Semigroup Representations of the Poincare Group and Relativistic Gamow Vectors

TL;DR

This paper develops a mathematical framework using semigroup representations of the Poincaré group to describe relativistic quasistable states, known as Gamow vectors, with complex invariant mass and lifetime.

Contribution

It introduces a novel application of Poincaré semigroup representations to model relativistic Gamow vectors with complex mass parameters.

Findings

Relativistic Gamow vectors accurately describe quasistable particles.

The framework captures resonance mass and lifetime properties.

Provides a consistent mathematical description of relativistic resonances.

Abstract

Gamow vectors are generalized eigenvectors (kets) of self-adjoint Hamiltonians with complex eigenvalues $(E_{R}\mp i\Gamma/2)$ describing quasistable states. In the relativistic domain this leads to Poincar\'e semigroup representations which are characterized by spin $j$ and by complex invariant mass square ${\mathsf{s}}={\mathsf{s}}_{R}=(M_{R}-\frac{i}{2}\Gamma_{R})^{2}$. Relativistic Gamow kets have all the properties required to describe relativistic resonances and quasistable particles with resonance mass $M_{R}$ and lifetime $\hbar/\Gamma_{R}$.