  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, N, of electrons, and hence describes a
true many-body system. In addition to the electrons there are a number, K, of
fixed nuclei with charges \leq Z. If the fields are not quantized but are
classical, it was shown earlier that such a model is always unstable (the
ground state energy E=-\infty) if one uses the customary D(0) to define the
electron space, but is stable (E > - const.(N+K)) if one uses D(A) itself
(provided the fine structure constant \alpha and Z are not too large). This
result is extended to quantized fields here, and stability is proved for \alpha
=1/137 and Z \leq 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.
