Construction of Wedge-Local Nets of Observables Through Longo-Witten Endomorphisms

Bischoff, Marcel; Tanimoto, Yoh

doi:10.1007/s00220-012-1462-7

Cited by 25 publications

(93 citation statements)

References 32 publications

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“…More precisely, we will show that the deformations presented in [24], starting from a chiral field theory, are equivalent to the deformations in terms of deformed field operators, presented in [17], for mass m = 0 and dimension d = 1 + 1 (Section 2). In the special case of the so-called warped convolution deformation [8], such an equivalence was already observed in [24].…”

Section: Deformations Of Qfts By Inner Functions and Their Rootsmentioning

confidence: 96%

“…In recent years, there has been a lot of interest in deformations of quantum field theories [1,3,[8][9][10]12,[17][18][19][20][21][22]24] in the sense of specific procedures modifying quantum field theoretic models on Minkowski space, mostly motivated by the desire to construct new models in a non-perturbative manner. Various constructions have been invented, relying on different methods such as smooth group actions, noncommutative geometry, chiral conformal field theory, boundary quantum field theory, and inverse scattering theory.…”

Section: Deformations Of Qfts By Inner Functions and Their Rootsmentioning

confidence: 99%

“…Symmetric inner functions provide the input into deformations making use of Longo-Witten endomorphisms [3,18,19,24], whereas scattering functions are used in inverse scattering approaches such as [5,15]. For convenience, the latter are usually defined with the additional requirement of extending continuously to the closure of S(0, π).…”

Section: Definition 11 (I) a Symmetric Inner Function Is A Functionmentioning

confidence: 99%

“…In the following, we will introduce for each R ∈ R two von Neumann algebras M R,m , N R with this property, obtained by (generalizations of) the deformation procedures in [17] and [24], respectively. For R = 1, both families reduce to the undeformed situation, i.e.…”

Section: Definition 11 (I) a Symmetric Inner Function Is A Functionmentioning

confidence: 99%

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On the Equivalence of Two Deformation Schemes in Quantum Field Theory

2012

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Abstract. Two recent deformation schemes for quantum field theories on two-dimensional Minkowski space, making use of deformed field operators and Longo-Witten endomorphisms, respectively, are shown to be equivalent. Mathematics Subject Classification (2010). 81T05, 81T40.Keywords. deformations of quantum field theories, two-dimensional models, modular theory. Deformations of QFTs by Inner Functions and Their RootsIn recent years, there has been a lot of interest in deformations of quantum field theories [1,3,[8][9][10]12,[17][18][19][20][21][22]24] in the sense of specific procedures modifying quantum field theoretic models on Minkowski space, mostly motivated by the desire to construct new models in a non-perturbative manner. Various constructions have been invented, relying on different methods such as smooth group actions, noncommutative geometry, chiral conformal field theory, boundary quantum field theory, and inverse scattering theory.In many situations, it is possible to set up the deformation in such a way that Poincaré covariance is completely preserved and locality partly. More precisely, often the deformation introduces operators which are no longer localized in arbitrarily small regions of spacetime, but rather in unbounded regions like a Rindler wedge W := {x ∈ R d : x 1 > |x 0 |}. In the operator-algebraic framework of quantum field theory [14], such a wedge-local Poincaré covariant model can be conveniently described by a so-called Borchers triple (M, U, ) [4,8], consisting of a GL and JS supported by FWF project P22929-N16 "Deformations of quantum field theories". Y. Tanimoto supported by Deutscher Akademischer Austauschdienst.

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Section: Deformations Of Qfts By Inner Functions and Their Rootsmentioning

confidence: 96%

Section: Deformations Of Qfts By Inner Functions and Their Rootsmentioning

confidence: 99%

Section: Definition 11 (I) a Symmetric Inner Function Is A Functionmentioning

confidence: 99%

Section: Definition 11 (I) a Symmetric Inner Function Is A Functionmentioning

confidence: 99%

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On the Equivalence of Two Deformation Schemes in Quantum Field Theory

2012

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“…2,15 The novel result in this review is some progress in models with bound states, or with S-matrices which have poles in the physical strip. …”

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confidence: 99%

Towards construction of integrable QFT with bound states

Tanimoto¹

2017

The Fourteenth Marcel Grossmann Meeting

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We review recent operator-algebraic constructions of quantum field theories, especially of two-dimensional integrable models. In the operator-algebraic approach, a model of quantum field theory is realized as a net of von Neumann algebras associated to spacetime regions. A key idea in the recent developments is to construct first the observables localized in wedge-shaped regions, then to define the algebras for double cones by intersection.Up to now, these constructions are limited to the class of S-matrices whose components are analytic in rapidity in the physical strip. We present candidates for observables in wedge regions for scalar factorizing S-matrices with poles in the physical strip. We discuss the self-adjointness of these candidate operators and strong commutativity between them.Keywords: Quantum field theory; operator algebras; integrable models; scattering theory; Rindler wedge. Operator-algebraic approach to QFTA mathematically consistent construction of Quantum Field Theories (QFT) is a hard task: On the one hand, the conventional perturbation theory has been phenomenologically very successfully and is well-understood as formal power series. 3On the other hand, it has been argued that Quantum Electrodynamics, a representative QFT, has divergent perturbative series in coupling constant. 7A program called Constructive QFT 8 has been quite successful in constructing models of QFT in two-and three-dimensional spacetimes. These models fit into mathematically sound frameworks (Wightman, Osterwalder-Schrader and HaagKastler axioms) and are shown to be different from the physically trivial free theories. However, no nontrivial model has been constructed in the physical four dimensions.Several progresses have been made in the last decade in the operator-algebraic framework (Haag-Kastler nets). A central idea is that, while an interacting local quantum fields can be very complicated objects, an infinitely-extended spacetime region can accommodate observables which have moderate behaviors in the momentum space. Especially, Schroer proposed the use of so-called wedge-local fields for a class of integrable QFT in two dimensions.14 Lechner took these wedge-local fields as a starting point and constructed Haag-Kastler nets for a family of integrable QFT.11 See Ref. 12 a more detailed review. This program has been extended to several directions. Models with multiple particle spectrum have been also studied, 13 but the existence proof of local observables is at the moment still missing.1 Given a model with global gauge symmetry,

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Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques

Lechner

2015

Advances in Algebraic Quantum Field Theory

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Several related operator-algebraic constructions for quantum field theory models on Minkowski spacetime are reviewed. The common theme of these constructions is that of a Borchers triple, capturing the structure of observables localized in a Rindler wedge. After reviewing the abstract setting, we discuss in this framework i) the construction of free field theories from standard pairs, ii) the inverse scattering construction of integrable QFT models on two-dimensional Minkowski space, and iii) the warped convolution deformation of QFT models in arbitrary dimension, inspired from non-commutative Minkowski space.

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Construction of Wedge-Local Nets of Observables Through Longo-Witten Endomorphisms

Cited by 25 publications

References 32 publications

On the Equivalence of Two Deformation Schemes in Quantum Field Theory

On the Equivalence of Two Deformation Schemes in Quantum Field Theory

Towards construction of integrable QFT with bound states

Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques

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