Elliott H. Lieb and Michael Loss
Stability of a Model of Relativistic Quantum Electrodynamics
(85K, latex 2e)
ABSTRACT. The relativistic ``no pair'' model of quantum
electrodynamics uses the Dirac operator, D(A) for the electron
dynamics together with the usual self-energy of the quantized
ultraviolet cutoff electromagnetic field A --- in the Coulomb gauge.
There are no positrons because the electron wave functions are
constrained to lie in the positive spectral subspace of some Dirac
operator, D, but the model is defined for any number of electrons and
hence describes a true many-body system. If the fields are not
quantized but are classical, it was shown earlier that such a model is
unstable if one uses the customary D(0) to define the electron space,
but is stable (if the fine structure constant alpha is not too large)
if one uses D(A) itself. This result is extened to quantized fields
here, and stability is proved for alpha =1/137 and Z <_ 42. This
formulation of QED is somewhat unusual because it means that the
electron Hilbert space is inextricably linked to the photon Fock space.
But such a linkage appears to better describe the real world of photons
and electrons.