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Fuzzy Geometry of Phase Space and Quantization of Massive Fields

Published Aug 5, 2013• 0 citations
OpenAlex

Abstract

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

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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
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.
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.
Commutative fuzzy geometry considered as the possible mathematical framework for reformulation of quantum-mechanical formalism in strictly geometric terms. In that approach the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. Due to manifold weak topology, in 1-dimensional case m space coordinate x acquires principal uncertainty σ x and described by positive, normalized density ω ( x,t ), the analogous uncertainties appear for 3-dimensional manifold. It's shown that m states on such manifold are equivalent to Dirac vectors of complex Hilbert space, their evolution correspond to Shroedinger dynamics of massive quantum particle.