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A regular approach to accounting for initial correlations, which allows to gobeyond the unrealistic random phase (initial product state) approximation inderiving the evolution equations, is suggested. An exact homogeneous equationfor a two-time equilibrium correlation function for the dynamical variables ofa subsystem interacting with a boson field (heat bath) is obtained. Noconventional approximation like RPA or Bogoliubov's principle of weakening ofinitial correlations is used. The obtained equation takes into account theinitial correlations in the kernel governing its evolution. The solution tothis equation is found in the second order of the kernel expansion in theelectron-phonon interaction, which demonstrates that generally the initialcorrelations influence the correlation function's evolution in time. It isshown that this influence vanishes on a large timescale. The developed approachis applied to the Fr\"ohlich polaron and the low-temperature polaron mobility(which was under a long-time debate) is found with a correction due to initialcorrelations.

Influence of initial correlations on evolution of correlation function  of a subsystem interacting with a quantum field (heat bath) and polaron  mobility