The \Phi^4 quantum field in a scale invariant random metric

TL;DR

This paper investigates a scalar quantum field interacting with a scale invariant random metric, revealing improved short-distance behavior and showing that in certain models, especially -dimensional  theories, no renormalization or divergences occur.

Contribution

It introduces a model of a scalar field coupled to a scale invariant random metric and demonstrates the absence of coupling constant renormalization in -dimensional  theories.

Findings

Quantum fields exhibit more regular short-distance behavior than free fields.

Explicit bounds are derived for the perturbation series in -dimensional  theory.

Certain models in four dimensions are free of divergences without renormalization.

Abstract

We discuss a D-dimensional Euclidean scalar field interacting with a scale invariant quantized metric. We assume that the metric depends on d-dimensional coordinates where d<D. We show that the interacting quantum fields have more regular short distance behaviour than the free fields. A model of a Gaussian metric is discussed in detail. In particular, in the \Phi^4 theory in four dimensions we obtain explicit lower and upper bounds for each term of the perturbation series. It turns out that there is no coupling constant renormalization in the \Phi^4 model in four dimensions. We show that in a particular range of the scale dimension there are models in D=4 without any divergencies.