  We study the low-energy collective properties of a 1D spin-1 Bose gas using
bosonization. After giving an overview of the technique, emphasizing the
physical aspects, we apply it to the $S=1$ Bose-Hubbard Hamiltonian and find a
novel separation of the quadrupole-spin-charge sectors, confirmed by time-MPS
numerical simulations. Additionally, through the single particle spectrum, we
show the existence of the superfluid-Mott insulator transition and the point at
which the physics are described by a Heisenberg-like Hamiltonian. The magnetic
phase diagrams are found for both the superfluid and insulating regimes; the
latter is determined by decomposing the complete Heisenberg
bilinear-biquadratic Hamiltonian, which describes the Mott insulator, into
simpler, effective Hamiltonians. This allows us to keep our methods flexible
and transferable to other interesting interacting condensed matter systems.
