Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/9511001 · International Journal of Modern Physics B

CLASSICAL BEHAVIOR IN QUANTUM MECHANICS: A TRANSITION PROBABILITY APPROACH

Nicolaas P. Landsman

Published Jun 30, 1996• 12 citations
arXivHTMLPDF

Abstract

A formalism is developed for describing approximate classical behavior in finite (but possibly large) quantum systems. This is done in terms of a structure common to classical and quantum mechanics, viz. a Poisson space with a transition probability. Both the limit where ħ→0 in a fixed finite system and the limit where the size of the system goes to infinity are incorporated. In either case, classical behavior is seen only for certain observables and in a restricted class of states.

DOI
10.1142/s0217979296000647

Related papers

Poisson Spaces with a Transition Probability

Nicolaas P. Landsman

The common structure of the space of pure states ℘ of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p:℘×℘→[0,1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context). In classical mechanics, where p(ρ,σ)=δρσ, unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of p, and by the property that the irreducible components of ℘ as a transition probability space coincide with the symplectic leaves of ℘ as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck's constant). Motivated by E. M. Alfsen, H. Hanche-Olsen and F. W. Shultz (Acta Math.144 (1980) 267–305) and F.W. Shultz (Commun. Math. Phys.82 (1982) 497–509), we give axioms guaranteeing that ℘ is the space of pure states of a unital C*-algebra. We give an explicit construction of this algebra from ℘.

arXiv:quant-ph/9603005•Reviews in Mathematical Physics•Jan 1, 1997•14 citations

THE POSSIBILITY OF RECONCILING QUANTUM MECHANICS WITH CLASSICAL PROBABILITY THEORY ∗

D. A. Slavnov

We describe a scheme for constructing quantum mechanics in which a quantum system is considered as a collection of open classical subsystems. This allows using the formal classical logic and classical probability theory in quantum mechanics. Our approach nevertheless allows completely reproducing the standard mathematical formalism of quantum mechanics and identifying its applicability limits. We especially attend to the quantum state reduction problem.

•Jan 1, 2007•14 citations

The Relation Between Classical and Quantum Mechanics

Peter Taylor

Familiar formulations of classical and quantum mechanics are shown to follow from a general theory of mechanics based on pure states with an intrinsic probability structure. This theory is developed to the stage where theorems from quantum logic enable expression of the state geometry in Hilbert space. Quantum and classical mechanics are then elaborated and applied to subsystems and the measurement process. Consideration is also given to space-time geometry and the constraints this places on the dynamics. Physics and Mathematics, it is argued, are growing apart; the inadequate treatment of approximations in general and localisation in quantum mechanics in particular are seen as contributing factors. In the description of systems, the link between localisation and lack of knowledge shows that quantum mechanics should reflect the domain of applicability. Restricting the class of states provides a means of achieving this goal. Localisation is then shown to have a mathematical expression in terms of compactness, which in turn is applied to yield a topological theory of bound and scattering states. Finally, the thesis questions the validity of classical limits and quantisations in intertheoretic reduction, and demonstrates that a widely accepted classical limit does not constitute a proof of reduction. It proposes a procedure for determining whether classical and quantum mechanics are weakly equivalent over a domain of applicability, and concludes that, in this restricted sense, classical mechanics reduces to quantum mechanics.

arXiv:1806.08827•Jan 1, 1984•0 citations

THE QUANTUM-CLASSICAL TRANSITION: THE FATE OF THE COMPLEX STRUCTURE

Giuseppe Marmo, G. Scolarici, Alberto Simoni +1 more

According to Dirac, fundamental laws of Classical Mechanics should be recovered by means of an "appropriate limit" of Quantum Mechanics. In the same spirit it is reasonable to enquire about the fundamental geometric structures of Classical Mechanics which will survive the appropriate limit of Quantum Mechanics. This is the case for the symplectic structure. On the contrary, such geometric structures as the metric tensor and the complex structure, which are necessary for the formulation of the Quantum theory, may not survive the Classical limit, being not relevant in the Classical theory. Here we discuss the Classical limit of those geometric structures mainly in the Ehrenfest and Heisenberg pictures, i.e. at the level of observables rather than at the level of states. A brief discussion of the fate of the complex structure in the Quantum-Classical transition in the Schrödinger picture is also mentioned.

arXiv:hep-th/0501094•International Journal of Geometric Methods in Modern Physics•Feb 1, 2005•18 citations

Quantum Mechanics As A Limiting Case of Classical Mechanics

Partha Ghose

In spite of its popularity, it has not been possible to vindicate the conventional wisdom that classical mechanics is a limiting case of quantum mechanics. The purpose of the present paper is to offer an alternative point of view in which quantum mechanics emerges as a limiting case of classical mechanics in which the classical system is decoupled from its environment.

arXiv:quant-ph/0001025•Jan 10, 2000•0 citations

CLASSICAL AND NONCLASSICAL REPRESENTATIONS IN PHYSICS II: QUANTUM MECHANICS

Francis Heylighen

The conceptual and formal structure of quantum mechanics is analyzed from the point of view of the dynamics of distinctions, occurring during the observation process. The Hilbert space formalism is simplified with the help of the concept of closure: Closure of an eigenstate under an operator is generalized to the linear closure of a subset of slates, and this is further simplified to orthogonal closure, meaning that a set of states can be distinguished by a single observation. Quantum states can be seen as (overlapping) subsets of unobservable infrastates, with the transition probability between two states proportional to the number of infra-stales they have in common. This makes it possible to reconstruct the superposition principle. An analysis of the observation process leads to the interpretation of closed sets of infrastates as attractors of the dynamics induced by the interaction with the observation apparatus. This interaction is always partially undeterminate, because of the unobservable microstate of the apparatus.

Cybernetics & Systems•Sep 1, 1990•9 citations
A formalism is developed for describing approximate classical behavior in finite (but possibly large) quantum systems. This is done in terms of a structure common to classical and quantum mechanics, viz. a Poisson space with a transition probability. Both the limit where ħ→0 in a fixed finite system and the limit where the size of the system goes to infinity are incorporated. In either case, classical behavior is seen only for certain observables and in a restricted class of states.
The common structure of the space of pure states ℘ of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p:℘×℘→[0,1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context). In classical mechanics, where p(ρ,σ)=δρσ, unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of p, and by the property that the irreducible components of ℘ as a transition probability space coincide with the symplectic leaves of ℘ as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck's constant). Motivated by E. M. Alfsen, H. Hanche-Olsen and F. W. Shultz (Acta Math.144 (1980) 267–305) and F.W. Shultz (Commun. Math. Phys.82 (1982) 497–509), we give axioms guaranteeing that ℘ is the space of pure states of a unital C*-algebra. We give an explicit construction of this algebra from ℘.
Familiar formulations of classical and quantum mechanics are shown to follow from a general theory of mechanics based on pure states with an intrinsic probability structure. This theory is developed to the stage where theorems from quantum logic enable expression of the state geometry in Hilbert space. Quantum and classical mechanics are then elaborated and applied to subsystems and the measurement process. Consideration is also given to space-time geometry and the constraints this places on the dynamics. Physics and Mathematics, it is argued, are growing apart; the inadequate treatment of approximations in general and localisation in quantum mechanics in particular are seen as contributing factors. In the description of systems, the link between localisation and lack of knowledge shows that quantum mechanics should reflect the domain of applicability. Restricting the class of states provides a means of achieving this goal. Localisation is then shown to have a mathematical expression in terms of compactness, which in turn is applied to yield a topological theory of bound and scattering states. Finally, the thesis questions the validity of classical limits and quantisations in intertheoretic reduction, and demonstrates that a widely accepted classical limit does not constitute a proof of reduction. It proposes a procedure for determining whether classical and quantum mechanics are weakly equivalent over a domain of applicability, and concludes that, in this restricted sense, classical mechanics reduces to quantum mechanics.
According to Dirac, fundamental laws of Classical Mechanics should be recovered by means of an "appropriate limit" of Quantum Mechanics. In the same spirit it is reasonable to enquire about the fundamental geometric structures of Classical Mechanics which will survive the appropriate limit of Quantum Mechanics. This is the case for the symplectic structure. On the contrary, such geometric structures as the metric tensor and the complex structure, which are necessary for the formulation of the Quantum theory, may not survive the Classical limit, being not relevant in the Classical theory. Here we discuss the Classical limit of those geometric structures mainly in the Ehrenfest and Heisenberg pictures, i.e. at the level of observables rather than at the level of states. A brief discussion of the fate of the complex structure in the Quantum-Classical transition in the Schrödinger picture is also mentioned.
The conceptual and formal structure of quantum mechanics is analyzed from the point of view of the dynamics of distinctions, occurring during the observation process. The Hilbert space formalism is simplified with the help of the concept of closure: Closure of an eigenstate under an operator is generalized to the linear closure of a subset of slates, and this is further simplified to orthogonal closure, meaning that a set of states can be distinguished by a single observation. Quantum states can be seen as (overlapping) subsets of unobservable infrastates, with the transition probability between two states proportional to the number of infra-stales they have in common. This makes it possible to reconstruct the superposition principle. An analysis of the observation process leads to the interpretation of closed sets of infrastates as attractors of the dynamics induced by the interaction with the observation apparatus. This interaction is always partially undeterminate, because of the unobservable microstate of the apparatus.