Mixed Models: Combining incompatible scalar models in any space–time dimension

Abstract

Traditionally, covariant scalar field theory models are either super renormalizable, strictly renormalizable, or nonrenormalizable. The goal of “Mixed Models” is to make sense of sums of these distinct examples, e.g. [Formula: see text], which includes an example of each kind for space–time dimension [Formula: see text]. We show how the several interactions such mixed models have may be turned on and off in any order without any difficulties. Analogous results are shown for [Formula: see text], etc. for all [Formula: see text]. Different categories hold for [Formula: see text] such as, e.g. [Formula: see text], that involve polynomial [Formula: see text] and suitable nonpolynomial [Formula: see text] interactions, etc. Analogous situations for [Formula: see text] (time alone) offer simple “toy” examples of how such mixed models may be constructed. As a general rule, if the introduction of a specific interaction term reduces the domain of the free classical action, we invariably find that the introduction of the associated quantum interaction leads, effectively, to a “nonrenormalizable” quantum theory. However, in special cases, a classical interaction that does not reduce the domain of the classical free action may generate an “unsatisfactory” quantum theory, which generally involves a model-specific, different approach to become “satisfactory.” We will encounter both situations in our analysis.