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. Using this quantization we study the Casimir effect in our coupled system. Specifically, we compute the renormalized local state polarization and the local Casimir energy, which we can define for both the bulk field and the boundary observable of our system. Numerical examples in which the integrated Casimir energy is positive or negative are presented.
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…
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…
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 quantum field theories with boundary on a codimension one hyperplane. Using 1+1 dimensional examples, we clarify the relation between three parameters characterising one-point functions, finite size corrections to the ground state energy and the singularity structure of scattering amplitudes, respectively. We then develop the formalism of boundary states in general D+1 spacetime dimensions and relate the cluster expansion of the boundary state to the correlation functions using reduction formulae. This allows us to derive the cluster expansion in terms of the boundary scattering am…
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…
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.
The ground state energy of a boundary quantum field theory is derived in planar geometry in D+1 dimensional spacetime. It provides a universal expression for the Casimir energy which exhibits its dependence on the boundary conditions via the reflection amplitudes of the low energy particle excitations. We demonstrate the easy and straightforward applicability of the general expression by analyzing the free scalar field with Robin boundary condition and by rederiving the most important results available in the literature for this geometry.
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,…
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.
In this work we analyze the Casimir energy and force for a {\it thick} piston configuration. This study is performed by utilizing the spectral zeta function regularization method. The results we obtain for the Casimir energy and force depend explicitly on the parameters that describe the general self-adjoint boundary conditions imposed. Numerical results for the Casimir force are provided for specific types of boundary conditions and are also compared to the corresponding force on an infinitely thin piston.
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.
We study the self adjoint extensions of a class of non maximal multiplication operators with boundary conditions. We show that these extensions correspond to singular rank one perturbations (in the sense of \cite{AK}) of the Laplace operator, namely the formal Laplacian with a singular delta potential, on the half space. This construction is the appropriate setting to describe the Casimir effect related to a massless scalar field in the flat space time with an infinite conducting plate and in the presence of a point like "impurity". We use the relative zeta determinant (as defined in \cite{Mul…
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…
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…
We study massless Dirac fields in a portion of $AdS_3$, where one of the boundaries coincides with the "boundary at infinity" of the Anti-de-Sitter space. We evaluate the vacuum energy arising when the local boundary conditions dictated by boundary chirality are imposed, in different combinations, at both one-branes.
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.
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 derive a master equation for a mirror interacting with the vacuum field via radiation pressure. The dynamical Casimir effect leads to decoherence of a 'Schroedinger cat' state in a time scale that depends on the degree of 'macroscopicity' of the state components, and which may be much shorter than the relaxation time scale. Coherent states are selected by the interaction as pointer states.
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 consider a real massless scalar field in 1+1 dimensions satisfying time-dependent Robin boundary condition at a static wall. This condition can simulate moving reflecting mirrors whose motions are determined by the time-dependence of the Robin parameter. We show that particles can be created from vacuum, characterizing in this way a dynamical Casimir effect.
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…
After reviewing some essential features of the Casimir effect and, specifically, of its regularization by zeta function and Hadamard methods, we consider the dynamical Casimir effect (or Fulling-Davis theory), where related regularization problems appear, with a view to an experimental verification of this theory. We finish with a discussion of the possible contribution of vacuum fluctuations to dark energy, in a Casimir like fashion, that might involve the dynamical version.
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…
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 propose a generalization of the superconducting circuit simulation of the dynamical Casimir effect where we consider relativistically moving boundary conditions following different trajectories. We study the feasibility of the setup used in the past to simulate the dynamical Casimir effect to reproduce richer relativistic trajectories differing from purely sinusoidal ones. We show how different relativistic oscillatory trajectories of the boundaries of the same period and similar shape produce a rather different spectrum of particles characteristic of their respective motions.
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.
We study a quantum model of dynamical Casimir effect in an optical cavity enclosed by a freely moving mirror attached to a harmonic spring. The quantum fluctuations of the friction force exerted by the dynamical Casimir emission onto the moving mirror are investigated, as well as their consequences on the quantum state of the mirror oscillation. Observable signatures of the interplay of the nonlinear nature of the effective mirror-cavity coupling and of the discreteness of the emitted photons are pointed out, in particular as a fast diffusion of the mirror oscillation phase. These results are …
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 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 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.
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 $d=2,3,4,5$.
Motivated by experiments in which moving boundaries are simulated by time-dependent properties of static systems, we discuss the model of a massless scalar field submitted to a time-dependent Robin boundary condition (BC) at a static mirror in 1+1 dimensions. Using a perturbative approach, we compute the spectral distribution of the created particles and the total particle creation rate, considering a thermal state as the initial field state.
Although Casimir, or quantum vacuum, forces between distinct bodies, or self-stresses of individual bodies, have been calculated by a variety of different methods since 1948, they have always been plagued by divergences. Some of these divergences are associated with the volume, and so may be more or less unambiguously removed, while other divergences are associated with the surface. The interpretation of these has been quite controversial. Particularly mysterious is the contradiction between finite total self-energies and surface divergences in the local energy density. In this paper we clarif…
It has been suggested by various authors that the `dynamical Casimir effect' might prove responsible for the production of visible-light photons in the bubble collapse which occurs in sonoluminescence. Previously, I have argued against this point of view based on energetic considerations, in the adiabatic approximation. Those arguments have recently been strengthened by the demonstration of the equivalence between van der Waals and Casimir energies. In this note I concentrate on the other extreme possibility, that of the validity of the `sudden approximation' where in effect the bubble instant…
We study the Casimir effect for scalar fields with general curvature coupling subject to mixed boundary conditions $(1+β_{m}n^μ\partial_μ)ϕ=0$ at $x=a_{m}$ on one ($m=1$) and two ($m=1,2$) parallel plates at a distance $a\equiv a_{2}-a_{1}$ from each other. Making use of the generalized Abel-Plana formula previously established by one of the authors \cite{Sahrev}, the Casimir energy densities are obtained as functions of $β_{1}$ and of $β_{1}$,$β_{2}$,$a$, respectively. In the case of two parallel plates, a decomposition of the total Casimir energy into volumic and superficial contributions is…
We present an approach to studying the Casimir effects by means of the effective theory. An essential point of our approach is replacing the mirror separation into the size of space S^1 in the adiabatic approximation. It is natural to identify the size of space S^1 with the scale factor of the Robertson-Walker-type metric. This replacement simplifies the construction of a class of effective models to study the Casimir effects. To check the validity of this replacement we construct a model for a scalar field coupling to the two-dimensional gravity and calculate the Casimir effects by the effect…
We show that even small nonlinearities significantly affect particle production in the dynamical Casimir effect at large evolution times. To that end, we derive the effective Hamiltonian and resum leading loop corrections to the particle flux in a massless scalar field theory with time-dependent Dirichlet boundary conditions and quartic self-interaction. To perform the resummation, we assume small deviations from the equilibrium and employ a kind of rotating wave approximation. Besides that, we consider a quantum circuit analog of the dynamical Casimir effect, which is also essentially nonline…
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…
Schwinger's Dynamical Casimir Effect is one of several candidate explanations for sonoluminescence. Recently, several papers have claimed that Schwinger's estimate of the Casimir energy involved is grossly inaccurate. In this letter, we show that these calculations omit the crucial volume term. When the missing term is correctly included one finds full agreement with Schwinger's result for the Dynamical Casimir Effect. We have nothing new to say about sonoluminescence itself except to affirm that the Casimir effect is energetically adequate as a candidate explanation.
In this paper we have considered the particle creation in the spatially closed Robertson-Walker space-time. We considered a real massive scalar field which conformally coupled to the Robertson-Walker background. With the dependence of the scale factor on time, the case under consideration is a dynamical Casimir effect with moving boundaries.
In [hep-th/0009173] Milton has presented a brief review of some interesting aspects of the theory of the Casimir effect. This comment is aimed at correcting some imprecise statements in that work with respect to the relevance of the Casimir effect for explaining the phenomenon of sonoluminescence.
Dirichlet boundary conditions on a surface can be imposed on a scalar field, by coupling it quadratically to a $δ$-like potential, the strength of which tends to infinity. Neumann conditions, on the other hand, require the introduction of an even more singular term, which renders the reflection and transmission coefficients ill-defined because of UV divergences. We present a possible procedure to tame those divergences, by introducing a minimum length scale, related to the non-zero `width' of a {\em nonlocal} term. We then use this setup to reach (either exact or imperfect) Neumann conditions,…
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…
We consider a quantum massless fermionic field in (1+1) dimensions in the case of moving boundaries. We work in the canonical approach in order to find a Hamiltonian describing the dynamics of the field. Thus, we study the statistics of work and particles produced from the vacuum by a linear driving of a boundary.
We apply the background field method and the effective action formalism to describe the four-dimensional dynamical Casimir effect. Our picture corresponds to the consideration of quantum cosmology for an expanding FRW universe (the boundary conditions act as a moving mirror) filled by a quantum massless GUT which is conformally invariant. We consider cases in which the static Casimir energy is repulsive and attractive. Inserting the simplest possible inertial term, we find, in the adiabatic (and semiclassical) approximation, the dynamical evolution of the scale factor and the dynamical Casimir…
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…
The Casimir energy for massless scalar field which satisfies priodic boundary conditions in two-dimensional domain wall background is calculated by making use of general properties of renormalized stress-tensor. The line element of domain wall is time dependent, the trace anomaly which is the nonvanishing $T^μ_μ$ for a conformally invariant field after renormalization, represent the back reaction of the dynamical Casimir effect.
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.
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 …
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.
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.
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…
