Advances and Prospects in Casimir Physics

Abstract

In the present introductory to the Special Issue “75 Years of the Casimir Effect: Advances and Prospects”, we summarize the state of the art in this field of physics, briefly describe the topics of the contributing papers, formulate several unresolved problems, and outline possible pathways towards their resolution. Special attention is given to experiments on measuring the Casimir force, to the known problem of the dissipation of conduction electrons when one compares experiment with theory, and to the Casimir effect in novel materials and non-traditional situations. We conclude that in the future, this multidisciplinary quantum effect will continue to play a crucial role in both fundamental physics and its applications.

1. Introduction

The Casimir effect [1] was discovered 75 years ago, and now is an appropriate time to summarize its role in different physical phenomena, the results thus far obtained, and the unsolved problems, as well as to outline possible pathways towards their resolution.

At first sight, the Casimir prediction of the attractive force acting between two parallel, uncharged ideal metal planes kept at zero temperature could be considered to have a rather modest physical significance. The reason is that this force takes noticeable values only at extremely short separations between the plates, and both the ideal metal and the zero temperature are idealizations which are literally unrealizable in physical experiments.

The importance of the Casimir discovery greatly exceeded these expectations. The Casimir force is determined by the vacuum fluctuations of the electromagnetic and other quantum fields. These fluctuations are inherent to all physical phenomena in which Casimir forces may play some role. A few years after its discovery, the Casimir effect was generalized for the case of ideal metal planes kept at non-zero temperatures [2,3,4] and, within the framework of the Lifshitz theory [5,6,7], for two thick plates made of any material. It was shown that the Casimir force is a generalization of the van der Waals force [8] for separations where relativistic effects come into play, and to any temperature. The forces in question also act between atoms, molecules, and material surfaces, and in this case they are called Casimir–Polder forces [9].

There are a great number of applications of the Casimir force caused by the zero-point and thermal fluctuations of the electromagnetic field in condensed matter physics and atomic physics. In condensed matter physics, the Casimir force acts between any closely spaced surfaces made of metallic, dielectric, and semiconductor materials (see, e.g., Refs. [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31], reviews [32,33], and monographs [34,35]). In atomic physics, the Casimir–Polder force has been calculated in many systems [36,37,38,39,40,41,42,43,44,45,46,47]. It plays a primary role in the phenomena of quantum reflection [48,49,50,51,52,53,54,55] and Bose–Einstein condensation [56,57,58,59,60,61] (see also the monograph [62,63]).

Many measurements of the Casimir force have been performed by means of an atomic force microscope, where the sharp tip was replaced with a relatively large sphere (see Refs. [32,34,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78]), and by means of a micromechanical torsional oscillator [32,34,79,80,81,82,83,84,85]. Based on these measurement results, the Casimir force was applied for the creation of various next-generation micro- and nanoelectromechanical devices [86,87,88,89,90,91,92,93,94,95,96,97,98,99]. The Casimir–Polder force has been measured in experiments on quantum reflection [49,50,51,52] and Bose–Einstein condensation [56,57,60,61].

The Casimir effect for other than electromagnetic (scalar, spinor, gluon, etc.) fields has found prospective applications in elementary particle physics, for instance, in the bag model of hadrons (see Refs. [100,101,102,103,104,105,106] and the monograph [107]). It has been demonstrated that the Casimir effect is also of high importance in topologically non-trivial cosmological models, where the identification conditions play the same role as the materials boundaries, changing the spectrum of the vacuum fluctuations and leading to non-zero Casimir energy density [108,109,110,111,112,113]. A similar effect was found in multi-dimensional physics, where the extra spatial dimensions are compactified at some energy scale [114,115,116].

All of the above permits us to conclude that the Casimir force, regarded initially as an interesting but toy and somewhat exotic example, has developed over time into a broad research area, which is often called Casimir physics.

2. The Topics Highlighted in This Special Issue

This Special Issue, entitled “75 Years of the Casimir Effect: Advances and Prospects”, presents several scientific directions within the wide research area of Casimir-related phenomena. After the creation of the Lifshtz theory, which allows for the calculation of the Casimir force in plane-parallel configurations only, it was generalized for the case of arbitrary-shaped bodies [117,118,119,120,121,122,123,124,125,126] and, specifically, for the experimentally important configuration of a sphere above a plate [127,128,129,130,131,132,133,134,135]. In this Special Issue, this line of research is represented by Ref. [136], which is devoted to the application of the scattering approach for calculating the Casimir–Polder interaction with magnetodielectric bodies, by Ref. [137], which reviews an application of the method of derivative expansion in Casimir physics, by Ref. [138], which considers the Casimir forces with periodic structures, and by Ref. [139], calculating the Casimir–Polder force for a conducting cone.

Special attention in this Special Issue is devoted to recent progress in measuring the Casimir force. Reference [140] reviews the last experiments performed by means of an atomic force microscope. This includes measuring the normal Casimir force between the smooth surfaces of both non-magnetic and magnetic metals, normal and lateral Casimir forces between the corrugated surfaces, and the thermal Casimir force in graphene systems. The comparison of the experimental results with theory for graphene systems required the development of a novel approach to describing the response of graphene to the electromagnetic field because the previously used semi-phenomenological approaches, based on the Kubo formula, the two-dimensional Drude model, and density–density correlation functions [141,142,143,144], turned out to be insufficient. The new approach, which was found to be in agreement with the measurement data, uses the polarization tensor of graphene with non-zero values of the energy gap and the chemical potential found within the framework of thermal quantum field theory in (2 + 1) dimensions [145,146,147,148].

Two other experimental papers are devoted to the dynamical sensitivity of three-layer microelectromechanical systems exploiting the Casimir force to the optical properties of the intervening liquid layer [149] and to the planned experiment on measuring the Casimir pressure between two parallel plates spaced at micrometer separations [150]. Realization of the last experiment will allow to strengthen the constraints on the Yukawa-type corrections to the Newton law of gravitation and on the hypothetical constituents of dark matter and dark energy, such as axions, chameleons, symmetrons, and environment-dependent dilatons [151,152,153,154,155], which are often constrained from measuring the Casimir force [156,157,158,159,160,161,162]. One more experimentally oriented paper considers the possibility of compensating the electrostatic interaction between dielectric and metallic test bodies [163]. This investigation is directed towards solving the problem of surface patches, which complicates measurements of the Casimir force [164,165].

The major problem of Casimir physics, which has remained unresolved over the last 25 years, is the question of how to describe the free charge carriers correctly when calculating the Casimir force. It has been shown Refs. [32,34,72,74,75,80,81,85] that for metallic test bodies the Casimir force, calculated by means of the Lifshitz theory using the dissipative Drude model at low frequencies, is excluded by the measurement data. If the dissipationless plasma model is used in the calculations, the theoretical results agree with all precise experiments on measuring the Casimir force [32,34,72,74,75,80,81,85]. For the dielectric test bodies, theory comes into agreement with the measurement data only if the role of free charge carriers, which are present in all real dielectrics at any non-zero temperature, is omitted in the computations [32,34,65,69,70,166].

According to the results of Ref. [167] published in this Special Issue, the roots of the problem are not in accounting for or disregarding the dissipation properties of the conduction electrons in calculations of the Casimir force, but in the necessity of accounting for these properties correctly. It is shown that an account of the relaxation properties of conduction electrons at low frequencies by means of the Drude model in the region of propagating waves with any polarization and transverse magnetic evanescent waves does not lead to contradictions with the measurement data. The contradiction between the calculated and measured Casimir forces arises only when the Drude model is used to describe the response of metals to the low-frequency evanescent waves with transverse electric polarization.

It is common knowledge that the Drude model has numerous experimental confirmations in the region of propagating waves. It has also been confirmed by special experiments in the area of transverse magnetic evanescent waves [168]. As to the area of transverse electric evanescent waves, the Drude model lacks any reliable experimental confirmation. On this basis, it was concluded [167] that the experiments on measuring the Casimir force invalidate the Drude model in the area of transverse electric evanescent waves. An alternative experiment in the field of classical electrodynamics was proposed, which can independently confirm this important conclusion [169,170]. Reference [171] of this Special Issue suggests another experimental means of distinguishing between the Drude and plasma models, which is based on measuring the Lorentz force originating from thermal fluctuations.

It is interesting that for the Casimir force between two graphene sheets considered in Ref. [172] of this Special Issue, theory is in good agreement with the measurement data. The reason for this is that the electromagnetic response of graphene is described on the rigorous basis of quantum electrodynamics at non-zero temperature do not using any phenomenological approach like the Drude model. This is reached by employing the polarization tensor of graphene with any energy gap and the chemical potential found in the framework of thermal quantum field theory [145,146,147,148].

Several papers belonging to this Special Issue are devoted to the investigation of the Casimir effect in various specific configurations. Thus, in Ref. [173] the Casimir energy in (2 + 1)-dimensional field theories is considered which is interesting in connection with its application to novel two-dimensional materials, such as graphene, silicene, stanene, phosphorene, and others [174,175,176,177]. The Casimir forces in conformal field theories with defects and boundaries are discussed in Ref. [178]. The normal Casimir force for the planes with isotropic conductivity in the state of lateral motion is found in Ref. [179]. In Ref. [180], the Casimir–Lifshitz force of friction, which arises due to the relative motion of interacting bodies, and resulting heating are considered in the framework of fluctuational electrodynamics. Finally, it is explained in Ref. [181] how the Casimir force can be used to stabilize the levitation of a graphene sheet lifted by the repulsive force arising in an inhomogeneous magnetic field.

In a few papers included in this Special Issue, the Casimir effect is considered in rather non-standard situations and using some alternative approaches. For instance, in Ref. [182] the Casimir effect in axion electrodynamics is investigated where, due to the presence of an additional pseudoscalar quantity, the relationship between the vectors of the electric field, magnetic induction, electric displacement, and the magnetic field becomes more complicated than in the standard electrodynamics of continuous media. This subject is closely related to new materials called topological insulators, which are of great practical interest.

An interesting approach to the Casimir effect based on semi-classical electrodynamics is discussed in Ref. [183] in this Special Issue. This paper includes a preface discussing different concepts of intermolecular forces in the early history of physics. It is shown how the semi-classical approach results in the familiar Lifshits formula for Casimir free energy. One more study [184] is devoted to a derivation of the Casimir pressure between two Chern–Simons boundary layers deposited on dielectric substrates. For this purpose, gauge-invariant formalism was developed using the electric and magnetic Green functions. Two more papers are devoted to the dynamical Casimir effect. One of them [185] considers an analogy between the dynamical Casimir effect, black holes, and the radiation temperature of an accelerated electron. The other one [186] investigates an asymmetric force acting on a moving mirror modeled by the potential, which is equal to the difference between the delta function and its derivative in two-dimensional space–time. We also list one more paper published in this Special Issue which is devoted to the one-loop correction to the mass of an electron in a homogeneous magnetic field [187]. This issue has been considered by several authors but with somewhat differing results. Keeping in mind that the radiative corrections are similar in their physical nature to the Casimir effect, it is necessary to resolve all existing discrepancies.

As was discussed in Section 1, the Casimir effect arises not only in configurations with material boundaries but also in spaces with non-trivial topology. Because of this, it plays a significant role in gravitation and cosmology and in multi-dimensional theories of elementary particle physics. Four papers in this Special Issue represent this scientific direction in the field of Casimir physics. It is known that anti-de Sitter space–time plays a significant role in cosmology. Generally speaking, the braneworld model contains the fields propagating in the bulk or localized on the branes. The boundary conditions on the branes induce Casimir-type contributions to the expectation values of physical observables. In Ref. [188], the vacuum expectation of the surface stress–energy tensor for a scalar field is calculated in the configuration of two parallel branes orthogonal to the boundary of anti-de Sitter space–time.

Other objects of importance to cosmology are the so-called cosmic strings, i.e., the topological defects which could have been created in the early Universe during cosmological phase transitions. The Casimir interaction between two cosmic strings arising due to vacuum fluctuations of the scalar field with minimal coupling is considered in Ref. [189] in the cases both small and large separation distances, taking into account the transverse size of a string.

Finally, the Casimir effect for two parallel plates in a weak gravitational field and the wormholes determined by the Casimir energy densities of the Yang–Mills field are discussed in Ref. [190]. The same paper studies the Casimir energy density in Euclidean space–time with a non-trivial topology, equivalent to imposing the so-called helix identification conditions.

3. Future Prospects

As has been demonstrated above, the Casimir effect is a wide research area, with implications for practically all branches of modern physics. It is actively investigated both theoretically and experimentally by many research groups working in many countries. Over the last few years, a number of new breakthrough results have been obtained. Below, we outline the most crucial problems in this research area to be solved in the future.

Although for metallic test bodies at separations below several micrometers, the Casimir force was already measured with high precision, at larger separations, and for the test bodies made of semiconductor and dielectric materials, new breakthrough experimental results are expected in near future. Progress in precise force measurements can be stimulated using the traditional and novel techniques (see Refs. [140,150] published in this Special Issue) and by compensating for the spurious electric forces [163,164,165]. The obtained results are considered to be used for the creation of next-generation micro- and nanoelectromechanical devices driven by the Casimir force (see Ref. [149] of this Special Issue, elaborating this scientific direction).

One can expect that the problem of disagreement between experiment and theory, taking into account the dissipation of conduction electrons by means of the Drude model (this problem is often called the Casimir puzzle [32,34,75]), will be solved soon. In this Special Issue, the problem has already been narrowed down to the inapplicability of the Drude model in the region of transverse electric evanescent waves [167]. It has been shown [191,192,193] that if to modify the Drude model in this area phenomenologically by adding a spatially non-local contribution, the theoretical predictions come to an agreement with the measurement data. It remains to put the phenomenology on a solid fundamental basis as has already been made for the electromagnetic response of graphene [77,78,145,146,147,148].

In near future, Casimir physics is going also to find new applications in modern theoretical approaches beyond the Standard Model, e.g., in the brane models, multi-dimensional physics with compacted extra dimensions, the theory of topological defects, etc. It is to be used for obtaining stronger constraints on the Yukawa-type corrections to Newtonian gravity and the hypothetical particle constituents of dark matter and dark energy.

We hope that the papers published in this Special Issue, “75 Years of the Casimir Effect: Advances and Prospects”, will be helpful by stimulating further developments in this prospective field of physics.