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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]

A bosonic coherent state obeys $a|\alpha ⟩=\alpha |\alpha ⟩$ and has expansion $e^{-|\alpha {|}^2/2}{\sum }_n{\alpha }^n|n⟩/\sqrt{n!}$. Its occupation is Poisson distributed, with mean and variance $|\alpha {|}^2$. Its nonzero field amplitude makes phase coherence natural, although physically observable phase comparisons require a reference.

For paired fermions, the corresponding Cooper-pair pseudospin coherent state isIt coherently superposes empty and occupied pair states and has an anomalous average $⟨c_{-\mathbf{k}\downarrow }{c}_{\mathbf{k}\uparrow }⟩\ne 0$. It is not an ordinary complex-eigenvalue coherent state of a single fermion annihilation operator. The broken-symmetry state has indefinite particle number; number projection can restore a fixed-number description without discarding the underlying pairing correlations.

Hartree-Fock-Bogoliubov theory approximates a two-body interaction using both the normal density ${\rho }_{ij}=⟨c_j^{†}{c}_i⟩$ and anomalous pairing tensor ${\kappa }_{ij}=⟨c_j{c}_i⟩$. Direct and exchange contractions determine a normal one-particle field, while anomalous contractions determine the pair potential. The resulting quadratic Hamiltonian is diagonalized by a fermionic Bogoliubov transformation. In a homogeneous singlet case its Nambu block isThe fields are determined self-consistently, rather than selected independently of the quasiparticle state. For example,together with the particle-number equation. The pairing field extends ordinary Hartree-Fock theory to states with pair correlations. This mean-field treatment omits fluctuations beyond its self-consistent quadratic description.

For bosons, write the field as a coherent condensate plus fluctuations, $\hat{\psi }=\varphi +\delta \hat{\psi }$. The same normal/anomalous strategy retains $\tilde{n}=⟨\delta {\hat{\psi }}^{†}\delta \hat{\psi }⟩$ and $\tilde{m}=⟨\delta \hat{\psi }\delta \hat{\psi }⟩$, with a bosonic, commutation-preserving Bogoliubov transformation. In the leading weakly depleted contact-gas limit, the condensate density $n_0$ and effective coupling $g>0$ give $E_k=\sqrt{{ϵ}_k({ϵ}_k+2g{n}_0)}$, with ${ϵ}_k={\hslash }^2{k}^2/(2m)$. This Bogoliubov theory connects low-momentum sound to high-momentum particle excitations. More elaborate self-consistent truncations must respect the symmetry constraints needed for a gapless condensate mode.

A complex order parameter captures both the amplitude and phase rigidity of a superfluid. A local isotropic Ginzburg-Landau theory writes its free-energy functional, to fourth order in amplitude and second order in gradients, asThe uniform minimum has $\Psi =0$ for $\alpha >0$ and $|{\Psi }_0{|}^2=-\alpha /\beta$ for $\alpha <0$. Its phase remains arbitrary under a global $U(1)$ transformation. Minimizing with respect to ${\Psi }^{\ast }$ gives $-K{\nabla }^2\Psi +\alpha \Psi +\beta |\Psi {|}^2\Psi =0$. The phenomenological description specifies the allowed structures and coefficients; microscopic input is needed to identify the normalization of $\Psi$ and calculate those coefficients. A static functional by itself does not fix the dynamical equation or mode frequencies.

For a dilute three-dimensional Bose gas with short-range repulsion and particle mass $m_B$, the microscopic low-energy Hamiltonian isThe interaction parameter is expressed through the low-energy s-wave scattering length $a_s$, rather than an unrenormalized bare delta potential. In a symmetry-selected description, $\Psi =⟨\hat{\psi }⟩$ and $|\Psi {|}^2=n_0$ is condensate density. Equivalently, fixed-number off-diagonal long-range order identifies the macroscopically occupied natural orbital and its amplitude without a nonzero one-point expectation.

Evaluating $\hat{H}-\mu \hat{N}$ in a coherent-field condensate variational state gives the Ginzburg-Landau functional of a dilute Bose gas, with the mean-field microscopic identificationsIts stationary variational equation is the Gross–Pitaevskii equation. The time-dependent variational action additionally contains $i\hslash {\Psi }^{\ast }{\partial }_t\Psi$, givingThis term expresses density-phase conjugacy. It is the missing microscopic dynamical input; a purely dissipative relaxation law would describe a different approximation.

Bogoliubov theory writes $\hat{\psi }=\Psi +\delta \hat{\psi }$ and retains quadratic fluctuations. In a uniform condensate, $\mu =g_B{n}_0$ and the fluctuation matrix isIts eigenvalues are $\pm \sqrt{{ϵ}_k({ϵ}_k+2g_B{n}_0)}$. The positive branch is the gapless sound mode, continuously crossing over to particle-like excitations at larger $k$. The same result follows by linearizing the time-dependent Gross–Pitaevskii equation about its uniform solution. Thus the microscopic condensate amplitude, scattering interaction and mass determine the mean-field static coefficients and the collective mode. Quadratic quantum fluctuations also produce quantum depletion of a Bose condensate, which a pure coherent-state product omits.

These identifications are controlled for a dilute weakly interacting gas away from a fluctuation-dominated critical region. They are not a universal statement that the exact finite-temperature GL coefficient is always $-\mu$ or that its quartic coefficient never receives corrections. Integrating out noncondensed and short-wavelength modes produces effective temperature-dependent coefficients and can change their near-critical behavior. The phase rigidity described by $K$ and the thermodynamic order-parameter expansion remain the shared organizing structure.

For BCS theory, the microscopic order parameter is instead an anomalous pair amplitude, commonly normalized as an energy gap $\Delta \propto ⟨{\hat{\psi }}_{\downarrow }{\hat{\psi }}_{\uparrow }⟩$. It is not an ordinary single-electron field expectation or automatically the square root of a pair number density. Integrating out fermions and expanding in small $\Delta$ and slow gradients near $T_c$ gives the microscopic derivation of Ginzburg-Landau theory. The BCS pairing-field effective action supplies its coefficients through normal-state pair-response functions and thermal fermion sums.

For example, a clean isotropic three-dimensional weak-coupling s-wave model, with $N(0)$ the density of states per spin at the Fermi energy and $\Delta$ measured in energy units, giveswith microscopic Ginzburg-Landau coefficients from BCS theoryThe numerical factors arise from fermionic Matsubara sums and the isotropic velocity average; they depend on the stated gap and density-of-states conventions. Rescaling the order parameter rescales all three coefficients. The expansion applies near the transition with sufficiently slow spatial variation; disorder, anisotropic pairing and strong coupling alter the coefficients or require additional components. Microscopic derivation also fixes the pair charge in the covariant gradient, producing the superconducting screening current and the Meissner effect.^[ref]

The neutral Bose and charged Fermi theories therefore share a complex order parameter and a phase-stiffness functional, while deriving that field from different microscopic objects. The Bogoliubov transformation mixes particle and hole fluctuations of the bosonic condensate in one case and paired fermionic excitations in the other. The static GL limit organizes long-distance behavior; the full microscopic theories retain excitation spectra and temperature dependence beyond that local near-transition expansion.