Time Asymmetric Quantum Theory - II. Relativistic Resonances from S-Matrix Poles

TL;DR

This paper develops a relativistic quantum framework for describing resonances and decaying states using Poincaré representations and Hardy subspaces, leading to precise resonance lineshapes and decay descriptions.

Contribution

It introduces a new hypothesis replacing Hilbert space assumptions to associate resonance poles with decaying states via Hardy subspaces and Gamow vectors.

Findings

Resonance amplitudes are separated into background and Breit-Wigner components.

Breit-Wigner amplitudes have a well-defined lineshape.

Gamow vectors describe exponential decay within Poincaré representations.

Abstract

Relativistic resonances and decaying states are described by representations of Poincar\'e transformations, similar to Wigner's definition of stable particles. To associate decaying state vectors to resonance poles of the $S$-matrix, the conventional Hilbert space assumption (or asymptotic completeness) is replaced by a new hypothesis that associates different dense Hardy subspaces to the in- and out-scattering states. Then one can separate the scattering amplitude into a background amplitude and one or several ``relativistic Breit-Wigner'' amplitudes, which represent the resonances per se. These Breit-Wigner amplitudes have a precisely defined lineshape and are associated to exponentially decaying Gamow vectors which furnish the irreducible representation spaces of causal Poincar\'e transformations into the forward light cone.