Quantum Lattice Model: Quantum Phase Transport and Wave Dynamics in the Quantum Lattice Model

This paper develops the quantum dynamical sector of the Quantum Lattice Model (QLM) by showing how the standard wave equations of quantum mechanics arise as continuum limits of coherent phase–action transport on a discrete Planck-scale lattice. The framework is constructed from the per-radian Planck primitives ℏ, ℓP, tP, where each lattice tick advances physical phase by one radian and transports one quantum of reduced action according to the phase-flow relationE = ℏ dθ/dτ. The invariant transport identityc = ℓP / tPfixes the maximal causal phase-transport speed of the lattice. Starting from a symmetric nearest-neighbor phase-transport rule between Planck cells, the continuum limit yields the relativistic wave operator (1/c²) ∂t² − ∇². Introducing intrinsic rest-phase rotation with frequency ω₀ = mc²/ℏ produces the Klein–Gordon equation. A first-order relativistic factorization of this operator yields the Dirac equation, while removing the rapidly rotating rest-phase carrier produces the nonrelativistic Schrödinger equation governing the slow envelope dynamics. Within this framework the quantum wavefunction represents a coherent phase envelope describing phase–action transport across the lattice, with the conserved density |φ|² arising from the continuity equation of Schrödinger evolution. The same transport framework provides geometric interpretations of interactions: • Electromagnetic binding is organized by the fine-structure constant α, interpreted as the fractional routing of phase transport relative to the vacuum baseline. • Hydrogenic structure emerges as a closed phase-transport configuration controlled by this routing fraction. • Gravitational coupling arises through a routing-suppression field Y (r) = 1 − rₛ / r that modulates the rest-phase carrier and produces the weak-field potential V = mΦ. These results show that the Schrödinger, Klein–Gordon, and Dirac equations emerge as successive continuum descriptions of discrete phase–action transport on the Planck lattice, providing the quantum dynamical layer of the Quantum Lattice Model.