Index Theory and Adiabatic Limit in QFT

TL;DR

This paper explores the relationship between different rigorous formulations of quantum field theory, aiming to identify obstructions to the adiabatic limit which relate to the confinement problem in the standard model.

Contribution

It extends deformation methods and Fedosov's index construction to analyze the adiabatic limit in QFT, providing new mathematical tools for the problem.

Findings

Extended deformation methods for QFT

Applied Fedosov's index to quantum field theory

Initial steps towards identifying obstructions in the adiabatic limit

Abstract

The paper has the form of a proposal concerned with the relationship between the three mathematically rigorous approaches to quantum field theory: 1) local algebraic formulation of Haag, 2) Wightman formulation and 3) the perturbative formulation based on the microlocal renormalization method. In this project we investigate the relationship between 1) and 3) and utilize the known relationships between 1) and 2). The main goal of the proposal lies in obtaining obstructions for the existence of the adiabatic limit (confinement problem in the phenomenological standard model approach). We extend the method of deformation of D\"utsch and Fredenhagen (in the Bordeman-Waldmann sense) and apply Fedosov construction of the formal index -- an analog of the index for deformed symplectic manifolds, generalizing the Atiyah-Singer index. We present some first steps in realization of the proposal.