Abstract
We discuss different formal frameworks for the description of generalized probabilities in statistical theories. We analyze the particular cases of probabilities appearing in classical and quantum mechanics and the approach to generalized probabilities based on convex sets. We argue for considering quantum probabilities as the natural probabilistic assignments for rational agents dealing with contextual probabilistic models. In this way, the formal structure of quantum probabilities as a non-Boolean probabilistic calculus is endowed with a natural interpretation.
Highlights
In the year 1900, the great mathematician David Hilbert presented a famous list of problems at a Conference in Paris
In Hilbert’s own words ([1], p. 454): “The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics
As an example, states of models of quantum mechanics can be described as measures over the orthomodular lattice of projection operators acting on a separable Hilbert space
Summary
In the year 1900, the great mathematician David Hilbert presented a famous list of problems at a Conference in Paris. This calculus can be considered as an extension of classical measure theory to a non-commutative setting (see, for example [12,21]; see [22] for a study of quantum measure theory) In this way, the axiomatization of probabilities arising in QM (and more general probabilistic models) can be viewed as a continuation of the Hilbert’s program with regard to probability theory. As an example (as we will discuss in Section 3.2), states of models of quantum mechanics can be described as measures over the orthomodular lattice of projection operators acting on a separable Hilbert space. It is interesting, for several reasons, to study more general models (that could describe, for example, alternative physical theories). Given that lattice theory is so central to the discussions presented here, we have included a short review about its elementary notions in Appendix A
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