DOCUMENTA MATHEMATICA, Extra Volume ICM III (1998), 153-162

Gian Michele Graf

Title: 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 {\it 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 {\it and\/} the magnetic moments associated to their spin. Stability of non-relativistic matter in presence of magnetic fields requires that $Z\alpha^2$ (where $Z$ is the largest nuclear charge in the system) as well as the fine structure constant $\alpha$ 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 $\alpha,\,Z\alpha^2$. 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.

1991 Mathematics Subject Classification: 81-02

Keywords and Phrases: Stability of matter

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