Lieb E.H., Siedentop H.
Renormalization of the Regularized Relativistic Electron-Positron Field
(36K, LATeX)
ABSTRACT. 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.