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Resources
The most comprehensive textbook covering the Casimir effect and related vacuum
phenomena is The Quantum Vacuum, by Peter Milonni, Academic Press(1994).
The Casimir effect is covered in Quantum Mechanics, Leslie Ballentine,
Prentice-Hall (1990). The most recent resource paper with many references on
Casimir forces was published by S. Lamoroux, American Journal of Physics, vol.
67, pp. 850-861, October, 1999. A good review describing the breadth of Casimir
phenomena is "Casimir Forces" by Peter Milonni and Mei-Li Shis,
Contemporary Physics, vol. 33, 313-322 (1992). Another
informative review is E.Elizalde and A.Romero, "Essentials of the Casimir
effect and its computation," Am. J. Phys. 59, 711-719 (1991). A long
and detailed review is given by P. Plunian, B. Muller, W. Greiner, "The
Casimir Effect," Physics Reports (Review Section of Physics Letters) 134,
2&3, pp. 87-193 (1986). Much information is contained in the text
"The Casimir Effect and its Applications" by V. Mostepanenko and N.
Trunov, published by Oxford University Press, 1997.
A few general comments on vacuum energy
Meaning of Zero-Point Fluctuations
"We cannot get something for nothing—this would violate the conservation
of energy and the second law of thermodynamics. But maybe we can find a
convenient way to pay for the vacuum energy and thereby use it profitably."
(JM)
If you have a hot body in equilibrium with its surroundings, it is radiating
and reabsorbing photons (quantized bundles of electromagnetic radiation) of all
wavelengths. The energy of a photon of frequency f is Planck's constant h times
the frequency hf . The energy distribution of these photons is given by the
Planck blackbody equation. If you imagine cooling the body and surroundings to
absolute zero, you would find that a temperature independent electromagnetic
field remained. This electromagnetic field, which is always present even if no
matter or charged particles are present and the temperature is absolute zero,
represents the lowest state of the electromagnetic field. The
corresponding fluctuating electromagnetic field is called the zero-point (ZP)
field of the electromagnetic field. The presence of this zero-point field is
predicted by QED (Quantum Electrodynamics). QED is the most precise physical
theory we have; its predictions have been verified to 1 part in 10 billion!
The zero-point field is the "ground state" of the electromagnetic
field. In this ground state, the equations indicate that no ordinary
physical photons are present, yet electromagnetic energy is present. The energy
for a given frequency is ½ hf , one half of the usual energy of a photon.
Sometimes the zero-point field is described as consisting of "virtual"
or very short-lived photons, that appear and disappear before it is possible to
detect them. The presence of zero-point fluctuations has been verified
experimentally with very accurate measurements of the Lamb Shift, other atomic
energy level shifts, the magnetic moment of the electron, and the Casimir force.
QED predicts that the number of ZP quanta (½ hf ) of frequency f is
proportional to the square of the frequency. This gives an energy density for
the vacuum that goes as the cube of the frequency.
Special relativity requires that any observer going through space cannot tell
how fast she is going in an absolute sense. Thus the zero-point fluctuations
must look the same, independent of her velocity as she travels through space.
Therefore the Doppler shifted frequency spectrum must look the same as the
unshifted frequency spectrum. This requirement of special relativity results in
an energy density of the zero-point fluctuations identical to that predicted by
QED, namely an energy density proportional to the cube of the frequency. Summing
over all the frequencies present, gives a total energy density in the vacuum of
which is proportional to 1/L4
where L is the shortest wavelength of the ZP fluctuations allowed. If we take L
as zero, then we obtain an infinite energy. Applying quantum principles to
general relativity (geometrodynamics) suggests that at lengths shorter than the
Planck length (10-35 m), the nature of space-time fluctuates, and
therefore no meaning can be ascribed to a length shorter than the Planck length.
Thus we could use the Planck length as a cutoff.
The energy density of the ZP fluctuations in empty space (according to QED)
is about 10114 joules/cubic meter if we use the Planck length (10-35
m) as a cut-off.
General Relativity and Vacuum Energy
In general relativity, any form of energy has an equivalent mass, given by E =
mc2, and is therefore coupled to gravity. This enormous zero-point
energy density is equivalent to a mass density of about 1092 kg/cc,
and would be expected to cause an enormous gravitational field. This large field
leads to some major problems with general relativity, such as the collapse of
the universe into a region of space that is about 1 Planck length across. Thus
we have an inconsistency in two very important and well-verified theories, QED
and General Relativity. A brief discussion of this problem is given in the
excellent book "Lorentzian Wormholes" (Springer-Verlag, 1996, p. 82)
by Matt Visser.
As a brief aside, it is amusing to compute the equivalent mass for a region
of the vacuum about the size of a proton, which is approximately a sphere about
10-13 cm across, using the enormous energy density formally predicted
above. This process yields an equivalent mass of about 1053
kg. This means the vacuum energy contained within a region of space the
size of a proton is equivalent to a mass of about 1053 kg. A very
rough estimate of the number of nucleons in the universe is 1080.
This number is based on the statistical distribution of stars in galaxies and
the number of galaxies. Most of the mass of matter is in nucleons, so the mass
of the universe is roughly the weight of a proton times 1080 or about
1053 kg, which is the same as the mass equivalent of the vacuum
energy in a region the size of a proton. Conclusion: A volume the size of a
proton in empty space contains about the same amount of vacuum energy as all the
matter in the entire universe!!!!!
This simple-minded computation is interesting when we think about the big
bang theories, in which ultra dense matter in a small region of space explodes
and develops into all the matter in the known universe. Could it be possible
that the vacuum energy in a small region or space underwent a transformation
into mass? This simple-minded computation does not include other forms of
energy, such as black holes, or gravitational potential energy.
As mentioned above, the enormous density of the vacuum energy appears to
cause severe conceptual problems in general relativity since the energy couples
with gravity. We can reduce the zero-point energy by the use of a larger cutoff
wavelength. One choice for the short wavelength limit is the Compton wavelength
of the proton as suggested by Nobel Laureate physicist Richard Feynman. The
energy density is then reduced to about 1035 joules/cubic-meter
corresponding to an equivalent mass density of 1012 kg/cc. To
appreciate the enormity of this number, compare it to the chemical energy of a
fuel, 1015 joules/cubic meter, to the energy density of matter, 1020
joules/cubic meter, and to the energy density of a nucleus, 1030
joules/cubic meter.
This energy density is still very large, but does look better from the
viewpoint of general relativity. However, now the bad news. The use of a cut-off
is actually in conflict with special relativity, because the value of the cutoff
will depend on the velocity of the rest frame of the observer. This would mean
that the vacuum energy density would depend on your relative motion, which is a
violation of special relativity. So again we have an inconsistency between well
verified theories.
A variety of "solutions" have been proposed to resolve the
inconsistencies between general relativity and quantum theory, including the use
of a "renormalized" vacuum energy. In this approach, the vacuum
energy of empty space is set equal to zero, and only changes in vacuum energy,
such as those that occur in Casimir effects, are included in the formulation of
general relativity. Other approaches include supersymmetry (SUSY)
and the use of extra dimensions (Kaluza-Klein theory), and superstrings.
Another approach might lie in a reinterpretation of Mach's principle. In
the spirit of Mach’s principle the mass of an object is interpreted as the
effective gravitational attraction of the object to all the rest of the matter
or energy in the universe. Perhaps the vacuum energy needs to be included
in this formulation. Many think that a new theory of quantum gravity is
needed to resolve these conflicts.
At this time there is no consistent interpretation of the zero-point energy
density in empty space. … It’s an embarrassment to physicists to have such a
conflict between such well verified and accepted theories. In fact it is so
painful, that most physicists don’t even want to think about it…
An Equivalent Interpretation of the Effects of Vacuum Energy
It should be mentioned that there is an alternative but completely equivalent
way to speak of vacuum energy, called the source theory. In this approach, the
fluctuation field is interpreted as arising from the matter in the region. For
example, in the parallel plate Casimir geometry, the vacuum field can be
interpreted as arising from the matter from which the plates are made. These
approaches are equivalent. This equivalence is covered very well in the text The
Quantum Vacuum by Peter Milonni, mentioned above. H. G. B. Casimir,
who originally brought attention to the reality of vacuum energy with his
seminal calculation of the "Casimir" force between two uncharged metal
plates over 50 years ago, described these two viewpoints very nicely:
"The [measurement of the attractive force between metal plates] has been
shown by clever experiments and I think we can claim that the existence of
electromagnetic zero-point energy in vacuum has been established beyond doubt.
But one can take a more modest point of view. Inside a metal there are
forces of cohesion and if you take two metal plates and press them together
these forces of cohesion begin to act. On the other hand you can start
from one piece and split it. Then you have first to break chemical bonds
and next to overcome van der Waals forces of classical type and if you separate
the two pieces even further there remains a curious little tail. The
Casimir force is the last but also the most elegant trace of cohesion
energy."
(H. B. G. Casimir, "Some Remarks on the HIstory of the So Called Casimir
Effect" in "The Casimir Effect 50 Years Later", edited by Michael
Bordag, World Scientific(1999).
Casimir's comments suggest one of the seemingly paradoxical observations
about vacuum energy phenomena: although the vacuum energy density is predicted
to be nearly infinite, all the effects that have been observed are quite small.
Indeed 50 years elapsed between Casimir's original prediction of the vacuum
force between two parallel metal plates and its' careful measurement by Lamoroux
and Mohideen.
Can we make use of the energy in the vacuum?? A few comments on
Conservative fields
The gravitational field is a conservative field. This means that the amount of
gravitational potential energy stored in a mass at the top of a hill only
depends on how high the hill is, not on how the mass got there. In this
conservative field, the potential energy depends only on the height. How does
one use gravitational energy in a machine?? One example is a waterfall. Here the
potential energy of the water is converted into kinetic energy as it falls, and
the kinetic energy is used to drive a turbine. In a conservative field energy is
conserved. What does this mean?? If you first raised the water to the top of the
falls, and then let it fall down to operate your turbine, the amount of energy
you got out (in the ideal case of 100% efficiency) would equal the energy you
put in.
The electromagnetic field is a conservative field also, with its rules. All
the devices we make are constrained to follow these rules. The vacuum energy
might be seen as analogous to the gravitational potential energy. The question
is now how can we take advantage of the energy in the vacuum fluctuations in a
useful way? In the conservative gravitation field, just raising a ball to
the top of a hill and letting it fall down is not very useful. This might
correspond to a vacuum energy battery, in which we have some charged plates
that, when released, move toward each other due to the Casimir force, converting
electrical energy into work or kinetic energy. When the plates are as close as
they get, the power has all been extracted. To restore the battery to its
original state would require the at least the same amount of power to be applied
to separate the plates. We get out of this reversible battery only what we put
into it (this idea of a ZP fluctuation battery is discussed by Robert L. Forward
in Phys Rev. B, 30, p. 1700-1702, 1982).
The question is, can we build or find something, for example, analogous to a
waterfall, or at least waves, that has a region with a high vacuum energy
density and a region with a low vacuum energy density, and make something useful
from this difference in vacuum energy? Or perhaps we need additional
thermodynamic variables such as discussed by F. Pinto (Engine cycle of an
optically controlled vacuum energy transducer, Physical Review B, volume 60,
pages 14740-14755, December, 1999). We cannot get something for nothing—this
would violate the conservation of energy and the second law of thermodynamics.
But maybe we can find a convenient way to pay for the vacuum energy and thereby
use it profitably.
Perhaps, we need to look for wholly new types of devices and structures that
are based on the unique properties of vacuum energy. That is the purpose of our
present effort in the Breakthrough Propulsion program, to explore this new
terrain of vacuum energy, and make a map we can use to get somewhere of
interest.
A few notes on our experiment with a resonant cavity
We want to explore the properties of a sub-micron rectangular cavity that is
vibrating due to its interaction with the zero point fluctuations. We can pick a
cavity with dimensions (2 x 0.1 x 0.15, with all dimensions in micrometers) such
that at equilibrium the Casimir force on the moveable walls (1 x .1) is zero,
and the forces on the remaining walls are negative and positive. When this 2 x
0.1 wall is deflected slightly outward, then the Casimir force on this wall
becomes positive or outward, and tends to increase the deflection. Conversely,
when the wall is deflected inward from its equilibrium location, then the
Casimir force tends to increase the inward deflection. Thus the Casimir force
always tends to increase the deflection from equilibrium. This is a
"destabilizing" or "non-restoring" force since it tends to
push the system away from the equilibrium state. On the other hand, the elastic
properties of the materials from which the cavity is made cause
"restoring" forces that always tend to move the deflected wall back to
the undeflected equilibrium position. Thus the Casimir force and the elastic
restoring force always oppose each other. If they were always exactly equal,
then the wall would simply retain its present deflection, whatever that might
be. If the Casimir force were greater, then the deflection would become
unstable, and possibly break the cavity wall. If the restoring force were
bigger, the membrane would undergo stable oscillations with a frequency that
would depend on the magnitude of the Casimir force. The ratio of the sides
determines the magnitude of the Casimir force. Therefore cavities with different
ratios of sides should have different frequencies of oscillation.
During the oscillation, energy from the vacuum is changed to potential energy
of the deflected membrane and to the kinetic energy of the oscillating membrane.
The total of these three energies is constant during the motion of the membrane.
This exchange of energy can be used to indirectly measure the Casimir force on
the moving wall of this cavity. Note that there is no net transfer
of energy from the vacuum to mechanical motion in this vibrating cavity.
It is about the simplest machine we can make, and one of the first micro-machine
designed to explore the link between our world and the world of vacuum
fluctuations, with its enormous energy density. In the next decade or two, we
hope to learn a lot more about vacuum energy, and how to control it, and make
use of vacuum energy in selected applications.
In any real physical system of this type, there are dissipative forces that
will be present that cause the total mechanical energy to slowly decrease, and
become converted into heat. There will be forces due to the residual air in the
vacuum chamber in which the oscillator is placed, due to the resistance of the
thermal radiation present, due to imperfections in the crystal lattice from
which the oscillator is made etc…All these dissipative forces are expected to
slowly damp the amplitude of oscillation, eventually stopping the oscillations
altogether.
Copyright - Quantum Fields LLC 2000
Last Update 01/23/01
