Infrared phase fluctuations of a homogeneous Bose gas ($⟨|{\varphi }_{\mathbf{q}}{|}^2⟩\propto T/q^2(T>0),\propto 1/|q|(T=0)$)

The Bose-gas phase-only action gives a thermal infrared integral $\int d^{d}q/q^2$ and a zero-temperature equal-time integral $\int d^{d}q/|q|$. Thus an infinite uniform short-range interacting gas has no true finite-temperature condensate in $d\le 2$, and the one-dimensional ground state also lacks true condensate long-range order. This does not exclude local density coherence, a finite-size condensate description or two-dimensional algebraic order and superfluidity.^[ref]