On the macroscopic theory of retarded Van der Waals forces

Electromagnetic Eigenmodes in Matter. Van Der Waals-London and Casimir Forces

“…The difference originates in the circumstance, usually overlooked, that the equivalent of our dispersion Equations (28) and (33) in the fluctuations theory have no solutions in some cases, as, for instance, for distinct dielectrics. The usual theorem of meromorphic functions, applied within the framework of the fluctuations theory [4][5][6], gives then a finite result, but it does not represent the energy of the eigenmodes. The problem does not appear in the non-retarded regime, where our results coincide with those of the fluctuations theory.…”

“…The matter polarization is usually represented in this case by a dielectric function. Recently, there is a renewed interest in this subject, motivated, on one hand, by the role played by plasmons, polaritons and other surface effects arising from the interaction between the electromagnetic field and matter and, on the other hand, by the querries related to the applicability of a dielectric function for discontinuous bodies [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. We report here on a different investigation of these forces, based on the calculation of the eigenfrequencies of the electromagnetic field interacting with matter.…”

The Casimir effect: medium and geometry

“…To obtain the Casimir energy and free energy one has to determine eigenfrequencies of normal modes of the electromagnetic field. Eigenfrequencies of normal modes can be summed up by making use of the argument principle [19,20], which states:…”

Casimir effect as a sum over modes in dissipative systems

“…This condition is sufficient to derive the Casimir energy using a sum-over-mode approach similar to the one used by Casimir in his seminal paper. To do this we will follow the procedure proposed by Schram in [7] which avoids problems with branch cuts (see Fig.1). For the development described here it will suffice to point out the main steps of [7]; a more thorough reading of Schram's work [7] is recommended for the reader in search of more details.…”

“…However, the first complete mathematical proof of the connection between Lifshitz and Casimir's approaches must be attributed to Schram [7] who showed the equivalence of the fully retarded Lifshitz formula and a sum over coupled cavity modes. The main limitation of Schram's work is that the medium composing the planes is allowed to be dispersive in a very general way but not dissipative so that the mode frequencies of the EM field are real functions.…”