From euclidean field theory to quantum field theory

In order to construct examples for interacting quantum field theory models, the methods of euclidean field theory have turned out to be a powerful tools since they make use of the techniques of classical statistical mechanics. Starting from an appropriate set of euclidean 3⁄4 -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 3⁄4 -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.