J (cid:2) − l χ ′ − il χ ′ (cid:3) , (A.11)ˆ Λ (+) σ = √ J (cid:2) − l χ ′ + il χ ′ (cid:3) , (A.12)and: ˆ ξ ( τ ) = i J [ χ χ + χ χ ] + J [ − χ ˙ χ − χ ˙ χ + χ ˙ χ + χ ˙ χ ] , (A.13)ˆ ξ ( τ ) = − il Λ ( − ) τ = − i √ J [ l χ − il χ ] − √ J [ l ˙ χ + il ˙ χ ] , (A.15)ˆ Λ (+) τ = i √ J [ l χ + il χ ] − √ J [ l ˙ χ − il ˙ χ ] . (A.16)Here we have denoted: χ = (cid:18) χ χ (cid:19) , χ ≡ χ ∗ , χ ≡ χ ∗ . (A.17)The dependence on the spectral parameter µ is encoded in the functions l i [1, 61, 62]: l = l = cosh ( µ ) , l = − sinh ( µ ) , l = cosh ( µ ) , l = sinh ( µ ) . (A.18)The constant k = √ λ ′ . B Leibnitz rule for Sklyanin’s product
In this appendix we show that the Leibnitz rule is valid for Sklyanin’s product. Namely,for two operator-valued functions A ( x ) and B ( x ) , one has: ∂ x [ A ( x ) ◦ B ( x )] = ∂ x [ A ( x )] ◦ B ( x ) + A ( x ) ◦ ∂ x [ B ( x )] . (B.1)26o prove this formula, we write Sklyanin’s product (3.12) in the following form: A ( x ) ◦ B ( x ) = lim ∆ → ǫ ( ∆ − ǫ ) x + ∆ ˆ x − ∆ + ǫ du A ( u ) u − ǫ ˆ x − ∆ dv B ( v ) + x + ∆ − ǫ ˆ x − ∆ du A ( u ) x + ∆ ˆ u + ǫ dv B ( v ) .(B.2)Then, it is easy to show that: ∂ x [ A ( x ) ◦ B ( x )] = lim ∆ → ǫ ( ∆ − ǫ ) A ( x + ∆ / ) x + ∆ − ǫ ˆ x − ∆ dv B ( v ) − A ( x − ∆ / ) x + ∆ ˆ x − ∆ + ǫ dv B ( v )+ x + ∆ − ǫ ˆ x − ∆ du A ( u ) B ( x + ∆ / ) − x + ∆ ˆ x − ∆ + ǫ du A ( u ) B ( x − ∆ / ) .(B.3)Next, we find for the first term in the right hand side of (B.1): ∂ x [ A ( x )] ◦ B ( x ) = lim ∆ → ǫ ( ∆ − ǫ ) x + ∆ ˆ x − ∆ + ǫ du ∂ u A ( u ) u − ǫ ˆ x − ∆ dv B ( v ) + x + ∆ − ǫ ˆ x − ∆ du ∂ u A ( u ) x + ∆ ˆ u + ǫ dv B ( v ) = lim ∆ → ǫ ( ∆ − ǫ ) A ( x + ∆ / ) x + ∆ − ǫ ˆ x − ∆ dv B ( v ) − A ( x − ∆ / ) x + ∆ ˆ x − ∆ + ǫ dv B ( v ) − x + ∆ ˆ x − ∆ + ǫ du A ( u ) B ( u − ǫ ) + x + ∆ − ǫ ˆ x − ∆ du A ( u ) B ( u + ǫ ) . (B.4)Similarly, for the second term in the right hand side of (B.1), we find: A ( x ) ◦ ∂ x [ B ( x )] = lim ∆ → ǫ ( ∆ − ǫ ) x + ∆ ˆ x − ∆ + ǫ du A ( u ) u − ǫ ˆ x − ∆ dv ∂ v B ( v ) + x + ∆ − ǫ ˆ x − ∆ du A ( u ) x + ∆ ˆ u + ǫ dv ∂ v B ( v ) = lim ∆ → ǫ ( ∆ − ǫ ) − x + ∆ ˆ x − ∆ + ǫ duA ( u ) B ( x − ∆ / ) + x + ∆ − ǫ ˆ x − ∆ duA ( u ) B ( x + ∆ / )+ x + ∆ ˆ x − ∆ + ǫ du A ( u ) B ( u − ǫ ) − x + ∆ − ǫ ˆ x − ∆ du A ( u ) B ( u + ǫ ) . (B.5)Finally, summing the terms in (B.4) and (B.5) we obtain (B.3), thus verifying the Lebnitz rule(B.1). 27 Jacobi identity
In this appendix, we consider the Jacobi identity for the quantum algebra (6.14) and(6.15), and address its relation to the consistency conditions for the classical algebra (6.1)derived in [7, 15].The Jacobi identity: (cid:2) T ( u , u ′ ; λ ) , (cid:2) T ( v , v ′ ; µ ) , T ( w , w ′ ; ρ ) (cid:3)(cid:3) + P P (cid:2) T ( w , w ′ ; ρ ) , (cid:2) T ( u , u ′ ; λ ) , T ( v , v ′ ; µ ) (cid:3)(cid:3) P P + P P (cid:2) T ( v , v ′ ; µ ) , (cid:2) T ( w , w ′ ; ρ ) , T ( u , u ′ ; λ ) (cid:3)(cid:3) P P =
0, (C.1)with P denoting the permutation operator acting on the auxiliary spaces, is clearly well-defined with respect to the usual operator product for the case where all points ( u , u ′ , v , v ′ , w , w ′ ) are different. On the other hand, if some of the points ( u , u ′ , v , v ′ , w , w ′ ) coincide, it involvesproducts of operators at the same point, and is, therefore, a singular expression. Hence, toformulate a well-defined Jacobi identity for the case of possibly coinciding points, one needsto regularize such products.To this end, we can use Sklyanin’s product (3.13) and the following property: [ A ( x ) ◦ , [ A ( x ) ◦ , A ( x )]] = lim ∆ / → ǫ ∆ V ∑ i = ˆ ∆ V i d ζ d ζ d ζ [ A ( ζ ) , [ A ( ζ ) , A ( ζ )]] , (C.2)to extend the well-defined expression (C.1) to the case where some arbitrary subset of pointsmay coincide. Thus, we obtain a formula valid for an arbitrary set of points ( x , y , x , y , x , y ) : [ T ( x , y ; λ ) ◦ , [ T ( x , y ; µ ) ◦ , T ( x , y ; ρ )]]+ P P [ T ( x , y ; ρ ) ◦ , [ T ( x , y ; λ ) ◦ , T ( x , y ; µ )]] P P + P P [ T ( x , y ; µ ) ◦ , [ T ( x , y ; ρ ) ◦ , T ( x , y ; λ )]] P P =
0. (C.3)It is clear that when all points ( x , y , x , y , x , y ) are different, the expression (C.3) triviallyreduces to (C.1). We emphasize that, although not explicitly indicated, the formula (C.3) de-pends on the regularization parameter ǫ (see the definitions (3.12) and (3.13)) and, therefore,is a well-defined expression.The general conditions imposed by the Jacobi identity (C.3) on the quantum algebra oftransition matrices (6.14) and (6.15) are not very enlightening. However, as we show inthe following, they lead in the classical limit to the same consistency conditions obtainedin [7,15]. For simplicity, we restrict the analysis to the simpler case [7] involving only bosonicfields, and for which the matrices a , d and b = c encoding the quantum algebra (6.14)and (6.15) depend only on the spectral parameters. In this case, using the quantum algebra(6.14) and (6.15) to evaluate the Jacobi identity (C.3) for all possible combinations of intervals,i.e., equal, adjacent and mixed, we can easily derive in the classical limit the following Yang-Baxter-like constraints: [ a ( λ , µ ) , a ( λ , ν )] + [ a ( λ , µ ) , a ( µ , ν )] + [ a ( λ , ν ) , a ( µ , ν )] =
0, (C.4) [ d ( λ , µ ) , d ( λ , ν )] + [ d ( λ , µ ) , d ( µ , ν )] + [ d ( λ , ν ) , d ( µ , ν )] =
0, (C.5) [ b ( λ , µ ) , d ( λ , ν )] + [ b ( ν , µ ) , d ( λ , ν )] + [ b ( ν , µ ) , b ( λ , µ )] =
0, (C.6) [ a ( ν , µ ) , c ( µ , λ )] + [ a ( ν , µ ) , c ( ν , λ )] + [ c ( ν , λ ) , c ( µ , λ )] =
0. (C.7)28oreover, we also obtain the classical relation b ( λ , µ ) = c ( µ , λ ) and the antisymmetryof the parameters a ( λ , µ ) and d ( λ , µ ) in the description of the classical algebra under thepermutation of the auxiliary spaces corresponding to the spectral parameters λ and µ .Finally, we note that two properties enjoyed by Sklyanin’s product render the aforemen-tioned calculations a mere repetition of the computation originally performed in [7]. Namely,the commutators of operator-valued functions endowed with Sklyanin’s product (3.13) sat-isfy the following standard relations: [ A ( x ) ◦ , A ( x ) ◦ A ( x )] = A ( x ) ◦ [ A ( x ) ◦ , A ( x )] + [ A ( x ) ◦ , A ( x )] ◦ A ( x ) , (C.8) [ A ( x ) ◦ , [ A ( x ) ◦ , A ( x )]] = [ A ( x ) ◦ , α A ( x ) ◦ A ( x ) + A ( x ) ◦ A ( x ) β ] , (C.9)where (C.9) holds provided the commutator of two operator-valued functions is of the form: [ A ( x ) ◦ , A ( x )] = α A ( x ) ◦ A ( x ) + A ( x ) ◦ A ( x ) β ,with α , β being arbitrary constants. Therefore, we omit tedious computational details. References [1] L. F. Alday, G. Arutyunov, and S. Frolov, “New integrable system of 2dim fermionsfrom strings on
AdS × S ,” JHEP (2006) 078, arXiv:hep-th/0508140 .[2] G. Arutyunov and S. Frolov, “Uniform light-cone gauge for strings in
AdS × S :Solving su ( | ) sector,” JHEP (2006) 055, arXiv:hep-th/0510208 .[3] J. M. Maillet, “New integrable canonical structures in two-dimensional models,”
Nucl.Phys.
B269 (1986) 54.[4] L. D. Faddeev and N. Y. Reshetikhin, “Integrability of the principal chiral field modelin (1+1) - dimension,”
Ann. Phys. (1986) 227.[5] A. Alekseev, L. D. Faddeev, and M. Semenov-Tian-Shansky, “Hidden quantum groupsinside Kac-Moody algebra,”
Commun. Math. Phys. (1992) 335–345.[6] A. Yu. Alekseev, L. D. Faddeev, J. Frohlich, and V. Schomerus, “Representation theoryof lattice current algebras,”
Commun. Math. Phys. (1998) 31–60, arXiv:q-alg/9604017 [q-alg] .[7] L. Freidel and J. Maillet, “On classical and quantum integrable field theories associatedto Kac-Moody current algebras,”
Phys. Lett.
B263 (1991) 403–410.[8] N. Beisert et al. , “Review of AdS/CFT Integrability: An Overview,”
Lett.Math.Phys. (2012) 3–32, arXiv:1012.3982 [hep-th] .[9] F. Delduc, M. Magro, and B. Vicedo, “Alleviating the non-ultralocality of coset sigmamodels through a generalized Faddeev-Reshetikhin procedure,” JHEP (2012) 019, arXiv:1204.0766 [hep-th] .[10] F. Delduc, M. Magro, and B. Vicedo, “Alleviating the non-ultralocality of the
AdS × S superstring,” arXiv:1206.6050 [hep-th] .2911] N. Dorey and B. Vicedo, “A Symplectic Structure for String Theory on IntegrableBackgrounds,” JHEP (2007) 045, arXiv:hep-th/0606287 [hep-th] .[12] R. Benichou, “Fusion of line operators in conformal sigma-models on supergroups,and the Hirota equation,”
JHEP (2011) 066, arXiv:1011.3158 [hep-th] .[13] R. Benichou, “First-principles derivation of the AdS/CFT Y-systems,”
JHEP (2011) 112, arXiv:1108.4927 [hep-th] .[14] R. Benichou, “The Hirota equation for string theory in
AdS × S from the fusion ofline operators,” Fortsch.Phys. (2012) 896–900, arXiv:1202.0084 [hep-th] .[15] L. Freidel and J. Maillet, “Quadratic algebras and integrable systems,” Phys.Lett.
B262 (1991) 278–284.[16] F. Delduc, M. Magro, and B. Vicedo, “A lattice Poisson algebra for the Pohlmeyerreduction of the
AdS × S superstring,” Phys. Lett.
B713 (2012) 347–349, arXiv:1204.2531 [hep-th] .[17] A. Melikyan and G. Weber, “The r-matrix of the Alday-Arutyunov-Frolov model,”
JHEP (2012) 165, arXiv:1209.6042 [hep-th] .[18] A. Melikyan and G. Weber, “Integrable theories and generalized graded Mailletalgebras,”
Journal of Physics A: Mathematical and Theoretical no. 6, (2014) 065401.[19] A. Melikyan, E. Pereira, and V. Rivelles, “On the equivalence theorem for integrablesystems,” J.Phys.
A48 no. 12, (2015) 125204, arXiv:1412.1288 [hep-th] .[20] T. Klose and K. Zarembo, “Bethe ansatz in stringy sigma models,”
J.Stat.Mech. (2006) P05006, arXiv:hep-th/0603039 .[21] A. Melikyan, A. Pinzul, V. Rivelles, and G. Weber, “Quantum integrability of theAlday-Arutyunov-Frolov model,”
JHEP (2011) 092, arXiv:1106.0512[hep-th] .[22] E. K. Sklyanin, “Quantization of the continuous Heisenberg ferromagnet,”
Lett. Math.Phys. (1988) 357–368.[23] M. Semenov-Tian-Shansky and A. Sevostyanov, “Classical and quantum nonultralocalsystems on the lattice,” arXiv:hep-th/9509029 [hep-th] .[24] F. Delduc, M. Magro, and B. Vicedo, “Generalized sine-Gordon models and quantumbraided groups,” JHEP (2013) 031, arXiv:1212.0894 [math-ph] .[25] T. J. Hollowood, J. L. Miramontes, and D. M. Schmidtt, “S-Matrices and QuantumGroup Symmetry of k-Deformed Sigma Models,” arXiv:1506.06601 [hep-th] .[26] A. Alekseev, L. D. Faddeev, M. Semenov-Tian-Shansky, and A. Volkov, “TheUnraveling of the quantum group structure in the WZNW theory.” 1991.[27] A. Melikyan and A. Pinzul, “On quantum integrability of the Landau-Lifshitz model,” J. Math. Phys. (2009) 103518, arXiv:0812.0188 [hep-th] .3028] A. Melikyan, A. Pinzul, and G. Weber, “Higher charges and regularized quantum traceidentities in su(1,1) Landau–Lifshitz model,” J. Math. Phys. no. 12, (2010) 123501, arXiv:1008.1054 [hep-th] .[29] F. Berezin, Introduction to Superanalysis . Mathematical Physics and AppliedMathematics. Springer, 1987.[30] P. Kulish and E. Sklyanin, “On the solution of the Yang-Baxter equation,”
J. Sov. Math. (1982) 1596–1620.[31] P. Kulish, “Integrable graded magnets,” J. Sov. Math. (1986) 2648–2662.[32] F. G ¨ohmann and S. Murakami, “Fermionic representations of integrable latticesystems,” Journal of Physics A: Mathematical and General no. 38, (1998) 7729.[33] F. Gohmann and V. Korepin, “Solution of the quantum inverse problem,” J.Phys.
A33 (2000) 1199–1220, arXiv:hep-th/9910253 [hep-th] .[34] F. G ¨ohmann and V. Korepin, “A quantum version of the inverse scatteringtransformation,”
Physics of Atomic Nuclei no. 6, (2002) 968–975.[35] F. H. L. Essler, H. Frahm, F. G ¨ohmann, A. Kl ¨umper, and V. E. Korepin, TheOne-Dimensional Hubbard Model . Cambridge University Press, Cambridge, 2005.[36] L. Faddeev and R. Jackiw, “Hamiltonian reduction of unconstrained and constrainedsystems,”
Phys. Rev. Lett. (1988) 1692–1694.[37] S. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov, Theory of Solitons : TheInverse Scattering Method, 276 p.
Contemporary Soviet Mathematics. New York, USA:Consultants Bureau, 1984.[38] V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin,
Quantum Inverse Scattering Methodand Correlation Functions . Cambridge Monographs on Mathematical Physics.Cambridge University Press, 1997.[39] R. Streater and A. Wightman,
PCT, Spin and Statistics, and All that . Princeton UniversityPress, 2000.[40] L. D. Faddeev and L. A. Takhtajan,
Hamiltonian Methods in the Theory of Solitons .Springer Series in Soviet Mathematics, 592 p., 1987.[41] S. Weinberg,
The Quantum Theory of Fields , vol. 1. Cambridge University Press, 1995.[42] V. Vladimirov,
Methods of the Theory of Generalized Functions . Analytical Methods andSpecial Functions. Taylor & Francis, 2002.[43] J. Novotny and M. Schnabl, “Point - splitting regularization of composite operatorsand anomalies,”
Fortsch. Phys. (2000) 253–302, arXiv:hep-th/9803244[hep-th] .[44] V. Moretti, “Local zeta function techniques versus point splitting procedure: A Fewrigorous results,” Commun. Math. Phys. (1999) 327–363, arXiv:gr-qc/9805091[gr-qc] . 3145] M. E. Peskin and D. V. Schroeder,
An Introduction To Quantum Field Theory (Frontiers inPhysics) . Westview Press, 1995.[46] M. Bergere and Y.-M. P. Lam, “Equivalence theorem and Faddeev-Popov ghosts,”
Phys.Rev.
D13 (1976) 3247–3255.[47] W. Zimmermann, “Local operator products and renormalization in quantum fieldtheory,” in
Brandeis Lectures , S. Deser et al., ed., vol. 1, p. 395. M.I.T. Press, Cambridge,1970.[48] K. G. Wilson and W. Zimmermann, “Operator product expansions and composite fieldoperators in the general framework of quantum field theory,”
Commun. Math. Phys. (1972) 87–106.[49] K. G. Wilson, “Operator product expansions and anomalous dimensions in theThirring model,” Phys. Rev. D2 (1970) 1473.[50] J. H. Lowenstein, “Normal products in the Thirring model,” Comm. Math. Phys. no. 4, (1970) 265–289.[51] H. B. Thacker, “Bethe’s hypothesis and Feynman diagrams: Exact calculation of a threebody scattering amplitude by perturbation theory,” Phys. Rev.
D11 (1975) 838.[52] H. B. Thacker, “Many body scattering processes in a one-dimensional boson system,”
Phys. Rev.
D14 (1976) 3508.[53] E. Zeidler,
Quantum field theory. I: Basics in mathematics and physics. A bridge betweenmathematicians and physicists . Springer-Verlag Berlin Heidelberg, 2006.[54] E. Zeidler,
Quantum field theory. II: Quantum electrodynamics. A bridge betweenmathematicians and physicists . Springer-Verlag Berlin Heidelberg, 2009.[55] C. Brouder, N. Viet Dang, and F. H´elein, “A smooth introduction to the wavefront set,”
Journal of Physics A Mathematical General (Nov., 2014) 3001, arXiv:1404.1778[math-ph] .[56] J. F. Colombeau, “Multiplication of distributions,” Bull. Amer. Math. Soc. (N.S.) no. 2,(10, 1990) 251–268.[57] J. Colombeau, New Generalized Functions and Multiplication of Distributions .North-Holland Mathematics Studies. Elsevier Science, 2000.[58] M. Sato, “Theory of hyperfunctions I,”
Journal of the Faculty of Science, University ofTokyo no. 1, (1959) 139–193.[59] M. Sato, “Theory of hyperfunctions II,” Journal of the Faculty of Science, University ofTokyo no. 2, (1959) 387–437.[60] J. Holland and S. Hollands, “Associativity of the operator product expansion,” arXiv:1507.07730 [math-ph] .[61] L. F. Alday, G. Arutyunov, and A. A. Tseytlin, “On integrability of classicalsuperstrings in AdS × S ,” JHEP (2005) 002, arXiv:hep-th/0502240 .3262] G. Arutyunov and S. Frolov, “Foundations of the
AdS × S Superstring. Part I,”
J.Phys.
A42 (2009) 254003, arXiv:0901.4937 [hep-th]arXiv:0901.4937 [hep-th]