We consider the Casimir interaction between two spheres corresponding to massless Dirac fields with MIT-bag boundary conditions. Using operator approach, we derive the TGTG-formula for the Casimir interaction energy between the two spheres. A byproduct is the explicit formula for the translation matrix that relates the fermionic spherical waves in different coordinate systems. In the large separation limit, it is found that the order of the Casimir interaction energy is $L^{-5}$, where $L$ is the separation between the centers of the spheres. This order is intermediate between that of two Dirichlet spheres (of order $L^{-3}$) and two Neumann spheres (of order $L^{-7}$). In the small separation limit, we derive analytically the asymptotic expansion of the Casimir interaction energy up to the next-to-leading order term. The leading term agrees with the proximity force approximation. The result for the next-to-leading order term is compared to the corresponding results for scalar fields and electromagnetic fields.
The Casimir energies for plate-sphere system and sphere-sphere system under PFA in the presence of one extra compactified universal dimension are analyzed. We find that the Casimir energy between a plate and a sphere in the case of sphere-based PFA is divergent. The Casimir energy of plate-sphere system in the case of plate-based PFA is finite and keeps negative. The extra-dimension corrections to the Casimir energy will be more manifest if the sphere is larger or farther away from the plate. It is shown that the negative Casimir energy for two spheres is also associated with the sizes of sphe…
For the configuration of a sphere in front of a plane we calculate the first two terms of the asymptotic expansion for small separation of the Casimir force. We consider both Dirichlet and Neumann boundary conditions.
A computation of the Casimir effect for a real scalar field in four situations: on a segment of a line, on a circle and on both standard commutative and noncommutative two-spheres is given in this paper. The main aim of this paper is to discuss the Casimir energy on the noncommutative sphere within the theory with commutative time. The comparison with the noncommutative cylinder is also done.
Using a path integral approach we rederive a recently found representation of the Casimir energy for a sphere and a cylinder in front of a plane and derive the first correction to the proximity force theorem.
The Casimir stress on two concentric spherical shell in de Sitter background for massless scalar field is calculated. The scalar field satisfies Dirichlet boundary conditions on the spheres. The metric is written in conformally flat form to make maximum use of Minkowski space calculations. Then the Casimir stress is calculated for inside and outside of the shell with different backgrounds. This model may be used to study the effect of the Casimir stress on the dynamics of the domain wall formation in inflationary models of early universe.
A simple method for calculating the Casimir energy for a sphere is developed which is based on a direct mode summation and counter integration in a complex plane of eigenfrequencies. The method uses only classical equations determining the eigenfrequencies of the quantum field under consideration. Efficiency of this approach is demonstrated by calculation of the Casimir energy for a perfectly conducting spherical shell and for a massless scalar field obeying the Dirichlet and Neumann boundary conditions on sphere. The possibility of rationalizing the removal of divergences in this problem as a…
A technique for evaluating the electromagnetic Casimir energy in situations involving spherical or circular boundaries is presented. Zeta function regularization is unambiguously used from the start and the properties of Bessel and related zeta functions are central. Nontrivial results concerning these functions are given. While part of their application agrees with previous knowledge, new results like the zeta-regularized electromagnetic Casimir energy for a circular wire are included.
A Casimir--type analysis of the effect of dividing the two--sphere by several lines of latitude is done for conformally invariant Dirichlet and Neumann scalars and for spinors. An effective action combination is shown to have minima for symmetrical arrangements of the circles. For a domain of two caps and a slice, the Dirichlet slice expands to fill the whole sphere while the Neumann one adjusts to an angular separation of 57.92 degrees. The fermion expression is written in terms of the Weber class function, $\gf$, and a connection is noted with an earlier calculation of twisted scalar effecti…
We examine the Casimir energy of 5D electro-magnetism in the recent standpoint. Z$_2$ symmetry is taken into account. After confirming the consistency with the past result, we do new things based on a new regularization. The regularization is based on the minimal area principle and the regularized configuration is the {\it sphere lattice}. We do it not in the Kaluza-Klein expanded form but in the closed form. The formalism is based on the heat-kernel approach using the position/momentum propagator. A useful expression of the Casimir energy, in terms of the P/M propagator, is obtained. Renormal…
We extend previous work on the vacuum energy of a massless scalar field in the presence of singular potentials. We consider a single sphere denoted by the so-called "delta-delta prime" interaction. Contrary to the Dirac delta potential, we find a nontrivial one-parameter family of potentials such that the regularization procedure gives an unambiguous result for the Casimir self-energy. The procedure employed is based on the zeta function regularization and the cancellation of the heat kernel coefficient a_2. The results obtained are in agreement with particular cases, such as the Dirac delta o…
The Casimir energy or stress due to modes in a D-dimensional volume subject to TM (mixed) boundary conditions on a bounding spherical surface is calculated. Both interior and exterior modes are included. Together with earlier results found for scalar modes (TE modes), this gives the Casimir effect for fluctuating ``electromagnetic'' (vector) fields inside and outside a spherical shell. Known results for three dimensions, first found by Boyer, are reproduced. Qualitatively, the results for TM modes are similar to those for scalar modes: Poles occur in the stress at positive even dimensions, and…
We consider the Casimir interaction between two spheres in $(D+1)$-dimensional Minkowski spacetime due to the vacuum fluctuations of scalar fields. We consider combinations of Dirichlet and Neumann boundary conditions. The TGTG formula of the Casimir interaction energy is derived. The computations of the T matrices of the two spheres are straightforward. To compute the two G matrices, known as translation matrices, which relate the hyper-spherical waves in two spherical coordinate frames differ by a translation, we generalize the operator approach employed in [IEEE Trans. Antennas Propag. \tex…
Casimir energy is calculated in the 5D warped system. It is compared with the flat one. The position/ momentum propagator is exploited. A new regularization, called {\it sphere lattice regularization}, is introduced. It is a direct realization of the geometrical interpretation of the renormalization group. The regularized configuration is closed-string like. We do {\it not} take the KK-expansion approach. Instead the P/M propagator is exploited, combined with the heat-kernel method. All expressions are closed-form (not KK-expanded form). Rigorous quantities are only treated (non-perturbative t…
We consider the finite temperature Casimir interaction between two Dirichlet spheres in $(D+1)$-dimensional Minkowski spacetime. The Casimir interaction free energy is derived from the zero temperature Casimir interaction energy via the Matsubara formalism. In the high temperature region, the Casimir interaction is dominated by the term with zero Matsubara frequency, and it is known as the classical term since this term is independent of the Planck constant $\hbar$. Explicit expression of the classical term is derived and it is computed exactly using appropriate similarity transforms of matric…
In this paper we compute the leading order of the Casimir energy for a free massless scalar field confined in a sphere in three spatial dimensions, with the Dirichlet boundary condition. When one tabulates all of the reported values of the Casimir energies for two closed geometries, cubical and spherical, in different space-time dimensions and with different boundary conditions, one observes a complicated pattern of signs. This pattern shows that the Casimir energy depends crucially on the details of the geometry, the number of the spatial dimensions, and the boundary conditions. The dependenc…
We consider the high temperature limit of the Casimir interaction between a Dirichlet sphere and a Dirichlet plate due to the vacuum fluctuations of a scalar field in $(D+1)$-dimensional Minkowski spacetime. The high temperature leading term of the Casimir free interaction energy is known as the classical term since it does not depend on the Planck constant $\hbar$. From the functional representation of the zero temperature Casimir interaction energy, we use Matsubara formalism to derive the classical term. It can be expressed as a weighted sum over logarithms of determinants. Using similarity…
In this paper, we derive the formula for the Casimir interaction energy between a sphere and a plate in $(D+1)$-dimensional Minkowski spacetime. It is assumed that the scalar field satisfies the Dirichlet or Neumann boundary conditions on the sphere and the plate. As in the $D=3$ case, the formula is of TGTG type. One of our main contributions is deriving the translation matrices which express the change of bases between plane waves and spherical waves for general $D$. Using orthogonality of Gegenbauer polynomials, it turns out that the final TGTG formula for the Casimir interaction energy can…
The first radiative correction to the Casimir energy of a perfectly conducting spherical shell is calculated. The calculation is performed in the framework of covariant perturbation theory with the boundary conditions implemented as constraints. The formalism is briefly reviewed and its use is explained by deriving the known results for two parallel planes. The ultraviolet divergencies are shown to have the same structure as those for a massive field in zeroth order of $α$. In the zeta-functional regularization employed by us no divergencies appear. If the radius of the sphere is large compare…
In a recent work Brevik \emph{et al.} have offered formal proofs of two results which figure prominently in calculations of the Casimir pressure on a sphere. It is shown by means of simple counterexamples that each of those proofs is necessarily incorrect.
The first heat kernel coefficients are calculated for a dispersive ball whose permittivity at high frequency differs from unity by inverse powers of the frequency. The corresponding divergent part of the vacuum energy of the electromagnetic field is given and ultraviolet divergencies are seen to be present. Also in a model where the number of atoms is fixed the pressure exhibits infinities. As a consequence, the ground-state energy for a dispersive dielectric ball cannot be interpreted easily.
The attractive Casimir force acting on a micrometer-sphere suspended in a spherical dip, close to the wall, is discussed. This setup is in principle directly accessible to experiment. The sphere and the substrate are assumed to be made of the same perfectly conducting material.
The electromagnetic vacuum energy is considered in the presence of a perfectly conducting plane and a ball with dielectric permittivity $\varepsilon$ and magnetic permeability $μ$, $μ\ne1$. The attention is focused on the Casimir repulsion in this system caused by magnetic permeability of the sphere. In the case of perfectly permeable sphere, $μ=\infty$, the vacuum energy is estimated numerically. The short and long distance asymptotes corresponding to the repulsive force and respective low temperature corrections and high temperature limits are found for a wide range of $μ$. The constraints o…
We consider the Casimir interaction between two spheres at zero and finite temperature, for both scalar fields and electromagnetic fields. Of particular interest is the asymptotic expansions of the Casimir free energy when the distance between the spheres is small. The scenario where one sphere is inside the other is discussed in detail. At zero temperature, we compute analytically the leading and the next-to-leading order terms from the functional determinant representation of the Casimir energy. As expected, the leading order term agrees with the proximity force approximation. The results fo…
We study the Casimir interaction between a sphere and a cylinder both subjected to Dirichlet, Neumann or perfectly conducting boundary conditions. Generalizing the operator approach developed by Wittman [IEEE Trans. Antennas Propag. 36, 1078 (1988)], we compute the scalar and vector translation matrices between a sphere and a cylinder, and thus write down explicitly the exact TGTG formula for the Casimir interaction energy. In the scalar case, the formula shows manifestly that the Casimir interaction force is attractive at all separations. Large separation leading term of the Casimir interacti…
The Casimir effect giving rise to an attractive force between the closely spaced two concentric spheres that confine the massless scalar field is calculated by using a direct mode summation with contour integration in the complex plane of eigenfrequencies. We devoleped a new approach appropriate for the calculation of the Casimir energy for spherical boundary conditions. The Casimir energy for a massless scalar field between the closely spaced two concentric spheres coincides with the Casimir energy of the parallel plates for a massless scalar field in the limit when the dimensionless paramete…
We calculate the low temperature corrections to the free energy for a sphere in front of a plane. First, the scalar field obeying Dirichet or Neumann boundary conditions is considered. Second, the electromagnetic field is studied, the sphere being perfectly conducting and being a dielectric ball with both, constant permittivity and permittivity of the plasma model.
The Casimir energy for a massless scalar field between the closely spaced two concentric D-dimensional (for D>3) spheres is calculated by using the mode summation with contour integration in the complex plane of eigenfrequencies and the generalized Abel-Plana formula for evenly spaced eigenfrequency at large argument. The sign of the Casimir energy between closely spaced two concentric D-dimensional spheres for a massless scalar field satisfying the Dirichlet boundary conditions is strictly negative. The Casimir energy between D-1 dimensional surfaces close to each other is regarded as interes…
We consider the finite temperature Casimir effect between two concentric spheres due to the vacuum fluctuations of the electromagnetic field in the $(D+1)$-dimensional Minkowski spacetime. Different combinations of perfectly conducting and infinitely permeable boundary conditions are imposed on the spheres. The asymptotic expansions of the Casimir free energies when the dimensionless parameter $\vep$, the ratio of the distance between the spheres to the radius of the smaller sphere, is small are derived in both the high temperature region and the low temperature region. It is shown that the le…
In this paper aiming to obtain the Casimir energy between a sinusoidally corrugated sphere and a plate, we first present a derivation of the sphere-plate Casimir force obtained by applying proximity force approximation (PFA). One may investigate this problem by using the scattering matrix method. Despite that this method achieves the exact result, but difficulties of applying this formalism make the procedure complicated. We use the PFA to achieve the purpose in a truly explicit manner. Considering the sphere to have different kinds of sinusoidally corrugations, we obtain the effect of deforma…
We present a simple formalism for the evaluation of the Casimir energy for two spheres and a sphere and a plane, in case of a scalar fluctuating field, valid at any separations. We compare the exact results with various approximation schemes and establish when such schemes become useful. The formalism can be easily extended to any number of spheres and/or planes in three or arbitrary dimensions, with a variety of boundary conditions or non-overlapping potentials/non-ideal reflectors.
Casimir interactions (due to the massless scalar field fluctuations) of two surfaces which are close to each other are studied. After a brief general presentation, explicit calculations for co-axial cylinders, co-centric spheres and co-axial cones are performed.
In this paper, dedicated to the career of Tom Erber, we consider the Casimir interaction between weakly coupled bodies at nonzero temperature. For the case of semitransparent bodies, that is, ones described by delta-function potentials, we first examine the interaction between an infinite plane and an arbitrary curved surface. In weak coupling, such an interaction energy coincides with the exact form of the proximity force approximation obtained by summing the interaction between opposite surface elements at arbitrary temperature. This result generalizes a theorem proved recently by Decca et a…
We discuss non-equilibrium extensions of the Casimir force (due to electromagnetic fluctuations), where the objects as well as the environment are held at different temperatures. While the formalism we develop is quite general, we focus on a sphere in front of a plate, as well as two spheres, when the radius is small compared to separation and thermal wavelengths. In this limit the forces can be expressed analytically in terms of the lowest order multipoles, and corroborated with results obtained by diluting parallel plates of vanishing thickness. Non-equilibrium forces are generally stronger …
We investigate the nontrivial interplay between geometry and temperature in the Casimir effect for the sphere-plate and cylinder-plate configurations. At low temperature, the thermal contribution to the Casimir force is dominated by this interplay, implying that standard approximation techniques such as the PFA are inapplicable even in the limit of small surface separation. Thermal fluctuations on scales of the thermal wavelength lead to a delocalization of the thermal force density at low temperatures. As a consequence, the temperature dependence strongly differs from naive expectations. Most…
In our paper [Ann. Phys. (NY) 395, 326 (2018)] we calculate the Casimir stress on a sphere immersed in a homogeneous background, assuming dispersionless dielectrics. Our results appear to challenge the conventional picture of Casimir forces. The paper [arXiv:1909.05721] criticises our approach without offering an alternative. In particular, the paper [arXiv:1909.05721] claims that we have made an unjustified mathematical step. This brief comment clarifies the matter.
Casimir energy changes are investigated for geometries obtained by small but arbitrary deformations of a given geometry for which the vacuum energy is already known for the massless scalar field. As a specific case, deformation of a spherical shell is studied. From the deformation of the sphere we show that the Casimir energy is a decreasing function of the surface to volume ratio. The decreasing rate is higher for less smooth deformations.
A multiple scattering formulation is used to calculate the force, arising from fluctuating scalar fields, between distinct bodies described by $δ$-function potentials, so-called semitransparent bodies. (In the limit of strong coupling, a semitransparent boundary becomes a Dirichlet one.) We obtain expressions for the Casimir energies between disjoint parallel semitransparent cylinders and between disjoint semitransparent spheres. In the limit of weak coupling, we derive power series expansions for the energy, which can be exactly summed, so that explicit, very simple, closed-form expressions a…
In this paper we sum over the spherical modes appearing in the expression for the Casimir energy of a conducting sphere and of a dielectric ball (assuming the same speed of light inside and outside), before doing the frequency integration. We derive closed integral expressions that allow the calculations to be done to all orders, without the use of regularization procedures. The technique of mode summation using a contour integral is critically examined.
We study the vacuum interaction of a scalar field and two concentric spheres defined by a singular potential on their surfaces. The potential is a linear combination of the Dirac-$δ$ and its derivative. The presence of the delta prime term in the potential causes that it behaves differently when it is seen from the inside or from the outside of the sphere. We study different cases for positive and negative values of the delta prime coupling, keeping positive the coupling of the delta. As a consequence, we find regions in the space of couplings, where the energy is positive, negative or zero. M…
It is familiar that the Casimir self-energy of a homogeneous dielectric ball is divergent, although a finite self-energy can be extracted through second order in the deviation of the permittivity from the vacuum value. The exception occurs when the speed of light inside the spherical boundary is the same as that outside, so the self-energy of a perfectly conducting spherical shell is finite, as is the energy of a dielectric-diamagnetic sphere with $\varepsilonμ=1$, a so-called isorefractive or diaphanous ball. Here we re-examine that example, and attempt to extend it to an electromagnetic $δ$-…
The possibility of repulsive Casimir forces between small metal spheres and a dielectric half-space is discussed. We treat a model in which the spheres have a dielectric function given by the Drude model, and the radius of the sphere is small compared to the corresponding plasma wavelength. The half-space is also described by the same model, but with a different plasma frequency. We find that in the retarded limit, the force is quasi-oscillatory. This leads to the prediction of stable equilibrium points at which the sphere could levitate in the Earth's gravitational field. This seems to lead t…
Recently, the topic of Casimir repulsion has received a great deal of attention, largely because of the possibility of technological application. The general subject has a long history, going back to the self-repulsion of a conducting spherical shell and the repulsion between a perfect electric conductor and a perfect magnetic conductor. Recently it has been observed that repulsion can be achieved between ordinary conducting bodies, provided sufficient anisotropy is present. For example, an anisotropic polarizable atom can be repelled near an aperture in a conducting plate. Here we provide new…
We investigate the Casimir effect, due to the confinement of a scalar field in a $D$-dimensional sphere, with Lorentz symmetry breaking. The Lorentz-violating part of the theory is described by the term $λ(u \cdot \partial ϕ) ^{2}$, where the parameter $λ$ and the background vector $u^μ$ codify the breakdown of Lorentz symmetry. We compute, as a function of $D$, the Casimir stress by using Green's function techniques for two specific choices of the vector $u ^μ$. In the timelike case, $u ^μ = (1,0,...,0)$, the Casimir stress can be factorized as the product of the Lorentz invariant result time…
Negative entropy has been known in Casimir systems for some time. For example, it can occur between parallel metallic plates modeled by a realistic Drude permittivity. Less well known is that negative entropy can occur purely geometrically, say between a perfectly conducting sphere and a conducting plate. The latter effect is most pronounced in the dipole approximation, which is reliable when the size of the sphere is small compared to the separation between the sphere and the plate. Therefore, here we examine cases where negative entropy can occur between two electrically and magnetically pol…
We derive analytically the asymptotic behavior of the Casimir interaction between a sphere and a plate when the distance between them, $d$, is much smaller than the radius of the sphere, $R$. The leading order and next-to-leading order terms are derived from the exact formula for the Casimir interaction energy. They are found to depend nontrivially on the dielectric functions of the objects. As expected, the leading order term coincides with that derived using the proximity force approximation. The result on the next-to-leading order term complements that found by Bimonte, Emig and Kardar [App…
In this paper we compute the Casimir energy and force for massless scalar fields endowed with hybrid boundary conditions, in the setting of the bounded generalized cone. By using spectral zeta function regularization methods, we obtain explicit expressions for the Casimir energy and force in arbitrary dimensions in terms of the zeta function defined on the piston. Our general formulas are, subsequently, specialized to the case in which the piston is modelled by a $d$-dimensional sphere. In this particular situation, explicit results are given for $d=2,3,4,5$.
We study the vacuum polarization (Casimir) energy in renormalizable, continuum quantum field theory in the presence of a background field, designed to impose Dirichlet boundary conditions on the fluctuating quantum field. In two and three spatial dimensions the Casimir energy diverges as a background field becomes concentrated on the surface on which the Dirichlet boundary condition would eventually hold. This divergence does not affect the force between rigid bodies, but it does invalidate calculations of Casimir stresses based on idealized boundary conditions.
We give an exact series expansion of the Casimir force between plane and spherical metallic surfaces in the non trivial situation where the sphere radius $R$, the plane-sphere distance $L$ and the plasma wavelength $λ_¶$ have arbitrary relative values. We then present numerical evaluation of this expansion for not too small values of $L/R$. For metallic nanospheres where $R, L$ and $λ_¶$ have comparable values, we interpret our results in terms of a correlation between the effects of geometry beyond the proximity force approximation (PFA) and of finite reflectivity due to material properties. …
Multiple scattering formulations have been employed for more than 30 years as a method of studying the quantum vacuum or Casimir interactions between distinct bodies. Here we review the method in the simple context of $δ$-function potentials, so-called semitransparent bodies. (In the limit of strong coupling, a semitransparent boundary becomes a Dirichlet one.) After applying the method to rederive the Casimir force between two semitransparent plates and the Casimir self-stress on a semitransparent sphere, we obtain expressions for the Casimir energies between disjoint parallel semitransparent…
We develop a mathematically precise framework for the Casimir effect. Our working hypothesis, verified in the case of parallel plates, is that only the regularization-independent Ramanujan sum of a given asymptotic series contributes to the Casimir pressure. As an illustration, we treat two cases: parallel plates, identifying a previous cutoff free version (by G. Scharf and W. W.) as a special case, and the sphere.We finally discuss the open problem of the Casimir force for the cube. We propose an Ansatz for the exterior force and argue why it may provide the exact solution, as well as an expl…
We investigate the vacuum expectation value of the energy-momentum tensor associated with a massive fermionic field obeying the MIT bag boundary condition on a spherical shell in the global monopole spacetime. The asymptotic behavior of the vacuum densities is investigated near the sphere center and surface, and at large distances from the sphere. In the limit of strong gravitational field corresponding to small values of the parameter describing the solid angle deficit in global monopole geometry, the sphere-induced expectation values are exponentially suppressed.
The vacuum expectation values for the energy-momentum tensor of a massive scalar field with general curvature coupling and obeying the Robin boundary condition on spherically symmetric boundaries in D-dimensional space are investigated. The expressions are derived for the regularized vacuum energy density, radial and azimuthal stress components (i) inside and outside a single spherical surface and (ii) in the intermediate region between two concentric spheres. Regularization procedure is carried out by making use of the generalized Abel-Plana formula for the series over zeros of cylinder funct…
The interaction between a polarizable particle and a reflecting wall is examined. A macroscopic approach is adopted in which the averaged force is computed from the Maxwell stress tensor. The particular case of a perfectly reflecting wall and a sphere with a dielectric function given by the Drude model is examined in detail. It is found that the force can be expressed as the sum of a monotonically decaying function of position and of an oscillatory piece. At large separations, the oscillatory piece is the dominant contribution, and is much larger than the Casimir-Polder interaction that arises…
Casimir energy is calculated for the 5D electromagnetism and 5D scalar theory in the {\it warped} geometry. It is compared with the flat case(arXiv:0801.3064). A new regularization, called {\it sphere lattice regularization}, is taken. In the integration over the 5D space, we introduce two boundary curves (IR-surface and UV-surface) based on the {\it minimal area principle}. It is a {\it direct} realization of the geometrical approach to the {\it renormalization group}. The regularized configuration is {\it closed-string like}. We do {\it not} take the KK-expansion approach. Instead, the posit…
We introduce a useful approach to find asymptotically explicit expressions for the Casimir free energy at large temperature. The resulting expressions contain the classical terms as well as the few first terms of the corresponding heat-kernel expansion, as expected. This technique works well for many familiar configurations in Euclidean as well as non-Euclidean spaces. By utilizing this approach, we provide some new numerically considerable results for the Casimir pressure in some rectangular ideal-metal cavities. For instance, we show that at sufficiently large temperature, the Casimir pressu…
In this paper we continue our program of computing Casimir self-entropies of idealized electrical bodies. Here we consider an electromagnetic $δ$-function sphere ("semitransparent sphere") whose electric susceptibility has a transverse polarization with arbitrary strength. Dispersion is incorporated by a plasma-like model. In the strong coupling limit, a perfectly conducting spherical shell is realized. We compute the entropy for both low and high temperatures. The TE self-entropy is negative as expected, but the TM self-entropy requires ultraviolet and infrared subtractions, and, surprisingly…
