  This paper is concerned with the well-posedness analysis of the Hartree-Fock
system modeling the time evolution of a quantum system comprised of fermions.
We consider quantum states with finite mass and finite kinetic energy, and the
self-consistent potential is the unbounded Coulomb interaction. This model is
first formulated as a semi-linear evolution problem for the one-particle
density matrix operator lying in the space of Hermitian trace class operators.
Using semigroup techniques and generalized Lieb-Thierring inequalities we then
prove global existence and uniqueness of mild and classical solutions. To this
end we prove that the quadratic Hartree-Fock terms are locally Lipschitz in the
space of trace class operators with finite kinetic energy. Technically, the
main challenge stems from considering the model as an evolution problem for
operators. Hence, many standard tools of PDE-analysis (density results, e.g.)
are not readily available for the density matrix formalism.
