Glocal Observables Encode Geometry in Background Independent Discrete Gravity Models

The fundamental challenge of defining measurable quantities in physics, known as the problem of observables, receives a novel resolution in new work led by Emil Broukal and Andrea Di Biagio of the Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences, alongside Eugenio Bianchi and Marios Christodoulou. Researchers demonstrate that a complete set of observables can be constructed for theories defined on discrete, network-like structures, achieving a form of background independence through invariance to relabeling the network’s connections. The team reveals that these observables possess a surprising “glocal” character, simultaneously probing the entire network globally while also focusing on local correlations within connected sub-structures. This discovery not only provides a physically meaningful framework for discrete gravity theories, but also suggests new avenues for understanding the complex state space of loop quantum gravity and establishes a deep connection to the mathematical problem of determining when two networks are fundamentally the same.

This work investigates the complete resolution of the problem of defining measurable quantities for theories of gravity formulated on finite networks. The researchers argue that the correct analogue of coordinate independence is invariance under changes of network labels, representing a form of permutation invariance. Invariants are formed through a group average that probes the entire network, establishing them as global quantities. Remarkably, complete sets of observables can be constructed that each seek a connected subgraph structure, thereby capturing local correlations. Geometrical information is fully encoded in these background independent observables through a subtle interplay between these global and local network notions

Weighted Graph Invariant Basis Construction

This research details the construction of a basis for describing the symmetries of a weighted graph, a network where each connection has an associated value. Understanding these symmetries is crucial for defining invariant properties, characteristics that remain unchanged even when the network is transformed. The study focuses on a specific weighted graph with four nodes, identifying 31 polynomials that form a complete basis for describing all its invariant properties. These polynomials combine the weights of the nodes and the connections between them in specific ways, yet effectively represent only eight distinct underlying network structures, suggesting a degree of redundancy and making the description more manageable. The identified basis provides a complete and explicit way to calculate any invariant property of the weighted graph, valuable for applications in network analysis, physics, and computer science.

Network Relabeling Defines Discrete Spacetime Observables

Researchers have made significant progress in defining measurable quantities for theories of gravity formulated on discrete structures, such as networks of interconnected nodes, moving closer to a truly background-independent approach. This work establishes a novel understanding of background independence, framing it not as reliance on continuous coordinates, but as invariance under the relabeling of nodes within a network. This allows for a natural mathematical framework where weighted graphs represent spacetime, with the weights defining a discrete metric and all points initially being indistinguishable. The core breakthrough lies in the development of “glocal” observables, functions that analyze the entire network but reveal information about local correlations.

These observables are constructed as averages across the network, systematically capturing local features, even unique ones, despite their global nature. Remarkably, researchers demonstrate that complete sets of these glocal observables can be algorithmically constructed for finite networks, guaranteeing the capture of all label-independent geometrical information. While constructing complete sets of observables for infinite networks proves mathematically challenging, the difficulty stems not from missing information, but from the inherent complexities of infinite systems. Researchers reveal that attempting to define complete observables on infinite structures encounters obstructions similar to those found in descriptive set theory.

However, by focusing on specific sectors of solutions, researchers show that vast sets of glocal observables can still be defined, offering a pathway to circumvent these limitations. This work has profound implications for loop quantum gravity and other approaches to discrete gravity, potentially offering a strategy for restricting theories to well-defined sectors where glocal observables can be meaningfully applied. The findings also suggest a deeper connection between the problem of defining observables and the mathematical problem of graph isomorphism.

Glocal Observables Define Discrete Spacetime Invariance

This research demonstrates that a fully background-independent formulation is achievable in discrete spacetime, offering a resolution to the longstanding problem of defining measurable quantities. The work establishes that invariance under permutations of graph labels corresponds to general covariance in this discrete setting. Crucially, the team shows how global observables can systematically capture local correlations, a behaviour termed ‘glocal’, and that complete sets of such observables can be algorithmically constructed for finite graphs. The formalism provides a constructible resolution of the problem of observables in Lorentzian Regge calculus, an approximation of general relativity, and offers a means to define a background-independent state space for spin networks, the foundational building blocks of loop quantum gravity. The authors acknowledge that the completeness of these observables relies on restricting the sector of solutions, and that further research is needed to fully explore this limitation.