We derive the quantum Boltzmann equation (QBE) by using generalized Landau-interaction parameters, obtained through the nonequilibrium Greens function technique. This is a generalization of the usual QBE formalism to non-Fermi liquid (NFL) systems, which do not have well-defined quasiparticles. We apply this framework to a controlled low-energy effective field theory for the Ising-nematic quantum critical point, in order to find the collective excitations of the critical Fermi surface in the collisionless regime. We also compute the nature of the dispersion after the addition of weak Coulomb interactions. The zero angular momentum longitudinal vibrations of the Fermi surface show a linear-in-wavenumber dispersion, which corresponds to the zero sound of Landaus Fermi liquid theory. The Coulomb interaction modifies it to a plasmon mode in the long-wavelength limit, which disperses as the square-root of the wavenumber. Remarkably, our results show that the zero sound and plasmon modes show the same behaviour as in a Fermi liquid, although an NFL is fundamentally different from the former.
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