Divergences in Scalar Quantum Field Theory: The Cause and the Cure

TL;DR

This paper identifies the root cause of divergences in covariant scalar quantum fields as due to mutually singular measures and proposes a counterterm method to transform these into equivalent measures, leading to divergence-free formulations.

Contribution

It introduces a novel approach that addresses divergences by modifying measures with counterterms, offering a divergence-free formulation for scalar quantum fields in all dimensions n ≥ 2.

Findings

Divergences stem from mutually singular measures in quantization.

Introducing an O(ħ) counterterm makes measures equivalent and removes divergences.

The new method provides a divergence-free formulation for all n ≥ 2.

Abstract

Covariant scalar fields exhibit divergences when quantized in two or more spacetime dimensions: n \geg 2. Does perturbation theory, effective theories, the renormalization group, etc., tell us all there is to know about these problems? An alternative approach identifies the cause of divergences as due to the effort to multiplicatively relate two measures that are mutually singular, while the cure for those divergences is to introduce an O(\hbar) counterterm that converts mutually singular measures into equivalent measures. This procedure leads to a nontrivial, divergence-free formulation for all n \geq 2. Finally, a critical comparison of the new methods with traditional procedures is given.