  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.
