Observables are glocal

TL;DR

This paper introduces the concept of 'glocal' observables in background independent theories on finite graphs, combining global invariance with local subgraph structures to encode geometrical information, impacting quantum gravity and graph theory.

Contribution

It proposes a novel framework for observables that unifies global and local graph properties, advancing understanding in background independent theories and quantum gravity.

Findings

Invariants are formed by group averaging over graph labels.

Complete observables can target connected subgraphs, capturing local correlations.

The approach links the problem of observables to the graph isomorphism problem.

Abstract

We study how the problem of observables is fully resolved for background independent theories defined on finite graphs. We argue the correct analogue of coordinate independence is the invariance under changes of graph labels, a kind of permutation invariance. Invariants are formed by a group average that probes the entire graph -- they are global. Strikingly, sets of complete observables can be constructed so that each seeks a connected subgraph structure -- local correlations. Geometrical information is fully encoded in background independent observables through this subtle interplay of global and local graph notions, a behavior we term glocal. This provides physically meaningful complete sets of observables for discrete general relativity, suggests a reformulation of the spin networks state space of loop quantum gravity, and reveals deep connections between the problem of observables and the graph isomorphism problem.