Canonical Realizations of the Galilei Group

Abstract

The general theory of the realizations of finite Lie groups by means of canonical transformations in classical mechanics, which has been developed in a preceding paper and already applied to the rotation group, is now applied to the Galilei group. Some complements to the general theory are introduced; in particular, a new kind of possible canonical realizations connected with the singularity surfaces of the functions Q(y),P(y),J(y) are discussed (singular realizations). In agreement with the situation encountered in quantum mechanics, the constants dρσ appearing in the fundamental Poisson bracket relations among the infinitesimal generators ({yρ,yσ}=cρστyτ+dρσ) cannot all be reduced to zero. There remains a single independent constant m, which, in the physically significant cases (m > 0), represents the mass of the system. No physical interpretation seems to be attachable to the realizations corresponding to m = 0. For m ≠ 0, two different kinds of irreducible realizations exist: one of a singular type which describes the free mass-point, and another of a regular type which describes a classical particle spin. A number of physical significant examples corresponding to nonirreducible realizations are thereafter discussed and the related typical forms are constructed: specifically, the cases dealt with are the rigid rod (linear rotator), the rigid body, and a system of two interacting mass points. It is shown that the problem of the construction of the variables of the typical form is equivalent to the determination of an appropriate solution of the time-independent Hamilton-Jacobi equation.