Thesis Defense: The Maxwell-Pauli Equations

PDE Seminar
Tuesday, March 10, 2020 - 3:00pm for 1 hour (actually 50 minutes)
Skiles 006
Thomas Kieffer – Georgia Tech –
Michael Loss

Energetic stability of matter in quantum mechanics, which refers to the question of whether the ground state energy of a

many-body quantum mechanical system is finite, has long been a deep question of mathematical physics. For a system of many
non-relativistic electrons interacting with many nuclei in the absence of electromagnetic fields this question traces back
to the seminal works of Tosio Kato in 1951 and Freeman Dyson and Andrew Lenard in 1967/1968. In particular, Dyson and Lenard
showed the ground state energy of the many-body Schrödinger Hamiltonian is bounded below by a constant times the total particle
number, regardless of the size of the nuclear charges. This situation changes dramatically when electromagnetic fields and spin
interactions are present in the problem. Even for a single electron with spin interacting with a single nucleus of charge
$Z > 0$ in an external magnetic field, Jurg Fröhlich, Elliot Lieb, and Michael Loss in 1986 showed that there is no ground state
energy if $Z > Z_c$ and the ground state energy exists if $Z < Z_c$.
Another notion of stability in quantum mechanics is that of dynamic stability. Dynamic stability refers to the question of global
well-posedness for a system of partial differential equations that models the dynamics of many electrons coupled to their
self-generated electromagnetic field and interacting with many nuclei. The central motivating question of our PhD thesis is
whether energetic stability has any influence on the global well-posedness of the corresponding dynamical equations. In this regard,
we study the quantum mechanical many-body problem of $N$ non-relativistic electrons with spin interacting with their self-generated classical electromagnetic field and $K$ 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 presented is the construction
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 sufficiently small to ensure energetic stability of this system. If time permits, we
will discuss several open problems that remain.