Communications in Mathematical Physics

, Volume 213, Issue 3, pp 673–683

Renormalization of the Regularized Relativistic Electron-Positron Field

  • Elliott H. Lieb
  • Heinz Siedentop


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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  • Heinz Siedentop
    • 2
  1. 1.Departments of Mathematics and Physics, Jadwin Hall, Princeton University, P.O.B. 708, Princeton,¶NJ 08544-0708, USA. E-mail: lieb¶princeton.eduUS
  2. 2.Mathematik, Universität Regensburg, 93040 Regensburg, GermanyDE

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