  We investigate a bosonic Josephson junction by using the path-integral
formalism with relative phase and population imbalance as dynamical variables.
We derive an effective only-phase action performing functional integration over
the population imbalance. We then analyze the quantum effective only-phase
action, which formally contains all the quantum corrections. To the second
order in the derivative expansion and to the lowest order in $\hbar$, we obtain
the quantum correction to the Josephson frequency of oscillation. Finally, the
same quantum correction is found by adopting an alternative approach. Our
predictions are a useful theoretical tool for experiments with atomic or
superconducting Josephson junctions.
