Explicit mass renormalization and consistent derivation of radiative response of classical electron

Abstract

The radiative response of the classical electron is commonly described by the Lorentz-Abraham-Dirac (LAD) equation. Dirac's derivation of this equation is based on energy and momentum conservation laws and on regularization of the field singularities and infinite energies of the point charge by subtraction of certain quantities: "We ... shall try to get over difficulties associated with the infinite energy of the process by a process of direct omission or subtraction of unwanted terms". To substantiate Dirac's approach and clarify the mass renormalization, we introduce the point charge as a limit of extended charges contracting to a point; the fulfillment of conservation laws follows from the relativistic covariant Lagrangian formulation of the problem. We derive the relativistic point charge dynamics described by the LAD equation from the extended charge dynamics in a localization limit by a method which can be viewed as a refinement of Dirac's approach in the spirit of Ehrenfest theorem. The model exhibits the mass renormalization as the cancellation of Coulomb energy with the Poincaré cohesive energy. The value of the renormalized mass is not postulated as an arbitrary constant, but is explicitly calculated. The analysis demonstrates that the local energy-momentum conservation laws yield dynamics of a point charge which involves three constants: mass, charge and radiative response coefficient θ. The value of θ depends on the composition of the adjacent potential which generates Poincaré forces. The classical value of the radiative response coefficient is singled out by the global requirement that the adjacent potential does not affect the radiated energy balance and affects only the local energy balance involved in the renormalization.

Scalar Field Theory for Mass Determination

“…The most famous development has been made for electrodynamics. Several relations can be proposed based on different approximations: with or without relativistic assumption, with or without quantum assumption [36][37][38][39]. For some of those relations, the mass depends on the inverse of the length scale r:…”

“…A strong divergence problem occurs for non-quantum approach (relativistic or not) [36,37] or for quantum and non-relativistic approach [38]. It is only with a simultaneous relativistic and quantum theory that the mass of particles can be renormalized with a logarithmic dependence on the length scale [39]: …”