The Casimir effect

Vacuum is what remains when there is nothing left in space. But this "nothing" is still able to act on objects placed in vacuum. The archetype of such an action of vacuum is the so-called Casimir force.In fact vacuum consists in fluctuating electromagnetic fields propagating through space with the speed of light, and having the minimal fluctuation energy allowed by quantum theory. Since vacuum energy is a minimum, it cannot be used to extract work. But the fluctuations have observable consequences in atomic and subatomic physics. An atom interacting only with vacuum fields suffers spontaneous emission processes which can be considered as induced by these fields. When fallen in its ground state, the atom no longer emits photons but its coupling to vacuum results in a measurable Lamb shift of the absorption frequencies.Two atoms located at different places experience an attractive van der Waals force which plays an important role in physico-chemical and biological processes. Casimir was studying this effect when he discovered in 1948 that two mirrors placed in vacuum are attracted towards each other.There are many reasons for the growing interest in this Casimir force. Progress in experimental techniques have recently allowed for accurate measurements and have then led to tests of this crucial prediction of quantum electrodynamics. These tests have required theoretical developments for dealing satisfactory with the description of the optical response of mirrors or of geometry of the experiments. They have also made it possible to extend the search for hypothetical new weak forces into the distance range where the Casimir force dominates over other forces.Nevertheless, in spite of all this progress and renewal interest on the subject there are still basic questions that have not been answered and extensions that have not been achieved, for example a consistent treatment of temperature effects or a full theory of geometrical effects between mirrors made of real materials.Recently it has been recognized that vacuum induced forces played a considerable role in micro- and nano-electromechanical systems (MEMS or NEMS) with typical sizes in the µm range or below. The emerging interface between quantum Casimir forces and nanophysics is already clearly visible in this Focus Issue, and it will certainly gain more and more importance in the forthcoming years.The topics covered by the articles in this Focus Issue of New Journal of Physics reflect the increasing activity and broadening spectrum of the studies connected to the Casimir forces. We hope that the reader will find in this collection of papers new research results or proposals as well as reviews by experts of a large set of open issues which are currently addressed by an expanding community. We also hope that the issue will interest the non-expert readers and draw their attention to this exciting and interdisciplinary research field.The articles below represent the first contributions and further additions will appear.Focus on Casimir Forces ContentsCasimir energy with a Robin boundary: the multiple-reflection cylinder-kernel expansion Z Liu and S A Fulling On the use of hydrogen switchable mirrors in Casimir force experiments Sven de Man and Davide Iannuzzi Thermal corrections to the Casimir effect Iver Brevik, Simen Ellingsen and K Milton Casimir forces and non-Newtonian gravitation R Onofrio Sample dependence of the Casimir force I Pirozhenko, A Lambrecht and V B Svetovoy Casimir effect: a novel experimental approach at large separation P Antonini, G Bressi, G Carugno, Giuseppe Galeazzi, Giuseppe Messineo and G Ruoso Exact zero-point interaction energy between cylinders F D Mazzitelli, D A R Dalvit and F C Lombardo Stability and the proximity theorem in Casimir actuated nano devices R Esquivel-Sirvent, Luis Reyes and Jeffrey Bárcenas Casimir effect for arbitrary materials: contributions within and beyond the light cone W L Mochán and C Villarreal The Casimir effect within scattering theory A Lambrecht, Neto P A Maia and S Reynaud Quantum electrodynamical torques in the presence of Brownian motion Jeremy N Munday, Davide Iannuzzi and F Capasso Rubén G Barrera, Universidad Nacional Autónoma de México, Mexico Serge Reynaud, Université Pierre et Marie Curie, Paris, France

Dirac's hole theory (HT) and quantum field theory (QFT) are generally considered equivalent. However, it was recently shown by several investigators that this is not necessarily the case because when the change in the vacuum energy was calculated for a time-independent perturbation, HT and QFT yielded different results. In this paper, we extend this discussion to include a time-dependent perturbation for which the exact solution to the Dirac equation is known. We show that for this case also, HT and QFT yield different results. In addition, we offer some discussion of the problem of anomalies in QFT. PACS Nos.: 03.65–w, 11.10–z

An important problem in Quantum Field Theory (QFT) is to understand the structures of observables on spacetime manifolds of nontrivial topology. Such observables arise naturally when studying physical systems at finite temperature and/or finite volume and encode subtle properties of the underlying microscopic theory that are often obscure on the flat spacetime. Locality of the QFT implies that these observables can be constructed from more basic building blocks by cutting-and-gluing along a spatial slice, where a crucial ingredient is the Hilbert space on the spatial manifold. In Conformal Field Theory (CFT), thanks to the operator-state correspondence, we have a non-perturbative understanding of the Hilbert space on a spatial sphere. However it remains a challenge to consider more general spatial manifolds. Here we study CFTs in spacetime dimensions d > 2 on the spatial manifold T2 × ℝd−3 which is one of the simplest manifolds beyond the spherical topology. We focus on the ground state in this Hilbert space and analyze universal properties of the ground state energy, also commonly known as the Casimir energy, which is a nontrivial function of the complex structure moduli τ of the torus. The Casimir energy is subject to constraints from modular invariance on the torus which we spell out using PSL(2, ℤ) spectral theory. Moreover we derive a simple universal formula for the Casimir energy in the thin torus limit using the effective field theory (EFT) from Kaluza-Klein reduction of the CFT, with exponentially small corrections from worldline instantons. We illustrate our formula with explicit examples from well-known CFTs including the critical O(N) model in d = 3 and holographic CFTs in d ≥ 3.

We calculate the Casimir force between a perfect reflective wall and a semitransparent wall in the laser cavity. Using the Fox—Li quasimode theory to describe the electromagnetic field in the laser cavity, the vacuum energy and the Casimir force are calculated. We compare our results to the force in the ideal situation and find it smaller in the dissipative cavity. We also find that the Casimir force decreases with the increase of the wall-wall distance and the decay rate of the quasimodes in the laser cavity.

The Casimir effect is usually interpreted as arising from the modification of the zero point energy of QED when two perfectly conducting plates are put very close to each other, and as a proof of the “reality” of this zero point energy. The Dark Energy, necessary to explain the acceleration of the expansion of the Universe is sometimes viewed as another proof of the same reality. Recently, several physicists have challenged the usual interpretation, arguing that the Casimir effect should rather be considered as a “giant” van der Waals effect. All these aspects are shortly reviewed.

We present a quantum framework grounded in micro-causality and spacetime quantization, where spacetime is composed of discrete, causally ordered units rather than a continuous manifold. This intrinsic structure naturally generates fundamental quantum features—wave–particle duality, energy quantization, and the Heisenberg uncertainty principle—without invoking canonical quantization, matter-wave postulates, or harmonic-oscillator models. Consequently, this paradigm eliminates the long-standing paradoxes of wavefunction collapse, self-interference, and non-local spooky correlations in entanglement.Within this causal lattice spacetime, quantum commutation relations emerge from finite shift operators, and the uncertainty principle arises geometrically from non-commutativity at the fundamental level. Unlike standard quantum field theory (QFT), which predicts infinite vacuum energy and requires renormalization, the present framework removes zero-point energy, ultraviolet divergences, and fine-tuning. Moreover, U(1) symmetry is broken geometrically on the lattice, leading to mass generation without a Higgs mechanism.The Casimir effect is reinterpreted as a temperature-dependent imbalance in thermal radiation pressure rather than the consequence of vacuum fluctuations, predicting that the Casimir force should vary with ambient temperature. Experimental confirmation of such dependence would directly support the lattice interpretation and challenge the QFT view of vacuum energy.By unifying key quantum behaviors through discrete causal geometry, this model offers a divergence-free and physically grounded alternative to continuum-based QFT. It opens new avenues for quantum unification, cosmology, and the understanding of fundamental constants, without speculative constructs, singularities, or infinities.

We study the adiabatic evolution of a two-level model in the presence of an external classical electric field. The coupling between the quantum model and the classical field is taken in the electric dipole approximation. In this regime, we show the absence of geometric phases in the interacting two-level model in the presence of any periodic real time-dependent classical electric field. We obtain a conservative scalar potential in the calculation of Berry's phases of the instantaneous eigenstates of the model. For complex electric fields, we recover the existence of geometric phases. In particular, the geometric phases of the instantaneous eigenstates of the model in the presence of a positive or of a negative frequency component of the monochromatic electric field differ by an overall sign. As a check on our results, we map this interacting two-level model onto a spin-1/2 model under the action of a classical magnetic field. We confirm that the first one acquires Berry's phase only in the rotating wave approximation [RWA].

In this paper we take a deeper look at the technically elementary but physically robust viewpoint in which the Casimir energy in dielectric media is interpreted as the change in the total zero point energy of the electromagnetic vacuum summed over all states. Extending results presented in previous papers [hep-th/9609195; hep-th/9702007] we approximate the sum over states by an integral over the density of states including finite volume corrections. For an arbitrarily-shaped finite dielectric, the first finite-volume correction to the density of states is shown to be proportional to the surface area of the dielectric interface and is explicitly evaluated as a function of the permeability and permitivity. Since these calculations are founded in an elementary and straightforward way on the underlying physics of the Casimir effect they serve as an important consistency check on field-theoretic calculations. As a concrete example we discuss Schwinger's suggestion that the Casimir effect might be the underlying physical basis behind sonoluminescence}. The recent controversy concerning the relative importance of volume and surface contributions is discussed. For sufficiently large bubbles the volume effect is always dominant. Furthermore we can explicitly calculate the surface area contribution as a function of refractive index.

The renormalization program in every renormalized theory should run consistently with the type of boundary condition imposed on quantum fields. To maintain this consistency, the counterterms usually appear in the position-dependent form. In this study, using such counterterms, we calculated the radiative correction to the Casimir energy for massive and massless Lorentz-violating scalar field constrained with Dirichlet boundary condition between two parallel plates in [Formula: see text] spatial dimensions. In the calculation procedure, to remove infinities appearing in the vacuum energies, the box subtraction scheme (BSS) supplemented by the cutoff regularization technique and analytic continuation technique was employed. Normally, in the BSS, two similar configurations are defined and their vacuum energies are subtracted from each other in the appropriate limits. Our final results regarding all spatial dimensions were convergent and consistent with the expected physical basis. We further plotted the Casimir energy density for the time-like and space-like Lorentz-violating systems in a number of odd and even dimensions; multiple aspects of the obtained results were ultimately discussed.

The Casimir effect is a fundamental quantum phenomenon induced by the zero-point energy for a quantum field. It is well-known for relativistic fields with a linear dispersion relation, while its existence or absence for nonrelativistic fields with a quadratic dispersion is an unsettled question. Here, we investigate the Casimir effects for various dispersion relations on the lattice. We find that Casimir effects for dispersions proportional to an even power of momentum are absent in a long distance but a remnant of the Casimir effect survives in a short distance. Such a remnant Casimir effect will be experimentally observed in materials with quantum fields on the lattice, such as thin films, narrow nanoribbons, and short nanowires. In terms of this effect, we also give a reinterpretation of the Casimir effect for massive fields.

This theoretical work aims to explore the possibility of extracting in electrical form and in accordance with EMMY NOETHER's theorem, part of quantum vacuum energy. Cyclic deformations of an elastic piezoelectric bridge, embedded at both ends, are generated by the Casimir force (FCA) between two near electrodes. This deformation of the piezoelectric bridge induces an automatic electrical charge which can in turn induce a possible Coulomb repulsive force (FCO). This force FCO is automatically triggered by the closing of a switch n°1 on a third electrode. The resultant force applied on the elastic piezoelectric bridge, FCO - FCA, straightens the bridge, imparting kinetic energy. This kinetic energy plus the deformation energy memorized in the elastic bridge is dissipated by the Casimir force FCA. The force FCO cancels out when a switch n°2 in series connects this third electrode to the ground shortly after closing the switch n°1. We return to the initial situation with just the Casimir force applied on the bridge. Thus, the system constantly exploits the energy of the quantum vacuum vibrates, creating peaks of usable electrical power with each vibration. This electrical could be of some microwatts per vibration depending on the chosen amplification FCO/FCA.

A pull-in parameter analysis for the cantilever NEMS actuator model including surface energy, fringing field and Casimir effects

The last decades have witnessed an unprecedented advancement in our knowledge of the large scale universe. In particular, increasingly accurate cosmological observations have allowed us to discover a form of "dark energy", which presently dominates the expansion of the universe -making it accelerated. On the other hand, fundamental problems in the standard (CDM) cosmological model point towards the possibility of a primordial inflationary period. Both these expansion phases have in common the fact that they should be governed by forms of energy with properties much similar to those of vacuum energy of classical or quantum fields. In the meanwhile, quantum field theory in curved spaces (QFTCS) has proved a rich framework to analyze phenomena of a quantum nature in regimes where spacetime curvature is relevant, but not too extreme, and, particularly, it yields novel insights on the structure and dynamics of quantum vacuum. In this dissertation, we make a thorough exposition of the fundamentals of QFTCS and present some of its applications in cosmological spacetimes. Particular attention is given to the construction of an empirical notion of particles through an idealized model of particle detectors, and to the phenomenon of particle creation in expanding FLRW spacetimes. Further, we develop the procedure of adiabatic renormalization, and use it to compute the renormalized stress tensor in these spacetimes. For a noninteracting scalar field in exponentially expanding (de Sitter) spaces, we find that these results take the form of a cosmological constant, although a quantitatively self-consistent value with the background expansion can only be found at Planckian densities. We also present a construction of a simple inflationary model, driven by a self-interacting classical scalar field, and show how the quantized fluctuations of this field could give rise to a nearly scale-invariant power spectrum, like the one that is currently observed in the Cosmic Microwave Background.

In this paper, we study the effect of detuning parameters and driven external classical field parameters on a quantum system consisting of a ♢-configuration four-level atom inside a quantized cavity field. Under some canonical conditional of dressed states, the exact solution of the Schrödinger equation is obtained. The occupation of atomic levels and statistical population inversion is studied. Our results show that the classical field parameter dissolved the collapse periods and increased the maximum bounds of the upper state, while decreasing the lower bounds of the lower state. The detuning parameters reduce the minimum bounds of atomic levels and their inversion. On the other hand, the linear entropy and l1 norm of coherence are employed to measure the temporal evolution of the mixedness and coherence. It is found that the driven classical field improves the temporal evolution of the mixedness and lower bounds of coherence. However, the detuning parameters have a destructive effect on the mixedness and lower bounds of coherence. The intensity of the external classical field is regarded as a control parameter with different values of detuning parameters.

In this paper, I study the Casimir effect caused by a charged and massive scalar field that breaks Lorentz invariance in a CPT-even, aether-like manner. I investigate the case of a scalar field that satisfies Dirichlet or mixed (Dirichlet–Neumann) boundary conditions on a pair of very large plane parallel plates. The case of Neumann boundary conditions is straightforward and will not be examined in detail. I use the [Formula: see text]-function regularization technique to study the effect of a constant magnetic field, orthogonal to the plates, on the Casimir energy and pressure. I investigate the cases of a timelike Lorentz asymmetry, a spacelike Lorentz asymmetry in the direction perpendicular to the plates, and a spacelike asymmetry in the plane of the plates and, in all those cases, derive simple analytic expressions for the zeta function, Casimir energy and pressure in the limits of small plate distance, strong magnetic field and large scalar field mass. I discover that the Casimir energy and pressure, and their magnetic corrections, all strongly depend on the direction of the unit vector that implements the breaking of the Lorentz symmetry.