Classical scattering matrix for hard and soft excitations in a plasma with non-abelian interaction

Within the framework of the Zakharov–Shulman approach, we determine the classical scattering matrix for the simplest process of interaction between hard and soft excitations in a quark-gluon plasma. Calculations are performed in close analogy with the methods of quantum field theory, with the replacement of the quantum commutator of quantum field operators by the so-called Lie–Poisson bracket of classical variables. The classical $\mathcal{S}$-matrix is determined in the form of the most general integro-power series in asymptotic values of the normal bosonic variables $c^{a}_{\boldsymbol{k}}(t)$ and $c^{\ast a}_{\boldsymbol{k}}(t)$ describing the soft gluon excitations of the system and the color charge $\mathcal{Q}^{a}(t)$ of the hard particle at $t\rightarrow\infty$. The first nontrivial contribution to the given $\mathcal{S}$-matrix is obtained.