Published November 18, 2025
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Kirchhoff Network Formulation of Quantum Gravity: Emergent Spacetime from Information Flow
Description
We formulate quantum gravity as information flow on discrete Kirchhoff networks, where spacetime emerges from graph connectivity. Key results: (1) Metric tensor = network conductance matrix G_ij in continuum limit, with Lorentz signature from timelike (capacitive) vs. spacelike (resistive) edges, (2) Einstein's field equations emerge from Kirchhoff current conservation: Lφ = ρ, with network Laplacian L encoding geometry, (3) Rigorously prove R = 6Tr(L)/ε² - 48N/ε² relating Ricci scalar to graph Laplacian trace, verified for sphere (χ=2), torus (χ=0), hyperbolic space (R<0), (4) Spectral geometry: eigenvalue spectrum {λ_k} encodes volume (Tr(L)), dimension (Weyl asymptotics), curvature (spectral gap), topology (Euler characteristic), (5) Complete Fock space construction for topology changes: edge creation/annihilation operators â†_ij implement wormhole dynamics, (6) Black hole entropy = boundary entanglement entropy, reproducing Bekenstein-Hawking S=A/4, (7) Quantum fluctuations as network topology changes with amplitude ~ exp(-A/4Għ). This provides a discrete, information-theoretic foundation for quantum gravity unifying GR with quantum mechanics through network theory.
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