Integrable QFT and Longo–Witten Endomorphisms

Our previous constructions of Borchers triples are extended to massless scattering with nontrivial left and right components. A massless Borchers triple is constructed from a set of left–left, right–right and left–right scattering functions. We find a correspondence between massless left–right scattering S-matrices and massive block diagonal S-matrices. We point out a simple class of S-matrices with examples. We study also the restriction of two-dimensional models to the lightray. Several arguments for constructing strictly local two-dimensional nets are presented and possible scenarios are discussed.


Introduction
Here we further study our operator-algebraic approach to constructing quantum field models in the two-dimensional spacetime. In the previous works we have established the general theory of (wedge-local) massless excitations [19,42] and constructed several families of examples [7,42]. It has been revealed that from a pair of chiral components of conformal field theory and an appropriate S-matrix one can construct the von Neumann algebra corresponding to the wedge-shaped region. The operators in strictly local regions are to be determined through the intersection of such wedges [8]. In our previous result, we considered only simple particle spectrum. Here we allow multiple particle spectrum. Given a set of massless S-matrices, we construct a Borchers triple, which is a weakened notion of Haag-Kastler net. A corresponding massive result has been obtained in [28]. We show also that given a set of massless inside: A(O) := D a,b ⊂O A(D a,b ) . Furthermore, one can extend the representation of the translation group to a representation of the whole Poincaré group using the Tomita-Takesaki theory of von Neumann algebras. Namely, one defines the representation guided by the Bisagnono-Wichmann property above and Borchers' theorem ensures that this really defines a representation of the Poincaré group [8]. More precisely, the one-parameter unitary group {Δ it } canonically associated with the pair of a von Neumann algebra M and Ω represents the Lorentz boosts.
Then one can show that this "net" (A, U, Ω) satisfies almost all of the properties of Haag-Kastler net. But, while the wedge algebras are by definition always sufficiently large, i.e. they generate the whole Hilbert space H from the vacuum Ω, it is in general difficult to show that for local algebras A(D a,b ) and it can actually fail [42,Theorem 4.16]. But if it is the case, then the triple indeed defines a Haag-Kastler net by the above structure. This program has been accomplished in some cases and obtained families of interacting models [26,44]. (2) The joint spectrum of T is contained in the closed forward lightcone V + := {(a 0 , a 1 ) ∈ R 2 : a 0 ≥ |a 1 |}. (3) Ω is cyclic and separating for M. In the sense explained above, a Borchers triple gives a Poincaré covariant, wedge-local net defined by Eq. (1) and can be considered to be a "net of observables localized in wedges". If Ω is cyclic for the von Neumann algebra M ∩ Ad T (a)(M ) for any a ∈ W R , one can construct a Haag-Kastler net on the original Hilbert space H and in this case we say that the Borchers triple (M, T, Ω) is strictly local. In Sects. 3 and 4 we construct Borchers triples and Sect. 5 is concerned with strictly local triples.
The Massive Scalar Free Field. The simplest Borchers triple is constructed from the simplest quantum field. The one-particle Hilbert space of the free scalar field of mass m > 0 is given by H m := L 2 (R, dθ) and the translation acts by (T m (a)ψ)(θ) = e ipm(θ)·a ψ(θ), where p m (θ) := (m cosh(θ), m sinh(θ)) parametrizes the mass shell. We need the unsymmetrized Hilbert space H Σ m := H ⊗n m and the symmetrized Hilbert space H r := P n,sym H ⊗n m , where P n,sym is the projection onto the symmetric subspace.
Let a † r and a r be the creation and annihilation operators as usual (see [44,Section 2.3]. In our notation, a † r (ψ) is linear and a r (ψ) is antilinear with respect to ψ). The (real) free field φ r is defined by where f is a test function in S (R 2 ) and J m ψ(θ) = ψ(θ). Our von Neumann algebra is The translation on the full space is the second quantized representation T r := Γ(T m ) and there is the Fock vacuum vector Ω r ∈ H r . This triple (M r , T r , Ω r ) is the Borchers triple of the free field. Of course this is strictly local and the corresponding net is the familiar free field net. A more abstract definition of this free field construction starting from a general positive energy representation of the Poincaré is given in [11].
Examples From Integrable Models. The form factor bootstrap program, an approach to integrable quantum field theory, can be briefly summarized as follows [40]: first a model with infinitely many conserved current is considered.
The scattering matrix turns out to be factorizing; then the explicit form of it is speculated by a symmetry argument. Finally, one finds solutions of the so-called form factor equation, which is given in terms of the two-particle scattering function. A solution of the form factor equation is a series of functions.
It is supposed to serve as the matrix coefficients of a local observable. The convergence of the series as an operator is expected in a wide class of models but remains open. An alternative approach has been initiated by Schroer [37,38] and worked out by Lechner [26]. In this approach, given an S-matrix, the operators localized in a wedge are constructed and the local observables are obtained as the intersection of left and right wedges. The determination of the intersection, which in the form factor program would correspond to finding form factors (and proving the convergence), has been done with the help of operator algebraic methods including the Tomita-Takesaki theory of von Neumann algebras [13].
The one-particle space H 1 is the same as that of the free field. On n-particle space one defines the S 2 -permutation by This time P n,S2 is the orthogonal projection onto the subspace of H ⊗n We take the Hilbert space H S2 := P n,S2 H ⊗n 1 ; the representation T S2 is the second quantized promotion of T 1 and the Fock vacuum is denoted by Ω S2 . The creation and annihilation operators are given by (z † S2 (ψ)Φ) n = √ nP n,S2 (ψ ⊗ Φ n−1 ) and z S2 (ψ) = z † S2 (ψ) * . For a test function f on R 2 , the wedge-local field is defined also as greatly reduces the problem [19,42,43]. Of particular importance is the result [42, Section 3] that a Haag-Kastler net which is asymptotically complete with respect to waves (the corresponding notion of massless particles in the twodimensional spacetime) can be easily reconstructed from its asymptotic (free) behavior and the S-matrix. In this paper we are concerned only with such models.
Borchers Triples by Tensor Product. A (two-dimensional) Borchers triple can be constructed out of a pair of one-dimensional Borchers triples (M ± , T ± , Ω ± ) as follows: let (t + , t − ) be the lightray coordinates of R 2 , where t + = t 0 − t 1 and t − = t 0 + t 1 (the indices might look unnatural, but are consistent with the scattering theory [12,19]). One takes a triple (M, T, Ω) where Then it is immediate to see that this is a Borchers triple. The representation T is said to contain waves, in the sense that there are nontrivial spectral projections concentrated in the lightrays. This triple naturally turns out not to interact, namely the S-matrix is the identity operator I [42].
How to Construct Interacting Models. We do not repeat the definition of asymptotic completeness for waves [12,19]

dimensional Borchers triples and S is a unitary operator on
The correspondence is given by Indeed, the properties of net (strict locality) are used only to show the Möbius covariance of the one-dimensional components, which we do not claim here and the rest of the proofs works. Our program to construct massless Borchers triples is now split into two steps: first prepare a pair of one-dimensional Borchers triples; then find an appropriate operator S to make them interact. We carry out this program in Sect. 3. We do not investigate strict locality in the present paper.

Massless Models With Nontrivial Scattering
Here we construct massless Borchers triples following the program described in Sect. 2.3. As an input we take so-called left-left, right-right and left-right scattering matrices (c.f. [4]).
Usually the form factor bootstrap program is carried out for massive models. Massless limit makes worse the behavior of the form factors in the momentum space and even the fundamental "local commutativity theorem" [40] fails when applied to concrete cases. As for the operator algebraic approach, the modular nuclearity has been proved through a careful estimate [26], which will no longer be valid for the massless case.
Yet in operator-algebraic approach, half of the program can be carried out: one can construct certain operators to be interpreted as observables in a wedge. This has been done in [28] for the massive case with multiple particle spectrum and in [7,42] for the massless case with simple spectrum. In this Section we exhibit a massless construction which includes several kinds of particles. It is also interesting to observe at which point the Yang-Baxter equation enters.

Scattering Matrices and Operators
As in massless bootstrap program, we need two kinds of input: left-left and right-right scattering and left-right scattering. While the former governs the asymptotic behavior of the model, the latter is directly related to the S-matrix.

Scattering Matrices for Chiral Parts.
One-dimensional Borchers triples can be obtained by second quantization of so-called standard pairs, similarly to the algebraic construction of massive models with factorizing S-matrices [28] and the free field construction in [11]. This will be done on a R-symmetric Fock space (defined in Sect. 3.2), where R is a certain operator. We give an abstract definition for suitable operators R and characterize them in terms of usual scattering matrices. They are called left-left or right-right scattering operator in physics literature from a formal similarity to S-matrix, but the physical meaning of R remains unclear, c.f. [10].
Let H be a Hilbert space. For operator A ∈ B(H ⊗ H) we denote by A ij the operator on B(H ⊗n ) (n ≥ i, j) which acts by A on the product of the i-th and the j-th tensor factors. For example, if  [29].
Let H be a standard subspace of a Hilbert space H and let us assume that there exists a one-parameter group T (t) = e itP on H such that • T (t)H ⊂ H for all t ≥ 0, • P is positive and P has no point spectrum in 0.
Then we call the pair (H, T ) a (non-degenerate) standard pair. A standard pair is called irreducible if it cannot be written as a non-trivial direct sum of two standard pairs. There exists a unique (up to unitary equivalence) irreducible standard pair (H 0 , T 0 ) whose "Schrödinger representation" is given as follows: we realize (H 0 , T 0 ) on H 0 = L 2 (R) and T 0 (t) = e itP0 , where Q 0 = ln P 0 with (e itQ0 f )(q) = e itq f (q). A function f ∈ L 2 (R) is in H 0 if and only if f admits an analytic continuation on the strip R + i(0, π), such that for every a ∈ (0, π) it is: f ( · + ia) ∈ L 2 (R) with boundary value f (q + iπ) = (J H0 f )(q) := f (q). One defines (Δ −is H0 f )(q) = f (q + 2πs) and it can be easily checked that (J H0 , Δ it H0 ) are indeed the modular objects for H 0 [29].
For a standard pair (H, T ) we give an abstract definition of an operator R, which encodes the two-particle scattering process. (1) Reflection property: R 21 = R * .
(2) Yang-Baxter equation: H and ξ, η ∈ H or equivalently: the operator A R f,g defined by with {e k } an orthonormal basis of H, is self-adjoint for all f ∈ H, g ∈ H .
We will see that the locality assumption follows from the requirement that, on two-particle level, certain generators of the wedge-algebra fulfill halfline locality in Lemma 3.11.
We remember that each (non-degenerate) standard pair (H, T ) is a direct sum of the unique irreducible standard pair (H 0 , T 0 ) [30]. A standard pair with multiplicity n can be given as follows: we can choose a Hilbert space K with dim K = n and H = H 0 ⊗ K ∼ = L 2 (R, C n ) and T (t) = e itP := T 0 (t) ⊗ 1, To make contact with the physics literature, we choose some orthonormal basis indexed by {α} of K and an involution α →ᾱ on the index set and define the antiunitary involution J H to be Then a function f = (f α ) ∈ L 2 (R, C n ) is in H if and only if f admits an analytic continuation on the strip R + i(0, π), such that for every a ∈ (0, π) it is f α ( · + ia) ∈ L 2 (R) with boundary value f α (q + iπ) = fᾱ(q). Every standard pair with finite multiplicity is of this form. Due to unitarity, translation covariance and the fact that R commutes with Δ it H ⊗ Δ it H , a two-particle scattering operator is given by the spectral calculus by R(Q 1 − Q 2 ), where Q 1 = Q ⊗ 1, Q 2 = 1 ⊗ Q, Q = ln P , P is the generator of T and q → R(q) is a operator-valued function from R to B(K⊗K) which is unitary almost everywhere. By fixing a basis on K, we can represent R(q) as a matrix R αβ γδ (q) (almost everywhere). In the above representation this reads where it is sometimes common to use the matrix valued function q → S(q) with interchanged indices, c.f. [28].
Note. In the following, symbols with underline denote matrix-valued functions or equivalently functions with operator-value on a finite dimensional Hilbert space.
Let us define the operator-valued function where ξ ∈ H. The partial disintegration of R reads where dE β δ = dE 0 ⊗ E β δ and dE 0 is the spectral measure of Q 0 = ln P 0 and E β δ is the operator corresponding on the fixed basis {ξ α } to the matrix which has the value 1 in (β, δ)-component and 0 in the others.
Before giving a characterization of the operators R ∈ S(H, T ) we prove the following Lemma, which will reduce the argument of half-line locality to two-particle processes: Lemma 3.2. Let (H, T ) be a standard pair, R ∈ S(H, T ) andR = R 1,n+1 R 1,n · · · R 1,2 on H ⊗n+1 . Then the operator AR f,g ∈ B(H ⊗n ) given by Proof. Because every standard pair is just a direct sum of the irreducible standard pairs, we may assumeᾱ = α in the above decomposition. We can write R as Then by the assumption that R ∈ S(H, T ), for all f ∈ H, g ∈ H we have which is equivalent to R β δ (q)S H ⊂ S H R δ β (q) for almost all q by Lemma A.1. But this implies that also R β1 δ1 (q 1 ) · · · R βn δn (q n )S H ⊂ S H R δ1 β1 (q 1 ) · · · R δn βn (q n ) holds; hence using again Lemma A.1 the equality of the following two operators follows: and which proves the claim.
We characterize the two-particle scattering operators R in terms of matrix-valued function and show that they indeed come from two-particle scattering matrices (c.f. [28]). (1) Unitarity: S(q) is an unitary matrix for almost all q ∈ R.
(3) Yang-Baxter equation: As discussed above the ansatz in Eq. (3) is equivalent to unitarity, translation covariance and the fact that R commutes with Δ it H ⊗ Δ it H . It is straightforward to check that hermitian analyticity of S( · ) is equivalent to the reflection property of R; the property R(J H ⊗J H ) = (J H ⊗J H )R * is equivalent to TCP, and Yang-Baxter equation of R with the one for the matrices S(q).
Using R δ β (s) defined in Eq. (4) we write A R f,g as [30] this is equivalent to R( · − q) being a bounded analytic matrix valued function on

Two-Particle Left-Right Scattering Matrices.
In this section we give an operator definition for two-particle scattering functions which describe the scattering behavior of a left and right moving particle in the sense of Fock space excitations. Bernard remarked that, for the left-right scattering, two of the conditions can be combined and thus weakened [4]. The following is our precise rendition in terms of standard subspaces: Using the physicists' notation, we will define the operator is an orthonormal basis of H − and analogously for "bra" on the second component. Left/right locality is with this notation equivalent to self-adjointness of the operators We use the same parametrization as before for the standard pairs where by abuse of notation S( · ) = (S αβ γδ ( · )) is a matrix valued function. The operators S ∈ S(R + , R − ) are characterized as follows: (2) Left mixed Yang-Baxter identity: For almost all q, q ∈ R following holds: (3) Right mixed Yang-Baxter identity: For almost all q, q ∈ R the following holds: (4) Analyticity: q → S(q) is boundary value of a bounded analytic function on R + i(0, π).
(5) Mixed unitary-crossing relation: S αβ γδ (q + iπ) = Sᾱ δ γβ (q) = S γβ αδ (q) holds. Proof. The above ansatz by a matrix-valued function is the most general ansatz Then the two notions of unitarity and Yang-Baxter identities can be checked to be pairwise equivalent. The proof that left and right locality are equivalent to the analyticity and mixed unitary-crossing relation is completely analogous to the proof of Proposition 3.3. So left and right locality is equivalent to (S αβ γδ (q)) being a bounded analytic matrix valued function on R + i(0, π) with boundary values S αβ γδ (q + iπ) = S γβ αδ (q) and S αβ γδ (q + iπ) = Sᾱ δ γβ (q), respectively, almost everywhere.

Examples.
One can see that the conditions in our Proposition 3.3 and [28, Definition 2.1] are essentially the same: the mass parameters and the global gauge action can be added by hand. They assume continuity at the boundary, but it is clear from the proof that their proof works with noncontinuous boundary values.
Hence, as for S(H + , T + ), we have the same set of examples as [28]. We point out that the S-matrix of the O(N ) σ-models satisfies our conditions, where H + has multiplicity N and S αα [1,28] for detail) and δ is the Kronecker Delta.
As for left-right scattering, we present a class of examples. The O(N ) σmodels can be used to construct examples of this class. Let us take R ∈ S(H, T ) and assume that F RF = R, where F ξ 1 ⊗ ξ 2 = ξ 2 ⊗ ξ 1 . For the corresponding matrix-valued function R, this means R αβ γδ = R βα δβ . Let us say in this case that R satisfies the flip symmetry. It is clear that the S-matrices of the O(N ) σmodels satisfy this. We claim that R itself can play the role of the left-left, right-right and left-right scatterings. Proof. From Proposition 3.3 we know that the matrix-valued function S αβ γδ := R βα γδ satisfies the conditions listed there and the necessary properties ofS = S(Q + ⊗ 1 + 1 ⊗ Q + ) in Proposition 3.5 can be read off: Unitarity is trivial. SinceS is defined through the same function R, the left and right Yang-Baxter equations follow trivially from the Yang-Baxter equation for R (note that Proposition 3.3 is written in S and must be translated in R). Analyticity foȓ S is exactly the analyticity of R. Finally, the mixed unitary-crossing relation can be shown as follows: , where we used the definition ofS, the definition of S, the crossing symmetry for S, Hermitian analyticity of S and the definitions of S andS in this order. This is the first of the Mixed unitary-crossing relation. The second relation is obtained by applying the flip symmetry to the both sides of the first relation and replacing the labels as α ↔ β, γ ↔ δ.
Hence we obtain a concrete family of left-right scattering operators out of O(N ) σ-models. We do not know whether there are Lagrangians for our new S-matrices. We will construct corresponding massless Borchers triples in Sect. 3.3 and massive Borchers triples in Sect. 4. This in turn gives again another family of left-left scattering. In order to repeat this procedure, it is necessary that the starting R satisfies further symmetry R(q) = R(iπ − q). We do not know any such example except constant matrices or scalar case [44].

Second Quantization of Standard Pairs
where D n (π) i1···in acts on i 1 · · · i n -th tensor components. The correspondence given by Φ = F R and D n is defined by D n (τ j ) = Φ j,j+1 for 1 ≤ j ≤ n − 1 and τ j is the transposition of j ↔ j + 1.
Proof. Given unitary R with F RF = R * , define Φ := F R and, therefore, It is obvious that F 12 F 23 F 12 = F 23 F 12 F 23 holds. For F and S we get the commutation relation S 12 F 23 = F 23 S 13 and, therefore, From this, the equivalence between the 1. and 2. is clear.
For τ i the transposition of the i-th and (i + 1)-th element, we define D n (τ i ) = Φ i,i+1 , which gives a representation of Bischoff and Y. Tanimoto Ann. Henri Poincaré by the properties of Φ. Given {D n } we set Φ := D 2 (τ 1 ) and we observe that the family is already fixed by D n (τ i ) = Φ i,i+1 , because the transpositions generate S n .
For a pair (H, R) of a Hilbert space and a unitary R ∈ U(H⊗H) fulfilling R 21 ≡ F RF = R * and the Yang-Baxter identity, i.e. Properties (1) and (2) of Definition 3.1, we associated the Fock space F H,R given by and The construction is functiorial, from the additive (by taking direct sums) category with Objects. Pairs (H, R) of a Hilbert space and a unitary R ∈ U(H ⊗H) fulfilling R 21 = F RF = R * and the Yang-Baxter relation.

Morphisms. Contractions
to the multiplicative (by taking tensor products) category of Hilbert spaces with contractions, which is given by namely they are preserved on the full Fock space and where for an antilinear operator A we define A 0 as the complex conjugation on C and F 1···n (f 1 ⊗ · · · ⊗ f n ) = f n ⊗ · · · ⊗ f 1 . This is well-defined, namely we have . This can also be formulated aŝ where in the product the operators are lexicographically ordered from left to right (or equivalently from right to left by YBE). Namely, for ψ ∈ H ⊗n the restricted vector P R ψ is R-symmetric in the sense that we have From this one can show that on H ⊗n it holds that F 1···n P R = 1≤i<j≤n R ij P R .

Second Quantization on R-Symmetric
Let D be the vectors with finite particle number, i.e. Ψ ∈ F H,R where n-th component vanishes for sufficiently large n. We define on F H,R the compressed operators a(f ) = P R b(f )P R and define the Segal type field φ(f ) = a(f ) + a(f ) * on D which is symmetric. We note that f → φ(f ) is just real linear.

If H is cyclic then Ω is cyclic for the polynomial algebra of
Proof. We proceed as in [25,Proposition 4.2.2]. For Ψ n ∈ D with N Ψ n = nΨ n we get with c f = 2 f with the help of the bounds of Lemma 3.8 the estimate = a(Uf)ξ and Γ(U )a(f ) * Γ(U * ) = (Γ(U )a(f )Γ(U * )) * = a(Uf) * ; hence we obtain 4. The cyclicity can be shown inductively, namely by applying φ(f ) on Ω one can show that one obtain a total set in P R H ⊗n .
We define for every real subspace H ⊂ H the von Neumann algebra This can be seen as a generalization of the CCR and CAR algebra.
Proof. The first statement is clear and the second follows from continuity. The covariance with respect to unitaries with [U ⊗ U, R] = 0 follows from the covariance of φ(f ). Let f 1 , · · · , f n ∈ H and let E k (t) be the spectral projection of the self-adjoint operator φ(f k ) on the spectral values [−t, t]. Then The cyclicity of Ω for M then follows from the cyclicity of Ω for φ.

R-Symmetric Second Quantization of Standards Pairs and Modular
Theory. In this section we are interested in the construction of one-dimensional Borchers triples from a standard pair (H, T 1 ) on H. It turns out that for all R ∈ S(H, T 1 ) it is possible to construct a one-dimensional Borchers triple on the "twisted Fock space" F H,R . Before we turn to the von Neumann algebras we first need commutation relation of the Segal field φ(f ) with the "reflected Segal field" Jφ(f )J. One can think of φ(f ) for f ∈ T 1 (a)H as a field localized in a right half-ray R + + a and of φ (g) := Jφ(J H g)J as a field localized in the left half-ray R − + b for g ∈ T 1 (b)H . Proof. Note that h| 1 and h| n , operators on F Σ H , preserve P R H ⊗n because P R H ⊗n is characterized by R-symmetry (see Sect. 3.2.1) and h| 1 and h| n do not affect the decomposition of a permutation into transpositions. For Ψ n ∈ P R H ⊗n we get Therefore, we have holds, where X 11 = 1 by convention and X 1i : 12 . In other words, this amounts to R-symmetrizing the first component since the rest is already 588 M. Bischoff and Y. Tanimoto Ann. Henri Poincaré R-symmetric. Therefore, the creation operator acts on Ψ n ∈ P R H ⊗n by a(f )Ψ n = 1 √ n+1 n+1 i=1 X 1i (f ⊗ Ψ n ) and we calculate: Finally, restricted to P H ⊗n withR = R 1,n+1 R 1,n · · · R 12 , holds for all f, g ∈ H because of Lemma 3.2. To show that M and M 2 commute we need to use energy bounds. Let P 0 = dΓ(P 1 + 1/P 1 ) ≥ 2 with domain D 0 be the generator of Γ(e it(P1+1/P1) ). We get P 0 ≥ 2N . We will see in Sect. 5.2 (only for the irreducible case, but reducible cases are just parallel) that P 1 and 1/P 1 can be identified with the generators of positive and negative lightlike translations in a massive representation. Hence P 1 + 1/P 1 is the generator of the timelike translations. Real Schwartz test functions with support in W R are mapped densely into H as we will see in Sect. 5.2. We get bounds from the proof of Lemma 3.8 and because the multiplicity is finite, it holds that (1 + P 0 ) − 1 2 φ(f ) < ∞ on D 0 and similar for the commutator [P 0 , φ(h)] = φ(∂ 0 h), where h is a test function with support in W R , ∂ 0 h is the timelike derivative and φ(h) is defined through the mapping mentioned above, and for Jφ(h)J, [P 0 , Jφ(h)J] (see also the argument in [13, Proposition 3.1]). By the commutator theorem [18] one can conclude that e iφ(h) and e iJφ(g)J commute for all such h, g which by continuity implies that M and M 2 commute.
The property of the modular operators (3) is proved as in [13, Proposition 3.1] and Ω is the unique translation invariant vector, because we assume that standard pairs are non-degenerate. Special cases of such models were constructed in [10] and were proposed as scaling limits of two-dimensional models with factorizing S-matrices. We will present a direct relation to massive models in two dimensions via a class of Longo-Witten unitaries like in Sect. 5; in other words, via the idea of lightfront holography.  [30] by matrices of analytic function, we get that these are exactly constant matrices in U(C n ) commuting with R in the above sense where n is the multiplicity of H. Therefore, we can associate with (H, T 1 , R) a compact group G ⊂ U(n) acting by internal symmetries.

Construction of Massless Wedge-Local Models From Scattering Operators
Given two standard pairs (H ± , T 1 ± ) on H ± , respectively, and two operators R ± ∈ S(H ± , T 1 ± ) we obtain two one-dimensional Borchers triples (M ± , T ± , Ω ± ) by the construction of Sect. 3.2.
We show that every S ∈ S(R + , R − ) gives rise to a wave-scattering matrix S as in Proposition 2.1.
Let us define the operatorS = m,n S (m,n) on full Fock space F Σ where we denote for 1 ≤ i ≤ m and 1 ≤ j ≤ n by S i|j ≡ S m|n i|j the operator on H ⊗m + ⊗ H ⊗n − given by S i,j+m (we omit m|n when no confusion arises). We will use notation as f | 1| , R + ij| and R − |ij as well. Namely, if one side of | is empty, then the operator acts trivially on that side.

Lemma 3.16.
Let (H ± , T 1 ± ) be two standard pairs on H ± , respectively, and R ± ∈ S(H ± , T 1 ± ). Given an operator S fulfilling the properties (1) and (2) is given by the sum of products of Φ i,i+1 and because S i+1|• always appears next to S i|• in the definition of S (m,n) , we can conclude that S (m,n) commutes with P (m) for all n. Similarly, one proves from the right YBE that 1 H ⊗m The converse holds because the left and right mixed YBE are equivalent to the commutation ofS with Therefore,S canonically restricts to an operator on F H+,R + ⊗ F H−,R − if and only if the left and right YBE are fulfilled. By abuse of notation we denote the restricted operator also byS. If left locality holds, then forR + = R + 1,m+1| · · · R + 12| S 1|1 · · · S 1|n and the operator AR Proof. The proof is analogous to the proof of Lemma 3.2. For example, for the left case we write and from left locality it holds that W β δ (q)S H+ ⊂ S H+ W δ β (q). Together with (R + ) β δ (q)S H+ ⊂ S H+ (R + ) δ β (q) it follows like in the above-mentioned proof that AR + is self-adjoint.

of Definition 3.4. And letS be the operator on
We calculate on the full tensor product space H ⊗m where we again used that h| • preserves the R ± -symmetric Fock space (see Lemma 3.11) as doS due to Lemma 3.16. Therefore, we get [Ja(g) * J ⊗ 1,S(a(f ) * ⊗ 1)S * ] = 0 and [Ja(g)J ⊗ 1,S(a(f ) ⊗ 1)S * ] = 0 by taking the adjoint on D. To compute mixed commutators we proceed as follows, noting thatS commutes with R + -symmetrization: and it holds on finite particle states that where we use that the operator AR + f,JH + g is self-adjoint for all f, g ∈ H + by Lemma 3.17.
For the second statement similar calculation leads to and the same arguments as above hold.
For the only if part we realize that the commutation of x ⊗ 1 with AdS(x ⊗ 1) implies that [Jφ(g)J ⊗ 1,S(φ(f ) ⊗ 1)S * ] = 0 on a dense domain. The above calculation for the case m = 0 and n = 1 shows that left locality holds and right locality is analogous. where the S-matrix is not of this form. Namely, for the case H ± the irreducible standard pair and R ± = 1 a more general family of wave S-matrix, not compatible with the Fock structure, has been implicitly constructed in [7].
We summarize the construction. Remark 3.21. We recall that in [7,42] we proved the corresponding commutation by decomposing the S-matrix into Longo-Witten unitaries. In this paper we took a slightly different strategy. This was necessary for nondiagonal S-matrix, which is more complicated and does not admit a simple decomposition into Longo-Witten unitaries. On the other hand, the commutation relation we needed is [x ⊗ 1, AdS(x ⊗ 1)] = 0 and it is sufficient that AdS(x ⊗ 1) ∈ M ⊗ B(F H−,R − ); hence on the B(F H−,R − ) side one has a greater freedom. One has to consider not Longo-Witten endomorphisms of M but commutation relations on a larger space. After this observation one can follow the same line of the proofs in [42].
The connection of these extended commutation relations to nets with boundary [30] is unclear.
We showed in [42, Section 3] that the asymptotic chiral components are conformal if the two-dimensional Borchers triple is strictly local. Conversely, in order to construct strictly local Borchers triples, one has to take strictly local one-dimensional components from the beginning. The question whether one-dimensional Borchers triples can be strictly local has been considered in [10], which largely remains open.
From the bootstrap approach, there have been found form factors of some local operators in certain massless models [17,32]. However, the existence of form factors by no means implies the existence of the corresponding Haag-Kastler net. Indeed, we showed [42,44] that in massless models with a prescribed S-matrix, the strict locality can fail. This should be connected with the well-known problem of the convergence of form factors, which is clearly worse in massless cases. Yet, the possibility that one-dimensional Borchers triples can be strictly local is a very interesting problem. We will discuss this point later in Sect. 5.

Massive Models From Left-Right Scattering
In this short section we construct massive Borchers triples. For a given standard pair (H + , T 1 + ), we define the opposite standard pair as follows: Let P 1 + be the generator of T 1 + . We put T 1 + (t) = e it/P 1 + and T + (t) = Γ(T 1 + ).  Proof. A standard pair admits the direct sum decomposition as in [30]. With this decomposition, our claim follows from the result for the irreducible pair [30, Theorem 2.6], namely T 1 + (t)H + ⊂ H + for t ≤ 0. One can use the converse of the one-particle Borchers theorem as well [29,Theorem 2.2.3].
is defined by a functional calculus of T 1 + . Hence it is clear that the second quantization T + = Γ(T 1 + ) restricts to the R + -symmetrized Fock space F H+,R + . Let us recall that one can construct a Borchers triple (M + , T + , Ω + ) (Sect. 3.2). From Lemma 4.1 it follows that Ad T + (t) preserves M + for t ≤ 0. This is equivalent to that Ad T + (t) preserves M + for t ≥ 0. Two representations T + and T + obviously commute; both have the positive generator. Hence the joint spectrum of the combined representation T + (t + )T + (t − ) of R 2 is contained in V + . Furthermore, if (t + , t − ) ∈ W R , or equivalently if t + ≤ 0 and t − ≥ 0 (see Fig. 1 and note an unusual definitions of t + , t − ), then Ad T + (t + )T + (t − )(M + ) ⊂ M + . Namely (M + , T + T + , Ω + ) is a two-dimensional Borchers triple.
By a parallel reasoning, one sees that (M − , T − T − , Ω − ) is a twodimensional Borchers triple, where T − is constructed analogously, but here t + -lightlike translations are given by T − and t − -translations by T − .
Proof. The properties forT andΩ are obvious. It follows from the properties of their two-particle components thatT andS commute; hence AdT (t + , t − ) (M) ⊂M for (t + , t − ) ∈ W R . The cyclicity and separating property ofΩ have been already proven in Proposition 3.20.
We will see in Sect. 5.2 that if (H + , T 1 + ) is irreducible, then T + (t + ) T + (t − ) is a massive representation. It follows immediately that for a reducible pair (H + , T 1 + ) the representation T + (t + ) T + (t − ) is just the massive representation with the same multiplicity. Accordingly, we can call (M S ,T ,Ω) a massive Borchers triple.
It can be easily realized that the construction here is a generalization of [44,Section 6]. Indeed, the present construction takes two standard pairs, not only irreducible ones, and promotes them by R ± -symmetric second quantization, not only by symmetric or antisymmetric second quantization. Finally, the operator S is allowed to have matrix-value, not only scalar. It is also a generalization of [44,Section 3], because S can depend on the rapidity. However, here we will not investigate the strict locality.
One may wonder if the S-matrices from our previous work [7] can be used, which does not preserve the two-particle space. This does not work, at least straightforwardly, because it is not clear whether the S-matrix commutes with the opposite translation T + ⊗ 1.
Finally, we remark that our construction in this section is a special case of [28]. To see this, it is enough to extract a Zamolodchikov-Fadeev algebra from our algebra. This can be done exactly as in [44,Section 6] and we omit the proof. As in [44], our von Neumann algebra is a tensor product twisted byS; hence the scattering inside a component remains the same. We just illustrate how the two-particle S-matrix looks like: As one sees from the construction, the first component is parity-transformed (c.f. Sect. 5); hence the scattering is determined by R + (q) = R + (iπ − q). If both the multiplicities of (H ± , T 1 ± ) are two, then understanding the q-dependence implicitly, it is given by Using the convention of [28] and with an appropriate basis, an S-matrix of this form could be called block diagonal. Of course, such an S-matrix has been already treated in [28] in more generality. The point here is that one can obtain concrete examples from massless left-right scattering.

Further Construction of Massive Models
Here we investigate another connection between two-and one-dimensional Borchers triples. In Sects. 3, 4 our construction has always been carried out on the tensor product Hilbert space. In this Section we work on a single Hilbert space.
A similar connection has been proposed under the name of algebraic lightfront holography [39]. There has been also an effort to reconstruct a full QFT net from a set of a few von Neumann algebras and some additional structure [46] where, however, strict locality remains open. We present a simple sufficient condition in order to reconstruct a strictly local Borchers triple out of a conformal net. This sufficient condition turns out to be hard to satisfy, but we believe that it is of some interest, because techniques to construct models are rather scarce.
The idea to recover the massive free field from the U(1)-current through the endomorphisms associated with the functions e it/p is due to Roberto Longo. Some of the results in this Section have already appeared in the Ph.D. thesis of the author (M.B.) [6].

Holographic Projection and Reconstruction
Let (M, T, Ω) be a (two-dimensional) Borchers triple. As we explained in Sect. 2.2, T can be restricted to the lightray t + = 0, the restriction we denote by T + , and the triple (M, T + , Ω) is a one-dimensional Borchers triple. We observe that the negative lightlike translation T + is now reinterpreted as a one-parameter semigroup of Longo-Witten endomorphisms. Indeed, T + obviously commutes with T + and Ad T + (t + ) preserves M for t + ≤ 0. Furthermore, T + (·) has the positive generator. These properties of T + are actually very rare if we exclude the massless asymptotically complete case which we considered in Sect. 3. Now let us reformulate the situation the other way around. Let (M, T + , Ω) be a one-dimensional Borchers triple and V (t) be a one-parameter semigroup of Longo-Witten endomorphism for t ≤ 0 with positive generator. Let T (t + , t − ) = V (t + )T + (t − ). By assumption T + and V commute; hence T is a representation of R 2 . By the assumed spectral conditions, sp T ⊂ V + . Then we have the following: Proof. The first statement is clear from the definition.
We assume that (M, T + , Ω) is strictly local. Let (t + , t − ) ∈ W R , in other words t + < 0 and t − > 0. One observes that Ad T and Ω is cyclic for the right-hand side by assumption.
As a strictly local Borchers triple corresponds to a Haag-Kastler net, this Theorem gives a simple construction strategy. However, as a natural consequence of difficulty in constructing Haag-Kastler nets, examples of such Longo-Witten endomorphisms seem very rare.
Let us take a closer look at this phenomenon. We take the Borchers triple (M, T + , Ω) associated with the U(1)-current net. Among the endomorphisms found by Longo and Witten, the only one-parameter family with positive generator (negative in their convention [30]) is given by the function ϕ(p) = e it/p with t ≤ 0. As we will see, if we take V ϕ = Γ(ϕ(P 1 )), the above prescription gives just the free massive field net; hence is not very interesting. However, this endomorphisms is expected not to extend to any extension of the U(1)-current net, due to the failure of Hölder continuity of the function e it/p at p = 0 for t < 0. We found another family of such endomorphisms in [7]. We will discuss it in Sect. 5.3.
General properties of such endomorphisms have been studied in [9]. It is very interesting to find out how to construct more examples of one-parameter semigroup of Longo-Witten endomorphisms with the semibounded generator, which would immediately lead to Haag-Kastler nets.

Examples Standard Pairs and Two-Dimensional Wigner Representations.
First we show that from a irreducible standard pair we can obtain a representation of the two-dimensional Poincaré group. Everything could be done abstractly by using Borchers commutation relations, but we rather give a proof using an explicit representation to get in contact with models constructed in the literature.
Let U m be the irreducible positive-energy representation of the twodimensional proper Poincaré group P + with mass m > 0 on a Hilbert space denoted by H m . We can identify H m = L 2 (R, dθ) and the action is given by p m (θ) = (m cosh θ, m sinh θ) where J m = U m (−I) is the anti-unitary representation of (a 0 , a 1 ) → (−a 0 , −a 1 ). We remind that we can associate a standard space H m (W R ) with the right wedge using modular localization [11], namely H m (W R ) = ker(1 − S m ) is the standard space associated with S m = J m Δ H 0 = H 0 (R + ) defined again through modular localization [29]. It can be represented on H 0 = L 2 (R + , p dp) by such that (J 0 , Δ 0 ) are the modular objects for H 0 . Proof. We show using the explicit parametrization. First we note that p dp = (f, g) L 2 (R+,p dp) shows unitarity. Then using we get in particular, one has R m T 0 ( 1 2 (a 0 + a 1 ))V m ( 1 2 (a 0 − a 1 ))Δ On the other hand, the scaling limit of the models [26] has been investigated and some one-dimensional Borchers triples (half local quantum fields, in their terminology) have been introduced [10, Section 4]. Here we observe that they simply coincide. As a special case, the lightray-restriction of the massive free net corresponds to the U(1)-current net, which we used in [44].
The one-dimensional Borchers triples in [10] are given as follows: let us fix S 2 . The Hilbert space is the same S 2 -symmetric Fock space H S2 based on the irreducible one-particle space L 2 (R, dθ). The representation T is restricted to the positive lightray, which acts on the one-particle space as T (t)(ξ)(θ) = e ite θ ξ(θ). For a test function g on R, the von Neumann algebra is in our notation given by wheref ± (θ) = ±ie θ f (t)e ite ±θ dt = ±ie θf (±θ), wheref is the Fourier transform of f . Note that in our notation z S2 (·) is antilinear, while [10] it is linear. T S1 and Ω S2 are same as before.
Let us compare this with the von Neumann algebra of the twodimensional Borchers triple. It is almost the same: Let us consider a function f (t + , t − ) = g 1 (t + )g 2 (t − ). Then f ± (θ) = g 1 (−e θ )g 2 (e θ ). If we take g 1 which is the derivative of g above and g 2 which approximates the delta function, it is clear that f ± approximateĝ ± ; hence we obtain N S2 ⊂ M S2 . By the standard argument using the cyclicity of Ω S2 and Takesaki's theory (see, e.g. the final paragraph of [27, Theorem 2.4]) one can conclude that N S2 = M S2 . Namely, the one-dimensional Borchers triples coincide. Finally, we observe that the case S 2 (θ) = 1 corresponds to the U(1)current net. The one-particle Hilbert spaces are identified as above and the full spaces are the symmetric Fock spaces; thus they coincide. Translations are also identified. In this case one can directly take M r := {e iφr(f ) : suppf ⊂ W R } If one takes f + as in the previous paragraph where f is a test function supported in W R , then as shown in [24], f + (θ −λ) has an analytic continuation in R + i(0, π) and f + (θ − iπ) = f − (θ) and it is clear that J m f − = f + . In other words, f + ∈ ker(1 − J m Δ where T (t) = Γ(T 1 (t)) and T 1 (t) = e itM 2 P −1 . The one-particle spaces can be identified with a direct sum of the spaces H mα like in Proposition 5.2. Each α corresponds, therefore, to a massive particle with mass m α . It is clear that the former example is just a special case and one obtains in this way the models in [28], namely from these assumptions about the particle spectrum and the two-particle scattering operator one can construct the needed data (H, T 1 , T 1 , R).
Conjecture on the SU(2)-Current Algebra. Zamolodchikov and Zamolodchikov conjectured [47] that, in our terminology, the one-dimensional Borchers triples constructed out of the S-matrix of the SU(2)-symmetric Thirring model is equivalent to the SU(2)-current algebra, the chiral component of a conformal field theory. This conjecture, if it turns out to be true, would imply that the SU(2)-current net admits a one-parameter semigroup of Longo-Witten endomorphisms with positive generator, which comes from the negative lightlike translation in the SU(2)-Thirring model. As remarked before, no such semigroup is so far known for the SU(2)-current net; hence this would be already new. Furthermore, as we see in the next Section, if we have two such semigroups, under suitable technical conditions we can "mix" them to obtain new strictly local Borchers triples, or equivalently Haag-Kastler nets, which would be a striking consequence.
As far as the authors understand, the conjecture remains open. Nakayashiki found a quite large family of form factors of the SU(2)-Thirring model which have the same character as the SU(2)-current algebra at level 1 [33]. However, it is not known whether the current algebra itself is appropriately represented. As another evidence, it has been revealed that both the SU(2)-Thirring model and the SU(2)-current algebra admit the same symmetry, so-called Yangian symmetry [5,31]. Yet the equivalence of the two-models is unknown.

Mixing Models by the Trotter Formula
Here we present a novel idea to construct strictly local Borchers triples. This has not led to any new example, but the authors expect that there should be concrete situations where it can apply, as we explain later. Proof. The Trotter product formula (proved in [14] under the assumption here) tells us that V (t) = lim n (V 1 (t/n)V 2 (t/n)) n . Then it is clear that We show that the common domain does not contain the bosonic oneparticle space. As we calculated in [7,Section 3.3], the bosonic one-particle space L 2 (R + , p dp) can be embedded in the fermionic "two-particle space" L 2 (R + , dq + ) ⊕ L 2 (R − , dq − ) ⊗2 as follows: for Ψ ∈ L 2 (R + , p dp), there corresponds a function ι(Ψ)(q 1 , q 2 ) = − 1 2π Ψ(q 1 − q 2 ), for q 1 > 0, q 2 < 0, ι(Ψ) = 0 if q 1 and q 2 have the same sign and on the region q 1 < 0, q 2 > 0 it is determined by antisymmetry (note the slight modification of notation from [7]). The generator of fermionic one-particle translation P 1 acts as the multiplication by |q|; hence 1 P1 acts by 1 |q| . Now we see that any function Ψ ∈ L 2 (R + , p dp) is not in the domain of 1 P1 . Indeed, we may assume that the support of Ψ contains some p 0 > 0. The multiplication by 1 q1 − 1 q2 in the fermionic two-particle space gives the function which has divergences like 1 q1 and 1 q2 around (0, −p 0 ) and (p 0 , 0), respectively, hence is clearly not in L 2 (R + , dq 1 ) ⊗ L 2 (R − , dq 2 ). This implies that ι(Ψ) is not in the domain of Λ( 1 P1 ). Therefore, we cannot find a common domain in such an elementary way to apply Proposition 5.3. There are still weaker conditions which enable such an addition of two generators [16], but we are so far not able to check them in this situation. To the authors' opinion, it is curious that the very existence of Haag-Kastler net is immediately related to such a domain problem.

Outlook
We constructed families of Borchers triples, massless ones with multiple particle components and nontrivial left-left, right-right and left-right scatterings and massive ones with block diagonal S-matrix. Strict locality of these models remains open. One should note that integrable models with bound states (S-matrix has poles in the strip [34,40]) have not been treated in the operatoralgebraic framework (c.f. [28]).
We presented also relations between massive models and onedimensional Borchers triples accompanied with a one-parameter semigroup of Longo-Witten endomorphisms with the semibounded generator. Many open problems in integrable models are relevant with this observation. We discussed the conjectured relations between the SU(2)-current algebra and the SU(2)symmetric Thirring model and asymptotic freedom in integrable models. We argued that an affirmative solution of any of these conjectures could lead to further new constructions of strictly local Borchers triples.
Another correspondence between an integrable model and a conformal field theory has been recently proposed, e.g. [20], in connection with higher dimensional gauge theory [2]. The authors would like to see possible consequences in the operator-algebraic framework.