Conformal extensions of the Galilei group and their relation to the Schrödinger group

Abstract

Various authors have considered a conformal extension CG0 of the Galilei group which in some sense is the nonrelativistic limit of the conformal extension of the Poincaré group, and have also established an invariance group for the free-particle Schrödinger equation, the ’’Schrödinger group.’’ Here we establish the most general conformal extension CG of the Galilei group, which is found to be identical to the group of the most general coordinate transformations that permit the use of noninertial frames of reference and of curvilinear coordinates in Galilei-invariant theories, which was considered by one of us some time ago, and is a gauge group containing a number of arbitrary functions. Both CG0 and the Schrödinger group are subgroups of CG containing the Galilei group, but otherwise they do not overlap. The Hamilton–Jacobi and Schrödinger equations for particles which are free or interact via inverse-square potentials are shown to be invariant under the Schrödinger group, and a further invariance of the Hamilton–Jacobi equation is established.