## Casimir Potentials,

by Darrell Moffitt

Electromagnetic Density Oscillations

and Gravitation

Among the less researched topics of classical and quantum physicists is that of electromagnetic density oscillations (e.d.o.s), i.e., acoustic-form longitudinal waves in electromagnetic media. Though current literature does address phonon interactions, and their influence on conductivity in solids, longitudinal plasma waves and their quantum relatives are largely ignored. While acoustic waves may admit transverse polarities, they are fundamentally a longitudinal phenomenon. Plasma oscillations possess up to four longitudinal modes; the alternating compression and rarefaction zones comprising a wave may consist of either particles or fields, and the field itself may be electric or magnetic. Still another form of e.d.o. results from interacting charge densities; the square of charge density is dimensionally identical to that of sound, (M/(R^3*t^2)), and van der Waals forces both derive from such couplings and contribute the restoring force responsible for ordinary sound. Quantum mechanics describes van der Waals forces via Casimir potentials, simple, non-relativistic equations which take their name from Hendrick B. G. Casimir. Casimir's polar-polar potential, applied to any two polarizable bodies, is a prototype of quantum-level e.d.o.s. This equation reads E=((h/(2pi*c^5))(P1*P2)(w^6/6))(1/R) where (P1,P2) are the respective (volume) polarizations, "w" is the characteristic frequency, and "R" is the separation of the bodies in question. ("h" is, of course, Planck's constant, and "c" is the speed of light..) The frequency is evaluated over a cut-off determined by the size of the system, "r", and roughly equal to (c/r). Here, the factor (1/6) represents an integral over w^5, and r may be taken as an average atomic radius. This equation is accurate over a wide range of scales, with corrections on the order of unity being found in most situations of interest. The factor (hw^6/(12pi*c^5)) is the e.d.o. term, and yields large exponents in the case of molecular or atomic systems. These large exponents are deceptive however, as average polarization in an atomic system is on the order of ((10^-27)cm^3) or less. Polar-polar potentials thus tend to produce energies which are small compared to the total energy of a given system. This low-energy behavior suggests some resemblance to gravitation, which a sample calculation will make more evident. Consider first the physical dimensions of Newton's constant, (R^3/(M*t^2)). It can be written (w^2/d), where "d" is a volume mass density, and could as well be written ((M/(R^3*t^2))(1/d)^2, the ratio of an e.d.o.divided by the square of a density. The Casimir interpretation of this expression takes the form (hw^6/(2pi*c^5))(1/d)^2.Œ To get an estimate of the frequency-to-density ratios which are relevant, one should look at a simple system with well-known parameters. Monoatomic hydrogen is just such a system. Assume that the atom in question is in its ground state, with a minimal volume polarization equal roughly to one electron volume (approx.(2.818*10^-13)^3). Then, the polar-polar potential of this system will be proportional to the orbital frequency of the electron (a^3*c/4pi*r#) where "a" is the electromagnetic coupling constant (1/137.036), and"r#" is the electron radius. The e.d.o. of this system ((h/(2pi*c^5))(a^3*c/(4pi*r#)^6) is 5.52107*10^13(gm/(cm^3*t^2)), (factor of (1/6) suppressed). When this quantity is multiplied by the electron volume squared, the result, 2.76449*10^-62(gm*cm^3/t^2), is a tiny value indeed. Herein lies a surprise. Recall that Newton's constant has multiple definitions in dimensional analysis. In this case, dividing the Casimir potential value just given by the electron mass squared creates a fascinating "coincidence", namely, a figure which is just shy of one-half Newton's constant. Properly reduced, and doubled to account for two interacting systems, the full expression will read G=((hc/(pi*m#^2))(a^3/4pi)^6) which equals Newton's constant to within 99.85%("m#"=electron mass). A more accurate derivation requires evaluation of the Lambshift contribution, and estimates of such factors as vacuum polarization, charge screening, relativistic corrections (long-range) and higher order interactions. The model just presented is not, arguably, an accurate "definition" of gravitation, as other models are much more precise and detailed. It does, however, illustrate the utility of alternative conceptualization made feasible by the use of e.d.o.s and Casimir potentials. The value of this approach lies in its ability to reveal new phenomena and relationships between seemingly well-known processes. A similar treatment might have been performed on quark-gluon coupling, say, where longitudinal virtuals have been found to play an active role. What is more important is that consideration be given to a broader range of qualitative issues in physics. Quantitative methods are only as good as the qualitative concepts they address, for one cannot calculate our planet's circumference without first asking if it is round. Add 1: A good introduction to Casimir potentials will be found in "Physics Today", 11/86, p.p.37-45, titled "Retarded, or long-range, Casimir potentials", by Larry Spruch, which contains a very complete bibliography.

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