Non-perturbative Quantization of Phantom and Ghost Theories: Relating Definite and Indefinite Representations

We investigate the non-perturbative 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 non-perturbative 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 non-perturbative 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 iǫ convergence factors in the path integral.