Speaker
Dimitrios Giannakis, Dartmouth College
Abstract
Fock spaces are widely used in quantum theory to model many-particle systems using direct sums of tensor product Hilbert spaces. In this talk, we discuss a framework for representing the evolution of observables of measure-preserving ergodic flows as infinite-dimensional rotation systems associated with a Fock space. This approach is based on an embedding of state space dynamics into the character spectrum of a commutative Banach algebra built as a symmetric weighted Fock space with reproducing kernel Hilbert space structure. Unitary approximations of the Koopman (composition) operator associated with the dynamical system are lifted to rotation systems on tori in the character spectrum akin to the topological models of ergodic systems with pure point spectra in the Halmos-von Neumann theorem. We also employ a procedure for representing observables of the original system by polynomial functions on the character spectrum, leading to models for the Koopman evolution of observables built from tensor products of finite collections of approximate Koopman eigenfunctions.