Noncommutative Unification of Dynamics and Probability

Abstract

Noncommutative geometry is quickly developing branch of mathematics finding important application in physics, especially in the domain of the search for the fundamental physical theory. It comes as a surprise that noncommutative generalizations of probabilistic measure and dynamics are unified into the same mathematical structure, i.e., noncommutative von Neumann algebra with a distinguished linear form on it. The so-called free probability calculus and the Tomita-Takesaki theorem, on which this unification is based, are briefly presented. It is argued that the unitary evolution, known from quantum mechanics, could be a trace of noncommutativity on a deeper level. Philosophical significance of these results is also discussed.