HIGH-TEMPERATURE EXPANSION FOR NONRENORMALIZABLE QUANTUM FIELD THEORIES*) J. R. Klauder Bell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974, USA By exploiting nontranslation invariant basic functional integral measures it is argued on the basis of a high-temperature series analysis that nontrivial results may be obtained for scalar model nonrenormalizable quantum field theories. 1. INTRODUCTION The lattice-space formulation of the generating functional for a Euclidean-space (4P),, p even, covariant scalar field theory may be taken as [t] (1) N f... j" exp [~hJpkA -- 89 ~(4~* - 4~)2 A/~2 -- 89 ~c~2A - - 2o Z,/2 EC)~ A] I-[ d~Jl4kl ~" Here k = (kl,..., k,) denotes a point of an n-dimensional (hyper-) cubic lattice, k* denotes 89 the nearest neighbours to k, e is the lattice spacing, A - s", the celt volume, N is a normalization factor, and B is an auxiliary parameter, the role of which will become clear below. The continuum limit (e ~ 0) of this expression along with a suitable choice of the various parameters characterizes the Euclidean-space quantum field theory. For p = 4 and B = 0 this model represents the usual (44), model and certain things are known about such models. For n = 2, 3 the model ks super-renormalizable and such constructions lead to genuine nontrivial quantum field theories. For n > 5 the model is nonrenormalizable and such constructions lead to trivial (free or gen- eralized-free) field theories. For n = 4 the theory is considered renormalizable, but there are plausible arguments and numerical evidence from high-temperature series that the result is also trivial. The triviality of the results for some of these models stands in stark contrast to the multiple divergences which such models exhibit in perturbation theory, and both results seem inconsistent with the fact that nontrivial, well-defined classical theories exist for all such models. We believe that this paradoxical situation arises from the tacit assumption of canonical commutation relations for the fields implicit in the usual choice of a translation invariant elementary measure in (1), i.e., B = 0. Non- asymptotically-free fields for which local products are defined by an operator product *) Invited talk presented at the International Symposium "Selected Topics in Quantum Field Theory and Mathematical Physics", Bechyn6, Czechoslovakia, June 14--19 1981. 494 Czech, J. Phys. B 32 [!982]