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

Journal of Physics A Mathematical and General

On quantisation ambiguity

Alex Kovner, B. Rosenstein

Published Jul 11, 1987• 3 citations
Publisher

Abstract

The correspondence between classical and quantum theories is considered. The authors investigate the set of linear quantisations of classical Hamiltonians written in various canonical variables. The inverse problem of defining the classical limit of a quantum theory is also studied.
DOI
10.1088/0305-4470/20/10/015

Related papers

Ambiguities in Quantizing a Classical System

Ian H. Redmount, Wai-Mo Suen, K. Young

One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.

arXiv:gr-qc/9904042•Apr 19, 1999•1 citation

The Basic Formalism of Field Theory

A.N. Vasiliev

The classical system is usually quantized using the canonical formalism, i.e., in the language of coordinates and momenta. In addition to this “purely algebraic” ambiguity, there is also an ambiguity in the choice of realization of the operators p^ and q^: the commutation relations do not determine it uniquely, but up to a unitary equivalence (the well known uniqueness theorem of von Neumann pertains only to the Weyl form of the commutation relations). The realization must be chosen on the basis of additional considerations regarding the nature of the canonical variables p and q. In this book the term “theory” is used in various ways: in a broad sense (quantum field theory, relativistic theory) and in a narrow sense, to refer to a particular system or class of systems with certain dynamics. In quantum theory there are also systems whose operators corresponding to canonical variables satisfy not commutation relations, but anticommutation relations.

Routledge eBooks•Jan 22, 2019•0 citations

Canonical Quantisation

K. Kong Wan

Many important quantum observables, e.g., angular momentum, have their origin in corresponding classical observables. Postulate 26.1(OV) does not tell us how to find the operators to describe these observables. We can tackle this problem by formulating a process of quantisation to establish the operators for the corresponding quantum observables. Before we can formulate a quantisation process we must choose an appropriate description of classical observables. This is done in the following section.

•Jul 9, 2019•0 citations

Noncommutative configuration space: classical and quantum mechanical aspects

Franciscus Jozef Vanhecke, Cassio Sigaud, A. R. da Silva

In this work we examine noncommutativity of position coordinates in classical symplectic mechanics and its quantisation.In coordinates {q i , p k } the canonical symplectic two-form is ω 0 = dq i ∧ d p i .It is well known in symplectic mechanics [5, 6,9] that the interaction of a charged particle with a magnetic field can be described in a Hamiltonian formalism without a choice of a potential.This is done by means of a modified symplectic twoform ω = ω 0 -eF, where e is the charge and the (time-independent) magnetic field F is closed: dF = 0.With this symplectic structure, the canonical momentum variables acquire non-vanishing Poisson brackets: {p k , p l } = e F kl (q).Similarly a closed two-form in p-space G may be introduced.Such a dual magnetic field G interacts with the particle's dual charge r.A new modified symplectic two-form ω = ω 0 -eF + rG is then defined.Now, both p-and q-variables will cease to Poisson commute and upon quantisation they become noncommuting operators.In the particular case of a linear phase space R 2N , it makes sense to consider constant F and G fields.It is then possible to define, by a linear transformation, global Darboux coordinates: {ξ i , π k } = δ i k .These can then be quantised in the usual way [ ξ i , π k ] = i δ i k .The case of a quadratic potential is examined with some detail when N equals 2 and 3.

arXiv:math-ph/0502003•Brazilian Journal of Physics•Mar 1, 2006•18 citations

Canonical scattering transformation in quantum mechanics

F. Guillod, P. Huguenin

The authors quantise the classical canonical scattering transformation of Hunziker (1968) with the representation method of Moshinsky and Seligmann (1977). This leads to a sheeted phase space characterised by the number of turns around the scatterer. The usual detection device projects on the trivial representation of the corresponding ambiguity group. This operation extracts the integer values of the angular momentum.

Journal of Physics A Mathematical and General•Dec 1, 1982•2 citations

Classical limit of quantum theories

Iwo Białynicki‐Birula

A lecture is given on the correspondence between quantum and classical theories. The novelty lies in the systematic use of coherent states which enable one to discuss quantitatively the transition from quantum to classical theories for particles and for fields as well. (JFP)

•Jan 1, 1972•0 citations
One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.
The classical system is usually quantized using the canonical formalism, i.e., in the language of coordinates and momenta. In addition to this “purely algebraic” ambiguity, there is also an ambiguity in the choice of realization of the operators p^ and q^: the commutation relations do not determine it uniquely, but up to a unitary equivalence (the well known uniqueness theorem of von Neumann pertains only to the Weyl form of the commutation relations). The realization must be chosen on the basis of additional considerations regarding the nature of the canonical variables p and q. In this book the term “theory” is used in various ways: in a broad sense (quantum field theory, relativistic theory) and in a narrow sense, to refer to a particular system or class of systems with certain dynamics. In quantum theory there are also systems whose operators corresponding to canonical variables satisfy not commutation relations, but anticommutation relations.
Many important quantum observables, e.g., angular momentum, have their origin in corresponding classical observables. Postulate 26.1(OV) does not tell us how to find the operators to describe these observables. We can tackle this problem by formulating a process of quantisation to establish the operators for the corresponding quantum observables. Before we can formulate a quantisation process we must choose an appropriate description of classical observables. This is done in the following section.
In this work we examine noncommutativity of position coordinates in classical symplectic mechanics and its quantisation.In coordinates {q i , p k } the canonical symplectic two-form is ω 0 = dq i ∧ d p i .It is well known in symplectic mechanics [5, 6,9] that the interaction of a charged particle with a magnetic field can be described in a Hamiltonian formalism without a choice of a potential.This is done by means of a modified symplectic twoform ω = ω 0 -eF, where e is the charge and the (time-independent) magnetic field F is closed: dF = 0.With this symplectic structure, the canonical momentum variables acquire non-vanishing Poisson brackets: {p k , p l } = e F kl (q).Similarly a closed two-form in p-space G may be introduced.Such a dual magnetic field G interacts with the particle's dual charge r.A new modified symplectic two-form ω = ω 0 -eF + rG is then defined.Now, both p-and q-variables will cease to Poisson commute and upon quantisation they become noncommuting operators.In the particular case of a linear phase space R 2N , it makes sense to consider constant F and G fields.It is then possible to define, by a linear transformation, global Darboux coordinates: {ξ i , π k } = δ i k .These can then be quantised in the usual way [ ξ i , π k ] = i δ i k .The case of a quadratic potential is examined with some detail when N equals 2 and 3.