The Hilbert space of the g = 0 theory is the Fock space of the modes { a n † † } . That is to say that there exists a vacuum state | Ω Ω ⟩ ⟩ obeying a n | Ω Ω ⟩ ⟩ = 0 for all n, and on top of that there are single- and multi-particle states. Explicitly, a general eigenstate of H 0 is labeled by a tuple { k n } of occupation numbers: