  Within the Schwinger-Keldysh formalism we derive a Ginzburg-Landau theory for
the Bose-Hubbard model which describes the real-time dynamics of the complex
order parameter field. Analyzing the excitations in the vicinity of the quantum
phase transitions it turns out that particle/hole dispersions in the Mott phase
map continuously onto corresponding amplitude/phase excitations in the
superfluid phase. Furthermore, in the superfluid phase we find a sound mode,
which is in accordance with recent Bragg spectroscopy measurements in the
Bogoliubov regime, as well as an additional gapped mode, which seems to have
been detected via lattice modulation.
