  The translationally invariant diagrammatic quantum perturbation theory (TPT)
is applied to the polaron problem on the 1D lattice, modeled through the
Holstein Hamiltonian with the phonon frequency omega0, the electron hopping t
and the electron-phonon coupling constant g. The self-energy diagrams of the
fourth-order in g are calculated exactly for an intermittently added electron,
in addition to the previously known second-order term. The corresponding
quadratic and quartic corrections to the polaron ground state energy become
comparable at t/omega0>1 for g/omega0~(t/omega0)^{1/4} when the electron
self-trapping and translation become adiabatic. The corresponding non
adiabatic/adiabatic crossover occurs while the polaron width is large, i.e. the
lattice coarsening negligible. This result is extended to the range
(t/omega0)^{1/2}>g/omega0>(t/omega0)^{1/4}>1 by considering the scaling
properties of the high-order self-energy diagrams. It is shown that the polaron
ground state energy, its width and the effective mass agree with the results
found traditionally from the broken symmetry side, kinematic corrections
included. The Landau self trapping of the electron in the classic
self-consistent, localized displacement potential, the restoration of the
translational symmetry by the classic translational Goldstone mode and the
quantization of the polaronic translational coordinate are thus all encompassed
by a quantum theory which is translationally invariant from the outset.
