Symmetries and ergodic properties in quantum probability

Vitonofrio Crismale¹ ; Francesco Fidaleo²

¹ Dipartimento di Matematica Università degli studi di Bari Via E. Orabona, 4 70125 Bari, Italy
² Dipartimento di Matematica Università degli studi di Roma Tor Vergata Via della Ricerca Scientifica 1 00133 Roma, Italy

Colloquium Mathematicum, Tome 149 (2017) no. 1, pp. 1-20.

Résumé

We deal with the general structure of (noncommutative) stochastic processes by using the standard techniques of operator algebras. Any stochastic process is associated to a state on a universal object, i.e. the free product $C^*$-algebra, in a natural way. In this setting, one recovers the classical (i.e. commutative) probability scheme and many others, like those associated to the monotone, boolean and $q$-deformed canonical commutation relations including the Bose/Fermi and Boltzmann cases. Natural symmetries like stationarity and exchangeability, as well as the ergodic properties of the stochastic processes are reviewed in detail for many interesting cases arising from quantum physics and probability.

Détail

DOI : 10.4064/cm6863-9-2016

Keywords: general structure noncommutative stochastic processes using standard techniques operator algebras stochastic process associated state universal object product * algebra natural setting recovers classical commutative probability scheme many others those associated monotone boolean q deformed canonical commutation relations including bose fermi boltzmann cases natural symmetries stationarity exchangeability ergodic properties stochastic processes reviewed detail many interesting cases arising quantum physics probability