Abstract
Fuzzy topology and geometry considered as the possible mathematical framework for novel quantum-mechanical formalism. In such formalism the states of massive particle m correspond to the elements of fuzzy manifold called fuzzy points. Due to the manifold weak topology, m space coordinate x acquires principal uncertainty σx and described by the positive, normalized density w(, t) in 3-dimensional case. It’s shown that the evolution of m state on such 3-dimensional manifold corresponds to Shroedinger dynamics of massive quantum particle.
Highlights
It’s well known that quantum mechanics (QM) can be consistently described by several alternative formalisms such as, Shroedinger or standard one, algebraic QM, functional integral, etc. [1]
Successful applications of fuzzy methods to particular QM problems demonstrate the close connection between fuzzy mathematics and QM formalism
The similar features possess the formalism of algebraic QM where the state space is defined by the observable algebra and system dynamics, as the result, the state space can differ from QM Hilbert space [1]
Summary
It’s well known that quantum mechanics (QM) can be consistently described by several alternative formalisms such as, Shroedinger or standard one, algebraic QM, functional integral, etc. [1]. From the early days of its formulation and development the significant similarity of that theory and QM formalism was noticed [4,5,6]. It was argued that the generic parameter uncertainties which is the cornerstone of fuzzy mathematics are, similar to QM observable uncertainties [4, 5]. The analysis of quantum particle coordinate measurement evidences that such coordinate can be described as the fuzzy observable [7]. Successful applications of fuzzy methods to particular QM problems demonstrate the close connection between fuzzy mathematics and QM formalism. In its essence the considered formalism is geometric; the different approaches to QM geometrization studied extensively in the last years, first of all, for the application to quantum gravity theory [13, 14]
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