Global wellposedness in the energy space for the Maxwell-Schr\"odinger system

We prove that the Maxwell-Schr"odinger system in ${\mathbb{R}}^{3+1}$ is globally well-posed in the energy space. The key element of the proof is to obtain a short time wave packet parametrix for the magnetic Schr"odinger equation, which leads to linear, bilinear and trilinear estimates. These, in turn, are extended to larger time scales via a bootstrap argument.