Operator-algebraic construction of quantum integrable models with bound states

Abstract: The rigorous construction of quantum field theories with self-interaction is one of the longstanding problems of Mathematical Physics. Progress in this respect has been made in integrable field theories in 1+1 spacetime dimensions. These are characterized by a factorizing scattering matrix, where two-particle interaction determines scattering completely.

Specifically, some of these theories (so-called scalar models without bound states) have been successfully treated in the operator-algebraic approach, which is based on quantum fields localized in infinitely extended wedge regions. The existence of strictly localized observables is then obtained by abstract W*-algebraic arguments. This avoids dealing with the functional analytic properties of pointlike interacting fields, which are difficult to control due to the convergence problem of the infinite series of their form factors.

In extension of these results, we consider S-matrices with poles in the physical strip (corresponding to the notion of `bound states’ in the quantum mechanical sense). We exhibit wedge-local fields in these models, which arise as a deformation of those in the non-boundstate models by an additive term, the so called ``bound state operator''. This technique applies to a variety of theories, e.g., the Bullough-Dodd model, the Z(N)-Ising model, the affine Toda field theories and the Sine-Gordon model. The link between these wedge-local fields and strictly local operator algebras is subject to ongoing research.

mathematical physics

Audience: researchers in the discipline

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