# The Casimir effect: Recent controversies and progress

###### Abstract

The phenomena implied by the existence of quantum vacuum fluctuations, grouped under the title of the Casimir effect, are reviewed, with emphasis on new results discovered in the past four years. The Casimir force between parallel plates is rederived as the strong-coupling limit of -function potential planes. The role of surface divergences is clarified. A summary of effects relevant to measurements of the Casimir force between real materials is given, starting from a geometrical optics derivation of the Lifshitz formula, and including a rederivation of the Casimir-Polder forces. A great deal of attention is given to the recent controversy concerning temperature corrections to the Casimir force between real metal surfaces. A summary of new improvements to the proximity force approximation is given, followed by a synopsis of the current experimental situation. New results on Casimir self-stress are reported, again based on -function potentials. Progress in understanding divergences in the self-stress of dielectric bodies is described, in particular the status of a continuing calculation of the self-stress of a dielectric cylinder. Casimir effects for solitons, and the status of the so-called dynamical Casimir effect, are summarized. The possibilities of understanding dark energy, strongly constrained by both cosmological and terrestrial experiments, in terms of quantum fluctuations are discussed. Throughout, the centrality of quantum vacuum energy in fundamental physics in emphasized.

###### type:

Review Article###### pacs:

11.10.Gh, 11.10.Wx, 42.50.Pq, 78.20.Ci## 1 Introduction

The essence of quantum physics is fluctuations. That is, knowing the position of a particle precisely means losing all knowledge about its momentum, and vice versa, and generally the product of uncertainties of a generalized coordinate and its corresponding momentum is bounded below:

(1.1) |

which reflects the fundamental commutation relation

(1.2) |

The Hamiltonian commutes with neither nor in general; this means that in an energy eigenstate the fluctuations in and are both nonzero:

(1.3) |

Moreover, a harmonic oscillator has correspondingly a ground-state energy which is nonzero:

(1.4) |

The apparent implication of this is that a crystal, which may be thought of, roughly, as a collection of atoms held in harmonic potentials, should have a large zero-point energy at zero temperature:

(1.5) |

being the characteristic frequency of each potential.

The vacuum of quantum field theory may similarly be regarded as an enormously large collection of harmonic oscillators, representing the fluctuations of, for quantum electrodynamics, the electric and magnetic fields at each point in space. (Canonically, the momentum-coordinate pair correspond to the electric field and the vector potential.) Put otherwise, the QED vacuum is a sea of virtual photons. Thus the zero-point energy density of the vacuum is

(1.6) |

where is the wavevector of the photon, and the factor of 2 reflects the two polarization states of the photon.

This is an enormously large quantity. If we say that the largest wavevector appearing in the integral is , say GeV, the Planck scale, then GeV/cm. So it is no surprise that Dirac suggested that this zero-point energy be simply discarded, as some irrelevant constant [1] (yet he became increasingly concerned about the inconsistency of doing so throughout his life [2]). Pauli recognized that this energy surely coupled to gravity, and it would then give rise to a large cosmological constant, so large that the size of the universe could not even reach the distance to the moon [3, 4]. This cosmological constant problem is with us to the present [5, 6]. But this was not the most perplexing issue confronting quantum electrodynamics in the 1930s.

Renormalization theory, that is, a consistent theory of quantum electrodynamics, was invented first by Schwinger [7] and then Feynman [8] in 1948; yet remarkably, across the Atlantic, Casimir in the same year predicted the direct macroscopically observable consequence of vacuum fluctuations that now bears his name [9]. This is the attraction between parallel uncharged conducting plates that has been so convincingly demonstrated by many experiments in the last few years [10]. Lifshitz and his group generalized the theory to include dielectric materials in the 1950s [11, 12, 13, 14]. There were many experiments to detect the effect in the 1950s and 1960s, but most were inconclusive, because the forces were so small, and it was very difficult to keep various interfering phenomena from washing out the effect [15]. However, there could be very little doubt of the reality of the phenomenon, since it was intimately tied to the theory of van der Waals forces between molecules, the retarded version of which had been worked out by Casimir [16] just before he discovered (with a nudge from Bohr [17]) the force between plates. Finally, in 1973, the Lifshitz theory was vindicated by an experiment by Sabisky and Anderson [18].

But by and large field theorists were unaware of the effect until Glashow’s student Boyer carried out a remarkable calculation of the Casimir self-energy of a perfectly conducting spherical shell in 1968 [19]. Glashow was aware of Casimir’s proposal [20] that a classical electron could be stablized by zero-point attraction, and thought the calculation made a suitable thesis project. Boyer’s result was a surprise: The zero-point force was repulsive for the case of a sphere. Davies improved on the calculation [21]; then a decade later there were two independent reconfirmations of Boyer’s result, one based on multiple scattering techniques [22] (now undergoing a renaissance, for example, see [23]) and one on Green’s functions techniques [24] (dubbed source theory [25]). Applications to hadronic physics followed in the next few years [26, 27, 28], and in the last two decades, there has been something of an explosion of interest in the field, with many different calculations being carried out [29, 10].

However, fundamental understanding has been very slow in coming. Why is the cosmological constant neither large nor zero? Why is the Casimir force on a sphere repulsive, when it is attractive between two plates? And is it possible to make sense of Casimir force calculations between two bodies, or of the Casimir self-energy of a single body, in terms of supposedly better understood techniques of perturbative quantum field theory [30]? As we will see, none of these questions yet has a definitive answer, yet progress has been coming. Even the temperature corrections to the Casimir effect, which were considered by Sauer [31], Mehra [32], and Lifshitz [11] in the 1950s and 1960s, have become controversial [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. Thus recent conferences on the Casimir effect have been quite exciting events [48, 49]. It is the aim of the present review to bring the various issues into focus, and suggest paths toward the solutions of the difficulties. It is a mark of the vitality and even centrality of this field that such a review is desirable on the heels of two significant meetings on the subject, and less than three years after the appearance of two major monographs [10, 29] on Casimir phenomena. There are in addition a number of earlier, excellent reviews [50, 51, 52], as well as more specialized treatments [53, 54, 55, 56]. Throughout this review Gaussian units are employed.

This review is organized in the following manner. In section 2 we compute Casimir energies and pressures between parallel function planes, which in the limit of large coupling reproduce the results for a scalar field satisfying Dirichlet boundary conditions on those surfaces. Although these results have been described before, clarification of the nature of surface energy and divergences is provided. TM modes are also discussed here for the first time. Then, in section 3 we rederive the Lifshitz formula for the Casimir force between parallel dielectric slabs using a multiple reflection technique. The Casimir-Polder forces between two atoms, and between an atom and a plate, are rederived. After reviewing roughness and conductivity corrections, a detailed discussion of the temperature controversy is given, with the conclusion that the TE zero-mode absence must be taken seriously, which will imply that large temperature corrections should be seen experimentally. New approaches to moving beyond the proximity approximation in computing forces between nonparallel plane surface are reviewed. A discussion of the remarkable progress experimentally since 1997 is provided. In section 4 after a review of the general situation with respect to surface divergences, TE and TM forces on -function spheres are described in detail, which in the limit of strong coupling reduce to the corresponding finite electromagnetic contributions. For weak coupling, Casimir energies are finite in second order in the coupling strength, but divergent in third order, a fact which has been known for several years. This mirrors the corresponding result for a dilute dielectric sphere, which diverges in third order in the deviation of the permittivity from its vacuum value. Self-stresses on cylinders are also treated, with a detailed discussion of the status of a new calculation for a dielectric cylinder, which should give a vanishing self-stress in second order in the relative permittivity. Section 5 briefly summarizes recent work on quantum fluctuation phenomena in solitonic physics, which has provided the underlying basis for much of the interest in Casimir phenomena over the years. Dynamical Casimir effects, ranging from sonoluminescence through the Unruh effect, are the subject of section 6. The presumed basis for understanding the cosmological dark energy in terms of the Casimir fluctuations is treated in section 7, where there may be a tight constraint emerging between terrestrial measurements of deviations from Newtonian gravity and the size of extra dimensions. The review ends with a summary of perspectives for the future of the field.

## 2 Casimir Effect Between Parallel Plates: A -Potential Derivation

In this section, we will rederive the classic Casimir result for the force between parallel conducting plates [9]. Since the usual Green’s function derivation may be found in monographs [29], and was recently reviewed in connection with current controversies over finiteness of Casimir energies [57], we will here present a different approach, based on -function potentials, which in the limit of strong coupling reduce to the appropriate Dirichlet or Robin boundary conditions of a perfectly conducting surface, as appropriate to TE and TM modes, respectively. Such potentials were first considered by the Leipzig group [58, 59], but recently have been the focus of the program of the MIT group [60, 61, 30]. The discussion here is based on a recent paper by the author [62]. We first consider two -function potentials in dimensions.

### 2.1 dimensions

We consider a massive scalar field (mass ) interacting with two -function potentials, one at and one at , which has an interaction Lagrange density

(2.1) |

where we have chosen the coupling constants and to be dimensionless. (But see the following.) In the limit as both couplings become infinite, these potentials enforce Dirichlet boundary conditions at the two points:

(2.2) |

The Casimir energy for this situation may be computed in terms of the Green’s function ,

(2.3) |

which has a time Fourier transform,

(2.4) |

Actually, this is a somewhat symbolic expression, for the Feynman Green’s function (2.3) implies that the frequency contour of integration here must pass below the singularities in on the negative real axis, and above those on the positive real axis [63, 64]. The reduced Green’s function in (2.4) in turn satisfies

(2.5) |

Here . This equation is easily solved, with the result

(2.6a) | |||

for both fields inside, , while if both field points are outside, , | |||

For , | |||

Here, the denominator is

(2.6g) |

Note that in the strong coupling limit we recover the familiar results, for example, inside

(2.6h) |

Evidently, this Green’s function vanishes at and at .

We can now calculate the force on one of the -function points by calculating the discontinuity of the stress tensor, obtained from the Green’s function (2.3) by

(2.6i) |

Writing a reduced stress tensor by

(2.6j) |

we find inside

(2.6k) |

Let us henceforth simplify the considerations by taking the massless limit, . Then the stress tensor just to the left of the point is

(2.6la) | |||

From this we must subtract the stress just to the right of the point at , obtained from (2.6), which turns out to be in the massless limit | |||

(2.6lb) |

which just cancels the 1 in braces in (2.6la). Thus the force on the point due to the quantum fluctuations in the scalar field is given by the simple, finite expression

(2.6lm) |

This reduces to the well-known, Lüscher result [65, 66] in the limit ,

(2.6ln) |

and for is plotted in Fig. 1.

Recently, Sundberg and Jaffe [67] have used their background field method to calculate the Casimir force due to fermion fields between two -function spikes in dimension. Apart from quibbles about infinite energies, in the limit they recover the same result as for scalar, (2.6ln), which is as expected [68], since in the ideal limit the relative factor between scalar and spinor energies is in spatial dimensions, i.e., 7/4 for three dimensions and 1 for one.

We can also compute the energy density. In this simple massless case, the calculation appears identical, because (reflecting the conformal invariance of the free theory). The energy density is constant [(2.6k) with ] and subtracting from it the -independent part that would be present if no potential were present, we immediate see that the total energy is , so . This result differs significantly from that given in Refs. [61, 60, 69], which is a divergent expression in the massless limit, not transformable into the expression found by this naive procedure. However, that result may be easily derived from the following expression for the total energy,

(2.6lo) | |||||

if we integrate by parts and omit the surface term. Integrating over the Green’s functions in the three regions, given by (2.6a), (2.6), and (LABEL:gleft), we obtain for ,

(2.6lp) |

where the first term is regarded as an irrelevant constant ( is constant), and the second is the same as that given by equation (70) of Ref. [60] upon integration by parts.

The origin of this discrepancy with the naive energy is the existence of a surface contribution to the energy. Because , we have, for a region bounded by a surface ,

(2.6lq) |

Here , so we conclude that there is an additional contribution to the energy,

(2.6lra) | |||||

(2.6lrb) |

where the derivative is taken at the boundaries (here , ) in the sense of the outward normal from the region in question. When this surface term is taken into account the extra terms in (2.6lp) are supplied. The integrated formula (2.6lo) automatically builds in this surface contribution, as the implicit surface term in the integration by parts. (These terms are slightly unfamiliar because they do not arise in cases of Neumann or Dirichlet boundary conditions.) See Fulling [70] for further discussion. That the surface energy of an interface arises from the volume energy of a smoothed interface is demonstrated in Ref. [62], and elaborated in section 2.4.

It is interesting to consider the behavior of the force or energy for small coupling . It is clear that, in fact, (2.6lm) is not analytic at . (This reflects an infrared divergence in the Feynman diagram calculation.) If we expand out the leading term we are left with a divergent integral. A correct asymptotic evaluation leads to the behavior

(2.6lrs) |

This behavior indeed was anticipated in earlier perturbative analyses. In Ref. [57] the general result was given for the Casimir energy for a dimensional spherical -function potential (a factor of was inadvertently omitted)

(2.6lrt) |

This possesses an infrared divergence as :

(2.6lru) |

which is consistent with the nonanalytic behavior seen in (2.6lrs).

### 2.2 Parallel Planes in Dimensions

It is trivial to extract the expression for the Casimir pressure between two function planes in three spatial dimensions, where the background lies at and . We merely have to insert into the above a transverse momentum transform,

(2.6lrv) |

where now . Then has exactly the same form as in (2.6a)–(LABEL:gleft). The reduced stress tensor is given by, for the massless case,

(2.6lrw) |

so we immediately see that the attractive pressure on the planes is given by ()

(2.6lrx) |

which coincides with the result given in Refs. [30, 71]. The leading behavior for small is

(2.6lrya) | |||

while for large it approaches half of Casimir’s result [9] for perfectly conducting parallel plates, | |||

(2.6lryb) |

The Casimir energy per unit area again might be expected to be

(2.6lryz) |

because then . In fact, however, it is straightforward to compute the energy density is the three regions, , , and , and then integrate it over to obtain the energy/area, which differs from (2.6lryz) because, now, there exists transverse momentum. We also must include the surface term (2.6lra), which is of opposite sign, and of double magnitude, to the term. The net extra term is

(2.6lryaa) |

If we regard as constant (so that the strength of the coupling is independent of the separation between the planes) we may drop the first, divergent term here as irrelevant, being independent of , because , and then the total energy is

(2.6lryab) |

which coincides with the massless limit of the energy first found by Bordag et al [58], and given in Refs. [30, 71]. As noted in section 2.1, this result may also readily be derived through use of (2.6lo). When differentiated with respect to , (2.6lryab), with fixed, yields the pressure (2.6lrx).

In the limit of strong coupling, we obtain

(2.6lryac) |

which is exactly one-half the energy found by Casimir for perfectly conducting plates [9]. Evidently, in this case, the TE modes (calculated here) and the TM modes (calculated in the following subsection) give equal contributions.

### 2.3 TM Modes

To verify this claim, we solve a similar problem with boundary conditions that the derivative of is continuous at and ,

(2.6lryada) | |||

but the function itself is discontinuous, | |||

(2.6lryadb) | |||

and similarly at . These boundary conditions reduce, in the limit of strong coupling, to Neumann boundary conditions on the planes, appropriate to electromagnetic TM modes: | |||

(2.6lryadc) |

It is completely straightforward to work out the reduced Green’s function in this case. When both points are between the planes, ,

(2.6lryadaea) | |||

while if both points are outside the planes, , | |||

(2.6lryadaeb) |

where the denominator is

(2.6lryadaeaf) |

It is easy to check that in the strong-coupling limit, the appropriate Neumann boundary condition (2.6lryadc) is recovered. For example, in the interior region, ,

(2.6lryadaeag) |

Now we can compute the pressure on the plane by computing the component of the stress tensor, which is given by (2.6lrw),

(2.6lryadaeah) |

The action of derivatives on exponentials is very simple, so we find

(2.6lryadaeaia) | |||

(2.6lryadaeaib) |

so the flux of momentum deposited in the plane is

(2.6lryadaeaiaj) |

and then by integrating over frequency and transverse momentum we obtain the pressure:

(2.6lryadaeaiak) |

In the limit of weak coupling, this behaves as follows:

(2.6lryadaeaial) |

which is to be compared with (2.6lrya). In strong coupling, on the other hand, it has precisely the same limit as the TE contribution, (2.6lryb), which confirms the expectation given at the end of the previous subsection. Graphs of the two functions are given in Fig. 2.

For calibration purposes we give the Casimir pressure in practical units between ideal perfectly conducting parallel plates at zero temperature:

(2.6lryadaeaiam) |

### 2.4 Surface energy as bulk energy of boundary layer

Here we show that the surface energy can be interpreted as the bulk energy of the boundary layer. We do this by considering a scalar field in dimensions interacting with the background

(2.6lryadaeaian) |

where

(2.6lryadaeaiao) |

with the property that . The reduced Green’s function satisfies

(2.6lryadaeaiap) |

This may be easily solved in the region of the slab, ,

(2.6lryadaeaiaq) |

Here , and

(2.6lryadaeaiar) |

This result may also easily be derived from the multiple reflection formulas given in section 3.1, and agrees with that given by Graham and Olum [72]. The energy of the slab now is obtained by integrating the energy density

(2.6lryadaeaias) |

over frequency and the width of the slab. This gives the vacuum energy of the slab

(2.6lryadaeaiat) | |||||

If we now take the limit and so that , we immediately obtain

(2.6lryadaeaiau) |

which precisely coincides with one-half the constant term in (2.6lp), with there replaced by here.

There is no surface term in the total Casimir energy as long as the slab is of finite width, because we may easily check that is continuous at the boundaries . However, if we only consider the energy internal to the slab we encounter not only the energy (2.6lo) but a surface term from the integration by parts. It is only this boundary term that gives rise to , (2.6lryadaeaiau), in this way of proceeding.

Further insight is provided by examining the local energy density. In this we follow the work of Graham and Olum [72, 73]. However, let us proceed here with more generality, and consider the stress tensor with an arbitrary conformal term,

(2.6lryadaeaiav) |

in dimensions, being the number of transverse dimensions. Applying the corresponding differential operator to the Green’s function (2.6lryadaeaiaq), introducing polar coordinates in the plane, with , , and

(2.6lryadaeaiaw) |

we get the following form for the energy density within the slab,

(2.6lryadaeaiax) |

From this we can calculate the behavior of the energy density as the boundary is approached from the inside:

(2.6lryadaeaiay) |

For for example, this agrees with the result found in Ref. [72] for :

(2.6lryadaeaiaz) |

Note that, as we expect, this surface divergence vanishes for the conformal stress tensor [74], where . (There will be subleading divergences if .)

We can also calculate the energy density on the other side of the boundary, from the Green’s function for ,

(2.6lryadaeaiba) |

and the corresponding energy density is given by

(2.6lryadaeaibb) |

which vanishes if the conformal value of is used. The divergent term, as , is just the negative of that found in (2.6lryadaeaiay). This is why, when the total energy is computed by integrating the energy density, it is finite for , and independent of . The divergence encountered for may be handled by renormalization of the interaction potential [72]. In the limit as , , we recover the divergent expression (2.6lryadaeaiau) for , or in general

(2.6lryadaeaibc) |

Therefore, surface divergences have an illusory character.

For further discussion on surface divergences, see section 4.1.

## 3 Casimir Effect Between Real Materials

### 3.1 The Lifshitz Formula Revisited

As a prolegomena to the derivation of the Lifshitz formula for the Casimir force between parallel dielectric slabs, let us note that the results in the previous section may be easily derived geometrically, in terms of multiple reflections. Suppose we have translational invariance in the and directions, so in terms of reduced Green’s functions, everything is one-dimensional. Suppose at and we have discontinuities giving rise to reflection and transmission coefficients. That is, if we only had the interface, the reduced Green’s function would have the form

(2.6lryadaeaiaa) | |||

for , while for , | |||

(2.6lryadaeaiab) |

Similarly, if we only had the interface at , we would have similarly defined reflection and transmission coefficients and . Transmission and reflection coefficients defined for a wave incident from the left instead of the right will be denoted with tildes. If both interfaces are present, we can calculate the Green’s function in the region to the right of the rightmost interface in the form

(2.6lryadaeaiaba) | |||

where may be easily computed by summing multiple reflections: | |||

(2.6lryadaeaiabb) |

For the TE -function potential (2.1), , and , and we immediately recover the result (2.6). But the same formula applies to electromagnetic modes in a dielectric medium with two parallel interfaces, where the permittivity is

(2.6lryadaeaiabc) |

In that case [75]

(2.6lryadaeaiabda) | |||

and | |||

(2.6lryadaeaiabdb) |

where . Substituting these expressions into (2.6lryadaeaiabb) we obtain

(2.6lryadaeaiabde) |

which coincides with the formula (3.16) given in Ref. [29].

However, to calculate most readily the force between the slabs, we need the corresponding formula for the reduced Green’s function between the interfaces. This may also be readily derived by multiple reflections:

Indeed, this reduces to (2.6a) when the appropriate reflection coefficients are inserted. The pressure on the planes may be computed from the discontinuity in the stress tensor, or

(2.6lryadaeaiabdg) |

from which the -potential results (2.6la) and
(2.6lb) follow immediately.
For the case of parallel dielectric slabs the TE modes therefore contribute
the following expression for the pressure^{1}^{1}1For the case of
dielectric slabs, the propagation constant is different on the two
sides; we omit the term corresponding to the free propagator, however.
In the energy, the omitted terms are proportional to the volume of each
slab, and therefore correspond to the volume or bulk energy of the material.:

(2.6lryadaeaiabdh) |

The contribution from the TM modes are obtained by the replacement

(2.6lryadaeaiabdi) |

except in the exponentials [75]. This gives for the force per unit area at zero temperature

(2.6lryadaeaiabdj) |

with the denominators here being []

(2.6lryadaeaiabdk) |

which correspond to the TE and TM Green’s functions, respectively. This is the celebrated Lifshitz formula [11, 12, 13, 14], which we shall discuss further in the following subsections. We merely note here that if we take the limit , and set , we recover Casimir’s result for the attractive force between parallel, perfectly conducting plates (2.6lryadaeaiam).

Henkel et al[76] have computed the Casimir force at short distances ( nm) from interactions between polaritons. Their result agrees with the Lifshitz formula with the plasma formula (2.6lryadaeaiabdag) employed, see Ref. [77, 78].

### 3.2 The Relation to van der Waals Forces

Now suppose the central slab consists of a tenuous medium and the surrounding medium is vacuum, so that the dielectric constant in the slab differs only slightly from unity,

(2.6lryadaeaiabdl) |

Then, with a simple change of variable,

(2.6lryadaeaiabdm) |

we can recast the Lifshitz formula (2.6lryadaeaiabdj) into the form

(2.6lryadaeaiabdn) |

If the separation of the surfaces is large compared to the wavelength characterizing , , we can disregard the frequency dependence of the dielectric constant, and we find

(2.6lryadaeaiabdo) |

For short distances,