  We present a canonical mapping transforming physical boson operators into
quadratic products of cluster composite bosons that preserves matrix elements
of operators when a physical constraint is enforced. We map the 2D lattice
Bose-Hubbard Hamiltonian into $2\times 2$ composite bosons and solve it at mean
field. The resulting Mott insulator-superfluid phase diagram reproduces well
Quantum Monte Carlo results. The Higgs boson behavior along the particle-hole
symmetry line is unraveled and in remarkable agreement with experiment. Results
for the properties of the ground and excited states are competitive with other
state-of-the-art approaches, but at a fraction of their computational cost. The
composite boson mapping here introduced can be readily applied to frustrated
many-body systems where most methodologies face significant hurdles.
