E.H. Lieb, H. Siedentop, J.P. Solovej
Stability and Instability of Relativistic Electrons in
Classical Electromagnetic Fields
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ABSTRACT.  The stability of matter composed of electrons and static nuclei is
investigated for a relativistic dynamics for the electrons given by
a suitably projected Dirac operator and with Coulomb interactions.
In addition there is an arbitrary classical magnetic field of finite
energy. Despite the previously known facts that ordinary
nonrelativistic matter with magnetic fields, or relativistic matter
without magnetic fields is already unstable when $\alpha$, the fine
structure constant, is too large it is noteworthy that the
combination of the two is still stable {\it provided} the projection
onto the positive energy states of the Dirac operator, which {\it
defines} the electron, is chosen properly.  A good choice is to
include the magnetic field in the definition. A bad choice, which
always leads to instability, is the usual one in which the positive
energy states are defined by the free Dirac operator. Both
assertions are proved here. (To appear in Jour. Stat. Phys.)