Time, irreversibility, and unstable systems in quantum physics

The recently developed quantum theory utilizing the ideas and results of Lax and Phillips for the description of scattering and resonances, or unstable systems, is reviewed. The framework for the construction of the Lax-Phillips theory is given by a functional space which is the direct integral over time of the usual quantum mechanical Hilbert spaces, defined at each t. It has been shown that quantum scattering theory can be formulated in this way. The theory of Lax and Phillips, however, also obtains a simple relation between the poles of the S-matrix and the spectrum of the generator of the semigroup corresponding to the reduced motion of the resonant state. In this Chapter, we show that to obtain such a relation in the quantum mechanical case, the evolution operator must act as an integral operator on the time variable. The structure required appears naturally in the Liouville space formulation of the evolution of the state of the system. The resulting S-matrix is a function, as an integral operator, of t - t′ (i.e., homogeneous), and the semigroup is contractive. A physical interpretation of this structure may be introduced, from which we obtain a quantitative description of the expected age of a created system, and the expected time of decay of an unstable system. The superselection rule which distinguishes between the unstable system and its decay products is realized in this way. It is also shown that, from this point of view, one has a natural mechanism for the dynamical mixing of the quantum mechanical states as observed by means of time-translation invariant operators. In particular, this provides a model for certain types of irreversible processes, and for the measurement process for closed and open systems.