Mixed Models: Combining Incompatible Scalar Models in Any Spacetime Dimension

TL;DR

This paper introduces 'Mixed Models' that combine incompatible scalar field theories across various spacetime dimensions, demonstrating their consistent construction and manipulation without difficulties.

Contribution

It develops a framework for combining different scalar field models, including super renormalizable, renormalizable, and nonrenormalizable types, in any spacetime dimension.

Findings

Mixed models can be turned on and off in any order.

Applicable to all dimensions n ≥ 3, including polynomial and nonpolynomial interactions.

Simplified examples for n=1 illustrate the concept.

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., $g\varphi^4_3+g'\varphi^6_3+g''\varphi^8_3$, which includes an example of each kind for spacetime dimension $n=3$. 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 $g\varphi^4_n+g'\varphi^{138}_n$, etc., for all $n\ge3$. Different categories hold for $n=2$ such as, e.g., ${g P(\varphi)_2+g' N\!P}(\varphi)_2$, that involve polynomial ($P$) and suitable nonpolynomial ($N\!P$) interactions, etc. Analogous situations for $n=1$ (time alone) offer simple `toy' examples of how such mixed models may be constructed.