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Notes to Quantum Logic and Probability Theory. 1. A few qualifications are in order already: In a more general formulation, one considers the lattice of projections of a von Neumann algebra. Only in the context of non-relativistic quantum mechanics, and then only absent superselection rules, is this algebra a type I factor.
In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory. The formal system takes as its starting point an observation of Garrett Birkhoff and John von Neumann, that the structure of experimental tests in ...
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Can quantum logic generalize the event spaces of classical probability?
Can quantum probability theory be imbedded in classical probability theory?
Can quantum mechanics be regarded as a non-classical probability calculus?
When did quantum probability theory become a noncommutative measure theory?
The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of
- Miklos Redei, Stephen J. Summers
- 2006
Softcover ISBN 978-3-662-13735-2 Published: 17 April 2014. eBook ISBN 978-3-540-46070-1 Published: 13 November 2005. Series ISSN 0075-8450. Series E-ISSN 1616-6361. Edition Number 1. Number of Pages IX, 210. Topics Quantum Physics, Probability Theory and Stochastic Processes, Quantum Information Technology, Spintronics.
The topic of probability in quantum mechanics is rather vast. In this chapter it is discussed from the perspective of whether and in what sense quantum mechanics requires a generalization of the usual (Kolmogorovian) concept of probability.
What we intend to do, in the following, is to contribute some novel points of view to the “foundations of quantum mechanics”, using mathematical tools from “quantum probability theory” (such as the theory of operator algebras). The “foundations of quantum mechanics” represent a notoriously thorny and enigmatic subject.
