2025/01/01 by Jurgen Wollbold, Wollbold, Jurgen
Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Noncommutative and Quantum Gravity Theories #Quantum Electrodynamics and Casimir Effect
paper · doi:10.17605/osf.io/y3mq5
This work demonstrates that the Dirac operator arises unavoidably as the continuum limit of the Methane Metauniverse (MMU), a geometric and elastic model of spacetime based on a dual tetrahedral lattice. Starting from a discrete ur cell structure with internal torsional degrees of freedom, the analysis shows that symmetry, locality, and stability constraints force the emergence of a first order linear spinor operator at long wavelengths. The derivation does not rely on quantum postulates, relativistic axioms, or imposed spinor structures. Instead, the two state nature of an internal torsional mode leads naturally to a Pauli algebra, while symmetry protected band degeneracy enforces linear dispersion. Mass arises through an allowed coupling between complementary internal modes, and the resulting completion uniquely yields the Dirac equation. The central result is a proof of uniqueness: under minimal and physically unavoidable assumptions such as isotropy, locality, and stability, no alternative first order continuum operator is compatible with the MMU geometry. The Dirac operator is therefore not an optional construct, but a necessary geometric consequence of the underlying dual tetrahedral structure. This article situates the Dirac equation within a broader program that interprets quantum and relativistic structures as emergent projections of discrete spacetime geometry. It contributes to the MMU corpus by providing a rigorous geometric foundation for spinors, linearity, and relativistic dynamics, and by clarifying the structural origin of fermionic behavior without phenomenological input.