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

(where Z is the largest

nuclear charge in the system) as well as the fine structure constant

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

,

.

An important tool is given by Lieb-Thirring type

inequalities for the sum of the eigenvalues of a one-particle Pauli opera

tor

with an arbitrary inhomogeneous magnetic field.