Instantaneous spreading versus space localization for nonrelativistic quantum systems

A theorem of Hegerfeldt (Instantaneous spreading and Einstein causality in quantum theory, Ann. Phys. Leipzig vol. 7, 716-725 (1998)) establishes, for a class of quantum systems, a dichotomy between those which are permanently localized in a bounded region of space, and those exhibiting instantaneous spreading. We analyse in some detail the physical inconsistencies which follow from both of these options, and formulate which, in our view, are the basic open problems.