Title: Nakanishi 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, taking into account 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. Remarkably, the establishment of a second-class nature for the primary constraints of such models demands a higher derivative structure for the auxiliary field Lagrangian following previous claims. Regarding the first model, it presents a reducible gauge symmetry implying the necessity of two sets of auxiliary fields. We also discuss its massless limit. For the case of the generalized QED$_4$, we derive a set of suitable definitions for the positive-definite Hilbert subspace in order to eliminate contributions from spurious modes and also the problematic negative norm transverse excitation. We show that the Hamiltonian operator taken within the domain of this subspace presents no instabilities. Finally, a set of discussions on the interacting regime are developed to ensure that the scattering processes restricted to the physical Hilbert subspace remain unitary even at this context.