Calculation of the Scattering Amplitude for Resonance Production using Relativistic Gamow Vectors

Relativistic Gamow vectors were originally obtained from the Smatrix pole term for resonance formation experiments. They furnish an irreducible representation of the Poincare semi-group in the forward light cone and are characterized by spin and the complex pole position, very much like stable relativistic particles are labeled by spin and mass, and defined by Wigner’s basis vectors of unitary Poincare group representations. Gamow vectors have an exponential semi-group time evolution and represent relativistically decaying states whose Born probabilities fulfill the exponential law and Einstein causality. Resonance bumps suggesting a Breit-Wigner amplitude are also observed in production experiments. In this paper it will be shown that the Breit-Wigner line-shape for production experiments follows from using Gamow vectors as intermediate interacting state vectors. The results demonstrate that stable and unstable state vectors integrate in consistent framework.