Integral representations of the Schwinger functions for wick polynomials in the free field

We obtain integral representations of Schwinger functions for Wick polynomials in the free field, in other words, we obtain Euclidean realizations of Wightman quantum fields given by Wick polynomials in the free field. Using these Euclidean realizations, a new non-perturbative mathematically rigorous approach to constructing quantum field theory with the polynomial interaction in a finite volume of $d$-dimensional spacetime $(d\geq 2)$ and without ultraviolet cut-offs is proposed. In particular, for imaginary values of the coupling constant the generating functional of Schwinger functions is constructed. The theory constructed by this method takes explicitly into account the presence of ultraviolet divergences and its expansion in powers of the coupling constant gives the renormalized series.