State space as projective space. The case of massless particles

The fact that the space of states of a quantum mechanical system is a projective space (as opposed to a linear manifold) has many consequences. We develop some of these here. First, the space is nearly contractible, namely all the finite homotopy groups (except the second) vanish (i.e., it is the Eilenberg-MacLane space K(, 2)). Moreover, there is strictly speaking no superposition principle in quantum mechanics as one cannot add rays; instead, there is adecomposition principle by which a given ray has well-defined projections in other rays. When the evolution of a system is cyclic, any representativevector traces out an open curve, defining an element of the holonomy group, which is essentially the (geometrical) Berry phase. Finally, for the massless case of the representations of the Poincaré group (the so-called Wigner program), there could be in principle arbitrarily multivalued representations coming from the Lie algebra of the Euclidean plane group. In fact they are at most bivalued (as commonly admitted).Presented at the International Symposium on Space-Time Symmetries, University of Maryland, Baltimore, Maryland, May 1988.