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:2207.10326

On quantum complex flows

Thierry Paul

Published Jul 21, 2022• 0 citations
arXivHTMLPDF

Abstract

We study the propagation of quantum T{ö}plitz observables through quantized complex linear canonical transformation of one degree of freedom systems. We associate to such a propagated observable a non local "T{ö}plitz'' expression involving off diagonal terms. We study the link of this constrauction with the usual Weyl symbolic paradigm.

DOI
10.48550/arxiv.2207.10326

Related papers

Husimi, Wigner, T{ö}plitz, quantum statistics and anticanonical transformations

Thierry Paul

We study the behaviour of Husimi, Wigner and T{ö}plitz symbols of quantum density matrices when quantum statistics are tested on them, that is when on exchange two coordinates in one of the two variables of their integral kernel. We show that to each of these actions is associated a canonical transform on the cotangent bundle of the underlying classical phase space. Equivalently can one associate a complex canonical transform on the complexification of the phase-space. In the off-diagonal T{ö}plitz representation introduced in [P], the action considered is associated to a complex aanticanonical relation.

arXiv:2207.10322•Jul 21, 2022•0 citations

Paths of canonical transformations and their quantization

Maurice A. de Gosson

In their simplest formulations, classical dynamics is the study of Hamiltonian flows and quantum mechanics of propagators. Both are linked, and emerge from the datum of a single classical concept, the Hamiltonian function. We study and emphasize the analogies between Hamiltonian flows and quantum propagators; this allows us to verify G. Mackey's observation that quantum mechanics (in its Weyl formulation) is a refinement of Hamiltonian mechanics. We discuss in detail the metaplectic representation, which very explicitly shows the close relationship between classical mechanics and quantum mechanics; the latter emerging from the first by lifting Hamiltonian flows to the double covering of the symplectic group. We also give explicit formulas for the factorization of Hamiltonian flows into simpler flows, and prove a quantum counterpart of these results.

arXiv:1501.03137•Reviews in Mathematical Physics•Jul 1, 2015•8 citations

Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

M. I. Krivoruchenko, Amand Faessler

The knowledge of quantum phase flow induced under Weyl’s association rule by the evolution of Heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic Heisenberg operators. The quantum phase flow curves obey the quantum Hamilton equations and play the role of characteristics. At any fixed level of accuracy of semiclassical expansion, quantum characteristics can be constructed by solving a coupled system of first-order ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phase-space analytic geometry based on the star-product operation can hardly be visualized. The statement “quantum trajectory belongs to a constraint submanifold” can be changed, e.g., to the opposite by a unitary transformation. Some of the relations among quantum objects in phase space are, however, left invariant by unitary transformations and support partly geometric relations of belonging and intersection. Quantum phase flow satisfies the star-composition law and preserves Hamiltonian and constraint star functions.

arXiv:quant-ph/0604075•Journal of Mathematical Physics•May 1, 2007•15 citations

Canonical transformations and non-unitary evolution

E. D. D. Davis, G. I. Ghandour

We test the idea that transformations which, at the classical level, can be interpreted as evolutions are represented within quantum mechanics by unitary operators. To this end, we consider non-trivial canonical transformations which leave invariant the form of the Hamilton function of a system. We demonstrate that infinite families of such transformations exist for a variety of familiar conservative systems of one degree of freedom. We show how the precise form of integral equations for the stationary state wavefunctions implied by the existence of these canonical transformations can be pinned down by exploiting the algebra of the transformations and a symmetry of their generating functions. We recover several integral equations found in the literature on standard special functions of mathematical physics. We find that when one of the classical canonical transformations we consider is non-linear, its quantum implementation is non-unitary. We end with some comments on the implications of our findings for semiclassical studies and a brief discussion relevant to string theory of the generalization to scalar field theories in 1 + 1 dimensions.

arXiv:quant-ph/9905002•Journal of Physics A Mathematical and General•Jul 5, 2002•3 citations

Linear canonical transformations in quantum mechanics

Andrew J. Bordner

We find explicit unitary operators that implement linear canonical transformations of the quantum mechanical operators for a system with N degrees of freedom. We then relate the operators effecting this transformation to the previous formulation of quantum canonical transformations in terms of an effective generating function introduced by Ghandour.

Journal of Mathematical Physics•Jul 1, 1997•3 citations

Mathematical Aspects of the Weyl Correspondence

James C. T. Pool

The Weyl correspondence between classical and quantum observables is rigorously formulated for a linear mechanical system with a finite number of degrees of freedom. A multiplication of functions and a *-operation are introduced to make the Hilbert space of Lebesgue square-integrable complex-valued functions on phase space into a H*-algebra. The Weyl correspondence is realized as a *-isomorphism f → W(f) of this H*-algebra onto the H*-algebra of Hilbert-Schmidt operators on the Hilbert space of Lebesgue square-integrable complex-valued functions on configuration space. Moreover, the kernel of W(f) is exhibited in terms of a Fourier-Plancherel transform of f. Elementary properties of the Wigner quasiprobability density function and its characteristic function are deduced and used to obtain these results.

Journal of Mathematical Physics•Jan 1, 1966•203 citations
We study the propagation of quantum T{ö}plitz observables through quantized complex linear canonical transformation of one degree of freedom systems. We associate to such a propagated observable a non local "T{ö}plitz'' expression involving off diagonal terms. We study the link of this constrauction with the usual Weyl symbolic paradigm.
We study the behaviour of Husimi, Wigner and T{ö}plitz symbols of quantum density matrices when quantum statistics are tested on them, that is when on exchange two coordinates in one of the two variables of their integral kernel. We show that to each of these actions is associated a canonical transform on the cotangent bundle of the underlying classical phase space. Equivalently can one associate a complex canonical transform on the complexification of the phase-space. In the off-diagonal T{ö}plitz representation introduced in [P], the action considered is associated to a complex aanticanonical relation.
In their simplest formulations, classical dynamics is the study of Hamiltonian flows and quantum mechanics of propagators. Both are linked, and emerge from the datum of a single classical concept, the Hamiltonian function. We study and emphasize the analogies between Hamiltonian flows and quantum propagators; this allows us to verify G. Mackey's observation that quantum mechanics (in its Weyl formulation) is a refinement of Hamiltonian mechanics. We discuss in detail the metaplectic representation, which very explicitly shows the close relationship between classical mechanics and quantum mechanics; the latter emerging from the first by lifting Hamiltonian flows to the double covering of the symplectic group. We also give explicit formulas for the factorization of Hamiltonian flows into simpler flows, and prove a quantum counterpart of these results.
The knowledge of quantum phase flow induced under Weyl’s association rule by the evolution of Heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic Heisenberg operators. The quantum phase flow curves obey the quantum Hamilton equations and play the role of characteristics. At any fixed level of accuracy of semiclassical expansion, quantum characteristics can be constructed by solving a coupled system of first-order ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phase-space analytic geometry based on the star-product operation can hardly be visualized. The statement “quantum trajectory belongs to a constraint submanifold” can be changed, e.g., to the opposite by a unitary transformation. Some of the relations among quantum objects in phase space are, however, left invariant by unitary transformations and support partly geometric relations of belonging and intersection. Quantum phase flow satisfies the star-composition law and preserves Hamiltonian and constraint star functions.
We test the idea that transformations which, at the classical level, can be interpreted as evolutions are represented within quantum mechanics by unitary operators. To this end, we consider non-trivial canonical transformations which leave invariant the form of the Hamilton function of a system. We demonstrate that infinite families of such transformations exist for a variety of familiar conservative systems of one degree of freedom. We show how the precise form of integral equations for the stationary state wavefunctions implied by the existence of these canonical transformations can be pinned down by exploiting the algebra of the transformations and a symmetry of their generating functions. We recover several integral equations found in the literature on standard special functions of mathematical physics. We find that when one of the classical canonical transformations we consider is non-linear, its quantum implementation is non-unitary. We end with some comments on the implications of our findings for semiclassical studies and a brief discussion relevant to string theory of the generalization to scalar field theories in 1 + 1 dimensions.
The Weyl correspondence between classical and quantum observables is rigorously formulated for a linear mechanical system with a finite number of degrees of freedom. A multiplication of functions and a *-operation are introduced to make the Hilbert space of Lebesgue square-integrable complex-valued functions on phase space into a H*-algebra. The Weyl correspondence is realized as a *-isomorphism f → W(f) of this H*-algebra onto the H*-algebra of Hilbert-Schmidt operators on the Hilbert space of Lebesgue square-integrable complex-valued functions on configuration space. Moreover, the kernel of W(f) is exhibited in terms of a Fourier-Plancherel transform of f. Elementary properties of the Wigner quasiprobability density function and its characteristic function are deduced and used to obtain these results.