  In the present work we show, analytically and numerically, that the variance
of many-particle operators and their uncertainty product for an
out-of-equilibrium Bose-Einstein condensate (BEC) can deviate from the outcome
of the time-dependent Gross-Pitaevskii dynamics, even in the limit of infinite
number of particles and at constant interaction parameter when the system
becomes 100% condensed. We demonstrate our finding on the dynamics of the
center-of-mass position--momentum uncertainty product of a freely expanding as
well as of a trapped BEC. This time-dependent many-body phenomenon is explained
by the existence of time-dependent correlations which manifest themselves in
the system's reduced two-body density matrix used to evaluate the uncertainty
product. Our work demonstrates that one has to use a many-body propagation
theory to describe an out-of-equilibrium BEC, even in the infinite particle
limit.
