We derive van der Waals-London and Casimir forces by calculating the eigenmodes of the electromagnetic field interacting with two semi-infinite bodies (two halves of space) with parallel surfaces separated by distance d. We adopt simple models for metals and dielectrics, well-known in the elementary theory of dispersion. In the non-retarded (Coulomb) limit we get a d-3-force (van der Waals-London force), arising from the zero-point energy (vacuum fluctuations) of the surface plasmon modes. When retardation is included we obtain a d-4-(Casimir) force, arising from the zero-point energy of the surface plasmon-polariton modes (evanescent modes) for metals, and from propagating (polaritonic) modes for identical dielectrics. The same Casimir force is also obtained for "fixed surfaces" boundary conditions, irrespective of the pair of bodies. The approach is based on the equation of motion of the polarization and the electromagnetic potentials, which lead to coupled integral equations. These equations are solved, and their relevant eigenfrequencies branches are identified.
2. Lifshitz, E., "The theory of molecular attractive forces between solids," ZhETF, Vol. 29, 94-105, 1956. (Sov. Phys. JETP, Vol. 2, 73-83, 1956)..
3. Dzyaloshinskii, I. E., E. M. Lifshitz, and L. P. Pitaevskii, "The general theory of van der Waals forces," Adv. Phys., Vol. 10, 165-209, 1961.
4. Van Kampen, N. G., B. R. A. Nijboer, and K. Schram, "On the macroscopic theory of van der Waals forces," Phys. Lett. A, Vol. 26, 307-308, 1968.
5. Gerlach, E., "Equivalence of van der Waals forces between solids and the surface-plasmon interaction," Phys. Rev. B, Vol. 4, 393-396, 1971.
6. Schram, K., "On the macroscopic theory of retarded van der Waals forces ," Phys. Lett. A, Vol. 43, 282-284, 1973.
7. Heinrichs, J., "Theory of van der Waals interaction between metal surfaces," Phys. Rev. B, Vol. 11, 3625-3636, 1975.
8. Milloni, P. W. and The Quantum Vacuum, , Academic Press, San Diego, 1994.
9. Mostepanenko, V. M. and N. N. Trunov, "The Casimir Effect and Its Applications," Clarendon, Oxford, 1997.
10. Lamoreaux, S. K., "Demonstration of the Casimir force in the 0.6 to 6 μm range," Phys. Rev. Lett., Vol. 78, 5-8, 1997.
11. Lambrecht, A. and S. Reynard, "Comment on ``Demonstration of the Casimir force in the 0.6 to μm range"," Phys. Rev. Lett., Vol. 84, 5672-5672, 2000.
12. Lamoreaux, S. K., "Calculation of the Casimir force between imperfectly conductiong plates," Phys. Rev. A, Vol. 59, R3149-R3153, 1999.
13. Bordag, M., U. Mohideen, and V. Mostepanenko, "New developments in the Casimir e®ect," Phys. Reps., Vol. 353, 1-205, 2001.
14. Milton, K. A. and The Casimir Effect, , World Scientific, Singapore, 2001.
15. Genet, C., A. Lambrecht, and S. Reynaud, "Casimir force and the quantum theory of lossy optical cavities," Phys. Rev. A, Vol. 67.
16. Chen, F., U. Mohideen, G. L. Klimchitskaya, and V. M. Mostepanenko, "Investigation of the Casimir force between metal and semiconductor test bodies," Phys. Rev. A, Vol. 72, No. 2, 2005.
17. Lamoreaux, S. K., "The Casimir force: Background, experiments and applications," Reps. Progr. Phys., Vol. 65, 201-236, 2005.
18. Intravaia, F., Effet Casimir et interaction entre plasmons de surface, These de Doctorat de l'Universite Paris VI, 1-177, Jun. 2005.
19. Obrecht, J. M., R. J. Wild, M. Antezza, L. P. Pitaevskii, S. Stringari, and E. A. Cornell, "Measurement of the temperature dependence of the Casimir-Polder force," Phys. Rev. Lett., Vol. 98, 063201, 1-4, 2007.
20. Intravaia, F., C. Henkel, and A. Lambrecht, "Role of surface plasmons in the Casimir effect," Phys. Rev. A, Vol. 76, 033820, 1-11, 2007.
21. Lorentz, H. A., The Theory of Electrons, Leipzig, Teubner, 1916.
22. Born, M. and E. Wolf, Principles of Optics, Pergamon, London, 1959.
23. Apostol, M. and G. Vaman, "Plasmons and polaritons in a semiinfinite plasma and a plasma slab," Physica B, Vol. 404, 3775-3781, 2009.
24. Ritchie, R. H., "Plasma losses by fast electrons in thin films," Phys. Rev., Vol. 106, 874-881, 1957.
25. Stern, E. A. and R. A. Ferrell, "Surface plasma oscillations of a degenerate electron gas," Phys. Rev., Vol. 120, 130-136, 1960.
26. Eguiluz, A. and J. J. Quinn, "Hydrodynamic model for surface plasmons in metals and degenerate semiconductors," Phys. Rev. B, Vol. 14, 1347-1361, 1976.
27. DasSarma, S. and J. J. Quinn, "Hydrodynamic model of linear response for a jellium surface: Non-retarded limit," Phys. Rev. B, Vol. 20, 4872-4882, 1979.
28. Glass, N. E. and A. A Maradudin, "Surface plasmons on a large-amplitude grating," Phys. Rev. B, Vol. 24, 595-602, 1981.
29. DasSarma, S. and J. J. Quinn, "Collective excitations in semiconductor superlattices," Phys. Rev. B, Vol. 25, 7603-7618, 1982.
30. Schaich, W. L. and J. F. Dobson, "Excitation modes of neutral jellium slabs," Phys. Rev. B, Vol. 49, 14700-14707, 1994.
31. Link, G. and R. V. Baltz, "Hydrodynamic description of surface plasmons: Nonexistence of the unrestricted half-space solution," Phys. Rev. B, Vol. 60, 16157-16163, 1999.
32. Landau, L. and E. Lifshitz, Course of Theoretical Physics, No. 5, (Statistical Physics), Part 2, Butterworth-Heinemann, Oxford, 2003.
33. Galkina, E. G., B. A. Ivanov, S. Savelev, V. A. Yampolskii, and F. Nori, "Drastic change of the Casimir force at the metal-insulator transition," Phys. Rev. B, Vol. 80, 125119, 1-11, 2009.
34. Gradshteyn, I. S. and I. M. Ryzhik, Table of Integrals, Series and Products, 714-715, Academic Press, 2000.
35. Whittaker, E. T. and G. N.Watson, Course of Modern Analysis, Cambridge, 2004.
36. Casimir, H. B. G. and D. Polder, "The influence of retardation on the London-van der Waals forces," Phys. Rev., Vol. 73, 360-372, 1948.