Abstract
Integral representations for the Schwinger functions of products of Wick polynomials in the free field are obtained, i.e., Euclidean realizations of Wightman quantum fields specified by Wick polynomials in the free field are obtained. These Euclidean realizations are used to propose a new mathematically rigorous approach to the construction of quantum field theory with polynomial interaction in a finite volume of d-dimensional space-time even without ultraviolet cutoff. In particular, the generating functional of the Schwinger functions is constructed for purely imaginary values of the coupling constant. The theory constructed in this manner takes into account explicitly the presence of ultraviolet divergences and if expanded with respect to the coupling constant gives a renormalized perturbation-theory series.
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