Stability of matter in classical and quantized fields

In recent years considerable activity was directed to the issue of stability in the case of matter interacting with an electromagnetic field. We shall review the results which have been established by various groups, in different settings: relativistic or non-relativistic matter, classical or quantized electromagnetic fields. Common to all of them is the fact that electrons interact with the field both through their charges and the magnetic moments associated to their spin. Stability of non-relativistic matter in presence of magnetic fields requires that Zα² (where Z is the largest nuclear charge in the system) as well as the fine structure constant α itself, do not exceed some critical value. If one imposes an ultraviolet cutoff to the field, as it occurs in unrenormalized quantum electrodynamics, then stability no longer implies a bound on α, Zα². An important tool is given by Lieb-Thirring type inequalities for the sum of the eigenvalues of a one-particle Pauli operator with an arbitrary inhomogeneous magnetic field.