Construction of Nontrivial Quantum Gauge Theories: The Continuum limit in a Finite Volume

TL;DR

This paper constructs and rigorously analyzes the continuum limit of quantum gauge theories, including Abelian and non-Abelian types, in a finite volume, demonstrating convergence of the renormalized theories.

Contribution

It provides a rigorous construction of quantum gauge theories in four dimensions with finite volume and proves the convergence of their renormalized perturbation series.

Findings

Successful construction of quantum gauge theories in finite volume.

Proof of convergence of ultraviolet limits and renormalized perturbation series.

Explicit construction of the renormalized Lagrangian and Schwinger functions.

Abstract

We construct several quantum gauge theories in 4 dimensional space time, including both Abelian and non Abelian gauge groups, with the Abelian gauge fields coupled to zero mass matter fields. The construction occurs in a fixed finite Euclidean spatial domain. The construction begins with the doubly cutoff bare field theory (with a finite space time volume and a mesh based ultraviolet cutoff) constructed in a separate paper. We use a limited range of renormalized perturbation theory, just sufficient to cancel all the divergences. We demonstrate convergence of the ultraviolet limit and of the renormalized perturbation theory, when summed to all orders. We construct the fully renormalized Lagrangian and Schwinger functions.