Quantum mechanics with only positive-time evolution for an isolated system

Physical consequences of a more general theory of symmetries, based on the Galilei semi-group, are investigated for an isolated quantum system. The existence of the fundamental constants of motion is no longer a consequence of the symmetry, but it is an additional requirement. The usual description of a free particle is fully recovered, but a more general way to introduce interactions between particles appears. Unstable bound states arise in the theory which can be interpreted in a very satisfactory way as unstable particles. Scattering theory is reformulated, new terms in the cross-section expression are pointed out and general perhaps observable consequences of the non-Hamiltonian character of the interaction are discussed