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

AIP conference proceedings

Noether’s theorem and dynamical algebras for stationary and non-stationary quantum systems

Octavio Castaños, Alejandro Frank, Ramón López–Peña

Published Jan 1, 1992• 1 citation
Publisher

Abstract

Following work of Malkin, Man’ko and Lipkin, Dothan proposed a general definition for the dynamical group generators associated to a given Hamiltonian, based on the symmetry group of the corresponding time‐dependent quantum mechanical equation of motion for the system. In this paper we establish a procedure to derive the dynamical symmetry algebra of a quantum system, which arises naturally from Noether’s theorem by allowing time dependent symmetry transformations. The method is applicable to non‐stationary systems, as illustrated for the varying‐mass harmonic oscillator.

DOI
10.1063/1.42848

Related papers

Noether's theorem and dynamical groups in quantum mechanics

Octavio Castaños, Alejandro Frank, Ramón López–Peña

In contrast with the case of symmetry groups, no unique definition for the dynamical group generators of a given quantum Hamiltonian has been adopted in the literature. Following some work by Malkin and Man'ko (1965), Lipkin (1969), Dothan (1970) proposed a general definition based on the symmetry group of the corresponding time-dependent quantum mechanical equation of motion for the system. The authors show that this definition arises naturally from the quantum version of time dependent Noether symmetry transformations and derive the dynamical algebras for some simple examples.

Journal of Physics A Mathematical and General•Nov 21, 1990•21 citations

Noether's theorem and time-dependent quantum invariants

Octavio Castaños, Ramón López–Peña, Vladimir I. Man’ko

The time-dependent integrals of motion, linear in position and momentum operators, of a quantum system are extracted from Noether's theorem prescription by means of special time-dependent variations of coordinates. For the stationary case of the generalized two-dimensional harmonic oscillator, the time-independent integrals of motion are shown to correspond to special Bragg-type symmetry properties. A detailed study for the nonstationary case of this quantum system is presented. The linear integrals of motion are constructed explicitly for the case of varying mass and coupling strength. They are also obtained from Noether's theorem. The general treatment for a multi-dimensional quadratic system is indicated and it is shown that the time-dependent variations that give rise to the linear invariants, as conserved quantities, satisfy the corresponding classical homogeneous equations of motion for the coordinates.

arXiv:hep-th/9404016•Journal of Physics A Mathematical and General•Mar 7, 1994•21 citations

Quantum basic invariants and classical Noether's theorem

Xiao-Chun Gao, Jing‐Bo Xu, Tiezheng Qian +1 more

Taking the generalized time-dependent harmonic oscillator (GTHO) and singular time-dependent harmonic oscillator (STHO) as illustrative examples, we discuss the connection between the quantum basic invariants and classical Noether's theorem and, for the system, find: (1) The classical basic invariants and general form of classical invariants. (2) The current algebra and corresponding operator algebra. (3) A kind of converse of the Noether's theorem. (4) The connection between the so-called symmetry algebra ({ G k } algebra) and the classical current algebra through the operator algebra.

Physica Scripta•Apr 1, 1993•4 citations

Getting to the Bottom of Noether’s Theorem

John C. Baez

This chapter examines the assumptions behind Noether’s theorem connecting symmetries and conservation laws, taking an algebraic approach to compare classical and quantum versions of this theorem. In both classical and quantum mechanics, observables are naturally elements of a Jordan algebra, while generators of one-parameter groups of transformations are naturally elements of a Lie algebra. Noether’s theorem holds whenever we can map observables to generators in such a way that each observable generates a one-parameter group that preserves itself. In ordinary complex quantum mechanics, this mapping is multiplication by the square root of ?1. In the more general framework of unital JB-algebras, Alfsen and Shultz call such a mapping a ‘dynamical correspondence’ and show its presence allows us to identify the unital JB-algebra with the self-adjoint part of a complex C*-algebra. However, to prove their result, they impose a second, more obscure, condition on the dynamical correspondence. This expresses a relation between quantum and statistical mechanics, closely connected to the principle that ‘inverse temperature is imaginary time’.

arXiv:2006.14741•Cambridge University Press eBooks•Aug 31, 2022•10 citations

OBTAINING DYNAMICAL GROUPS AND QUANTUM DEGREES OF FREEDOM BY USING NOETHER’S THEOREM: THE PLANAR LANDAU SYSTEM

T. J. M. Simões

The noncompact quantum dynamical group SU(2, 1) and the associated su(2, 1) spectrum-generating algebra of the nonrelativistic planar Landau system are constructively obtained from the classical theory, by considering time-dependent Noether symmetry transformations and along the lines of a procedure recently proposed by Castaños, Frank and López-Peña. For this nonconstrained system in the circular gauge, we have found an explicitly time-dependent Hermitian invariant operator, which is crucial for the correct identification of the number of quantum degrees of freedom.

International Journal of Modern Physics A•Nov 20, 1994•1 citation

Group-theoretical approach to the classical and quantum oscillator with time-dependent mass and frequency

Gabriella Profilo, Giulio Soliana

We investigate the classical and quantum oscillator with time-dependent mass and frequency, in the framework of the Lie group theory. Our analysis shows that both systems possess five time-dependent Noether invariants, where only two of them are functionally independent. In the classical case, three of these invariants form the su(1,1) Lie algebra under the Poisson bracket operation, while in the quantum case the corresponding invariants form the su(1,1) Lie algebra under the commutator operation. Since this algebra underlies the noncompact group SU(1,1), we are thus able to define unambiguously the dynamical group related to the generalized quantum oscillator. Furthermore, the Hamiltonian of this system can be expressed in terms of the Noether invariant operators satisfying the su(1,1) Lie algebra. This form is useful in the evaluation of the energy spectrum and in the study of coherence and squeezing. Some particular cases are briefly illustrated. The other two Noether invariants can be interpreted as the quadrature-phase amplitudes involved in the quantum nondemolition measurements and in the problem of the generation of coherent and squeezed states.

Physical Review A•Aug 1, 1991•41 citations
Following work of Malkin, Man’ko and Lipkin, Dothan proposed a general definition for the dynamical group generators associated to a given Hamiltonian, based on the symmetry group of the corresponding time‐dependent quantum mechanical equation of motion for the system. In this paper we establish a procedure to derive the dynamical symmetry algebra of a quantum system, which arises naturally from Noether’s theorem by allowing time dependent symmetry transformations. The method is applicable to non‐stationary systems, as illustrated for the varying‐mass harmonic oscillator.
In contrast with the case of symmetry groups, no unique definition for the dynamical group generators of a given quantum Hamiltonian has been adopted in the literature. Following some work by Malkin and Man'ko (1965), Lipkin (1969), Dothan (1970) proposed a general definition based on the symmetry group of the corresponding time-dependent quantum mechanical equation of motion for the system. The authors show that this definition arises naturally from the quantum version of time dependent Noether symmetry transformations and derive the dynamical algebras for some simple examples.
The time-dependent integrals of motion, linear in position and momentum operators, of a quantum system are extracted from Noether's theorem prescription by means of special time-dependent variations of coordinates. For the stationary case of the generalized two-dimensional harmonic oscillator, the time-independent integrals of motion are shown to correspond to special Bragg-type symmetry properties. A detailed study for the nonstationary case of this quantum system is presented. The linear integrals of motion are constructed explicitly for the case of varying mass and coupling strength. They are also obtained from Noether's theorem. The general treatment for a multi-dimensional quadratic system is indicated and it is shown that the time-dependent variations that give rise to the linear invariants, as conserved quantities, satisfy the corresponding classical homogeneous equations of motion for the coordinates.
Taking the generalized time-dependent harmonic oscillator (GTHO) and singular time-dependent harmonic oscillator (STHO) as illustrative examples, we discuss the connection between the quantum basic invariants and classical Noether's theorem and, for the system, find: (1) The classical basic invariants and general form of classical invariants. (2) The current algebra and corresponding operator algebra. (3) A kind of converse of the Noether's theorem. (4) The connection between the so-called symmetry algebra ({ G k } algebra) and the classical current algebra through the operator algebra.
This chapter examines the assumptions behind Noether’s theorem connecting symmetries and conservation laws, taking an algebraic approach to compare classical and quantum versions of this theorem. In both classical and quantum mechanics, observables are naturally elements of a Jordan algebra, while generators of one-parameter groups of transformations are naturally elements of a Lie algebra. Noether’s theorem holds whenever we can map observables to generators in such a way that each observable generates a one-parameter group that preserves itself. In ordinary complex quantum mechanics, this mapping is multiplication by the square root of ?1. In the more general framework of unital JB-algebras, Alfsen and Shultz call such a mapping a ‘dynamical correspondence’ and show its presence allows us to identify the unital JB-algebra with the self-adjoint part of a complex C*-algebra. However, to prove their result, they impose a second, more obscure, condition on the dynamical correspondence. This expresses a relation between quantum and statistical mechanics, closely connected to the principle that ‘inverse temperature is imaginary time’.
The noncompact quantum dynamical group SU(2, 1) and the associated su(2, 1) spectrum-generating algebra of the nonrelativistic planar Landau system are constructively obtained from the classical theory, by considering time-dependent Noether symmetry transformations and along the lines of a procedure recently proposed by Castaños, Frank and López-Peña. For this nonconstrained system in the circular gauge, we have found an explicitly time-dependent Hermitian invariant operator, which is crucial for the correct identification of the number of quantum degrees of freedom.
We investigate the classical and quantum oscillator with time-dependent mass and frequency, in the framework of the Lie group theory. Our analysis shows that both systems possess five time-dependent Noether invariants, where only two of them are functionally independent. In the classical case, three of these invariants form the su(1,1) Lie algebra under the Poisson bracket operation, while in the quantum case the corresponding invariants form the su(1,1) Lie algebra under the commutator operation. Since this algebra underlies the noncompact group SU(1,1), we are thus able to define unambiguously the dynamical group related to the generalized quantum oscillator. Furthermore, the Hamiltonian of this system can be expressed in terms of the Noether invariant operators satisfying the su(1,1) Lie algebra. This form is useful in the evaluation of the energy spectrum and in the study of coherence and squeezing. Some particular cases are briefly illustrated. The other two Noether invariants can be interpreted as the quadrature-phase amplitudes involved in the quantum nondemolition measurements and in the problem of the generation of coherent and squeezed states.