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

Question

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 $(5.3.12)$.

Answer 1

I think what we are dealing with here is more of an educated guess than a rigorous derivation - both in case of extracting the Hamiltonian from the Boltzmann equation and in case of quantizing hydrodynamic equations (or, more generally, any equations for a continuous media - hydrodynamic, elasticity, plasma, magnetization, etc.)

There are however a few more points to make:

• The OP mentions that Boltzmann equation can be derived as an equatioin for a Wigner function - the approach described in the classical Kadanoff&Baym's Quantum statistical mechanics, although there are more recent treatments. I suggest taking a look at Rammer&Smith's Quantim field-theoretical methods in transport theory of metals for rigorous derivation of the quantum corrections to the Boltzmann equation and, in particular, their diagrammatic representation.
• I don't know well the context in the Wen's book, but when the treatment is geared twoards treatment of critical phenomena, the underlying quantum nature is often irrelevant (There is a discussion in Goldenfeld's Lectures on phase transitions and the renormalization group. In fact, critical phenomena are often correctly captured by ad-hoc models such as Ising model, Heisenberg model, Hubbard model - which have little to do with the details of the underlying interactions and geometries (which typically vanish during renormalization).