Casimir Potentials,
Electromagnetic Density Oscillations
and Gravitation
by Darrell Moffitt
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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