  We develop a quantum many-body theory of the Bose-Hubbard model based on the
canonical quantization of the action derived from a Gutzwiller mean-field
ansatz. Our theory is a systematic generalization of the Bogoliubov theory of
weakly-interacting gases. The control parameter of the theory, defined as the
zero point fluctuations on top of the Gutzwiller mean-field state, remains
small in all regimes. The approach provides accurate results throughout the
whole phase diagram, from the weakly to the strongly interacting superfluid and
into the Mott insulating phase. As specific examples of application, we study
the two-point correlation functions, the superfluid stiffness, the density
fluctuations, for which quantitative agreement with available quantum Monte
Carlo data is found. In particular, the two different universality classes of
the superfluid-insulator quantum phase transition at integer and non-integer
filling are recovered.

Quantum Gases;Strongly Correlated Electrons;Superconductivity