Relativistic Fermion Dynamics in Noncommutative Space Time: The Fuzzy Dirac Equation

Overview

We present the derivation of a wave equation for the relativistic description of massive spin 1/2 particles in finite-noncommutative spacetime. We first construct a complex projective space CP^n and define a Kähler structure, extending the assumptions of Affine Quantum Mechanics to a complex coordinate space and establish the isomorphic embedding of CP^n to the relativistic spinning top. A worldsheet effective non-abelian action in the 11-D configuration space is described then made conformally invariant by replacing particle mass with scalar pseudo-Weyl curvature. As a beautiful result, the Lagrangians are shown to be sheet fibrations between the complex and real space. Characterizing separable coordinates in terms of the first homology group, we obtain the bundles of extremals belonging to the family of equidistant hypersurfaces, yielding a parity-invariant semilinear eigenvalue equation. We then formalize the transition to Hydrodynamical Quantum Mechanics then modern Quantum Field Theory via the facile transformation of complex Poisson structures to Dirac brackets and quantization of antiparticles via a Foldy-Wouthuysen transformation of the Newton-Wigner representation of the inhomogeneous coordinates, permitting the discussion of CPT symmetry, respectively. Finally, we show complete physical equivalence with the ordinary Dirac equation at sufficiently large scales beyond the Planck length. This theory is compared with quantum loop gravity and the noncommutative Yang Mills framework of M-theory in respect to gauge invariances. Holistically, the presented theory leverages insights from contemporary superstring theories to provide a direct analytical framework for the dynamics of fermions at minimal scales, a unique perspective in the ongoing pursuit of unification.

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• AJAS 2025 Category not listed

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