On the Anticipatory Aspects of the Four Interactions: what the Known Classical and Semi-Classical Solutions Teach us

TL;DR

This paper explores the classical and semi-classical solutions of the four fundamental interactions, highlighting how anticipatory effects like the Feynman propagator and radiative corrections extend beyond semi-classical approximations.

Contribution

It analyzes the role of gauge variables and Dirac observables in classical and quantum theories, emphasizing the emergence of anticipatory effects in the context of fundamental interactions.

Findings

Classical causality favors retarded solutions.

Quantum regularization introduces anticipatory effects.

Radiative corrections extend anticipatory effects beyond semi-classical level.

Abstract

The four (electro-magnetic, weak, strong and gravitational) interactions are described by singular Lagrangians and by Dirac-Bergmann theory of Hamiltonian constraints. As a consequence a subset of the original configuration variables are {\it gauge variables}, not determined by the equations of motion. Only at the Hamiltonian level it is possible to separate the gauge variables from the deterministic physical degrees of freedom, the {\it Dirac observables}, and to formulate a well posed Cauchy problem for them both in special and general relativity. Then the requirement of {\it causality} dictates the choice of {\it retarded} solutions at the classical level. However both the problems of the classical theory of the electron, leading to the choice of ${1\over 2} (retarded + advanced)$ solutions, and the regularization of quantum field teory, leading to the Feynman propagator, introduce {\it anticipatory} aspects. The determination of the relativistic Darwin potential as a semi-classical approximation to the Lienard-Wiechert solution for particles with Grassmann-valued electric charges, regularizing the Coulomb self-energies, shows that these anticipatory effects live beyond the semi-classical approximation (tree level) under the form of radiative corrections, at least for the electro-magnetic interaction.