Casimir effect

 ( Quantum Field Theory ) 

Consider, for example, the calculation of the vacuum expectation value of the electromagnetic field inside a metal cavity, such as, for example, a Cavity magnetron 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 of the cavity. To each and every possible standing wave corresponds an energy; say the energy of the th standing wave is . The vacuum expectation value of the energy of the electromagnetic field in the cavity is then

$$\backslash langle\; E\; \backslash rangle=\backslash tfrac12\; \backslash sum\_n\; E\_n$$

with the sum running over all possible values of enumerating the standing waves. The factor of is present because the zero-point energy of the th mode is , where is the energy increment for the th mode. (It is the same 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 of the cavity. Each energy level depends on the shape, and so one should write for the energy level, and for the vacuum expectation value. At this point comes an important observation: The force at point on the wall of the cavity is equal to the change in the vacuum energy if the shape of the wall is perturbed a little bit, say by , at . That is, one has

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

This value is finite in many practical calculations.For a brief summary, see the introduction in

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 from one of two widely separated plates (distance apart). With , the states within the slot of width are highly constrained so that the energy of any one mode is widely separated from that of the next. This is not the case in the large region where there is a large number of states (about ) with energy evenly spaced between and the next mode in the narrow slot, or in other words, all slightly larger than . Now on shortening by an amount (which is negative), the mode in the narrow slot shrinks in wavelength and therefore increases in energy proportional to , whereas all the states that lie in the large region lengthen and correspondingly decrease their energy by an amount proportional to (note the different denominator). The two effects nearly cancel, but the net change is slightly negative, because the energy of all the modes in the large region are slightly larger than the single mode in the slot. Thus the force is attractive: it tends to make slightly smaller, the plates drawing each other closer, across the thin slot.

$$\backslash psi\_n(x,y,z;t)=e^\{-i\backslash omega\_nt\}\; e^\{ik\_xx+ik\_yy\}\; \backslash sin(k\_n\; z)\; \backslash ,,$$

where stands for the electric component of the electromagnetic field, and, for brevity, the polarization and the magnetic components are ignored here. Here, and are the in directions parallel to the plates, and

$$k\_n=\backslash frac\{n\backslash pi\}\{a\}$$

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

$$\backslash omega\_n=c\; \backslash sqrt\; \backslash ,,$$

where 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 -space. The assumption of periodic boundary conditions yields,

$$\backslash langle\; E\; \backslash rangle=\backslash frac\{\backslash hbar\}\{2\}\; \backslash cdot\; 2\; \backslash int\; \backslash frac\{A\; \backslash ,dk\_x\; \backslash ,dk\_y\}\{(2\backslash pi)^2\}\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash omega\_n\; \backslash ,,$$

where 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

$$\backslash frac\{\backslash langle\; E(s)\; \backslash rangle\}\{A\}=\backslash hbar\; \backslash int\; \backslash frac\{dk\_x\; \backslash ,dk\_y\}\{(2\backslash pi)^2\}\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash omega\_n\; \backslash left|\; \backslash omega\_n\; \backslash right|^\{-s\}\; \backslash ,.$$

In the end, the limit is to be taken. Here is just a complex number, not to be confused with the shape discussed previously. This integral sum is finite for real number and larger than 3. The sum has a pole at , but may be analytically continued to , where the expression is finite. The above expression simplifies to:

$$\backslash frac\{\backslash langle\; E(s)\; \backslash rangle\}\{A\}=\; \backslash frac\{\backslash hbar\; c^\{1-s\}\}\{4\backslash pi^2\}\; \backslash sum\_n\; \backslash int\_0^\backslash infty\; 2\backslash pi\; q\; \backslash ,dq\; \backslash left\; |\; q^2\; +\; \backslash frac\{\backslash pi^2\; n^2\}\{a^2\}\; \backslash right|^\backslash frac\{1-s\}\{2\}\; \backslash ,,$$

where polar coordinates were introduced to turn the double integral into a single integral. The in front is the Jacobian, and the comes from the angular integration. The integral converges if , resulting in

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

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

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

But and so one obtains

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

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 for idealized, perfectly conducting plates with vacuum between them is

$$\backslash frac\{F\_\backslash mathrm\{c\}\}\{A\}=-\backslash frac\{d\}\{da\}\; \backslash frac\{\backslash langle\; E\; \backslash rangle\}\{A\}\; =\; -\backslash frac\; \{\backslash hbar\; c\; \backslash pi^2\}\; \{240\; a^4\}$$

where

• is the reduced Planck constant,
• is the speed of light,
• 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 is very small, and that furthermore, the force is inherently of quantum-mechanical origin.

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

where

• is the reduced Planck constant,
• is the speed of light,
• is the area of one of the plates,
• is the distance between the two plates

In Casimir's original derivation, a moveable conductive plate is positioned at a short distance from one of two widely separated plates (distance apart). The zero-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 Euler–Maclaurin summation with a regularizing function (e.g., exponential regularization) not so anomalous as in the above.

Lifshitz's result was subsequently generalized to arbitrary multilayer planar geometries as well as to anisotropic and magnetic materials, but for several decades the calculation of Casimir forces for non-planar geometries remained limited to a few idealized cases admitting analytical solutions. For example, the force in the experimental sphere–plate geometry was computed with an approximation (due to Derjaguin) that the sphere radius is much larger than the separation , in which case the nearby surfaces are nearly parallel and the parallel-plate result can be adapted to obtain an approximate force (neglecting both skin-depth and higher-order curvature effects).B. V. Derjaguin, I. I. Abrikosova, and E. M. Lifshitz, Quarterly Reviews, Chemical Society, vol. 10, 295–329 (1956). However, in the 2010s a number of authors developed and demonstrated a variety of numerical techniques, in many cases adapted from classical computational electromagnetics, that are capable of accurately calculating Casimir forces for arbitrary geometries and materials, from simple finite-size effects of finite plates to more complicated phenomena arising for patterned surfaces or objects of various shapes. Review article.

The Casimir effect was measured more accurately in 1997 by Steve K. Lamoreaux of Los Alamos National Laboratory, and by Umar Mohideen and Anushree Roy of the University of California, Riverside. 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. Numerous variations of these experiments are summarized in the 2009 review by Klimchitskaya.

In 2013, 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 demonstrated a compact integrated silicon chip that can measure the Casimir force. The integrated chip defined by electron-beam lithography does not need extra alignment, making it an ideal platform for measuring Casimir force between complex geometries. In 2017 and 2021, the same group from Hong Kong University of Science and Technology demonstrated the non-monotonic Casimir force and distance-independent Casimir force, respectively, using this on-chip platform.

The heat kernel or exponentially regulated sum is $$\backslash langle\; E(t)\; \backslash rangle=\backslash frac12\; \backslash sum\_n\; \backslash hbar\; |\backslash omega\_n\; |\; \backslash exp\; \backslash bigl(-t\; |\backslash omega\_n\; |\backslash bigr)\backslash ,,$$

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

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

for three-dimensional cavities. The infinite part of the sum is associated with the bulk constant 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

$$\backslash langle\; E(t)\; \backslash rangle=\backslash frac12\; \backslash sum\_n\; \backslash hbar\; |\backslash omega\_n\; |\; \backslash exp\; \backslash left(-t^2\; |\backslash omega\_n\; |^2\backslash right)$$

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

$$\backslash langle\; E(s)\; \backslash rangle=\backslash frac12\; \backslash sum\_n\; \backslash hbar\; |\backslash omega\_n\; |\; |\backslash omega\_n\; |^\{-s\}$$

is completely unsuited for numerical calculations, but is quite useful in theoretical calculations. In particular, divergences show up as poles in the complex plane, with the bulk divergence at . This sum may be analytically continued past this pole, to obtain a finite part at .

Not every cavity configuration necessarily leads to a finite part (the lack of a pole at ) 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 (such as ), 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".)

More interesting is the understanding that the sums over the energies of standing waves should be formally understood as sums over the 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.

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.

Timothy Boyer showed in his work published in 1968 that a conductor with spherical symmetry will also show this repulsive force, and the result is independent of radius. Further work shows that the repulsive force can be generated with materials of carefully chosen dielectrics.

In 1995 and 1998 Maclay et al. published the first models of a microelectromechanical system (MEMS) with Casimir forces. While not exploiting the Casimir force for useful work, the papers drew attention from the MEMS community due to the revelation that Casimir effect needs to be considered as a vital factor in the future design of MEMS. In particular, Casimir effect might be the critical factor in the stiction failure of MEMS.