Abstract:
The talk will discuss a theorem stating that any quantum dynamical system is integrable. It will be explained how this theorem is consistent with ergodicity, mixing and other properties of quantum chaos.
In more detail, a quantum dynamical system is given by a unitary representation of the additive group of real numbers, called the evolution operator. By means of the spectral theorem we prove that this unitary representation is unitary equivalent to a system of harmonic oscillators which is integrable. As a criterion for the efficiency of constructing integrals of motion, we consider the solvability of the corresponding Galois group, by analogy with the efficiency of computing the roots of a polynomial in the main theorem of algebra.
A brief discussion of integrability of classical dynamical systems, wave operators in scattering theory, open quantum systems, the hypothesis of thermalization of eigenstates, and a categorical formulation will also be given.