Some Results for the Exponential Interaction in Two or More Dimensions

We show that for the regularized exponential interaction λ : e : in d space-time dimensions the Schwinger functions converge to the Schwinger functions for the free field if d>2 for all α or if d = 2 for all α such that |α| >α0. 1. Notations and Results In this paper we study the space-time cut-off exponential interaction in d spacetime dimensions VΛ = λ§:d* :dx, where A is a bounded subset of R, λ>0 and Λ the corresponding Euclidean measure dμΛ(ξ) = Z^ le~VΛ(ξ)dμ0(ξ), μ0 being the free Euclidean field of mass 1 on IR [1], αeR and : : being the Wick ordering (see below for details on notation). Such models of quantum fields were introduced in [2], and in [2, 3] it was shown that if d=2,|α| < |/4π then VΛeL2(dμ0) and μA is a (non Gaussian) probability measure. The existence of a measure μΛ of the above form was shown for all d ̂ 2 and arbitrary α in [4], see also [5] and, for a different proof, [6] . In [5] it was shown that in the case d ̂ 4 the regularized (ultraviolet cut off) version of the measure μΛ converges as the regularization is removed to μ0. In the present paper we tackle, using a modification of the basic idea of [5] together with methods of [7], the case d^3 and also the case d = 2 for |α| large. The results of the present paper were announced in [10]. Let us now give the notations and state the results. We define the free field on IR with ultraviolet cut off at distance y~, y > 1, ΛΓ a positive number, as the Gaussian field ξN (1.1) where AN is the kernel of the operator