A quantum mechanical approach to a fuzzy theory

Jarosław Pykacz

Tofigh Allahviranloo, Sovan Samanta, Prasenjit Mandal +1 more

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.

Jarosław Pykacz

Rudolf Seising

In this paper we discuss the principles of two fundamental theories of physics: mechanics and quantum mechanics. First, we consider two philosophical positions of the German physicist Heinrich Hertz. He established one of the both in the introduction of his well known Principles of Mechanics. This view - Hertz's "picture" concept - which has a long tradition in philosophy, particularly with regard to physics and later to other empirical sciences, serves as the starting point of an interpretation of the relationship between real systems and theoretical structures of modern science using the concept of fuzziness. This "fuzzy view" on philosophy of science gives particularly an interpretation of Hertz's earlier lecture on The Constitution of matter. Second we consider some principles of quantum mechanics, particularly Werner Heisenberg's uncertainty relation of subatomic object's position and momentum, and Max Born's probabilistic interpretation of the quantum mechanical wave function. These principles are parts of the so-called Copenhagen interpretation of quantum mechanics. Third, we argue that the "fuzzy approach" is fruitful for a "fuzzy" interpretation of quantum mechanics because fuzziness fills the gap between systems and phenomena in reality and the scientific theory of quantum mechanics - as every other scientific theory.

Wenbing Qiu

The goal of this study is to extend quantum theory to the field of fuzzy phenomena. Using the idea and method of fuzzy mathematics, the quantum probability theory of fuzzy event is proposed, with the form of fuzzy event route integral, membership degree amplitude, fuzzy field function, Green function, physical quantity and fuzzy diagram. This theory changes quantum mechanics, making it a special instance of the former. Under some conditions, this theory breaches unitarity, gauge invariance, probability conservation, and information conservation, rendering these concepts approximate. This new theory, which does not require renormalization and can naturally produce meaningful results that are consistent with the experiments, is the proper theory for describing microscopic high-speed phenomena, whereas quantum mechanics is only a proper theory for describing microscopic low-speed phenomena. Because this theory is not divergent under the conditions that there is no renormalization and an infinite number of offsetting terms, it can constitute the theoretical framework required for gravity quantization.