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:2005.01876 · Modern Physics Letters A

Lagrangian description of Heisenberg and Landau–von Neumann equations of motion

Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, G. Marmo, Luca Schiavone, Alessandro Zampini

Published Apr 27, 2020• 3 citations
arXivHTMLPDF

Abstract

An explicit Lagrangian description is given for the Heisenberg equation on the algebra of operators of a quantum system, and for the Landau–von Neumann equation on the manifold of quantum states which are isospectral with respect to a fixed reference quantum state.
DOI
10.1142/s0217732320501618

Related papers

Derivation of equations of motion of noncanonical operators from an action principle

J. K. Bhagat, N. Solimene

The Lagrangian description of a quantum system with a finite number of noncanonical operators is presented. Using this description the dynamics of the operators of the system are derived. The class of variations, commutation rules, and the form of Lagrangian are shown to be self-consistent.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jul 15, 1977•0 citations

A quantum route to the classical Lagrangian formalism

Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort +3 more

Using the recently developed groupoidal description of Schwinger’s picture of Quantum Mechanics, a new approach to Dirac’s fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function [Formula: see text] on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call q-Lagrangian, can be described in terms of a new function [Formula: see text] on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold M, the quadratic expansion of [Formula: see text] will reproduce the standard Lagrangians on TM used to describe the classical dynamics of particles.

arXiv:2106.00998•Modern Physics Letters A•Apr 30, 2021•9 citations

On integrability of wave propagator arising from von Neumann-Landau equations

Youngwoo Koh, Yoonjung Lee, Ihyeok Seo

The von Neumann-Landau equation describes the change in the density matrix with time. Interestingly, this equation was recently regarded as a wave equation for wave functions but not a equation for density functions. This setting leads to an extended form of the Schrodinger wave equation governing the motion of a quantum particle. In this paper we obtain integrability of wave propagator arising from the von Neumann-Landau equation in this setting.

•Sep 26, 2019•0 citations

Dissipative Quantum Mechanics: The Generalization of the Canonical Quantization and von Neumann Equation

Vasily Evgenevich Tarasov

The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system. But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems.

arXiv:hep-th/9410025•Oct 5, 1994•3 citations

Equations of Motion for Operators

Gordon Baym

The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.

CRC Press eBooks•Mar 5, 2018•0 citations

Lagrangian Approach to Quantum Mechanics

Yong Gwan Yi

The Lagrangian approach of Dirac is presented in a complete form. This suggests to identify the Schrödinger equation as the Euler-Lagrange equation rather than the Hamiltonian operator equation.

arXiv:physics/0005044•May 17, 2000•1 citation
Using the recently developed groupoidal description of Schwinger’s picture of Quantum Mechanics, a new approach to Dirac’s fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function [Formula: see text] on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call q-Lagrangian, can be described in terms of a new function [Formula: see text] on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold M, the quadratic expansion of [Formula: see text] will reproduce the standard Lagrangians on TM used to describe the classical dynamics of particles.
The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system. But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems.
The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.