In this paper we study the system of a scalar quantum field confined between two plane, isotropic, and homogeneous parallel plates at thermal equilibrium. We represent the plates by the most general lossless and frequency-independent boundary conditions that satisfy the conditions of isotropy and homogeneity and are compatible with the unitarity of the quantum field theory. Under these conditions we compute the thermal correction to the quantum vacuum energy as a function of the temperature and the parameters encoding the boundary condition. The latter enables us to obtain similar results for the pressure between plates and the quantum thermal correction to the entropy. We find out that our system is thermodynamically stable for any boundary conditions, and we identify a critical temperature below which certain boundary conditions yield attractive, repulsive, and null Casimir forces.
The geometry dependence of Casimir forces is significantly more pronounced in the presence of thermal fluctuations due to a generic geometry-temperature interplay. We show that the thermal force for standard sphere-plate or cylinder-plate geometries develops a non-monotonic behavior already in the simple case of a fluctuating Dirichlet scalar. In particular, the attractive thermal force can increase for increasing distances below a critical temperature. This anomalous behavior is triggered by a reweighting of relevant fluctuations on the scale of the thermal wavelength. The essence of the phen…
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…
We recently proposed a new approach to the Casimir effect based on classical ray optics (the "optical approximation"). In this paper we show how to use it to calculate the local observables of the field theory. In particular we study the energy-momentum tensor and the Casimir pressure. We work three examples in detail: parallel plates, the Casimir pendulum and a sphere opposite a plate. We also show how to calculate thermal corrections, proving that the high temperature `classical limit' is indeed valid for any smooth geometry.
We discuss Casimir phenomena which are dominated by long-range fluctuations. A prime example is given by "geothermal" Casimir phenomena where thermal fluctuations in open Casimir geometries can induce significantly enhanced thermal corrections. We illustrate the underlying mechanism with the aid of the inclined-plates configuration, giving rise to enhanced power-law temperature dependences compared to the parallel-plates case. In limiting cases, we find numerical evidence even for fractional power laws induced by long-range fluctuations. We demonstrate that thermal energy densities for open ge…
A new solution for an analytic spectrum of particle creation by an accelerating mirror (dynamical Casimir effect) is given. It is the first model to simultaneously radiate thermally and emit a finite number of particles.
We review recent results obtained in the physics of the thermal Casimir force acting between two dielectrics, dielectric and metal, and between metal and semiconductor. The detailed derivation for the low-temperature behavior of the Casimir free energy, pressure and entropy in the configuration of two real dielectric plates is presented. For dielectrics with finite static dielectric permittivity it is shown that the Nernst heat theorem is satisfied. Hence, the Lifshitz theory of the van der Waals and Casimir forces is demonstrated to be consistent with thermodynamics. The nonzero dc conductivi…
We examine recent prescriptions for estimating the thermal Casimir force between two metallic plates from the point of view of their consistency with the Bohr-van Leeuwen theorem of classical statistical physics. We find that prescriptions including the effect of ohmic dissipation satisfy the theorem, while prescriptions that neglect ohmic dissipation do not.
We develop an exact method for computing Casimir forces and the power of radiative heat transfer between two arbitrary nanostructured surfaces out of thermal equilibrium. The method is based on a generalization of the scattering approach recently used in investigations on the Casimir effect. Analogously to the equilibrium case, we find that also out of thermal equilibrium the shape and composition of the surfaces enter only through their scattering matrices. The expressions derived provide exact results in terms of the scattering matrices of the intervening surfaces.
We give a short review on the static and dynamical Casimir effects, recalling their historical prediction, as well as their more recent experimental verification. We emphasise on the central role played by so-called {\it dynamical boundary conditions} (for which the boundary condition depends on a second time derivative of the field) in the experimental verification of the dynamical Casimir effect by Wilson et al. We then go on to review our previous work on the static Casimir effect with dynamical boundary conditions, providing an overview on how to compute the so-called local Casimir energy,…
The Casimir and Casimir-Polder interactions are investigated in a stack of equally spaced graphene layers. The optical response of the individual graphene is taken into account using gauge invariant components of the polarization tensor extended to the whole complex frequency plane. The planar symmetry for the electromagnetic boundary conditions is further used to obtain explicit forms for the Casimir energy stored in the stack and the Casimir-Polder energy between an atom above the stack. Our calculations show that these fluctuation induced interactions experience strong thermal effects due t…
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.
One of the most important and still unresolved problems in the field of dispersion forces, is that of determining the influence of temperature on the Casimir force between two metallic plates. While alternative theoretical approaches lead to contradictory predictions for the magnitude of the effect, no experiment has yet detected the thermal correction to the Casimir force. In this paper we show that a superconducting cavity provides a new opportunity to investigate the problem of the thermal dependence of the Casimir force in real materials, by looking at the change of the Casimir force deter…
The effect of edges and apertures on the Casimir energy of an arrangement of plates and boundaries can be calculated in terms of an effective nonlocal lower-dimensional field theory that lives on the boundary. This formalism has been developed in a number of previous papers and applied to specific examples with Dirichlet boundary conditions. Here we generalize the formalism to arbitrary boundary conditions. As a specific example, the geometry of a flat plate and a half-plate placed parallel to it is considered for a number of different boundary conditions and the area-dependent and edge depend…
Following the derivation of the Green function for the massless scalar field satisfying the Dirichlet boundary condition on the Plane (x > 0, y = 0), we calculate the Casimir energy.
When one studies the Casimir effect, the periodic (anti-periodic) boundary condition is usually taken to mimic a periodic (anti-periodic) structure for a scalar field living in a flat space with a non-Euclidean topology. However, there could be an arbitrary phase difference between the value of the scalar field on one endpoint of the unit structure and that on the other endpoint, such as the structure of nanotubes. Then, in this paper, a periodic condition on the ends of the system with an additional phase factor, which is called the ``quasi-periodic" condition, is imposed to investigate the c…
We study the interplay of thermal and diffractive effects in Casimir energies. We consider plates with edges, oriented either parallel or perpendicular to each other, as well as a single plate with a slit. We compute the Casimir energy at finite temperature using a formalism in which the diffractive effects are encoded in a lower dimensional non-local field theory that lives in the gap between the plates. The formalism allows for a clean separation between direct or geometric effects and diffractive effects, and makes an analytic derivation of the temperature dependence of the free energy poss…
A diffraction problem for a flat Chern-Simons layer at plane boundary of a dielectric half space is solved. The Casimir energy of two dielectric half spaces with Chern-Simons layers at plane-parallel boundaries separated by a vacuum slit is derived. Crossing from the repulsive to the attractive Casimir force is analyzed for two Au and two Si half spaces with boundary Chern-Simons layers. Boundary quantum Hall layers in external magnetic field lead to Casimir repulsion at nanoscales. We discuss features that make systems with boundary quantum Hall layers unique for force measurements and search…
The Casimir effect giving rise to an attractive or repulsive force between the configuration boundaries that confine the massless scalar field is reexamined for one to three-dimensional pistons in this paper. Especially, we consider Casimir pistons with hybrid boundary conditions, where the boundary condition on the piston is Neumann and those on other surfaces are Dirichlet. We show that the Casimir force on the piston is always repulsive, in contrast with the same problem where the boundary conditions are Dirichlet on all surfaces.
In this paper, we consider the fermionic Casimir effect under a new type of space-time topology using the concept of quotient topology. The relation between the new topology and that in Ref. \cite{Feng,Zhai3} is something like that between a Möbius strip and a cylindric. We obtain the exact results of the Casimir energy and force for the massless and massive Dirac fields in the ($D+1$)-dimensional space-time. For both massless and massive cases, there is a $Z_2$ symmetry for the Casimir energy. To see the effect of the mass, we compare the result with that of the massless one and we found that…
Casimir effect in the planar setting is described using the boundary state formalism, for general partially reflecting boundaries. It is expressed in terms of the low-energy degrees of freedom, which provides a large distance expansion valid for general interacting field theories provided there is a non-vanishing mass gap. The expansion is written in terms of the scattering amplitudes, and needs no ultraviolet renormalization. We also discuss the case when the quantum field has a nontrivial vacuum configuration.
The Casimir force between two perfectly reflecting parallel plates is considered. In a recent paper we presented generalised physical boundary conditions describing perfectly reflecting parallel plates. These boundary conditions are applicable to a field possessing any spin, and include the well-known spin-1/2 and spin-1 boundary conditions as special cases. Here we use these general boundary conditions to show that the allowed values of energy-momentum turn out to be the same for any massless fermionic field and the same for any massless bosonic field. As a result one expects to obtain only t…
We study a coupled system that describes the interacting dynamics between a bulk field, confined to a finite region with timelike boundary, and a boundary observable. In our system the dynamics of the boundary observable prescribes dynamical boundary conditions for the bulk field. We cast our classical system in the form of an abstract linear Klein-Gordon equation, in an enlarged Hilbert space for the bulk field and the boundary observable. This makes it possible to apply to our coupled system the general methods of quantization. In particular, we implement the Fock quantization in full detail…
The stress tensor of a massless scalar field satisfying Robin boundary conditions on two one-dimensional wall in two-dimensional Schwarzschild background is calculated. We show that vacuum expectation value of stress tensor can be obtained explicitly by Casimir effect, trace anomaly and Hawking radiation.
In this paper, the finite temperature Casimir force acting on a two-dimensional Casimir piston due to electromagnetic field is computed. It was found that if mixed boundary conditions are assumed on the piston and its opposite wall, then the Casimir force always tends to restore the piston towards the equilibrium position, regardless of the boundary conditions assumed on the walls transverse to the piston. In contrary, if pure boundary conditions are assumed on the piston and the opposite wall, then the Casimir force always tend to pull the piston towards the closer wall and away from the equi…
We evaluate the Casimir energy and force for a massive scalar field with general curvature coupling parameter, subject to Robin boundary conditions on two codimension-one parallel plates, located on a $(D+1)$-dimensional background spacetime with an arbitrary internal space. The most general case of different Robin coefficients on the two separate plates is considered. With independence of the geometry of the internal space, the Casimir forces are seen to be attractive for special cases of Dirichlet or Neumann boundary conditions on both plates and repulsive for Dirichlet boundary conditions o…
The infrared behavior of quantum field theories confined in bounded domains is strongly dependent on the shape and structure of space boundaries. The most significant physical effect arises in the behaviour of the vacuum energy. The Casimir energy can be attractive or repulsive depending on the nature of the boundary. We calculate the vacuum energy for a massless scalar field confined between two homogeneous parallel plates with the most general type of boundary conditions depending on four parameters. The analysis provides a powerful method to identify which boundary conditions generate attra…
Thermal Casimir Effect in A Schwarzschild-like Wormhole Spacetime
Sepuluh Nopember Institute of Technology（十十一月理工大学）；Universitas Islam Negeri Maulana Malik Ibrahim Malang（穆拉纳马利克伊布拉欣伊斯兰国立大学）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；National Research and Innovation Agency (BRIN)（国家研究与创新署）；Universitas Indonesia（印度尼西亚大学）
The thermal correction to the energy of Casimir-Polder interaction of atoms with a suspended graphene membrane described by the Dirac model is investigated. We show that a major impact on the thermal correction is made by the size of the gap in the energy spectrum of graphene quasiparticles. Specifically, if the temperature is much smaller than the gap parameter (alternatively, larger or of the order of the gap parameter), the thermal correction is shown to be relatively small (alternatively, large). We have calculated the free energy of the thermal Casimir-Polder interaction of atoms of He, N…
This work focuses on the analysis of the Casimir effect for pistons subject to transmittal boundary conditions. In particular we consider, as piston configuration, a direct product manifold of the type $I\times N$ where $I$ is a closed interval of the real line and $N$ is a smooth compact Riemannian manifold. By utilizing the spectral zeta function regularization technique, we compute the Casimir energy of the system and the Casimir force acting on the piston. Explicit results for the force are provided when the manifold $N$ is a $d$-dimensional ball.
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.
We compute the first radiative correction to the Casimir energy in the $(d+1)$-dimensional $λ|ϕ|^{4}$ model submitted to quasi-periodic boundary conditions in one spatial direction. Our results agree with the ones found in the literature for periodic and anti-periodic boundary conditions, special cases of the quasi-periodic boundary conditions.
In this work we study the Casimir effect for massless scalar fields propagating in a piston geometry of the type $I\times N$ where $I$ is an interval of the real line and $N$ is a smooth compact Riemannian manifold. Our analysis represents a generalization of previous results obtained for pistons configurations as we consider all possible boundary conditions that are allowed to be imposed on the scalar fields. We employ the spectral zeta function formalism in the framework of scattering theory in order to obtain an expression for the Casimir energy and the corresponding Casimir force on the pi…
We demonstrate that the static ground state atom, which interacts with a conformally coupled massless scalar field in the de Sitter invariant vacuum, can obtain a position-dependent energy-level shift and this shift could cause a Casimir-Polder-like force on it. Interestingly no such force arises on the inertial atom bathed in a thermal radiation in the Minkowski universe. Thus, although the energy-level shifts of the static atom for these two cases are structurally the same, whether the energy-level shift causes the Casimir-Polder-like force, in principle, could be as an indicator to distingu…
In this paper, a massless scalar field coupled to gravity is considered. Then the Casimir effect at finite temperature is calculated. Such development is carried out in the Thermo Field Dynamics formalism. This approach presents a topological structure that allows for investigating the effects of temperature and the size effect in a similar way. These effects are calculated considering Gödel-type solutions as a gravitational background. The Stefan-Boltzmann law and its consistency are analyzed for both causal and non-causal Gödel-type regions. In this space-time and for any region, the Casimir…
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 $d=2,3,4,5$.
In this paper, we investigate Casimir effect, Weyl anomaly and displacement operator for boundary conformal field theory in general dimensions. We find universal relations between them. In particular, they are all determined by the central charge of boundary conformal field theory. We verify these relations by studying free BCFTs and holographic BCFTs. As a byproduct, we obtain the holographic two point function of stress tensor when the bulk boundary is perpendicular to the AdS boundary.
This paper studies the finite temperature Casimir force acting on a rectangular piston associated with a massless fractional Klein-Gordon field at finite temperature. Dirichlet boundary conditions are imposed on the walls of a $d$-dimensional rectangular cavity, and a fractional Neumann condition is imposed on the piston that moves freely inside the cavity. The fractional Neumann condition gives an interpolation between the Dirichlet and Neumann conditions, where the Casimir force is known to be always attractive and always repulsive respectively. For the fractional Neumann boundary condition,…
In this article we consider the Casimir energy and force of massless Majorana fermions and vector bosons between parallel plates. The vector bosons satisfy perfect electric conductor boundary conditions while the Majorana fermions satisfy bag and chiral boundary conditions. We consider various piston configurations containing one vector boson and one fermion. We present a new regularization mechanism the piston offers. In our case regularization occurs explicitly at the Casimir energy density and not at the Casimir force level, as in usual pistons. We make use of boundary broken supersymmetry …
In this article, we study the finite temperature Casimir effect for scalar field with Robin boundary conditions on two parallel plates in a background spacetime that has a compact internal manifold with arbitrary geometry. The finite temperature Casimir force acting on the plates is derived using the piston approach, with which we show that the divergences of the Casimir forces from the region between the plates and the region outside the plates cancel each other. The nature and asymptotic behaviors of the Casimir force at different limits such as small and large plate separation, low and high…
It is well known that, beginning in 2000, the behavior of the thermal correction to the Casimir force between real metals has been hotly debated. As was shown by several research groups, the Lifshitz theory, which provides the theoretical foundation for the calculation of both the van der Waals and Casimir forces, leads to different results depending on the model of metal conductivity used. To resolve these controversies, the theoretical considerations based on the principles of thermodynamics and new experimental tests were invoked. We analyze the present status of the problem (in particular,…
The Casimir effect for parallel plates satisfying the Dirichlet boundary condition in the context of effective QED coming from a six-dimensional Nielsen-Olesen vortex solution of the Abelian Higgs model with fermions coupled to gravity is studied at finite temperature. We find that the sign of the Casimir energy remains negative under the thermal influence. It is also shown that the Casimir force between plates will be weaker in the higher-temperature surroundings while keeps attractive. This Casimir effect involving the thermal influence is still inconsistent with the known experiments. We fi…
Casimir forces are conventionally computed by analyzing the effects of boundary conditions on a fluctuating quantum field. Although this analysis provides a clean and calculationally tractable idealization, it does not always accurately capture the characteristics of real materials, which cannot constrain the modes of the fluctuating field at all energies. We study the vacuum polarization energy of renormalizable, continuum quantum field theory in the presence of a background field, designed to impose a Dirichlet boundary condition in a particular limit. We show that in two and three space dim…
We calculate the Casimir energy for scalar fields in interaction with finite-width mirrors, described by nonlocal interaction terms. These terms, which include quantum effects due to the matter fields inside the mirrors, are approximated by means of a local expansion procedure. As a result of this expansion, an effective theory for the vacuum field emerges, which can be written in terms of generalized $δ$-potentials. We compute explicitly the Casimir energy for these potentials and show that, for some particular cases, it is possible to reinterpret them as imposing imperfect Dirichlet boundary…
In the present paper we investigate thermal fluctuation corrections to the vacuum energy at zero temperature of a conformally coupled massless scalar field whose modes propagate in the Einstein universe with a spherical boundary, characterized by both Dirichlet and Neumann boundary conditions. Thus, we generalize the results found in literature in this scenario, which has considered only the vacuum energy at zero temperature. To do this, we use the generalized zeta function method plus Abel-Plana formula and calculate the renormalized Casimir free energy as well as other thermodynamics quantit…
We consider the vacuum energy for a scalar field subject to a frequency dependent boundary condition. The effect of a frequency cut-off is described in terms of an {\it incomplete} $ζ$-function. The use of the Debye asymptotic expansion for Bessel functions allows to determine the dominant (volume, area, >...) terms in the Casimir energy. The possible interest of this kind of models for dielectric media (and its application to sonoluminescence) is also discussed.
We have studied in a previous work the quantization of a mixed bulk-boundary system describing the coupled dynamics between a bulk quantum field confined to a spacetime with finite space slice and with timelike boundary, and a boundary observable defined on the boundary. Our bulk system is a quantum field in a spacetime with timelike boundary and a dynamical boundary condition-the boundary observable's equation of motion. Owing to important physical motivations, in such previous work we have computed the renormalized local state polarization and local Casimir energy for both the bulk quantum f…
We consider a massless scalar field in 1+1 dimensions that satisfies a Robin boundary condition at a non-relativistic moving boundary. Using the perturbative approach introduced by Ford and Vilenkin, we compute the total force on the moving boundary. In contrast to what happens for the Dirichlet and Neumann boundary conditions, in addition to a dissipative part, the force acquires also a dispersive one. Further, we also show that with an appropriate choice for the mechanical frequency of the moving boundary it is possible to turn off the vacuum dissipation almost completely.
For scalar and electromagnetic fields we evaluate the vacuum expectation value of the energy-momentum tensor induced by a curved boundary in the Robertson--Walker spacetime with negative spatial curvature. In order to generate the vacuum densities we use the conformal relation between the Robertson-Walker and Rindler spacetimes and the corresponding results for a plate moving by uniform proper acceleration through the Fulling--Rindler vacuum. For the general case of the scale factor the vacuum energy-momentum tensor is presented as the sum of the boundary free and boundary induced parts.
In this paper, we investigate the thermal effect on the Casimir energy associated with a massive scalar quantum field confined between two large parallel plates in a CPT-even, aether-like Lorentz-breaking scalar field theory. In order to do that we consider a nonzero chemical potential for the scalar field assumed to be in thermal equilibrium at some finite temperature. The calculations of the energies are developed by using the Abel-Plana summation formula, and the corresponding results are analyzed in several asymptotic regimes of the parameters of the system, like mass, separations between …
The local Casimir energy is investigated for a wedge with and without a circular outer boundary due to the confinement of a massless scalar field with general curvature coupling parameter and satisfying the Dirichlet boundary conditions. Regularization procedure is carried out making use of a variant of the generalized Abel-Plana formula, previously established by one of the authors. The surface divergences in the vacuum expectation values of the energy density near the boundaries are considered. The corresponding results can be applied to the cosmic strings.
We obtain new expressions for the Casimir energy between plates that are mimicked by the most general possible boundary conditions allowed by the principles of quantum field theory. This result enables to provide the quantum vacuum energy for scalar fields propagating under the influence of a one-dimensional crystal represented by a periodic potential formed by an infinite array of identical potentials with compact support.
In this work we analyze the Casimir energy and force for a scalar field endowed with general self-adjoint boundary conditions propagating in a higher dimensional piston configuration. The piston is constructed as a direct product $I\times N$, with $I=[0,L]\subset\mathbb{R}$ and $N$ a smooth, compact Riemannian manifold with or without boundary. The study of the Casimir energy and force for this configuration is performed by employing the spectral zeta function regularization technique. The obtained analytic results depend explicitly on the spectral zeta function associated with the manifold $N…
