Description of Unstable Systems in Relativistic Quantum Mechanics in the Lax-Phillips Theory

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.