  This paper gives a review of our recent work on the quantum Frenkel-Kontorova
model. Using the squeezed state theory and the quantum Monte Carlo method, we
have studied the effects of quantum fluctuations on the Aubry transition and
the behavior of the ground state wave function. We found that quantum
fluctuations renormalize the sinusoidal standard map to a sawtooth map.
Although quantum fluctuations have smeared the Aubry transition, the remnants
of this transition are still discernible. The ground state wave function also
changes from an extended state to a localized state. The squeezed state results
agree very well with those from the Monte Carlo and mean field studies.
