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Casimir effect

2007 Schools Wikipedia Selection. Related subjects: General Physics

   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.

   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.

   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.

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.

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.

   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

          {E} = \begin{matrix} \frac{1}{2} \end{matrix} \hbar \omega \ .

   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.

The Casimir effect

   Casimir forces on parallel plates.
   Enlarge
   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.

   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

          \langle E \rangle = \frac{1}{2} \sum_n E_n

   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 E=\hbar \omega/2 ). 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 \langle E(s)
   \rangle 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

          F(p) = - \left. \frac{\delta \langle E(s) \rangle} {\delta s}
          \right\vert_p\,

   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

          \psi_n(x,y,z,t) = e^{-i\omega_nt} e^{ik_xx+ik_yy} \sin \left(
          k_n z \right)

   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

          k_n = \frac{n\pi}{a}

   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

          \omega_n = c \sqrt{{k_x}^2 + {k_y}^2 + \frac{n^2\pi^2}{a^2}}

   where c is the speed of light. The vacuum energy is then the sum over
   all possible excitation modes

          \langle E \rangle = \frac{\hbar}{2} \cdot 2 \int \frac{dk_x
          dk_y}{(2\pi)^2} \sum_{n=1}^\infty A\omega_n

   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

          \frac{\langle E(s) \rangle}{A} = \hbar \int \frac{dk_x
          dk_y}{(2\pi)^2} \sum_{n=1}^\infty \omega_n \vert
          \omega_n\vert^{-s}

   In the end, the limit s\to 0 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

          \frac{\langle E(s) \rangle}{A} = \frac{\hbar c^{1-s}}{4\pi^2}
          \sum_n \int_0^\infty 2\pi qdq \left \vert q^2 + \frac{\pi^2
          n^2}{a^2} \right\vert^{(1-s)/2}

   where polar coordinates q^2 = k_x^2+k_y^2 were introduced to turn the
   double integral into a single integral. The q in front in the Jacobian,
   and the 2π comes from the angular integration. The integral is easily
   performed, resulting in

          \frac{\langle E(s) \rangle}{A} = -\frac {\hbar c^{1-s}
          \pi^{2-s}}{2a^{3-s}} \frac{1}{3-s} \sum_n \vert n\vert ^{3-s}

   The sum may be understood to be the Riemann zeta function, and so one
   has

          \frac{\langle E \rangle}{A} = \lim_{s\to 0} \frac{\langle E(s)
          \rangle}{A} = -\frac {\hbar c \pi^{2}}{6a^{3}} \zeta (-3)

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

          \frac{\langle E \rangle}{A} = \frac {-\hbar c \pi^{2}}{3 \cdot
          240 a^{3}}

   The Casimir force per unit area F[c] / A for idealized, perfectly
   conducting plates with vacuum between them is

          {F_c \over A} = - \frac{d}{da} \frac{\langle E \rangle}{A} =
          -\frac {\hbar c \pi^2} {240 a^4}

   where

          \hbar (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 \hbar 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.

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.

   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.

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

          \langle E(t) \rangle = \frac{1}{2} \sum_n \hbar |\omega_n| \exp
          (-t|\omega_n|)

   where the limit t\to 0^+ is taken in the end. The divergence of the sum
   is typically manifested as

          \langle E(t) \rangle = \frac{C}{t^3} + \textrm{finite}\,

   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

          \langle E(t) \rangle = \frac{1}{2} \sum_n \hbar |\omega_n| \exp
          (-t^2|\omega_n|^2)

   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

          \langle E(s) \rangle = \frac{1}{2} \sum_n \hbar |\omega_n|
          |\omega_n|^{-s}

   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.

   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".)

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".

   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 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.

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).

   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.
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