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Ehrenfest Theorems, Deformation Quantization, and the Geometry of Inter-Model Reduction

Rosaler, Joshua (2018) Ehrenfest Theorems, Deformation Quantization, and the Geometry of Inter-Model Reduction. Generalized Ehrenfest Relations, Deformation Quantization, and the Geometry of Inter-Model Reduction.

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Abstract

This study attempts to spell out more explicitly than has been done previously the connection between two types of formal correspondence that arise in the study of quantum–classical relations: one the one hand, deformation quantization and the associated continuity between quantum and classical algebras of observables in the limit ℏ→0, and, on the other, a certain generalization of Ehrenfest’s Theorem and the result that expectation values of position and momentum evolve approximately classically for narrow wave packet states. While deformation quantization establishes a direct continuity between the abstract algebras of quantum and classical observables, the latter result makes in-eliminable reference to the quantum and classical state spaces on which these structures act — specifically, via restriction to narrow wave packet states. Here, we describe a certain geometrical re-formulation and extension of the result that expectation values evolve approximately classically for narrow wave packet states, which relies essentially on the postulates of deformation quantization, but describes a relationship between the actions of quantum and classical algebras and groups over their respective state spaces that is non-trivially distinct from deformation quantization. The goals of the discussion are partly pedagogical in that it aims to provide a clear, explicit synthesis of known results; however, the particular synthesis offered aspires to some novelty in its emphasis on a certain general type of mathematical and physical relationship between the state spaces of different models that represent the same physical system, and in the explicitness with which it details the above-mentioned connection between quantum and classical models.


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Item Type: Published Article or Volume
Creators:
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Rosaler, Joshua
Keywords: reduction, geometry, physics, dynamical systems, quantum, classical, quantum-classical correspondence, Ehrenfest's Theorem, deformation quantization, limits
Subjects: Specific Sciences > Physics > Classical Physics
Specific Sciences > Physics > Quantum Mechanics
General Issues > Reductionism/Holism
General Issues > Structure of Theories
Specific Sciences > Physics > Symmetries/Invariances
General Issues > Theory Change
Depositing User: Dr. Joshua Rosaler
Date Deposited: 24 Aug 2022 12:55
Last Modified: 24 Aug 2022 12:55
Item ID: 21068
Journal or Publication Title: Generalized Ehrenfest Relations, Deformation Quantization, and the Geometry of Inter-Model Reduction
Publisher: Springer
Subjects: Specific Sciences > Physics > Classical Physics
Specific Sciences > Physics > Quantum Mechanics
General Issues > Reductionism/Holism
General Issues > Structure of Theories
Specific Sciences > Physics > Symmetries/Invariances
General Issues > Theory Change
Date: 24 February 2018
URI: http://philsci-archive.pitt.edu/id/eprint/21068

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