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 Mathematical Physics

Canonical transformations in quantum mechanics: A canonically invariant path integral

Anwar Y. Shiekh

Published Apr 1, 1988• 8 citations
Publisher

Abstract

A particular form of the path integral is presented that allows the implementation of general canonical transformations in quantum mechanics, as well as a consistent quantization procedure.
DOI
10.1063/1.527989

Related papers

Quantum canonical transformations revisited

A. Y. Shiekh

A preferred form for the path integral discretization is suggested that allows the implementation of canonical transformations in quantum theory.

arXiv:hep-th/9411199•Journal of Mathematical Physics•Dec 1, 1995•2 citations

Canonical transformations and path integrals

Robert L. Fanelli

A limited class of canonical transformations is introduced into the Lagrangian path integral method of quantization. Path integral quantization in different representations is discussed and a simple example is given.

Journal of Mathematical Physics•Sep 1, 1975•8 citations

Quantum canonical transformations revisited

Anwar Y. Shiekh, International Centre for Theoretical Physics, Trieste (Italy)

Abstract A preferred form for the path integral discretization is suggested that allows the implementation of canonical transformations in quantum theory.

•Jan 1, 1994•2 citations

Canonical transformations and Hamiltonian path integrals

Прохоров Лев Васильевич

The behavior of Hamiltonian path integrals under canonical transformations defined by a generator is investigated. The exact form of the kernel of the unitary operator which implements the quantum transformation is indicated. Equivalence rules are found (for the Hamiltonian formalism in the one-dimensional case) allowing one to get rid of the nonstandard terms in the action. It is shown that under canonical transformations the Hamiltonian path integral changes its form: in the transformed expression one cannot get along only with the corresponding classical Hamiltonian function.

•Feb 1, 1982•0 citations

Selecta from a Life-Long Obsession with Path Integrals

John R. Klauder, Christopher Casenas Bernido, M. Victoria Carpio-Bernido

The definition and interpretation of canonical, phase space path integrals has evolved over many years to achieve a form that now admits a correct and rigorous formulation, which is also covariant under canonical coordinate transformations. Such formulations involve coherent state representations, which, in their modern version, were originally introduced as an alternative tool to construct phase space path integrals. Moreover, coherent state representations lead to physical interpretations that are more natural than those afforded by more traditional representations. Suitable continuous time regularization procedures lead to a covariant phase space path integral formulation that greatly clarifies the vague phrase that canonical quantization requires Cartesian coordinates.

AIP conference proceedings•Jan 1, 2008•0 citations

Quantum canonical transformations as integral transformations

Jarmo Hietarinta

We discuss how the Hamiltonian changes in quantum canonical transformations. To the operator $\stackrel{^}{\mathcal{H}}(\stackrel{^}{p},\stackrel{^}{q})$ one can associate (in a given ordering rule) a $c$-number function $\mathcal{H}(p,q)$. It is this function that appears in the action of the phase-space path integral. A quantum canonical transformation $\stackrel{^}{\mathcal{H}}\ensuremath{\rightarrow}{\stackrel{^}{\mathcal{H}}}^{\ensuremath{'}}$ can now be expressed as an integral transformation ${\mathcal{H}}^{\ensuremath{'}}(\overline{p},\overline{q})=\ensuremath{\int}\mathrm{dp}\mathrm{dq}\mathcal{T}(\overline{p},\overline{q};p,q)\mathcal{H}(p,q)$. The kernel $\mathcal{T}$ is constructed explicitly for point transformations and for the $p=\ensuremath{-}\overline{q}$, $q=\overline{p}$ reflection by studying changes of variables in the path integral. The ordering dependence of $\mathcal{T}$ is displayed. The invariance of commutation rules is also discussed.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 15, 1982•18 citations
The behavior of Hamiltonian path integrals under canonical transformations defined by a generator is investigated. The exact form of the kernel of the unitary operator which implements the quantum transformation is indicated. Equivalence rules are found (for the Hamiltonian formalism in the one-dimensional case) allowing one to get rid of the nonstandard terms in the action. It is shown that under canonical transformations the Hamiltonian path integral changes its form: in the transformed expression one cannot get along only with the corresponding classical Hamiltonian function.
The definition and interpretation of canonical, phase space path integrals has evolved over many years to achieve a form that now admits a correct and rigorous formulation, which is also covariant under canonical coordinate transformations. Such formulations involve coherent state representations, which, in their modern version, were originally introduced as an alternative tool to construct phase space path integrals. Moreover, coherent state representations lead to physical interpretations that are more natural than those afforded by more traditional representations. Suitable continuous time regularization procedures lead to a covariant phase space path integral formulation that greatly clarifies the vague phrase that canonical quantization requires Cartesian coordinates.
We discuss how the Hamiltonian changes in quantum canonical transformations. To the operator $\stackrel{^}{\mathcal{H}}(\stackrel{^}{p},\stackrel{^}{q})$ one can associate (in a given ordering rule) a ccc-number function H(p,q)\mathcal{H}(p,q)H(p,q). It is this function that appears in the action of the phase-space path integral. A quantum canonical transformation $\stackrel{^}{\mathcal{H}}\ensuremath{\rightarrow}{\stackrel{^}{\mathcal{H}}}^{\ensuremath{'}}$ can now be expressed as an integral transformation ${\mathcal{H}}^{\ensuremath{'}}(\overline{p},\overline{q})=\ensuremath{\int}\mathrm{dp}\mathrm{dq}\mathcal{T}(\overline{p},\overline{q};p,q)\mathcal{H}(p,q)$. The kernel T\mathcal{T}T is constructed explicitly for point transformations and for the $p=\ensuremath{-}\overline{q}$, q=p‾q=\overline{p}q=p​ reflection by studying changes of variables in the path integral. The ordering dependence of T\mathcal{T}T is displayed. The invariance of commutation rules is also discussed.