Scalar Field Quantization Without Divergences In All Spacetime Dimensions

TL;DR

This paper introduces a novel quantization method for scalar quantum field theories that eliminates divergences across all spacetime dimensions by using a local counterterm, enabling consistent solutions for models previously deemed problematic.

Contribution

A nontraditional, divergence-free quantization approach for covariant scalar field theories using a local counterterm and an alternative perturbation model, applicable in all spacetime dimensions.

Findings

Achieves divergence-free perturbation analysis in all spacetime dimensions.

Provides solutions for models like ^4_n in higher dimensions where traditional methods fail.

Supports similar properties for multicomponent scalar models, including the Higgs model.

Abstract

Covariant, self-interacting scalar quantum field theories admit solutions for low enough spacetime dimensions, but when additional divergences appear in higher dimensions, the traditional approach leads to results, such as triviality, that are less than satisfactory. Guided by idealized but soluble {\it non}renormalizable models, a nontraditional proposal for the quantization of covariant scalar field theories is advanced, which achieves a term-by-term, divergence-free, perturbation analysis of interacting models expanded about a suitable pseudofree theory, which differs from a free theory by an O(\hbar^2) counterterm. These positive features are realized within a functional integral formulation by a local, nonclassical, counterterm that effectively transforms parameter changes in the action from generating mutually singular measures, which are the basis for divergences, to equivalent measures, thereby removing all divergences. The use of an alternative model about which to perturb is already supported by properties of the classical theory, and is allowed by the inherent ambiguity in the quantization process itself. This procedure not only provides acceptable solutions for models for which no acceptable, faithful solution currently exists, e.g., \phi^4_n, for spacetime dimensions n\ge4, but offers a new, divergence-free solution, for less-singular models as well, e.g., \phi^4_n, for n=2,3. Our analysis implies similar properties for multicomponent scalar models, such as those associated with the Higgs model.