Quantum entropy couples matter with geometry

Abstract: We propose a theory for coupling matter fields with discrete geometry on higher-order networks, i.e. cell complexes. The key idea of the approach is to associate to a higher-order network the quantum entropy of its metric. Specifically we propose an action having two contributions. The first contribution is proportional to the logarithm of the volume associated to the higher-order network by the metric. In the vacuum this contribution determines the entropy of the geometry. The second contribution is the quantum relative entropy between the metric of the higher-order network and the metric induced by the matter and gauge fields. The induced metric is defined in terms of the topological spinors and the discrete Dirac operators. The topological spinors, defined on nodes, edges and higher-dimensional cells, encode for the matter fields. The discrete Dirac operators act on topological spinors, and depend on the metric of the higher-order network as well as on the gauge fields via a discrete version of the minimal substitution. We derive the coupled dynamical equations for the metric, the matter and the gauge fields, providing an information theory principle to obtain the field theory equations in discrete curved space.

• The paper introduces a novel framework linking quantum entropy and discrete geometry by coupling matter and gauge fields via an action principle.
• It defines an induced metric using topological spinors and discrete Dirac operators, yielding motion equations parallel to Klein-Gordon and Dirac equations.
• The work offers insights into quantum gravity and complex systems, highlighting applications in network topology and machine learning.

Quantum Entropy Couples Matter with Geometry: An Overview

This paper presents a theoretical framework that connects quantum entropy with the coupling of matter and geometry on higher-order networks, specifically cell complexes. The work proposes an action principle with two primary components: the geometric entropy related to the network's volume, and the quantum relative entropy that couples matter and gauge fields to the network's geometry.

Key Contributions

1. Higher-Order Networks and Quantum Entropy: The research introduces a method for associating quantum entropy with the metric of higher-order networks, such as cell complexes. This connection is crucial for understanding discrete geometries and their role in quantum gravity.
2. Induced Metric and Action Formulation: A central idea is to define an induced metric based on topological spinors—mathematical objects representing matter fields across various dimensions—and the discrete Dirac operators acting on these spinors. The proposed action includes a term for the entropy of geometry and a term for quantum relative entropy, guiding the dynamical interactions between the network's metric and the matter and gauge fields.
3. Dynamical Equations: The paper derives equations of motion for the metric, matter, and gauge fields within discrete curved space. These equations resemble the Klein-Gordon and Dirac equations, adapted for the discrete geometry of higher-order networks.
4. 3D Lattice Topologies: The study explores the theory within the context of 3-dimensional lattice topologies. By introducing gamma matrices and defining discrete curvature and gauge field interactions, the authors extend the framework to encompass more complex structures. This section elaborates on how combinatorial and geometrical properties emerge from these lattice models, revealing potential physical insights.

Implications and Future Directions

• Implications for Quantum Gravity and Geometry: The framework provides a novel perspective on coupling quantum mechanics and geometry, particularly in a discrete setting. This approach is different from traditional quantum gravity theories and may offer insights into the fundamental nature of space-time.
• Topology-Dependent Dynamics: The study's basis in complex network topologies suggests possible applications in understanding how topological changes affect physical dynamics. This aspect is crucial for both theoretical explorations and practical simulations.
• Extension to Lorentzian and More General Geometries: The methodology can be extended to Lorentzian geometries, relevant for models considering temporal dimensions alongside spatial ones.
• Complex Systems and Machine Learning: Beyond physics, the principles could be applied to complex systems, including brain networks and machine learning models. The interplay of geometry and information in these systems might be influenced by concepts introduced in this paper.

Conclusion

This investigation paves the way for innovatively linking information theory with network topology and quantum field theories. The integration of quantum entropy with discrete geometric structures proposes a unique theoretical framework that may guide future research in quantum gravity, discrete network geometry, and beyond. The potential to address longstanding issues like defining curvature in complex networks also suggests promising interdisciplinary applications.