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

Source abstract

The true irreducible unitary representations of central extensions GM 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 GM and restricted to their values at one representative of each left coset of GM 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.