# Macroscopic observables from the comparison of local reference systems

## TL;DR

This paper introduces a discretization method for gauge fields using local reference systems and cellular decompositions, leading to a framework for macroscopic observables relevant to quantum gravity.

## Contribution

It proposes a novel discretization of gauge fields based on a cellular decomposition, defining macroscopic observables and extending lattice gauge fields for quantum gravity applications.

## Key findings

- Defines a set of macroscopic observables from local reference systems.
- Constructs an extended lattice gauge field incorporating a G-bundle and lattice gauge field.
- Provides initial insights into regularization and coarse graining in gauge theories.

## Abstract

Parallel transport as dictated by a gauge field determines a collection of local reference systems. Comparing local reference systems in overlapping regions leads to an ensemble of algebras of relational kinematical observables for gauge theories including general relativity. Using an auxiliary cellular decomposition, we propose a discretization of the gauge field based on a decimation of the mentioned ensemble of kinematical observables. The outcome is a discrete ensemble of local subalgebras of ``macroscopic observables'' characterizing a measuring scale. A set of evaluations of those macroscopic observables is called an extended lattice gauge field because it determines a $G$-bundle over $M$ (and over submanifolds of $M$ that inherit a cellular decomposition) together with a lattice gauge field over an embedded lattice. A physical observable in our algebra of macroscopic observables is constructed. An initial study of aspects of regularization and coarse graining, which are special to this description of gauge fields over a combinatorial base, is presented. The physical relevance of this extension of ordinary lattice gauge fields is discussed in the context of quantum gravity.