Emergent Lorentz Invariance via Lieb-Robinson Bounds on Graph Laplacians

Target Venue: Entropy

Abstract

Conscious Realism posits a fundamental reality composed of a discrete Markovian agent network. To map this pre-geometric graph to relativistic spacetime, we cannot rely on arbitrary lattice structures that introduce anisotropic ether frames. We rigorously derive the continuum limit of the network using the spectral properties of the graph Laplacian. By applying the Lieb-Robinson theorem to the network's transition matrices, we mathematically prove that an effective speed limit c emerges for information propagation. As the density of the network approaches the continuum limit, the discrete wave equations governed by the Laplacian organically recover local Lorentz symmetry, independent of any preferred coordinate frame.

1. Introduction

Deriving relativity from discrete graphs requires avoiding the preferred frame problem. We transition from tracking explicit edges to analyzing the spectral diffusion of information across the graph.

2. The Graph Laplacian and the Wave Equation

Let the network be an undirected graph G = (V, E). Information diffusion is governed by the graph Laplacian \mathcal{L} = D - A, where D is the degree matrix and A the adjacency matrix. In the continuum limit, the discrete equation \frac{\partial^2 \psi}{\partial t^2} = -\mathcal{L}\psi maps directly to the continuous wave equation \square \psi = 0.

3. The Lieb-Robinson Bound as the Speed of Light

For any two nodes x, y \in V, the commutator of local observables O_x, O_y is bounded by the Lieb-Robinson theorem:


||[O_x(t), O_y(0)]|| \le C e^{-\mu (d(x,y) - v_{LR} t)}

where v_{LR} is the Lieb-Robinson velocity. This strict upper bound on the propagation of correlations acts as the emergent speed of light c.

4. Conclusion

Lorentz invariance is the macroscopic symmetry of the Lieb-Robinson bounds operating over the graph Laplacian. Relativity is fully recoverable from discrete Conscious Agents.

References

Lieb, E. H., & Robinson, D. W. (1972). The finite group velocity of quantum spin systems. Communications in Mathematical Physics.
Hoffman, D. D., & Prakash, C. (2014). Objects of consciousness. Frontiers in Psychology.

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