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Projective limits in Euclidean quantum field theory, II: Abelian gauge theory

Svetoslav Zahariev

TL;DR

The paper develops two projective-limit frameworks to realize continuum and infinite-volume limits of Abelian polyhedral gauge theories: (i) a pushforward heat-kernel construction on the image of the coboundary and (ii) an infinite-dimensional heat kernel built from projective limits of Hilbert spaces. It proves the existence of limit measures and shows that a translation-invariant, massless infinite-volume Abelian gauge theory emerges for $d>2$, contrasting with the mass-gap behavior of standard lattice thermodynamic limits. By uniting harmonic analysis, heat-kernel techniques, and projective-limit methods, the work provides a robust approach to defining and analyzing continuum Abelian gauge theories on arbitrary polyhedral complexes. The results generalize Villain-action constructions and yield invariant, massless continuum limits in higher dimensions, offering a new perspective on infrared behavior in Abelian lattice gauge theories.

Abstract

We present two constructions of continuum and thermodynamic limits of Abelian polyhedral gauge theories in arbitrary spacetime dimension. The first construction relies on the existence of projective systems of heat kernel measures, while the second involves an infinite dimensional heat kernel measure defined using projective limit of Hilbert spaces. As a special case, we obtain a model of Abelian gauge theory on the infinite cubical lattice, which, in contrast to the standard one, is massless for arbitrary values of the coupling parameter.

Projective limits in Euclidean quantum field theory, II: Abelian gauge theory

TL;DR

The paper develops two projective-limit frameworks to realize continuum and infinite-volume limits of Abelian polyhedral gauge theories: (i) a pushforward heat-kernel construction on the image of the coboundary and (ii) an infinite-dimensional heat kernel built from projective limits of Hilbert spaces. It proves the existence of limit measures and shows that a translation-invariant, massless infinite-volume Abelian gauge theory emerges for , contrasting with the mass-gap behavior of standard lattice thermodynamic limits. By uniting harmonic analysis, heat-kernel techniques, and projective-limit methods, the work provides a robust approach to defining and analyzing continuum Abelian gauge theories on arbitrary polyhedral complexes. The results generalize Villain-action constructions and yield invariant, massless continuum limits in higher dimensions, offering a new perspective on infrared behavior in Abelian lattice gauge theories.

Abstract

We present two constructions of continuum and thermodynamic limits of Abelian polyhedral gauge theories in arbitrary spacetime dimension. The first construction relies on the existence of projective systems of heat kernel measures, while the second involves an infinite dimensional heat kernel measure defined using projective limit of Hilbert spaces. As a special case, we obtain a model of Abelian gauge theory on the infinite cubical lattice, which, in contrast to the standard one, is massless for arbitrary values of the coupling parameter.
Paper Structure (18 sections, 19 theorems, 67 equations)

This paper contains 18 sections, 19 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fourier transform on compact Abelian groups
  4. Heat kernels and heat kernel measures
  5. Pushforward measure properties
  6. Group actions and projective limits
  7. Villain action in Abelian polyhedral gauge theory
  8. Basic notions
  9. Gauge equivalence and pushforward measures
  10. Projective systems of measures
  11. The general construction
  12. Examples
  13. Limit measures via projective limits of Hilbert spaces
  14. Limit heat kernel measures
  15. The infinite lattice limit
  16. ...and 3 more sections

Key Result

Lemma 2.1

One has $F(\mu_1) = \mu_2$ if and only if $\widehat{\mu_2} \,=\, \widehat{\mu_1} \circ \widehat{F}$.

Theorems & Definitions (45)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 35 more