Retarded, or Casimir, Long‐Range Potentials

NOV 01, 1986

Even if two systems consist entirely of slowly moving constituents, nonrelativistic theory may not be adequate to determine the interaction energy of the systems when they are very far apart.

DOI: 10.1063/1.881043

Larry Spruch

Only someone with a short‐range view could fail to be aware of the great importance of long‐range interactions. Indeed, from the late 18th century, when Coulomb discovered that the electrostatic interaction has the same ${1/r}^{2}$ force law that Newton had found for the gravitational interaction, until perhaps the 1930s, when the strong and weak interactions began to be understood, long‐range interactions largely were the subject of physics. By long‐range interactions I mean not only those for which the potential behaves as $1/r$ for all r but those whose potentials behave asymptotically as some power of $1/r.$ These originate in $1/r$ potentials and include, for example, the van der Waals ${1/r}^{6}$ interaction (as calculated nonrelativistically) between two spherically symmetric atoms at a large separation r, and multipole interactions between charge distributions. Long‐range potentials therefore not only play a vital role in astrophysics via Newton’s law of gravitation and a significant role in nuclear physics via Coulomb’s law, but determine almost all of atomic, molecular and condensed‐matter physics.

References

1. L. Spruch, E. J. Kelsey, Phys. Rev. A 18, 845 (1978). https://doi.org/PLRAAN
See also the reference therein by T. Boyer.
2. M. Born, M. Goeppert‐Mayer, in Handbuch der Physik, vol. 24/2, S. Flügge, ed., Springer‐Verlag, Berlin (1933), p. 623.
3. J. A. Wheeler, Phys. Rev. 59, 928 (1941) (abstract).https://doi.org/PHRVAO
4. The original calculations of the different potentials are in H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 60, 793 (1948) (wall‐wall);
H. B. G. Casimir, D. Polder, Phys. Rev. 73, 360 (1948) (atom‐atom); https://doi.org/PHRVAO
H. B. G. Casimir, J. Chim. Phys. 46, 407 (1949) (atom‐wall); https://doi.org/JCPBAN
E. Salpeter, Phys. Rev. 87, 328 (1952); https://doi.org/PHRVAO
T. Fulton, P. Martin, Phys. Rev. 95, 811 (1954) (electron‐electron); https://doi.org/PHRVAO
G. Barton, J. Phys. A 10, 601 (1977); https://doi.org/JPHAC5
L. Spruch, E. J. Kelsey, Phys. Rev. A 18, 845 (1978) (electron‐wall); https://doi.org/PLRAAN
J. Bernabeu, R. Tarrach, Ann. Phys. (N.Y.) 102, 323 (1976) (electron‐atom); https://doi.org/APNYA6
and reference 5 (electron‐ion).
5. E. J. Kelsey, L. Spruch, Phys. Rev. A 18, 15 (1978); https://doi.org/PLRAAN
E. J. Kelsey, L. Spruch, Phys. Rev. A 18, 1055 (1978).https://doi.org/PLRAAN
6. E. A. Power, Introductory Quantum Electrodynamics, Elsevier, New York (1965),
contains some excellent material on retarded potentials. For historical background, see S. J. Brush, Statistical Physics of Atomic Theory of Matter, from Boyle and Newton to Landau and Onsager, Princeton U.P., Princeton, N.J. (1983), p. 215.
7. H. B. G. Casimir, Haphazard Reality, Harper and Row, New York (1983).
8. G. Feinberg, J. Sucher, J. Chem. Phys. 48, 3333 (1968).https://doi.org/JCPSA6
9. B. Chu, Molecular Forces: Based on the Baker Lectures of P. W. Debye, Interscience, New York (1967);
section 5.2 contains a list of references.
10. R. Shakeshaft, L. Spruch, Phys. Rev. A 22, 811 (1980).https://doi.org/PLRAAN
11. S. L. Palfrey, S. R. Lundeen, Phys. Rev. Lett. 53, 1141 (1984).https://doi.org/PRLTAO
12. C. K. Au, G. Feinberg, J. Sucher, Phys. Rev., to be published November 1986.
See also R. J. Drachman, Phys. Rev. A 31, 1253 (1985).https://doi.org/PLRAAN
13. H. B. G. Casimir, Physica (Utrecht) 19, 846 (1959).https://doi.org/PHYSAG
14. T. H. Boyer, Phys. Rev. 174, 1764 (1968). https://doi.org/PHRVAO
See also K. A. Milton, L. L. DeRaad, J. Schwinger, Ann. Phys. (N.Y.) 115, 388 (1978).https://doi.org/APNYA6
15. K. A. Milton, Phys. Rev. D 22, 1441 (1980); https://doi.org/PRVDAQ
K. A. Milton, Phys. Rev. D 22, 1444 (1980); and references therein.https://doi.org/PRVDAQ

More about the Authors

Larry Spruch. New York University, New York, New York.