Quasiclassical theory of spin dynamics in superfluid ${}^3$He: kinetic equations in the bulk and spin response of surface Majorana states

We develop a theory based on the formalism of quasiclassical Green’s functions to study the spin dynamics in superfluid ${}^3$He. First, we derive kinetic equations for the spin-dependent distribution function in the bulk superfluid reproducing the results obtained earlier without quasiclassical approximation. Then we consider a spin dynamics near the surface of fully gapped ${}^3$He-B phase taking into account spin relaxation due to the transitions in the spectrum of localized fermionic states. The lifetime of longitudinal and transverse spin waves is calculate taking into account the Fermi-liquid corrections which lead to the crucial modification of fermionic spectrum and spin responses.