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Journal of Physics A Mathematical and General

Noether's theorem and dynamical groups in quantum mechanics

Octavio Castaños, Alejandro Frank, Ramón López–Peña

Published Nov 21, 1990• 21 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/23/22/009

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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.
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.
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.
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’.
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 introduces the general concept of symmetry in both classical and quantum physics. The part devoted to classical physics stresses the importance of conservation laws and the role of Noether’s theorem for their understanding. A fairly general discussion of Noether’s theorem is offered, including ambiguities in the definition of the Noether current, possible existence of generalized, tensor conservation laws, and the inverse of Noether’s theorem. The Hamiltonian formalism is then used to motivate the transition from classical to quantum physics, and the definition of symmetry in the latter. In anticipation of the discussion of spontaneous symmetry breaking in the following chapter, the text here emphasizes the difference between implementing symmetry in terms of transformations on the algebra of physical observables, and in terms of unitary mappings on the Hilbert space of states.
The aim of this thesis is to provide a definition of dynamical symme- try and to study its properties within simple quantum systems. In particular, I investigate Kepler's problem and the isotropic harmonic oscillator. Dynamical sy- mmetry is a kind of higher symmetry which is broken in a specific way. Definition of dynamical group and quantum mechanical system is presented. Subsequently, a definition of quantum degrees of freedom and quantum integrability is propo- sed. I mention briefly a possibility of finding the generators of dynamical group by considering time dependent constants of motion.