We investigate the nonperturbative quantization of phantom and ghost degrees of freedom by relating their representations in definite and indefinite inner product spaces. For a large class of potentials, we argue that the same physical information can be extracted from either representation. We provide a definition of the path integral for these theories, even in cases where the integrand may be exponentially unbounded, thereby removing some previous obstacles to their nonperturbative study. We apply our results to the study of ghost fields of Pauli–Villars and Lee–Wick type, and we show in the context of a toy model how to derive, from an exact nonperturbative path integral calculation, previously ad hoc prescriptions for Feynman diagram contour integrals in the presence of complex energies. We point out that the pole prescriptions obtained in ghost theories are opposite to what would have been expected if one had added conventional i∊ convergence factors in the path integral.
Abstract The orthogonality and completeness properties of the resonant states as defined by Humblet and Rosenfeld are investigated using a simple regularization method first suggested by Zel'dovich. It is found that at leat for finite-range potentials, the set of bound states and any finite number of proper resonant states can be completed by a set of continuum states. This makes it possible to expand the scattering and reaction amplitudes in such a way that their resonance behabiour is exhibited, and the dependence of the corresponding partial widths on the interaction becomes more explicit.