Casimir Effect and

Fluctuation-Induced Phenomena:An Overview

Ali Naji (DAMTP, University of Cambridge)

25 July 2010, IPM Workshop

fluctuations

fluctuations

mesoscopic scale “Bio”membranes, colloids and polymers

(~10nm- 1 micron)classical fields (water waves, acoustic waves, etc)

cosmological scale

quantum vacuum

atoms & molecules(0.1nm-1nm) nano structures

(~1nm-10nm)

fluctuations vs “boundary conditions”,

“inhomogeneity”

inclusions or proteins in biological membranes

(M.M. Mueller, Dissertation, 2007)

ships in ocean

mirrors in vacuum

atoms & molecules

carbon nanotubes

5.1 Forces and torques on particles

4

1

2

3

l tx

yz

Figure 5.2: Two identical particles bound to an interface. As described in the text,it is possible to deform the contour of integration in order to exploit the availablesymmetries (see also Ref. [Mul04]).

We will discuss two possible symmetries: mirror symmetry in the (y, z) plane (thesymmetric case) or a twofold rotational symmetry with respect to the y axis (theantisymmetric case). The line joining corresponding points on the particles liesparallel to the (x, z) plane. We place the origin of the coordinate system in themiddle between the two particles on the intersection line of the asymptotic andsymmetry plane (symmetric case) or the line of symmetry (antisymmetric case),respectively.In the previous section we have seen that external horizontal torques can be bal-anced by the outer boundary. This is necessary for a stable configuration: weconsider configurations where the separation between the particles is fixed by hor-izontal constraining forces. In the antisymmetric case, the two particles do notgenerally lie on the same line of action. Thus the forces will apply a horizontaltorque My to the surface which has to be balanced by the outer boundary. Inthe following, we will consider situations in which this torque is the only externaltorque on the entire surface.This restriction does not exclude external vertical torques M (i) on the individualparticles ; the symmetry will not be broken as long as all these torques cancel.Think, for instance, of a symmetric configuration consisting of two spheres on asoap film with a saddle-shaped (quadrupolar) line of contact [SDJ00, FG02]. Avertical torque Mz on one of the particles is consistent with the symmetry so longas a torque !Mz is applied to its partner. Such possibilities may, in principle,be accommodated within the formalism. Here, however, we will only considersituations in which the particle orientations have equilibrated and these torquesvanish.Before considering torques more closely, let us first derive analytical expressionsfor the pair force.

113

“Casimir effect”

electromagnetic fluctuations

The Quantum Vacuum

- Consider metal box at room temperature

Removing particles and radiation leaves quantum vacuum fluctuations - can not be removed any further !!

- Some actual vacuums:Earth atmosphere 250 km above surface 10-7 mbar, 109 molecules/cm3

Large Electron–Positron Collider 10-9 mbar, 107molecules/cm3

concept of zero-point energy!

Quantum field theory treats each mode of the electromagnetic (EM) field as a quantum harmonic oscillator. Thus for a monochromatic EM wave, we get the corresponding energy spectrum as:

Electromagnetic (EM) zero-point energy

finite zero-point energy!

En = (n + 1/2)!!

E0 = !!/2

- Summing over all possible modes gives an infinite quantity!

- zero-point energy can not be observed by measurements within the quantum system, and theoretically, it is considered to define the reference of the energy.

- one can expect that “differences” in zero-point energy could be observed!

Imposing “boundary conditions”glass plates

(nearly transparent)

metal plates

only certain wave-lengths (colors) are allowed

attractive force from “nothing”!

Casimir interaction energy:

d

k = !n/d

H. B. G. Casimir , Proc. Nederl. Akad. Wetensch. 60, 793 (1948).

Poutside > Pinside

Casimir force can be extremely large!

But is this force real?

weight of water droplet 0.5 mm in diameter.

Surface area = 1 cm^2

distance = 1 micron Pressure = 0.001 Padistance = 10 nm Pressure = 100 KPa = 1 atm !

Casimir Force experiments- First attempt to measure the Casimir force: 1958 by M.J.Sparnaay but had systematic errors (100% uncertainty!)

(Sparnaay, 1958) Physica, 24 751-764

- First “successful” attempt nearly 50 years later: 1996 by Steven Lamoreaux (agreement with the theory to within uncertainty of 5%). Phys. Rev. Lett. 78, 5–8 (1997)

Several other successful experiments since.

Range (µm) Precision (%)

Van Blokland and Overbeek (1978)

0.13-0.67 25 at small distances50 average

Lamoreaux (1997) 0.6-6 5 at very small distance, larger elsewhere

Mohideen et al (1998) 0.1-0.8 1

Chan et al (2001) 0.075-2.2 1 and more....

‣ There has been a few dozen published experimental measurements of the Casimir force but hundreds of theoretical papers

‣ Citations of Casimir’s 1948 paper:

otherfluctuation-induced forces

vs“Casimir force”

Casimir forces

- ideal metals- “universal” (no material properties)

van der Waals forces

- real materials!- “non-universal”

Lifshitz forces

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6 VAN DER WAALS FORCES / PR.1. THE DANCE OF THE CHARGES

1. Keesom interactions of permanent dipoles whose mutual angles are, on average, inattractive orientations:

! CKeesom

r 6

Dipole–dipole electrostatic interaction perturbs the randomness of orientation. Ifthe dipole on the left is pointing “up,” then there is a slightly greater chance that thedipole on the right will point “down” (or vice versa; both particles are equivalent inmutual perturbation). By increasing the chances of an attractive mutual orientation,the perturbation creates a net-attractive interaction energy.

2. Debye interactions in which a permanent dipole induces a dipole in another non-polar molecule, with the induction necessarily in an attractive direction:

!CDebye

r 6

The relatively sluggish permanent dipole polarizes the relatively frisky electrons onthe nonpolar molecule and induces a dipole of opposite orientation. The directionof the induced dipole is such as to create attraction.

3. London dispersion interactions between transient dipoles of nonpolar but polariz-able bodies:

! CLondon

r 6

Here, the electrons on each molecule create transient dipoles. They couple the direc-tions of their dipoles to lower mutual energy. “Dispersion” recognizes that naturalfrequencies of resonance, necessary for the dipoles to dance in step, have the samephysical cause as that of the absorption spectrum—the wavelength-dependent dragon light that underlies the dispersion of white light into the spectrum of a rainbow.

There is an easy way to remember why these interaction free energies go as aninverse-sixth power. Think of the interaction between a “first” dipole pointing in aparticular direction and a “second” dipole that has been oriented or induced by the1/r 3 electric field of the first. The degree of its orientation or induction, favorable forattraction, is proportional to the strength of the orienting or inducing electric field.The oriented or induced part of the second dipole then interacts back with the first.Because the interaction energy of two dipoles goes as 1/r 3, we have

1/r 3 (for induction or orientation force)

" 1/r 3 (for interaction between the two dipoles) = 1/r 6.

This is not an explanation of the inverse-sixth-power energy in gases; it is only amnemonic.

In quantum mechanics, we think of each atom or molecule as having its own wavefunctions that describe the distribution of its electrons. The expected basis of interactionis that two atoms or molecules react to each other as dipoles, each atom’s or molecule’sdipolar electric field shining out as 1/r 3 with distance r from the center. This dipoleinteraction averages to zero when taken over the set of electron positions predictedfor the isolated atoms. However, when the isolated-atom electron distributions are

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6 VAN DER WAALS FORCES / PR.1. THE DANCE OF THE CHARGES

1. Keesom interactions of permanent dipoles whose mutual angles are, on average, inattractive orientations:

− CKeesom

r 6

Dipole–dipole electrostatic interaction perturbs the randomness of orientation. Ifthe dipole on the left is pointing “up,” then there is a slightly greater chance that thedipole on the right will point “down” (or vice versa; both particles are equivalent inmutual perturbation). By increasing the chances of an attractive mutual orientation,the perturbation creates a net-attractive interaction energy.

2. Debye interactions in which a permanent dipole induces a dipole in another non-polar molecule, with the induction necessarily in an attractive direction:

−CDebye

r 6

The relatively sluggish permanent dipole polarizes the relatively frisky electrons onthe nonpolar molecule and induces a dipole of opposite orientation. The directionof the induced dipole is such as to create attraction.

3. London dispersion interactions between transient dipoles of nonpolar but polariz-able bodies:

− CLondon

r 6

Here, the electrons on each molecule create transient dipoles. They couple the direc-tions of their dipoles to lower mutual energy. “Dispersion” recognizes that naturalfrequencies of resonance, necessary for the dipoles to dance in step, have the samephysical cause as that of the absorption spectrum—the wavelength-dependent dragon light that underlies the dispersion of white light into the spectrum of a rainbow.

There is an easy way to remember why these interaction free energies go as aninverse-sixth power. Think of the interaction between a “first” dipole pointing in aparticular direction and a “second” dipole that has been oriented or induced by the1/r 3 electric field of the first. The degree of its orientation or induction, favorable forattraction, is proportional to the strength of the orienting or inducing electric field.The oriented or induced part of the second dipole then interacts back with the first.Because the interaction energy of two dipoles goes as 1/r 3, we have

1/r 3 (for induction or orientation force)

× 1/r 3 (for interaction between the two dipoles) = 1/r 6.

This is not an explanation of the inverse-sixth-power energy in gases; it is only amnemonic.

In quantum mechanics, we think of each atom or molecule as having its own wavefunctions that describe the distribution of its electrons. The expected basis of interactionis that two atoms or molecules react to each other as dipoles, each atom’s or molecule’sdipolar electric field shining out as 1/r 3 with distance r from the center. This dipoleinteraction averages to zero when taken over the set of electron positions predictedfor the isolated atoms. However, when the isolated-atom electron distributions are

Verwey & Overbeek(1948)

Deryaguin & Landau(1941)

dispersion forces

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PAIRWISE SUMMATION, LESSONS LEARNED FROM GASES 7

themselves perturbed by each other’s dipolar fields, “second-order perturbation” in theparlance, the resulting position-averaged mutual perturbation makes the extra energygo as 1/r 6 at separations r much greater than dipole size.

Pairwise summation, lessons learned from gases appliedto solids and liquidsFor practical and fundamental reasons, there was a need to learn about the interactionsof bodies much larger than the atoms and small molecules in gases. What interestedpeople were systems we now call mesoscopic, with particles whose finite size WilhelmOstwald famously termed “the neglected dimension”: 100-nm to 100-µm colloids sus-pended in solutions, submicrometer aerosols sprayed into air, surfaces and interfacesbetween condensed phases, films of nanometer to millimeter thickness. What to do?

In 1937 H. C. Hamaker,5,6 following the work of Bradley, DeBoer, and others inthe Dutch school, published an influential paper investigating the properties of vander Waals interactions between large bodies, as distinct from the small-molecule inter-actions that had been considered previously. Hamaker used the pairwise-summationapproximation. The idea of this approximation was to imagine that incremental partsof large bodies could interact by !C/r 6 energies as though the remaining material wereabsent. The influence of the intermediate material was included as the electromagneticequivalent of Archimedean buoyancy.

De Boer had shown that, summed over the volumes of two parallel planar blockswhose separation l was smaller than their depth and lateral extent, the !C/r 6 energybecame an energy per area that varied as the inverse square of the separation l (1/ l2 forsmall l):

l

6

Cr

!

In the Hamaker summation over the volumes of two spheres, the interaction energyapproaches inverse-first-power variation near contact (1/ l when l " R) and reverts tothe expected inverse-sixth-power dependence of point particles when the spheres arewidely separated compared with their size (1/r 6 when r # R):

R Rl

r

The newly recognized possibility that van der Waals forces could be of much longerrange than the 1/r 6 reach previously expected from Keesom, Debye, and London forces

colloidal and bio-matter“softness”

E. M. Lifshitz, Dokl. Akad. Nauk. SSSR, 97: 643 (1954); 100: 879 (1955); Zh. Eksp.Teor. Fiz., 29:94 (1955)

Lifshitz theory

1961 classic!

Some fun phenomena....

K. Autumn, W.-P. Chang, R. Fearing, T. Hsieh, T. Kenny, L. Liang, W. Zesch, R.J. Full. Nature 2000.

van der Waals interactions!

How does Gecko manage to walk on vertical smooth walls?

TextText

A single seta can lift the weight of an ant 200 µN = 20 mg. A million setae (1 square cm) could lift the weight of a child (20kg, 45lbs). Maximum potential force of 2,000,000 setae on 4 feet of a gecko = 2,000,000 x 200 micronewton = 400 newton = 40788 grams force, or about 90 lbs! Weight of a Tokay gecko is approx. 50 to

150 grams.

- The Casimir force will affect the operation of any nano-scale mechanical device

“Stiction”

9

FIG. 5: Contributions of specific reflections to the optical approximation.

C. Casimir Pendulum

In this section we treat a problem for which the exact answer is unknown. The configuration is shown in Fig. 6. Thebase plate is taken to be infinite. The upper plate is held at its midpoint a distance a above the base plate. The widthof the upper plate is w and its depth, d (out of the page), is assumed to be infinite. We define the Casimir energyper unit depth, ! = E/d. " is the angle of inclination of the upper plate. It will be convenient to use z = 1

2w sin "as a variable as well. It is also possible to view this configuration as a finite slice between #1 = a/ sin " ! w/2 and#2 = a/ sin " + w/2 of a wedge of opening angle ". In this section we will discuss both the Casimir energy and the“Casimir torque”, $ = 1

ddEdθ , per unit depth. We are aware of two ad hoc approximate approaches to this problem. The

FIG. 6: Geometry for the Casimir Pendulum.

first is the PFA which treats each element of the system perpendicular to the lower plate as an infinitesmal parallelplate Casimir system. It is easy to show that

!PFA = !%2!c

1440

w cos "

a3

!

1 ! w2 sin2 "

4a2

"!2

= !%2!ca

1440

"w2 ! 4z2

(a2 ! z2)2(II.22)

which gives a torque,

$PFA(a, w, z) = !%2!c

720

az(w2 ! a2 ! 3z2)

(a2 ! z2)3(II.23)

where the minus sign denotes that the torque is destabilizing: z = 0 is a point of unstable equilibrium. As in the caseof the sphere and the plane, the PFA is ambiguous. A more symmetric treatment of the two planes in the presentgeometry would integrate over the surface that bisects the wedge and take the distance normal to that surface. Theresult is the replacement of cos " by cos4("/2) in eq. (II.22) and a similar modification of the torque. A second“approximate” treatment of the Casimir Pendulum can be extracted from the known exact solution for the Casimir

- Repulsive casimir-vdW forces in vacuum are possible for (meta)materials with specific dielectric properties

- Designing micro- and nano-(electro)mechanical systems

Casimir pendulum Casimir anharmonic oscillatorRack and pinion

Chan et al, PRL 87, 211801 (2001)Scardicchio et al, Nucl. Phys. B 704, 552 (2005)Ashourvan et al, PRL 98, 140801 (2007)

“Casimir-like”(non-electromagnetic)

fluctuation-induced phenomena

(Goulian, Bruinsma, Pincus 1993)

• Charged soft Matter: Plasma fluctuationsCharged colloids

counterions

co-ions

+

-

-

Charged polymers(Polyelectrolytes)

+-- --

Charged membranes

- -

-- -- - -

++

++

++ ++

+

--

-- -- - -(Podgornik and Zeks, 1989)

• Liquid crystals

• Membrane fluctuations

A. Larraza and B. Denardo 1998.

Acoustic “Casimir effect”

Results depend on the nature of the noise spectrum.

This effect is not driven by thermal fluctuations but by the artificially generated accoustic noise.

Non-monotonic interactions!

A water wave analog of the Casimir effectBruce C. Denardo,a! Joshua J. Puda, and Andrés LarrazaDepartment of Physics, Naval Postgraduate School, Monterey, California 93943

!Received 28 October 2008; accepted 3 August 2009"

Two rigid plates are vertically suspended by thread such that they are parallel to and opposite eachother. The plates are partially submerged in a dish of liquid that is attached to the top of a verticalshake table. When the shake table is driven with noise in a frequency band, random surface wavesare parametrically excited, and the plates move toward each other. The reason for this attraction isthat the waves carry momentum, and the wave motion between the plates is visibly reduced. Thebehavior is analogous to the Casimir effect, in which two conducting uncharged parallel platesattract each other due to the zero-point spectrum of electromagnetic radiation. The water waveanalog can be readily demonstrated and offers a visual demonstration of a Casimir-type effect.Measurements of the force agree with the water wave theory even at large wave amplitudes, wherethe theory is expected to break down. The water wave analog applies to side-by-side ships in a roughsea and is distinct from the significant attraction that can be caused by a strong swell.#DOI: 10.1119/1.3211416$

I. INTRODUCTION

In the Casimir effect two conducting uncharged parallelplates attract each other due to the ground state or “zero-point” spectrum of electromagnetic radiation at zero absolutetemperature.1 The effect can be understood and calculated asan imbalance in the radiation force on the inside and theoutside surfaces of the plates.2 A radiation force on a body isthe time-averaged force due to waves that are incident on thebody. The force arises due to the momentum of the wavesand is proportional to the energy density. For the electromag-netic zero-point spectrum the ground state energy of !" /2for each normal mode leads to the classical spectral energydensity !energy per unit frequency per unit volume"!"3 /2#2c3 in empty space, where ! is the reduced Planck’sconstant, " is the angular frequency, and c is the speed oflight. Although this energy cannot be directly observed, thepresence of the plates discretizes the spectrum between andtransverse to the plates, which causes the imbalance of theradiation force. The energy density !the integral of the spec-tral energy density over frequency" is infinite, and thus theforce on either side of a plate is infinite. The use of a regu-larization procedure2,3 yields the net attractive force per unitarea of #2!c /240d4, where d is the distance between theplates.

The Casimir effect is not restricted to photons but is ex-pected to occur for any waves that carry momentum. Forexample, two rigid conducting uncharged parallel plates thatare vertically and partially submerged in liquid helium areexpected to attract each other not only due to zero-point pho-tons but also due to zero-point phonons !acoustic excitations"and zero-point ripplons !liquid surface wave excitations". Toour knowledge, this effect has not yet been observed.

AnalogCasimir effects are similarly not restricted. By analog wemean that the force between bodies arises from driven wavesrather than a zero-point spectrum. Analog Casimir effectshave been investigated for two side-by-side ships in a strongswell !long-wavelength waves",4 acoustic waves,5,6 twobeads on a vibrating string,7 and other systems.8 As in theCasimir effect, the behavior in these driven systems can beunderstood and calculated as an imbalance in the radiationforce.

In this paper we investigate an analog Casimir effect inwhich two rigid parallel plates are vertically suspended andpartially submerged in a dish of liquid !Fig. 1". The dish isattached to the top of a vertical shake table that is drivenwith noise in a finite band of frequencies. Random surfacewaves are parametrically excited, and the plates are observedto move toward each other as a result of the waves. Thelongest wavelength in the spectral band is sufficiently smallcompared to the size of the dish so that the waves are ap-proximately homogeneous and isotropic outside the plates!see Fig. 1". We are primarily interested in the simple case inwhich the plate separation distance is sufficiently small sothat the wave motion is negligible between and transverse tothe plates, as shown in Fig. 1, which can occur because asmallest wavelength exists in the dish. This situation yieldsan attractive force that is independent of the distance and isproportional to the energy density and thus the mean-squareamplitude of the waves.

The analogy of our water wave system to the Casimireffect is not exact. Because the water waves are driven, theenergy density of the spectrum is not infinite, so a regular-ization procedure is not needed. Furthermore, we are prima-rily concerned with the case of closely spaced plates, whichyields a force that is independent of the separation distanced. This behavior is in contrast to the Casimir force, whichhas a 1 /d4 dependence due to the divergence of the "3 spec-trum at high frequencies.

An example of an analog Casimir system that is similar, inprinciple, to our water wave system is two conducting paral-lel plates in a microwave cavity, where the microwave radia-tion is driven in a finite band of frequencies. The maximumwavelength should be small compared to the size of the cav-ity so that the electromagnetic field is approximately homo-geneous and isotropic. As in the water wave case, a suffi-ciently small plate separation distance yields an attractiveforce that is independent of the distance and is proportionalto the mean-square amplitude of the waves. The microwaveforce can be easily estimated. The radiation pressure due tohomogeneous and isotropic electromagnetic waves incidentupon a perfectly reflecting surface is I /3c,9 where I is theaverage intensity $0Erms

2 , $0 is the vacuum permittivity, andErms is the root-mean-square electric field. The attractiveforce per unit area for closely spaced plates with a negligible

1095 1095Am. J. Phys. 77 !12", December 2009 http://aapt.org/ajp

A water wave analog of the Casimir effectBruce C. Denardo,a! Joshua J. Puda, and Andrés LarrazaDepartment of Physics, Naval Postgraduate School, Monterey, California 93943

!Received 28 October 2008; accepted 3 August 2009"

Two rigid plates are vertically suspended by thread such that they are parallel to and opposite eachother. The plates are partially submerged in a dish of liquid that is attached to the top of a verticalshake table. When the shake table is driven with noise in a frequency band, random surface wavesare parametrically excited, and the plates move toward each other. The reason for this attraction isthat the waves carry momentum, and the wave motion between the plates is visibly reduced. Thebehavior is analogous to the Casimir effect, in which two conducting uncharged parallel platesattract each other due to the zero-point spectrum of electromagnetic radiation. The water waveanalog can be readily demonstrated and offers a visual demonstration of a Casimir-type effect.Measurements of the force agree with the water wave theory even at large wave amplitudes, wherethe theory is expected to break down. The water wave analog applies to side-by-side ships in a roughsea and is distinct from the significant attraction that can be caused by a strong swell.#DOI: 10.1119/1.3211416$

I. INTRODUCTION

In the Casimir effect two conducting uncharged parallelplates attract each other due to the ground state or “zero-point” spectrum of electromagnetic radiation at zero absolutetemperature.1 The effect can be understood and calculated asan imbalance in the radiation force on the inside and theoutside surfaces of the plates.2 A radiation force on a body isthe time-averaged force due to waves that are incident on thebody. The force arises due to the momentum of the wavesand is proportional to the energy density. For the electromag-netic zero-point spectrum the ground state energy of !" /2for each normal mode leads to the classical spectral energydensity !energy per unit frequency per unit volume"!"3 /2#2c3 in empty space, where ! is the reduced Planck’sconstant, " is the angular frequency, and c is the speed oflight. Although this energy cannot be directly observed, thepresence of the plates discretizes the spectrum between andtransverse to the plates, which causes the imbalance of theradiation force. The energy density !the integral of the spec-tral energy density over frequency" is infinite, and thus theforce on either side of a plate is infinite. The use of a regu-larization procedure2,3 yields the net attractive force per unitarea of #2!c /240d4, where d is the distance between theplates.

The Casimir effect is not restricted to photons but is ex-pected to occur for any waves that carry momentum. Forexample, two rigid conducting uncharged parallel plates thatare vertically and partially submerged in liquid helium areexpected to attract each other not only due to zero-point pho-tons but also due to zero-point phonons !acoustic excitations"and zero-point ripplons !liquid surface wave excitations". Toour knowledge, this effect has not yet been observed.

AnalogCasimir effects are similarly not restricted. By analog wemean that the force between bodies arises from driven wavesrather than a zero-point spectrum. Analog Casimir effectshave been investigated for two side-by-side ships in a strongswell !long-wavelength waves",4 acoustic waves,5,6 twobeads on a vibrating string,7 and other systems.8 As in theCasimir effect, the behavior in these driven systems can beunderstood and calculated as an imbalance in the radiationforce.

In this paper we investigate an analog Casimir effect inwhich two rigid parallel plates are vertically suspended andpartially submerged in a dish of liquid !Fig. 1". The dish isattached to the top of a vertical shake table that is drivenwith noise in a finite band of frequencies. Random surfacewaves are parametrically excited, and the plates are observedto move toward each other as a result of the waves. Thelongest wavelength in the spectral band is sufficiently smallcompared to the size of the dish so that the waves are ap-proximately homogeneous and isotropic outside the plates!see Fig. 1". We are primarily interested in the simple case inwhich the plate separation distance is sufficiently small sothat the wave motion is negligible between and transverse tothe plates, as shown in Fig. 1, which can occur because asmallest wavelength exists in the dish. This situation yieldsan attractive force that is independent of the distance and isproportional to the energy density and thus the mean-squareamplitude of the waves.

The analogy of our water wave system to the Casimireffect is not exact. Because the water waves are driven, theenergy density of the spectrum is not infinite, so a regular-ization procedure is not needed. Furthermore, we are prima-rily concerned with the case of closely spaced plates, whichyields a force that is independent of the separation distanced. This behavior is in contrast to the Casimir force, whichhas a 1 /d4 dependence due to the divergence of the "3 spec-trum at high frequencies.

An example of an analog Casimir system that is similar, inprinciple, to our water wave system is two conducting paral-lel plates in a microwave cavity, where the microwave radia-tion is driven in a finite band of frequencies. The maximumwavelength should be small compared to the size of the cav-ity so that the electromagnetic field is approximately homo-geneous and isotropic. As in the water wave case, a suffi-ciently small plate separation distance yields an attractiveforce that is independent of the distance and is proportionalto the mean-square amplitude of the waves. The microwaveforce can be easily estimated. The radiation pressure due tohomogeneous and isotropic electromagnetic waves incidentupon a perfectly reflecting surface is I /3c,9 where I is theaverage intensity $0Erms

2 , $0 is the vacuum permittivity, andErms is the root-mean-square electric field. The attractiveforce per unit area for closely spaced plates with a negligible

1095 1095Am. J. Phys. 77 !12", December 2009 http://aapt.org/ajp

field between the plates is thus !0Erms2 /3c. The electric field

due to a magnetron in a microwave oven is limited by thedielectric breakdown strength of air, which is 3"106 V /m.For two 10"10 cm2 closely spaced plates, this field yields aforce that has a mass equivalent of the order of 1 #g, whichis four orders of magnitude smaller than the order-of-magnitude value of 10 mg in our water wave experiment. Toour knowledge, the microwave analog Casimir effect has notyet been observed.

Our water wave analog has application to two side-by-sideships in a rough sea and is distinct from the effect due to aswell.4 Due to its long wavelength, the swell causes the shipsto roll side-to-side in phase. This motion generates secondarywaves outside the ships. Between the ships, however, the

waves are 180° out-of-phase and thus tend to cancel. A wavepropulsion effect consequently occurs in which either shipdrives itself toward the other due to the emission of waves.As shown in Ref. 4, this attraction can be significant.

Casimir forces can be repulsive in zero-point10 and analogcases. When the band of waves has a nonzero lower fre-quency limit, which often occurs in analog Casimir effects, arepulsive force can occur between two parallel plates forsome separation distances. This force has been calculatedand observed in the acoustics case.5

Our water wave system offers a visual demonstration of ananalog Casimir effect. All three aspects are directly ob-served: The generation of waves, a substantial reduction inwave motion between the plates, and the attraction of theplates. We describe the demonstration in Sec. II. Gravity-capillary waves are discussed in Sec. III, and the theory ofthe radiation force due to random gravity waves is developedin Sec. IV, where we also estimate the attractive force be-tween two side-by-side ships in a rough sea. The theory as-sumes linear !sinusoidal" waves, whereas the waves in thedemonstration can be noticeably nonlinear, so the validity ofthe theory is questionable in this case. In addition, othereffects could be playing a role in the observed attraction.Accordingly, we describe a quantitative experiment in Secs.V and VI and compare the data to the theory. Concludingremarks are made in Sec. VII.

II. DEMONSTRATION

It is not difficult to produce a demonstration of the waterwave analog of the Casimir effect !Fig. 1". We have arrangedthe entire apparatus to be on a table that was rolled into aclassroom. We use 9.5 cm square acrylic or PVC plates witha thickness of 1.6 mm. Each plate is suspended with two 80cm lengths of thread whose upper ends are secured byclamps. As shown in Fig. 1!a", opposing pendulum clamps11

are very convenient for the purpose of the suspension. Theclamps are attached to a common horizontal support rod thatis held by two vertical rods which are clamped to the table.The plates are separated by approximately 1.7 cm in equilib-rium !in the absence of wave motion".

We use a glass dish with diameter of 19 cm and height of10 cm as a container.12 The dish is filled to a depth of 7 cmwith ethyl alcohol or water, and a small amount of the fluo-rescein dye is added so that the liquid is easily observable.The dish rests on a vertical shake table that produces waves.We use a commercial shaker,13 but a loudspeaker can beadapted for this purpose.14 We attach the dish to the shakerbecause the acceleration amplitude of the shaker can exceedthe acceleration due to gravity for the generation of higher-amplitude waves. As shown in Fig. 1!b", the plates are par-tially but deeply submerged !approximately 2/3 of a plate".The deep placement serves two purposes: To prevent thewave motion from passing under the plates and into the re-gion between the plates and to increase the viscous dampingto reduce the motion of the plates due to fluctuations of therandom waves.

The shaker causes the effective acceleration due to gravityto be modulated in the frame of reference of the dish, whichparametrically excites surface waves. A similar excitationcan occur for a pendulum whose pivot is verticallyoscillated.15 To produce random waves, we drive the shakerwith noise from a function generator that is bandpassfiltered16 in the octave band of 10–20 Hz and then power

(b)

(a)

(c)

Fig. 1. Apparatus for demonstrating a water wave analog of the Casimireffect. !a" Support arrangement for the plates with thread and pendulumclamps. The thread length is 80 cm. !b" Parallel plates partially but deeplysubmerged in ethyl alcohol with the fluorescein dye. The dish has the diam-eter of 19 cm and is attached to a vertical shake table. !c" Shaker produceswaves that cause a substantial attraction of the plates. There is very littlewave motion between the plates.

1096 1096Am. J. Phys., Vol. 77, No. 12, December 2009 Denardo, Puda, and Larraza

Two rigid plates are vertically suspended by thread such that they are parallel to and opposite each other. The plates are partially submerged in a dish of liquid that is attached to the top of a vertical shake table. When the shake table is driven with noise in a frequency band, random surface waves are parametrically excited, and the plates move toward each other. The reason for this attraction is that the waves carry momentum, and the wave motion between the plates is visibly reduced. The behavior is analogous to the Casimir effect, in which two conducting uncharged parallel plates attract each other due to the zero-point spectrum of electromagnetic radiation. The water wave analog can be readily demonstrated and offers a visual demonstration of a Casimir-type effect. Measurements of the force agree with the water wave theory even at large wave amplitudes, where the theory is expected to break down. The water wave analog applies to side-by-side ships in a rough sea and is distinct from the significant attraction that can be caused by a strong swell.

In 1996 by Dutch scientist Sipko Boersma (A maritime analogy of the Casimir effect Am.J.Phys. 64. 539-541 (1996)) dug up the French nautical writer P. C. Caussée and his 1836 book The Album of the Mariner that two ships should not be moored too close together because they are attracted one towards the other by a certain force of attraction. Boersma suggested that this early observation could be described by a phenomenon analogous to the Casimir effect.

“Une force certaine d’attraction“P.C. Causee: L'Album du Marin, (Mantes, Charpentier, 1836)

In the age of great sailboats it was noted that at certain conditions of the sea the ships attract mysteriously, leading often to major damage.

The ships are free floating, not moored. I was told that the effect was also reported in the world literature: Herman Melville's "Moby Dick" and Philip Roth's "Rites of passage". It is not a myth, the original paper

Am.J.Phys. 64. 539-541 (1996) gives the quantitative theory to calculate the attractive force, given Ships rolling amplitude, weight, metacentric height, "Q" oscillator quality factor and wave period. An example for two 700

ton clipper ships gives 2000 Newton, quite reasonable. The theory gives also another effect: Repulsion. An atom is attracted to a conducting plate but a ship in a wave field is repelled from a steep cliff. This is due to a

difference in boundary conditions between Electromagnetic waves and Seawaves. This repulsion was already known to the Cape Horn sailors of the Cape Horn Society, Hoorn Holland. Caussé's error: Caussé put his ships

in a "Flat Calm" without any waves. That won't work. However, already a small swell suffices if its period matches the natural period of the ships and we have resonance magnification. A long light swell can easily have

been overlooked by the mariners on board. The second possibility is that Caussé should have put his ships In "Calm with Big Swell" which after all to me seems less likely.

S.L.Boersma Delft The Netherlands

Fabrizio Pinto thinks that the whole tale is symptomatic of physicists' approach to the history of their subject. "Physicists love lore about their own science," he says. "There are other stories that

are unfounded historically." (Nature, 4 may 2006).You may read about this in Nature blog.

http://blogs.nature.com/news/blog/2006/05/popular_physics_myth_is_all_at.html

some fundamental questions remain....

Cosmological Implications of zero-point energy

‣ In the presence of “gravity”, the vacuum energy can not be discarded simply as the reference of the energy! All energy gravitates and thus the energy density of the vacuum enter Einstein’s equation

‣ Vacuum energy is “infinite”!‣ Cosmological constant measured to be very

small! (accounting for the observation that the expansion of the universe is accelerating)

‣ The problem of cosmological constant and the “dark energy”.... could vacuum energy be this dark energy...?!

Casimir effect and the quantum vacuum

R. L. JaffeCenter for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA(Received 3 May 2005; published 12 July 2005)

In discussions of the cosmological constant, the Casimir effect is often invoked as decisive evidencethat the zero-point energies of quantum fields are ‘‘real.’’ On the contrary, Casimir effects can beformulated and Casimir forces can be computed without reference to zero-point energies. They arerelativistic, quantum forces between charges and currents. The Casimir force (per unit area) betweenparallel plates vanishes as !, the fine structure constant, goes to zero, and the standard result, whichappears to be independent of !, corresponds to the ! ! 1 limit.

DOI: 10.1103/PhysRevD.72.021301 PACS numbers: 98.80.Es, 11.10.2z

I. INTRODUCTION

In quantum field theory as usually formulated, the zero-point fluctuations of the fields contribute to the energy ofthe vacuum. However this energy does not seem to beobservable in any laboratory experiment. Nevertheless,all energy gravitates, and therefore the energy density ofthe vacuum, or more precisely the vacuum value of thestress tensor, hT"#i ! "Eg"# [1], appears on the right-hand side of Einstein’s equations,

R"# "1

2g"#R # "8$G$ ~T"# " Eg"#% (1)

where it affects cosmology. ( ~T"# is the contribution ofexcitations above the vacuum.) It is equivalent to addinga cosmological term, % # 8$GE, on the left-hand side.

A small, positive cosmological term is now required toaccount for the observation that the expansion of theUniverse is accelerating. Recent measurements give [3]

% # $2:14& 0:13' 10"3 eV%4 (2)

at the present epoch. This observation has renewed interestin the idea that the zero-point fluctuations of quantumfields contribute to the cosmological constant, % [4].However, estimates of the energy density due to zero-pointfluctuations exceed the measured value of % by manyorders of magnitude. Caution is appropriate when an effect,for which there is no direct experimental evidence, is thesource of a huge discrepancy between theory andexperiment.

As evidence of the ‘‘reality’’ of the quantum fluctuationsof fields in the vacuum, theorists often point to the Casimireffect [7]. For example, Weinberg, in his introduction tothe cosmological constant problem, writes [6], ‘‘Perhapssurprisingly, it was a long time before particle physicistsbegan seriously to worry about (quantum zero-point fluc-tuation contributions to %) despite the demonstration in theCasimir effect of the reality of zero-point energies.’’ Morerecent examples can be found in the widely read reviews byCarroll [8], ‘‘. . .And the vacuum fluctuations themselvesare very real, as evidenced by the Casimir effect,’’ and by

Sahni and Starobinsky [9,10] ‘‘The existence of zero-pointvacuum fluctuations has been spectacularly demonstratedby the Casimir effect.’’

In 1997 Lamoreaux opened the door to precise measure-ment of Casimir forces [11]. The Casimir force (per unitarea) between parallel conducting plates,

F # " @c$2

240d4(3)

has now been measured to about 1% precision. Casimirphysics has become an active area of nanoscopic physics inits own right [12]. Not surprisingly, every review and texton the subject highlights the supposed special connectionbetween the Casimir effect and the vacuum fluctuations ofthe electromagnetic field [13].

The object of this paper is to point out that the Casimireffect gives no more (or less) support for the reality of thevacuum energy of fluctuating quantum fields than anyother one-loop effect in quantum electrodynamics, likethe vacuum polarization contribution to the Lamb shift,for example. The Casimir force can be calculated withoutreference to vacuum fluctuations, and like all other observ-able effects in QED, it vanishes as the fine structure con-stant, !, goes to zero.

There is a long history and large literature surroundingthe question whether the zero-point fluctuations of quan-tized fields are ‘‘real’’ [14]. Schwinger, in particular, at-tempted to formulate QED without reference to zero-pointfluctuations [15]. In contrast Milonni has recently refor-mulated all of QED from the point of view of zero-pointfluctuations [14]. The question of whether zero-point fluc-tuations of the vacuum are or are not real is beyond thescope of this paper. Instead I address only the narrowerquestion of whether the Casimir effect can be consideredevidence in their favor.

For a noninteracting quantum field the vacuum (or zero-point) energy is given by E # & 1

2

P

@!0, where the f@!0gare the eigenvalues of the free Hamiltonian and the plus orminus sign applies to bosons or fermions, respectively. Inthree dimensions the sum over frequencies diverges quarti-

PHYSICAL REVIEW D 72, 021301(R) (2005)

RAPID COMMUNICATIONS

1550-7998=2005=72(2)=021301(5)$23.00 021301-1 ! 2005 The American Physical Society

But After all, is zero-point energy “real”?!Casimir effect and the quantum vacuum

R. L. JaffeCenter for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA(Received 3 May 2005; published 12 July 2005)

In discussions of the cosmological constant, the Casimir effect is often invoked as decisive evidencethat the zero-point energies of quantum fields are ‘‘real.’’ On the contrary, Casimir effects can beformulated and Casimir forces can be computed without reference to zero-point energies. They arerelativistic, quantum forces between charges and currents. The Casimir force (per unit area) betweenparallel plates vanishes as !, the fine structure constant, goes to zero, and the standard result, whichappears to be independent of !, corresponds to the ! ! 1 limit.

DOI: 10.1103/PhysRevD.72.021301 PACS numbers: 98.80.Es, 11.10.2z

I. INTRODUCTION

In quantum field theory as usually formulated, the zero-point fluctuations of the fields contribute to the energy ofthe vacuum. However this energy does not seem to beobservable in any laboratory experiment. Nevertheless,all energy gravitates, and therefore the energy density ofthe vacuum, or more precisely the vacuum value of thestress tensor, hT"#i ! "Eg"# [1], appears on the right-hand side of Einstein’s equations,

R"# "1

2g"#R # "8$G$ ~T"# " Eg"#% (1)

where it affects cosmology. ( ~T"# is the contribution ofexcitations above the vacuum.) It is equivalent to addinga cosmological term, % # 8$GE, on the left-hand side.

A small, positive cosmological term is now required toaccount for the observation that the expansion of theUniverse is accelerating. Recent measurements give [3]

% # $2:14& 0:13' 10"3 eV%4 (2)

at the present epoch. This observation has renewed interestin the idea that the zero-point fluctuations of quantumfields contribute to the cosmological constant, % [4].However, estimates of the energy density due to zero-pointfluctuations exceed the measured value of % by manyorders of magnitude. Caution is appropriate when an effect,for which there is no direct experimental evidence, is thesource of a huge discrepancy between theory andexperiment.

As evidence of the ‘‘reality’’ of the quantum fluctuationsof fields in the vacuum, theorists often point to the Casimireffect [7]. For example, Weinberg, in his introduction tothe cosmological constant problem, writes [6], ‘‘Perhapssurprisingly, it was a long time before particle physicistsbegan seriously to worry about (quantum zero-point fluc-tuation contributions to %) despite the demonstration in theCasimir effect of the reality of zero-point energies.’’ Morerecent examples can be found in the widely read reviews byCarroll [8], ‘‘. . .And the vacuum fluctuations themselvesare very real, as evidenced by the Casimir effect,’’ and by

Sahni and Starobinsky [9,10] ‘‘The existence of zero-pointvacuum fluctuations has been spectacularly demonstratedby the Casimir effect.’’

In 1997 Lamoreaux opened the door to precise measure-ment of Casimir forces [11]. The Casimir force (per unitarea) between parallel conducting plates,

F # " @c$2

240d4(3)

has now been measured to about 1% precision. Casimirphysics has become an active area of nanoscopic physics inits own right [12]. Not surprisingly, every review and texton the subject highlights the supposed special connectionbetween the Casimir effect and the vacuum fluctuations ofthe electromagnetic field [13].

The object of this paper is to point out that the Casimireffect gives no more (or less) support for the reality of thevacuum energy of fluctuating quantum fields than anyother one-loop effect in quantum electrodynamics, likethe vacuum polarization contribution to the Lamb shift,for example. The Casimir force can be calculated withoutreference to vacuum fluctuations, and like all other observ-able effects in QED, it vanishes as the fine structure con-stant, !, goes to zero.

There is a long history and large literature surroundingthe question whether the zero-point fluctuations of quan-tized fields are ‘‘real’’ [14]. Schwinger, in particular, at-tempted to formulate QED without reference to zero-pointfluctuations [15]. In contrast Milonni has recently refor-mulated all of QED from the point of view of zero-pointfluctuations [14]. The question of whether zero-point fluc-tuations of the vacuum are or are not real is beyond thescope of this paper. Instead I address only the narrowerquestion of whether the Casimir effect can be consideredevidence in their favor.

For a noninteracting quantum field the vacuum (or zero-point) energy is given by E # & 1

2

P

@!0, where the f@!0gare the eigenvalues of the free Hamiltonian and the plus orminus sign applies to bosons or fermions, respectively. Inthree dimensions the sum over frequencies diverges quarti-

PHYSICAL REVIEW D 72, 021301(R) (2005)

RAPID COMMUNICATIONS

1550-7998=2005=72(2)=021301(5)$23.00 021301-1 ! 2005 The American Physical Society

‣ Casimir effect can be formulated without any reference to zero-point energy only as forces between charges and currents (being thus simply a retarded relativistic vdW force)!! [Schwinger 1975]

‣ The effect vanishes as the “fine structure constant” goes to zero!‣ The standard “universal” result corresponds to

the universality of this force (independence from e etc) is an illusion!

‣ Boundaries were idealized as “perfect metal” - assumptions were made implicitly about properties of matter!!

‣ So far no known phenomenon demonstrates that zero-point energies are real BUT WORSE, Casimir effect is NO EVIDENCE of their reality either!!

!!" U = ! !cπ2

240d4

! = e2/!c

Back to square one....??