Semigroup Representations of the Poincare Group and Relativistic Gamow Vectors

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 $\hbar/\Gamma_{R}$.