Ze → Twistor → Spin Network

Abstract

This paper presents the Ze → Twistor → Spin Network framework, a unified conceptual pathway from a primitive discrete ontology to the emergence of relativistic spacetime. The framework begins with fundamental events, denoted ΔC_i, each characterized by dual aspects: a temporal component C_i^{temporal} governing participation in sequential causal chains, and a spatial component C_i^{spatial} governing participation in parallel structural configurations. The first transition, Ze → Twistor, encodes these aspects in a complex representation Z_i = C_i^{temporal} + i C_i^{spatial}, drawing on Penrose's twistor theory. The Hermitian norm |Z_i|^2 = (ΔC_i^{spatial})^2 - γ(ΔC_i^{temporal})^2 carries the Minkowski signature intrinsically, emerging from the SU(2,2) invariant structure of twistor space rather than being inserted by hand. This addresses the fundamental question of how a discrete substrate can give rise to a Lorentzian manifold without violating Lorentz invariance. The second transition, Twistor → Spin Network, discretizes the twistor representation into a labeled graph. Nodes correspond to events (antichains representing spatial slices), edges correspond to causal links, and each edge carries a spin label j determined by j(j+1) ∝ |Z_i|^2, connecting directly to loop quantum gravity where spin networks provide an orthonormal basis for kinematical states. From this structure, relativistic spacetime emerges in the continuum limit: proper time along a worldline is given by the sum of spin labels τ = Σ √[j(j+1)] × τ_Planck, velocity emerges from the ratio of accumulated spatial to temporal increments, and the twin paradox resolves combinatorially through different total spin sums along distinct worldlines. The framework synthesizes insights from causal set theory, twistor theory, and loop quantum gravity, demonstrating how these approaches complement rather than compete with one another. It offers resolutions to long-standing puzzles including the origin of the Lorentzian signature, the compatibility of discreteness with Lorentz invariance, and the combinatorial definition of proper time. Open questions regarding dynamics, the quantum measure, and phenomenological predictions are discussed as directions for future investigation.