Infrared qualification of condensate fluctuation counting

A small noncondensed fraction provides a useful global expansion parameter for a weakly interacting Bose gas, but does not bound every long-wavelength mode individually. In Bogoliubov theory, mode amplitudes can grow as momentum approaches zero, while their integrated contribution remains finite in some dimensions and regimes.

A fluctuation integral includes both the small-momentum behavior and the momentum-space measure. For example, an integrand proportional to $1/k$ contributes ${\int }_{0}dk{k}^{d-2}$ in $d$ spatial dimensions, which converges at zero only for $d>1$. Thermal factors can make the infrared behavior more singular. Thus condensate fluctuation power counting must be supplemented by checking the relevant integrals, system size and temperature. The uniform phase mode and fixed particle number constraint also require their own treatment.