Fuzzy Analysis of Particle movement in Quantum Mechanics

Sergey Nikolaevich Mayburov

Quantum Space-Time and Phase Space with fuzzy geometric structure are studied as possible formalism for quantization of massive particles and fields. In this approach the state of nonrelativistic particle m described by the fuzzy point of fuzzy (weakly) ordered Manifold X. Due to m partial ordering in X m coordinate x principal uncertainty is generic in this ansatz. It's shown that in 1+1 dimension such Fuzzy Mechanics (FM) is equivalent to Path Integral formalism of Quantum Mechanics (QM).

Beloslav Riečan

Christophe Tzara

Hongxing Li

"This unique compendium represents important action of fuzzy systems to quantum mechanics. From fuzzy sets to fuzzy systems, it also gives clear descriptions on the development on fuzzy logic, where the most important result is the probability presentation of fuzzy systems. The important conclusions on fuzzy systems are used in the study of quantum mechanics, which is a very new idea. Eight important conclusions are obtained. The author has proved that mass-point motions in classical mechanics must have waves, which means that any mass-point motion in classical mechanics has wave mass-point dualism as well as any microscopic particle motion must have wave-particle dualism. Based on this conclusion, it has been proven that classical mechanics and quantum mechanics are unified."--Publisher's website

Alex Granik, H. John Caulfield

It is shown that quantum mechanics can be regarded as what one might call a "fuzzy" mechanics whose underlying logic is the fuzzy one, in contradistinction to the classical "crisp" logic.Therefore classical mechanics can be viewed as a crisp limit of a "fuzzy" quantum mechanics.Based on these considerations it is possible to arrive at the Schroedinger equation directly from the Hamilton-Jacobi equation.The link between these equations is based on the fact that a unique ( "crisp") trajectory of a classical particle emerges out of a continuum of possible paths collapsing to a single trajectory according to the principle of least action.This can be interpreted as a consequence of an assumption that a quantum "particle" "resides" in every path of the continuum of paths which collapse to a single( unique) trajectory of an observed classical motion.A wave function then is treated as a function describing a deterministic entity having a fuzzy character.As a consequence of such an interpretation, the complimentarity principle and wave-particle duality can be abandoned in favor of a fuzzy deterministic microoobject.

Jarosław Pykacz