Fuzzy Phase Space and Space-time Structure as Approach to Quantization

The quantum space-time and the phase space of massive particles with Fuzzy Geometry structure investigated as the possible quantization formalism. In this model the state of nonrelativistic particle m corresponds to the element of fuzzy ordered set (Foset)-fuzzy point. Due to its partial ordering m space coordinate x acquires principal uncertainty σ x. It's shown that Fuzzy Mechanics (FM) in 1+1 dimension is equivalent to the path integral formalism of nonrelativistic Quantum Mechanics. 1 Introduction The properties of quantum space-time and its relation to Quantum Mechanics (QM) axiomatics is actively discussed now from the different angles [1, 2, 3]. The interest to it enforced by the indications that its structure at small (Plank) scale can be quite nontrivial [4, 5]. In particular, it was proposed that such fundamental geometric features like the metrics or topology are modified significantly in this case [5, 6]. Our work motivated largely by this ideas and we shall explore the possible insights prompted by Sets Theory, exploring the various set structures of space-time manifold M. For example, in 1-dimensional Euclidean Geometry, the elements of its manifold X-the points x i constitute the ordered set. Yet Sets Theory includes other kinds of sets which also permit to construct the consistent geometries. In this context we shall investigate Posets and the fuzzy ordered sets (Fosets); in this case their 1