Moving mirrors are submitted to reaction forces by vacuum fields. The motional force is known to vanish for a single mirror uniformly accelerating in vacuum. We show that inertial forces (proportional to accelerations) arise in the presence of a second scatterer, exhibiting properties expected for a relative inertia: the mass corrections depend upon the distance between the mirrors, and each mirror experiences a force proportional to the acceleration of the other one. When the two mirrors move with the same acceleration, the mass correction obtained for the cavity represents the contribution to inertia of Casimir energy. Accounting for the fact that the cavity moves as a stressed rigid body, it turns out that this contribution fits Einstein's law of inertia of energy.
We combine linear response theory and dimensional regularization in order to derive the dynamical Casimir force in the low frequency regime. We consider two parallel plates moving along the normal direction in $D-$dimensional space. We assume the free-space values for the mass of each plate to be known, and obtain finite, separation-dependent mass corrections resulting from the combined effect of the two plates. The global mass correction is proportional to the static Casimir energy, in agreement with Einstein's law of equivalence between mass and energy for stressed rigid bodies.
It has been demonstrated, using variational methods, that quantum vacuum energy gravitates according to the equivalence principle, at least for the finite Casimir energies associated with perfectly conducting parallel plates. This conclusion holds independently of the orientation of the plates. We review these arguments and add further support to this conclusion by considering parallel semitransparent plates, that is, $δ$-function potentials, acting on a massless scalar field, in a spacetime defined by Rindler coordinates. We calculate the force on systems consisting of one or two such plates …
We discuss several approaches to determine the Casimir force in inertial frames of reference in different dimensions. On an example of a simple model involving mirrors in Rindler spacetime we show that Casimir's and Lifschitz's methods are inequivalent and only latter can be generalized to other spacetime geometries. For conformally coupled fields we derive the Casimir force in conformally flat spacetimes utilizing an anomaly and provide explicit examples in the Friedmann-Lemaitre-Robertson-Walker $(k=0)$ models.
Vacuum field fluctuations exert a radiation pressure which induces mechanical effects on scatterers. The question naturally arises whether the energy of vacuum fluctuations gives rise to inertia and gravitation in agreement with the general principles of mechanics. As a new approach to this question, we discuss the mechanical effects of quantum field fluctuations on two mirrors building a Fabry-Perot cavity. We first put into evidence that the energy related to Casimir forces is an energy stored on field fluctuations as a result of scattering time delays. We then discuss the forces felt by the…
The Casimir-Polder force between two atoms with equal uniform acceleration and separated by a constant distance R is considered. We show that, in the low-acceleration limit, while the near-zone R^{-6} behavior of the interatomic interaction energy is not changed by the acceleration of the atoms, the far-zone interaction energy decreases as R^{-5} instead of the well-known R^{-7} behavior for inertial atoms. Possibility of an indirect detection of the Unruh effect through measurements of the Casimir-Polder force between the two accelerating atoms is also suggested. We also consider a heuristic …
We show the influence of surface plasmons on the Casimir effect between two plane parallel metallic mirrors at arbitrary distances. Using the plasma model to describe the optical response of the metal, we express the Casimir energy as a sum of contributions associated with evanescent surface plasmon modes and propagative cavity modes. In contrast to naive expectations, the plasmonic modes contribution is essential at all distances in order to ensure the correct result for the Casimir energy. One of the two plasmonic modes gives rise to a repulsive contribution, balancing out the attractive con…
Doubt continues to linger over the reality of quantum vacuum energy. There is some question whether fluctuating fields gravitate at all, or do so anomalously. Here we show that for the simple case of parallel conducting plates, the associated Casimir energy gravitates just as required by the equivalence principle, and that therefore the inertial and gravitational masses of a system possessing Casimir energy $E_c$ are both $E_c/c^2$. This simple result disproves recent claims in the literature. We clarify some pitfalls in the calculation that can lead to spurious dependences on coordinate syste…
We explore the non-linear dynamics of two parallel periodically patterned metal surfaces that are coupled by the zero-point fluctuations of the electromagnetic field between them. The resulting Casimir force generates for asymmetric patterns with a time-periodically driven surface-to-surface distance a ratchet effect, allowing for directed lateral motion of the surfaces in sizeable parameter ranges. It is crucial to take into account inertia effects and hence chaotic dynamics which are described by Langevin dynamics. Multiple velocity reversals occur as a function of driving, mean surface dist…
We take dissipation into account in the derivation of the Casimir energy formula between two objects placed in a surrounding medium. The dissipation channels are considered explicitly in order to take advantage of the unitarity of the full scattering processes. We demonstrate that the Casimir energy is given by a scattering formula expressed in terms of the scattering amplitudes coupling internal channels and taking dissipation into account in an implicit way. We prove that this formula is also valid when the surrounding medium is dissipative.
We investigate the thermal Casimir effect of a massless scalar field for two parallel plates moving in the equatorial orbit in Kerr space-time. Under the assumption that the typical cavity size is much smaller than the orbital radius, proposed by Sorge, we deduce the analytical expression of the renormalized free energy in this curved space-time. We also get the analytical representation for the renormalized internal energy, and find that there is a thermal correction to the Casimir energy, which depends on the proper temperature and the proper geometrical parameters of the plates. The asympto…
We investigate Casimir effect as well as thermal Casimir effect for a pair of parallel perfectly plates placed in general stationary space-time background. It is found that the Casimir energy is influenced by the 00-component of metric and the corresponding quantity in dragging frame. We give a scheme to renormalize thermal correction to free energy in curved space-time. It is shown that the thermal corrections to Casimir thermodynamic quantities not only depend on the proper temperature and proper geometrical parameters of the plates, but also on the determinant of space-time metric.
We consider the vacuum energy of the electromagnetic field in systems characterized by a constant conductivity using the zeta-regularization approach. The interaction in two cases is investigated: two infinitely thin parallel sheets and an infinitely thin spherical shell. We found that the Casimir energy for the planar system is always attractive and it has the same characteristic distance dependence as the interaction for two perfect semi-infinite metals. The Casimir energy for the spherical shell depends on the inverse radius of the sphere, but it maybe negative or positive depending on the …
We discuss the Casimir effect between two identical, parallel slabs, emphasizing the role of dissipation and temperature. Starting from quite general assumptions, we analyze the behavior of the Casimir entropy in the limit T->0 and link it to the behavior of the slab's reflection coefficients at low frequencies. We also derive a formula in terms of a sum over modes, valid for dissipative slabs that can be interpreted in terms of a damped quantum oscillator.
In a recent paper [1] the Casimir energy was calculated for a massive dirac field in (1+1) dimensional space-time in the presence of an inverse square well potential and shown to be positive. It will be shown that this result violates a key assumption of quantum field theory which is that the vacuum state is the state of minimum energy. The reason for this discrepancy is examined and is shown to be related to the way the charge density operator is defined. If the charge density operator is defined using point splitting then an extra term will be added to the charge which will result in the Cas…
We consider the Casimir-Polder interaction between two atoms, one in the ground state and the other in its excited state. The interaction is time-dependent for this system, because of the dynamical self-dressing and the spontaneous decay of the excited atom. We calculate the dynamical Casimir-Polder potential between the two atoms using an effective Hamiltonian approach. The results obtained and their physical meaning are discussed and compared with previous results based on a time-independent approach which uses a non-normalizable dressed state for the excited atom.
Effects due to vacuum fluctuations in a semi-classical model of a massless scalar field interacting with a rotating ring are investigated by introducing a collective coordinate for the motion of the background potential. The model is solved for a repulsive periodic $δ$-distribution background of arbitrary strength. The Casimir energy of this system is calculated in the co-rotating and, by Legendre transformation, in the stationary laboratory frame. The zero-point contribution to the angular momentum in this model is bounded below by $|\ell_{ZP}|\leq\hbar/24$ and the ground state of the entire …
We study quantum friction and Casimir forces with a full-relativistic formalism for atoms modelled as Unruh-DeWitt detectors in the presence of arbitrary macroscopic objects. We consider the general case of atoms with arbitrary relativistic trajectories in arbitrary quantum states (including coherent superpositions) close to objects that impose arbitrary boundary conditions. Particularizing for conducting plates, we show that, for relative velocities close to the speed of light, the quantum friction diverges while the Casimir force is almost independent of the velocity. Since we include the ef…
We show that Casimir-Polder forces between two relativistic uniformly accelerated atoms exhibit a transition from the short distance thermal-like behavior predicted by the Unruh effect, to a long distance non-thermal behavior, associated with the breakdown of a local inertial description of the system. This phenomenology extends the Unruh thermal response detected by a single accelerated observer to an accelerated spatially extended system of two particles, and we identify the characteristic length scale for this crossover with the inverse of the proper acceleration of the two atoms. Our resul…
We consider here, a two-atom system is uniformly moving through a circular ring at an ultra-relativistic speed and weakly interacting with common external fields. The vacuum fluctuations of the quantum fields generate the entanglement between the atoms. Hence an effective energy shift is originated, which depends on the inter-atomic distance. This is commonly known as resonance Casimir-Polder interaction (RCPI). It is well known that, for a linearly accelerated system coupled with a massless scalar field, we get a thermal response when the local inertial approximation is valid. On the contrary…
We study the lateral dependence of the Casimir energy for different corrugated gratings of arbitrary periodic profile. To this end we model the profiles as stacks of horizontal rectangular slices following the profiles' shape and evaluate numerically the Casimir energy between them for different relative lateral displacements of the two corrugated plates. We compare our results with predictions obtained within the proximity force approximation (PFA). At comparable separation of the corrugated plates and geometric parameters, we find a strong dependence of the Casimir energy on the shape of the…
The regularized total Casimir energy in spacetimes with boundaries is not in general equal to the integral of the regularized energy density. This paradoxical phenomenon is most transparently analyzed in the simple example of a massless scalar field in 1+1 dimensions confined to a line element of length $L$ and obeying Dirichlet boundary conditions.
The attractive force between metallic surfaces, predicted by Casimir in 1948, seems to indicate the physical existence and measurability of the quantized electromagnetic field's zero-point energy. It is shown in this article, that the measurements of that force do not confirm Casimir's model, but in fact disprove it's foundational assumption that metal plates may be represented in the theory by quantum-field-theoretical boundaries. The consequences for the cosmological constant problem are discussed.
We extend our previous work on a derivative expansion for the Casimir energy, to the case of the electromagnetic field coupled to two thin, imperfect mirrors. The latter are described by means of vacuum polarization tensors localized on the mirrors. We apply the results so obtained to compute the first correction to the proximity force approximation to the static Casimir effect.
We compute the Casimir Energy of a spherical region using a Surface Impedance approach. We characterize the Surface Impedance of the boundary using plasma model. Exact analytical formulae are obtained by means of the zeta function regularization method and all the divergencies are explicitly computed. We find that it is possible to have negative Casimir energy for some range of values of the relevant parameter. Limits of applicability of the model are discussed.
This paper has been withdrawn because a much better version, G. Gazzola, M. C. Nemes and W. F. Wreszinski - On the Casimir Energy for a massive quantum scalar field and the cosmological constant- Annals of Physics (N.Y.) vol. 324, 2095-2107 (2009) has appeared, which corrects several conceptual errors in the present one.
In this paper we discuss the computation of Casimir energy on a quantum computer. The Casimir energy is an ideal quantity to calculate on a quantum computer as near term hybrid classical quantum algorithms exist to calculate the ground state energy and the Casimir energy gives physical implications for this quantity in a variety of settings. Depending on boundary conditions and whether the field is bosonic or fermionic we illustrate how the Casimir energy calculation can be set up on a quantum computer and calculated using the Variational Quantum Eigensolver algorithm with IBM QISKit. We compa…
The Casimir energy of a dilute homogeneous nonmagnetic dielectric ball at zero temperature is derived analytically within a microscopic realistic model of dielectrics for an arbitrary physically possible frequency dispersion of dielectric permittivity. Divergences are absent in calculations, a minimum interatomic distance is a physical cut-off. Casimir surface force is proved to be attractive. A physical definition of the Casimir energy is discussed.
We derive exact expressions for the Casimir scalar interaction energy between media-separated eccentric dielectric cylinders and for the media-separated cylinder-plane geometry using a mode-summation approach. Similarly to the electromagnetic Casimir-Lifshitz interaction energy between fluid-separated planar plates, the force between cylinders is attractive or repulsive depending on the relative values of the permittivities of the three intervening media.
We use a point-matching approach to numerically compute the Casimir interaction energy for a two perfect-conductor waveguide of arbitrary section. We present the method and describe the procedure used to obtain the numerical results. At first, our technique is tested for geometries with known solutions, such as concentric and eccentric cylinders. Then, we apply the point-matching technique to compute the Casimir interaction energy for new geometries such as concentric corrugated cylinders and cylinders inside conductors with focal lines.
We study the Casimir energy due to a quantum real scalar field coupled to two planar, infinite, zero-width, parallel mirrors with non-homogeneous properties. These properties are represented, in the model we use, by scalar functions defined on each mirror's plane. Using the Gelfand-Yaglom's theorem, we construct a Lifshitz-like formula for the Casimir energy of such a system. Then we use it to evaluate the energy perturbatively, for the case of almost constant scalar functions, and also implementing a Derivative Expansion, under the assumption that the spatial dependence of the properties is s…
The Casimir energy, free energy and Casimir force are evaluated, at arbitrary finite temperature, for a dilute dielectric ball with uniform velocity of light inside the ball and in the surrounding medium. In particular, we investigate the classical limit at high temperature. The Casimir force found is repulsive, as in previous calculations.
The leading semiclassical estimates of the electromagnetic Casimir stresses on a spherical and a cylindrical metallic shell are within 1% of the field theoretical values. The electromagnetic Casimir energy for both geometries is given by two decoupled massless scalars that satisfy conformally covariant boundary conditions. Surface contributions vanish for smooth metallic boundaries and the finite electromagnetic Casimir energy in leading semiclassical approximation is due to quadratic fluctuations about periodic rays in the interior of the cavity only. Semiclassically the non-vanishing Casimir…
We introduce a general, simple and effective method of evaluating the zero point energy of a quantum field under the influence of arbitrary boundary conditions imposed on the field on flat surfaces perpendicular to a chosen spatial direction. As an example we apply the method to the Casimir effect associated with a massive fermion field on which MIT bag model type of boundary conditions are imposed.
Casimir pistons are models in which finite Casimir forces can be calculated without any suspect renormalizations. It has been suggested that such forces are always attractive, but we present several counterexamples, notably a simple type of quantum graph in which the sign of the force depends upon the number of edges. We also show that Casimir forces in quantum graphs can be reliably computed by summing over the classical orbits, and study the rate of convergence of the periodic orbit expansion. In generic situations where no analytic expression is available, the sign and approximate magnitude…
We study the role of surface polaritons in the zero-temperature Casimir effect between two graphene layers that are described by the Dirac model. A parametric approach allows us to accurately calculate the dispersion relations of the relevant modes and to evaluate their contribution to the total Casimir energy. The resulting force features a change of sign from attractive to repulsive as the distance between the layers increases. Contrary to similar calculations that have been performed for metallic plates, our asymptotic analysis demonstrates that at small separations the polaritonic contribu…
We consider a dielectric medium with an ultraviolet behavior as it follows from the Drude model. Compared with dilute models, this has the advantage that, for large frequencies, two different media behave the same way. As a result one expects the Casimir energy to contain less divergencies than for the dilute media approximation. We show that the Casimir energy of a spherical dielectric ball contains just one divergent term, a volume one, which can be renormalized by introducing a contact term analogous to the volume energy counterterm needed in bag models. PACS: 12.20.Ds, 03.70.+k, 77.22.Ch
One of the very first applications of the quantum field theoretic vacuum state was in the development of the notion of Casimir energy. Now field theoretic Casimir energies, considered individually, are always infinite. But differences in Casimir energies (at worst regularized, not renormalized) are quite often finite --- a fortunate circumstance which luckily made some of the early calculations, (for instance, for parallel plates and hollow spheres), tolerably tractable. We shall explore the extent to which this observation can be made systematic. For instance: What are necessary and sufficien…
It was recently found that thermodynamic anomalies which arise in the Casimir effect between metals described by the Drude model can be attributed to the interaction of fluctuating Foucault (or eddy) currents [Phys. Rev. Lett. 103, 130405 (2009)]. We show explicitly that the two leading terms of the low-temperature correction to the Casimir free energy of interaction between two plates, are identical to those pertaining to the Foucault current interaction alone, up to a correction which is very small for good metals. Moreover, a mode density along real frequencies is introduced, showing that t…
Casimir entropy is an important aspect of casimir effect.In this paper,we employ the path integral method to derive the total relation for casimir entropy and internal energy of arbitrary shaped objects in the presence of two,three and four dimensions scalar fields and electromagnetic field.We obtain the casimir entropy and internal energy of two nanoribbon immersed in scalar field and two nanospheres immersed in scalar field and electromagnetic field.The casmir entropy of two nanospheres immersed in the electromagnetic field in small interval of temperature variations,shown a different behavi…
The Casimir energy for a conducting spherical shell of radius $a$ is computed using a direct mode summation approach. An essential ingredient is the implementation of a recently proposed method based on Cauchy's theorem for an evaluation of the eigenfrequencies of the system. It is shown, however, that this earlier calculation uses an improper set of modes to describe the waves exterior to the sphere. Upon making the necessary corrections and taking care to ensure that no mathematically ill-defined expressions occur, the technique is shown to leave numerical results unaltered while avoiding a …
We derive an exact analytic expression for the high-temperature limit of the Casimir interaction between two Drude spheres of arbitrary radii. Specifically, we determine the Casimir free energy by using the scattering approach in the plane-wave basis. Within a round-trip expansion, we are led to consider the combinatorics of certain partitions of the round trips. The relation between the Casimir free energy and the capacitance matrix of two spheres is discussed. Previously known results for the special cases of a sphere-plane geometry as well as two spheres of equal radii are recovered. An asy…
The Casimir interaction energy for a class of discrete self-similar configuration of parallel plates is evaluated using existing methods. The similarities to characteristics of an attractive Casimir force is deduced only at infinite range of configuration. Further, the emergence of Casimir-like energy is qualitatively described for a Gaussian model of Landau-Ginzburg scalar field. Its relevance to self-similarity in the statistical field is shown at infinite range of fluctuations.
We consider the finite temperature Casimir free energy acting on a spherical shell in (D+1)-dimensional Minkowski spacetime due to the vacuum fluctuations of scalar and electromagnetic fields. Dirichlet, Neumann, perfectly conducting and infinitely permeable boundary conditions are considered. The Casimir free energy is regularized using zeta functional regularization technique. To renormalize the Casimir free energy, we compute the heat kernel coefficients $c_n$, $0\leq n\leq D+1$, from the zeta function $ζ(s)$. After renormalization, the high temperature leading term of the Casimir free ener…
We evaluate the difference between the Casimir free energies corresponding to either grounded or isolated perfect conductors, at high temperatures. We show that a general and simple expression for that difference can be given, in terms of the electrostatic capacitance matrix for the system of conductors. For the case of close conductors, we provide approximate expressions for that difference, by evaluating the capacitance matrix using the proximity force approximation. Since the high-temperature limit for the Casimir free energy for a medium described by a frequency-dependent conductivity dive…
A general, exact formula is derived for the expectation value of the electromagnetic energy density of an inhomogeneous absorbing and dispersive dielectric medium in thermal equilibrium, assuming that the medium is well approximated as a continuum. From this formula we obtain the formal expression for the Casimir force density. Unlike most previous approaches to Casimir effects in which absorption is either ignored or admitted implicitly through the required analytic properties of the permittivity, we include dissipation explicitly via the coupling of each dipole oscillator of the medium to a …
We use the mode summation method together with zeta-function regularization to compute the Casimir energy of a dilute dielectric cylinder. The method is very transparent, and sheds light on the reason the resulting energy vanishes.
It is shown that recent criticism by C. R. Hagen (hep-th/9902057) questioning the validity of stress tensor treatments of the Casimir energy for space divided into two parts by a spherical boundary is without foundation.
We develop a diagrammatic representation of the Casimir energy of a multibody configuration. The diagrams represent multiple reflections between the objects and can be organized by a few simple rules. The lowest-order diagrams (or reflections) give the main contribution to the Casimir interaction which proves the usefulness of this expansion. Among some applications of this, we find analytical formulae describing the interaction between "edges", i.e. semi-infinite plates, where we also give a first example of blocking in the context of the Casimir energy. We also find the interaction of edges …
Zero-point energy is generally known to be unphysical. Casimir effect, however, is often presented as a counterexample, giving rise to a conceptual confusion. To resolve the confusion we study foundational aspects of Casimir effect at a qualitative level, but also at a quantitative level within a simple toy model with only 3 degrees of freedom. In particular, we point out that Casimir vacuum is not a state without photons, and not a ground state for a Hamiltonian that can describe Casimir force. Instead, Casimir vacuum can be related to the photon vacuum by a non-trivial Bogoliubov transformat…
The Casimir energy of a dilute dielectric cylinder, with the same light-velocity as in its surrounding medium, is evaluated exactly to first order in $ξ^2$ and numerically to higher orders in $ξ^2$. The first part is carried out using addition formulas for Bessel functions, and no Debye expansions are required.
We study the singularities that the Casimir energy of a scalar field in spacetimes with Lifshitz dimensions exhibits, and provide expressions of the energy in terms of multidimensional zeta functions for the massless case. Using the zeta-regularization method, we found that when the 4-dimensional spacetime has Lifshitz dimensions, then for specific values of the critical exponents, the Casimir energy is singular, in contrast to the non-Lifshitz case. Particularly we found that when the value of the critical exponent is $z=2$, the Casimir energy is singular, while for $z\geq 3$ the Casimir ener…
We compute the Casimir energy of the photon field in a de Sitter space and find it to be proportional to the size of the horizon, the same form of the holographic dark energy. We suggest to make metamaterials to mimic de Sitter space in laboratory and measure the predicted Casimir energy.
We develop the worldline formalism for computations of composite operators such as the fluctuation induced energy-momentum tensor. As an example, we use a fluctuating real scalar field subject to Dirichlet boundary conditions. The resulting worldline representation can be evaluated by worldline Monte-Carlo methods in continuous spacetime. We benchmark this worldline numerical algorithm with the aid of analytically accessible single-plate and parallel-plate Casimir configurations, providing a detailed analysis of statistical and systematic errors. The method generalizes straightforwardly to arb…
We consider a Casimir cavity, one plate of which is a thin superconducting film. We show that when the cavity is cooled below the critical temperature for the onset of superconductivity, the sharp variation (in the far infrared) of the reflection coefficient of the film engenders a variation in the value of the Casimir energy. Even though the relative variation in the Casimir energy is very small, its magnitude can be comparable to the condensation energy of the superconducting film, and this gives rise to a number of testable effects, including a significant increase in the value of the criti…
The Casimir effect is investigated in cylindrical and toroidal carbon nanotubes within the framework of the Dirac-like model for the electronic states. The topological Casimir energy is positive for metallic cylindrical nanotubes and is negative for semiconducting ones. The toroidal compactification of a cylindrical nanotube along its axis increases the Casimir energy for metallic-type (periodic) boundary conditions along its axis and decreases the Casimir energy for the semiconducting-type compactifications. For finite length metallic nanotubes the Casimir forces acting on the tube edges are …
The Casimir energy is the first-order-in-\hbar correction to the energy of a time-independent field configuration in a quantum field theory. We study the Casimir energy in a toy model, where the classical field is replaced by a separable potential. In this model the exact answer is trivial to compute, making it a good place to examine subtleties of the problem. We construct two traditional representations of the Casimir energy, one from the Greens function, the other from the phase shifts, and apply them to this case. We show that the two representations are correct and equivalent in this mode…
Performing functional integration of the free Lagrangian, we find the vacuum energy of a field. The functional integration is performed in a way which easily generalizes to systems at non-zero temperature. We use this technique to obtain the Casimir energy density and pressure at arbitrary temperatures.
