Euclidean Field Theory

In this review, we consider Euclidean field theory as a formulation of quantum field theory which lives in some Euclidean space, and is expressed in probabilistic terms. Methods arising from Euclidean field theory have been introduced in a very successful way in the study of the concrete models of Constructive Quantum Field Theory. Euclidean field theory was initiated by Schwinger [1] and Nakano [2], who proposed to study the vacuum expectation values of field products analytically continued into the Euclidean region (Schwinger functions), where the first three (spatial) coordinates of a world point are real and the last one (time) is purely imaginary (Schwinger points). The possibility of introducing Schwinger functions, and their invariance under the Euclidean group are immediate consequences of the by now classic formulation of quantum field theory in terms of vacuum expectation values given by Wightman [3]. The convenience of dealing with the Euclidean group, with its positive definite scalar product, instead of the Lorentz group is evident, and has been exploited by several authors, in different contexts. The next step was made by Symanzik [4], who realized that Schwinger functions for Boson fields have a remarkable positivity property, allowing to introduce Euclidean fields on their own sake. Symanzik also pointed out an analogy between Euclidean field theory and classical statistical mechanics, at least for some interactions [5]. This analogy was successfully extended, with a different interpretation, to all Boson interaction by Guerra, Rosen and Simon [6], with the purpose