Translation invariance and the problem of the bipolaron

Differences between translation-invariant and broken-symmetry bipolaron theories are analyzed in detail. It is shown that the Bogolyubov—Tyablikov canonical transformation allows collective coordinates to be introduced in a regular way for two particles in a quantum field and that for the case of the bipolaron the resulting electron—electron interaction in a phonon field depends on the electron coordinate difference alone. Predictions using a revised solution of the nonlinear differential equations for a bipolaron are given. It is shown that solving bipolaron equations numerically reduces the total bipolaron energies compared to known variational results.