  We present an extension of the well-known Bogoliubov theory to treat low
dimensional degenerate Bose gases in the limit of weak interactions and low
density fluctuations. We use a density-phase representation and show that a
precise definition of the phase operator requires a space discretisation in
cells of size $l$. We perform a systematic expansion of the Hamiltonian in
terms of two small parameters, the relative density fluctuations inside a cell
and the phase change over a cell. The resulting macroscopic observables can be
computed in one, two and three dimensions with no ultraviolet or infrared
divergence. Furthermore this approach exactly matches Bogoliubov's approach
when there is a true condensate. We give the resulting expressions for the
equation of state of the gas, the ground state energy, the first order and
second order correlations functions of the field. Explicit calculations are
done for homogeneous systems.
