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Feb 4, 2002 · A substantial literature has grown up around the programme of giving some independent motivation for this structure—ideally, by deriving it from more primitive and plausible axioms governing a generalized probability theory. 1. Quantum Mechanics as a Probability Calculus. 1.1 Quantum Probability in a Nutshell. 1.2 The “Logic” of Projections.
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Quantum Probability: An Introduction Guido Bacciagaluppiy 14 February 2014 The topic of probabilty in quantum mechanics is rather vast, and in this article, we shall choose to discuss it from the perspective of whether and in what sense quantum mechanics requires a generalisation of the usual (Kolmogorovian) concept of probability.
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Is quantum mechanics a generalization of classical probability?
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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
is probability theory. Since the early 1930-ies, in particular since the monograph of Kolmogorov [Kol33], probability theory is based on measure theory. Incomplete knowledge about a physical system is described by a probability space (;F; ), where is a set, called the state space, Fis a ˙-algebra on , and is a probability measure on F.
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Feb 4, 2002 · 1. Quantum Mechanics as a Probability Calculus. It is uncontroversial (though remarkable) that the formal apparatus of quantum mechanics reduces neatly to a generalization of classical probability in which the role played by a Boolean algebra of events in the latter is taken over by the “quantum logic” of projection operators on a Hilbert space. []
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.
This fact outlines a deep departure from the realm of classical physics where the events pertaining to a physical system carry the structure of a Boolean algebra, hence an algebraic model of classical logic. In Sect. 2 we shall review the structure of the events of classical and of quantum events and we will recall the main branching point.
