Relativistic resonances, semigroup representation of Poincar transformations, the exponential decay law and deviations thereof

Abstract

Starting from the phenomenological lineshape, relativistic Gamow vectors are defined. They span an irreducible representation ([j, sR]) of the causal Poincare semigroup. Their transformation properties are presented, from which follow the exponential time evolution of the (relativistic) Gamow states (of spin j and mass sR = (M − iΓ/2)2). The preparation and analysis of decay data are complicated by the presence of a continuous background integral—always omitted in the Weisskopf–Wigner approximation—in the complex basis vector expansion for the in-state of a resonance scattering experiment. This background integral, which is related to the background term in the scattering amplitude, gives rise to deviations from the exponential time evolution. To what extent a prepared state decays exponentially depends on the experiment and not the resonance per se.