Poincare wave equations as Fourier transforms of Galilei wave equations

It is well known that the Galilei algebra is a sub algebra of Poincare algebra in one space dimension more. 1 This fact allows us to relate relativistic Poincare and Galilean theories. An interesting point is that Galilei transformations in two space dimensions are contained in the usual Poincare transformations? This enables us to present Poincare spin zero wavefunctions as Fourier transforms of Galilean ones. In the same way it is possible to see the Klein-Gordon equation as the Fourier transform of the Schrodinger equation in one space dimension less. On the other hand, due to the fact that the Poincare algebra is a subalgebra of the complex Galilei algebra in one space dimension more,3 it is possible to do a similar analysis as in the preceding case, i.e., the Schrodinger equation can be obtained as a Fourier transform of the Klein-Gordon equation. c The aim of this paper is to extend the results above quoted to the arbitrary spin case and study the possible relations between the Lagrangian formulations of Poincare and Galilei theories. The organization of this paper is as follows: In Sec. 2 we give a summary of the results of Ref. 2, in Sec. 3 we extend these results to the arbitrary spin case; in Sec. 4 we study some aspects of the Lagrangian formulation; Sec. 5 is devoted to conclusions.