  We present a fully comprehensive multi-mode quantum treatment based on the
truncated Wigner approximation (TWA) to study many-body effects and quantum
fluctuations on the formation of a discrete time crystal (DTC) in a
Bose-Einstein condensate (BEC) bouncing resonantly on an atom mirror, driven at
period T. Our theoretical approach avoids the restrictions both of mean-field
theory, where all bosons are assumed to remain in a single mode, and of
time-dependent Bogoliubov theory, which assumes boson depletion from the
condensate mode is small. For realistic initial conditions corresponding to a
harmonic trap condensate mode function, our TWA calculations performed for
period-doubling agree broadly with recent mean-field calculations for times out
to at least 2000 T, except at interaction strengths very close to the threshold
value for DTC formation where the position probability density differs
significantly from that determined from mean-field theory. For typical
attractive interaction strengths above the threshold value for DTC formation
and for the chosen trap and driving parameters, the TWA calculations indicate a
quantum depletion due to quantum many-body fluctuations of less than about two
atoms out of 600 atoms at times corresponding to 2000 T, in agreement with
time-dependent Bogoliubov theory calculations. On the other hand, for
interaction strengths very close to the threshold value for DTC formation, the
TWA calculations predict a large quantum depletion - as high as about 260 atoms
out of 600. We also show that the mean energy of the DTC does not increase
significantly for times out to at least 2000 mirror oscillations, so TWA theory
predicts that thermalisation is absent. Finally, we find that the dynamical
behaviour is similar for attractive or repulsive boson-boson interactions, and
that a stable DTC based on repulsive interactions can be created.
