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• W1749803513 abstract "We study the algebraic consequences of the traditional descriptions of dynamics based on differential equations. We thus show how C∗-dynamical systems consisting of an action of a locally compact group on a C∗-algebra provide a common framework for studying the time evolution and symmetry groups of systems in classical and quantum physics. One thing which emerged clearly at the workshop in March is that there are many different ideas about what constitutes a “dynamical system.” In my talk, therefore, I tried to explain how the dynamical systems operator-algebraists study arise from the more classical notion of dynamical system as a system of ordinary differential equations. In discussions afterwards, I became aware that many workers in my own area are unaware of the circle of ideas that led to the general acceptance of C∗-dynamical systems as an important field of research. So I have included in this written version a discussion of how these also arise in quantum mechanics. Dynamical systems were originally mathematical formulations of dynamics — how physical systems change in time. By exploring the algebraic properties of dynamics, we shall arrive at a notion of dynamical system which provides a common framework for studying time evolutions and symmetry groups in classical mechanics. We then seek a parallel framework for Hilbertspace models of quantum mechanics; carrying out this program raises a variety of interesting mathematical issues. We finish by showing how C∗-algebraic dynamical systems bring these ideas together in one neat conceptual package. 1. Classical dynamical systems We consider a physical system in which the different states of the system are described by points of a state space X. The time evolution is given by an ordinary differential equation x′ = f(x), in which the unknown is a function x : R → X, and the value x(t) represents the state of the system at time t. To see what happens to a system which starts in a given state x0, we solve the initial-value problem x′ = f(x), x(0) = x0. (1) We know from experience that the general solution of x′ = f(x) will involve an arbitrary constant c ∈ X, which should be determined by the initial condition; thus we expect the initial-value problem (1) to have a unique solution. More formally, if f : X → R satisfies a Lipschitz condition |f(x)− f(y)| ≤ K|x− y|, then the global existence and uniqueness theorem implies that (1) has a unique solution valid on all of R. Example 1. Consider a particle of mass m moving in R3 according to Newton’s laws; we assume that there are no external influences, so that the system is conservative in the sense that energy is conserved. Then the position of the particle is described by a vector-valued 1991 Mathematics Subject Classification. 46L55; 20C35; 81P10. This research was supported by the Australian Research Council." @default.
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• W1749803513 date "1999-01-01" @default.
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• W1749803513 title "Dynamical systems and operator algebras" @default.
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