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We consider the electric and magnetic energy densities (or equivalently fieldfluctuations) in the space around a point-like field source in its groundstate, after having subtracted the spatially uniform zero-point energy terms,and discuss the problem of their singular behavior at the source's position. Weshow that the assumption of a point-like source leads, for a simple Hamiltonianmodel of the interaction of the source with the electromagnetic radiationfield, to a divergence of the renormalized electric and magnetic energy densityat the position of the source. We analyze in detail the mathematical structureof such singularity in terms of a delta function and its derivatives. We alsoshow that an appropriate consideration of these singular terms solves anapparent inconsistency between the total field energy and the space integral ofits density. Thus the finite field energy stored in these singular terms givesan important contribution to the self-energy of the source. We then considerthe case of an extended source, smeared out over a finite volume and describedby an appropriate form factor. We show that in this case all divergences inlocal quantities such as the electric and the magnetic energy density, as wellas any inconsistency between global and space-integrated local self-energies,disappear.

Vacuum local and global electromagnetic self-energies for a point-like and an extended field source