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EPJ Web of Conferences

Fuzzy space-time topology and geometric quantization

Sergey Nikolaevich Mayburov

Published Jan 1, 2015• 0 citations
Publisher

Abstract

Dodson-Zeeman fuzzy topology considered as the possible mathematical framework of geometric quantum formalism. In such approach the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. Due to their weak (partial) ordering, m space coordinate x acquires principal uncertainty σx. It’s shown that m evolution on such manifold is described by Schroedinger or Dirac formalisms in norelativistic and relativistic cases correspondingly.

DOI
10.1051/epjconf/20159505010

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Dodson-Zeeman fuzzy topology considered as the possible mathematical framework of geometric quantum formalism. In such approach the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. Due to their weak (partial) ordering, m space coordinate x acquires principal uncertainty σx. It’s shown that m evolution on such manifold is described by Schroedinger or Dirac formalisms in norelativistic and relativistic cases correspondingly.
Dodson-Zeeman fuzzy topology considered as the possible mathematical framework of geometric quantum formalism. In such formalism the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. Due to their weak (partial) ordering, m space coordinate x acquires principal uncertainty σ x . It’s shown that m evolution with minimal number of additional assumptions obeys to schroedinger and dirac formalisms in norelativistic and relativistic cases correspondingly. It’s argued that particle’s interactions on such fuzzy manifold should be gauge invariant.
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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.
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