  We present a generalized quasi-particle theory for bosonic lattice systems,
which naturally contains all relevant collective modes, including the Higgs
amplitude in the strongly correlated superfluid. In contrast to Bogoliubov
theory, this non-perturbative method does not rely on a small condensate
depletion and is valid for any interaction strength in three spatial
dimensions. It is based on an expansion around the bosonic Gutzwiller ground
state in terms of appropriately chosen fluctuation operators and lays the
foundation for the description of real-time dynamics in terms of the natural,
weakly interacting quasi-particles. Furthermore, it provides a systematic
framework for efficiently calculating observables beyond the Gutzwiller
approximation and for including external perturbations, as well as higher order
decay and interactions in terms of quasi-particle operators. It allows for the
construction of an alternative path integral approach in terms of
quasi-particle coherent states.
