Dirk Schlingemann
From Euclidean Field Theory to Quantum Field Theory
The paper is published: Rev. Math. Phys. 11, No. 9 (1999) 1151-1178

MSC:

81T05 Axiomatic quantum field theory; operator algebras

81T13 Yang-Mills and other gauge theories, See also {53C07,

46L60 Applications of selfadjoint operator algebras to physics, See also {46N50, 46N55, 47D45, 81T05, 82B10, 82C10}

46N50 Applications in quantum physics

Abstract: 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 $n$-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 {\em 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.


Keywords: euclidean field theory, quantum field theory, Bargmann-Hall-Wightman theorem, Osterwalder-Schrader reconstruction theorem, gauge theory, functional integral