Decaying states as complex energy eigenvectors in generalized quantum mechanics

Abstract

The problem of particle decay is reexamined within the Hamiltonian formalism. By deforming contours of integration, the survival amplitude is expressed as a sum of purely exponential contributions arising from the simple poles of the resolvent on the second sheet plus a background integral along a complex contour GAMMA running below the location of the poles. One observes that the time dependence of the survival amplitude in the small time region is strongly correlated to the asymptotic behaviour of the energy spectrum of the system; one computes the small time behavior of the survival amplitude for a wide variety of asymptotic behaviors. In the special case of the Lee model, using a formal procedure of analytic continuation, it is shown that a complete set of complex energy eigenvectors of the Hamiltonian can be associated with the poles of the resolvent of the background contour GAMMA. These poles and points along GAMMA correspond to the discrete and the continuum states respectively. In this context, each unstable particle is associated with a well defined object, which is a discrete generalized eigenstate of the Hamiltonian having a complex eigenvalue, with its real and negative imaginary parts being the mass and half width ofmore » the particle respectively. Finally, one briefly discusses the analytic continuation of the scattering amplitude within this generalized scheme, and notes the appearance of ''redundant poles'' which do not correspond to discrete solutions of the modified eigenvalue problem.« less