Nonrenormalizability and Nontriviality

A redesigned starting point for covariant φn, n ≥ 4, models is suggested that takes the form of an alternative lattice action and which may have the virtue of leading to a nontrivial quantum field theory in the continuum limit. The lack of conventional scattering for such theories is understood through an interchange of limits. Despite being perturbatively nonrenormalizable, the quantum theory of covariant scalar φn models has been shown to be trivial for all space-time dimensions n ≥ 5, while for n = 4 it is widely believed to be trivial as well [1]. Triviality follows by showing that the conventionally lattice-regularized, Euclidean-space functional integral tends to a Gaussian distribution in the continuum limit independent of any choice of renormalizations for the mass, coupling constant, and field strength. Although mathematically sound, a trivial result is inconsistent in the sense that the classical limit of the quantized theory differs from the original (nontrivial) classical theory. In this Letter we reexamine this problem once again, and suggest an alternative formulation whereby quantum models for φn may be nontrivial. Generally, in what follows, we set ~ = 1. We start with a lattice-regularized, Euclidean-space functional integral expressed as Sa(h) ≡ 〈exp(Σ hkφka)〉 ≡ Na ∫ exp{Σ hkφka − 12Z Σ(φk∗ −−φk)2an−2 − 12Zmo Σ φka −Zgo Σ φka − Σ P (Zφk, a) an}Π dφk . (1)