Modular Constructions of Quantum Field Theories with Interactions

TL;DR

This paper advances the construction of interacting quantum field theories using modular methods, clarifying quantum localization, and establishing rigorous wedge and double cone algebra structures in 1+1 dimensions.

Contribution

It introduces a nonperturbative modular approach to QFT, linking quantum localization with algebraic structures and crossing symmetry via PFG generators and Zamolodchikov-Faddeev algebra.

Findings

Constructed wedge algebras rigorously for 1+1 factorizing theories.

Linked crossing symmetry to KMS properties of PFG generators.

Revealed particle content through double cone algebra analysis.

Abstract

We extend the previously introduced constructive modular method to nonperturbative QFT. In particular the relevance of the concept of ``quantum localization'' (via intersection of algebras) versus classical locality (via support properties of test functions) is explained in detail, the wedge algebras are constructed rigorously and the formal aspects of double cone algebras for d=1+1 factorizing theories are determined. The well-known on-shell crossing symmetry of the S-Matrix and of formfactors (cyclicity relation) in such theories is intimately related to the KMS properties of new quantum-local PFG (one-particle polarization-free generators) of these wedge algebras. These generators are ``on-shell'' and their Fourier transforms turn out to fulfill the Zamolodchikov-Faddeev algebra. As the wedge algebras contain the crossing symmetry informations, the double cone algebras reveal the particle content of fields. Modular theory associates with this double cone algebra two very useful chiral conformal quantum field theories which are the algebraic versions of the light ray algebras.