Eur. Phys. J. C (2026) 86: 752
https://doi.org/10.1140/epjc/s10052-026-16018-y

Regular Article - Theoretical Physics

Chrono-Grid Dynamics: a discrete unitary framework for emergent gravity

Independent Researcher, Batočina, Serbia

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Received: 16 March 2026
Accepted: 1 June 2026
Published online: 3 July 2026

Abstract

We present a comprehensive and self-contained formulation of Chrono-Grid Dynamics (CGD), a theory in which spacetime and general relativity emerge from a discrete, unitary evolution of a quantum spin network on a spatial lattice. The fundamental postulate is a universal quantum of proper time $\tau$ (the Takt), which drives the unitary step $|{\psi }_{n+1}⟩=e^{-i\widehat{H}\tau }|{\psi }_n⟩$. Starting from a 3D cubic or tetrahedral lattice with spin-1/2 degrees of freedom on sites and dynamical SU(2) link variables, we construct a local Hamiltonian that generates the dynamics. In the continuum limit, the link variables become the Ashtekar connection and their conjugate momenta become the densitised triad, leading to the canonical formulation of general relativity and the fundamental relation $8\pi G\beta ={\tau }^2$. We define the Gauss, diffeomorphism and Hamiltonian constraints on the lattice and show that their algebra closes, reproducing the Dirac algebra in the continuum. Edge modes appear on boundaries and carry topological degrees of freedom. Upon quantization we obtain discrete spectra for area, volume and length operators, expressed in terms of $\tau$ and spin labels. The 6j symbol of SU(2) emerges as the central object describing the quantum geometry of a tetrahedron, linking CGD to spin-foam models and the Ponzano–Regge asymptotic. When a cosmological constant is introduced, the curvature constraint regularizes to the quantum group $S{U}_q(2)$, connecting CGD to topological field theories such as the Turaev–Viro model. The discrete time step automatically provides a natural ultraviolet cutoff, and the lattice dispersion relation yields a minimal length of order the Planck scale. This work provides the mathematical foundation for the CGD program and paves the way for applications in quantum cosmology and numerical studies of lattice quantum gravity.