From euclidean field theory to quantum field theory

TL;DR

This paper develops a C*-algebraic approach to reconstructing quantum field theories from Euclidean field theory data, enabling direct construction of operator nets and including non-point-like objects like Wilson loops.

Contribution

It introduces a C*-algebraic version of the Osterwalder-Schrader theorem, facilitating direct reconstruction of Haag-Kastler nets from Euclidean data, including gauge-invariant objects.

Findings

Reconstruction of Haag-Kastler nets from Euclidean functions

Inclusion of Wilson loop variables in the framework

Potential applications to gauge theories

Abstract

In order to construct examples for interacting quantum field theory models, the methods of euclidean field theory turned out to be powerful tools since they make use of the techniques of classical statistical mechanics. Starting from an appropriate set of euclidean n-point functions (Schwinger distributions), a Wightman theory can be reconstructed by an application of the famous Osterwalder-Schrader reconstruction theorem. This procedure (Wick rotation), which relates classical statistical mechanics and quantum field theory, is, however, somewhat subtle. It relies on the analytic properties of the euclidean n-point functions. We shall present here a C*-algebraic version of the Osterwalder-Scharader reconstruction theorem. We shall see that, via our reconstruction scheme, a Haag-Kastler net of bounded operators can directly be reconstructed. Our considerations also include objects, like Wilson loop variables, which are not point-like localized objects like distributions. This point of view may also be helpful for constructing gauge theories.