A Pythagorean-Fuzzy Nonlocal Reformulation of Quantum Electrodynamics
Abstract
Quantum Electrodynamics (QED) is the most precise theory in physics, yet its assumption of pointlike interactions between charged particles and photons leads to ultraviolet divergences that require renormalization. This paper proposes a Pythagorean-Fuzzy Nonlocal Reformulation of QED, embedding structured uncertainty directly into the interaction framework. Each spacetime region is described by a Pythagorean fuzzy field with degrees of membership, non-membership, and hesitation, quantifying how strongly an event participates in an interaction and how precisely it can be localized. The conventional point vertex is replaced by a smooth, gauge-covariant nonlocal coupling modulated by a Lorentz-invariant kernel and the fuzzy field’s defuzzified weight. This structure preserves all symmetries of QED while automatically suppressing short-distance divergences. Ultraviolet divergences are suppressed at their origin, yielding finite self-energy and vacuum-polarization contributions within the nonlocal framework, without the appearance of divergent counter terms. Physically, the formulation interprets quantum interactions as finite “fuzzy” processes distributed over regions of limited definability. Mathematically, it unites the logic of Pythagorean fuzzy sets with the geometry of field theory, providing a natural regularization mechanism that remains fully consistent with standard QED in the sharp-local limit.
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