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

TL;DR

This paper explores the Lax-Phillips theory for unstable systems in relativistic quantum mechanics, highlighting how semigroup decay laws and resonances are modeled within this framework, with potential for computing physical properties of resonant states.

Contribution

It demonstrates that the relativistic quantum theory provides a natural setting for applying the Lax-Phillips theory to describe unstable systems and their decay laws.

Findings

Decay of unstable systems is modeled by semigroups.

Resonances correspond to discrete spectrum and exponential decay.

Physical properties of resonant states can be computed within this framework.

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.