# Talk / Accardi

SEMINARIO MATEMATICO E FISICO DI MILANO

AVVISO DI CONFERENZA

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

"LE RADICI CLASSICHE DELLA TEORIA QUANTISTICA"

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:

http://www.mate.polimi.it/smf