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.

## REFERENCES

*Group Theory*(Addison‐Wesley Publishing Company, Inc., Reading, Massachusetts, 1962), Sec. 12‐7.

*G*, they are said to be equivalent if $Ur\u2032\u2009=\u2009VUrV\u22121$ holds between operator rays ($Ur$ is the operator ray generated by $Ur,$ i.e. the set of all operators $\tau Ur,\tau \u2208,|\tau |\u2009=\u20091$). For the operators themselves, we have $Ur\u2032\u2009=\u2009\phi (r)VUrV\u22121,$ where $\phi (r)$ is some complex function of modulus 1 on the group. We see that $Ur\u2032$ is indeed a projective representation of the group with a factor system $\omega \u2032(r,s)\u2009=\u2009[\phi (r)\phi (s)/\phi (rs)]\omega (r,s)$ equivalent to the factor system $\omega (r,s)$ of ${Ur}.$

*Les Houches 1960 Summer School Proceedings*(Hermann et Cie., Paris, 1960), pp. 159–226.