# The continuum approach to the BF vacuum: the U(1) case

## TL;DR

This paper presents a continuum quantum representation for a U(1) gauge theory that naturally incorporates the Pullin-Dittrich-Geiller vacuum, with discreteness arising from the spectrum of quantum operators, avoiding discretization.

## Contribution

It introduces a continuum quantization approach for U(1) gauge theories that reproduces the DG vacuum and naturally explains discreteness without discretizing the theory.

## Key findings

- Quantum holonomy operators have discrete spectra.
- The approach reproduces the DG vacuum in a continuum setting.
- A natural cylindrical consistency condition emerges.

## Abstract

A quantum representation of holonomies and exponentiated fluxes of a $U(1)$ gauge theory that contains the Pullin-Dittrich-Geiller (DG) vacuum is presented and discussed. Our quantization is performed manifestly in a continuum theory, without any discretization. The discretness emerges on the quantum level as a property of the spectrum of the quantum holonomy operators. The new type of a cylindrical consistency present in the DG approach, now follows easily and naturally. A generalization to the non--Abelian case seems possible.