Jarosław Pykacz

Fuzzy quantum logics and infinite-valued ?ukasiewicz logic

A new, physically more plausible definition of a fuzzy quantum logic is proposed. It is shown that this definition coincides with the previously studied definition of a fuzzy quantum logic; therefore it defines objects which are traditional quantum logics with ordering sets of states. The new definition is expressed exclusively in terms of fuzzy set operations which are generated by connectives of multiple-valued logic studied by Łukasiewicz at the beginning of the 20th century. Therefore, the logic of quantum mechanics is recognized as a version of infinite-valued Łukasiewicz logic.

Locality and Measurements Within the SR Model for an Objective Interpretation of Quantum Mechanics

One of the authors has recently propounded an SR (semantic realism) model which shows, circumventing known no-go theorems, that an objective (noncontextual, hence local) interpretation of quantum mechanics (QM) is possible. We consider here compound physical systems and show why the proofs of nonlocality of QM do not hold within the SR model, which is slightly simplified in this paper. We also discuss quantum measurement theory within this model, note that the objectification problem disappears since the measurement of any property simply reveals its unknown value, and show that the projection postulate can be considered as an approximate law, valid FAPP (for all practical purposes). Finally, we provide an intuitive picture that justifies some unusual features of the SR model and proves its consistency.

Bell-type inequalities in fuzzy probability calculus

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 Franks 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 .

Łukasiewicz Operations in Fuzzy Set and Many-Valued Representations of Quantum Logics

It, is shown that Birkhoff –von Neumann quantum logic (i.e., an orthomodular lattice or poset) possessing an ordering set of probability measures S can be isomorphically represented as a family of fuzzy subsets of S or, equivalently, as a family of propositional functions with arguments ranging over S and belonging to the domain of infinite-valued Łukasiewicz logic. This representation endows BvN quantum logic with a new pair of partially defined binary operations, different from the order-theoretic ones: Łukasiewicz intersection and union of fuzzy sets in the first case and Łukasiewicz conjunction and disjunction in the second. Relations between old and new operations are studied and it is shown that although they coincide whenever new operations are defined, they are not identical in general. The hypothesis that quantum-logical conjunction and disjunction should be represented by Łukasiewicz operations, not by order-theoretic join and meet is formulated and some of its possible consequences are considered.

Bell's theorem: Proposition of realizable experiment using linear momenta

Abstract A proposition of a realizable experiment using linear momenta for testing the Bell inequalities is presented. The proposed experimental setup employs only standard modern optical devices.

On Some New Operations on Orthomodular Lattices

Kotas conditionals are used to define six pairs of disjunction- andconjunction-like operations on orthomodular lattices. Although five of them necessarily differfrom the lattice operations on elements that are not compatible, they coincidewith the lattice operations on all compatible elements of the lattice and theydefine on the underlying set a partial order relation that coincides with the originalone. Some of the new operations are noncommutative on noncompatible elements,but this does not exclude the possibility to endow them with a physicalinterpretation. The new operations are in general nonassociative, but for someof them a Foulis—Holland-type theorem concerning associativity instead ofdistributivity holds. The obtained results suggest that these new operations canserve as alternative algebraic models for the logical operations of disjunctionand conjunction.

On Bell-Type Inequalities in Quantum Logics

Bell-type inequalities are studied within the framework of quantum logic approach. It is shown that violation of Bell’s inequalities indicates that pure states are not dispersion-free, whenever they are Jauch-Piron states which is true both in classical and in quantum mechanics. This result completes the results obtained by Santos [4] who has shown that violation of Bell’s inequalities implies that a lattice of propositions for a physical system is not distributive. Connections between Jauch-Piron properties of states, non-dispersive character of pure states and distributivity of a logic are studied and it is shown that if a logic is finite the former two properties imply the latter.

Fuzzy quantum logic I

The paper gives a review of the application of fuzzy set ideas in quantum logics. After a brief introduction to the fuzzy set theory, the historical development of the main attempts to utilize fuzzy set ideas in quantum logics are presented. Results of investigations of all major researchers (except the Italian group discussed elsewhere), who work or worked in the field, are discussed.

Quantum Machine and Semantic Realism Approach: a Unified Model

The Geneva–Brussels approach to quantum mechanics (QM) and the semantic realism (SR) nonstandard interpretation of QM exhibit some common features and some deep conceptual differences. We discuss in this paper two elementary models provided in the two approaches as intuitive supports to general reasonings and as a proof of consistency of general assumptions, and show that Aerts’ quantum machine can be embodied into a macroscopic version of the microscopic SR model, overcoming the seeming incompatibility between the two models. This result provides some hints for the construction of a unified perspective in which the two approaches can be properly placed.

Affine Maczyński logics on compact convex sets of states

Sets of affine functions satisfying Maczyński orthogonality postulate and defined on compact convex sets of states are examined. Relations between affine Maski logics and Boolean algebras when the set of states is a Bauer simplex (classical mechanics, some models of nonlinear quantum mechanics) are studied. It is shown that an affine Maczyński logic defined on a Bauer simplex is a Boolean algebra if it is a sublattice of a lattice consisting of all bounded affine functions defined on the simplex.