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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.

Use a real, even interaction $V_{-\mathbf{k}}=V_{\mathbf{k}}$. Let $\mathcal{V}=L^3$ and $N_{\mathrm{ex}}={\sum }_{\mathbf{k}\ne 0}{b}_{\mathbf{k}}^{†}{b}_{\mathbf{k}}$. In the Bogoliubov theory expansion, keep terms with zero or two noncondensed operators. The zero-momentum term gives $V_0{N}_0(N_0-1)/(2\mathcal{V})$. The terms with two noncondensed operators giveHere $V_0$ is the direct condensate interaction and $V_{\mathbf{k}}$ is its exchange contribution. At fixed total particle number, $N_0=N-N_{\mathrm{ex}}$, soThe second term cancels the direct normal term. Adding the kinetic energy yields the number-conserving Bogoliubov quadratic HamiltonianThe constant is the condensate mean-field theory interaction energy; the normal term describes moving a particle out of the condensate, while the anomalous term creates or annihilates an opposite-momentum pair by exchanging two particles with the condensate. Total particle number remains fixed in the underlying theory; the condensate amplitudes in this notation are treated as classical amplitudes.

For condensate fluctuation power counting, put $\delta =N_{\mathrm{ex}}/N\ll 1$. Formal interaction-amplitude counting gives $H_2-E_{\mathrm{mf}}\sim E_{\mathrm{mf}}\delta$, cubic terms $\sim E_{\mathrm{mf}}{\delta }^{3/2}$ and quartic terms $\sim E_{\mathrm{mf}}{\delta }^2$; these are not uniform bounds on individual modes. Thus the latter are down by $\sqrt{\delta }$ and $\delta$ relative to the quadratic interaction terms. The $N_0-1$ correction is relative $1/N$. Homogeneity eliminates terms linear in nonzero-momentum operators. Small depletion and weak interactions, not low temperature alone, justify the truncation. ^[ref]