Analytic continuation of quantum systems and their temporal evolution

Abstract

The generalized vector space of quantum states is used to study the correspondence between the physical state space scrH and its continuation scrG. Consider the integral representation defined by the scalar product between an arbitrary vector in the dense subset of analytic vectors in scrH and its dual vector, where the integration is along the real axis. Keeping the scalar product fixed, the analytic vectors may be continued through the deformation of the integration contour. The deformed contour defines the generalized spectrum of the operator in the continued theory, which typically consists of a deformed contour in the fourth quadrant and the exposed singularities, if any, between the real axis and the deformed contour. Several models are studied with special attention to the unfolding of the generalized spectrum. The two-body models studied are the Lee model in the lowest sector and the Yamaguchi potential model, where the exposed singularities, if present, are simple poles. The three-body model studied is the cascade model, where the exposed singularities may be poles and the branch cuts associated with the quasi-two-body states. We demonstrate that the generalized spectrum obtained leads to the correct extended unitarity relation for the scattering amplitudes. The possibility of having mismatches between poles in the S matrix and the discrete states in the Hamiltonian, which exists in the scrH space, obtains also in the generalized scrG space. Finally, two distinct views on what constitutes an unstable particle are contrasted. One view is to identify it as a physical state of the system which ceases to exist as a discrete eigenstate in scrH. Here the survival amplitude of the unstable particle cannot be ever strictly exponential in time. The other view is to identify the unstable particle as a discrete state in the generalized space. It has a pure exponential time dependence. So the corresponding time evolution is realized by a semigroup. While the latter approach appears to be elegant, it is obtained at the expense of giving up the very starting premise of the lower boundedness of the energy spectrum and therefore we consider it to be the less desirable choice.