Time Asymmetric Quantum Theory - III. Decaying States and the Causal Poincare Semigroup

TL;DR

This paper develops a relativistic quantum framework for decaying states using Gamow kets, establishing a Poincaré semigroup structure that clarifies resonance properties and addresses causality issues.

Contribution

It introduces a Poincaré semigroup representation for relativistic decaying states, providing a rigorous definition of resonance mass and width, and explores their transformation properties.

Findings

Gamow kets form an irreducible Poincaré semigroup representation.

Resonance mass and width are unambiguously defined relativistically.

Transformation properties shed light on causality in relativistic quantum physics.

Abstract

A relativistic resonance which was defined by a pole of the $S$-matrix, or by a relativistic Breit-Wigner line shape, is represented by a generalized state vector (ket) which can be obtained by analytic extension of the relativistic Lippmann-Schwinger kets. These Gamow kets span an irreducible representation space for Poincar\'e transformations which, similar to the Wigner representations for stable particles, are characterized by spin (angular momentum of the partial wave amplitude) and complex mass (position of the resonance pole). The Poincar\'e transformations of the Gamow kets, as well as of the Lippmann-Schwinger plane wave scattering states, form only a semigroup of Poincar\'e transformations into the forward light cone. Their transformation properties are derived. From these one obtains an unambiguous definition of resonance mass and width for relativistic resonances. The physical interpretation of these transformations for the Born probabilities and the problem of causality in relativistic quantum physics is discussed.