Mathematics Colloquia and Seminars

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The flow of time in quantum theory is represented by a one-paramter group of automorphisms of a noncommutative algebra of observables - a *-subalgebra of the algebra B(H) of all bounded operators on a Hilbert space H. In 1938, the one-parameter groups of automorphisms of B(H) were exhibited by Eugene Wigner; and that result leads to a classification of these dynamical groups. By contrast, if one introduces a natural notion of causality into the dynamics of B(H) - that is, an appropriate notion of past and future - one encounters entirely new phenomena. For example, there is a definite "state of the past" and a definite "state of the future", as well as two semigroups of endomorphisms, one for the past and one for the future. We discuss the qualitative nature of such dynamical systems, emphasizing the differences between noncommutative dynamics and the dynamics of flows on spaces. Despite encouraging evidence that a classification of causal dynamical systems should be possible, they are still only partially understood: we have surely not seen all of them, and while we have an effective index invariant for the simplest of them, we lack invariants for the others. There are opportunities here. We describe the rough classification of these semigroups into types and bring out the role of of cocycle perturbations and continuous tensor products of Hilbert spaces, emphasizing unsolved problems. We'll avoid technicalities and try to make the talk comprehensible to anyone with some affinity for operators on Hilbert space.