  In this paper we study the Cauchy problem associated to the
Maxwell-Schr\"odinger system with a defocusing pure-power non-linearity. This
system has many applications in physics, for instance in the description of a
charged non-relativistic quantum plasma, interacting with its self-generated
electromagnetic potential. One consequence of our analysis is to demonstrate
that the Lorentz force associated with the electromagnetic field is
well-defined for solutions slightly more regular than the finite energy class.
This aspect is of fundamental importance since all the related physical models
require the observability of electromagnetic effects. The well-posedness of the
Lorentz force still seems to be a major open problem in the class of solutions
which are only finite energy. We show the global well-posedness at high
regularity for the cubic and sub-cubic case, and we provide polynomial bounds
for the growth of the Sobolev norm of the solutions, for a certain range of
non-linearities. An important role is played by appropriate a priori dispersive
estimates, obtained by means of Koch-Tzvetkov type bounds for the
non-homogeneous Schr\"odinger equation, which overcome the lack of Strichartz
estimates for the magnetic Schr\"odinger flow. Because of the power-type
non-linearity, the propagation of higher regularity, globally in time, cannot
be achieved via a bootstrap argument as done in [44]. Our approach then
exploits the analysis of a modified energy functional, combined with the a
priori bounds coming from the dispersive estimates obtained previously.
