Effective theories of connections and curvature: abelian case

TL;DR

This paper develops a framework for measuring scales and coarse graining in quantum abelian gauge theories, capturing curvature and holonomy information across scales, and establishing measure theory foundations for quantum gauge models.

Contribution

It introduces a novel scale-based structure for quantum abelian gauge systems, enabling effective theories and continuum limits with measure-theoretic foundations.

Findings

Framework captures curvature evaluation on all piecewise linear surfaces.

Incorporates holonomy evaluation along loops into the scale structure.

Provides measure theory tools for quantum abelian gauge theories.

Abstract

We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play the role of coarse graining maps. This structure induces a continuum limit space which contains information regarding curvature evaluation on all piecewise linear surfaces with boundary. The evaluation of holonomies along loops is also encoded in the spaces introduced here; thus, our framework is closely related to loop quantization and it allows us to discuss effective theories in a sensible way. We develop basic elements of measure theory on the introduced spaces which are essential for the applicability of the framework to the construction of quantum abelian gauge theories.