Fuzzy physics

Wawrzyniec Guz

Diederik Aerts, Thomas Durt, Bruno Van Bogaert

We present an explicit physical example of an experimental situation on a physical entity that gives rise to a fuzzy set. The fuzziness in the example is due to ∞uctuations of the experimental apparatus, and not to an indeterminacy about the states of the physical entity, and is described by a varying parameter †. For zero value of the parameter (no ∞uctuations), the example reduces to a classical mechanics situation, and the corresponding fuzzy set is a quasi-crisp set. For maximal value (maximal ∞uctuations), the example gives rise to a quantum fuzzy set, more precisely a spin- model. In between, we have a continuum of fuzzy situations, neither classical, nor quantum. We believe that the example can make us understand the nature of the quantum mechanical fuzziness and probability, and how these are related to the classical situation.

Alfonso Barbas Espa

Rudolf Seising

In this paper we consider physical systems and the concept of their states in the context of the theory of fuzzy sets and systems. In section 1 we give a brief sketch on the fundamental difference between the theories of classical physics and quantum mechanics. In section 2 and 3 we introduce very shortly systems and their states in classical and quantum mechanics, respectively. Section 4 presents the concept of fuzzy systems. We propose to fuzzify the classical systems in section 5 and quantum systems in section 6. In section 7 we start to consider a fuzzy interpretation of the uncertainty principle.

Wolfgang W. Osterhage

Jarosław Pykacz

Bell-type inequalities, used in mathematical physics as a criterion to check whether a physical situation allows description in terms of classical (Kolmogorovian) or quantum probability calculus are applied to various fuzzy probability models. It occurs that the standard set of Bell-type inequalities does not allow to distinguish Kolmogorovian probabilities from fuzzy probabilities based on the most frequently used Zadeh intersection or probabilistic intersection, but it allows to distinguish all these models from fuzzy probability models based on Giles (Łukasiewicz) intersection. It is proved that if we use fuzzy set intersections pointwisely generated by Frank's fundamental triangular norms Ts(x,y), then the borderline between fuzzy probability models that can be distinguished from Kolmogorovian ones and these fuzzy probability models that cannot be distinguished is for .