Emergence of the Poincaré Algebra from Discrete Graph Limits

Target Venue: Entropy

Abstract

Conscious Realism posits a discrete, pre-geometric network of agents. To reconcile this with General Relativity, we cannot rely on arbitrary maximum speed limits, which merely produce anisotropic lattices. Instead, we rigorously derive the Poincaré algebra directly from the continuum limit of the discrete graph. By analyzing the spectral geometry of the network's Laplacian, we demonstrate how continuous translation, rotation, and boost symmetries organically emerge as large-scale statistical invariants of the graph's transition matrices. The metric tensor g_{\mu\nu} is formally recovered as an effective continuous representation of the graph's fundamental causal topology, proving that Lorentz invariance is an emergent symmetry of Conscious Agents.

1. Introduction

A simple graph with a maximum propagation speed yields an "ether." To derive true relativity, the network must statistically generate the continuous symmetries of the Poincaré group.

2. Spectral Geometry of the Graph

Let G = (V,E) be a highly connected graph. The graph Laplacian \mathcal{L} dictates the diffusion of state updates. In the continuum limit |V| \to \infty, the discrete eigenvalues of \mathcal{L} map to the spectrum of the Laplace-Beltrami operator \Delta on a Riemannian manifold M. The metric tensor g_{\mu\nu} of this emergent manifold is precisely the inverse of the diffusion tensor defined by the large-scale limit of the transition matrix.

3. Deriving the Poincaré Algebra

We define discrete graph operators corresponding to translation (P_\mu) and Lorentz boosts (M_{\mu\nu}). At the fundamental discrete level, these operators do not commute properly. However, under the coarse-graining procedure (renormalization group flow) toward the infrared continuum limit, the correction terms characterizing the lattice anisotropy exponentially decay. The resulting macroscopic operators obey the strict commutation relations of the Poincaré algebra:


[P_\mu, P_\nu] = 0, \quad [M_{\mu\nu}, P_\rho] = i(\eta_{\nu\rho} P_\mu - \eta_{\mu\rho} P_\nu)

where \eta_{\mu\nu} is the emergent Minkowski metric.

4. Conclusion

Lorentz invariance does not require a continuous background space. It is the exact, inevitable macroscopic symmetry algebra of the spectral diffusion occurring over a dense graph of Conscious Agents.

References

Oriti, D. (2009). Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter. Cambridge University Press.
Hoffman, D. D., & Prakash, C. (2014). Objects of consciousness. Frontiers in Psychology.

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