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As a person majoring in condensed matter physics, I frequently encounter Landau Fermi-liquid theory. Almost every literature says that the concept of the adiabatic continuity (to the non-interacting ground state) is the heart of the theory so that the notion of a $\it{quasiparticle}$ is well-defined. For the practical aspect, the energy of the electronic system (relative to that of the ground state) is given in Landau Fermi-liquid theory as $$E[\delta n_\vec{k}]=\sum_{\vec{k}}\varepsilon_{\vec{k}}\delta n_\vec{k} + \frac{1}{2}\sum_{\vec{k}\sigma,\vec{p}\sigma'}f(\vec{k}\sigma,\vec{p}\sigma')\delta n_{\vec{k},\sigma }\delta n_{\vec{k},\sigma'},$$ where $\delta n_\vec{k}$ is the deviation of the distribution of the $\it{quasiparticle}$ from the value at the ground state.

What makes me astnonished is that the energy functional is just expressed with classical functions $\delta n_\vec{k}$ rather than quantum-mechanical operators. Could I or we say that the Landau Fermi liquid theory a classical theory?

Also, why (or how) could (or should) an interacting ground state of a Fermi liquid be completely or well described by $\delta n_\vec{k}$? Is this feature associated with the adiabatic continuity?

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    $\begingroup$ The distribution of quasiparticles is still similar to Fermi distribution. Also, fermi liquid quasiparticles can be derived consistently as poles of interacting green's function, see AGD for example. $\endgroup$
    – Roger V.
    Commented Mar 12, 2023 at 8:19
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    $\begingroup$ Landau's formulation is not classical, bit it is phenomenological - it includes only observable quantities. The essence however is that the interacting system can be described using the methods for non-interacting Fermi gas. $\endgroup$
    – Roger V.
    Commented Mar 12, 2023 at 8:23
  • $\begingroup$ Thanks for reply, Roger Vadim. After reading your reply, I feel it necessary to clarify my question. I completely agree with you that the qausiparticles in Landau's phenomenological description of Fermi liquid obeys the Fermi-Dirac statistics, which could be considered as a signature of a quantum phenomena, Pauli exclusion. Other than the Fermi-Dirac statistics, is there any other quantum mechanical phenomena that I have to take into account of when I study Landau Fermi liquid theory? $\endgroup$
    – Changhee Lee
    Commented Mar 14, 2023 at 14:28
  • $\begingroup$ Also, a simple but naive distinction between quantum mechanics and classical mechanics is whether Planck constant $\hbar$ appears. In this regards, does an equation of motion for the quasiparticle derived from $E[\delta n_{k}]$ involve $\hbar$? $\endgroup$
    – Changhee Lee
    Commented Mar 14, 2023 at 14:28
  • $\begingroup$ It really depends on what you are going to use it for. E.g., if you want to write a kinetic equation to study transport properties, you may treat it as a classical theory. If you want to study things like a cross-over to non-Fermi-liquid behavior in one dimension (Luttinger liquid) or in presence of magnetic impurities, then you have to view it as a quantum theory. One of classical texts is Theory Of Quantum Liquids by Nozieres $\endgroup$
    – Roger V.
    Commented Mar 14, 2023 at 14:34

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