Covariant operator formalism for higher derivative systems: Vector spin-$0$ dual model as a prelude to generalized QED$_4$

Abstract: In this work we extend the Kugo-Ojima-Nakanishi covariant operator formalism to quantize two higher derivative systems, considering their extended phase space structures. More specifically, the one describing spin-$0$ particles by a vector field and the generalized electrodynamics. We investigate the commutator structure of these theories and present the definition of their physical Hilbert subspaces. The first model presents a reducible gauge symmetry, implying the necessity of two sets of auxiliary fields. The massless limit is also carefully analyzed. After this prelude, the generalized QED$_4$ can be investigated with such machinery. Regarding the interacting regime, the positive norm subspace is no longer time invariant, since the interaction can create negative norm states from an initially ghost-free one. Then, we furnish an alternative description of the situation by analyzing a set of spectral representations highlighting the lack of positivity associated with the well-known ultraviolet improvement. Finally, based on these efforts and also on recent discussions about Lee-Wick like models, we prove that it is possible to establish a specific higher derivative interacting model compatible with establishing a time-invariant positive norm subspace.