The quantum description of unstable systems has been the subject of many investigations since the early days of quantum mechanics. As is well known such a description gives rise to several conceptual problems, arising from the difficulty of finding a satisfactory general characterisation of these systems, and from the unavoidable difference which exists between the quantum and the classical predictions about the decay. For these reasons interest in this subject has increased recently and some contributions which have provided a new insight into the problem have appeared.The present review article is aimed at a clear formulation of the basic problematics of the decay and at a description of the various recent attempts to solve this delicate problem, illustrating both their successes and limitations. From this review should emerge a clear view of the present theoretical situation. The organisation of the article is as follows. After a short summary of the classical description of the decay, a detailed study of the quantum non-decay probability is given. Then the peculiar dynamical situation leading to the formation of an unstable system is identified and the formal and physical aspects of the process are discussed. A detailed analysis is then made of the preparation procedure of an unstable system which is shown to amount to a localisation of the decay fragments within a distance of the order of the range of the forces acting between them. After this the discussion on the general properties of the non-decay probability at all times is completed.I n the first part of the review the basic elements for a theoretical description of decay processes are formulated and all mathematical aspects as well as conceptual implications are clarified. It is then possible to undertake a discussion on the crucial points of the theory. First, the various attempts to characterise unstable systems are reviewed, in particular those based on the association of unstable particles with poles of the S matrix and those based on the consideration of the time-translation semigroup. Finally, the most recent contributions to the problem are reviewed. They take more properly into account the actual physical situation, i.e. the fact that an unstable system unavoidably interacts with its environment. As a consequence one is led to the formulation of a better dynamical description which allows one to overcome, in a very natural way, all the relevant difficulties of the previous approaches. The problem raised by the new description, such as the possible dependence of the lifetime on the measuring apparatus, are discussed. I n conclusion it is shown how the new approach makes it possible to associate the unfolding of the decay process with a semi-group law of evolution with time.
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