Vacuum local and global electromagnetic self-energies for a point-like and an extended field source
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Regular Article - Theoretical Physics
Vacuum local and global electromagnetic self-energies for a point-like and an extended field source Roberto Passantea , Lucia Rizzutob , Salvatore Spagnoloc Dipartimento di Fisica e Chimica, Università degli Studi di Palermo and CNISM, Via Archirafi 36, 90123 Palermo, Italy
Received: 8 February 2013 / Revised: 25 March 2013 / Published online: 1 May 2013 © Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013
Abstract We consider the electric and magnetic energy densities (or equivalently field fluctuations) in the space around a point-like field source in its ground state, after having subtracted the spatially uniform zero-point energy terms, and discuss the problem of their singular behavior at the source’s position. We show that the assumption of a point-like source leads, for a simple Hamiltonian model of the interaction of the source with the electromagnetic radiation field, to a divergence of the renormalized electric and magnetic energy density at the position of the source. We analyze in detail the mathematical structure of such a singularity in terms of a delta function and its derivatives. We also show that an appropriate consideration of these singular terms solves an apparent inconsistency between the total field energy and the space integral of its density. Thus the finite field energy stored in these singular terms gives an important contribution to the self-energy of the source. We then consider the case of an extended source, smeared out over a finite volume and described by an appropriate form factor. We show that in this case all divergences in local quantities such as the electric and the magnetic energy density, as well as any inconsistency between global and space-integrated local self-energies, disappear.
1 Introduction Vacuum fluctuations and the existence of the zero-point energy of the electromagnetic field are a remarkable prediction of quantum electrodynamics [1]. They have been extensively investigated in the literature, especially in relation to Casimir and Casimir–Polder forces, which are long-range electromagnetic interactions between neutral macroscopic a e-mail:
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objects (metals or dielectrics), between atoms and surfaces or among neutral atoms or molecules [2–5]. The Casimir effect, in particular, is related to the modification of the zeropoint energy as a consequence of a change of the boundary conditions on the electromagnetic field. Despite its oddity, the idea of a force generated by a change of vacuum fluctuations in the presence of boundary conditions has stimulated great interest in the literature, both from theoretical and experimental point of view. These purely quantum effects have now been experimentally confirmed with remarkable accuracy [6–9] and many efficient methods have been proposed to calculate the Casimir force between bodies for real materials [6] and nontrivial geometries [10–16], even if some controver
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