#### Is bosonic state space a proper subspace of the original Hilbert space on which we have an N-particle free Hamiltonian?

Consider the single particle Hilbert space, $\mathcal{H}$. For $N$ particles, the total Hilbert space is simply given by $\mathcal{H}_{total} = \otimes_{i=1}^{N} \mathcal{H}$, i.e., the tensor product of the single particle Hilbert spaces. In this language, your Hamiltonian ab...