Casimir effect: Difference between revisions

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In

The Casimir effect can be understood by the idea that the presence of conducting metals and dielectrics alters the vacuum expectation value of the energy of the second-quantized electromagnetic field.^[2][3] 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.

Any medium supporting oscillations has an analogue of the Casimir effect. For example, beads on a string^[4][5] as well as plates submerged in turbulent water^[6] or gas^[7] illustrate the Casimir force.

In modern

The typical example is of the two uncharged conductive plates in a vacuum, placed a few nanometers apart. In a classical description, the lack of an external field means that there is no field between the plates, and no force would be measured between them.^[9] When this field is instead studied using the quantum electrodynamic vacuum, it is seen that the plates do affect the virtual photons which constitute the field, and generate a net force^[10] – either an attraction or a repulsion depending on the specific arrangement of the two plates. Although the Casimir effect can be expressed in terms of virtual particles interacting with the objects, it is best described and more easily calculated in terms of the zero-point energy of a quantized field in the intervening space between the objects. This force has been measured and is a striking example of an effect captured formally by second quantization.^[11][12]

The treatment of boundary conditions in these calculations has led to some controversy. In fact, "Casimir's original goal was to compute the

Dutch physicists Hendrik Casimir and Dirk Polder at Philips Research Labs proposed the existence of a force between two polarizable atoms and between such an atom and a conducting plate in 1947; this special form is called the Casimir–Polder force. After a conversation with Niels Bohr, who suggested it had something to do with zero-point energy, Casimir alone formulated the theory predicting a force between neutral conducting plates in 1948 which is called the Casimir effect in the narrow sense.

Predictions of the force were later extended to finite-conductivity metals and dielectrics, and recent calculations have considered more general geometries. Experiments before 1997 had observed the force qualitatively, and indirect validation of the predicted Casimir energy had been made by measuring the thickness of liquid helium films. However it was not until 1997 that a direct experiment by S. Lamoreaux quantitatively measured the force to within 5% of the value predicted by the theory.^[1] Subsequent experiments approach an accuracy of a few percent.

Possible causes

Vacuum energy

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The causes of the Casimir effect are described by 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 simplified view, 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. At the most basic level, 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, even the vacuum has a vastly complex structure, so 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,^[14] or polarization in the case of light, energy, and so on. On average, most 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

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

Summing over all possible oscillators at all points in space gives an infinite quantity. Since only differences in energy are physically measurable (with the notable exception of gravitation, which remains beyond the scope of quantum field theory), this infinity may be considered a feature of the mathematics rather than of the physics. This argument is the underpinning of the theory of renormalization. Dealing with infinite quantities in this way was a cause of widespread unease among quantum field theorists before the development in the 1970s of the renormalization group, a mathematical formalism for scale transformations that provides a natural basis for the process.

When the scope of the physics is widened to include gravity, the interpretation of this formally infinite quantity remains problematic. There is currently no compelling explanation as to why it should not result in a cosmological constant that is many orders of magnitude larger than observed.^[15] However, since we do not yet have any fully coherent quantum theory of gravity, there is likewise no compelling reason as to why it should instead actually result in the value of the cosmological constant that we observe.^[16]

The Casimir effect for fermions can be understood as the spectral asymmetry of the fermion operator ${\displaystyle (-1)^{F}}$, where it is known as the Witten index.

Relativistic van der Waals force

Alternatively, a 2005 paper by Robert Jaffe of MIT states that "Casimir effects can be formulated and Casimir forces can be computed without reference to zero-point energies. They are relativistic, quantum forces between charges and currents. The Casimir force (per unit area) between parallel plates vanishes as alpha, the fine structure constant, goes to zero, and the standard result, which appears to be independent of alpha, corresponds to the alpha approaching infinity limit," and that "The Casimir force is simply the (relativistic, retarded) van der Waals force between the metal plates."^[17] Casimir and Polder's original paper used this method to derive the Casimir-Polder force. In 1978, Schwinger, DeRadd, and Milton published a similar derivation for the Casimir Effect between two parallel plates.^[18] In fact, the description in terms of van der Waals forces is the only correct description from the fundamental microscopic perspective,^[19][20] while other descriptions of Casimir force are merely effective macroscopic descriptions.

Effects

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 ${\displaystyle E_{n}}$. The vacuum expectation value of the energy of the electromagnetic field in the cavity is then

$${\displaystyle \langle E\rangle ={\frac {1}{2}}\sum _{n}E_{n}}$$

with the sum running over all possible values of n enumerating the standing waves. The factor of 1/2 is present because the zero-point energy of the n'th mode is ${\displaystyle E_{n}/2}$, where ${\displaystyle E_{n}}$ is the energy increment for the n'th mode. (It is the same 1/2 as appears in the equation ${\displaystyle E=\hbar \omega /2}$.) 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 ${\displaystyle E_{n}}$ depends on the shape, and so one should write ${\displaystyle E_{n}(s)}$ for the energy level, and ${\displaystyle \langle E(s)\rangle }$ 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 ${\displaystyle \delta s}$, at point p. That is, one has

$${\displaystyle F(p)=-\left.{\frac {\delta \langle E(s)\rangle }{\delta s}}\right\vert _{p}.\,}$$

This value is finite in many practical calculations.^[21]

Attraction between the plates can be easily understood by focusing on the one-dimensional situation. Suppose that a moveable conductive plate is positioned at a short distance a from one of two widely separated plates (distance L apart). With a << L, the states within the slot of width a are highly constrained so that the energy E of any one mode is widely separated from that of the next. This is not the case in the large region L, where there is a large number (numbering about L / a) of states with energy evenly spaced between E and the next mode in the narrow slot – in other words, all slightly larger than E. Now on shortening a by da (< 0), the mode in the narrow slot shrinks in wavelength and therefore increases in energy proportional to −da/a, whereas all the L /a states that lie in the large region lengthen and correspondingly decrease their energy by an amount proportional to da/L (note the denominator). The two effects nearly cancel, but the net change is slightly negative, because the energy of all the L/a modes in the large region are slightly larger than the single mode in the slot. Thus the force is attractive: it tends to make a slightly smaller, the plates attracting each other across the thin slot.

Derivation of Casimir effect assuming zeta-regularization

• See Wikiversity for an elementary calculation in one dimension.

In the original calculation done by Casimir, he considered the space between a pair of conducting metal plates at distance ${\displaystyle a}$ apart. In this case, the standing waves are particularly easy to calculate, because 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 plates lie parallel to the xy-plane, the standing waves are

$${\displaystyle \psi _{n}(x,y,z;t)=e^{-i\omega _{n}t}e^{ik_{x}x+ik_{y}y}\sin(k_{n}z)}$$

where ${\displaystyle \psi }$ stands for the electric component of the electromagnetic field, and, for brevity, the polarization and the magnetic components are ignored here. Here, ${\displaystyle k_{x}}$ and ${\displaystyle k_{y}}$ are the

$${\displaystyle k_{n}={\frac {n\pi }{a}}}$$

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

$${\displaystyle \omega _{n}=c{\sqrt {{k_{x}}^{2}+{k_{y}}^{2}+{\frac {n^{2}\pi ^{2}}{a^{2}}}}}}$$

where c is the speed of light. The vacuum energy is then the sum over all possible excitation modes. Since the area of the plates is large, we may sum by integrating over two of the dimensions in k-space. The assumption of periodic boundary conditions yields,

$${\displaystyle \langle E\rangle ={\frac {\hbar }{2}}\cdot 2\int {\frac {Adk_{x}dk_{y}}{(2\pi )^{2}}}\sum _{n=1}^{\infty }\omega _{n}}$$

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

$${\displaystyle {\frac {\langle E(s)\rangle }{A}}=\hbar \int {\frac {dk_{x}dk_{y}}{(2\pi )^{2}}}\sum _{n=1}^{\infty }\omega _{n}\vert \omega _{n}\vert ^{-s}.}$$

In the end, the limit ${\displaystyle s\to 0}$ is to be taken. Here s is just a

to s=0, where the expression is finite. The above expression simplifies to:

$${\displaystyle {\frac {\langle E(s)\rangle }{A}}={\frac {\hbar c^{1-s}}{4\pi ^{2}}}\sum _{n}\int _{0}^{\infty }2\pi qdq\left\vert q^{2}+{\frac {\pi ^{2}n^{2}}{a^{2}}}\right\vert ^{(1-s)/2},}$$

where polar coordinates ${\displaystyle q^{2}=k_{x}^{2}+k_{y}^{2}}$ were introduced to turn the double integral into a single integral. The ${\displaystyle q}$ in front is the Jacobian, and the ${\displaystyle 2\pi }$ comes from the angular integration. The integral converges if Re[s] > 3, resulting in

$${\displaystyle {\frac {\langle E(s)\rangle }{A}}=-{\frac {\hbar c^{1-s}\pi ^{2-s}}{2a^{3-s}}}{\frac {1}{3-s}}\sum _{n}\vert n\vert ^{3-s}=-{\frac {\hbar c^{1-s}\pi ^{2-s}}{2a^{3-s}(3-s)}}\sum _{n}{\frac {1}{\left|n\right|^{s-3}}}.}$$

The sum diverges at s in the neighborhood of zero, but if the damping of large-frequency excitations corresponding to analytic continuation of the Riemann zeta function to s=0 is assumed to make sense physically in some way, then one has

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

But

$${\displaystyle \zeta (-3)={\frac {1}{120}}}$$

and so one obtains

$${\displaystyle {\frac {\langle E\rangle }{A}}={\frac {-\hbar c\pi ^{2}}{720a^{3}}}.}$$

The analytic continuation has evidently lost an additive positive infinity, somehow exactly accounting for the zero-point energy (not included above) outside the slot between the plates, but which changes upon plate movement within a closed system. The Casimir force per unit area ${\displaystyle F_{c}/A}$ for idealized, perfectly conducting plates with vacuum between them is

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

where

• ${\displaystyle \hbar }$ (hbar, ħ) is the
  reduced Planck constant
  ,
• ${\displaystyle c}$ is the speed of light,
• ${\displaystyle a}$ 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 ${\displaystyle \hbar }$ shows that the Casimir force per unit area ${\displaystyle F_{c}/A}$ is very small, and that furthermore, the force is inherently of quantum-mechanical origin.

By integrating the equation above it is possible to calculate the energy required to separate to infinity the two plates as:

${\displaystyle U_{E}(a)=\int F(a)\,da=\int -\hbar c\pi ^{2}{\frac {A}{240a^{4}}}\,da}$

${\displaystyle U_{E}(a)=-\hbar c\pi ^{2}{\frac {A}{720a^{3}}}}$

where

• ${\displaystyle \hbar }$ (hbar, ħ) is the
  reduced Planck constant
  ,
• ${\displaystyle c}$ is the speed of light,
• ${\displaystyle A}$ is the area of one of the plates,
• ${\displaystyle a}$ is the distance between the two plates

NOTE: In Casimir's original derivation [1], a moveable conductive plate is positioned at a short distance a from one of two widely separated plates (distance L apart). The 0-point energy on both sides of the plate is considered. Instead of the above ad hoc analytic continuation assumption, non-convergent sums and integrals are computed using

${\displaystyle \vert \omega _{n}\vert ^{-s}}$

More recent theory

Casimir's analysis of idealized metal plates was generalized to arbitrary dielectric and realistic metal plates by

Measurement

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One of the first experimental tests was conducted by Marcus Sparnaay at Philips in Eindhoven (Netherlands), in 1958, in a delicate and difficult experiment with parallel plates, obtaining results not in contradiction with the Casimir theory,^[29][30] but with large experimental errors. Some of the experimental details as well as some background information on how Casimir, Polder and Sparnaay arrived at this point^[31] are highlighted in a 2007 interview with Marcus Sparnaay.

The Casimir effect was measured more accurately in 1997 by Steve K. Lamoreaux of Los Alamos National Laboratory,^[1] and by Umar Mohideen and Anushree Roy of the University of California, Riverside.^[32] 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 very large radius.

In 2001, a group (Giacomo Bressi, Gianni Carugno, Roberto Onofrio and Giuseppe Ruoso) at the University of Padua (Italy) finally succeeded in measuring the Casimir force between parallel plates using microresonators.^[33]

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

regulated sum is

$${\displaystyle \langle E(t)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}|\exp(-t|\omega _{n}|)}$$

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

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

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

$${\displaystyle \langle E(t)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}|\exp(-t^{2}|\omega _{n}|^{2})}$$

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

$${\displaystyle \langle E(s)\rangle ={\frac {1}{2}}\sum _{n}\hbar |\omega _{n}||\omega _{n}|^{-s}}$$

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

A "pseudo-Casimir" effect can be found in liquid crystal systems, where the boundary conditions imposed through anchoring by rigid walls give rise to a long-range force, analogous to the force that arises between conducting plates.^[34]

Dynamical Casimir effect

The dynamical Casimir effect is the production of particles and energy from an accelerated moving mirror. This reaction was predicted by certain numerical solutions to quantum mechanics equations made in the 1970s.^[35] In May 2011 an announcement was made by researchers at the Chalmers University of Technology, in Gothenburg, Sweden, of the detection of the dynamical Casimir effect. In their experiment, microwave photons were generated out of the vacuum in a superconducting microwave resonator. These researchers used a modified SQUID to change the effective length of the resonator in time, mimicking a mirror moving at the required relativistic velocity. If confirmed this would be the first experimental verification of the dynamical Casimir effect.^{[36] [37]} In March 2013 an article appeared on the PNAS scientific journal describing an experiment that demonstrated the dynamical Casimir effect in a Josephson metamaterial.^[38]

Analogies

A similar analysis can be used to explain Hawking radiation that causes the slow "evaporation" of black holes (although this is generally visualized as the escape of one particle from a virtual particle-antiparticle pair, the other particle having been captured by the black hole).^[39]

Constructed within the framework of quantum field theory in curved spacetime, the dynamical Casimir effect has been used to better understand acceleration radiation such as the Unruh effect.^{[citation needed]}

Repulsive forces

There are few instances wherein the Casimir effect can give rise to repulsive forces between uncharged objects. Evgeny Lifshitz showed (theoretically) that in certain circumstances (most commonly involving liquids), repulsive forces can arise.

It has been suggested that the Casimir forces have application in nanotechnology,[46] in particular silicon integrated circuit technology based micro- and nanoelectromechanical systems, and so-called Casimir oscillators.^[47]

The Casimir effect shows that quantum field theory allows the energy density in certain regions of space to be negative relative to the ordinary vacuum energy, and it has been shown theoretically that quantum field theory allows states where the energy can be arbitrarily negative at a given point.^[48] Many physicists such as Stephen Hawking,^[49] Kip Thorne,^[50] and others^[51][52][53] therefore argue that such effects might make it possible to stabilize a traversable wormhole.

Manipulation of the Casimir effect to extract energy from the vacuum has also been suggested as a way of fueling space propulsion.^[54] Theoretical methods of extraction include the manipulation of the sides of a Casimir box so as to perform a cycle with a net extraction of energy.^[55][56] Since extraction of this energy would come from the vacuum of space, this would allow for space travel without the restriction of carrying onboard fuel. However, theoretical and practical considerations have excluded the manufacture of a functioning apparatus of vacuum energy extraction for space travel.^[54][57]

On 4 June 2013 it was reported^[58] that a conglomerate of scientists from Hong Kong University of Science and Technology, University of Florida, Harvard University, Massachusetts Institute of Technology, and Oak Ridge National Laboratory have for the first time demonstrated a compact integrated silicon chip that can measure the Casimir force.^[59]

See also

• Casimir pressure
• Negative energy
• Scharnhorst effect
• Van der Waals force
• Squeezed vacuum

References