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Steklov Mathematical Institute Seminar
November 16, 2023 16:00, Moscow, online


Integrability, chaos and Galois groups in quantum dynamical systems

I. V. Volovich


https://youtu.be/hzjAA7DtLsg

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.

References
  1. I. V. Volovich, “Complete integrability of quantum and classical dynamical systems”, $p$-Adic Numbers Ultrametric Anal. Appl., 11:4 (2019), 328–334  mathnet  crossref  mathscinet  zmath
  2. I. V. Volovich, “Ob integriruemosti dinamicheskikh sistem”, Izbrannye voprosy matematiki i mekhaniki, Sbornik statei. K 70-letiyu so dnya rozhdeniya akademika Valeriya Vasilevicha Kozlova, Trudy MIAN, 310, MIAN, M., 2020, 78–85  mathnet  crossref  elib; I. V. Volovich, “On the integrability of dynamical systems”, Proc. Steklov Inst. Math., 310 (2020), 70–77  crossref  isi  scopus


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