Noether Symmetries, Dynamical Constants of Motion, and Spectrum Generating Algebras

TL;DR

This paper revisits Noether's theorem within the Hamiltonian framework, clarifying that conserved quantities can be explicitly time-dependent and still be associated with symmetries, expanding the traditional understanding of constants of motion.

Contribution

It provides a detailed analysis showing that dynamical constants of motion can be linked to symmetries with explicit time dependence, broadening the scope of Noether's theorem.

Findings

Conserved quantities may explicitly depend on time.

Dynamical constants of motion correspond to symmetries with total time derivatives.

Illustrations with three classes of simple systems.

Abstract

When discussing consequences of symmetries of dynamical systems based on Noether's first theorem, most standard textbooks on classical or quantum mechanics present a conclusion stating that a global continuous Lie symmetry implies the existence of a time independent conserved Noether charge which is the generator of the action on phase space of that symmetry, and which necessarily must as well commute with the Hamiltonian. However this need not be so, nor does that statement do justice to the complete scope and reach of Noether's first theorem. Rather a much less restrictive statement applies, namely that the corresponding Noether charge as an observable over phase space may in fact possess an explicit time dependency, and yet define a constant of the motion by having a commutator with the Hamiltonian which is nonvanishing, thus indeed defining a dynamical conserved quantity. Furthermore, and this certainly within the Hamiltonian formulation, the converse statement is valid as well, namely that any dynamical constant of motion is necessarily the Noether charge of some symmetry leaving the system's action invariant up to some total time derivative contribution. The present contribution revisits these different points and their consequences, straightaway within the Hamiltonian formulation which is the most appropriate for such issues. Explicit illustrations are also provided through three general but simple enough classes of systems.