Is is possible to extract an effective Hamiltonian from a Boltzmann equation (or any other kinetic theories)?

I got kind of confused when reading Xiaogang Wen's famous textbook Quantum Field Theory of Many-body Systems. In Section 5.3.3 the book claims that

• From a kinetic theory of Fermi liquid (a Boltzmann equation with a self-energy correction which in turn relies on the distribution function, which people sometimes call the "Landau's equation") we can derive a bosonized Hamiltonian of the Fermi liquid, by considering the kinetic equation $$\frac{\partial \tilde{\rho}}{\partial t}+v_{F}^{*} \hat{\boldsymbol{k}} \cdot \frac{\partial \tilde{\rho}}{\partial \boldsymbol{x}}+\frac{k_{F}}{(2 \pi)^{2}} \int \mathrm{d} \theta^{\prime} f\left(\theta, \theta^{\prime}\right) \hat{\boldsymbol{k}} \cdot \frac{\partial \tilde{\rho}\left(\theta^{\prime}, \boldsymbol{x}, t\right)}{\partial \boldsymbol{x}}=0 \tag{5.3.11}$$ as the equation of motion of the Hamiltonian \begin{aligned} H &=E_{g}+\int \mathrm{d}^{2} \boldsymbol{x} \mathrm{d} \theta \tilde{\rho} \frac{h v_{F}^{*}}{2}+\frac{1}{2} \int \mathrm{d}^{2} \boldsymbol{x} \mathrm{d} \boldsymbol{k} \mathrm{d} \boldsymbol{k}^{\prime} f\left(\boldsymbol{k}, \boldsymbol{k}^{\prime}\right) \delta n_{\boldsymbol{k}} \delta n_{\boldsymbol{k}^{\prime}} \\ &=E_{g}+\int \mathrm{d}^{2} \boldsymbol{x} \mathrm{d} \theta \frac{2 \pi^{2} v_{F}^{*}}{k_{F}} \tilde{\rho}^{2}+\frac{1}{2} \int \mathrm{d}^{2} \boldsymbol{x} \mathrm{d} \theta \mathrm{d} \theta^{\prime} f\left(\theta, \theta^{\prime}\right) \tilde{\rho}(\theta) \tilde{\rho}\left(\theta^{\prime}\right), \end{aligned} \tag{5.3.12} and
• The field in the bosonized Hamiltonian is labeled by $$\theta$$, which is the direction of the momentum in the distribution function appearing in the Boltzmann equation.

I understand that Boltzmann equation may be viewed as an equation about the Wigner function of the two-point correlation function, so it can be used to describe quantum systems. However it should then be noted that after all we are dealing with $$\langle \phi(x) \phi(y) \rangle$$ instead of $$\phi(x) \phi(y)$$. Treating the Boltzmann equation for Fermi liquid as a quantum EOM is therefore not well justified, as the real EOM may contain certain operators that vanishe when being averaged.

So my question is what he is doing in that section, and when the kinetic equations is nonlinear for collision, whether it is equivalent to adding a higher order term ("interaction channels between density modes") to $$(5.3.12)$$.

• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.
– Community Bot
Nov 18 '21 at 8:07