Classical and quantum probability

Streater, R. F.

doi:10.1063/1.533322

Cited by 83 publications

(37 citation statements)

References 121 publications

(190 reference statements)

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“…Note that contrary to classical mechanics different generalized observables have different probability distributions and not all observables can be described in terms of a joint probability distribution, leading to a structure known as quantum probability, generalizing the classical notion of probability theory [15]. In fact it has been argued that the passage from classical to quantum theory is actually a generalization of probability theory [16]. Note that inside Ludwig's point of view coexistence of observables is related to the actual possibility of constructing concrete measuring apparatuses.…”

Section: B Generalized Notion Of Observablementioning

confidence: 99%

Quantum theory: the role of microsystems and macrosystems

Lanz¹,

Vacchini²,

Melsheimer³

2007

J. Phys. A: Math. Theor.

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We stress the notion of statistical experiment, which is mandatory for quantum mechanics, and recall Ludwig's foundation of quantum mechanics, which provides the most general framework to deal with statistical experiments giving evidence for particles. In this approach particles appear as interaction carriers between preparation and registration apparatuses. We further briefly point out the more modern and versatile formalism of quantum theory, stressing the relevance of probabilistic concepts in its formulation. At last we discuss the role of macrosystems, focusing on quantum field theory for their description and introducing for them objective state parameters.

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Section: B Generalized Notion Of Observablementioning

confidence: 99%

Quantum theory: the role of microsystems and macrosystems

Lanz¹,

Vacchini²,

Melsheimer³

2007

J. Phys. A: Math. Theor.

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“…The standpoint according to which quantum mechanics actually is a probability theory is by now well understood, and even though it is still not in the spirit of typical textbook presentations, it has been developed and thoroughly investigated in various books and monographes (see e.g. [3][4][5][6][7][8][9][10]), to which we refer the reader for more rigorous and detailed presentations. A more concise account of similar ideas has also been given in [11].…”

Section: Quantum Mechanics As Quantum Probabilitymentioning

confidence: 99%

Theoretical Foundations of Quantum Information Processing and Communication

Brüning¹,

Petruccione²

2010

Lecture Notes in Physics

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Summary. The general formalism of quantum mechanics for the description of statistical experiments is briefly reviewed, introducing in particular position and momentum observables as POVM characterized by their covariance properties with respect to the isochronous Galilei group. Mappings describing state transformations both as a consequence of measurement and of dynamical evolution for a closed or open system are considered with respect to the general constraints they have to obey and their covariance properties with respect to symmetry groups. In particular different master equations are analyzed in view of the related symmetry group, recalling the general structure of mappings covariant under the same group. This is done for damped harmonic oscillator, two-level system and quantum Brownian motion. Special attention is devoted to the general structure of translation-covariant master equations. Within this framework a recently obtained quantum counterpart of the classical linear Boltzmann equation is considered, as well as a general theoretical framework for the description of different decoherence experiments, pointing to a connection between different possible behaviours in the description of decoherence and the characteristic functions of classical Lévy processes.

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“…Although there is a high amount of papers written on this topic, it seems that a framework like the one we are outlining here is not considered. As an example, in a quite recent general review on the subject [27], R. F. Streater pointed out that: "Though the classical axioms were yet to be written down by Kolmogorov, Heisenberg, with help of the Copenhagen interpretation, invented a generalisation of the concept of probability, and physicists showed that this was the model of probability chosen by atoms and molecules." However, the algebraic (W * -)approach envisaged therein appears less close than ours to the standard treatment of probability on topological measure spaces, where the Borel or Baire structure is determined by the topology, as B(A) is determined by A.…”

Section: Theorem 32 [8] If a Has A Faithful Representation π On A Smentioning

confidence: 99%

Quantum Logic and Non-Commutative Geometry

Marchetti

Rubele

2006

Int J Theor Phys

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We propose a general scheme for the "logic" of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non Commutative Geometry. It involves Baire*-algebras, the non-commutative version of measurable functions, arising as envelope of the C*-algebras identifying the topology of the (non-commutative) phase space. We outline some consequences of this proposal in different physical systems. This approach in particular avoids some problematic features appearing in the definition of the state of "initial conditions" in the standard (W * -)algebraic approach to classical systems.

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Classical and quantum probability

Cited by 83 publications

References 121 publications

Quantum theory: the role of microsystems and macrosystems

Quantum theory: the role of microsystems and macrosystems

Theoretical Foundations of Quantum Information Processing and Communication

Quantum Logic and Non-Commutative Geometry

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