In physics, the Casimir effect is a physical force exerted between separate objects, which is due to neither charge, gravity, nor the exchange of particles, but instead is due to resonance of all-pervasive energy fields in the intervening space between the objects. This is sometimes described in terms of virtual particles interacting with the objects, due to the mathematical form of one possible way of calculating the strength of the effect. Since the strength of the force falls off rapidly with distance it is only measurable when the distance between the objects is extremely small. On a submicron scale, this force becomes so strong that it becomes the dominant force between uncharged conductors. Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space and time. ... In physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... Gravity is a force of attraction that acts between bodies that have mass. ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... The Tacoma Narrows Bridge (shown twisting) in Washington collapsed spectacularly, under moderate wind, in part because of resonance. ... In physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess; it is the energy of the ground state of the system. ... In physics, a virtual particle is a particle-like abstraction used in some models of quantum field theory. ...

Dutch physicist Hendrik B. G. Casimir first proposed the existence of the force, and he formulated an experiment to detect it in 1948 while participating in research at Philips Research Labs. The classic form of his experiment used a pair of uncharged parallel metal plates in a vacuum, and successfully demonstrated the force to within 15% of the value he had predicted according to his theory. Physicists working in a government lab A physicist is a scientist who studies or practices physics. ... Hendrik Brugt Gerhard Casimir (July 15, 1909 – May 4, 2000) was a Dutch physicist. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Philips HQ in Amsterdam Koninklijke Philips Electronics N.V. (Royal Philips Electronics N.V.), usually known as Philips, (Euronext: PHIA, NYSE: PHG) is one of the largest electronics companies in the world. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ...

The van der Waals force between a pair of neutral atoms is a similar effect. In modern theoretical physics, the Casimir effect plays an important role in the chiral bag model of the nucleon, and in applied physics, it is becoming of increasing importance in development of the ever-smaller, miniaturised components of emerging micro- and nano- technologies. In chemistry, the term van der Waals force originally referred to all forms of intermolecular forces; however, in modern usage it tends to refer to intermolecular forces that deal with forces due to the polarization of molecules. ... Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle of a chemical element that retains its chemical properties. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand Nature. ... In physics a nucleon is a collective name for two baryons: the neutron and the proton. ... In physics a nucleon is a collective name for two baryons: the neutron and the proton. ... Cutout of the ITER project Applied physics is physics that is intended for a particular technological or practical use, as for example in engineering, as opposed to basic research. ... Microtechnology is technology with features near one micrometre (one millionth of a metre, or 10-6 metre, or 1μm). ... Molecular gears from a NASA computer simulation. ...

Overview

The Casimir effect can be understood by the idea that the presence of conducting metals and dielectrics alter the vacuum expectation value of the energy of the electromagnetic field. Since the value of this energy depends on the shapes and positions of the conductors and dielectrics, the Casimir effect manifests itself as a force between such objects. A dielectric, or electrical insulator, is a substance that is highly resistant to electric current. ... In quantum field theory the vacuum expectation value (also called condensate) of an operator is its average, expected value in the vacuum. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ...

Vacuum energy

The Casimir effect is an outcome of quantum field theory, which states that all of the various fundamental fields, such as the electromagnetic field, must be quantized at each and every point in space. In a naïve sense, a field in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field can be visualized as the displacement of a ball from its rest position. Vibrations in this field propagate, and are governed by the appropriate wave equation for the particular field in question. The second quantization of quantum field theory requires that each such ball-spring combination be quantized, that is, that the strength of the field be quantized at each point in space. Canonically, the field at each point in space is a simple harmonic oscillator, and its quantization places a quantum harmonic oscillator at each point. Excitations of the field correspond to the elementary particles of particle physics. However, as this picture shows, even the vacuum has a vastly complex structure. All calculations of quantum field theory must be made in relation to this model of the vacuum. Quantum field theory (QFT) is the application of quantum mechanics to fields. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ... Second quantization refers to quantizing fields by expressing them as operator-valued distributions The most elementary, or semiclassical treatments of quantum mechanics fix the number of particles and treat the field classically, including it as a parameter in the Hamiltonian or Lagrangian or whatever. ... A harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement : where is a positive constant. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. ... Particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Look up Vacuum in Wiktionary, the free dictionary. ...

The vacuum has, implicitly, all of the properties that a particle may have: spin, or polarization in the case of light, energy, and so on. On average, all of these properties cancel out: the vacuum is after all, "empty" in this sense. One important exception is the vacuum energy or the vacuum expectation value of the energy. The quantization of a simple harmonic oscillator states that the lowest possible energy or zero-point energy that such an oscillator may have is In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye (visible light) or, in a technical or scientific context, electromagnetic radiation of any wavelength [citation needed]. The elementary particle that defines light is the photon. ... It has been suggested that this article or section be merged with Zero-point energy. ... In quantum field theory the vacuum expectation value (also called condensate) of an operator is its average, expected value in the vacuum. ... In physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess; it is the energy of the ground state of the system. ...

Summing over all possible oscillators at all points in space gives an infinite quantity. To remove this infinity, one may argue that only differences in energy are physically measurable; this argument is the underpinning of the theory of renormalization. In all practical calculations, this is how the infinity is always handled. In a deeper sense, however, renormalization is unsatisfying, and the removal of this infinity presents a challenge in the search for a Theory of Everything. As of 2006, there is no compelling explanation for how this infinity should be treated as essentially zero; a non-zero value is essentially the cosmological constant and any large value causes trouble in cosmology. Figure 1. ... This article or section is in need of attention from an expert on the subject. ... 2006 is a common year starting on Sunday of the Gregorian calendar. ... The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) occurs in Einsteins theory of general relativity. ... Cosmology, as a branch of astrophysics, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. ...

The Casimir effect

Casimir forces on parallel plates.

Casimir's observation was that the second-quantized, quantum electromagnetic field, in the presence of bulk bodies such as metals or dielectrics, must obey the same boundary conditions that the classical electromagnetic field must obey. In particular, this affects the calculation of the vacuum energy in the presence of a conductor or dielectric. Image File history File links Casmir_plates. ... Image File history File links Casmir_plates. ... In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... A dielectric, or electrical insulator, is a substance that is highly resistant to electric current. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... In science and engineering, conductors are materials that contain movable charges of electricity. ...

Consider, for example, the calculation of the vacuum expectation value of the electromagnetic field inside a metal cavity, such as, for example, a radar cavity or a microwave waveguide. In this case, the correct way to find the zero point energy of the field is to sum the energies of the standing waves of the cavity. To each and every possible standing wave corresponds an energy; say the energy of the nth standing wave is E_n. The vacuum expectation value of the electromagnetic field in the cavity is then A cavity magnetron is a high-powered vacuum tube that generates coherent microwaves. ... Microwaves are electromagnetic waves with wavelengths longer than those of terahertz (THz) wavelengths, but relatively short for radio waves. ... Look up waveguide in Wiktionary, the free dictionary. ... A standing wave, also known as a stationary wave, is a wave that remains in a constant position. ...

with the sum running over all possible values of n enumerating the standing waves. The factor of 1/2 corresponds to the fact that the zero-point energies are being summed (it is the same 1/2 as appears in the equation ). Written in this way, this sum is clearly divergent; however, it can be used to create finite expressions.

In particular, one may ask how the zero point energy depends on the shape s of the cavity. Each energy level E_n depends on the shape, and so one should write E_n(s) for the energy level, and for the vacuum expectation value. At this point comes an important observation: the force at point p on the wall of the cavity is equal to the change in the vacuum energy if the shape s of the wall is perturbed a little bit, say by δs, at point p. That is, one has

This value is finite in many practical calculations.

Casimir's calculation

In the original calculation done by Casimir, he considered the space between a pair of conducting metal plates a distance a apart. In this case, the standing waves are particularly easy to calculate, since the transverse component of the electric field and the normal component of the magnetic field must vanish on the surface of a conductor. Assuming the parallel plates lie in the x-y plane, the standing waves are

where ψ stands for the electric component of the electromagnetic field, and, for brevity, the polarization and the magnetic components are ignored here. Here, k_x and k_y are the wave vectors in directions parallel to the plates, and In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ...

is the wave-vector perpendicular to the plates. Here, n is an integer, resulting from the requirement that ψ vanish on the metal plates. The energy of this wave is

where c is the speed of light. The vacuum energy is then the sum over all possible excitation modes The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation in a vacuum, not just visible light. ...

where A is the area of the metal plates, and a factor of 2 is introduced for the two possible polarizations of the wave. This expression is clearly infinite, and to proceed with the calculation, it is convenient to introduce a regulator (discussed in greater detail below). The regulator will serve to make the expression finite, and in the end will be removed. The zeta-regulated version of the energy per unit-area of the plate is In mathematics and theoretical physics, zeta function regularization is a summability method assign finite values to superficially divergent sums. ...

In the end, the limit is to be taken. Here s is just a complex number, not to be confused with the shape discussed previously. This integral/sum is finite for s real and larger than 3. The sum has a pole at s=3, but may be analytically continued to s=0, where the expression is finite. Expanding this, one gets In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the real numbers may be described informally in several different ways. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ...

where polar coordinates were introduced to turn the double integral into a single integral. The integral is easily performed, resulting in This article describes some of the common coordinate systems that appear in elementary mathematics. ... In mathematical analysis, there is a serious distinction between a double integral and an iterated integral. ...

The sum may be understood to be the Riemann zeta function, and so one has In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...

But ζ( − 3) = 1 / 120 and so one obtains

The Casimir force per unit area F_c / A for idealized, perfectly conducting plates with vacuum between them is

where

(hbar, ℏ) is the reduced Planck constant,

c is the speed of light,

a is the distance between the two plates.

The force is negative, indicating that the force is attractive: by moving the two plates closer together, the energy is lowered. The presence of shows that the Casimir force per unit area F_c / A is very small, and that furthermore, the force is inherently of quantum-mechanical origin. Plancks constant, denoted h, is a physical constant that is used to describe the sizes of quanta. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation in a vacuum, not just visible light. ... Distance is a numerical description of how far apart things lie. ...

Measurement

One of the first experimental tests was conducted by Marcus Spaarnay at Philips in Eindhoven, in 1958, in a delicate and difficult experiment with parallel plates, obtaining results not in contradiction with the Casimir theory, but with large experimental errors.

The Casimir effect was measured more accurately in 1997 by Steve K. Lamoreaux of Los Alamos National Laboratory and by Umar Mohideen of the University of California at Riverside and his colleague Anushree Roy. In practice, rather than using two parallel plates, which would require phenomenally accurate alignment to ensure they were parallel, the experiments use one plate that is flat and another plate that is a part of a sphere with a large radius of curvature. In 2001, a group at the University of Padua finally succeeded in measuring the Casimir force between parallel plates using microresonators. 1997 (MCMXCVII) was a common year starting on Wednesday of the Gregorian calendar. ... Los Alamos National Laboratory, aerial view from 1995. ... The University of California, Riverside is a public, coeducational university situated in Riverside, California beside Box Springs Mountain. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... Curvature is the amount by which a geometric object deviates from being flat. ... Gymnasivm Patavinum: The Universitys main Bo palace shown in a 1654 woodcut The University of Padua (Università degli Studi di Padova, UNIPD) is one of the most well-renowned universities in Italy. ...

Further research has shown that, with materials of certain permittivity and permeability, or with a certain configuration, the Casimir effect could be made repulsive instead of attractive, although there are no experimental demonstrations of these predictions. Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ... Permeability has several meanings: In electromagnetism, permeability is the degree of magnetisation of a material in response to a magnetic field. ...

Regularization

In order to be able to perform calculations in the general case, it is convenient to introduce a regulator in the summations. This is an artificial device, used to make the sums finite so that they can be more easily manipulated, followed by the taking of a limit so as to remove the regulator.

The heat kernel or exponentially regulated sum is In mathematics and theoretical physics, zeta-function regularization is a type of regularization or summability method that assigns finite values to superficially divergent sums. ... The term exponential may refer to any of several topics in mathematics: Exponential distribution Exponential function Exponential growth, exponential decay Exponential time Matrix exponential Exponential map (in differential geometry) All relate in some fashion to exponents. ...

where the limit is taken in the end. The divergence of the sum is typically manifested as

for three-dimensional cavities. The infinite part of the sum is associated with the bulk constant C which does not depend on the shape of the cavity. The interesting part of the sum is the finite part, which is shape-dependent. The Gaussian regulator Gaussian curves parametrised by expected value and variance (see normal distribution) A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ...

is better suited to numerical calculations because of its superior convergence properties, but is more difficult to use in theoretical calculations. Other, suitably smooth, regulators may be used as well. The zeta function regulator In mathematics and theoretical physics, zeta-function regularization is a type of regularization or summability method that assigns finite values to superficially divergent sums. ...

is completely unsuited for numerical calculations, but is quite useful in theoretical calculations. In particular, divergences show up as poles in the complex s plane, with the bulk divergence at s=4. This sum may be analytically continued past this pole, to obtain a finite part at s=0. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ...

Not every cavity configuration necessarily leads to a finite part (the lack of a pole at s=0) or shape-independent infinite parts. In this case, it should be understood that additional physics has to be taken into account. In particular, at extremely large frequencies (above the plasma frequency), metals become transparent to photons (such as x-rays), and dielectrics show a frequency-dependent cutoff as well. This frequency dependence acts as a natural regulator. There are a variety of bulk effects in solid state physics, mathematically very similar to the Casimir effect, where the cutoff frequency comes into explicit play to keep expressions finite. (These are discussed in greater detail in Landau and Lifshitz, "Theory of Continuous Media".) In physics, plasma oscillations, often referred to as Langmuir waves or plasma waves, are periodic oscillations of charge density in conducting media such as plasmas or metals. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ... In the NATO phonetic alphabet, X-ray represents the letter X. An X-ray picture (radiograph) taken by Röntgen An X-ray is a form of electromagnetic radiation with a wavelength approximately in the range of 5 pm to 10 nanometers (corresponding to frequencies in the range 30 PHz... Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ... The Butterworth filters frequency response, with cutoff frequency labeled. ...

Generalities

The Casimir effect can also be computed using the mathematical mechanisms of functional integrals of quantum field theory, although such calculations are considerably more abstract, and thus difficult to comprehend. In addition, they can be carried out only for the simplest of geometries. However, the formalism of quantum field theory makes it clear that the vacuum expectation value summations are in a certain sense summations over so-called "virtual particles". In physics, functional integration is integration over certain infinite-dimensional spaces. ... In physics, a virtual particle is a particle-like abstraction used in some models of quantum field theory. ...

More interesting is the understanding that the sums over the energies of standing waves should be formally understood as sums over the eigenvalues of a Hamiltonian. This allows atomic and molecular effects, such as the van der Waals force, to be understood as a variation on the theme of the Casimir effect. Thus one considers the Hamiltonian of a system as a function of the arrangement of objects, such as atoms, in configuration space. The change in the zero-point energy as a function of changes of the configuration can be understood to result in forces acting between the objects. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In physics, Hamiltonian has distinct but closely related meanings. ... In chemistry, the term van der Waals force originally referred to all forms of intermolecular forces; however, in modern usage it tends to refer to intermolecular forces that deal with forces due to the polarization of molecules. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ...

In the chiral bag model of the nucleon, the Casimir energy plays an important role in showing the mass of the nucleon is independent of the bag radius. In addition, the spectral asymmetry is interpreted as a non-zero vacuum expectation value of the baryon number, cancelling the topological winding number of the pion field surrounding the nucleon. In physics a nucleon is a collective name for two baryons: the neutron and the proton. ... In physics a nucleon is a collective name for two baryons: the neutron and the proton. ... In mathematics and physics, the spectral asymmetry is the asymmetry in the distribution of the spectrum of eigenvalues of an operator. ... In particle physics, the baryon number is an approximate conserved quantum number. ... It has been suggested that this article or section be merged with Soliton (topological). ... In particle physics, pion (short for pi meson) is the collective name for three subatomic particles: π0, π+ and π−. Pions are the lightest mesons and play an important role in explaining low-energy properties of the strong nuclear force. ...

Analogies

A similar analysis can be used to explain Hawking radiation that causes the slow "evaporation" of black holes (although this is generally explained as the escape of one particle from a virtual particle-antiparticle pair, the other particle having been captured by the black hole). In physics, Hawking radiation is thermal radiation thought to be emitted by black holes due to quantum effects. ... This article or section is in need of attention from an expert on the subject. ... This article is about the astronomical body. ...

A more practical analogy is to look at two ships in the open ocean, sailing alongside each other. As they come closer together, their hulls shield the space in between from more and more wave energy, both from the sides as well as from front and back, which increasingly cancel out waves of longer wavelengths than the distance between the hulls. This causes the hulls to be increasingly pushed by this difference in wave activity toward each other, as they get closer to each other, such that if both ships do not actively steer away from each other under power, they will eventually collide. It is for this reason that naval vessels, when resupplying or transferring personnel at sea, must use lines and maintain a minimum distance from each other, based on vessel length and wave height.

References

• H.B.G. Casimir, Proc. Kon. Nederland. Akad. Wetensch. B51, 793 (1948)
• Casimir effect description from University of California, Riverside's version of the Usenet physics FAQ.
• A. Lambrecht, "The Casimir effect: a force from nothing", Physics World, September 2002.
• G. Bressi, G. Carugno, R. Onofrio, G. Ruoso, "Measurement of the Casimir force between Parallel Metallic Surfaces", Phys. Rev. Lett. 88 041804 (2002)
• M. Bordag, U. Mohideen, V.M. Mostepanenko, "New Developments in the Casimir Effect", ArXiv quant-ph/0106045. (275 page review paper.)
• O. Kenneth, I. Klich, A. Mann and M. Revzen, Repulsive Casimir forces, Department of Physics, Technion - Israel Institute of Technology, Haifa, February 2002
• S. K. Lamoreaux, "Demonstration of the Casimir Force in the 0.6 to 6 µm Range", Phys. Rev. Lett. 78, 5–8 (1997)
• J. D. Barrow, "Much ado about nothing", (2005) Lecture at Gresham College. (Includes discussion of French naval analogy.)
• Barrow, John D. (2000). The book of nothing : vacuums, voids, and the latest ideas about the origins of the universe, 1st American Ed., New York: Pantheon Books. ISBN 0-09-928845-1. (Also includes discussion of French naval analogy.)
• G. Lang, The Casimir Force web site, 2002
• V.V. Nesterenko, G. Lambiase, G. Scarpetta, Calculation of the Casimir energy at zero and finite temperature: some recent results, arXiv:hep-th/0503100 v2 13 May 2005

The University of California, Riverside, is a public coeducational university whose main campus is in a suburban district of the city of Riverside, California. ... Gresham College is an unusual institution of higher learning in London which enrolls no students and grants no degrees. ... John David Barrow FRS (born November 29, 1952, London) is an English cosmologist, theoretical physicist, and mathematician. ...

External links

• Casimir effect on arxiv.org