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Combinatorial Spacetime from Loop Quantum Gravity

Mikhail Altaisky

TL;DR

This work proposes a Penrose-inspired reformulation of loop quantum gravity in which spacetime geometry emerges from matter fields alone, using a Lorentz-invariant, Regge-discretized framework. The key idea is to replace the dual spin-network with a triangulation graph Γ whose edges carry matter-spin networks and whose vertices act as interaction morphisms, thereby localizing curvature at vertices and relating geometric quantities to matter-state evolution. The approach yields a discrete, vertex-centric description of geometry via reconstructed tetrads and bivectors, linking matter dynamics directly to spacetime structure while avoiding a preferred time coordinate. A simple toy model demonstrates the construction, and the paper outlines future work on vertex renormalization and extensions to higher dimensions, suggesting avenues for RG-like evolution in quantum gravity.

Abstract

Loop quantum gravity is a perspective candidate for the quantum theory of gravity. However, there is a conceptual controversy in it: having started from the Einstein-Hilbert action and describing spacetime without matter, we can hardly define spacetime as anything other than a set of relations between matter fields. Here, following the Penrose idea of combinatorial spacetime we reformulate loop quantum gravity theory solely in terms of the matter fields.

Combinatorial Spacetime from Loop Quantum Gravity

TL;DR

This work proposes a Penrose-inspired reformulation of loop quantum gravity in which spacetime geometry emerges from matter fields alone, using a Lorentz-invariant, Regge-discretized framework. The key idea is to replace the dual spin-network with a triangulation graph Γ whose edges carry matter-spin networks and whose vertices act as interaction morphisms, thereby localizing curvature at vertices and relating geometric quantities to matter-state evolution. The approach yields a discrete, vertex-centric description of geometry via reconstructed tetrads and bivectors, linking matter dynamics directly to spacetime structure while avoiding a preferred time coordinate. A simple toy model demonstrates the construction, and the paper outlines future work on vertex renormalization and extensions to higher dimensions, suggesting avenues for RG-like evolution in quantum gravity.

Abstract

Loop quantum gravity is a perspective candidate for the quantum theory of gravity. However, there is a conceptual controversy in it: having started from the Einstein-Hilbert action and describing spacetime without matter, we can hardly define spacetime as anything other than a set of relations between matter fields. Here, following the Penrose idea of combinatorial spacetime we reformulate loop quantum gravity theory solely in terms of the matter fields.
Paper Structure (5 sections, 56 equations, 11 figures)

This paper contains 5 sections, 56 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Spin networks and quantum geometry
  3. Matter spin networks
  4. A toy-universe model
  5. Conclusion

Figures (11)

  • Figure 1: Transforming a qubit from $N$-block to $M$-block may result in either $(M\pm1)$-blocks. Redrawn from Penrose1971
  • Figure 2: Vertex structure in a spin network
  • Figure 3: Tetrahedron labelled by spin indices on its edges
  • Figure 4: Spin network graph $\tilde{\Gamma}$ crossing the surface $\Sigma$ in $n=3$ points
  • Figure 5: Partitioning of a 3d manifold into tetrahedra. Triangulation graph $\Gamma$, consisting of the edges of tetrahedra and their vertices, is shown by black solid lines. The dual graph $\tilde{\Gamma}$, which connects the centres of the tetrahedra, is shown by dashed lines.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Definition 1