  We study vacuum quantum fluctuations of simple Nambu-Goldstone bosons -
derivatively coupled single scalar-field theories possessing shift-symmetry in
field space. We argue that quantum fluctuations of the interacting field can be
drastically suppressed with respect to the free-field case. Moreover, the
power-spectrum of these fluctuations can soften to become red for sufficiently
small scales. In quasiclassical approximation, we demonstrate that this
suppression can only occur for those theories that admit such classical static
backgrounds around which small perturbations propagate faster than light. Thus,
a quasiclassical softening of quantum fluctuations is only possible for
theories which classicalize instead of having a usual Lorentz invariant and
local Wilsonian UV- completion. We illustrate our analysis by estimating the
quantum fluctuations for the DBI-like theories.
