RESONANCE STATES FROM POLES OF THE RELATIVISTIC S-MATRIX

Abstract

A state vector description for relativistic resonances is derived from the first order pole of the jth partial S-matrix at the invariant square mass value [Formula: see text] 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: (i) They are simultaneous generalized eigenvectors of the four velocity operators with real eigenvalues and of the self-adjoint invariant mass operator M =(Pμ Pμ)1/2 with complex eigenvalue [Formula: see text]. (ii) They have a Breit–Wigner distribution in the invariant square mass variable [Formula: see text] and lead to an exactly exponential law for the decay rates and probabilities.