International Journal of Modern Physics: Conference Series

Additional information is provided on the effect of the significant (up to 35%) reduction in the magnitude of the Casimir force between an Au-coated sphere and an indium tin oxide film which was observed after UV treatment of the latter. A striking feature of this effect is that the reduction is not accompanied with a corresponding variation of the dielectric permittivity, as confirmed by direct ellipsometry measurements. The measurement data are compared with computations using the Lifshitz theory. It is shown that the data for the untreated sample are in a very good agreement with theory taking into account the free charge carriers in the indium tin oxide. The data for the UV-treated sample exclude the theoretical results obtained with account of free charge carriers. These data are found to be in a very good agreement with theory disregarding the free charge carriers in an indium tin oxide film. A possible theoretical explanation of our observations as a result of phase transition of indium tin oxide from metallic to dielectric state is discussed in comparison with other related experiments.

In the van der Waals regime (neglecting relativity and retardation), we find the power P generated by friction between two Drude-modelled dissipative half-spaces, at fixed separation and relative speed u, admitting only low u and low temperatures. This requires only elementary quantum mechanics; but the results can serve as partial checks on calculations in the fully retarded Casimir regime. They also raise questions regarding (i) the frequency-distribution of P; (ii) the status of predictions about Casimir forces generally, insofar as they feature parameters like conductivities with their empirical temperature-dependence; and (iii) calculations of heat transfer, insofar as they assume fluctuations in the two bodies to be uncorrelated.

In a background of a very strong magnetic field a quantum vacuum may turn into a new phase characterized by anisotropic electromagnetic superconductivity. The phase transition should take place at a critical magnetic field of the hadronic strength (B_c ≈ 10¹⁶ Tesla or eB_c ≈ 0.6 GeV²). The transition occurs due to an interplay between electromagnetic and strong interactions: virtual quark–antiquark pairs popup from the vacuum and create – due to the presence of the intense magnetic field – electrically charged and electrically neutral spin-1 condensates with quantum numbers of ρ mesons. The ground state of the new phase is a complicated honeycomblike superposition of superconductor and superfluid vortex lattices surrounded by overlapping charged and neutral condensates. In this talk we discuss similarities and differences between the superconducting state of vacuum and conventional superconductivity, and between the magnetic–field–induced vacuum superconductivity and electric–field–induced Schwinger pair production.

On this 75th anniversary of the publication of the Heisenberg-Euler paper on the full non-perturbative one-loop effective action for quantum electrodynamics I review their paper and discuss some of the impact it has had on quantum field theory.

A summary of relevant contributions, ordered in time, to the subject of operator zeta functions and their application to physical issues is provided. The description ends with the seminal contributions of Stephen Hawking and Stuart Dowker and collaborators, considered by many authors as the actual starting point of the introduction of zeta function regularization methods in theoretical physics, in particular, for quantum vacuum fluctuation and Casimir effect calculations. After recalling a number of the strengths of this powerful and elegant method, some of its limitations are discussed. Finally, recent results of the so called operator regularization procedure are presented.

The quantum contributions to the gravitational action are relatively easy to calculate in the higher derivative sector of the theory. However, the applications to the post-inflationary cosmology and astrophysics require the corrections to the Einstein-Hilbert action and to the cosmological constant, and those we can not derive yet in a consistent and safe way. At the same time, if we assume that these quantum terms are covariant and that they have relevant magnitude, their functional form can be defined up to a single free parameter, which can be defined on the phenomenological basis. It turns out that the quantum correction may lead, in principle, to surprisingly strong and interesting effects in astrophysics and cosmology.

This is a short non-technical introduction to applications of the Quantum Field Theory methods to graphene. We derive the Dirac model from the tight binding model and describe calculations of the polarization operator (conductivity). Later on, we use this quantity to describe the Quantum Hall Effect, light absorption by graphene, the Faraday effect, and the Casimir interaction.

In this paper we study the Casimir energy and force for generalized pistons constructed from warped product manifolds of the type I ×_f N where I = [a, b] is an interval of the real line and N is a smooth compact Riemannian manifold either with or without boundary. The piston geometry is obtained by dividing the warped product manifold into two regions separated by the cross section positioned at R ∈ (a, b). By exploiting zeta function regularization techniques we provide formulas for the Casimir energy and force involving the arbitrary warping function f and base manifold N.

A finite ultraviolet cutoff near a reflecting boundary yields a stress tensor that violates the basic energy-pressure relation. Therefore, a "soft" wall described by a power-law potential, which needs no ad hoc cutoff, is being investigated by the collaboration centered at Texas A&M University and the University of Oklahoma. Progress is reported here.

This contribution presents an overview of fundamental QED processes in the presence of an external field produced by an ultra-intense laser. The discussion focusses on the basic intensity effects on vacuum polarisation and the prospects for their observation. Some historical remarks are added where appropriate.

Casimir friction is analyzed for a pair of dielectric particles in relative motion. We first adopt a microscopic model for harmonically oscillating particles at finite temperature T moving non-relativistically with constant velocity. We use a statistical-mechanical description where time-dependent correlations are involved. This description is physical and direct, and, in spite of its simplicity, is able to elucidate the essentials of the problem. This treatment elaborates upon, and extends, an earlier theory of ours back in 1992. The energy change ΔE turns out to be finite in general, corresponding to a finite friction force. In the limit of zero temperature the formalism yields, however, ΔE → 0, this being due to our assumption about constant velocity, meaning slowly varying coupling. For couplings varying more rapidly, there will also be a finite friction force at T = 0. As second part of our work, we consider the friction problem using time-dependent perturbation theory. The dissipation, basically a second order effect, is obtainable with the use of first order theory, the reason being the absence of cross terms due to uncorrelated phases of eigenstates. The third part of the present paper is to demonstrate explicitly the equivalence of our results with those recently obtained by Barton (2010); this being not a trivial task since the formal results are seemingly quite different from each other.

We analyze recent experiments on measuring the thermal Casimir force with account of possible background effects. Special attention is paid to the validity of the proximity force approximation (PFA) used in the comparison between the experimental data and computational results in experiments employing a sphere-plate geometry. The PFA results are compared with the exact results where they are available. The possibility to use fitting procedures in theory-experiment comparison is discussed. On this basis we reconsider experiments exploiting spherical lenses of centimeter-size radii.

The Casimir effect is a crucial prediction of Quantum Field Theory which has fascinating connections with open questions in fundamental physics. The ideal formula written by Casimir does not describe real experiments and it has to be generalized by taking into account the effects of imperfect reflection, thermal fluctuations, geometry as well as the corrections coming from surface physics. We discuss these developments in Casimir physics and give the current status in the comparison between theory and experiment after years of improvements in measurements as well as theory.

Like Casimir's original force between conducting plates in vacuum, Casimir forces are usually attractive. But repulsive Casimir forces can be achieved in special circumstances. These might prove useful in nanotechnology. We give examples of when repulsive quantum vacuum forces can arise with conducting materials.

Measurements of the Casimir force are used to obtain stronger constraints on the parameters of hypothetical interactions predicted in different unification schemes beyond the Standard Model. We review new strong constraints on the Yukawa-type interactions derived during the last two years from recent experiments on measuring the lateral Casimir force, Casimir force in configurations with corrugated boundaries and the Casimir-Polder force. Specifically, from measurements of the lateral Casimir force compared with the exact theory the strengthening of constraints up to a factor of 24 millions was achieved. We also discuss further possibilities to strengthen constraints on the Yukawa interactions from the Casimir effect.

We provide a thorough exposition of recent results on the quantum stabilization of cosmic strings. Stabilization occurs through the coupling to a heavy fermion doublet in a reduced version of the standard model. The study combines the vacuum polarization energy of fermion zero-point fluctuations and the binding energy of occupied energy levels, which are of the same order in a semi-classical expansion. Populating these bound states assigns a charge to the string. Strings carrying fermion charge become stable if the Higgs and gauge fields are coupled to a fermion that is less than twice as heavy as the top quark. The vacuum remains stable in the model, because neutral strings are not energetically favored. These findings suggest that extraordinarily large fermion masses or unrealistic couplings are not required to bind a cosmic string in the standard model.

New results for scalar Casimir self-energies arising from interior modes are presented for the three integrable tetrahedral cavities. Since the eigenmodes are all known, the energies can be directly evaluated by mode summation, with a point-splitting regulator, which amounts to evaluation of the cylinder kernel. The correct Weyl divergences, depending on the volume, surface area, and the corners, are obtained, which is strong evidence that the counting of modes is correct. Because there is no curvature, the finite part of the quantum energy may be unambiguously extracted. Dirichlet and Neumann boundary conditions are considered and systematic behavior of the energy in terms of geometric invariants is explored.

We study a family of local boundary conditions for the Dirac problem corresponding to the continuum limit of graphene, both for nanoribbons and nanodots. We show that, among the members of such family, MIT bag boundary conditions are the ones which are in closest agreement with available experiments. For nanotubes of arbitrary chirality satisfying these last boundary conditions, we evaluate the Casimir energy via zeta function regularization, in such a way that the limit of nanoribbons is clearly determined.

We present calculations of the Casimir interaction between a sphere and a plane, using a multipolar expansion of the scattering formula. This configuration enables us to study the nontrivial dependence of the Casimir force on the geometry, and its correlations with the effects of imperfect reflection and temperature. The accuracy of the Proximity Force Approximation (PFA) is assessed, and is shown to be affected by imperfect reflexion. Our analytical and numerical results at ambient temperature show a rich variety of interplays between the effects of curvature, temperature, finite conductivity, and dissipation.

It is shown that linearizing higher order theories of gravity, further gravitational massive modes emerge. Besides massless spin-2, also spin-0 and spin-2 massive and ghost fields have to be considered. We investigate the possible detectability of such additional modes on the stochastic cosmic background of gravitational waves by ground-based and space interferometric detectors and calculate the detectable energy density of the spectrum. In conclusion, these massive modes could be of interest for direct detection by the forthcoming LISA experiment.

The gradient of the Casimir force between carefully cleaned Au surfaces of a sphere and a plate is measured using a dynamic atomic force microscope in the frequency modulation regime in high vacuum. The electrostatic calibration of the setup did not reveal any effect of patches or surface contaminants. The experimental data for the force gradient are found to be consistent with theory using the plasma model approach over the entire measurement range. The Drude model approach is excluded by the data at separations from 235 to 400 nm at a 67% confidence level.

We show how the image method can be used together with a recent method developed by C. Eberlein and R. Zietal to obtain the dispersive van der Waals interaction between an atom and a perfectly conducting surface of arbitrary shape. We discuss in detail the case of an atom and a semi-infinite conducting plane. In order to employ the above procedure to this problem it is necessary to use the ingenious image method introduced by Sommerfeld more than one century ago, which is a generalization of the standard procedure. Finally, we briefly discuss other interesting situations that can also be treated by the joint use of Sommerfeld's image technique and Eberlein-Zietal method.

We present a microscopical approach to the electromagnetic vacuum energy of a random medium made of non-polar molecules. We evaluate the contribution of statistical fluctuations to the average total vacuum energy. While the Lamb shift is a function of the electrical susceptibility only, the vacuum energy is generally not, except in the quasicrystalline approximation. We comment on the possibility of testing experimentally our results. We clarify why the effective medium vacuum energy (i.e., that of long-wavelengths) does not account for the total vacuum energy of a molecular dielectric. Consequently, the Lamb shift does not derive from the effective medium vacuum energy except at leading order in the molecular density.

Motivated by experiments in which moving boundaries are simulated by time-dependent properties of static systems, we discuss the model of a massless scalar field submitted to a time-dependent Robin boundary condition (BC) at a static mirror in 1 + 1 dimensions. Using a perturbative approach, we compute the spectral distribution of the created particles and the total particle creation rate, considering a thermal state as the initial field state.

We discuss homogeneous and isotropic cosmological models driven by SU(2) gauge fields in the framework of Einstein gravity. There exists a Yang-Mills field configuration, parametrized by a single scalar function, which consists of parallel electric and magnetic fields and has the stress tensor mimicking an homogeneous and isotropic fluid. The unique SU(2) gauge theory with spontaneous symmetry breaking sharing the same property is the Yang-Mills coupled to the complex doublet Higgs, this exists only in the case of the closed universe. This model contains an intrinsic mechanism for inflation due to the Higgs potential. Our second goal is to show that a successful inflation can be achieved also within the pure Yang-Mills theory adding an appropriate theta-term.

Quantum Field Theory is plagued by divergences in the attempt to calculate physical quantities. Standard techniques of regularization and renormalization are used to keep under control such a problem. In this paper we would like to use a different scheme based on Modified Dispersion Relations (MDR) to remove infinities appearing in one loop approximation in contrast to what happens in conventional approaches. In particular, we apply the MDR regularization to the computation of the entropy of a Schwarzschild black hole from one side and the Zero Point Energy (ZPE) of the graviton from the other side. The graviton ZPE is connected to the cosmological constant by means of of the Wheeler-DeWitt equation.

We report development of a high-sensitivity torsion balance to measure the thermal Casimir force. Special emphasis is placed on experimental investigations of a possible surface electric force originating from surface patch potentials that have been recently noticed by several experimental groups. By gaining a proper understanding of the actual contribution of the surface electric force in real materials, we aim to undertake precision force measurements to resolve the Casimir force at finite temperature in real metals, as well as in other semiconducting materials, such as graphene.

Diagrammatic techniques are well-known in the calculation of dispersion interactions between atoms or molecules. The multipolar coupling scheme combined with Feynman ordered diagrams significantly reduces the number of graphs compared to elementary stationary perturbation theory. We review calculations of van der Waals-Casimir-Polder forces, focusing on two atoms or molecules, one of which is excited. In this case, calculations of the corresponding force are notorious for mathematical issues connected with the spontaneous decay of the excitation. Treating such unstable states in a full non-equilibrium theory provides a physical interpretation of apparent contradictions in previous results and underlines the importance of decay processes for the intermolecular potential. This may have important implications on reactions in biological systems, where excited states may be relatively long-lived and the resonant intermolecular force may result in directed Brownian motion.

We present an integral formulation of the vacuum energy of Schrödinger operators on finite metric graphs. Local vertex matching conditions on the graph are classified according to the general scheme of Kostrykin and Schrader. While the vacuum energy of the graph can contain finite ambiguities the Casimir force on a bond with compactly supported potential is well defined. The vacuum energy is determined from the zeta function of the graph Schrödinger operator which is derived from an appropriate secular equation via the argument principle. A quantum graph has an associated probabilistic classical dynamics which is generically both ergodic and mixing. The results therefore present an analytic formulation of the vacuum energy of this quasi-one-dimensional quantum system which is classically chaotic.

We explore the effects of radiation damping on electrons in a high intensity laser pulse. From classical simulations it is found that an electron can get captured in the beam focus, solely because of its radiation back reaction. This capture regime is considered in the context of an intensity-dependent mass shift of the electron. It is demonstrated that the mass shift is not just a classical effect, occurring also in the (strong field) QED theory. Using the QED process of nonlinear Compton scattering as an example, we show that the condition for the onset of electron capture is the same as the definition of the center-of-mass frame for the effective mass kinematics.

Casimir effect, in most general terms, is the backreaction of a quantum system responding to an adiabatic change of external conditions. This backreaction is expected to be quantitatively measured by a change in the expectation value of a certain energy observable of the system. However, for this concept to be applicable, the system has to retain its identity in the process. Most prevailing tendencies in the analysis of the effect seem to ignore this question.

In general, a quantum theory is defined by an algebra of observables, whose representations by operators in a Hilbert space define concrete physical systems described by the theory. A quantum system retains its identity if both the algebra as well as its representation do not change. We discuss the resulting restrictions for admissible models of changing external conditions. These ideas are applied to quantum field models. No infinities arise, if the algebraic demands are respected.

We review what is presently known about higher loop corrections to the Euler-Heisenberg Lagrangian and its Scalar QED analogue. The use of those corrections as a tool for the study of the properties of the QED perturbation series is outlined. As a further step in a long-term effort to prove or disprove the convergence of the N photon amplitudes in the quenched approximation, we present a parameter integral representation of the three-loop Euler-Heisenberg Lagrangian in 1+1 dimensional QED, obtained in the worldline formalism.

Intense laser light at facilities such as the Extreme Light Infrastructure and the European X-ray Free Electron Laser offer new prospects for probing fundamental physics. Here we examine the theory behind QED processes in background fields modelling ultra-strong, short-duration laser pulses. We focus on pair creation via the trident and photon-stimulated mechanisms.

The photon polarization tensor is the central building block of an effective theory description of photon propagation in the quantum vacuum. It accounts for the vacuum fluctuations of the underlying theory, and in the presence of external electromagnetic fields, gives rise to such striking phenomena as vacuum birefringence and dichroism. Standard approximations of the polarization tensor are often restricted to on-the-light-cone dynamics in homogeneous electromagnetic fields, and are limited to certain momentum regimes only. We devise two different strategies to go beyond these limitations: First, we aim at obtaining novel analytical insights into the photon polarization tensor for homogeneous fields, while retaining its full momentum dependence. Second, we employ wordline numerical methods to surpass the constant-field limit.

Combined effects on the Casimir-Polder potential due to a cosmic string and coaxial metallic cylindrical shell are investigated. For the both regions inside and outside the shell, the potential is decomposed into pure string and shell-induced parts. In a special case of an isotropic polarizability tensor, the Casimir-Polder force in the interior region is attractive with respect to the shell. In the exterior region the force is attractive near the shell and repulsive at large distances from the shell.

We present analytical and simple expressions to determine the free energy, internal energy, entropy, as well as the pressure acting at the interface of a perfectly conducting rectangular Casimir piston. We show that infrared divergencies linear in temperature become cancelled within the piston configuration, and show a continuous behavior consistent with intuitive expectations.

Theory of the Casimir effect for a flat graphene layer interacting with a parallel flat material is presented in detail. The high-temperature asymptotics of a free energy in a graphene-metal system coincides with a Drude high-temperature asymptotics of the metal-metal system. High-temperature behavior in the graphene-metal system is expected at separations of the order of 100 nm at temperature T = 300K.

We proposed a quantum-circuit analog of dynamical Casimir effect by using a superconducting quantum circuit. The proposed system can be regarded as a time-dependent harmonic oscillator, resulting in squeezing of a quantum flux trapped in the system. Here we develop our theory taking into account finite temperatures, in order to contact with experiments like circuit-QED. We present numerical results of occupation probability distributions produced by non-stationary boundary effect at various temperatures.

We review on the models of gravity with a constraint by the Lagrange multiplier field. The constraint breaks general covariance or Lorentz symmetry in the ultraviolet region. We report on the F(R) gravity model with the constraint and the proposal of the covariant (power-counting) renormalized gravity model by using the constraint and scalar projectors. We will show that the model admits flat space solution, its gauge-fixing formulation is fully developed, and the only propagating mode is (higher derivative) graviton, while scalar and vector modes do not propagate. The preliminary study of FRW cosmology indicates to the possibility of inflationary universe solution is also given.

Many calculations of dispersion interactions between atoms and macroscopic bodies or between two bodies make use of Green tensors. The expansion of this tensor in polarizations allows for an anatomic interpretation of the interaction. In planar systems with partial translation invariance, the Weyl representation of the Green tensor is often applied. Although it is transverse in this representation, we argue that the field it describes contains nonradiative parts as well. This can be seen by calculating the Green tensor in the momentum representation and observing certain cancellations among longitudinal and transverse contributions.

The phase diagram of planar fermionic systems, with four-fermion interactions, are studied when external magnetic fields in different directions are applied to the system. The application of external fields can lead to an asymmetric system, which with the combined effects due to temperature and chemical potential, can lead to a rich phase diagram with multiple critical points. The phase transition we are primarily interested in studying is that of the metal-insulator type, in which our model is particularly well suitable. The relation between the phase transition with chiral symmetry breaking and the emergence of a gap (or the presence of a nonvanishing chiral vacuum expectation value) in the planar fermionic system, as a function of the different magnetic fields applied, is analyzed in details. The applications of our results in the context of condensed matter and quantum field theory at low dimensions are discussed.

We calculate the Casimir energy and entropy for two perfect metal spheres in the large and short separation limit. We obtain nonmonotonic behavior of the Helmholtz free energy with separation and temperature, leading to parameter ranges with negative entropy, and also nonmonotonic behavior of the entropy with temperature and with the separation between the spheres. The appearance of this anomalous behavior of the entropy is discussed as well as its thermodynamic consequences.

Recently, a method based on stochastic quantization has been proposed to compute the Casimir force and its fluctuations in arbitrary geometries. It relies on the spectral decomposition of the Laplace operator in the given geometry. Both quantum and thermal fluctuations were considered. Here we use this method to compute the Casimir force on the plates of a finite piston of arbitrary cross section. Asymptotic expressions valid at low and high temperatures, as well as short and long distances are obtained. The case of a piston with triangular cross section is analyzed in detail. The regularization of the divergent stress tensor is described.

The fermionic condensate and current density are investigated in a (2 + 1)-dimensional conical spacetime in the presence of a circular boundary and a magnetic flux. On the boundary the fermionic field obeys the MIT bag boundary condition. For irregular modes, we consider a special case of boundary conditions at the cone apex, when the MIT bag boundary condition is imposed at a finite radius, which is then taken to zero. The condensate and current are periodic functions of the magnetic flux with the period equal to the flux quantum. For both exterior and interior regions, the expectation values are decomposed into boundary-free and boundary-induced parts. In the case of a massless field the boundary-free part in the vacuum expectation value of the charge density vanishes, whereas the presence of the boundary induces nonzero charge density. At distances from the boundary larger than the Compton wavelength of the fermion particle, the condensate and current decay exponentially, with the decay rate depending on the opening angle of the cone.

We define irreducible N-body spectral functions and Casimir energies and consider a massless scalar quantum field interacting locally by positive potentials with classical objects. Irreducible N-body spectral functions in this case are shown to be conditional probabilities of random walks. The corresponding irreducible contributions to scalar many-body Casimir energies are finite and positive/negative for an odd/even number of objects. The force between any two finite objects separable by a plane is always attractive in this case. Analytical and numerical world-line results for the irreducible four-body Casimir energy of a scalar with Dirichlet boundary conditions on a tic-tac-toe pattern of lines are presented. Numerical results for the irreducible three-body Casimir energy of a massless scalar satisfying Dirichlet boundary conditions on three intersecting lines forming an isosceles triangle are also reported. In both cases the symmetric configuration (square and isosceles triangle) corresponds to the minimal irreducible contribution to the Casimir energy.

We apply the worldline formalism and its numerical Monte-Carlo approach to computations of fluctuation induced energy-momentum tensors. For the case of a fluctuating Dirichlet scalar, we derive explicit worldline expressions for the components of the canonical energy-momentum tensor that are straightforwardly accessible to partly analytical and generally numerical evaluation. We present several simple proof-of-principle examples, demonstrating that efficient numerical evaluation is possible at low cost. Our methods can be applied to an investigation of positive-energy conditions.

Irreducible many-body contributions to Green's functions and Casimir energies are defined. We show that the irreducible three-body contribution to Casimir energies are significant and can be more than twenty percent of the total interaction energy. Irreducible three-body contribution for three parallel semitransparent plates in the limit when two plates overlap is obtained in terms of irreducible two-body contributions and shown to be finite and well defined in this limit.

We present calculations of the zero-temperature Casimir interaction between two freestanding graphene sheets as well as between a graphene sheet and a substrate. Results are given for undoped graphene and for a set of doping levels covering the range of experimentally accessible values. We describe different approaches that can be used to derive the interaction. We point out both the predicted power law for the interaction and the actual distance dependence.

A new approach to path integral quantum mechanics in curved space-time for a scalar particle is presented in terms of local curvature involving Fermi or Riemann normal co-ordinates. This approach involves use of a local time translation operator with Lie transport that, while strictly non-unitary in form, nonetheless yields the correct expression for the curved space-time free-particle Lagrangian in the sum-over-histories, with additional terms corresponding to a curvature-dependent violation of probability. These terms simultaneously induce a breakdown of time-reversal symmetry at the quantum mechanical level, and also a violation of the weak equivalence principle at the particle's Compton wavelength scale. Furthermore, the scalar propagator generates a gravitational analogue of the Aharonov-Bohm effect and Berry's phase through the appearance of an overall gauge-invariant phase factor. Future directions to follow from this initial research are presented.

Quantum-mechanical scattering of nonrelativistic charged particles by a magnetic vortex of nonzero transverse size is considered. The high-frequency limit of a scattered particle corresponds to the quasiclassical limit, and we show that the scattering Aharonov-Bohm effect persists in the quasiclassical limit owing to the Fraunhofer diffraction in the forward direction. Therefore, the flux of a magnetic vortex serves as a gate for the propagation of quasiclassical particles moving orthogonally to the vortex. The issue of the experimental detection of the Fraunhofer diffraction peak and the scattering Aharonov-Bohm effect is discussed.

Using nonstandard recursion relations for Fresnel coefficients involving successive stacks of layers, we extend the Lifshitz formula to configurations with an inhomogeneous, n-layered, medium separating two planar objects. The force on each object is the sum of a Lifshitz like force and a force arising from the inhomogeneity of the medium. The theory correctly reproduces very recently obtained results for the Casimir force/energy in some simple systems of this kind. As a by product, we obtain a formula for the force on an (unspecified) stack of layers between two planar objects which generalizes our previous result for the force on a slab in a planar cavity.

The Casimir pressure on the dielectric layer in metal-dielectric-semiconductor (MDS) structures is calculated in the framework of the Lifshitz theory at nonzero temperature. In this calculation the standard parameters of semiconductor devices with a thin dielectric layer are used. We consider the thickness of a layer decreasing from 40 to 1 nm. At the shortest thickness the Casimir pressure achieves 8 MPa. At small thicknesses the results are compared with the predictions of nonrelativistic theory.