Slawomir Bugajski

Fundamentals of fuzzy probability theory

The canonical classical extension of quantum mechanics studied recently by E. G. Beltrametti and S. Bugajski opens a new way toward generalizing the standard probability theory. The emerging fuzzy probability theory is able to give a full account of both classical and quantal probabilities, and—like the standard probability theory—could be of universal use, far outside the borders of physics. A specific feature of this hypothetical theory of probability is its mixed, classical-quanta character: classical as well as quantal random variables are described on an equal footing in a unified framework. Some new features of the fuzzy probability theory are shown on simple examples.

Fundamental principles of quantum theory

After introducing general versions of three fundamental quantum postulates—the superposition principle, the uncertainty principle, and the complementarity principle—we discuss the question of whether the three principles are sufficiently strong to restrict the general Mackey description of quantum systems to the standard Hubert-space quantum theory. We construct an example which shows that the answer must be negative. We introduce also an abstract version of the projection postulate and demonstrate that it could serve as the missing physical link between the general Mackey description and the standard quantum theory.

Fundamental principles of quantum theory. II. From a convexity scheme to the DHB theory

Some classical and quantum theories are characterized within the convexity approach to probabilistic physical theories. In particular, the structure of the so-called DHB quantum theory will be analyzed. It turns out that the natural generalization of the standard Hubert space quantum mechanics, the operational one, is such a theory. The operational Hilbert space quantum theory will be reconstructed from the (weak) projection postulate and the complementarity principle. This is then used to argue that the DHB quantum theory is identical with the operational Hilbert space quantum theory.

Delinearization of quantum logic

The algebraic structure of the set of elementary observables of a delinearized quantal theory is described. As the delinearization procedure provides a kind of classical representation for any quantal theory, its relation to the traditional hypothesis of hidden variables is discussed.

The Inner Language of Operational Quantum Mechanics

The development of the “logical approach” to the foundations of quantum theories shows that it is possible to extract a full-fledged language from a physical theory -- in this case from the standard quantum mechanics (SQM). The “extraction procedure”, leading from SQM to the corresponding quantum language, has been discussed long ago by Finkelstein,14 Putnam,38 and others.† It is, however, still an object of some attacks, caused presumably by misunderstandings (see, for example,42, 32).

What is quantum logic

The paper describes in detail the procedure of identification of the inner language and an inner logico of a physical theory. The procedure is a generalization of the original ideas of J. von Neuman and G. Birkhoff about quantum logic.

Generalized Borel Law and Quantum Probabilities

A version of the Borel law of large numbers is found which remains valid for nonclassical (Mackey-type) probability theories. Its relation to a frequency interpretation of quantum probabilities is discussed.

Fuzzy dynamical systems, fuzzy random fields

Abstract A general notions of fuzzy dynamical systems and that of fuzzy stochastic processes, which encompass both classical and quantumdashmechanical cases, are introduced. One of the interesting features of this approach is a natural possibility of dilating any fuzzy dynamical system as well as any fuzzy random field into an operational classical one, what in turn leads to a reduction of the original system to a standard classical random field.