New Characterization of the Ray Representations of the Galilei Group

Abstract

An elementary derivation of the system of phase factors defining the ray representations (those faithful to within a phase factor) of the Galilei group is presented. The proof employs the group elements themselves. In particular, the operation of conjugation (which corresponds to coordinate transformation) is used extensively to effect the desired result, i.e., in the notation of Levy‐Leblond, $$\mathrm{U(b\prime ,}\mathbf{a}\mathrm{\prime ,}\mathbf{v}\mathrm{\prime ,R\prime )U(b,}\mathbf{a,v}\mathrm{,R)=\pm }\exp \mathrm{\hspace{0.17em}[i(}\frac{1}{2}\mathrm{m)(}\mathbf{a}\mathrm{\prime ·R\prime }\mathbf{v}-\mathbf{v}\mathrm{\prime ·R\prime }\mathbf{a}\mathrm{+b}\mathbf{v}\mathrm{\prime ·R\prime }\mathbf{v}\mathrm{)]\times U(b\prime +b,}\mathbf{a}\mathrm{\prime +R\prime }\mathbf{a}\mathrm{+b}\mathbf{v}\mathrm{\prime ,}\mathbf{v}\mathrm{\prime +R\prime }\mathbf{v}\mathrm{,R\prime R)}$$ .