Towards construction of integrable QFT with bound states

Abstract

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,