Finite calculation of divergent selfenergy diagrams

Using dispersive techniques, it is possible to avoid ultraviolet divergences in the calculation of Feynman diagrams, making subsequent regularization of divergent diagrams unnecessary. We give a simple introduction to the most important features of such dispersive techniques in the framework of the so-called finite causal perturbation theory. The method is also applied to the 'divergent' general massive two-loop sunrise selfenergy diagram, where it leads directly to an analytic expression for the imaginary part of the diagram in accordance with the literature, whereas the real part can be obtained by a single integral dispersion relation. It is pointed out that dispersive methods have been known for decades and have been applied to several nontrivial Feynman diagram calculations.Comment: 15 pages, Latex, one figure, added reference

Non-Perturbative Mass and Charge Renormalization in Relativistic No-Photon Quantum Electrodynamics

Starting from a formal Hamiltonian as found in the physics literature -- omitting photons -- we define a renormalized Hamiltonian through charge and mass renormalization. We show that the restriction to the one-electron subspace is well-defined. Our construction is non-perturbative and does not use a cut-off. The Hamiltonian is relevant for the description of the Lamb shift in muonic atoms.Comment: Reformulation of main theorem, minor changes in the proo

Self-consistent solution for the polarized vacuum in a no-photon QED model

We study the Bogoliubov-Dirac-Fock model introduced by Chaix and Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989) which is a mean-field theory deduced from no-photon QED. The associated functional is bounded from below. In the presence of an external field, a minimizer, if it exists, is interpreted as the polarized vacuum and it solves a self-consistent equation. In a recent paper math-ph/0403005, we proved the convergence of the iterative fixed-point scheme naturally associated with this equation to a global minimizer of the BDF functional, under some restrictive conditions on the external potential, the ultraviolet cut-off $\Lambda$ and the bare fine structure constant $\alpha$. In the present work, we improve this result by showing the existence of the minimizer by a variational method, for any cut-off $\Lambda$ and without any constraint on the external field. We also study the behaviour of the minimizer as $\Lambda$ goes to infinity and show that the theory is "nullified" in that limit, as predicted first by Landau: the vacuum totally kills the external potential. Therefore the limit case of an infinite cut-off makes no sense both from a physical and mathematical point of view. Finally, we perform a charge and density renormalization scheme applying simultaneously to all orders of the fine structure constant $\alpha$, on a simplified model where the exchange term is neglected.Comment: Final version, to appear in J. Phys. A: Math. Ge

Ward Identities and Renormalization of General Gauge Theories

We introduce the concept of general gauge theory which includes Yang-Mills models. In the framework of the causal approach and show that the anomalies can appear only in the vacuum sector of the identities obtained from the gauge invariance condition by applying derivatives with respect to the basic fields. Then we provide a general result about the absence of anomalies in higher orders of perturbation theory. This result reduces the renormalizability proof to the study of lower orders of perturbation theory. For the Yang-Mills model one can perform this computation explicitly and obtains its renormalizability in all orders.Comment: 38 pages, LATEX2

Reverse Engineering Approach to Quantum Electrodynamics

The S matrix of e--e scattering has the structure of a projection operator that projects incoming separable product states onto entangled two-electron states. In this projection operator the empirical value of the fine-structure constant alpha acts as a normalization factor. When the structure of the two-particle state space is known, a theoretical value of the normalization factor can be calculated. For an irreducible two-particle representation of the Poincare group, the calculated normalization factor matches Wyler's semi-empirical formula for the fine-structure constant alpha. The empirical value of alpha, therefore, provides experimental evidence that the state space of two interacting electrons belongs to an irreducible two-particle representation of the Poincare group.Comment: 12 pages, minor change