Relativistic Gamow Vectors I Derivation from Poles of the S-Matrix

A state vector description for relativistic resonances is derived from the first order pole of the $j$-th partial $S$-matrix at the invariant square mass value ${\text{sm}}_R=(m-i\mathrm{\Gamma }\mathrm{/}2{)}^2$ in the second sheet of the Riemann energy surface. To associate a ket, called Gamow vector, to the pole, we use the generalized eigenvectors of the four-velocity operators in place of the customary momentum eigenkets of Wigner, and we replace the conventional Hilbert space assumptions for the in- and out-scattering states with the new hypothesis that in- and out-states are described by two different Hardy spaces with complementary analyticity properties. The Gamow vectors have the following properties:

• They are simultaneous generalized eigenvectors of the four velocity operators with real eigenvalues and of the self-adjoint invariant mass operator $M=(P_{\mu }{P}^{\mu }{)}^{1\mathrm{/}2}$ with complex eigenvalue $\sqrt{{\text{sm}}_R}$.
• They have a Breit-Wigner distribution in the invariant square mass variable $\text{sm}$ and lead to an exactly exponential law for the decay rates and probabilities.