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Talk / Accardi



Martedi' 27 gennaio 2014, alle ore 17:00
preso la Sala Rappresentanza (piano rialzato) del Dipartimento di
Matematica dell'Universita' di Milano
in via Saldini 50

Prof. Luigi Accardi
Universita' di Roma Tor Vergata

terra' la conferenza


Abstract: For several decades the mathematical model of Quantum
Probability (QP) has been considered as a generalization of classical
probability. However some discoveries of the past 15 years show that
the whole quantum theory, including quantum fields, is not a
generalization, but rather a deeper level of classical probability. In
fact, combining classical probability with the theory of orthogonal
polynomials in 1 or several real variables, it is possible to prove
that the canonical commutation relations, both Fermi and Bose (and in
fact even their q-deformations), arise canonically from the Bernoulli
and Gaussian random variables respectively. More generally one can
prove that there is a one-to-one correspondence between
Heisenberg-type commutation relations and equivalence classes of
probability measures on R with all moments. The equivalence relation
being defined, in the one-dimensional case, by the fact that all
measures in a class share the same principal Jacobi sequence. To each
of these equivalence classes it is canonically associated a free
evolution, generalizing the classical harmonic oscillator evolution.
The characterization of the equilibrium states with respect to any
such evolution naturally leads to a generalization of the Planck
factor. Similar arguments, applied to the recently introduced local
equilibrium states, lead to non-linear extensions of the Planck factor
and non-linear Gibbs states. Being functorial, the above construction
also provides a generalization of the second quantization procedure
both at Hilbert space (Fock) and -algebra level and in some special
cases (e.g. probability measures on Rd with compact support) even at
C-algebra level. However in general the class of morphisms will be
much narrower than in usual second quantization. This fact supports
the intuition that the new quantizations have a physical meaning in
terms of non-linear completely integrable classical systems. The
present talk is concentrated on the goal to illustrate the classical
roots of quantum theory, however if time allows it will be also
mentioned how these new ideas have allowed to solve a multiplicity of
long standing open problems both in classical probability and in the
theory of orthogonal polynomials.


Tutti gli interessati sono invitati a partecipare.

Il Direttore del Seminario
Franco Tomarelli

Per ulteriori informazioni sulla attivita' del seminario:

21 January 2015
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