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Mathematics of the classical and the quantum

Kryukov, Alexey (2000) Mathematics of the classical and the quantum. J. Math. Phys., 61, 082101.

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Abstract

Newtonian and Schrodinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrodinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed through the curvature of the space. The resulting embedding of the Newtonian and Schrodinger dynamics into a unified geometric framework is rigid in the sense that the Schrodinger dynamics is a unique extension of the Newtonian one. Under the embedding, the normal distribution of measurement results associated with a classical measurement implies the Born rule for the probability of transition of quantum states. The mathematics of the discovered relationship between the classical and the quantum is reviewed and investigated here in detail, and applied to the process of measurement of spin and position observables.


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Item Type: Published Article or Volume
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Kryukov, Alexey
Additional Information: Mathematics part of the paper: "The classical and the quantum" in the archive. Published in the AIP Journal of Mathematical Physics. To be followed by the physics part entitled "Physics in the space of quantum states"
Subjects: Specific Sciences > Physics
Specific Sciences > Physics > Quantum Mechanics
Depositing User: Alexey Kryukov
Date Deposited: 08 Aug 2020 02:30
Last Modified: 08 Aug 2020 02:30
Item ID: 17703
Journal or Publication Title: J. Math. Phys., 61, 082101
Publisher: AIP
Subjects: Specific Sciences > Physics
Specific Sciences > Physics > Quantum Mechanics
Date: 6 August 2000
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/17703

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