Basic notions and theWigner–Weisskopf theory

Although the notions of bound states, scattering and quantum transitions are well defined in the quantum theory, the description of an unstable system, involved in the process of decay, has remained an outstanding issue for many years. The problem is fundamental, since it concerns the nature of irreversible processes, one of the most important issues in statistical mechanics and the theme that is central to this book.

The theory of decay is intimately connected with scattering theory and necessarily contains mathematical ideas and methods. We shall try to explain these points carefully as we get to them.

We treat elsewhere in the book the ideas of Boltzmann, Van Hove and Prigogine on irreversible phenomena. The tools that are developed there are basically approximate, although very useful. One can argue that the basic rigorous characteristic of an irreversible process is that, as represented in terms of the evolution of a state in the Hilbert space of the quantum theory, it must be a semigroup. This type of evolution, resulting in an operation Z(t) on a state ψ, should satisfy the property

Z(t2) Z(t1) = Z(t1 + t2). (17.1)

The argument is as follows. If the system evolves in time t1 and is stopped, then evolves further at time t2, since the process has no memory, the total evolution should be as if the system evolved from the initial state to a state at t1 + t2 independently of the fact that it was done in two stages (Piron, 1976). Since the process is irreversible, the operator Z(t) may have no inverse. Such an evolution is called a semigroup. As we shall see, it is not possible to obtain such an evolution law in the framework of the standard quantum theory (Horwitz et al., 1971), but recently much work has been done, and methods have been developed, based on ideas of Sz.-Nagy and Foias (1976), such as the theory of Lax and Phillips (1967) and its extension to the quantum theory (Strauss et al., 2000) in which semigroup evolution can be achieved.