Wedge-local observables in the deformed sine-Gordon model
aa r X i v : . [ m a t h - ph ] D ec Wedge-local observables in the deformed sine-Gordonmodel
Daniela Cadamuro e-mail: [email protected]
Mathematisches Institut, Universit¨at G¨ottingenBunsenstrasse 3-5, D-37073 G¨ottingen, Germany.
Yoh Tanimoto e-mail: [email protected]
Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”Via della Ricerca Scientifica 1, 00133 Roma, Italy
Abstract
In the bootstrap approach to integrable quantum field theories in the (1 + 1)-dimensional Minkowski space, one conjectures the two-particle S-matrix and tries tostudy local observables. The massless sine-Gordon model is conjectured to be equiva-lent to the Thirring model, and its breather-breather S-matrix components (where thefirst breather corresponds to the scalar field of the sine-Gordon model) are closed underfusion. Yet, the residues of the poles in this breather-breather S-matrix have wrongsigns and cannot be considered as a separate model.We find CDD factors which adjust the signs, so that the breather-breather S-matrixalone satisfies reasonable assumptions. Then we propose candidates for observables inwedge-shaped regions and prove their commutativity in the weak sense.
Dedicated to Karl-Henning Rehren on the occasion of his 60th birthday
Recently there have been progresses in the construction of (1 + 1)-dimensional quantum fieldtheories with factorizing S-matrices in the operator algebraic approach [Lec03, Lec08, DT11,Tan12, BT13, LST13, LS14, Tan14, Ala14, BT15, AL16]. The basic idea is the following[Sch97]: while pointlike local observables are hard to construct, observables localized in aninfinitely extended wedge-shaped region might be tractable and have simple expressions. Ithas been first implemented for a scalar analytic factorizing S-matrix [Lec08, Ala14, AL16]and strictly local observables have been shown to exist using a quite indirect proof thatrelies on properties of the underlying modular operators (for double cones above the minimalsize). In this construction, the input is the particle spectrum of the theory, together with theS-matrix with certain properties. Construction of observables in wedges has been extended1o theories with several particle species by Lechner and Sch¨utzenhofer [LS14], including the O ( N )-invariant nonlinear σ -models.Recently, in [CT15, CT16, Tan16], we further generalized this construction to scalarmodels with S-matrices which have poles in the physical strip. The poles in the S-matrix arebelieved to correspond to the presence of bound states (e.g. the Bullough-Dodd model). Wealso extended this construction to models with several particle species, where the S-matrixis “diagonal” in a certain sense. They include, e.g. the Z ( N )-Ising model and the A N -affineToda field theories.In this work, we extend this last mentioned class of S-matrices to those which are modi-fications of the S-matrix of the massless sine-Gordon model by a CDD factor, and moreoverrestricted to a certain range of values for the coupling constant. This is again of diagonaltype.The massless sine-Gordon model has been conjectured to be equivalent to the Thirringmodel in a certain sense (Coleman’s equivalence). In [BFM09] Benfatto, Falco and Mat-ropietro proved the equivalence between the massless sine-Gordon model with finite volumeinteraction and the Thirring model with a finite volume mass term. The Thirring model hasbeen also constructed by the functional integral methods [BFM07]. On the other hand, themassless sine-Gordon model has been expected to be integrable and its S-matrix has beenconjectured [ZZ79]. Yet, in the rigorous constructions, the factorization of the S-matrix hasnot been proved (c.f. [BR16], where the perturbative S-matrix with IR cutoff is shown toconverge, yet its factorization has not been proved).The conjectured S-matrix of the massless sine-Gordon model has been studied in the formfactor programme [BFKZ99, BK02]. Certain matrix components of the pointlike local fields(“form factors”) have been computed, yet the existence of the Wightman field are currentlyout of reach, because the expansion of the n -point functions in terms of form factors is notunder control. Here, we are not dealing with the massless sine-Gordon model itself, but witha new model with the same fusion structure, that has not been considered before. It arises asa deformation of the “breather-breather” S-matrix of the massless sine-Gordon model by themultiplication of a CDD factor. The coupling constant is restricted here to a certain rangeof values, where there are only two species of particles involved (two breathers).Our goal is to attain a realization of this model associated with this new S-matrix in theoperator-algebraic framework, i.e. the Haag-Kastler axioms. In this framework, we constructcandidates for wedge-local observables by extending the construction carried out in [CT16].The question of strong commutativity remains open also in this model.With the presence of poles in the S-matrix, the construction of wedge-local observablesmust be studied in a case-by-case approach, in contrast to the homogeneous construction forthe analytic S-matrices [LS14]. This is due to the idea that simple poles in the S-matrixcorrespond to the bound states in the model, therefore, the wedge-local observables mustreflect such fusion processes. We do this by introducing the operators which we call thebound state operators. Furthermore, higher order poles bring further complications and weneed the existence of what we call elementary particles. Our proof of wedge-locality is basedon a number of properties of the two-particle scattering function, and there is actually ainfinite family of examples satisfying them, therefore, we have correspondingly an infinitefamily of candidates for quantum field theories.The paper is organized as follows. In Sec. 2, we will introduce the model and fix the input2cattering data, including the properties of the S-matrix. In Sec. 3, we exhibit our generalnotation for multi-particle Fock space, partially following Lechner-Sch¨utzenhofer [LS14]. InSec. 4 we introduce the bound state operators χ ( f ), χ ′ ( g ), we analyse their domains andsymmetry properties as quadratic forms. In Sec. 5 we construct the fields e φ ( f ) and e φ ′ ( g ) andshow the weak wedge-commutativity between the components for “elementary particles”. InSec. 6 we conclude our paper with some remarks. In the conjectured integrable sine-Gordon model, the particle spectrum consists of a familyof finitely many particles called breathers { b ℓ } [BK02]. It is also conjectured that, the sine-Gordon model is equivalent to the Thirring model, where the breathers are the bound statesof soliton and the anti-soliton (the anti-particle of the soliton).In the sine-Gordon model, the number of breathers depends on the coupling constant0 < ν < < ν < , and differently from the sine-Gordon model, we donot consider solitons and interpret that there are only two breathers b , b , by taking the maximal analyticity (see below) in a strict sense. The masses of the breathers are givenby m b ℓ = 2 m sin ℓνπ , where m > ℓ = 1 ,
2. These particles are neutral and hence thecharge conjugate of b ℓ (denoted with ¯ b ℓ in literature) is b ℓ itself.In this case, the elastic two-particle scattering processes are characterized by a matrix-valued function with only non-zero components S b b b b ( θ ), S b b b b ( θ ), S b b b b ( θ ) and S b b b b ( θ ), where θ is the difference of the rapidities of the incoming particles. We will give explicit expressionsfor them in Section 2.2. They are the breather-breather S-matrix components of the sine-Gordon model multiplied by so-called CDD factors.The particles b , b may form a bound state in a scattering process. We declare that thepossible fusion processes are only of three types, ( b b ) → b , ( b b ) → b and ( b b ) → b .On the other hand, ( b b ) is not a fusion. The corresponding imaginary rapidities of thefusing particles are denoted by θ b ( b b ) for the first fusion, and θ b ( b b ) , θ b ( b b ) for the secondtwo types of fusion. Correspondingly, we do not specify the rapidity θ ( b b ) , since there is nofusion ( b b ). The actual values will be given in Section 2.2.In the same way as in [CT16], to these fusion processes there correspond the so-called fusion angles , which determine the position of the simple poles in the components S b b b b ( ζ ), S b b b b ( ζ ) and S b b b b ( ζ ) in the physical strip < Im ζ < π . Specifically, for the fusion ( b b ) → b , S b b b b ( ζ ) has a simple pole at ζ = iθ b b b , where θ b b b := θ b ( b b ) + θ b ( b b ) (= 2 θ b ( b b ) ) . Similarly, S b b b b ( ζ ), corresponding to the fusion process ( b b ) → b , has a simple pole at In the form factor programme [BFKZ99], for a given 0 < ν <
1, there are K breathers, where K is thelargest integer such that Kν <
1. Especially, if < ν <
1, there is only one breather b , differently from ourcase (we are indeed not considering the Thirring model). = iθ b b b , where θ b b b := θ b ( b b ) + θ b ( b b ) , and the same holds for the S-matrix component S b b b b ( ζ ). In our construction, the poles in thecomponent S b b b b ( ζ ) do not matter. We will indeed introduce the additional concept of ele-mentary particle in analogy with [CT16], and we assume the so-called “maximal analyticity”only for the elementary particle b .These angles correspond to s -channel poles and in the model under investigation theyare explicitly given in Table 1. The S-matrix components S b b b b , S b b b b , S b b b b and S b b b b aremeromorphic functions on C , which we present below. In addition, we will introduce thebound state intertwiners η b b b , η b b b and η b b b (there is no corresponding matrix element for( b b ), as this is not a fusion.) In a general non-diagonal case, they formally diagonalizethe S-matrix components above at the corresponding pole, and their eigenvalues correspondto the residues. They were also defined in [BFKZ99] and more explicitly in [Que99, beforeEq.(1.13)] and here we adopt a slightly modified convention, as below. The input which specifies the S-matrix of our model is the following. • The coupling constant ν , which is a parameter such that < ν < and the massparameter m > ν in the range above, we consider two breathers, b , b . Indeed, K = 2 is thelargest integer such that Kν < • The S-matrix components: S b ℓ b k b k b ℓ ( ζ ) = S SG b ℓ b k b k b ℓ ( ζ ) S CDD b ℓ b k b k b ℓ ( ζ ), where S SG b b b b ( ζ ) = tan i ( ζ + iπν )tan i ( ζ − iπν ) ,S SG b b b b ( ζ ) = S SG b b b b ( ζ ) = tan i (cid:0) ζ + iπν (cid:1) tan i (cid:0) ζ − iπν (cid:1) tan i (cid:0) ζ + iπν (cid:1) tan i (cid:0) ζ − iπν (cid:1) ,S SG b b b b ( ζ ) = tan i ( ζ + 2 iπν )tan i ( ζ − iπν ) (cid:0) tan i ( ζ + iπν ) (cid:1) (cid:0) tan i ( ζ − iπν ) (cid:1) . are the breather-breather S-matrix components of the sine-Gordon model (see [BK02,Que99]), and S CDD b ℓ b k b k b ℓ are introduced as follows: S CDD b b b b ( ζ ) := sinh ( θ − iπ ( ν − ν − ))sinh ( θ + iπ ( ν − ν − )) · sinh ( θ − iπ ( ν + ν + ))sinh ( θ + iπ ( ν + ν + )) × sinh ( θ − iπ (1 − ν + ν − ))sinh ( θ + iπ (1 − ν + ν − )) · sinh ( θ − iπ (1 − ν − ν + ))sinh ( θ + iπ (1 − ν − ν + )) , (1)4nd expecting the bootstrap equation (see condition (S6) below), we also define S CDD b b b b ( ζ ) = S CDD b b b b ( ζ ):= S CDD b b b b ( ζ + iθ b ( b b ) ) S CDD b b b b ( ζ − iθ b ( b b ) )= sinh (cid:0) θ − iπ ( ν − ν − ) (cid:1) sinh (cid:0) θ + iπ ( ν − ν − ) (cid:1) · sinh (cid:0) θ − iπ ( ν + ν + ) (cid:1) sinh (cid:0) θ + iπ ( ν + ν + ) (cid:1) × sinh (cid:0) θ − iπ (1 − ν + ν − ) (cid:1) sinh (cid:0) θ + iπ (1 − ν + ν − ) (cid:1) · sinh (cid:0) θ − iπ (1 − ν − ν + ) (cid:1) sinh (cid:0) θ + iπ (1 − ν − ν + ) (cid:1) × sinh (cid:0) θ − iπ ( ν − ν − ) (cid:1) sinh (cid:0) θ + iπ ( ν − ν − ) (cid:1) · sinh (cid:0) θ − iπ ( ν + ν + ) (cid:1) sinh (cid:0) θ + iπ ( ν + ν + ) (cid:1) × sinh (cid:0) θ − iπ (1 − ν + ν − ) (cid:1) sinh (cid:0) θ + iπ (1 − ν + ν − ) (cid:1) · sinh (cid:0) θ − iπ (1 − ν − ν + ) (cid:1) sinh (cid:0) θ + iπ (1 − ν − ν + ) (cid:1) , (2) S CDD b b b b ( ζ ) := S CDD b b b b ( ζ + iθ b ( b b ) ) S CDD b b b b ( ζ − iθ b ( b b ) )= S CDD b b b b ( ζ + iθ b b b ) S CDD b b b b ( ζ ) S CDD b b b b ( ζ − iθ b b b ) . We do not need an explicit expression for S CDD b b b b , and we omit computing it. ν − and ν + are parameters satisfying the following set of conditions:(i) ν − , ν + > ν − ∈ (cid:0) ν − , ν (cid:1) .(iii) ν + ∈ (0 , − ν ).(iv) 1 − ν = ν − + ν + .For < ν < , there are such ν − , ν + . Indeed, by rewriting every condition (i)–(iii)only in terms of ν + through (iv) which is equivalent to ν − = 1 − ν − ν + , we obtain0 < ν + < − ν and 1 − ν < ν + < − ν , which have always a nontrivial intersection for < ν < (on the other hand, in the interval < ν < ν − , ν + . (iv) is equivalent to − (cid:0) ν − ν − (cid:1) = 1 − ν − ν + , therefore,from (2) and sinh ( ζ + 2 πi ) = − sinh ζ we have S CDD b b b b ( ζ ) = S CDD b b b b ( ζ )= sinh (cid:0) θ − iπ ( ν − ν − (cid:1) )sinh (cid:0) θ + iπ ( ν − ν − (cid:1) ) · sinh (cid:0) θ − iπ ( ν + ν + (cid:1) )sinh (cid:0) θ + iπ ( ν + ν + (cid:1) ) × − (cid:0) θ + iπ (1 − ν + ν − (cid:1) ) · sinh (cid:0) θ − iπ (1 − ν − ν + (cid:1) )1 × (cid:0) θ + iπ ( ν − ν − (cid:1) ) · sinh (cid:0) θ − iπ ( ν + ν + (cid:1) )1 × sinh (cid:0) θ − iπ (1 − ν + ν − (cid:1) )sinh (cid:0) θ + iπ (1 − ν + ν − (cid:1) ) · sinh (cid:0) θ − iπ (1 − ν − ν + (cid:1) )sinh (cid:0) θ + iπ (1 − ν − ν + (cid:1) ) , b b ) −→ b θ b ( b b ) = πν θ b b b = πν ( b b ) −→ b , ( b b ) −→ b θ b ( b b ) = π (1 − ν ) , θ b ( b b ) = πν θ b b b = θ b b b = π (cid:0) − ν (cid:1) ( b b ) not a fusion Table 1: Fusions and anglesand it is straightforward to see that these S-matrix components have no pole in thephysical strip 0 < Im ζ < π . • There are only three possible fusion processes ( b b ) → b , ( b b ) → b and ( b b ) → b .Note that ( b b ) is not a fusion. The corresponding rapidities of particles θ b ( b b ) , θ b ( b b ) and θ b ( b b ) are presented in the fusion table (Table 1). We define the fusion angles by θ γαβ := θ γ ( αβ ) + θ γ ( βα ) if ( αβ ) → γ is a two fusion process, where α, β, γ = b or b .The data collected above satisfy the following “axioms” (in general, these axioms involvethe charge conjugation, but for breathers it is trivial, ¯ b = b and ¯ b = b ). In thefollowing, k, ℓ = 1 , Meromorphy.
The functions S b ℓ b k b k b ℓ ( ζ ) are meromorphic on C .(S2) Parity symmetry. S b ℓ b k b k b ℓ ( ζ ) = S b k b ℓ b ℓ b k ( ζ ).(S3) Unitarity. S b ℓ b k b k b ℓ ( ζ ) − = S b k b ℓ b ℓ b k ( ¯ ζ ).(S4) Hermitian analyticity. S b ℓ b k b k b ℓ ( ζ ) = S b ℓ b k b k b ℓ ( − ζ ) − .(S5) Crossing symmetry. S b ℓ b k b k b ℓ ( iπ − ζ ) = S b k b ℓ b ℓ b k ( ζ ).(S6) Bootstrap equation.
Let α, β, γ, µ = b or b . If ( αβ ) → γ is a fusion processin Table 1, there holds S γµµγ ( ζ ) = S αµµα ( ζ + iθ γ ( αβ ) ) S βµµβ ( ζ − iθ γ ( βα ) ) . (S7) Value at zero. S b k b k b k b k (0) = − Regularity.
The components S b ℓ b k b k b ℓ have only finitely many zeros in the physicalstrip and there is κ > k S k κ := sup n | S b ℓ b k b k b ℓ ( ζ ) | : ζ ∈ R + i ( − κ, κ ) o < ∞ (the value of κ depends on the parameters ν, ν − , ν + ).(S9) Maximal analyticity (for b ). The component S b b b b ( ζ ) has only two simplepoles in the physical strip. They are at iθ b b b = iπν (called s -channel pole) and iθ ′ b b b := iπ − iθ b b b = iπ (1 − ν ) (called t -channel pole, whose existence followsfrom crossing symmetry). Similarly, the component S b b b b ( ζ ) has also only two We call this “maximal analyticity” because each s -channel pole at iθ b ℓ b b k has a corresponding entry( b b k ) → b ℓ in the fusion Table 1. It should be noted that this is required only for the S-matrix componentscontaining b , the “elementary particle” defined below. s -channel pole at iθ b b b = iπ (cid:0) − ν (cid:1) and a t -channel pole at iθ ′ b b b := iπ − iθ b b b = iπν .Furthermore, S b k b b b k have no double or higher poles in the physical strip, k = 1 , Positive residue (for b ) . If ( b b k ) → b ℓ is a fusion process, then R b ℓ b b k := Res ζ = iθ bℓb bk S b k b b b k ( ζ ) ∈ i R + . Proof of the axioms. • (S1)–(S6) and (S8). These properties are already satisfied by the S-matrix with com-ponents S SG b ℓ b k b k b ℓ of the sine-Gordon model (and well-known in the literature). It is alsostraightforward to check that S CDD b ℓ b k b k b ℓ ( ζ ) satisfy (S1)–(S5) and (S8). As for (S6), wehave by construction S CDD b b b b ( ζ ) = S CDD b b b b (cid:18) ζ + iπν (cid:19) S CDD b b b b (cid:18) ζ − iπν (cid:19) . By the properties mentioned above (in particular, hermitian analyticity), we have S CDD b b b b ( ζ ) = S CDD b b b b (cid:18) ζ + iπν (cid:19) S CDD b b b b ( ζ − iπ (1 − ν ))Similarly, the bootstrap for b can be satisfied by construction.Therefore, the products S b ℓ b k b k b ℓ ( ζ ) = S SG b ℓ b k b k b ℓ ( ζ ) S CDD b ℓ b k b k b ℓ ( ζ ) satisfy them as well. • (S7). It is easy to see that S SG b k b k b k b k (0) = −
1, while S CDD b k b k b k b k (0) = 1, therefore, we have S CDD b k b k b k b k (0) = − • (S9). The expression of S CDD b k b b b k ( ζ ) does not have poles in the physical strip, so the polestructure of S b k b b b k ( ζ ) is determined by S SG b k b b b k ( ζ ), which is easy to check (and known inthe literature). • (S10) is violated in the sine-Gordon model, indeed R SG b b b := Res ζ = iθ b b b S SG b b b b ( ζ ) = 2 i tan( πν ) ∈ − i R + , for our range of ν ∈ ( , ).On the other hand, by counting the zeros on the imaginary line and by recallingthat S CDD b b b b (0) = 1, one can see that S CDD b b b b ( iπν ) <
0, hence we obtain R b b b =Res ζ = iπν S b b b b ( ζ ) ∈ i R + as desired. From this it follows that R b b b ∈ i R + as well, sincewe will see below that R b b b = R b b b .The residues of S b k b b b k ( ζ ) will play an important role, so we give them symbols. R b ℓ b b k := Res ζ = iθ bℓb bk S b k b b b k ( ζ ) , R ′ b ℓ b b k := Res ζ = iθ ′ bℓb bk S b k b b b k ( ζ ) R b ℓ b k b := Res ζ = iθ bℓbkb S b b k b k b ( ζ ) , R ′ b ℓ b k b := Res ζ = iθ ′ bℓbkb S b b k b k b ( ζ )7nd it follows that R b ℓ b b k = R b ℓ b k b .As before, we also introduce the symbols η b b b and η b b b by the following formula: η b b b = i q π (cid:12)(cid:12) R b b b (cid:12)(cid:12) , η b b b = i q π (cid:12)(cid:12) R b b b (cid:12)(cid:12) , η b b b = i q π (cid:12)(cid:12) R b b b (cid:12)(cid:12) . (3)Furthermore, by convention, we set to zero any residues and matrix elements of the abovetype which do not correspond to a fusion in Table 1. From the properties (S2)–(S7) of theS-matrix, there is a number of other properties of the fusion angles and of the residues thatfollow, and we refer for the proofs to [CT16, Sec. 2.1]. We would mention here only thefollowing. The residue of the t-channel pole is related to the residue of the s-channel poleby R ′ b b b = − R b b b and R ′ b b b = − R b b b , and that by (S2), R b b b = R b b b . (S6) and (S7) implythat R b b b = R b b b . Furthermore, if ( b b k ) → b ℓ is a fusion process, the fusion angles are alsorelated by π − θ b ℓ b b k = θ b k ( b b ℓ ) , θ b k ( b ℓ b ) = θ b ℓ ( b k b ) . (4)From the equality R b b b = R b b b and the parity R b b b = R b b b , it also holds that η b b b = η b b b = η b b b . Particle spectrum.
Given the mass parameter m >
0, we define the masses of the particlesas m b = 2 m sin νπ , m b = 2 m sin 2 νπ . They satisfy the following “fusion” rule: m b = m b cos θ b ( b b ) + m b cos θ b ( b b ) , m b = m b cos θ b ( b b ) + m b cos θ b ( b b ) . (5)As b plays a special role in our methods, we call it an elementary particle as in [CT16,Sec. 2.1]. From the scattering data of Section 2.2, we construct basic mathematical structures forthe wedge-observables in the quantum field theory on the S -symmetric Fock space. Theconstruction can be thought of as a kind of deformation of a free field theory with the inputgiven by the S-matrix. The single-particle Hilbert space accommodates the two species ofparticles: H = M k =1 , H ,b k , H ,b k = L ( R , dθ ) . An element Ψ ∈ H can be identified as a vector valued function with components θ Ψ b k ( θ ). On the unsymmetrized n -particle space H ⊗ n , there is a unitary representation D n ofthe symmetric group G n which, with θθθ := ( θ , · · · , θ n ), acts as( D n ( τ j )Ψ n ) b k b k b k ( θθθ ) = S b kj b kj +1 b kj +1 b kj ( θ j +1 − θ j )Ψ b k ··· b kj +1 b kj ··· b kn n ( θ , · · · , θ j +1 , θ j , · · · , θ n ) , We use a slightly different convention from [Que99]: For a fusion process ( αβ ) → γ , we have η γαβ = √ π η γαβ (Quella) . k , . . . , k n ∈ { , } , θθθ := ( θ , . . . , θ n ), b k b k b k := ( b k , . . . , b k n ) and τ j ∈ G n is the transposi-tion ( j, j + 1) → ( j + 1 , j ).The full Hilbert space H is H := L ∞ n =0 H n with H = C Ω, where H n = P n H ⊗ n and P n := n ! P σ ∈ G n D n ( σ ) is an orthogonal projection. The elements of H are L -sequencesΨ = (Ψ , Ψ , . . . ), where Ψ n are S -symmetric functions , namely invariant under the actionof G n . Finally, we denote by D the linear hull (without closure) of {H n } .There is a unitary representation U of the proper orthochronous Poincar´e group P ↑ + on H which preserves each H n , U := M n U n , ( U ( a, λ )Ψ) b k b k b k n ( θθθ ) := exp i n X l =1 p b kl ( θ l ) · a ! Ψ b k b k b k n ( θ − λ, · · · , θ n − λ ) , where p b kl ( θ ) = ( m b kl cosh θ, m b kl sinh θ ). Additionally, there is an antiunitary representationof the CPT operator on H : J := M n J n , ( J Ψ) b k b k b k n ( θθθ ) := Ψ b kn ...b k n ( θ n , . . . , θ ) . We consider test functions with multi-components and are chosen as g ∈ L k =1 S ( R ) with g b k ∈ S ( R ), and we adopt the following convention: g ± b k ( θ ) := 12 π Z d x g b k ( x ) e ± ip bk ( θ ) · x . We note that if g b k is supported in W R , then g + b k ( θ ) has a bounded analytic continuationin R + i ( − π,
0) and | g + b k ( θ + iλ ) | decays rapidly as θ → ±∞ in the strip for λ ∈ ( − π, g + b k ( θ − iπ ) = g − b k ( θ ).There is a natural action of the proper Poincar´e group on R and on the space of testfunctions, denoted by g ( a,λ ) , and it is compatible with the action on the one-particle space:( g ( a,λ ) ) ± b k = U ( a, λ ) g ± b k . The CPT transformation acts also on multi-components test functions, which we denoteby j , as g g j , ( g j ) b k ( x ) := g b k ( − x ), and this is again compatible with J : ( g j ) ± b k ( θ ) = J g ± b k ( θ ) = g ± b k ( θ ).Moreover, we introduce the complex conjugate of a multi-component test function by( g ∗ ) b k ( x ) := g b k ( x ) and if g = g ∗ , then we say that g is real and it follows that g ± b k ( ζ ) = g ∓ b k ( ζ )(c.f. [LS14, Proposition 3.1]). Zamolodchikov-Faddeev algebra
Similarly to [LS14], creators and annihilators z † b k ( θ ) , z b k ( θ ) are introduced in the S -symmetricFock space H . For ϕ ∈ H , their actions on vectors Ψ = (Ψ n ) ∈ D are given by( z ( ϕ )Ψ) b k b k b k n ( θθθ ) = √ n + 1 X l =1 , Z dθ ′ ϕ b l ( θ ′ )Ψ b l b k b k b k n +1 ( θ ′ , θθθ ) ,z † ( ϕ ) = ( z ( ϕ )) ∗ Our convention of the Lorentz metric is a · b = a b − a b . z † b k ( θ ) z † b l ( θ ′ ) = S b l b k b k b l ( θ − θ ′ ) z † b l ( θ ′ ) z † b k ( θ ) ,z b k ( θ ) z b l ( θ ′ ) = S b l b k b k b l ( θ − θ ′ ) z b l ( θ ′ ) z b k ( θ ) ,z b k ( θ ) z † b l ( θ ′ ) = S b k b l b l b k ( θ ′ − θ ) z † b l ( θ ′ ) z b k ( θ ) + δ b k b l δ ( θ − θ ′ ) H . They are opereator-valued distributions defined on D and bounded on each n -particle space H n when smeared by a test function.Let f ∈ L k =1 , S ( R ), we define φ ( f ) := z † ( f + ) + z ( J f − ) = X k =1 , Z dθ (cid:16) f + b k ( θ ) z † b k ( θ ) + ( J f − ) b k ( θ ) z b k ( θ ) (cid:17)! . This multi-component quantum field is defined on the subspace D of H of vectors with finiteparticle number and the properties listed in [LS14, Proposition 3.1] are fulfilled, as long asthe analyticity in the physical strip is not used. We also introduce φ ′ , the reflected fielddefined for g ∈ S ( R ), φ ′ ( g ) := J φ ( g j ) J = z ′† ( g + ) + z ′ ( J g − ) , where z ′ , z ′† are the reflected creators and annihilators z ′ b k ( θ ) := J z b k ( θ ) J and z ′† b k ( θ ) := J z † b k ( θ ) J .For the class of two-particle S-matrices S ( θ ) with components which are not analytic in thephysical strip θ ∈ R + i (0 , π ), we have [ φ ( f ) , φ ′ ( g )] = 0, namely, even the weak commutativityfails for φ, φ ′ . The goal of the present paper is to find alternative wedge-observables for theS-matrix of the sine-Gordon model. We introduce an operator χ ( f ) similarly to [CT16], which we again call the “bound stateoperator”, whose mathematical structure corresponds to our fusion table, which is same asthe breather-breather fusion processes in the sine-Gordon model with two breathers. In thismodel, the “elementary particle” is b , and we restrict ourselves to the case where f b is theonly non-zero component of a test function f . We define it as an unbounded operator on the S -symmetric Fock space H . Recall that for s < t , H ( S s,t ) is the Hardy space of analytic functions Ψ in S s,t := R + i ( s, t ) such that If the S-matrix S ( ζ ) were analytic in the physical strip , φ ( f ) could be considered as an observable localizedin the standard left wedge W L and if furthermore S is diagonal with additional regularity conditions, onewould be able to obtain a Haag-Kastler net with minimal length [LS14, AL16]. In contrast, our S-matrix haspoles in the physical strip. θ + iλ ) is L ( R ) as a function of θ for each λ ∈ ( s, t ) and their L -norm is uniformlybounded for λ . For a multi-component test function f whose only non-zero component is f b and is supported in W L , its action on H is given as follows:Dom( χ ( f )) := H (cid:18) S − θ b b b , (cid:19) ⊕ H (cid:18) S − θ b b b , (cid:19) ( χ ( f ) ξ ) b k ( θ ) := − iη b b b f + b ( θ + iθ b ( b b ) ) ξ b ( θ − iθ b ( b b ) ) if k = 1 , − iη b b b f + b ( θ + iθ b ( b b ) ) ξ b ( θ − iθ b ( b b ) ) if k = 2 . (6)where η b b b , η b b b are the matrix elements introduced in Sec. 2.2, see Eq. (3). Actually, θ b ( b b ) = θ b ( b b ) = πν , hence Dom( χ ( f )) = H (cid:0) S − πν , (cid:1) ⊕ , but we often keep the notationabove for homogeneity.The full operator χ ( f ) is the direct sum of its components χ n ( f ) on H n : χ n ( f ) := nP n ( χ ( f ) ⊗ ⊗ · · · ⊗ ) P n , χ ( f ) = ∞ M n =0 χ n ( f ) . (7)Similarly, and as in [CT16], we introduce the reflected bound state operator χ ′ ( g ) for a testfunction g supported in the right wedge W R . Again, its one particle projection for g havingonly one non-zero component g b is given byDom( χ ′ ( g )) := H (cid:18) S ,θ b b b (cid:19) ⊕ H (cid:18) S ,θ b b b (cid:19) ( χ ′ ( g ) ξ ) b k ( θ ) := − iη b b b g + b ( θ − iθ b ( b b ) ) ξ b ( θ + iθ b ( b b ) ) if k = 1 , − iη b b b g + b ( θ − iθ b ( b b ) ) ξ b ( θ + iθ b ( b b ) ) if k = 2 . The full operator on H is given by χ ′ n ( g ) := nP n ( ⊗ · · · ⊗ ⊗ χ ′ ( g )) P n , χ ′ ( g ) = M n χ ′ n ( g ) . (8)This operator is related to χ by the CPT operator J : χ ′ ( g ) = J χ ( g j ) J. To see this, let us consider the one-particle components. By recalling the expression (6),(
J χ ( g j ) J ξ ) b ℓ ( θ ) = ( χ ( g j ) J ξ ) b ℓ ( θ )= − iη b ℓ b b k ( g j ) + b ( θ + iθ b ℓ ( b b k ) )( J ξ ) b k ( θ − iθ b ℓ ( b k b ) )= − iη b ℓ b b k g + b ( θ − iθ b ℓ ( b b k ) ) ξ b k ( θ + iθ b ℓ ( b k b ) )= ( χ ′ ( g ) ξ ) b ℓ ( θ ) , where l = 1 or 2 and k = 2 or 1, respectively, and we used that − iη γαβ ∈ R . As J n commuteswith P n , we have χ ′ n ( g ) = J n χ n ( g j ) J n . Since the whole operators χ ( g ) and χ ′ ( g ) are definedas the direct sum, the desired equality follows.11e give some more explicit expressions of Eq.ns (7) and (8) by applying them to a n -particle vector which we assume to be S -symmetric and in the domain of χ ( f ) ⊗ ⊗ · · · ⊗ and of ⊗ · · · ⊗ ⊗ χ ′ ( g ), respectively. We have, from (S6), (S2) and (S4) exactly as in[CT16, Section 3.2],( χ ( f )Ψ n ) b k ··· b kn ( θ , . . . , θ n )= − i X ≤ ℓ ≤ n, α ℓ = b ,b η b kℓ b α ℓ Y ≤ j ≤ ℓ − S b kj b b b kj ( θ ℓ − θ j + iθ b kℓ ( b α ℓ ) ) ! × f + b ( θ ℓ + iθ b kℓ ( b α ℓ ) )Ψ b k ...b kℓ − α ℓ b kℓ +1 ...b kn n (cid:16) θ , · · · , θ ℓ − , θ ℓ − iθ b kℓ ( α ℓ b ) , θ ℓ +1 , · · · θ n (cid:17) , (9)where k , . . . , k n = 1 , η γαβ = 0 if ( αβ ) → γ is not afusion, and terms containing such η γαβ should be ignored (even if it contains expressions suchas Ψ( · · · , θ − iθ γ ( βα ) , · · · ) which can be meaningless, as it might be outside the domain ofanalyticity).We have a similar expression for χ ′ ( g ):( χ ′ ( g )Ψ n ) b k ...b kn ( θ , . . . , θ n ) == − i X ≤ ℓ ≤ n, α ℓ = b ,b η b kℓ b α ℓ Y ℓ +1 ≤ j ≤ n S b b kj b kj b ( θ j − θ ℓ + iθ b kℓ ( b α ℓ ) ) ! × g + b ( θ ℓ − iθ b kℓ ( b α ℓ ) )Ψ b k ...b kℓ − α ℓ b kℓ +1 ...b kn n (cid:16) θ , · · · , θ ℓ − , θ ℓ + iθ b kℓ ( α ℓ b ) , θ ℓ +1 · · · θ n (cid:17) . (10) We remark here on some of the properties of χ ( f ), noting that analogous properties holdby construction for χ ′ ( g ). For a multi-component real test function f whose only non-zerocomponent is f b which is real, we can prove that χ ( f ) is densely defined and symmetric asfollows.By construction, χ ( f ) is clearly densely defined. To show that χ ( f ) is symmetric,we take two vectors ξ, ψ ∈ Dom( χ ( f )) whose components have compact inverse Fouriertransform. One can show that these vectors form a core for χ ( f ). By recalling that η b ℓ b b k = 0unless k = 1 , ℓ = 2 or k = 2 , ℓ = 1, we compute on vectors ξ, ψ from the core: h ψ, χ ( f ) ξ i = − X k,ℓ iη b ℓ b b k Z dθ ψ b ℓ ( θ ) f + b ( θ + iθ b ℓ ( b b k ) ) ξ b k ( θ − iθ b ℓ ( b k b ) )= − X k,ℓ iη b ℓ b b k Z dθ f + b ( θ + iπ − iθ b ℓ ( b b k ) ) ψ b ℓ ( θ ) ξ b k ( θ − iθ b ℓ ( b k b ) )= − X kℓ iη b ℓ b b k Z dθ f + b ( θ + iπ − iθ b ℓ b b k ) ψ b ℓ ( θ − iθ b ℓ ( b k b ) ) ξ b k ( θ )= − X kℓ iη b k b b ℓ Z dθ f + b ( θ + iθ b k ( b b ℓ ) ) ψ b ℓ ( θ − iθ b k ( b ℓ b ) ) ξ b k ( θ ) = h χ ( f ) ψ, ξ i , f + ( θ + iλ ) = f + ( iπ − θ − iλ ) explainedat the end of Sec. 3. In the third equality we used Cauchy theorem and performed theshift θ → θ + iθ b ℓ ( b k b ) , since the integrand is analytic, bounded and rapidly decreasing in thestrip R + i (0 , π ) due to ξ, ψ being the Fourier transforms of compactly supported functionsand the properties of f + . In the fourth equality we used the properties π − θ b b b = θ b ( b b ) , θ b ( b b ) = θ b ( b b ) and η b b b = η b b b from Sec. 2.2.We can show that χ n ( f ) is densely defined and symmetric by arguing as in [CT16, Propo-sition 3.1].Furthermore, the operator χ ( f ) is covariant with respect to the action U of the Poincar´egroup P ↑ + on H that we introduced in Section 3 in the following sense. For a test function f supported in W L and ( a, λ ) ∈ P ↑ + such that a ∈ W L , we can show that Ad U ( a, λ )( χ ( f )) ⊂ χ ( f ( a,λ ) ). The key to the proof are the relations (5), see [CT16, Proposition 3.2] for details. We introduce the field ˜ φ ( f ) = φ ( f ) + χ ( f )and its reflected field ˜ φ ′ ( g ) = φ ′ ( g ) + χ ′ ( g ) = J ˜ φ ( g j ) J in a similar manner as in [CT16]. For f with support in W L and such that f ∗ = f , the field ˜ φ ( f ) fulfills the properties listed in[CT16, Proposition 4.1], and a similar result also holds for the reflected field ˜ φ ′ ( g ). Regardingthe domain of ˜ φ , we note that, since the domain of χ ( f ) contains vectors with finite particlenumber and with certain analyticity and boundedness properties (see Sec. 4), its domain isincluded in the domain of φ ( f ), and therefore Dom( ˜ φ ( f )) = Dom( χ ( f )).As already mentioned in [CT16], the field ˜ φ ( f ) has very subtle domain properties. Inparticular, because of the poles of S , after applying this operator to a vector (not the vacuum)in its domain, it generates a vector which is no longer in the domain of ˜ φ ′ ( g ). For this reason,products of the form ˜ φ ( f ) ˜ φ ′ ( g ) and ˜ φ ′ ( g ) ˜ φ ( f ) are not well defined, and we need to computethe commutator [ ˜ φ ( f ) , ˜ φ ′ ( g )] between arbitrary vectors Φ , Ψ from a suitable space (see below).Moreover, the commutator is smeared with test functions f, g with only nonzero componentscorresponding to b .We start by considering vectors Ψ b k b k b k n in the domain discussed in Sec. 4.2. These vectorsadmit analytic continuation in the first variable, and actually a meromorphic continuationin each variable, to ± i πν . We also note that for certain components Ψ b k ··· b kn n ( θ , · · · , θ n ),specifically in the case where two of the indices are equal, b k j = b k ℓ = α , we can infer theexistence of zeros by the following computation:Ψ b k ··· α ··· α ··· b kn n ( θ , · · · , θ j , · · · , θ ℓ , · · · , θ n )= ℓ − Y p = j +1 S αb kp b kp α ( θ p − θ j ) S b kp ααb kp ( θ ℓ − θ p ) ! S αααα ( θ ℓ − θ j ) × Ψ b k ··· α ··· α ··· b kn n ( θ , · · · , θ ℓ , · · · , θ j , · · · , θ n ) . Hence, by (S7) and (S4), Ψ b k b k b k n has a zero at θ j − θ ℓ = 0. However, this does not imply existenceof zeros for other components. Furthermore, in the proof of Theorem 5.1, we will encounter13ertain poles of S in the computation. Hence, we consider vectors from the following space: D := Ψ ∈ D : Ψ b k b k b k n is analytic in R n + i ( − πν , πν ) n , Ψ b k b k b k n ( θθθ + iλλλ ) ∈ L ( R n ) for λλλ ∈ ( − πν , πν ) n , with a uniform bound andhas a zero at θ j − θ ℓ = 0 , ± iπ (1 − ν ) , ± iπ ( ν − , ± iπν for all j, ℓ , (11)where k j = 1 ,
2. Note that D ⊂ Dom( e φ ( f )) ∩ Dom( e φ ′ ( g )).One can see that D is dense as follows: we take C n ( θθθ ) := Y λ ∈ Λ Y ≤ j Let f and g be test functions supported in W L and W R , respectively, and withthe property that f = f ∗ and g = g ∗ . Furthermore, assume that f, g have components f b k = 0 and g b k = 0 for k = 1 . Then, for each Φ , Ψ in D , we have h e φ ( f )Φ , e φ ′ ( g )Ψ i = h e φ ′ ( g )Φ , e φ ( f )Ψ i . Proof. As in our previous works, we may assume that the vectors Φ and Ψ are already S -symmetric. Furthermore, we recall that the domains of e φ ( f ) , e φ ′ ( g ) coincide with those of χ ( f ) , χ ′ ( g ), respectively, hence we have the following equalities as operators: e φ ( f ) = φ ( f ) + χ ( f ) = z † ( f + ) + χ ( f ) + z ( J f − ) , e φ ′ ( g ) = φ ′ ( g ) + χ ′ ( g ) = z ′† ( g + ) + χ ′ ( g ) + z ′ ( J g − ) . Therefore, the (weak) commutator [ e φ ( f ) , e φ ′ ( g )] expands into several terms that we will com-pute individually. The commutator [ φ ( f ) , φ ′ ( g )]This commutator has been computed in [LS14] and then simplified in the case where S is diagonal in [CT16]. Here, we briefly recall its expression:([ φ ′ ( g ) , φ ( f )]Ψ n ) b k b k b k ( θ , · · · , θ n )= Z dθ ′ g − b ( θ ′ ) n Y p =1 S b kp b b b kp ( θ ′ − θ p ) ! f + b ( θ ′ ) − g + b ( θ ′ ) n Y p =1 S b kp b b b kp ( θ ′ − θ p ) ! f − b ( θ ′ ) ! × (Ψ n ) b k b k b k ( θ , . . . , θ n ) . 14y (S5) and the analytic properties of f ± , g ± explained in Section 3, the first term in theintegrand is equal to the second term up to a shift of + iπ in θ ′ . Since S has some poles inthe physical strip, we obtain residues from this difference.We are considering test functions f, g whose only non-zero components correspond to b .In this case, the factor S b k b b b k appearing in the expression of the commutator have exactly twosimple poles at ζ = iθ b k ′ b b k , iθ ′ b k ′ b b k with k = 1 , k ′ = 2 and k = 2 , k ′ = 1, as seen in the fusiontable in Sec. 2.2.With the notation R b k ′ b b k , R ′ b k ′ b b k which are nonzero only for k = 1 , k ′ = 2 and k = 2 , k ′ = 1,by applying the Cauchy theorem, we get the contributions from the above-mentioned poles:12 πi ([ φ ′ ( g ) , φ ( f )]Ψ n ) b k b k b k ( θ , . . . , θ n )= X k =1 , n X j =1 R b k b b kj g − b ( θ j + iθ b k b b kj ) f + b ( θ j + iθ b k b b kj ) n Y p =1 p = j S b kp b b b kp ( θ j + iθ b k b b kj − θ p ) + n X j =1 R ′ b k b b kj g − b ( θ j + iθ ′ b k b b kj ) f + b ( θ j + iθ ′ b k b b kj ) n Y p =1 p = j S b kp b b b kp ( θ j + iθ ′ b k b b kj − θ p ) × (Ψ n ) b k ...b kn ( θ , . . . , θ n ) . More explicitly, the possible terms from the above expression are given by the following.12 πi ([ φ ′ ( g ) , φ ( f )]Ψ n ) b k b k b k ( θ , . . . , θ n )= n X j =1 R b b b g − b ( θ j + iθ b b b ) f + b ( θ j + iθ b b b ) n Y p =1 p = j S b kp b b b kp ( θ j + iθ b b b − θ p ) × (Ψ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) (12a)+ n X j =1 R ′ b b b g − b ( θ j + iθ ′ b b b ) f + b ( θ j + iθ ′ b b b ) n Y p =1 p = j S b kp b b b kp ( θ j + iθ ′ b b b − θ p ) × (Ψ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) (12b)+ n X j =1 R b b b g − b ( θ j + iθ b b b ) f + b ( θ j + iθ b b b ) n Y p =1 p = j S b kp b b b kp ( θ j + iθ b b b − θ p ) × (Ψ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) (12c)15 n X j =1 R ′ b b b g − b ( θ j + iθ ′ b b b ) f + b ( θ j + iθ ′ b b b ) n Y p =1 p = j S b kp b b b kp ( θ j + iθ ′ b b b − θ p ) × (Ψ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) . (12d) The commutator [ χ ( f ) , χ ′ ( g )]We compute this commutator between vectors Ψ , Φ with only n -particle componentsand with f, g having only non-zero components of type b . Recall the expressions of χ ( f )and χ ′ ( g ) in Sec. 4, where they are written as the sum of n operators acting on differentvariables, therefore, there are n terms in each of the scalar products h χ ′ ( g )Φ , χ ( f )Ψ i and h χ ( f )Φ , χ ′ ( g )Ψ i . Of these, one can show that the n ( n − 1) terms in which the above-mentionedoperators act on different variables give exactly the same contribution, exactly as in [CT16](this time the operators χ ( f ) and χ ′ ( g ) are not positive, but χ ( f ) ⊗ ⊗ · · · ⊗ and ⊗· · ·⊗ ⊗ χ ′ ( g ) are strongly commuting, hence we may consider their polar decomposition),which we denote by C , therefore, they cancel in the commutator and hence are irrelevant.Following [CT16, P. 35], we exhibit the relevant parts ( kkk := k , . . . , k n where each k j cantake 1 , 2. Furthermore, if k j = 1, then we put k ′ j = 2 and if k j = 2, then k ′ j = 1): h χ ′ ( g )Φ , χ ( f )Ψ i − C = n X j =1 X kkkα j ,β j =1 , Z dθ . . . dθ n η b kj b b αj j − Y p =1 S b kp b b b kp (cid:16) θ j − θ p + iθ b kj ( b b αj ) (cid:17)! f + b (cid:16) θ j + iθ b kj ( b b αj ) (cid:17) × (Ψ n ) b k ...b αj ...b kn (cid:16) θ , . . . , θ j − iθ b kj ( b αj b ) , . . . , θ n (cid:17) η b kj b b βj n Y q = j +1 S b kq b b b kq (cid:16) θ j − θ q + iθ b kj ( b b βj ) (cid:17)! × g + b (cid:16) θ j + iθ b kj ( b b βj ) − iπ (cid:17) (Φ n ) b k ...b βj ...b kn (cid:16) θ , . . . , θ j + iθ b kj ( b βj b ) , . . . , θ n (cid:17) = n X j =1 X kkk Z dθ . . . dθ n η b kj b b k ′ j j − Y p =1 S b kp b b b kp (cid:18) θ j − θ p + iθ b kj ( b b k ′ j ) (cid:19)! f + b (cid:18) θ j + iθ b kj ( b b k ′ j ) (cid:19) × (Ψ n ) b k ...b k ′ j ...b kn (cid:18) θ , . . . , θ j − iθ b kj ( b k ′ j b ) , . . . , θ n (cid:19) η b kj b b k ′ j n Y q = j +1 S b kq b b b kq (cid:18) θ j − θ q + iθ b kj ( b b k ′ j ) (cid:19)! × g + b (cid:18) θ j + iθ b kj ( b b k ′ j ) − iπ (cid:19) (Φ n ) b k ...b k ′ j ...b kn (cid:18) θ , . . . , θ j + iθ b kj ( b k ′ j b ) , . . . , θ n (cid:19) = n X j =1 X kkk Z dθ . . . dθ n η b kj b b k ′ j j − Y p =1 S b kp b b b kp (cid:18) θ j − θ p + iθ b kj b b k ′ j (cid:19)! f + b (cid:18) θ j + iθ b kj b b k ′ j (cid:19) × (Ψ n ) b k ...b k ′ j ...b kn ( θ , . . . , θ j , . . . , θ n ) η b kj b b k ′ j n Y q = j +1 S b kq b b b kq (cid:18) θ j − θ q + iθ b kj b k ′ j b (cid:19)! × g + b (cid:18) θ j + iθ b kj b b k ′ j − iπ (cid:19) (Φ n ) b k ...b k ′ j ...b kn ( θ , . . . , θ j , . . . , θ n ) , η b b b , η b b b are the only nonzero combinations,then performed the shift θ j → θ j + iθ b kj ( b k ′ j b ) in the third equality and used θ γαβ = θ γ ( αβ ) + θ γ ( βα ) .This shift in θ j is allowed by the analyticity and decay properties of f + , g + at infinity inthe strip, [CT15, Lemma B.2] and the property of Ψ , Φ ∈ D explained before Theorem 5.1:more precisely, depending on whether b k p = b or b (respectively for b k q ), S b kp b b b kp ( ζ ) has apole at iπν and iπ (1 − ν ), or at i πν and i (1 − πν ). As θ j → θ j + iθ b kj ( b k ′ j b ) = θ j + i πν (thisdoes not depend on b k j : see Table 1), the integral contour might move across the pole when θ j → θ j + iπ (1 − ν ) , θ j → θ j + iπ ( ν − 1) or θ j → θ j + i πν , depending on the combinationof b k p and b k j . But these poles are cancelled by the zeros of Ψ n , Φ n ∈ D , hence the shift islegitimate and the result is L (the integral is the inner product of two L -functions).Similarly, we can compute the other term h χ ( f )Φ , χ ′ ( g )Ψ i in the commutator [ χ ( f ) , χ ′ ( g )]and obtain: h χ ( f )Φ , χ ′ ( g )Ψ i − C = n X j =1 X kkkα j ,β j =1 , η b kj b b αj Z dθ . . . dθ n j − Y p =1 S b kp b b b kp (cid:16) θ j − θ p + iθ b kj ( b b αj ) (cid:17)! × f + b (cid:16) θ j + iθ b kj ( b b αj ) (cid:17) (Φ n ) b k ...b αj ...b kn (cid:16) θ , . . . , θ k − iθ b kj ( b αj b ) , . . . , θ n (cid:17) × η b kj b b βj n Y q = j +1 S b b kq b kq b (cid:16) θ q − θ j + iθ b kj ( b b βj ) (cid:17)! × g + b (cid:16) θ j − iθ b kj ( b b βj ) (cid:17) (Ψ n ) b k ...b βj ...b kn (cid:16) θ , . . . , θ j + iθ b kj ( b βj b ) , . . . , θ n (cid:17) = n X j =1 X kkk η b kj b b k ′ j Z dθ . . . dθ n j − Y p =1 S b kp b b b kp (cid:18) θ j − θ p − iθ b kj b k ′ j b + iπ (cid:19)! × f + b (cid:18) θ j − iθ b kj b k ′ j b + iπ (cid:19) (Φ n ) b k ...b k ′ j ...b kn ( θ , . . . , θ j , . . . , θ n ) × η b kj b b k ′ j n Y q = j +1 S b kq b b b kq (cid:18) θ j − θ q − iθ b kj b b k ′ j + iπ (cid:19)! g + b (cid:18) θ j − iθ b kj b b k ′ j (cid:19) × (Ψ n ) b k ...b k ′ j ...b kn ( θ , . . . , θ j , . . . , θ n ) , where we used (9), (10) and θ γαβ = θ γ ( αβ ) + θ γ ( βα ) , we performed the shift θ j → θ j − iθ b kj ( b k ′ j b ) andwe used properties (S3)–(S5). As before, we can perform the shift in θ j using the analyticityand decay properties of f + , g − at infinity in the strip, [CT15, Lemma B.2] and the zeros ofthe vectors Ψ , Φ ∈ D . This also guarantees the fact that the result is still L .Since there are only two types of fusion processes ( b b ) → b and ( b b ) → b in themodel, the possible contributions to the expectation values above are h χ ′ ( g )Φ , χ ( f )Ψ i − C n X j =1 X kkk η b b b η b b b Z dθ . . . dθ n j − Y p =1 S b kp b b b kp (cid:0) θ j − θ p + iθ b b b (cid:1) × f + b (cid:0) θ j + iθ b b b (cid:1) (Ψ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) n Y q = j +1 S b kq b b b kq (cid:0) θ j − θ q + iθ b b b (cid:1) × g + b (cid:0) θ j + iθ b b b − iπ (cid:1) (Φ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) (13a)+ n X j =1 X kkk η b b b η b b b Z dθ . . . dθ n j − Y p =1 S b kp b b b kp (cid:0) θ j − θ p + iθ b b b (cid:1) × f + b (cid:0) θ j + iθ b b b (cid:1) (Ψ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) n Y q = j +1 S b kq b b b kq (cid:0) θ j − θ q + iθ b b b (cid:1) × g + b (cid:0) θ j + iθ b b b − iπ (cid:1) (Φ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) , (13b)and similarly, h χ ( f )Φ , χ ′ ( g )Ψ i − C = n X j =1 X kkk η b b b η b b b Z dθ . . . dθ n j − Y p =1 S b kp b b b kp (cid:0) θ j − θ p − iθ b b b + iπ (cid:1) × f + b (cid:0) θ j − iθ b b b + iπ (cid:1) (Φ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) × n Y q = j +1 S b kq b b b kq (cid:0) θ j − θ q − iθ b b b + iπ (cid:1) g + b ( θ j − iθ b b b )(Ψ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) (14a)+ n X j =1 X kkk η b b b η b b b Z dθ . . . dθ n j − Y p =1 S b kp b b b kp (cid:0) θ j − θ p − iθ b b b + iπ (cid:1) × f + b ( θ j − iθ b b b + iπ )(Φ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) × n Y q = j +1 S b kq b b b kq (cid:0) θ j − θ q − iθ b b b + iπ (cid:1) g + b ( θ j − iθ b b b )(Ψ n ) b k ...b ...b kn ( θ , . . . , θ j , . . . , θ n ) . (14b)Now, the commutator [ φ ′ ( g ) , φ ( f )] cancels the commutator [ χ ( f ) , χ ′ ( g )]: more precisely, (12a)cancels (13a), (12b) cancels (14b), (12c) cancels (13b), (12d) cancels (14a). This uses thefollowing properties: • The properties of fusion angles and residues, such as θ b b b := θ b ( b b ) + θ b ( b b ) , θ ′ b b b = π − θ b b b , θ ′ b b b = π − θ b b b , R ′ b b b = − R b b b and R ′ b b b = − R b b b . • Eq. (3) and R b b b , R b b b ∈ i R + , hence ( η b b b ) = − πiR b b b and ( η b b b ) = − πiR b b b . • f + b ( θ + iπ ) = f − b ( θ ) , g + b ( θ − iπ ) = g − b ( θ ).Most of these properties are from Section 2.2.18 he commutators [ χ ( f ) , z ′ ( J g − )] and [ z ( J f − ) , χ ′ ( g )]Using the expressions of χ ( f ) and χ ′ ( g ) in (9) and (10), we can also compute these com-mutators as in [CT16]. Noting that η b b b , η b b b are the only possible non-zero combinations,we find([ χ ( f ) , z ′ ( J g − )]Ψ n ) b k ...b kn − ( θ , · · · , θ n − )= √ n iη b b b Z dθ ′ g − b ( θ ′ ) f + b ( θ ′ + iθ b ( b b ) )(Ψ n ) b b k ...b kn − ( θ ′ − iθ b ( b b ) , θ . . . θ n − ) × n − Y j =1 S b kj b b b kj ( θ ′ − θ j ) ! , which it can be rewritten by shifting θ ′ → θ ′ + iθ b ( b b ) as follows([ χ ( f ) , z ′ ( J g − )]Ψ n ) b k ...b kn − ( θ , · · · , θ n − )= √ n iη b b b Z dθ ′ g − b ( θ ′ + iθ b ( b b ) ) f + b ( θ ′ + iθ b b b )(Ψ n ) b b k ...b kn − ( θ ′ , θ . . . θ n − ) × n − Y j =1 S b kj b b b kj ( θ ′ + iθ b ( b b ) − θ j ) ! . (15)For the shift in θ ′ , as it is based on an application of the Cauchy Theorem, it uses theanalyticity and decay properties of f + , g − at infinity in the strip, [CT15, Lemma B.2] andthe fact that the poles of the S -factors in the product above are cancelled by the zeros of thevector Ψ n ∈ D . More precisely, for b k j = b , S b b b b ( ζ ) has a pole at ζ = iπ − iθ b b b = iπ (1 − ν ).Noting that π (1 − ν ) < θ b ( b b ) = πν for < ν < , the zero of the factor C n at iπ (1 − ν )becomes relevant here (see below (11)), while the pole at ζ = iθ b b b = iπν is not reached bythe shift by iπν in θ ′ . The pole of S b b b b ( ζ ) at ζ = i πν = iθ b ( b b ) is cancelled by the zeros ofΨ n arising from S -symmetry (see the observations above (11)), as in this case b k j = b , whilethe pole at ζ = iπ (1 − ν ) is not reached by the shift by iπν in θ ′ .This also guarantees the fact that the result is still L . Similarly, we have([ z ( J f − ) , χ ′ ( g )]Ψ n ) b k ...b kn − ( θ , · · · , θ n − )= −√ n iη b b b Z dθ ′ f − b ( θ ′ ) g + b ( θ ′ − iθ b ( b b ) )(Ψ n ) b b k ...b kn − ( θ ′ + iθ b ( b b ) , θ . . . θ n − ) × n − Y j =1 S b b kj b kj b ( θ j − θ ′ + iθ b ( b b ) ) ! , and by shifting θ ′ → θ ′ − iθ b ( b b ) we can rewrite this expression as([ z ( J f − ) , χ ′ ( g )]Ψ n ) b k ...b kn − ( θ , · · · , θ n − )= −√ n iη b b b Z dθ ′ f − b ( θ ′ − iθ b ( b b ) ) g + b ( θ ′ − iθ b b b )(Ψ n ) b b k ...b kn − ( θ ′ , θ . . . θ n − )19 n − Y j =1 S b b kj b kj b ( θ j − θ ′ + iθ b b b ) ! = −√ n iη b b b Z dθ ′ f + b ( θ ′ + iπ − iθ b ( b b ) ) g − b ( θ ′ + iπ − iθ b b b )(Ψ n ) b b k ...b kn − ( θ ′ , θ . . . θ n − ) × n − Y j =1 S b b kj b kj b ( θ ′ − θ j + iπ − iθ b b b ) ! , (16)where we used the property of f − , g + under π -translation and (S5). As before, the shift in θ ′ is allowed as the poles of the S -factors in the product above are cancelled by the zeros ofΨ n ∈ D . More precisely, S b b b b ( ζ ) has a pole at i πν and this is crossed as θ ′ is shifted by iπ (1 − ν ), hence the zero of the factor C n at iπ (1 − ν ) becomes relevant, while the pole at ζ = iπ (1 − ν ) is cancelled by the zeros of Ψ n arising from S -symmetry. The pole of S b b b b ( ζ )at ζ = iπ (1 − ν ) is crossed when θ ′ is shifted by iπ (1 − ν ), hence we need the zero of thefactor C n at i πν to compensate it, while the pole at ζ = iπν is not reached by the shift.The commutators (15) and (16) cancel each other due to the property π − θ b b b = θ b ( b b ) (see Eq. (4)). The commutators [ z † ( f + ) , χ ′ ( g )] and [ χ ( f ) , z ′† ( g + )]These commutators are the adjoints of the previous ones, therefore, they cancel weaklyby the above computations.This shows the weak-commutativity property of the fields ˜ φ ( f ) and ˜ φ ′ ( g ). While beingalready a major step towards the construction of the model in the algebraic setting, it wouldbe important to obtain a proof of strong commutativity of these fields in order to constructthe corresponding wedge-algebras and to prove the existence of strictly local observablesthrough intersection of a shifted right and left wedge. The proof of strong commutativityis however a hard task because of the subtle domain properties of ˜ φ ( f ) as mentioned at thebeginning of Sec. 5. We are in fact able to show that ˜ φ ( f ) is a symmetric quadratic formon a suitable domain of vectors, but it is not self-adjoint. Therefore, for the proof of strongcommutativity, we would need not only to prove existence of self-adjoint extensions of thetwo fields, but also to select the ones that strongly commute. Some results in this directionare recently available in [Tan15, Tan16] in the case of scalar S-matrices with bound states(e.g. the Bullough-Dodd model), but these techniques are hard to extend to more generalS-matrices. Remark . Our proof depends only on the axioms and properties summarized in Section2.2 and not on the specific expressions of the S-matrix. This implies that our constructionand the proof of weak commutativity work as well if one considers S-matrix such as S b ℓ b k b k b ℓ ( ζ ) = S SG b ℓ b k b k b ℓ ( ζ ) N Y j =1 S j, CDD b ℓ b k b k b ℓ ( ζ ) , where S j, CDD b ℓ b k b k b ℓ ( ζ ) is a factor as in (1) with (possibly different) parameters ν j, ± , and N is anodd number (this is necessary to maintain (S10)). Therefore, we have abundant candidatesfor integrable QFT with the fusion structure considered in this paper.20 Concluding remarks We have investigated the construction of integrable models with bound states in a seriesof two papers [CT15, CT16]. In the second paper the construction methods introducedin [CT15] are extended to a class of models with several particle species and “diagonal” S-matrices with poles in the physical strip, which includes the Z ( N )-Ising model and the affine-Toda field theories as examples. This construction is based on finding observables localizedin unbounded wedge-shaped regions to avoid infinite series that characterize strictly localoperators. These strictly local observables, with some regularity condition on S , should berecovered by taking intersection of the algebras generated by observables in right and leftwedges (c.f. [Lec08, AL16]).Here we considered a model which arises as a deformation of the massless sine-Gordonmodel with a parameter ν which corresponds to a certain range of the coupling constant, < ν < , with an additional CDD factor. As for the proof of weak wedge-locality, we needonly some properties of the S-matrix components, and there are abundant examples, as wepointed out in Remark 5.2. As far as we know, that QFTs with such S-matrices have neverappeared in the literature. It is an interesting problem to find (or exclude) a Lagrangiandescription of them (note that the CDD factors appearing here are necessary and our S-matrix cannot be considered as a perturbation of the sine-Gordon model in the sense of, e.g.,[SZ16]). In this respect, let us observe that we could find the sign-adjusting CDD factor onlyfor the interval < ν < , while ν = 1 corresponds to the (doubled) Ising model. As thereis a gap ≤ ν < 1, this casts doubt that a naive perturbation argument should work.The resulting theory describes two breathers b , b subject to elastic scattering and withthe property that they can also fuse to form a bound state (the fusion processes are ( b b ) → b , ( b b ) → b and ( b b ) → b ). This model falls again into the class of “diagonal” S-matrices, and in this sense, it can be regarded as an extension of the previous techniquesinvestigated in [CT16]. This fusion table is the same as the restriction of the table of theThirring model [Smi92, BFKZ99] to the breather-breather sector (note that it is called “thesine-Gordon model” in the literature in the form factor programme, e.g. [BFKZ99], assumingthe equivalence between them). Yet, the original breather-breather S-matrix of the Thirringmodel does not satisfy the positivity of residues (see Section 2.2), hence cannot be consideredas a separate model. In this sense, the present paper highlights the really necessary propertiesof the S-matrix for wedge-locality and contains a new hint in the construction of interactingquantum field theories in the algebraic framework.An interesting problem would be an extension of such a construction to integrable modelswith “non-diagonal” S-matrices, e.g. the Thirring model [BFKZ99] or SU( N )-invariant S-matrices [BFK08]. It would be interesting to show that weak wedge-commutativity holds atleast for some of these models. They are currently under investigation. It should be notedthat commutation relations of pointlike fields have not been proved for these models . Ourmethods represent a complementary way of proving existence of local observables, whichmay work if the S-matrix components concerning elementary particles (solitons in the caseof the Thirring model) have only simple poles, yet here several analytic questions (such asthe domains of unbounded operators and the modular nuclearity) must be addressed. Michael Karowski, private communication. ν The residue of pole of S Comment4 / < ν < − i R + No adjusting CDD factor found2 / < ν < / − i R + Adjusting CDD factors found1 / < ν < / − i R + There are three breathers if one requires themaximal analyticity within breathers.No adjusting CDD factor found0 < ν < / i R + There is a breather b K for whichRes ζ = iθ bK +1 b bk S b K b b b K ( ζ ) ∈ − i R + Table 2: Ranges of the coupling constant ν in the sine-Gordon modelAs we mentioned in Section 2.2, the S-matrix studied in the present paper is a deformationof the S-matrix of the sine-Gordon model in the range of the coupling constant < ν < by a CDD factor. The reason for the CDD factor is the following: while the fusion table ofthe breather-breather S-matrix is closed under fusions, these S-matrix components cannotbe considered as a separate model because the residues of some poles in the physical stripare on − i R + (see comment before Eq. (3)), which is not compatible with our proof. We notethat also in the proof of local commutativity theorem in the form factor programme [Que99]this property is used, therefore, it must be adjusted in some way. Varying the range of thecoupling constant ν , the situation is as pictured in Table 2. In particular, as explained inSec. 2.2, for < ν < ν − and ν + which fulfill the required conditionsafter Eq. (2), and our simplest form for a CDD factor does not work. For < ν < thereare three breathers in the model (if we take the maximal analyticity literally), and both S SG1111 and S SG2112 have s -channel poles with residues in − i R + . We could not find a suitableCDD factor adjusting all the residues. Finally, in the range 0 < ν < there is an increasingnumber of breathers by maximal analyticity, and while Res ζ = iθ b b b S SG1111 ( ζ ) ∈ i R + , there areother S-matrix components whose residues are in − i R + . We could not find a suitable CDDfactor for this range as well.Finally, the domain of the operator χ ( f ) is considerably small, one can not only show thateven the one-particle components χ ( f ) is not self-adjoint, see [Tan15], but the domains of χ n ( f ) must be somehow enlarged compensating the factor C n . We believe that these domainissues are fundamentally related with the complicated fusion processes of the models, hencedeserve a separate study. Acknowledgements We thank Michael Karowski for informing us of the current status of the form factor pro-gramme. Y.T. thanks Sabina Alazzawi and Wojciech Dybalski for the discussion on therelations between Thirring and sine-Gordon models.Y.T. is supported by the JSPS overseas fellowship.22 eferences [Ala14] Sabina Alazzawi. Deformations of quantum field theories and the con-struction of interacting models. 2014. Ph.D. thesis, Universit¨at Wien, http://arxiv.org/abs/1503.00897 .[AL16] Sabina Alazzawi and Gandalf Lechner. Inverse scattering and locality in integrablequantum field theories. 2016. https://arxiv.org/abs/1608.02359 .[BFKZ99] H. Babujian, A. Fring, M. Karowski, and A. Zapletal. Exact form factors in integrablequantum field theories: the sine-Gordon model. Nuclear Phys. B , 538(3):535–586, 1999. http://arxiv.org/abs/hep-th/9805185 .[BK02] H. Babujian and M. Karowski. Sine-Gordon breather form factors and quantum fieldequations. J. Phys. A , 35(43):9081–9104, 2002.[BFK08] H. Babujian, A. Foerster, and M. Karowski. The nested SU( N ) off-shellBethe ansatz and exact form factors. J. Phys. A , 41(27):275202, 21, 2008. https://arxiv.org/abs/hep-th/0611012v1 .[BR16] Dorothea Bahns and Kasia Rejzner. The quantum sine Gordon model in perturbativeAQFT. 2016. https://arxiv.org/abs/1609.08530 .[BFM07] G. Benfatto, P. Falco, and V. Mastropietro. Functional integral construction of themassive thirring model: Verification of axioms and massless limit. Commun. Math.Phys , 273(1):67–118, 2007.[BFM09] G. Benfatto, P. Falco, and V. Mastropietro. Massless sine-Gordon and massive Thirringmodels: Proof of the Coleman’s equivalence. Commun. Math. Phys. , 285: 713–762, 2009.[BT13] Marcel Bischoff and Yoh Tanimoto. Construction of Wedge-Local Nets of Observablesthrough Longo-Witten Endomorphisms. II. Comm. Math. Phys. , 317(3):667–695, 2013. http://arxiv.org/abs/1111.1671 .[BT15] Marcel Bischoff and Yoh Tanimoto. Integrable QFT and Longo-Witten endomorphisms. Ann. Henri Poincar´e , 16(2):569–608, 2015. http://arxiv.org/abs/1305.2171 .[CT15] Daniela Cadamuro and Yoh Tanimoto. Wedge-Local Fields in IntegrableModels with Bound States. Comm. Math. Phys. , 340(2):661–697, 2015. http://arxiv.org/abs/1502.01313 .[CT16] Daniela Cadamuro and Yoh Tanimoto. Wedge-local fields in integrable models withbound states II. diagonal S-matrix. 2016. to appear in Ann. Henri Poincar´e, http://arxiv.org/abs/1601.07092 .[DT11] Wojciech Dybalski and Yoh Tanimoto. Asymptotic completeness in a class of mass-less relativistic quantum field theories. Comm. Math. Phys. , 305(2):427–440, 2011. http://arxiv.org/abs/1006.5430 .[Lec03] Gandalf Lechner. Polarization-free quantum fields and interaction. Lett. Math. Phys. ,64(2):137–154, 2003. http://arxiv.org/abs/hep-th/0303062 . Lec08] Gandalf Lechner. Construction of quantum field theories with factorizing S -matrices. Comm. Math. Phys. , 277(3):821–860, 2008. http://arxiv.org/abs/math-ph/0601022 .[LS14] Gandalf Lechner and Christian Sch¨utzenhofer. Towards an operator-algebraic construc-tion of integrable global gauge theories. Ann. Henri Poincar´e , 15(4):645–678, 2014. http://arxiv.org/abs/1208.2366 .[LST13] Gandalf Lechner, Jan Schlemmer, and Yoh Tanimoto. On the equivalence of two de-formation schemes in quantum field theory. Lett. Math. Phys. , 103(4):421–437, 2013. http://arxiv.org/abs/1209.2547 .[Que99] Thomas Quella. Formfactors and locality in integrable models of quantum field theoryin 1+1 dimensions (in German). 1999. Diploma thesis, Freie Universit¨at Berlin. .[Sch97] Bert Schroer. Modular localization and the bootstrap-formfactor program. Nuclear Phys.B , 499(3):547–568, 1997. http://arxiv.org/abs/hep-th/9702145 .[Smi92] F.A. Smirnov. Form factors in completely integrable models of quantum field theory ,volume 14 of Advanced Series in Mathematical Physics . World Scientific Publishing Co.Inc., River Edge, NJ, 1992.[SZ16] F.A. Smirnov and A.B. Zamolodchikov. On space of integrable quantum field theories.2016. https://arxiv.org/abs/1608.05499 .[Tan12] Yoh Tanimoto. Construction of Wedge-Local Nets of Observables ThroughLongo-Witten Endomorphisms. Comm. Math. Phys. , 314(2):443–469, 2012. http://arxiv.org/abs/1107.2629 .[Tan14] Yoh Tanimoto. Construction of two-dimensional quantum field modelsthrough Longo-Witten endomorphisms. Forum Math. Sigma , 2:e7, 31, 2014. http://arxiv.org/abs/1301.6090 .[Tan15] Yoh Tanimoto. Self-adjointness of bound state operators in integrable quantum fieldtheory. 2015. http://arxiv.org/abs/1508.06402 .[Tan16] Yoh Tanimoto. Bound state operators and wedge-locality in integrable quantum fieldtheories. SIGMA Symmetry Integrability Geom. Methods Appl. , 12:100, 39 pages, 2016. https://arxiv.org/abs/1602.04696 .[ZZ79] Alexander B. Zamolodchikov and Alexey B. Zamolodchikov. Factorized S -matrices intwo dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Physics , 120(2):253–291, 1979., 120(2):253–291, 1979.