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arXiv:hep-th/0210113

Fuzzy Phase Space Structure as Approach to Quantization

Sergey Nikolaevich Mayburov

Published Oct 11, 2002• 0 citations
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Abstract

Modification of nonrelativistic phase space structure based on fuzzy ordered sets (Fosets) structure investigated as a possible quantization framework. In this model particle's $m$ state corresponds to Foset element - fuzzy point. Due to fuzzy ordering its space coordinate $x$ acquires principal uncertainty $σ_x$. It's shown that proposed Mechanics on fuzzy phase space manifold reproduces the main quantum effects, in particular the interference of quantum states.

DOI
10.48550/arxiv.hep-th/0210113

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Modification of nonrelativistic phase space structure based on fuzzy ordered sets (Fosets) structure investigated as a possible quantization framework. In this model particle's mmm state corresponds to Foset element - fuzzy point. Due to fuzzy ordering its space coordinate xxx acquires principal uncertainty σxσ_xσx​. It's shown that proposed Mechanics on fuzzy phase space manifold reproduces the main quantum effects, in particular the interference of quantum states.
The quantum spacetime and the phase space with fuzzy structure are investigated as the possible quantization formalism. In this theory, the state of the nonrelativistic particle m corresponds to the element of fuzzy ordered set (Foset), i.e. the fuzzy point. Due to Foset partial (weak) ordering, the m space coordinate x acquires principal uncertainty σ x . It is shown that Schrödinger formalism of quantum mechanics can be completely derived from consideration of m evolution in fuzzy phase space with minimal number of axioms.
The quantum space-time and the phase space with fuzzy structure is investigated as the possible quantization formalism. In this theory the state of nonrelativistic particle m corresponds to the element of fuzzy ordered set (Foset)- fuzzy point. Due to Foset partial (weak) ordering, m space coordinate x acquires principal uncertainty σx. It’s shown that Shroedinger formalism of Quantum Mechanics can be completely derived from consideration of m evolution in fuzzy phase space with minimal number of axioms. Talk given at Workshop ’Quantum Field Theory under Influence of External conditions ’ Leipzig, sept. 2007 1
Fuzzy geometry considered as the possible mathematical framework for reformulation of quantum-mechanical formalism in geometric terms. In this approach the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. In 1-dimensional case, due to manifold ultraweak (fuzzy) topology, m space coordinate x acquires principal uncertainty dx and described by positive, normalized density w(x,t). Analogous uncertainties appear for fuzzy point on 3-dimensional manifold. It's shown that m states on such manifold are equivalent to vectors (rays) on complex Hilbert space, their evolution correspond to Shroedinger dynamics of nonrelativistic quantum particle.