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Applications of Mathematics

Exponential decay law and irreversibility of decay and collision processes

Václav Alda, V. Kundrát, M. V. Lokajíček

Published Jan 1, 1974• 4 citations
Publisher

Abstract

It is assumed that any decay process is described in a Hilbert space $\Cal H=\oplus \Cal H_D$, where $\Cal H_A$ corresponds to an unstable particle and $\Cal H_D$ to its decay products. It is shown that the generalized Weisskopf-Wigner condition (which guarantees an exponential decay law of the given unstable particle) has a close relation to the irreversibility of decay processes as well as of collision ones described in the same space $\Cal H$. If unitarity is added the principal structure of $\Cal H$ is identical with that onwhich the scattering theory of Lax and Phillips is based.

DOI
10.21136/am.1974.103547

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It is assumed that any decay process is described in a Hilbert space \CalH=⊕\CalHD\Cal H=\oplus \Cal H_D\CalH=⊕\CalHD​, where \CalHA\Cal H_A\CalHA​ corresponds to an unstable particle and \CalHD\Cal H_D\CalHD​ to its decay products. It is shown that the generalized Weisskopf-Wigner condition (which guarantees an exponential decay law of the given unstable particle) has a close relation to the irreversibility of decay processes as well as of collision ones described in the same space \CalH\Cal H\CalH. If unitarity is added the principal structure of \CalH\Cal H\CalH is identical with that onwhich the scattering theory of Lax and Phillips is based.
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 ( t 2 ) Z ( t 1 ) = Z ( t 1 + t 2 ). (17.1) The argument is as follows. If the system evolves in time t 1 and is stopped, then evolves further at time t 2 , since the process has no memory, the total evolution should be as if the system evolved from the initial state to a state at t 1 + t 2 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.
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Abstract: We discuss some of the experimental motivation for the need for semigroup decay laws, and the quantum Lax-Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax-Phillips S-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup, associated with resonances, the decay law is exactly exponential. The states corresponding to these resonances (eigenfunctions of the generator of the semigroup) lie in the Lax-Phillips Hilbert space, and therefore all physical properties of the resonant states can be computed. We show that the parametrized relativistic quantum theory is a natural setting for the realization of the Lax-Phillips theory. 1.
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