Mathematics Colloquia and Seminars
Return to Colloquia & Seminar listing
Causality in noncommutative dynamicsColloquium
|Speaker: ||William Arveson, UC Berkeley|
|Location: ||693 Kerr|
|Start time: ||Mon, Nov 25 2002, 4:10PM|
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.