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.
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References
Bach, V., Barbaroux, J.-M., Helffer, B. and Siedentop, H.: Stability of matter for the Hartree-Fock functional of the relativistic electron-positron field. Doc. Math. 3, 353–364 (electronic) (1998)
Bach, V., Barbaroux, J.-M., Helffer, B. and Siedentop, H.: On the stability of the relativistic electron-positron field. Commun. Math. Phys. 201, 445–460 (1999)
Bjorken, J.D. and Drell, S.D.: Relativistic Quantum Fields. International Series in Pure and Applied Physics. New York: McGraw-Hill, 1st edition, 1965
Helffer, B. and Siedentop, H.: Form perturbations of the second quantized Dirac field. Math. Phys. Electron. J. 4, Paper 4, 16 pp. (electronic) (1998)
Källén, G.: Quantenelektrodynamik. In: S. Flügge, editor, Prinzipien der Quantentheorie I, volume V/1 of Handbuch der Physik, Berlin: Springer-Verlag, 1st edition, 1958, pp. 169–364
Lieb, E.H. and Loss, M.: Self-energy of electrons in non-perturbation QED. In: R. Weihard and G. Weinstein (eds.): Differential equations and mathematical physics, (Birmingham, AL, 1999). Amer. Math Soc/International Press (2000), pp. 255–269
Thaller, B.: The Dirac Equation. Texts and Monographs in Physics. Berlin: Springer-Verlag, 1st edition, 1992
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Lieb, E.H., Siedentop, H. (2001). Renormalization of the Regularized Relativistic Electron-Positron Field. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04360-8_41
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DOI: https://doi.org/10.1007/978-3-662-04360-8_41
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