A review of old inconsistencies of Classical Electrodynamics (CED) and of some new ideas that solve them is presented. Problems with causality violating solutions of the wave equation and of the electron equation of motion, and problems with the non-integrable singularity of its self-field energy tensor are well known. The correct interpretation of the two (advanced and retarded) Lienard-Wiechert solutions are in terms of creation and annihilation of particles in classical physics. They are both retarded solutions. Previous work on the short distance limit of CED of a spinless point electron are based on a faulty assumption which causes the well known inconsistencies of the theory: a diverging self-energy (the non-integrable singularity of its self-field energy tensor) and a causalityviolating third order equation of motion (the Lorentz-Dirac equation). The correct assumption fixes these problems without any change in the Maxwell's equations and let exposed, in the zero-distance limit, the discrete nature of light: the flux of energy from a point charge is discrete in time. CED cannot have a true equation of motion but only an effective one, as a consequence of the intrinsic meaning of the Faraday-Maxwell concept of field that does not correspond to the classical description of photon exchange, but only to the smearing of its effects in the space around the charge. This, in varied degrees, is transferred to QED and to other field theories that are based on the same concept of fields as space-smeared interactions.

An interpretation of the causality implementation of the Lienard-Wiechert solution raises doubts against the validity of the Lorentz-Dirac equation and the limits of validity of the Minkowski structure of spacetime. I IntroductionEarly in this century, the search for the correct equation of motion for a pointlike c harged classical particle was a major problem in theoretical physics. The advent of quantum mechanics brought some hope that it could be properly understood in the framework of a quantum theory. The proposed third-order LorentzDirac equation could not be accepted because of its numerous problems which h a ve not been solved, but just forgotten. In our opinion, this was a bad point for theoretical physics: one has failed to see that the Minkowski space is not the appropriate underlying geometric structure for the description of close interacting elds. The solution of these problems is still of great relevance since it may signal steering corrections one has to make in eld theory for avoiding old problems of QED and the stalling situations in some areas as quantum gravity and QCD.In modern eld theories, Poincar e i n variance is imposed, and the Minkowski space-time is taken as the appropriate scenario for describing non-gravitational phenomena. For electromagnetic elds in vacuum, far from charges, this has received con rmation from a solid experimental basis , but not for elds in a close vicinity of their sources. Even from a theoretical viewpoint, the question is not so clear: the problems faced by quantum eld theories for dealing with elds de ned in close neighboring points are well known. These di culties are generally taken as indications of some failure in the quantum basis of the theories or at least as an indication of the existence of some limits of validity. In this paper, we wish to emphasize that the same problem occurs in classical physics disguised on this old controversy about the correct equation of motion for the classical electron. Having inherited the same spacetime structure of their classical predecessors, it is not surprising that the quantum theories also face a similar problem for de ning elds in a close vicinity. Therefore, the roots of this problem must be searched at deeper grounds, in the very foundation of the assumed structures of the space-time continuum.Assuming the validity of energy momentum con- a, is the cause of some pathological

We develop the General Theory of Relativity in a formalism with extended causality that describes physical interaction through discrete, transversal and localized point-like fields. The essence of this approach is of working with fields defined with support on straightlines and not on hypersurfaces as usual. The homogeneous field equations of General Relativity are then solved for a finite, singularity-free, point-like field that we associate to a "classical graviton". The standard Einstein's continuous formalism is retrieved by means of an averaging process, and its continuous solutions are determined by the chosen imposed symmetry. The Schwarzschild metric is obtained by imposing spherical symmetry on the averaged field.

Motivated by improving the understanding of the quantum-to-classical transition we use a simple model of classical discrete interactions for studying the discrete-to-continuous transition in the classical harmonic oscilator. A parallel is traced with gravity for stressing the relevance of such discrete interaction models. Fundamental interactions, according to quantum field theory, are realized through the exchange of interaction quanta-packets of matter-energy with defined quantum numbers, viz. momentum-energy, spin, electric charge, etc. Thei are discrete interactions, in contradistinction to the classical continuous picture. The Bohr's correspondence principle, a useful guideline in the early days of quantum mechanics, states that in the limit of very large quantum numbers the classical idea of continuity must result from the quantum discreteness as an effective description. It would be very interesting to see in a clear way how this transition discrete-to-continuous occurs. This is the objective of the present letter with the use of a simple model of discrete classical interaction for studying this transition in the classical simple harmonic oscillator. We should not forget, however, that the harmonic potential, although being an extremely useful tool in all branches of modern physics, is not itself a fundamental interaction, which, as well known, are just the gravitational, the electromagnetic, the weak and the strong interactions; actually it is just an effective description. This may just valorize the importance of studying how it can be understood as an effective * PIVIC-UFES

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