Quantum electrodynamics on the 3-torus -I

TL;DR

This paper investigates the ultraviolet behavior of quantum electrodynamics on a three-dimensional torus, employing renormalization group techniques to control singularities as the lattice spacing approaches zero.

Contribution

It introduces a detailed analysis of the first step in the renormalization group flow for QED on a 3-torus, focusing on controlling gauge field singularities.

Findings

Successful control of gauge field singularities at the first RG step

Framework for analyzing QED on a toroidal lattice

Foundation for further steps in the renormalization process

Abstract

We study the ultraviolet problem for quantum electrodynamics on a three dimensional torus. We start with the lattice gauge theory on a toroidal lattice and seek to control the singularities as the lattice spacing is taken to zero. This is done by following the flow of a sequence of renormalization group transformations. The analysis is facilitated by splitting the space of gauge fields into large and small fields at each step following Balaban. In this paper we explore the first step in detail.