Linear mode analysis from spin transport equation

We provide a linear analysis on normal modes of the spin Boltzmann equation proposed in \cite{Weickgenannt:2021cuo}, where the non-diagonal or polarized part of the transition rate is neglected to ensure the Hermitian property of linearized collision operator. As an instrumental element of spin kinetic theory, the conservation of total angular momentum is explicitly considered, thus our analysis is relevant to the recent investigation on the issue of local spin polarization. By treating the linearized collision operator as an evolution operator, solving the normal modes turns out a degenerate perturbation problem in quantum mechanics. The dispersion relations of spinless modes are in accordance with well-known calculations, while the frequencies of spin modes are also determined up to second-order in wave vector and the second order expressions are only formal solutions to be further determined. Moreover, the relaxation of spin density is related to our linear mode analysis, which shall play a big role in investigating the issues of the local spin polarization in the relativistic heavy-ion collisions.