Title: Limiting dynamics of large quantum systems and Egorov-type theorems
Abstract:
In certain limiting regimes, the microscopic Hamiltonian dynamics of a
quantum system may be replaced by a simpler, "emergent" description.
Well-known examples are the semiclassical limit of a quantum system of
finitely many degrees of freedom and the mean-field limit of a Bose
gas. Traditionally, the study of these limits is confined to a
particular class of initial states, such as coherent states. In the
spirit of Egorov's classical result, I derive results in terms of
Heisenberg-picture observables, thus removing any restrictions on the
initial state. In the first part of the talk, I review Egorov's result
and discuss the semiclassical and mean-field limits of quantum spin
systems. I also discuss the mean-field limits of Bose and Fermi gases.
In the second part of the talk, I give an outline of the proofs, and
in particular describe a new and simple method to deal with singular
interaction potentials (such as the Coulomb potential) in the
mean-field limit of a quantum gas.