Renormalization of the Regularized Relativistic Electron-Positron Field

TL;DR

This paper explores a non-perturbative renormalization approach for the relativistic electron-positron field interacting via Coulomb potential, aiming to develop a consistent quantum electrodynamics framework.

Contribution

It introduces a non-perturbative normal ordering method that leads to a mass, wave-function, and charge renormalization, providing a new perspective on QED formulation.

Findings

Non-perturbative redefinition of electron/positron states via a Bogolubov-Valatin transformation.

Equivalent to a renormalization of the Dirac operator with physical implications.

Potential foundation for a consistent quantum electrodynamics theory.

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.