  Quantum mechanics is essentially a statistical theory. Classical mechanics,
however, is usually not viewed as being inherently statistical. Nevertheless,
the latter can also be formulated statistically. Furthermore, a statistical
formulation of both can be expressed in terms of unital C*-algebras, in which
case it becomes clear that they have the same general structure, with quantum
mechanics a noncommutative generalization of the classical case. In purely
mathematical terms it is seen that quantum mechanics is a noncommutative
generalization of probability theory. The most important insight in this
respect is that the projection postulate of quantum mechanics is a
noncommutative conditional probability. This is the subject of Chapter 1 of the
thesis.
  As ergodic theory (the long term behaviour of a dynamical system) is done in
classical probability theory, it is then also done for noncommutative
probability theory in Chapter 2. In particular, generalizations of Khintchine's
recurrence theorem and a variation thereof for ergodic systems are proved, as
well as various characterizations of noncommutative ergodicity.
  Lastly, in Chapter 3, recurrence and ergodicity is then investigated from a
physical perspective in quantum and classical mechanics, by means of a quantum
mechanical analogue of Liouville's Theorem in classical mechanics which was
suggested in Chapter 1.
