Symmetry and Dynamical Lie Algebras in Classical and Quantum Mechanics

Abstract

A unified definition is given for the symmetry and dynamical Lie algebras of a Hamiltonian in both classical and quantum mechanics. It is stressed that the differences observed between classical symmetry and dynamical Lie algebras and quantum ones stem from the importance of the algebra unitary irreducible representations in quantum mechanics. These concepts are illustrated on some two-dimensional rotationally-invariant Hamiltonians. By using a set of canonically conjugate variables including the Hamiltonian and the angular momentum, universal symmetry and dynamical algebras are constructed in classical mechanics and shown to have in general no counterpart in quantum mechanics.