dorsal/arxiv
View SchemaThe general structure and ergodic properties of quantum and classical mechanics: A unified C*-algebraic approach
| Authors | Rocco Duvenhage |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0312051 |
| URL | https://arxiv.org/abs/quant-ph/0312051 |
Abstract
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.
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"abstract": "Quantum mechanics is essentially a statistical theory. Classical mechanics,\nhowever, is usually not viewed as being inherently statistical. Nevertheless,\nthe latter can also be formulated statistically. Furthermore, a statistical\nformulation of both can be expressed in terms of unital C*-algebras, in which\ncase it becomes clear that they have the same general structure, with quantum\nmechanics a noncommutative generalization of the classical case. In purely\nmathematical terms it is seen that quantum mechanics is a noncommutative\ngeneralization of probability theory. The most important insight in this\nrespect is that the projection postulate of quantum mechanics is a\nnoncommutative conditional probability. This is the subject of Chapter 1 of the\nthesis.\n As ergodic theory (the long term behaviour of a dynamical system) is done in\nclassical probability theory, it is then also done for noncommutative\nprobability theory in Chapter 2. In particular, generalizations of Khintchine\u0027s\nrecurrence theorem and a variation thereof for ergodic systems are proved, as\nwell as various characterizations of noncommutative ergodicity.\n Lastly, in Chapter 3, recurrence and ergodicity is then investigated from a\nphysical perspective in quantum and classical mechanics, by means of a quantum\nmechanical analogue of Liouville\u0027s Theorem in classical mechanics which was\nsuggested in Chapter 1.",
"arxiv_id": "quant-ph/0312051",
"authors": [
"Rocco Duvenhage"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"title": "The general structure and ergodic properties of quantum and classical mechanics: A unified C*-algebraic approach",
"url": "https://arxiv.org/abs/quant-ph/0312051"
},
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