  We study the noncommutative corrections on the time-dependent Aharonov-Bohm
effect when both the coordinate-coordinate and momentum-momentum
noncommutativities are considered. This study is motivated by the recent
observation that there is no net phase shift in the time-dependent AB effect on
the ordinary space, and therefore tiny derivation from zero can indicate new
physics. The vanishing of the time-dependent AB phase shift on the ordinary
space is preserved by the gauge and Lorentz symmetries. However, on the
noncomutative phase space, while the ordinary gauge symmetry can be kept by the
Seiberg-Witten map, but the Lorentz symmetry is broken. Therefore nontrivial
noncommutative corrections are expected. We find there are three kinds of
noncommutative corrections in general: 1) $\xi$-dependent correction which
comes from the noncommutativity among momentum operators; 2) momentum-dependent
correction which is rooted in the nonlocal interactions in the noncommutative
extended model; 3) momentum-independent correction which emerges become of the
gauge invariant condition on the nonlocal interactions in the noncommutative
model. We proposed two dimensionless quantities, which are based on the
distributions of the measured phase shift with respect to the external magnetic
field and to the cross section enclosed by the particle trajectory, to extract
the noncommutative parameters. We find that stronger (weaker) magnetic field
strength can give better bounds on the coordinate-coordinate
(momentum-momentum) noncommutative parameter, and large parameter space region
can be explored by the time-dependent AB effect.
