  Small Fermi energy effects are induced in MgB$_2$ by the low hole doping in
the $\sigma$ bands which are characterized by a Fermi energy $E_{\rm F}^\sigma
\sim 0.5$ eV. We show that, due to the particularly strong deformation
potential relative to the $E_{2g}$ phonon mode, lattice fluctuations are
reflected in strong fluctuations in the electronic band structure. Quantum
fluctuations associated to the zero-point lattice motion are responsible for an
uncertainty of the Fermi energy of the order of the Fermi energy itself,
leading to the breakdown of the adiabatic principle underlying the
Born-Oppenheimer approximation in MgB$_2$ even if $\omega_{\rm ph}/E_{\rm F}
\sim 0.1-0.2$, where $\omega_{\rm ph}$ are the characteristic phonon
frequencies. This amounts to a new nonadiabatic regime, which could be relevant
to other unconventional superconductors.
