  We investigate the role of quantum fluctuations in the dynamics of a bosonic
Josephson junction in $D$ spatial dimensions, by using beyond mean-field
Gaussian corrections. We derive some key dynamical properties in a systematic
way for $D=3, 2, 1$. In particular, we compute the Josephson frequency in the
regime of low population imbalance. We also obtain the critical strength of the
macroscopic quantum self-trapping. Our results show that quantum corrections
increase the Josephson frequency in spatial dimensions $D=2$ and $D=3$, but
they decrease it in the $D=1$ case. The critical strength of macroscopic
quantum self-trapping is instead reduced by quantum fluctuations in $D=2$ and
$D=3$ cases, while it is enhanced in the $D=1$ configuration. We show that the
difference between the cases of D = 2 and D = 3 on one side, and D = 1 on the
other, can be related to the qualitatively different dependence of the
interaction strength on the scattering length in the different dimensions.
