Complex mass in nonrelativistic quantum mechanics

Bernard H. Lavenda, Enrico Santamato

Bernard H. Lavenda, Enrico Santamato

Walter T. Grandy

Jean-Marc Lévy-Leblond

This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.

Daniel M. Greenberger

We show that the superselection rule in the Galilean transformation, forbidding the superposition of states of different mass, in inconsistent with the nonrelativistic limit of the Lorentz transformation. We also point out that the extra Galilean phase is merely the residue of the "twin-paradox" effect, which does not vanish nonrelativistically. In general, there are phase effects due to proper time differences, and effects due to mass-energy equivalence, that do not vanish nonrelativistically but that are not handled adequately by the Galilean transformation.

ANTON Z. CAPRI