  We study the quantum mechanical many-body problem of $N \geq 1$
non-relativistic electrons with spin interacting with their self-generated
classical electromagnetic field and $K \geq 0$ static nuclei. We model the
dynamics of the electrons and their self-generated electromagnetic field using
the so-called many-body Maxwell-Pauli equations. The main result of this thesis
is to construct time global, finite-energy, weak solutions to the many-body
Maxwell-Pauli equations under the assumption that the fine structure constant
$\alpha$ and the nuclear charges are not too large. The assumptions on the size
of $\alpha$ and the nuclear charges ensure that we have energetic stability for
this system, i.e., the absolute ground state energy exists. The work in this
thesis serves as an initial step towards understanding the connection between
the energetic stability of matter in quantum mechanics and the well-posedness
of the corresponding dynamical equations.
