  We consider the relativistic electron-positron field interacting with itself
via the Coulomb potential defined with the physically motivated, positive,
density-density quartic interaction. The more usual normal-ordered Hamiltonian
differs from the bare Hamiltonian by a quadratic term and, by choosing the
normal ordering in a suitable, self-consistent manner, the quadratic term can
be seen to be equivalent to a renormalization of the Dirac operator. Formally,
this amounts to a Bogolubov-Valatin transformation, but in reality it is
non-perturbative, for it leads to an inequivalent, fine-structure dependent
representation of the canonical anticommutation relations. This
non-perturbative redefinition of the electron/positron states can be
interpreted as a mass, wave-function and charge renormalization, among other
possibilities, but the main point is that a non-perturbative definition of
normal ordering might be a useful starting point for developing a consistent
quantum electrodynamics.
