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Quantum logic and the concept of probability in quantum mechanics

Keywords:

А.А. Pechenkin – Dr. Sc. (Phylos.), Professor, Philosophical Faculty, Lomonosov Moscow State University; Senior Research Scientist, S.I. Vavilov Institute of History of Natural Sciences and Equipment of RAS (Moscow)


In this article we will talk about the logic of the first type, developed in the 21st century, about the logical-algebraic approach to quantum theory [8-9]. G. Birkhoff and I. von Neumann saw in the structure of a Hilbert space (more precisely - in the algebra of projection operators acting in this space) an algebraic object - an orthomodular atomistic lattice. Kolmogorov formulated the axiomatic concept of probability, based on the theory of sets. This concept of probability became fundamental in the twentieth century. However, how to extend this concept to quantum mechanics? After all, quantum law does not apply the law of distributivity, from which the theory of sets proceeds. The Kolmogorov probability is defined on a distributive structure, called σ algebra. We must again turn to the lattice theory mentioned above. As noted, the lattice of quantum utterances is not distributive. However, it has the property of orthomodularity. This means that in the nondistributive lattice of quantum sentences there are many distributive sublattices. With their help, the concept of probability can be formulated which, on the one hand, will correspond to the Kolmogorov axiomatic concept, but it will be nonclassical, since it relies not on the lattice of quantum utterances, but on the sublattices of this lattice [10]. The concept of orthomodular lattice introduced by Birkhoff and von Neumann simplifies the real mathematical structure of quantum mechanics (see [2]). Consequently, the concept of probability, determined by means of this lattice, is a tentative concept, something like a mathematical experiment. However, it is possible that this concept will become in the future the mathematical basis of that quantum mechanics, which is presented in courses of mathematical physics.

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