  We first consider an exactly solvable classical field model to understand the
coherence properties and the density fluctuations of a one-dimensional (1D)
weakly interacting degenerate Bose gas with repulsive interactions at
temperatures larger than the chemical potential. In a second part, using a
lattice model for the quantum field, we explain how to carefully generalize the
usual Bogoliubov approach to study a degenerate and weakly interacting Bose gas
in 1D, 2D or 3D in the regime of weak density fluctuations. In the last part,
using the mapping to an ideal Fermi gas in second quantized formalism, we
calculate and discuss physically the density fluctuations and the coherence
properties of a gas of impenetrable bosons in 1D.
