Application of Tomita-Takesaki theory in algebraic euclidean field theories

Abstract

The construction of the known interacting quantum field theory models is mostly based on euclidean techniques. The expectation values of interesting quantities are usually given in terms of euclidean correlation functions from which one should be able to extract information about the behavior of the correlation functions of the Minkowskian counterpart. We think that the C*-algebraic approach to euclidean field theory gives an appropriate setup in order to study structural aspects model independently. A previous paper deals with a construction scheme which relates to each euclidean field theory a Poincaré covariant quantum field theory model in the sense of R. Haag and D. Kastler. Within the framework of R. Haag and D. Kastler, the physical concept of PCT symmetry and spin and statistics is related to the Tomita-Takesaki theory of von Neumann algebras and this important aspects has been studied by several authors. We express the PCT symmetry in terms of euclidean reflexions and we explicitly identify the corresponding modular operator and the modular conjugation of the related Tomita-Takesaki theory. Locality, wedge duality, and a geometric action of the modular group of the von Neumann algebra of observables, localized within a wedge region in Minkowski space, are direct consequences.