Casimir Puzzle and Casimir Conundrum: Discovery and Search for Resolution

We review complicated problems in the Lifshitz theory describing the Casimir force between real material plates made of metals and dielectrics including different approaches to their resolution. It has been shown that both for metallic plates with perfect crystal lattices and for any dielectric plates the Casimir entropy calculated in the framework of the Lifshitz theory violates the Nernst heat theorem when the well approved dielectric functions are used in computations. The respective theoretical Casimir forces are excluded by the measurement data of numerous precision experiments. In the literature this situation received the names of the Casimir puzzle and the Casimir conundrum for the cases of metallic and dielectric plates, respectively. The review presents a summary of the main facts on this subject on both theoretical and experimental sides. Next, we discuss the main approaches proposed in the literature in order to bring the Lifshitz theory in agreement with the measurement data and with the laws of thermodynamics. Special attention is paid to the recently suggested spatially nonlocal Drude-like response functions which take into account the relaxation properties of conduction electrons, as does the standard Drude model, but lead to the theoretical results in agreement with both thermodynamics and the measurement data through the alternative response to quantum fluctuations off the mass shell. Further advances and trends in this field of research are discussed.


Introduction
The Casimir effect is one of the counterintuitive physical phenomena. It is somewhat against expectations that two parallel, electrically neutral ideal metal plates at zero temperature spaced in vacuum at a distance a attract each other. However, Casimir [1] has proven that there is an attractive force per unit area of the plates in this situation which depends on a, the Planck constanth, and the speed of light c The Casimir force arises due to the zero-point oscillations of quantized electromagnetic field. In free space the energy density of these oscillations is infinitely large and unobservable. All the physical energies are measured from this infinity. In the presence of parallel plates, however, the tangential component of the electric field and the normal component of the magnetic induction vanish on their surfaces and the third component of the wave vector perpendicular to the plates becomes discrete. In so doing the vacuum energy density in the presence of plates remains infinitely large but if we, following the common procedure, subtract from it the vacuum energy density in free space, the result turns out to be finite This is the Casimir energy per unit area of the plates. In that case (1) follows from (2) by the negative differentiation with respect to a. Thus, the Casimir effect is directly connected with the concept of the quantum vacuum which is the most fundamental and not clearly understood type of physical reality. In spite of the elaborated in detail renormalization and regularization procedures, which help to obtain physically meaningful calculation results, the enormously large vacuum energy obtained after the momentum cut off is sometimes considered as catastrophic [2]. It should be mentioned also that the energy of the quantum vacuum is directly connected with the cosmological constant [3], dark energy and the acceleration of the Universe expansion [4]. This puts the Casimir effect in the list of closely connected greatest problems of modern physics. It is not surprising then that experimental and theoretical investigations of the Casimir force met with outstanding problems which received the names of the Casimir puzzle and the Casimir conundrum.
At first, no fundamental problems in the theory of the Casimir effect were anticipated. Lifshitz [5,6] developed the general theory expressing the Casimir free energy and force between two thick plates (semispaces) kept at any temperature T and made of real materials described by the frequency-dependent dielectric permittivities. This theory was generalized for the plates possessing magnetic properties described by the frequency-dependent magnetic permeability [7]. It was shown that the familiar van der Waals force is nothing but the Casimir force at separations between the plates below a few nanometers where the speed of light can be considered as infinitely large. The Casimir results (1) and (2) were obtained from the Lifshitz theory at T = 0 for the plates made of a perfectly reflecting material. The Lifshitz results for two thick plates were generalized for the case of planar systems containing any number of material layers with different dielectric permittivities [8].
First experiments on measuring the Casimir force were not enough precise and have been found only in qualitative agreement with theoretical predictions of the Lifshitz theory (see [9] for a review). It was shown [10], however, that at large separations the thermal Casimir force between metallic plates calculated using the Lifshitz theory and the dielectric permittivity of the dissipative Drude model is equal to one half of that found for ideal metal plates at nonzero temperature. This result generated a serious incomprehension. On the one hand, at low frequencies, which determine the force behavior at large separations, the Drude model was well tested in numerous experiments and even in technical applications. On the other hand, it was expected that with increasing separation the plate metal should behave more and more closer to the ideal one. That is why the special prescription was proposed [10] which is supplementary to the formalism of the Lifshitz theory (see below).
Wide public attention to the internal problems of this theory was attracted in 2000 after a publication of the following result. It was found [11] that at short separations the thermal correction to the Casimir force computed using the Drude model is relatively large and decreases the force magnitude, i.e., corresponds to a repulsion which is quite unexpected. By contrast, the Lifshitz theory combined with the dielectric permittivity of the dissipationless plasma model (which should not be applicable at low frequencies) predicted a smooth approach of the Casimir force to the ideal metal value at large separations and a very small thermal correction at short separation. The latter is of the same sign as the zero-temperature force and only slightly increases its magnitude [12]. These problems were dramatized by an inconsistency of the Lifshitz theory using the Drude model with the laws of thermodynamics. It was proven [13][14][15] that in the configuration of two metallic plates with perfect crystal lattices described by the Drude model the Casimir entropy goes with vanishing temperature to a nonzero negative constant depending on the parameters of a system in violation of the third law of thermodynamics (the Nernst heat theorem). It was also shown [13][14][15] that the Lifshitz theory using the plasma model leads to the positive Casimir entropy which goes to zero when the temperature vanishes in agreement with the Nernst heat theorem. More recently, the same results were obtained for several other Casimir configurations [16][17][18][19].
Precise measurements of the Casimir interaction between metallic test bodies added to the complexity of this situation. In a series of experiments performed by means of a micromechanical torsional oscillator and an atomic force microscope the Lifshitz theory using the Drude model at low frequencies was unambiguously excluded by the measurement data whereas the same theory using the plasma model was found to be in good agreement with the measurement results [20][21][22][23][24][25][26][27]. The dielectric permittivity of a metal in these experiments was found from the measured optical data for its complex index of refraction extrapolated by means of either the Drude or the plasma model to the region of low frequencies where the optical data are not available. Note that in the single experiment which was interpreted in favor of the Drude model [28] the Casimir force was not directly measured but extracted using the fitting procedure from up to an order of magnitude larger force presumably determined by the surface patches. It was shown [29,30] that the interpretation of this experiment suffers from serious uncertainties.
The situation with dielectric test bodies is closely parallel to that with metallic ones. It was found that the Casimir entropy calculated using the Lifshitz theory for ideal dielectrics (insulators possessing zero electrical conductivity) satisfies the Nernst heat theorem [31][32][33][34][35][36]. However, at any nonzero temperature the dielectric permittivity of real dielectric bodies includes the term describing small but nonzero electric conductivity at a constant current (the so-called dc conductivity). If this term is taken into account, as it should be done, the Casimir entropy calculated using the Lifshitz theory goes to a nonzero positive constant depending on the parameters of a system with vanishing temperature, i.e., violates the Nernst heat theorem [31][32][33][34][35][36].
The experimental results obtained for dielectric test bodies also resemble the case of metallic plates. If the dc conductivity of plate material is taken into account, the predictions of the Lifshitz theory are excluded by the measurement data [37][38][39][40][41]. If, however, the dc conductivity is simply omitted in calculations, the theoretical results are found in agreement with the data [37][38][39][40][41][42]. Once again, as in the case of metals, the thermodynamically consistent theory agrees with the experimental results. This is reached, however, by disregarding the real physical phenomena -the dissipation of conduction electrons for metals and the dc conductivity for dielectrics. That is why the above problems have been called in the literature the Casimir puzzle and the Casimir conundrum [43][44][45][46][47]. Taking into account that for metals and dielectrics the above inconsistencies originate from different sources (see below), it was suggested [43,47] to call them a puzzle and a conundrum, respectively.
In this article, we review different efforts to understand both the puzzle and the conundrum in Casimir physics undertaken in the literature during the last 20 years. We start with the more rigorous mathematical formulation of the essence of these problems and elucidate in more detail the theoretical and experimental parts of the Casimir puzzle for metallic and the Casimir conundrum for dielectric test bodies, respectively. The basically viable approaches to the resolution of these problems can be of two types. The approaches of the first type search for some physical effects which could make an impact on the measurement results but were not taken into account (or were accounted for improperly) in the comparison between experiment and theory. Among them one could mention the role of impurities of a crystal lattice and respective residual relaxation, a possibility of the alternative sets of optical data for the complex index of refraction, an impact of the surface roughness and patch potentials etc. (see below for the references to each of these approaches).
The approaches of the second type admit that some serious modifications of the Lifshitz theory might be necessary for a resolution of the above problems. These approaches deal with generalizations of the Lifshitz theory for the case of configurations with nonplanar boundaries, with taken into account spatial dispersion, screening effects etc. (the related references are provided below). In this review, we consider both types of attempts to find a resolution of the Casimir puzzle and the Casimir conundrum including a very recent one which shows considerable promise.
The review is organized as follows: In Section 2, we briefly present the formulations of the Lifshitz theory in terms of real and imaginary frequencies. In Section 3, a correlation is made between the thermal Casimir forces for ideal and real metals. The thermodynamic and experimental parts of the Casimir puzzle for real metals are considered in Section 4. Section 5 is devoted to the thermal Casimir force between ideal and real dielectrics and Section 6 -to the thermodynamic and experimental parts of the Casimir conundrum. In Section 7, we summarize the main approaches to a resolution of the Casimir puzzle and the Casimir conundrum. In Section 8, the new way toward resolving the Casimir puzzle is considered. Section 9 is devoted to the discussion and in Section 10 the reader will find our conclusions.

The Lifshitz Theory of the Casimir Force
There are different approaches to derivation of the Lifshitz theory starting from the fluctuation-dissipation theorem of statistical physics [5,6,48] and quantum electrodynamics with continuity boundary conditions on the boundary surfaces [9,49,50]. For a discussion of the Casimir puzzle and conundrum, we need only the expression for the Casimir free energy and force in the configuration of two thick plates (semispaces) separated by a gap of width a and kept at temperature T in thermal equilibrium with the environment. In the framework of the Lifshitz theory it is assumed that the material of the plates is described by the frequency-dependent dielectric permittivity ε(ω) and magnetic permeability µ(ω). This means that the original formulation of the Lifshitz theory does not take into account the spatial dispersion. If the plate material is nonmagnetic, one should put µ = 1.

Formulation in Terms of Real Frequencies
The immediate result of derivation using the fluctuation-dissipation theorem is the following Lifshitz formula for the Casimir free energy per unit area of the plates Here, k ⊥ = k 2 1 + k 2 2 is the projection of the wave vector on a plane of the plates (it is perpendicular to the Casimir force), ω is the frequency, k B is the Boltzmann constant and q = q(ω, k ⊥ ) = (k 2 ⊥ − ω 2 /c 2 ) 1/2 . The reflection coefficients for the transverse magnetic (TM) and transverse electric (TE) polarizations of the electromagnetic field are given by where It is seen that (4) are the familiar Fresnel reflection coefficients of classical electrodynamics. The Casimir force per unit area of the plates is obtained from (3) by the negative differentiation with respect to a The integration in k ⊥ in (3) and (6) is performed from 0 to ∞ at any fixed ω. In so doing the integration over k ⊥ > ω/c describes the contribution from the off-the-mass-shell electromagnetic fluctuations which are also often called the evanescent waves, as opposed to the propagating waves which are characterized by k ⊥ ω/c and correspond to the on-the-mass-shell fluctuations.
Note that the factor coth[hω/(2k B T)] in (3) and (6) implies an origin of the Casimir force at T = 0 from both the zero-point and thermal fluctuations of the electromagnetic field. This factor arises from the free energy of an oscillatorh after an application of the Cauchy theorem [9]. For the off-the-mass-shell fluctuations the quantity q is real. Because of this, the contribution of these fluctuations to the Casimir free energy and force can be calculated using (3) and (6). However, for the on-the-mass-shell fluctuations the quantity q becomes pure imaginary. As a result, both (3) and (6) become the integrals of rapidly oscillating functions which plagues their calculation. Because of this, another mathematically equivalent formulation of the expressions (3) and (6) was suggested [5].

Formulation in Terms of Imaginary Matsubara Frequencies
This formulation is based on the fact that the dielectric permittivity is a causal function which does not possess either poles or zeros in the upper half plane of complex frequency. As a consequence, the dielectric permittivity along the imaginary frequency axis takes the real values ε(iξ). Finally, the equivalent formulation of (3) is given by the Lifshitz formula where the prime on the summation sign divides the term with l = 0 by 2 and are the Matsubara frequencies. The quantities q l , r TM (iξ l , k ⊥ ) and r TE (iξ l , k ⊥ ) are obtained from q(ω, k ⊥ ), r TM (ω, k ⊥ ) and r TE (ω, k ⊥ ) defined above by a replacement of ω with iξ l .
In a similar way, the expression (6) for the Casimir force takes the form The expressions (8) and (10) are convenient for both analytical and numerical calculations because q l is a real positive number at any l.

Thermal Casimir Force between Ideal and Real Metals: First Surprise
The Casimir free energy (8) and force (10) are defined at nonzero temperature T and, therefore, are called thermal. In the limiting case T → 0 one has the discrete Matsubara frequencies ξ l are replaced with the continuous frequency ξ, q l is replaced with q = (k 2 ⊥ + ξ 2 /c 2 ) 1/2 and we obtain from (8) and (10) the expressions for the Casimir energy E(a) and force F(a) between real metal plates at zero temperature.
The ideal metal is perfect conductor at all frequencies including the zero frequency, so that This is in line with the prescription [10] which demands to consider the limit of ε to infinity in (4) first and the limit of frequency to zero second. Introducing into the Lifshitz formula (8) written at T = 0 with account of (11) the dimensionless variables y = 2aq and ζ = 2aξ/c, and taking into account (12), one finds the Casimir energy between ideal metal plates which is in agreement with the Casimir result (2). Similarly, the Lifshitz formula (10) written at T = 0 for ideal metal plates results in as was originally obtained by Casimir [1] considering the zero-point oscillations of quantized electromagnetic field. An application of the Lifshitz theory at nonzero temperature becomes more involved. We illustrate it first in the case of large separation distances between the plates. In this case all terms of (8) and (10) with l 1 are exponentially small and only the term with l = 0 determines the total result. The formal condition for the application of large separation limit is [9] a ≫h c At room temperature T = 300 K this gives a ≫ 0.64 µm, so that at a 6 µm one can safely use only the terms of (8) and (10) with l = 0.
Then, for ideal metal plates, i.e., under a condition (12), the Casimir free energy and force at large separations are given by where ζ(z) is the Riemann zeta function.
For real metals at large separations between the plates one might expect a similar result. This expectation, however, lacks of support from calculations. The point is that the low-frequency response of metals to the electromagnetic field is described by the dissipative Drude model where ω p is the plasma frequency and γ(T) ≪ ω p is the relaxation parameter.
For the nonmagnetic metals one has from (5) Then, from (4) we obtain which is in contradiction with (12) valid for an ideal metal. As a result, in the limiting case of large separations the terms of (8) and (10) with l = 0 are equal to i.e., by the factor of 2 smaller in magnitude than for an ideal metal in (16). It is interesting that if, instead of (17), the dissipationless plasma model is used, one finds This leads to the following values of the reflection coefficients at zero frequency: For the Casimir free energy and force defined using the plasma model at large separations, the Lifshitz theory leads to where r TE,p (0, k ⊥ ) is defined in (23). Taking into account that in the limiting case of ideal metal (ω p → ∞) it holds lim (25) in accordance to (23), we conclude that the results (24) obtained for real metal described by the plasma model join smoothly with the respective results (16) obtained for ideal metal plates. This could be considered as an advantage of the plasma model, as compared to the Drude one. It should be remembered, however, that in the region of quasistatic frequencies the relaxation properties of conduction electrons in metals disregarded by the plasma model play an important role and should be taken into account. Moreover, according to the Bohr-van Leeuwen theorem of classical statistical physics the TE electromagnetic fields should decouple from matter in the classical limit. According to the results of [51], this leads to the zero value of r TE in this limit (see, however, Section 7 concerning the role of quantum fluctuations off the mass shell at all separations).
The complexity to this problem was added by a calculation of the thermal correction to the Casimir force between metallic plates described by the Drude model [11]. It turned out that this correction takes relatively large in magnitude negative values and, thus, decreases the magnitude of the Casimir force which is a counterintuitive result. We recall that for the ideal metal plates at room temperature the thermal correction to the zero-temperature force at separations below 1 µm is negligibly small. This result is justified by a smallness of parameter T/T eff where k B T eff =hc/(2a).
To illustrate a fundamental difference between the thermal effects in the Casimir force predicted by the Drude and plasma models we consider the quantity computed by (10) using the Drude (17) and the plasma (21) dielectric permittivities of gold with ω p = 9.0 eV and γ(T) = 0.035 eV at T = 300 K. The computational results in percent are shown in Figure 1 as the functions of separation between the plates by the solid and dashed lines for the Drude and plasma models, respectively. As is seen in Figure 1, the Lifshitz theory using the Drude model predicts rather large in magnitude thermal correction to the Casimir force of the opposite sign to the zero-temperature contribution up to the separation of 6.3 µm where it vanishes [11]. For example, at a = 500 nm, 700 nm, and 1 µm this correction is equal to -6.4%, -9.4%, and -13.8%, respectively (see the solid line). In contrast to this, the Lifshitz theory using the plasma model predicts the thermal correction of the same sign as the zero-temperature Casimir force which magnitude is in qualitative agreement with that for ideal metal plates, i.e., very small at separations below 1 µm [12]. Thus, at separations a = 500 nm and 1 µm we have ∆ T F p = 0.058% and 0.29%, respectively. These results remain almost unchanged when one uses not the simple Drude and plasma models (17) and (21) but the dielectric permittivities obtained from the measured optical data for the complex index of refraction of gold [52], extrapolated by the Drude and plasma models down to zero frequency where the optical data are unavailable.
Starting from 2000, the problems in the Lifshitz theory have attracted considerable attention. The surprising thing is that the use of the well-tested Drude model leads to somewhat strange and unexpected results whereas the results obtained using the seemingly inapplicable plasma model are quite reasonable from the theoretical point of view. As is seen in the next section, at a later time this problem was further aggravated both theoretically and experimentally.

Thermodynamic and Experimental Parts of the Casimir Puzzle for Real Metals
As was mentioned in Section 1, first measurements of the Casimir force were not enough precise and demonstrated only a qualitative agreement with theoretical predictions of the Lifshitz theory. With an advent of micromechanical devices, it has become possible to perform precise and reproducible measurements of small forces at separations below a micrometer. At the same time, quick progress in analytical calculations using the powerful computer means made more accessible an investigation of the asymptotic behaviors of complicated mathematical expressions. Taken together, these achievements gave the possibility to compare in more detail the predictions of the Lifshitz theory with the measurement data and correlate its formalism with the fundamental laws of thermodynamics. Below we consider both the experimental and thermodynamic aspects of the Lifshitz theory. It is shown that they constitute two parts of the grave problem which received the name of the Casimir puzzle. We begin with the thermodynamic part.

The Casimir Entropy for Metallic Plates and the Nernst Heat Theorem
The Casimir free energy per unit area of two parallel metallic plates kept at temperature T in thermal equilibrium with the environment is expressed by the Lifshitz formula (8). Taking into consideration that there are problems in the Lifshitz theory considered in Section 3, it is interesting to perform some kind of thermodynamic checking for a behavior of the Casimir free energy and respective entropy at vanishing temperature when the Drude and the plasma models of the dielectric response are used. It is well known that in accordance to the third law of thermodynamics (the Nernst heat theorem) with vanishing temperature the entropy of a closed system in the state of thermal equilibrium must go to zero or to some universal constant which does not depend on the parameters of this system [53,54].
To check whether or not the Casimir entropy satisfies the Nernst heat theorem the metal of the plates with perfect crystal lattice was considered [13][14][15]. Perfect crystal lattice is a basic theoretical model of condensed matter physics possessing the nondegenerate ground state. For perfect crystal lattices the relaxation parameter of the Drude model (17) goes to zero with vanishing temperature as γ(T) ∼ T 2 . This dependence is caused by the electron-electron scattering and is followed at all T below the temperature of liquid helium [55]. In fact at higher T the relaxation parameter decreases following the power law with different powers but the condition γ(T) ≪ ξ 1 (T) is satisfies in all cases. This is shown in [15].
For obtaining the asymptotic behavior of the Casimir free energy F D when the Drude model is used one should substitute ε(iξ) = ε D (iξ) in (8). It is convenient to separate the term with l = 0 and expand the sum of all other terms in powers of the small parameter γ(T)/ξ 1 (T) ≪ 1. Then, calculating the Casimir entropy as one can present S D in the form [9,15] The Casimir entropy S p calculated using the plasma model was found [14,15] as a perturbation expansion in powers of the small parameter δ 0 /a ≪ 1 where δ 0 = c/ω p is the skin depth of a metal Note that the contribution of the zeroth order in δ 0 /a in (29) coincides with the Casimir entropy for the ideal metal plates at low temperature [56]. It is seen that i.e., the Lifshitz theory using the plasma model satisfies the Nernst heat theorem. This is not the case, however, when the Drude model is used. From (28) with the help of (30) one obtains Expanding this expression in powers of the small parameter δ 0 /a, we arrive at The quantities (31) and (32) depend on the parameters of a system such as the separation between the plates and the plasma frequency which means that the Nernst heat theorem is violated. Note that in [57,58] this violation was attributed to the role of eddy currents which explain big difference between theoretical predictions for the Casimir force when the Drude and the plasma models are used. According to [57,58], the nonzero entropy (32) at T = 0 is explained by an initiation of the correlated glassy (i.e., nonequilibrium) state.
Here we considered the parallel plates made of a nonmagnetic metal. The same results are, however, valid for other geometric configurations and for magnetic metals [16][17][18][19]. These results are really puzzling because the Lifshitz theory arrives to a conflict with the fundamental law of thermodynamics when using the Drude response function in the frequency region where it is well applicable. It is no less surprising that an agreement of the Lifshitz theory with thermodynamics is restored when the plasma response function is used in the region of quasistatic frequencies, i.e., outside of its application region.

The Lifshitz theory Confronts Experiments with Metallic Test Bodies
First precise measurements of the Casimir force allowing a discrimination between theoretical predictions using the plasma and Drude response functions have been performed by means of micromechanical torsional oscillator in the configuration of an Au-coated sapphire sphere above an Au-coated polysilicon plate [20][21][22][23]. The thicknesses of Au coatings were sufficiently large in order the test bodies could be considered as all-gold. The dielectric permittivity of Au along the imaginary frequency axis was found by means of the Kramers-Kronig relations using the available tabulated optical data of Au [52] extrapolated down to zero frequency either by the Drude or the plasma model [59].
As discussed in Section 2, the Lifshitz theory was formulated for a configuration of two parallel plates. It turned out, however, that the sphere-plate configuration is much more convenient for precise measurements of the Casimir interaction. Taking into account that in the early 21st century the exact expression for the Casimir force between a sphere and a plate was not available, the comparison between experiment and theory have been made using the so-called proximity force approximation (PFA) [9,59,60].
According to PFA, the Casimir force between a sphere and a plate F sp is approximately equal to where F (a, T) is the Casimir free energy per unit area of two parallel plates defined in (8), a is the minimum sphere-plate separation, and R in the sphere radius. The error of this approximation was not known precisely at that time but it was expected to be of the order of a/R. Taking into account that the measurements [20][21][22][23] were performed at a < 1 µm with the sphere radius R ≈ 150 µm, this error does not exceed 0.7%. The most precise experiments on measuring the Casimir interaction were performed in the dynamic mode where not the Casimir force but its gradient F ′ sp has been measured. By differentiating (33) with respect to a, the force gradient is expressed by the Lifshitz formula (10) for the force per unit area of two parallel plates Another point which was taken into account when comparing experiment with theory is an impact of the surface roughness. The roughness profiles on both interacting surfaces have been examined using an atomic force microscope. The heights of the maximum roughness peaks were found to be much less than the minimum separation between a sphere and a plate. In this case the Casimir force F(a, T) between two parallel plates with account of surface roughness was found using an additive method of geometrical averaging of the forces (10) calculated at the local separations a i over the profiles of both surfaces [9,59,61]. The gradient of the Casimir force with account of surface roughness was then calculated using (34).
In Figure 2, we present the typical comparison between the measurement data obtained using a micromechanical torsional oscillator in high vacuum [22] and theoretical predictions of the Lifshitz theory.
The predictions found with the Drude and plasma extrapolations of the optical data are shown by the blue and red bands, respectively, as the functions of separation. The widths of the bands reflect the total theoretical error which includes inaccuracies of the PFA, errors in the optical data and in the sphere radius. The mean experimental data are shown as crosses. The arms of the crosses undicate the total experimental errors. All errors are found at the 95% confidence level. Note that in [22] the comparison between experiment and theory was made in terms of the effective Casimir pressure between two parallel plates which is equivalently presented here in terms of the originally measured gradients of the Casimir force between a sphere and a plate.
As is seen in Figure 2, the Lifshitz theory using the Drude model for an extrapolation of the optical data is experimentally excluded. The same result was obtained over the separation region from 160 to 750 nm. From Figure 2 it is apparent that the Lifshitz theory using the plasma model for an extrapolation is in a very good agreement with the measurement data. This is the case also over the entire measurement range [22].
As the second example of comparison between experiment and theory, we consider experiment on measuring the gradient of the Casimir force between a sphere of R ≈ 62 µm radius and a plate with coated by a magnetic metal Ni surfaces [25,26]. These experiments were performed in high vacuum using an atomic force microscope operated in the dynamic mode. The metallic coatings of the test bodies were not magnetized, so that the magnetic force did not contribute to the measurement results. It should be recalled also that at room temperature the magnetic permeability µ(iξ) falls to unity at frequencies much below the first Matsubara frequency [62]. Because of this, the magnetic properties of Ni make an impact on the Casimir force only through the term of (10) with l = 0 where µ(0) = 110 whereas in all terms with l 1 one should put µ(iξ l ) = 1.
The comparison between an experimen and theory was made in the same way as above using the optical data of Ni [52] extrapolated to zero frequency either by the Drude or by the plasma model, the proximity force approximation, and the additive approach to an account of the surface roughness. In Figure 3, the predictions of the Lifshitz theory using the Drude and plasma models are shown as the a (nm) The gradients of the Casimir force between an Au-coated sphere and an Au-coated plate measured by means of micromechanical torsional oscillator [22] (crosses) are compared with theoretical predictions of the Lifshitz theory using the Drude and plasma extrapolations of the optical data of Au (blue and red bands, respectively).
blue and red bands, respectively, whereas the measurement data are shown as crosses with all errors determined at the 67% confidence level [25]. It is seen that the Lifshitz theory using the Drude model is excluded by the measurement data which are in good agreement with the same theory using the plasma model. Altogether the Lifshitz theory using the Drude model was excluded over the separation region from 223 nm (the minimum separation of this experiment) to 420 nm. It should be noted that measurements of the Casimir interaction between magnetic metals have an important difference as compared to experiments using the gold test bodies. As is seen in Figure 2, the predictions of the Lifshitz theory using the Drude model (the blue band) are situated below the experimental crosses. This means that in order to bring these predictions in agreement with the data some additional attractive force would be needed. By contrast, for the magnetic test bodies the predictions of the Lifshitz theory using the Drude model lie above the measurement data. An agreement between them could be reached only at the expense of some extra repulsive force. Thus, both sets of experiments make any attempt to bring the Lifshitz theory combined with the Drude model in agreement with measurements a challenging task (see Section 7).
From Figures 2 and 3 it is seen that, although the Lifshitz theory using the Drude model is clearly excluded by the measurement data, its predictions differ from these data (and from the predictions of the same theory using the plasma model) by only a few percent of the force gradient. It was shown, however, that using the differential force measurements, originally suggested for searching the Yukawa-type corrections to Newton's gravitational law [63,64], it is possible to increase a difference between the predictions of the Lifshitz theory using the Drude and plasma response functions by the several orders of magnitude [65,66]. Using this idea, the differential Casimir force was measured between a Ni-coated sphere and Au and Ni sectors covered with an Au overlayer [27]. A presence of the overlayer makes almost equal the contributions of all nonzero Matsubara frequencies in the positions of a sphere above the Au and Ni sectors. As a result, the differences in the Casimir forces acting on a sphere in these positions, calculated by the Lifshitz theory using the Drude and the plasma model extrapolations of the optical data, diverge by up to a factor of 1000. The measurement results for the differential Casimir force were found in good agreement with the predictions of the Lifshitz theory using the plasma model extrapolations of the optical data. The alternative predictions using the Drude model, which differ from the confirmed ones by up to a factor of 1000, were found to be out of all proportion to the actual size of the measured differential force. The comparison between the predictions of the Lifshitz theory and the measurement data of precise experiments presented above is puzzling because the commonly used and considered as correct Drude response function failed, whereas the evidently inapplicable plasma response function demonstrated that it can be used successfully.

The Casimir Force between Ideal and Real Dielectrics
The ideal dielectric or insulator is a material which does not conduct an electric current at any temperature. The dielectric permittivity of an ideal dielectric at the pure imaginary Matsubara frequencies can be presented in the oscillator form [67] where ω j are the oscillator frequencies (all of them are nonzero), γ j are the relaxation parameters, g j are oscillators strengths, and K is the number of oscillators. At zero Matsubara frequency, ξ 0 = 0, the permittivity of an ideal dielectric takes the finite value By putting ω = iξ = 0 in (4) for an ideal dielectric with no magnetic properties (µ = 1) one obtains As is seen in (37), the value of the TE reflection coefficient is the same as for the Drude model in (19) whereas the value of the TM one is different. According to Section 3, the zero-frequency terms of the Lifshitz formulas determine the behaviors of the Casimir free energy and force at large separations. Substituting (37) in (8) and integrating, we find the Casimir free energy per unit area of the plates made of an ideal dielectric where In a similar way, substituting (37) in (10), one finds the asymptotic behavior of the Casimir force per unit area between two ideal dielectric plates at large separations Real dielectrics are also characterized by the zero electric conductivity at zero temperature. This property differentiates them from metals which have a nonzero conductivity at any temperature. In this regard all materials are either dielectrics or metals [68]. There are several types of real dielectrics but all of them have some nonzero conductivity at T > 0.
The dielectric permittivity of a real dielectric at temperature T considered at the pure imaginary Matsubara frequencies can be presented in the form [52] where ε(iξ l ) is determined in (35) and σ 0 is the temperature-dependent static conductivity. This static conductivity of dielectric material vanishes with temperature exponentially fast where ∆ is the band gap. It is convenient to call the dielectric material an insulator if ∆ > 2 − 3 eV and an intrinsic semiconductor if ∆ < 2 − 3 eV [68]. Substituting (40) in (4) one obtains for real dielectrics the following expressions for the reflection coefficients at zero frequency: which coincide with (19) obtained for metals described by the Drude model but are different from (37) obtained for ideal dielectrics. Thus, the Casimir free energy and force for dielectric plates at sufficiently large separations are given by expressions (20) obtained in Section 3 for metallic plates described by the Drude model. In view of this results one can suggest that an inclusion of the dc conductivity in calculation of the Casimir free energy and force between dielectric plates plays the same role as an inclusion of the relaxation properties of conduction electrons for the metallic ones. This guess is confirmed by calculations of the relative thermal correction to the Casimir force between dielectric plates defined as Computations have been performed for a fused silica using the optical data for its complex index of refraction [52] with ε(0) = 3.81 (ideal dielectric) and with taken into account typical value of the dc conductivity of fused silica at room temperature σ 0 = 29.7 s −1 (real dielectric). Note that the computational results for real fused silica do not depend on the value of σ 0 but only on the fact that it is not equal to zero. In Figure 4, the predictions of the Lifshitz theory for the thermal correction (43) in percent for the real and ideal fused silica (i.e., with included and omitted dc conductivity) at T = 300 K are shown as the functions of separation between the plates by the dashed and solid lines, respectively. As is seen in this figure, at short separations the thermal correction is reasonably small if the dc conductivity is omitted in computations (see the solid line). Thus, at a = 1 and 2 µm it is equal to only 3.9% and 15.4%, respectively. The situation reverses if the dc conductivity is taken into account in computations (see the dashed line). In this case the relative thermal correction (43) is equal to 182% and 314% at the same respective separations.
It should be noted that so big difference between the predictions of the Lifshitz theory with omitted and included dc conductivity for dielectrics arises from a difference between the TM reflection coefficients in (37) and (42). We recall that a similar difference between predictions of the same theory for metallic plates described by the Drude and plasma models is due to a difference between the TE reflection coefficients in (19) and (23). In any case, the above results suggest that the grave problems met by the Lifshitz theory in the case of metals (see Section 4) may appear for dielectrics as well. This conclusion is confirmed in the next section.

Thermodynamic and Experimental Parts of the Casimir Conundrum for Dielectrics
Consideration of the thermal Casimir force between dielectric plates demonstrates that there is an ample evidence of problems between the Lifshitz theory using the dielectric response function for real dielectrics and thermodynamics, on the one hand, and measurement data, on the other hand. Experiments on measuring the Casimir interaction with dielectric test bodies are more complicated in comparison with metallic ones due to the problem of localized electric charges which are present on dielectric surfaces. Up to now, only three relatively precise measurements using the dielectric test bodies have been performed. However, the thermodynamic test similar to that elucidated in Section 4.1 for metals is even more conclusive because it does not require an assumption of the perfect crystal lattice. We begin with the thermodynamic argument and continue with the comparison between experiment and theory.

The Casimir Entropy for Dielectric Plates and the Nernst Heat Theorem
The thermodynamic test for two dielectric plates with either included or omitted dc conductivity can be performed in close analogy to that presented in Section 4.1 for metallic plates described either by the Drude or by the plasma model. One should substitute the dielectric permittivity of a real dielectric (40) with ε(iξ l ) given by (35) in (8) in order to obtain the Casimir free energy F RD for real dielectric plates and then find the low-temperature behavior of the Casimir entropy In a similar way, substituting the dielectric permittivity (35) in (8) where the reflection coefficient r TM (0, k ⊥ ) defined in (37) does not depend on k ⊥ : r TM (0, k ⊥ ) = r TM (0). Because of this, the integral in (45) can be calculated with the result The asymptotic behavior of the Casimir entropy for an ideal dielectric at arbitrary low temperature is given by [9,31,32] where the quantity G is expressed via the oscillator parameters introduced in (35) As is seen from (47), lim i.e., the Lifshitz theory satisfies the Nernst heat theorem in the case of Casimir plates made of ideal dielectric. In this sense the ideal dielectric plates are similar to the ideal metal plates and to the plates made of a metal described by the plasma model. The situation reverses when we consider the Casimir entropy of real dielectric plates made of real dielectric possessing some dc conductivity at any nonzero temperature. In the limiting case of zero temperature (47) where r TM (0) is given in (37). Thus, for the plates made of real dielectric possessing some dc conductivity at any nonzero temperature the Lifshitz theory leads to a positive Casimir entropy depending on the separation between the plates and the static dielectric permittivity, i.e., violates the Nernst heat theorem. Note that for both metallic and dielectric plates the Casimir entropy is not the entropy of a closed system which should also include, for instance, the entropy of plate materials. It should be stressed, however, that the Casimir entropy alone depends on the separation distance between the plates. Because of this it cannot be compensated by some other contribution in the total entropy of a closed system making the violation of the Nernst heat theorem unavoidable.
The presented results demonstrating that with taken into account dc conductivity of plate material the Lifshitz theory violates the Nernst heat theorem and satisfies it when the dc conductivity is disregarded are called the Casimir conundrum. These results are really strange if to take into account that the dc conductivity does exist and has been measured in numerous experiments.

The Lifshitz Theory Confronts Experiments with Dielectric Test Bodies
As mentioned in the beginning of Section 6, measurements of the Casimir force between dielectric test bodies are more complicated than between metallic ones due to the localized electric charges which are present on dielectric surfaces. Because of this, it is preferable to use dielectrics with sufficiently high concentration of charge carries (for instance, doped semiconductors with doping concentration below the critical value at which the dielectric-to-metal phase transition occurs).
We start with an experiment on the optically modulated Casimir force where the force difference between an Au-coated sphere of R = 98.9 µm radius and a silicon membrane illuminated with laser pulses has been measured in high vacuum in the presence and in the absence of light [37,38]. Measurements of the force difference have been performed by means of an atomic force microscope.
In the absence of a laser pulse on the membrane made of a p-type silicon the concentration of charge carriers in it was equal to approximately 5 × 10 14 cm −3 [52]. Thus, the membrane was in a dielectric state.
In the presence of a laser pulse the charge carrier concentration increased by the five orders of magnitude up to (1 − 2) × 10 19 cm −3 depending on the absorbed power. This means that in the bright phase silicon was in a metallic state.
The difference of the Casimir forces between an Au-coated sphere and a silicon membrane in the bright and dark phases was calculated using the Lifshitz theory and the proximity force approximation (33) with account of surface roughness by means of the geometrical averaging at the laboratory temperature T = 300 K. In Figure 5 the computational results for F diff are shown by the solid and dashed lines as the functions of sphere-plate separation (the absorbed power was 8.5 mW). The solid and dashed lines were computed with omitted and taken into account dc conductivity of silicon, respectively, in the absence of a laser pulse, i.e., in the dark phase. In the presence of a laser pulse (the bright phase) the charge carriers were taken into account by means of the Drude model (the use of the plasma model in the bright phase for silicon and for Au leads to only minor differences in this case which cannot be distinguished experimentally). The experimental data are shown as crosses plotted at the 95% confidence level.
As is seen in Figure 5, the predictions of the Lifshitz theory taking into account the dc conductivity of silicon in the dielectric state (the dashed line) are experimentally excluded whereas the predictions with omitted dc conductivity of dielectric silicon are in a very good agreement with the measurement data. Thus, the Lifshitz theory is experimentally consistent only if one ignores the real physical phenomenon -small but quite measurable electric conductivity. It is highly meaningful that just in this case the theory is in agreement with the requirements of thermodynamics and is in contradiction with them otherwise.
One more experiment performed by using an atomic force microscope was devoted to measurements of the Casimir force between an Au-coated sphere of R = 101.2 µm radius and an indium tin oxide (ITO) film deposited on a quartz substrate. These measurements have been performed for two times -before and after a UV irradiation of the ITO film [40,41]. At room temperature ITO is a transparent conductor. Based on this, it was suggested to use this material in measurements of the Casimir force [14]. First experiments of this kind have been performed earlier [69,70]. The main novelty of the experiments [40,41]  is a UV irradiation of the ITO film with subsequent measurement of the Casimir force. The point is that the UV irradiation of ITO leads to a lower mobility of charge carriers [71] with no sensible changes in the optical data and, as was hypothesized in [40,41], to a phase transition from a metallic to a dielectric state.
The experimental results and their comparison with theory confirmed this hypothesis. In Figure 6(a), the pairs of blue and red lines indicate the boundaries of theoretical bands for the Casimir force between an Au sphere and an ITO film computed before and after a UV irradiation of this film with taken into account and omitted contribution of free charge carriers, respectively. The experimental data in both cases are shown as crosses plotted at the 95% confidence level. It is seen that the theoretical predictions are in a very good agreement with the measurement data both before and after irradiation.
In Figure 6(b) we again plot the upper set of crosses obtained after a UV irradiation of the ITO film in comparison with the theoretical band computed with taken into account contribution of free charge carriers in the UV irradiated sample. It is seen that the predictions of the Lifshitz theory are excluded by the measurement data. This result is in line with the results found from measuring the optically modulated Casimir force which show that in the dielectric state an agreement between experiment and theory is reached by disregarding the contribution of free charge carriers to the dielectric response of a material.
The last experiment utilizing the dielectric test body was on measuring the thermal Casimir-Polder force between 87 Rb atoms belonging to the Bose-Einstein condensate and a dielectric fused silica plate at separations of a few micrometers [42]. In the first series of measurements the plate was kept at the same temperature as the environment and in the second and third at higher temperatures. This means that in the last two cases the Casimir-Polder force had two contributions -the equilibrium one given by the Lifshitz theory and another one from its generalization for a nonequilibrium case [72,73] (see also the case of phase-change wall material at nonequilibrium conditions [74]  the measurement data are in good agreement with theory when the dc conductivity of fused silica is not taken into account in computations. However, by repeating the same calculations with included dc conductivity of the plate material, one obtains the theoretical results excluded by the measurement data [39]. This happens due to exclusively the equilibrium contribution to the Casimir-Polder force given by the Lifshitz theory (the nonequilibrium one is almost independnet on whether or not the dc conductivity is taken into account in computations).
Thus, experiments on measuring the Casimir interaction with dielectric test bodies are consistent with theoretical predictions of the Lifshitz theory only under a condition that the dc conductivity of dielectric material is omitted in computations. In so doing, as shown in Section 6.1, the theory is consistent with the laws of thermodynamics, but at the sacrifice of the phenomenon of dc conductivity. The approaches to resolution of the Casimir conundrum originating from this situation are discussed below.

Different Approaches to Resolution of the Casimir Puzzle and Casimir Conundrum
The above problems attracted much attention of experts in the field during the last 20 years. A lot of different explanations and suggestions were put forward on why the predictions of the Lifshitz theory appear to be in disagreement with the laws of thermodynamics and the measurement data and how the agreement could be restored. Certain of the proposed approaches were seeking for some unaccounted systematic effects in the performed experiments, such as, for instance, an impact of the additional electric force due to electrostatic patches on metallic surfaces or the nonadditive effects in the surface roughness. According to other approaches, the roots of the problems could lay in simplifications used in the Lifshitz theory which does not take into account the spatial dispersion. There even was a suggestion [75] to modify the Planck distribution by including in it the special damping parameter. This parameter, however, should take different values for bringing theoretical predictions in agreement with the measurement data of different experiments. The separate line of investigation was a generalization of the Lifshitz theory for more complicated geometries used in experiments, including the configuration of a sphere above a plate, basing on the first principles of quantum field theory with no use of uncontrolled additive methods such as PFA. Below we consider several suggested approaches directed to the elimination of contradictions between the Lifshitz theory, on the one hand, and thermodynamics and/or experiment, on the other hand.

Variations of the Optical Data
Many precise experiments on measuring the Casimir force between metallic test bodies used the Au-coated surfaces of a sphere and a plate. The frequency-dependent dielectric permittivity of Au was found using the tabulated optical data for the complex index of refraction [52] extrapolated down to zero frequency by means of either the Drude or the plasma models as discussed above. It has been known, however, that the optical properties of Au film are sample-dependent and also depend on the used method of deposition. Most importantly, different sets of the optical data lead to different values of the Drude parameters ω p and γ used in extrapolation to zero frequency. This may influence on the values of the Casimir force calculated using the Lifshitz theory and on the comparison between experiment and theory. The question arises whether it is possible to bring the theoretical predictions in agreement with the experimental results if the optical data are not taken from the tables but measured for the laboratory bodies used in this specific experiment [76,77]. To answer this question, the Casimir force was computed using different sets of the optical data for Au available in the literature. It was concluded [76,77] that the variations in the optical data of Au with respective changes in the plasma frequency used in extrapolations to zero frequency may result in up to 5-10% differences in the Casimir force. These differences are of the same size as the discrepancies between experiment and theory found in [20][21][22][23][24][25][26].
To determine the actual role of the optical data variations in resolution of the Casimir puzzle in the experiment [23], the plasma frequency and the relaxation parameter were found for the specific Au coatings used in measurements of the Casimir force. The obtained values turned out to be very close to the tabulated ones [52]. In addition, a complete set of the optical data for experimental Au coatings was measured by means of ellipsometry and found [78] in agreement with the tabulated one. These results were confirmed by the method of the weighted Kramers-Kronig relations [79]. Moreover, all sets of the optical data of Au available in the literature with respective plasma frequencies varying from 6.85 to 9.0 eV have been used in calculations of the Casimir interaction for subsequent comparison with the measured values of [23]. The results of this comparison are shown in Figure 7 where the measured gradients of the Casimir force are shown as crosses (the same as in Figure 2) and the blue theoretical band is obtained using all sets of the optical data of Au extrapolated to zero frequency by means of the Drude model. From Figure 7 it is seen that the use of alternative sets of the optical data only increases a disagreement between experiment and theory. Thus, it was concluded [59] that it is not possible to bring the predictions of the Lifshitz theory using the Drude model in agreement with the measurement data at the cost of variation of the optical data.

Impact of the Surface Patches
The Au coatings on the test bodies in experiments on measuring the Casimir force have a polycrystal structure. In addition, even in high vacuum, there are some contaminants and dust on the surfaces. As a result, some spatial distribution of electrostatic potentials appear even on the grounded surfaces of a sphere and a plate which are called the patch potentials [80]. The patch potentials lead to some additional attractive force acting between a sphere and a plate and to a dependence on separation of the residual potential difference between them. The magnitude of this force depends on the size of surface patches or contaminants.
It was hypothesized [81] that an additional force gradient due to surface patches could compensate a difference between the blue and red bands in Figure 2 and, thus, bring the Lifshitz theory using the Drude model in agreement with the measurement data of [22]. Sufficiently large patches and contaminants up to 2 µm size can really lead to the force gradient of required magnitude but also result in strong dependence of the residual potential difference on the sphere-plate separation which was not present in the calibration measurements of [20][21][22][23][24][25][26] due to a special selection of samples. An investigation [82] of the patch potentials on Au surfaces of the samples similar to those used in the performed experiments [22,23] by means of Kelvin probe force microscopy demonstrated that the force of patch origin cannot explain a difference between the measured force gradients and theoretical predictions of the Lifshitz theory using the Drude model.
Further clarity to the role of patches in measurements of the Casimir force was added by the experiments with magnetic test bodies [25,26]. As noted in Section 4.2, for bringing the theoretical predictions obtained using the Drude model in agreement with the measurement data in this case, one needs some additional repulsive force which does not occur due to surface patches. It should be mentioned also that recently several experiments on measuring the Casimir interaction have been performed where the role of surface patches and contaminants was strongly suppressed by means of Ar-ion and UV-cleaning of the interacting surfaces [83][84][85]. The results of these experiments are consistent with theoretical predictions of the Lifshitz theory using the plasma model and exclude the predictions of the same theory using the Drude model up to the separation distance of 1.1 µm. Thus, by accounting the surface patches it is impossible to reconcile the Lifshitz theory using the Drude model with the measurement data.

The Role of Surface Roughness
The interacting surfaces in measurements of the Casimir force are not perfectly plane or spherical, but covered with some roughness. If there is a large-scale deviations from the perfect geometry, it can be described by a regular function. The short-scale roughness can be considered as stochastic and described by the random functions. In the comparison between precise experiments on measuring the Casimir force mentioned above and theory the roughness was taken into account in an additive way [9,59,61]. This means that in the case of a regular roughness the Casimir force was computed using the Lifshitz theory at all local separations and was then averaged over the actual geometrical profiles of interacting surfaces determined by means of an atomic force microscope. In the case of stochastic roughness with small dispersions δ 1,2 ≪ a the roughness correction can be found perturbatively in powers of δ 1,2 /a and taken into account in a multiplicative way. For example, a corrected for the presence of stochastic roughness gradient of the Casimir force between a sphere and a plate takes the form [9,59] Different approaches to the account of surface roughness in experiments on measuring the Casimir force are the subject of considerable literature [86][87][88][89][90]. One may doubt whether an additive approach used in comparison between experiment and theory is exact enough. Strictly speaking, the larger and smaller roughness corrections to the Casimir force could decrease a difference between the predictions of the Lifshitz theory using the Drude model and the measurement data in experiments with the nonmagnetic and magnetic surfaces, respectively.
The more fundamental scattering approach to the Casimir force between rough surfaces was developed in [91,92]. It was shown that the additive approach leads to sufficiently precise results at much smaller separations between the interacting bodies as compared with the roughness correlation length. In fact for a typical correlation length Λ c ≈ 200 nm the additive approach is already applicable at a < 2Λ c /3 ≈ 130 nm. This is the most important region where the roughness correction contributes up to a few percent fraction of the Casimir force. At larger separations the additive approach underestimates the role of surface roughness, so that the more accurate results are given by the scattering approach. For instance, at a = 2Λ c = 400 nm the scattering approach leads to by a factor 1.5 larger roughness correction than the additive one. This is, however, not damaging for the comparison between experiment and theory. The point is that with increasing separation the roughness correction decreases much faster than the Casimir force between perfectly shaped surfaces. Because of this for surfaces of sufficiently good quality at separations of a few hundred nanometers it does not play any role in the comparison between experiment and theory and can be simply neglected. This means that the surface roughness is not helpful for resolution of the Casimir puzzle.

Deviations from the Proximity Force Approximation
As was mentioned in Section 4.2, calculations of the Casimir force and its gradient in the sphere-plate geometry used in comparison between experiment and theory were performed by means of the Lifshitz theory and the proximity force approximation (33) and (34). Although it was believed that for small a/R the error introduced by using the PFA should be of the order of a/R, the exact information concerning the size of this error was missing. Because of this one could hope that a disagreement between theoretical predictions of the Lifshitz theory using the Drude model and the measurement data may be caused by some deviations from the PFA in the region of experimental separations and sphere radii.
A major step forward was made in 2006 when it was shown that the Casimir energy between nonplanar surfaces can be written in terms of functional determinants [93,94]. The obtained results were applied to calculate the Casimir energy of an ideal metal cylinder and a sphere in front of an ideal metal plane [93][94][95]. Later the Lifshitz theory was generalized for real material bodies of arbitrary shape kept at any temperature in thermal equilibrium with the environment [96][97][98][99]. Application of the developed theory to a sphere and a plate made of real metals spaced at separations below a micrometer turned out to be a complicated problem. First computations [100] produced an impression that in the limiting case of perfect reflectors the exact results rapidly depart from the PFA expectations and this fact should be taken into account in theory-experiment comparison. For real metals the computations have been performed in [101] but for not too small values of a/R > 0.2, as compared to the experimental values a/R < 0.01.
Calculations of the Casimir force in a sphere-plate geometry for the experimental values of a/R have been performed both analytically using the method of gradient expansion and numerically [102][103][104][105][106][107][108][109]. As only one example of the obtained result, we present an expansion of the exact gradient of the Casimir force between a sphere and a plate which includes the first-order correction to the PFA [107] where F ′ sp is the PFA result presented in (34). The values of the expansion coefficient β D,p were computed at different separations for the sphere radii varying from 10 to 100 µm using the tabulated optical data for the complex index of refraction of Au extrapolated to zero frequency by the Drude (β D ) and plasma (β p ) models. According to the obtained results [107], within the region of separations from 200 to 600 nm the coefficient β D varies from -0.35 to -0.45 whereas β p varies from -0.5 to -0.6. Note that in the interpretation of experiments on measuring the Casimir force [20][21][22][23][24][25][26] the conservative estimation |β D,p | 1 has been used. Thus, inaccuracies in the PFA are irrelevant to the Casimir puzzle and an exact computation of the Casimir force in the sphere-plate geometry does not help to resolve it.

Impurities in a Crystal Lattice and the Nernst Heat Theorem
The above consideration of the variations of the optical data, impact of surface patches, nonadditive effects in the surface roughness, and deviations of the Casimir force from PFA is aimed at resolving the second, experimental, part of the Casimir puzzle. All these research directions are unrelated to an inconsistency of the Lifshitz theory with thermodynamics discussed in Sections 4.1 and 6.1 (note that in the sphere-plate geometry the Lifshitz theory faces the same problems with violation of the Nernst heat theorem as for two parallel plates [16]). It should be stressed also that after the crucial experiment [27], where the theoretical predictions using the plasma and Drude models differed by up to a factor of 1000, an exclusion of the latter by the measurement data was established conclusively. This attaches particular significance to the thermodynamic parts of the Casimir puzzle and Casimir conundrum.
The historically first approach to the problem of violation of the Nernst heat theorem in the configuration of two parallel metallic plates was made with no modifications in the Lifshitz theory. It was suggested [110] to take into account that the crystal lattice of any real metal has some small fraction of impurities. As a result, the relaxation parameter γ of the Drude model (17) does not go to zero with vanishing temperature but to some residual value γ 0 which depends on the impurity concentration [55]. Based on this, it was shown numerically [110] that at sufficiently low temperature the Casimir entropy abruptly jumps to zero starting from the negative value (32), i.e., the Nernst heat theorem is formally satisfied.
It should be taken into account, however, that the numerical proof of vanishing cannot be considered as completely satisfactory because it is burdened with some computational error. Because of this, the case of metallic plates with impurities was also considered analytically [111,112]. As a result, it was shown that in the asymptotic limit of low T the Casimir entropy calculated using the Drude model with the residual relaxation γ 0 vanishes as [112] For the typical value of γ 0 = 5.32 × 10 10 rad/s, one finds from (54) that the Casimir entropy jumps to zero at about 0.001 K starting from the negative value of −2MeV m −2 K −1 [9].
Although an account of impurities provides an apparent resolution of the thermodynamic part of the Casimir puzzle, it cannot be considered as completely satisfactory. The point is that the perfect crystal lattice with no impurities serves as the basis for quantum condensed matter physics. It possesses the nondegenerate ground state and the Nernst heat theorem must be satisfied in this case as well [113,114]. In Section 8, we will return to this point in connection with the recently proposed Drude-like nonlocal response functions.
As discussed in Section 6.1, the thermodynamic part of the Casimir conundrum for dielectrics is due to the exponentially fast vanishing of the conductivity (41) with decreasing temperature. This leads to a violation of the Nernst heat theorem if the conductivity is taken into account in the Lifshitz theory. To avoid this conclusion in a similar way as for metals with impurities, it was suggested [115] to consider some imaginary dielectric material possessing a constant conductivity at low temperature. Although the proposed model does not carry the thermodynamic anomaly, it is incapable to solve the Casimir conundrum for real dielectric materials because they are characterized by the exponentially fast vanishing conductivity at low temperatures. Thus, neither the Casimir puzzle nor the Casimir conundrum can be solved without some radical changes in the Lifshitz theory and/or in the used response functions.

The Anomalous Skin Effect and Spatial Nonlocality
The original formulation of the Lifshitz theory assumes that the material of the plates is described by the dielectric permittivity depending on the frequency but not on the wave vector. In the case of metallic plates there is, however, the frequency region where a connection between the electric field and the current becomes nonlocal and the concept of the frequency-dependent dielectric permittivity loses its meaning. This is the region of the anomalous skin effect where the spatial nonlocality plays an important role. At room temperature for Au the frequency region of the anomalous skin effect is from 10 12 to 10 13 rad/s, i.e., it is rather narrow. With decreasing temperature, however, this frequency region widens at the cost of a decrease of its left boundary. The question arises what is an impact of the anomalous skin effect on the Casimir force and what is its possible role in the resolution of the Casimir puzzle.
In the presence of spatial nonlocality the reflection coefficients in the Lifshitz formulas (8) and (10) are expressed via the surface impedances for the TM and TE polarizations [116][117][118] The impedances in turn are expressed via the longitudinal ε L (iξ l , k) and transverse ε T (iξ l , k) dielectric permittivities which describe a dielectric response of metal to parallel and perpendicular to k electric fields, respectively [117,118] Here, k z is the third component of the wave vector, so that k 2 = k 2 ⊥ + k 2 z and The nonlocal generalizations ε L,T of the Drude dielectric function ε D for the description of the anomalous skin effect suggested in [117] were used in [119] to calculate the correction to the Casimir force due to the effect of nonlocality. It was shown that with increasing separation from 100 to 300 nm the relative nonlocal correction is negative and its magnitude decreases from 0.3% to 0.1%. Thus, an account of the effects of nonlocality arising in the dielectric response due to the anomalous skin effect cannot solve the experimental part of the Casimir puzzle.
Concerning the thermodynamic part of the Casimir puzzle, an account of the anomalous skin effect leads to the same conclusions as were discussed above for the simple Drude model. The Nernst heat theorem is formally restored if there is some nonzero value of the effective relaxation parameter at zero temperature and is violated otherwise [120]. In this sense the spatial dispersion can play the same role as the relaxation of conduction electrons [121].

Inclusion of the Screening Effects
One more attempt to solve the Casimir puzzle and Casimir conundrum was made by taking into account the screening of the electric field created by an external sourse in a medium containing free charge carriers. This effect leads to a penetration of the static electric field into a conducting material to a depth of the so-called screening length [122].
In the Casimir physics an account of the screening effects was first proposed in [123] in connection with an experiment on measuring the Casimir-Polder force between a 87 Rb atom and a dielectric fused silica plate [42] (see Section 6.2). If the dc conductivity of fused silica is taken into account in calculations of the Casimir-Polder force, the theoretical predictions of the standard Lifshitz theory are excluded by the measurement data. The Casimir-Polder entropy at zero temperature is equal to where r TM (0) is defined in (37) and α(0) is the static polarizability of 87 Rb atom, in violation of the Nernst heat theorem [39].
With account of screening, the TM reflection coefficient at zero frequency (37) is replaced with [123] r src where κ is the inverse quantity to the Debye-Hückel screening length and n = n(T) is the concentration of charge carriers in a dielectric material.
According to the results of [123], the Lifshitz theory using the screened reflection coefficient (59) taking into account the presence of free charge carriers is consistent with the measurement data. This solves the experimental part of the Casimir conundrum in application to this specific experiment. It was shown [124], however, that the same theory still remains in disagreement with the measurement data of an experiment on the optically modulated Casimir force (see Section 6.2).
The Lifshitz theory with the screened reflection coefficient (59) also provides a partial resolution of the thermodynamic part of the Casimir conundrum. It turns out that for the insulators and intrinsic semiconductors, whose concentration of charge carriers n(T) vanishes with temperature exponentially fast, the Nernst heat theorem is satisfied. There are, however, dielectric materials (dielectric-type semimetals, doped semiconductors with the dopant concentration below critical, dielectrics with ionic conductivity, etc) whose static conductivity σ(0) vanishes with temperature not due to the vanishing n but due to the vanishing mobility of charge carriers. For these dielectric materials the Lifshitz theory using the screened reflection coefficient (59) still leads to a violation of the Nernst heat theorem [124].
In the end of this section, we note that the screening effects were also taken into account in the reflection coefficients with any l. The developed formalism was presented in the form applicable to bodies with arbitrarily large charge carrier density including metals [125]. For this purpose the Debye-Hückel screening length should be replaced with the Thomas-Fermi one [122]. It was shown [124], however, that this approach leads to approximately the same Casimir forces between metallic plates, as given by the standard Lifshitz theory using the Drude model, and, thus, remains to be in contradiction with the measurement data. For metals with perfect crystal lattices the modified Lifshitz theory accounting for the screening effects violates the Nernst heat theorem. Thus, in spite of some encouraging results, an account of the screening effects did not resolve the Casimir puzzle and Casimir conundrum.

The nonlocal Drude-Like Response to Quantum Fluctuations off the Mass Shell and the Casimir Puzzle
Many unsuccessful attempts to solve the Casimir puzzle and conundrum undertaken for the last 20 years suggest that there should be some alternative approach to their resolution. In this respect our attention is attracted by graphene which is a 2D sheet of carbon atoms in the form of a hexagonal crystal lattice. The outstanding property of graphene is that at energies below a few eV it is well described by the Dirac model characterized by the linear dispersion relation where the speed of light c is replaced with the Fermi velocity v F [126]. This makes it possible to calculate many graphene properties, including its dielectric response to the electromagnetic field, on the basis of first principles of Quantum Electrodynamics at nonzero temperature. In so doing the phenomenological response functions given by the Drude or plasma models become unneeded.
Thus, it was found that graphene is described by the transverse and longitudinal dielectric permittivities depending on both the frequency and the wave vector which are expressed via the polarization tensor [127][128][129][130]. In doing so, the predictions of the Lifshitz theory using the exact graphene response functions are in good agreement with an experiment on measuring the Casimir force between an Au-coated sphere and a graphene-coated plate [131,132]. What is more, the Casimir and Casimir-Polder entropies calculated for both the pristine graphene sheets and for graphene possessing the nonzero energy gap and chemical potential satisfy the Nernst heat theorem [133][134][135][136][137]. This means that there is no Casimir puzzle for graphene.
The case of graphene suggests that the roots of the problems discussed above might be in the inadequate response functions of conventional 3D materials used in the Lifshitz theory. The point is that the Casimir free energy (3) and force (6) are obtained by the integration ober all k ⊥ from zero to infinity at each fixed ω. This means that both the propagating waves, which are on the mass shell, and evanescent waves, which are off the mass shell, contribute to the final result. The Drude response function is well checked experimentally and provides the correct response to real electromagnetic fields with a nonzero field strength. There is no direct experimental confirmation to this model, however, if we are seeking for the response to quantum fluctuations which are off the mass shell. What's more, the Casimir puzzle can be considered as an indirect indication that the Drude model describes the response to fluctuations of such kind incorrectly.
At this point it is reasonable to search for the nonlocal Drude-like response functions which provide an approximately the same response, as the standard Drude model, to the electromagnetic fluctuations and fields on the mass shell but describe adequately the response to the off-shell fluctuations. The response functions of this kind were recently suggested in [138]. They are given bỹ where the quantities v T,L are the constants of the order of the Fermi velocity v F . It is seen that in the local limit bothε T D andε L D coincide with the standard Drude model For the electromagnetic fields on the mass shell we have ck ⊥ ω and Thus, the dielectric functions (61) lead in this case to approximately the same results as the standard Drude model. This is quite different from the nonlocal response functions discussed in Section 7.6. For example, the nonlocal response describing the anomalous skin effect [117] used in theory of the Casimir force [119,120] (see Section 7.6) was introduced for a description of the physical phenomenon occurring in the fields on the mass shell. The nonlocal response functions describing the electron gas without and with account of collisions were introduced in [139,140]. In the static limit they describe the screening effects considered in Section 7.7. These again occur in real fields on the mass shell. By contrast, the suggested permittivities (61) are not intended for a description of small nonlocal effects in real fields and deviate from the standard Drude model only for the off-the-mass-shell fluctuations. Note also that the permittivities (61) depend only on k ⊥ and do not depend on k z as in the case of [117,139,140]. The reason is that in the presence of the Casimir plates the translational invariance along the z axis is violated and it is strictly speaking impossible to define the response functions depending on the 3D vector k [141,142].
For the permittivities depending only on k ⊥ (56) results in Then the reflection coefficients (55) valid in the nonlocal case are given by Now it is possible to substitute the specific Drude-like permittivities (61) taken at ω = iξ l in (65), calculate the Casimir force between two Au plates (10) and, by using the PFA (34), find the gradient of a Casimir force between a sphere and a plate. After introducing small corrections due to surface roughness and inaccuracies in the PFA, considered in Sections 7.3 and 7.4, one can compare the theoretical results obtained using the Lifshitz theory with the nonlocal Drude-like response functions (61) with the measurement data.
In Figure 8 the obtained theoretical predictions in the experimental configuration of the micromechanical torsional oscillator [22] are shown as a function of separation by the black band. The measurement results are shown as crosses (the same as in Figure 2). We also reproduce from Figure 2 the red and blue theoretical bands found using the extrapolations of the optical data by means of the plasma and Drude models. As is seen from Figure 8, the theoretical predictions using both the nonlocal Drude-like and plasma response functions are in agreement with the measurement data. An important advantage of the Drude-like response is that it takes the proper account of the relaxation properties of free charge carriers which are disregarded by the plasma model.
As one more example, in Figure 9 we plot by the black band the predictions of the Lifshitz theory using the optical data of Au extrapolated by the nonlocal Drude-like response functions (61) in the experiment measuring the gradient of the Casimir force between an Au-coated sphere and an Au-coated plate at larger separations by means of dynamic atomic force microscope [85] (see Section 7.2). The theoretical predictions using the Drude and plasma models are shown as the blue and red bands, respectively, whereas the measured gradients of the Casimir force are shown as crosses. It is again seen that the theoretical approach using the nonlocal Drude-like response functions is in agreement with the measurement data along with the plasma model. The advantages of the Drude-like functions are that they take the proper account of dissipation of conduction electrons and describe correctly the reflectances of the electromagnetic waves on the mass shell incident on an Au plate [138].
Based on the above one can conclude that the nonlocal Drude-like response functions (61) provide a resolution of the experimental part of the Casimir puzzle for nonmagnetic metals. Detailed examination of the asymptotic behavior of the Casimir entropy calculated using the Drude-like functions (61) at low temperature demonstrates that these response functions also solve the thermodynamic part of the Casimir puzzle. Thus, for nonmagnetic metals with perfect crystal lattices the low-temperature behavior of the Casimir entropy is given byS where the coefficient C 1 , in addition to separation, depends on ω p , v T and the fundamental constants c, h, and k B .
For metal with impurities possessing some residual relaxation γ 0 the low-temperature behavior of the Casimir entropy found using the nonlocal Drude-like response functions (61) takes the form S D (a, T) = C 2 (a)T, (67) where, in addition to the parameters indicated above, the coefficient C 2 also depends on γ 0 . From (66) and (67) it is seen that both for metals with perfect crystal lattices and for metals with impurities the Casimir entropy goes to zero with vanishing temperature. This means that the Lifshitz theory using the nonlocal Drude-like response functions is consistent with the laws of thermodynamics.

Discussion: The Present Status of the Casimir Puzzle and Casimir Conundrum
In the foregoing, we have considered several theoretical and experimental results which are not fully understood up to the present time and were called in the literature the Casimir puzzle and the Casimir conundrum. These results are related to the Casimir effect which is the physical phenomenon determined by the properties of the quantum vacuum and its interaction with matter. As discussed in Section 1, several basic challenges of modern physics are directly connected with the concept of vacuum. Because of this, it is probably not surprising that the Casimir puzzle and the Casimir conundrum have no an ultimate resolution until now in spite of much work made by many researches.
Several precise experiments on measuring the Casimir interaction between metallic and dielectric test bodies confirmed the fact that the standard dielectric functions commonly used for a description of the response of matter to real electromagnetic fields with a nonzero field strength lead to incorrect predictions in the framework of the Lifshitz theory. Over a period of time, there were serious doubts regarding the precision and interpretation of these measurements but after the experiments with magnetic test bodies [25,26] and differential force measurement, where the measured signal differed from theoretical prediction of the Lifshitz theory by the factor of 1000 (see Section 4.2), the facts have been established with certainty.
Taking into account that the Drude response function also has an unambiguous experimental confirmation in the area of real electromagnetic fields with a nonzero field strength, it is reasonable to suggest that it may describe incorrectly the response of metals to quantum fluctuations off the mass shell. This suggestion found support in the fact that some indirect experimental evidence can be obtained only about the form of the longitudinal permittivityε L in the off the mass shell fields, but not aboutε T [116] which should be derived theoretically like this is done for graphene using the polarization tensor.
No less difficulties are connected with the thermodynamic parts of the Casimir puzzle and the Casimir conundrum. Contradictions between the Lifshitz theory and thermodynamics arise when we use the response functions excluded by measurements of the Casimir force. Thus, it is reasonable to hope that an agreement will be restored if the used response to the off the mass shell quantum fluctuations, which is extrapolated from the area of real fields with a nonzero field strength and has no direct experimental confirmation, will be somehow corrected.
In the case of nonmagnetic metals possible realization of this program has already been suggested (see Section 8) in the form of nonlocal Drude-like response functions which lead to approximately the same results, as the standard Drude model, in the on the mass shell fields, but to significantly different response to the off-the-mass-shell quantum fluctuations. It was shown that the Lifshitz theory using the nonlocal Drude-like model is in agreement with the measurement data, as it does when using the plasma model which simply disregards the relaxation properties of conduction electrons. What is more, the Casimir entropy calculated using the Drude-like response functions satisfies the Nernst heat theorem for metals with both perfect crystal lattices and lattices with impurities. Thus, the suggested Drude-like model points the way to a resolution of the Casimir puzzle.
We emphasize, however, that, unlike the case of graphene, this is a phenomenological solution yet. It demonstrates the possibility to solve the Casimir puzzle by modifying the dielectric response to quantum fluctuations off the mass shell, but this solution might be not unique. Furthermore, it is necessary to extend the proposed approach to the case of magnetic metals and apply similar ideas to dielectric materials in order to find a resolution of the Casimir conundrum. Thus, further investigations of these challenging problems are necessary.

Conclusions
To conclude, we have elucidated current situation regarding the complicated problems faced by the Lifshitz theory during the last 20 years. It is shown that the experimental facts in this field of research are largely settled. However, their theoretical understanding is far from being complete. There are some ideas and even examples of considerable promise on how the Casimir puzzle and the Casimir conundrum could be solved but they need further theoretical justification.
The thermodynamic test of the Lifshitz theory using one or other type of the dielectric response confirmed its usefulness. It is not by chance that the Lifshitz theory using the Drude model for metals or the dc conductivity for dielectrics was found to violate the Nernst heat theorem. A development of the Casimir physics confirmed that the thermodynamic and experimental problems happen concurrently. This means that if the Lifshitz theory using some model of the dielectric response leads to a violation of the Nernst heat theorem for the Casimir entropy one should expect that theoretical predictions using this model will be found in disagreement with the measurement data.
The suggested nonlocal Drude-like response functions provide a supposed resolution for both the experimental and thermodynamic parts of the Casimir puzzle for nonmagnetic metals. Future trends should bring the final resolution of the complicated problems of Casimir physics for both metallic and dielectric materials and widen the scope of fluctuational electrodynamics by a more reliable description of the dielectric response of matter to quantum fluctuations off the mass shell.