A simple mathematical procedure is introduced which allows redefining in an exact way divergent integrals and limits that appear in the basic equations of classical electrodynamics with point charges. In this way all divergences are at once removed without affecting the locality and the relativistic covariance of the theory, and with no need for mass renormalization. The procedure is first used to obtain a finite expression for the electromagnetic energy-momentum of the system. We showthat the relativistic Lorentz-Dirac equation can be deduced from the conservation of this electromagnetic energy-momentum plus the usual mechanical term. Then we derive a finite lagrangian, which depends on the particle variables and on the actual electromagnetic potentials at a given time. From this lagrangian the equations of motion of both particles and fields can be derived via Hamilton’s variational principle. The hamiltonian formulation of the theory can be obtained in a straightforwardway. This leads to an interesting comparison between the resulting divergence-free expression of the hamiltonian functional and the standard renormalization rules for perturbative quantum electrodynamics.

Classical electrodynamics of point charges / Massimo Marino. - In: ANNALS OF PHYSICS. - ISSN 0003-4916. - 301:2(2002), pp. 85-127.

Classical electrodynamics of point charges

Massimo Marino
2002

Abstract

A simple mathematical procedure is introduced which allows redefining in an exact way divergent integrals and limits that appear in the basic equations of classical electrodynamics with point charges. In this way all divergences are at once removed without affecting the locality and the relativistic covariance of the theory, and with no need for mass renormalization. The procedure is first used to obtain a finite expression for the electromagnetic energy-momentum of the system. We showthat the relativistic Lorentz-Dirac equation can be deduced from the conservation of this electromagnetic energy-momentum plus the usual mechanical term. Then we derive a finite lagrangian, which depends on the particle variables and on the actual electromagnetic potentials at a given time. From this lagrangian the equations of motion of both particles and fields can be derived via Hamilton’s variational principle. The hamiltonian formulation of the theory can be obtained in a straightforwardway. This leads to an interesting comparison between the resulting divergence-free expression of the hamiltonian functional and the standard renormalization rules for perturbative quantum electrodynamics.
classical electrodynamics ; point charge ; mass renormalization ; Lorentz-Dirac equation ; Hamilton equations
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/28357
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