2003/12/05 by Rocco Duvenhage, Duvenhage, Rocco · 1 citation
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0312051
PhD thesis, University of Pretoria, May 2002, 87 pages
arxiv created 2003/12/05 · openalex publication_date 2003/12/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.