The axioms of probability

The main philosophical successes of quantum probability is the discovery that all the so-called quantum paradoxes have the same conceptual root and that such root is of probabilistic nature. This discovery marks the birth of quantum probability not as a purely mathematical (noncommutative) generalization of a classical theory, but as a conceptual turning point on the laws of chance, made necessary by experimental results.

Indice 2 O. The goal of any mathematical investigation of the foundations of a physical theory is to clarify to what extent the mathematical formalism of that theory is uniquely determined by some clearly and explicitely stated physical assumptions. The achievement of that goal is particularly relevant in the case of the quantum theory where the novelty of the formalism, its being far away from any immediate intuition, the substantial failure met, for many years, by any attempt to deduce the quantum formalism from plausible physical assumptions, intersected with the never solved problems concerning the interpretation of the theory. That with quantum theory a new kind of probability theory was involved, was clear since the very beginnings of quantum mechanics (cf. [28]), even if it was not so clear which of the axioms of classifical probability had to be substituted, which physically meaningful statement had to replace it, how and if a physically meaningful statement could justify the apparently strange quantum mechanical formalism. The lack of clear answers to these questions had a tremendous impact on the process of interpretation and misinterpretation of quantum theory. The attempts to answer these questions motivated the development of a new branch of probability theory -quantum probability-and led to definite mathematical answers to these questions. In the present paper we want to discuss how these mathematical results allow to solve in a rather natural way some old problems concerning the interpretation of quantum theory and its mathematical foundations.

Journal of Philosophical Logic, 1993

We study the origin of quantum probabilities as arising from non-boolean propositionaloperational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorvian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).

Foundations of Physics, 1992

In this paper the role of the mathematical probability models in the classical and quantum physics is shortly analyzed. In particular the formal structure of the quantum probability spaces (QPS) is contrasted with the usual Kolmogorovian models of probability by putting in evidence the connections between this structure and the fundamental principles of the quantum mechanics. The fact that there is no unique Kolmogorovian model reproducing a QPS is recognized as one of the main reasons of the paradoxical behaviors pointed out in the quantum theory from its early days.

Quantum Studies: Mathematics and Foundations

Here we continue with the ideas expressed in "On the strangeness of quantum mechanics" [1] aiming to demonstrate more concretely how this philosophical outlook might be used as a key for resolving the measurement problem. We will address in detail the problem of determining how the concept of undecidability leads to substantial changes to classical theory of probability by showing how such changes produce a theory that coincides with the principles underlying quantum mechanics.

Eprint Arxiv 1011 6331, 2010

In the following we revisit the frequency interpretation of probability of Richard von Mises, in order to bring the essential implicit notions in focus. Following von Mises, we argue that probability can only be defined for events that can be repeated in similar conditions, and that exhibit 'frequency stabilization'. The central idea of the present article is that the mentioned 'conditions' should be well-defined and 'partitioned'. More precisely, we will divide probabilistic systems into object, environment, and probing subsystem, and show that such partitioning allows to solve a wide variety of classic paradoxes of probability theory. As a corollary, we arrive at the surprising conclusion that at least one central idea of the orthodox interpretation of quantum mechanics is a direct consequence of the meaning of probability. More precisely, the idea that the "observer influences the quantum system" is obvious if one realizes that quantum systems are probabilistic systems; it holds for all probabilistic systems, whether quantum or classical.

It is argued that quantum mechanics does not have merely a predictive function like other physical theories; it consists in a formalisation of the conditions of possibility of any prediction bearing upon phenomena whose circumstances of detection are also conditions of production. This is enough to explain its probabilistic status and theoretical structure. Published in: Collapse, 8, 87-121, 2014

Quantum Probability Communications, 2001

We analyse the thesis of that classical probability is unable to model the the stochastic nature of the Aspect experiment, in which violation of Bell's inequality was experimentally demonstrated. According to these authors the experiment shows the need to introduce the extension of classical probability known as Quantum Probability. We show that their argument depends on hidden assumptions and a highly restrictive view of the scope of classical probability. A careful probabilistic analysis shows, on the contrary, that it is classical deterministic physical thinking which cannot cope with the Aspect experiment and therefore needs revision. The ulterior aim of the paper is to help mathematical statisticians and probabilists to find their way into the fascinating world of quantum probability (thus: the same aim as that of Kümmerer and Maassen) by dismantling the bamboo curtain between ordinary and quantum probability which over the years has been built up as physicists and pure mathematicians have repeated to one another Feynman's famous dictum 'quantum probability is a different kind of probability'.