Canonical Realizations of Lie Symmetry Groups

Abstract

A general theory of the realizations of Lie groups by means of canonical transformations in classical mechanics is given. The problem is the analog to that of the characterization of the projective representations in quantum mechanics considered by Wigner, Bargmann, and others in the case of the Galilei and the Lorentz group. However, no application to particular groups is given in this paper. It turns out that the generators yτ of the infinitesimal transformations in a canonical realization of a Lie group 𝒢 satisfy relations of the form {yρ, yσ} = cρστyτ + dρσ, where cρστ are the structure constands of 𝒢 and dρσ are constants depending on the particular realization. It also turns out that any canonical realization of 𝒢 can be reduced to a fundamental typical form by means of a constant canonical transformation in the phase space of the system. This typical form allows one to obtain a complete characterization of all the possible canonical realizations of 𝒢. Once a suitable definition of ``irreducible'' canonical realization is given, a simple classification can be obtained in terms of the values of certain functions of the generators yτ (canonical invariants). Recalling the correspondence {} → [], formal analogy appears to be achieved with the quantum mechanical case. Even more, a parallel development of the outlined theory in quantum mechanics is of interest in the construction of the invariant operators (even in case of non-semisimple groups) and of complete systems of commuting observables acting within the irreducible representations.