On Some Unitary Representations of the Galilei Group I. Irreducible Representations

Abstract

The true irreducible unitary representations of central extensions G_M of the Galilei universal covering group G and hence the physical representations of G are constructed by Mackey's method of induced representations. The elements of the representation space H are obtained from functions defined on G_M and restricted to their values at one representative of each left coset of G_M modulo K where K is the induction subgroup. The physical interpretation of these functions is in terms of wave functions and comes from the definition of a basis in H. This interpretation depends on the choice of a fundamental frame of reference in space‐time and on the physical meaning given to a fundamental state. To a change of the representatives corresponds a change of basis in H. By a suitable choice of these representatives, we obtain in particular the momentum‐spin representation and the momentum‐helicity representation. The zero mass case named class II by Inönü and Wigner is then obtained by the limit process M → 0 applied to the helicity representation.