A Local No-Go Theorem For The Rahman–Susskind Proca Stability Window

Can a vector boson with a tachyonic Lagrangian mass be stable in de Sitter space? Rahman and Susskind (arXiv:2412.14749) conjectured that fixing a static patch keeps the standard Proca theory well-defined throughout the naively tachyonic window −2(D−1) < m²ℓ² < 0 (with ℓ the de Sitter radius), including an explicit positive-energy quantization of the D = 3 s-wave sector.

This paper proves a local no-go theorem to the contrary: for the standard Proca action on any static spacetime, the static Killing Hamiltonian is unbounded below whenever m² < 0. Remarkably simple data suffice — smooth, compactly supported pure-gradient configurations with vanishing electric and magnetic fields carry energy (m²/2)∫|Dψ|², which becomes arbitrarily negative at large amplitude. In the de Sitter static patch the counterexample can be chosen spherically symmetric and supported strictly between the center and the horizon, landing directly inside the claimed D = 3 s-wave stability window. A second construction shows the instability is also ultraviolet in character: a genuinely local longitudinal failure rather than an endpoint or boundary effect.

The origin of the discrepancy is identified as a sign error: exact canonical reduction of the D = 3 s-wave sector yields an electric-field action with the opposite overall sign from Rahman–Susskind's equation (4.41), so the oscillator norm is proportional to m² rather than −m². Since the wave equation and quasinormal frequencies are insensitive to an overall sign, decay of solutions cannot certify positivity or unitarity — positive norm and a bounded-below Hamiltonian cannot both be retained in the proposed window.

Declaration of AI Assistance

The generative AI tools used in the preparation of this paper were Fable 5 and GPT 5.6 Pro.