Euclidean Field Theory

Abstract

In this review, we consider Euclidean field theory as a formulation of quan-tum field theory which lives in some Euclidean space, and is expressed inprobabilistic terms. Methods arising from Euclidean field theory have beenintroduced in a very successful way in the study of the concrete models ofConstructive 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 an-alytically continued into the Euclidean region (Schwinger functions), wherethe first three (spatial) coordinates of a world point are real and the lastone (time) is purely imaginary (Schwinger points). The possibility of intro-ducing Schwinger functions, and their invariance under the Euclidean groupare immediate consequences of the by now classic formulation of quantumfield theory in terms of vacuum expectation values given by Wightman [3].The convenience of dealing with the Euclidean group, with its positive def-inite scalar product, instead of the Lorentz group is evident, and has beenexploited by several authors, in different contexts.The next step was made by Symanzik [4], who realized that Schwingerfunctions for Boson fields have a remarkable positivity property, allowing tointroduce Euclidean fields on their own sake. Symanzik also pointed out ananalogy 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