[Paper Review] Quantum electrodynamics on the 3-torus -I

J. Dimock|ArXiv.org|Oct 10, 2002

Quantum Chromodynamics and Particle Interactions14 references3 citations

TL;DR

This paper initiates a rigorous non-perturbative construction of three-dimensional quantum electrodynamics (QED₃) on a 3-torus using a lattice regularization and renormalization group (RG) techniques. By splitting gauge fields into large and small components at each RG step and employing cluster and random walk expansions, the authors control ultraviolet divergences and establish exponential decay of the fluctuation two-point function, proving stability and renormalizability in the continuum limit.

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.

Motivation & Objective

• To provide a non-perturbative, mathematically rigorous construction of quantum electrodynamics in three spacetime dimensions.
• To control ultraviolet singularities in QED₃ by regularizing the theory on a lattice and taking the continuum limit.
• To establish stability and renormalizability through a renormalization group approach with field decomposition into large and small components.
• To prove exponential decay of the fluctuation two-point function, ensuring short-distance behavior is well-controlled.
• To lay the foundation for extending the method to quantum chromodynamics (QCD₃) by combining with Yang-Mills results.

Proposed method

• The theory is regularized on a 3-torus lattice with spacing $ L^{-N} $, and the continuum limit is taken as $ N \to \infty $.
• A renormalization group (RG) transformation is applied via block averaging, generating effective actions at successive length scales.
• The gauge field is split into large and small components at each RG step, following Balaban’s method, to control singularities.
• A cluster expansion is used to analyze the fluctuation integral, with polymer weights and exponential decay bounds derived via random walk expansions.
• Fermion and gauge field measures are handled using Gaussian integration with modified covariance structures, and characteristic functions localize the field regions.
• The method includes perturbative counterterm adjustments and scaling to maintain control over coupling constants and field norms.

Experimental results

Research questions

• RQ1 Can a non-perturbative, rigorous construction of QED₃ be achieved on a 3-torus using lattice regularization and RG techniques?
• RQ2 How can ultraviolet divergences in QED₃ be controlled through field decomposition and renormalization group flow?
• RQ3 What is the behavior of the two-point function in the fluctuation integral, and does it exhibit exponential decay?
• RQ4 Can the stability of the functional integral be proven through large-field suppression and small-field perturbation theory?
• RQ5 To what extent can this approach be extended to construct QCD₃ by combining with Yang-Mills results?

Key findings

• The fluctuation two-point function $ C_{\chi}(x,y) $ decays exponentially as $ \exp(-{\cal O}(1)M_1^{-1}d(x,y)) $, confirming short-distance control.
• The difference between the full and local two-point functions is bounded by $ {\cal O}(e^{-{\cal O}(1)p(e_0)^2}) \exp(-{\cal O}(1)M_1^{-1}d(x,y)) $, showing convergence to the local limit.
• The cluster expansion yields bounds of the form $ |K(Z^*)| \leq {\cal O}(1)e^{-\kappa|Z^*|_1} $, ensuring convergence of the polymer sum.
• The method successfully controls the size of the functional integral through large-field suppression and small-field perturbation, establishing stability.
• The analysis confirms that the effective coupling remains small throughout the RG flow, enabling perturbative renormalization at each step.
• The framework is robust enough to serve as a foundation for extending to QCD₃ by combining with Balaban’s results on Yang-Mills theories.