Electromagnetic field fluctuations near a point-like and an extended field source

We investigate the field fluctuations near a point-like and an extended field source, such as an atom or a polarisable body, and discuss the problem of their singular behaviour at the position of the source. We consider a point-like source interacting with the electromagnetic field, in its dressed ground-state and investigate the local and global properties of the electric and magnetic energy densities in the space around the point-like source, after that the zero-point energy has been subtracted. We show that the assumption of a point-like source leads to a divergence of the renormalized electric and magnetic energy densities at the position of the source. We investigate in detail the mathematical structure of this divergence and show that it is at the origin of a discrepancy between the value of total electromagnetic field energy calculated as expectation value on the dressed ground-state of the field Hamiltonian or as a space integral of the electromagnetic energy density. We discuss that such inconsistency is solved if the singularity at the position of the field source is correctly taken into account. We then consider the case of an extended source, with a finite size and described by an appropriate form factor and investigate in detail the behaviour of the electric and magnetic energy densities in the space around the source. We show that the proposed model of extended source eliminates divergences and singularities in local quantities (such as the electric and the magnetic energy density), and also solve the inconsistency between global and space-integrated local self-energies, for any nonvanishing size of the source.