Some Interesting Properties of Field theories with an Infinite Number of Fields

TL;DR

This paper explores the potential renormalizability of gravity coupled with infinitely many fields, analyzing a toy model with infinite interacting fermions that remains finite at all perturbation orders but has limited perturbative validity.

Contribution

It introduces a toy model with infinite fermionic fields demonstrating finiteness at all perturbation orders, suggesting possible renormalizability of such theories.

Findings

The toy model is finite at all perturbation orders.

Perturbation theory is valid only for low external momenta.

Gravity with infinite fields might be renormalizable.

Abstract

We give an indication that gravity coupled to an infinite number of fields might be a renormalizable theory. A toy model with an infinite number of interacting fermions in four-dimentional space-time is analyzed. The model is finite at any order in perturbation theory. However, perturbation theory is valid only for external momenta smaller than $\lambda ^{-\frac{1}{2}}$ , where $\lambda$ is the coupling constant.