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 σ x . It’s shown that m evolution on such manifold is described by Schroedinger or Dirac formalisms in norelativistic and relativistic cases correspondingly.
Highlights
Importance of geometric methods in Quantum Physics is duly acknowledged the reformulation of quantum mechanics (QM) in geometric terms is still an open problem ([1] and refs. therein). Such formalism can be applied primarily for quantum gravity problems, and for gauge field theory; its implications can be important for the analysis of QM foundations
The main impact of these studies is done on the exploit of Hilbert manifolds [1], alternatively in the approach considered here the basic structure is the real manifold equipped with fuzzy topology (FT) [2,3,4]
The general feature of such theories is that the space coordinates turn out to be principally fuzzy, the reason of that is the noncommutativity of coordinate observables x1,2,3
Summary
Importance of geometric methods in Quantum Physics is duly acknowledged the reformulation of quantum mechanics (QM) in geometric terms is still an open problem ([1] and refs. therein). It’s possible to detalize such smearing introducing the fuzzy relations, for that purpose one can put in correspondence to each a0, a j pair the weight w j ≥ 0 with the norm j w j = 1 In this case A is fuzzy ordered set (Foset), a0 called the fuzzy point (FP) [3, 10]. In the geometry with fundamental set CF the position of fuzzy point a j described by the positive normalized function w j(x) on X, which called the fuzzy value xj; w j dispersion σx characterizes a j coordinate uncertainty on X.
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