Velocity dependent interactions and a new sum rule in bcc He

Recent neutron scattering experiments [PRL,{\bf 88},p.195301 (2002)] on solid $^4$He, discovered a new optic-like mode in the bcc phase. This excitation was predicted by a recently proposed model that describes the correlated atomic zero-point motion in bcc Helium in terms of dynamic electric dipole moments. Modulations of the relative phase of these dipoles between different atoms describes the anomalously soft T$_1$(110) phonon and two new optic-like modes, one of which was recently found in the neutron scattering experiments. In this work we show that the correlated dipolar interactions can be written as a velocity dependent interaction. This then results in a modified f-sum rule for the T$_1$(110) phonon, in good agreement with the recent experimental data.