Deformations of Quantum Field Theories and the Construction of Interacting Models

TL;DR

This thesis develops rigorous methods for constructing interacting quantum field theories using inverse scattering and deformation techniques, enabling the creation of models with specific S-matrices in various spacetime dimensions.

Contribution

It introduces two novel approaches within AQFT for constructing nontrivial QFT models, including a new deformation method applicable in arbitrary dimensions.

Findings

Constructed integrable models with factorizing S-matrices in 1+1 dimensions.

Extended the inverse scattering approach to models with multiple particle species.

Demonstrated the deformation method can produce models like the Sinh-Gordon theory.

Abstract

The subject of this thesis is the rigorous construction of QFT models with nontrivial interaction. Two different approaches in the framework of AQFT are discussed. On the one hand, an inverse scattering problem is considered. A given factorizing S-matrix is thereby taken as the starting point of the construction. The particle spectrum taken into account involves an arbitrary number of massive particle species, transforming under a global gauge group. Starting from known wedge-local auxiliary fields, the transition to local theories is shown. To his end, we make use of the modular nuclearity condition and investigate certain maps from the wedge algebras, generated by the auxiliary fields, to the considered Hilbert space. Under a very plausible conjecture it is shown that these maps are nuclear, which implies the nontriviality of algebras associated with bounded regions in the sense that the Reeh-Schlieder property holds. A large class of integrable models with factorizing S-matrices in 1+1 dimensions can be constructed in this way. Among them are the O(N)-invariant nonlinear sigma-models. Deformation techniques, on the other hand, constitute a method of construction which may be applied in arbitrary spacetime dimensions. This approach starts from a known quantum field theoretic model which is subjected to a certain modification. Here, concretely, the model of a scalar massive Fermion was deformed. The emerging models are based on wedge-local fields, allowing for the computation of the two-particle S-matrix. The resulting scattering matrix depends on the deformation and differs from the one of the initial model. By restricting to 1+1 dimensions, the deformation method yields a large class of integrable models with factorizing S-matrices, including the Sinh-Gordon model.