  We investigate dynamical transport properties of interacting electrons moving
in a vibrating nanoelectromechanical wire in a magnetic field. We have built an
exactly solvable model in which electric current and mechanical oscillation are
treated fully quantum mechanically on an equal footing. Quantum mechanically
fluctuating Aharonov-Bohm phases obtained by the electrons cause nontrivial
contribution to mechanical vibration and electrical conduction of the wire. We
demonstrate our theory by calculating the admittance of the wire which are
influenced by the multiple interplay between the mechanical and the electrical
energy scales, magnetic field strength, and the electron-electron interaction.
