Quantizability of Lorentz-violating field theories with singular Hamiltonians

Spontaneous Lorentz-violating models of an antisymmetric tensor field are known to possess a singular Hamiltonian on the vacuum manifold, leading to unresolvable pathologies that render such theories unfit for cosmological studies. In this work, we introduce an auxiliary vector field inspired by the Stückelberg mechanism to investigate whether this resolves the fundamental phase space singularities. The total Hamiltonian is found to be globally ill-defined. For completeness, in an effective field theoretic sense where the phase space is restricted to near the vacuum manifold of Lorentz-violating models, we derive the singularity conditions near the vacuum manifold for an explicit model and find that the resulting constraint matrix acquires an explicit dependence on the gradients and conjugate momenta of the Stückelberg field components that can still lead to a singular Hamiltonian.