  Axions and axion-like particles are bosonic quantum fields. They are often
assumed to follow classical field equations due to their high degeneracy in the
phase space. In this work, we explore the disparity between classical and
quantum field treatments in the context of density and velocity fields of
axions. Once the initial density and velocity field are specified, the
evolution of the axion fluid is unique in the classical field treatment.
However, in the quantum field treatment, there are many quantum states
consistent with the given initial density and velocity field. We show that
evolutions of the density perturbations for these quantum states are not
necessarily identical and, in general, differ from the unique classical
evolution. To illustrate the underlying physics, we consider a system of large
number of bosons in a one-dimensional box, moving under the gravitational
potential of a heavy static point-mass. We ignore the self-interactions between
the bosons here. Starting with homogeneous number density and zero velocity
field, we determine the density perturbations in the linear regime in both
quantum and classical field theories. We find that classical and quantum
evolutions are identical in the linear regime if only one single-particle state
is occupied by all the bosons and the self-interaction is absent. If more than
one single-particle states are occupied, the density perturbations in quantum
evolutions differ from the classical prediction after a certain time which
depends upon the parameters of the system.
