Session 1: Probabilistic Quantization: Why Applied Mathematics Interested Should Check It Out Alkhwarizmi Applied Mathematics webinar 

Speaker: Luigi Accardi, University of Rome, Italy.

Date: March 31, 2026.

Description of the lecture:  The emergence of Heisenberg commutation rule [q, p] = iℏ for position and momentum in boson quantum mechanics (QM), and more generally of non–commutativity in QM, has been a mystery since the dawn of this theory and has remained so for almost 100 years. The discovery of the quantum decomposition of a classical random variable gave rise to a line of research that, in little more that 25 years, has produced a natural solution to this problem, namely: the whole quantum theory (QT) can be deduced from the combination of classical probability (CP) with the classical theory of orthogonal polynomials (OP). The importance of this new approach does not lie so much in the solution of the problem mentioned above, as in the fact that the QT used until now appears as a very special case of a much broader purely deductive theory, which shows that every classical random variable with all moments, is canonically associated to a new QT, which reduces to the standard boson one in the case of gaussian random variables and to the standard fermion one in the case of Bernoulli random variables. That’s why we speak of probabilistic quantization. In other words, for the first time in almost 100 years, the mathematical apparatus of QT appears in the perspective of a natural deduction and not as a strange, singular theory justified a posteriori by its enormous empirical success, but totally mysterious in its origins and meaning. The fruitfulness of this new point of view is demonstrated by the fact that it has already led to the solution of several open problems in classical probability, in the classical theory of orthogonal polynomials and in physics itself, where it has provided powerful new tools for the description of natural phenomena. This poses an exciting challenge for everybody interested in classical probability and its applications. In fact, since we now know that quantization is a universal classical probabilistic phenomenon and that every classical random variable with all moments (or process or field, since the theory applies also to infinite dimensions) canonically produces a quantum theory, it follows that every field of science in which classical probability is involved (i.e. almost all), will have to learn how to exploit the benefits of this duality between classical and quantum description. By exploiting the quantum mathematical formalism, the physicists have managed to produce fantastic results in the past 100 years: now this power is available to all those who use classical probability in any field of science and technology. We need to learn to use these new mathematical tools. 

Biography: Luigi Accardi was born in Naples on 13.8.1947. With his Degree in Mathematics cum laude, at University of Naples 31.10.1970, he won the Prize Vittorio Nobile for the best degree in Mathematics. After  a Scolarship at the Laboratorio di Cibernetica of Consiglio Nazionale delle Ricerche (CNR), Arco Felice, Napoli, in 1972 became researcher of the CNR at the same  laboratory. 

He spent 4 years in Moscow within the framework of the agreements CNR-URSS Academy of Sciences, and obtained  the title Candidat Nauk at the Faculty of Mathematics and Mechanics of  Moscow University, with a thesis on Noncommutative Markov theory and square roots of measures  under the supervision of prof. I.M. Gelfand. In this period he worked at the Institute of Control Sciences in Moscow at the Laboratory directed by prof. M.A. Aizerman and at the Cathedra of Functional Analysis of Moscow University. Until 1982 he worked as resercher for CNR in Italy and France (Luminy, CNRS). In 1982 became Professor of Probability theory at the University of Roma Tor Vergata and traveled as Visiting Professor worldwide. In 1996 he become Professor at the Graduate School of Polymathematics, Nagoya University. From 1998 to the present he is Professor at the Faculty of Economy of the University of Roma Torvergata.

He has been member of Organizing, Scientific or Program Committe of many national and international conferences. He has been organizer of the "Workshop on Information Security" (Tokyo, 15 September 2011). The Workshop is included in the session "Italy in Japan 2011: Science, Technology and Innovation". The topic of the workshop is the main research theme of the joint Italo-Japanese Research Laboratory for Information Security with venue in the Research Institute of Tokyo University of Science, co-founded with the Centro V. Volterra and presently involved in several industrial projects related to applications of cryptography.