We present a detailed description of a quantum scalar field theory within a flat spacetime confined to a cavity with perfectly reflecting moving boundaries. Moreover, we establish an equivalence between this time-dependent setting and a field theory on an acoustic metric with static Dirichlet boundary conditions. We discuss the classical and quantum aspects of the theory from the latter perspective, accompanied by the introduction of novel numerical techniques designed for the (nonperturbative) computation of particle production attributed to the Dynamical Casimir effect, applicable to arbitrary boundary trajectories. As an illustrative example of these methodologies, we compute the particle production for a massless field in 1+1 dimensions. Notably, our approaches readily extend to encompass scenarios involving massive fields and higher dimensions
The Casimir effect for spherical geometry is calculated using generalized Thermofield Dynamics for the case of scalar field. Casimir force and Casimir pressure are presented. It is found that for high temperatures the Casimir force does change sign.
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.
Calculations of the Casimir energy for spherical geometries which are based on integrations of the stress tensor are critically examined. It is shown that despite their apparent agreement with numerical results obtained from mode summation methods, they contain a number of serious errors. Specifically, these include (1) an improper application of the stress tensor to spherical boundaries, (2) the neglect of pole terms in contour integrations, and (3) the imposition of inappropriate boundary conditions upon the relevant propagators. A calculation which is based on the stress tensor and which av…
We perform the matching required to compute the leading effective boundary contribution to the QED lagrangian in the presence of a conducting surface, once the electron is integrated out. Our result resolves a confusion in the literature concerning the interpretation of the leading such correction to the Casimir energy. It also provides a useful theoretical laboratory for brane-world calculations in which kinetic terms are generated on the brane, since a lot is known about QED near boundaries.
This is a review of publications on classical and quantum electrodynamics in cavities with moving boundaries (in the quantum case this subject is labeled frequently as "nonstationary Casimir effect" or "dynamical Casimir effect"), from 1921 to October of 2000, with an emphasis on analytical results related to cavities with resonantly oscillating boundaries.
The thermal Casimir effect in ideal metal rectangular boxes is considered using the method of zeta functional regularization. The renormalization procedure is suggested which provides the finite expression for the Casimir free energy in any restricted quantization volume. This expression satisfies the classical limit at high temperature and leads to zero thermal Casimir force for systems with infinite characteristic dimensions. In the case of two parallel ideal metal planes the results, as derived previously using thermal quantum field theory in Matsubara formulation and other methods, are rep…
Compact U(1) gauge theory in 3+1 dimensions possesses the confining phase, characterized by a linear raise of the potential between particles with opposite electric charges at sufficiently large inter-particle separation. The confinement is generated by condensation of Abelian monopoles at strong gauge coupling. We study the properties of monopoles and the deconfining order parameter in zero-temperature theory in the presence of ideally conducting parallel metallic boundaries (plates) usually associated with the Casimir effect. Using first-principle numerical simulations in compact U(1) lattic…
Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor are investigated for a massive scalar field with general curvature coupling parameter in the region between two coaxial cylindrical boundaries. It is assumed that the field obeys general Robin boundary conditions on bounding surfaces. The application of a variant of the generalized Abel-Plana formula allows to extract from the expectation values the contribution from single shells and to present the interference part in terms of exponentially convergent integrals. The vacuum forces acting on the…
We evaluate the correlators for the vector potential and for the field strength tensor of the electromagnetic field in the geometry of two parallel planar plates in AdS spacetime. Two types of boundary conditions are considered on the plates. The first one is a generalization of perfect conductor boundary condition and the second one corresponds to the confining boundary conditions. By using the expressions for the correlators, the vacuum expectation values (VEVs) of the photon condensate and of the electric and magnetic fields squared are investigated. As another important local characteristi…
Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor are investigated for a massive scalar field with general curvature coupling parameter inside a wedge with two coaxial cylindrical boundaries. It is assumed that the field obeys Dirichlet boundary condition on bounding surfaces. The application of a variant of the generalized Abel-Plana formula enables to extract from the expectation values the contribution corresponding to the geometry of a wedge with a single shell and to present the interference part in terms of exponentially convergent integr…
The Casimir effect in an inhomogeneous dielectric is investigated using Lifshitz's theory of electromagnetic vacuum energy. A permittivity function that depends continuously on one Cartesian coordinate is chosen, bounded on each side by homogeneous dielectrics. The result for the Casimir stress is infinite everywhere inside the inhomogeneous region, a divergence that does not occur for piece-wise homogeneous dielectrics with planar boundaries. A Casimir force per unit volume can be extracted from the infinite stress but it diverges on the boundaries between the inhomogeneous medium and the hom…
We consider a massless scalar field obeying Dirichlet boundary conditions on the walls of a two-dimensional L x b rectangular box, divided by a movable partition (piston) into two compartments of dimensions a x b and (L-a) x b. We compute the Casimir force on the piston in the limit L -> infinity. Regardless of the value of a/b, the piston is attracted to the nearest end of the box. Asymptotic expressions for the Casimir force on the piston are derived for a << b and a >> b.
Schlömilch's formula is generalized and applied to the thermal Casimir effect of a fermionic field confined a three-dimensional rectangular box. The analytic expressions of the Casimir energy and Casimir force are derived for arbitrary temperature and edge sizes. The low and high temperature limits and finite temperature cases are considered for the entire parameter space spanned by edge sizes and/or temperature. In the low temperature limit, it is found that for typical rectangular box, the effective 2-dimensional parameter space spanned by the two edge size ratios can be split into four regi…
The prototypical Casimir effect arises when a scalar field is confined between parallel Dirichlet boundaries. We study corrections to this when the boundaries themselves have apertures and edges. We consider several geometries: a single plate with a slit in it, perpendicular plates separated by a gap, and two parallel plates, one of which has a long slit of large width, related to the case of one plate being semi-infinite. We develop a general formalism for studying such problems, based on the wavefunctional for the field in the gap between the plates. This formalism leads to a lower dimension…
Applying the general framework for local zeta regularization proposed in Part I of this series of papers, we compute the renormalized vacuum expectation value of several observables (in particular, of the stress-energy tensor and of the total energy) for a massless scalar field confined within a rectangular box of arbitrary dimension.
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.
The effects of reflecting boundaries on vacuum electric field fluctuations are treated. The presence of the boundaries can enhance these fluctuations and possibly lead to observable effects. The electric field fluctuations lead to voltage fluctuations along the worldline of a charged particle moving perpendicularly to a pair of reflecting plates, These voltage fluctuations in turn lead to fluctuations in the kinetic energy of the particle, which may enhance the probability of quantum barrier penetration by the particle. A recent experiment by Moddel {\it et al} is discussed as a possible examp…
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.
Casimir forces are a manifestation of the change in the zero-point energy of the vacuum caused by the insertion of boundaries. We show how the Casimir force can be computed by consideration of the vacuum fluctuations that are suppressed by the boundaries, and rederive the scalar Casimir effects for a series of geometries. For the planar case a finite universal force is automatically found. For curved geometries formally divergent expressions are encountered which we argue are largely due to the divergent self-energy of the boundary contributing to the force. This idea is supported by computing…
The vacuum expectation values of the field squared and the energy-momentum tensor are investigated for a scalar field with Dirichlet boundary conditions and for the electromagnetic field inside a wedge with a coaxial cylindrical boundary. In the case of the electromagnetic field perfectly conducting boundary conditions are assumed on the bounding surfaces. By using the Abel-Plana-type formula for the series over the zeros of the Bessel function, the vacuum expectation values are presented in the form of the sum of two terms. The first one corresponds to the geometry without a cylindrical bound…
We consider the Casimir energy in a geometry of an infinite magnetodielectric wedge closed by a circularly cylindrical, perfectly conducting arc embedded in another magnetodielectric medium, under the condition that the speed of light be the same in both media. An expression for the Casimir energy corresponding to the arc is obtained and it is found that in the limit where the reflectivity of the wedge boundaries tends to unity the finite part of the Casimir energy of a perfectly conducting wedge-shaped sheet closed by a circular cylinder is regained. The energy of the latter geometry possesse…
We develop a unified scattering approach to dynamical Casimir problems which can be applied to both accelerating boundaries, as well as dispersive objects in relative motion. A general (trace) formula is derived for the radiation from accelerating boundaries. Applications are provided for objects with different shapes in various dimensions, and undergoing rotational or linear motion. Within this framework, photon generation is discussed in the context of a modulated optical mirror. For dispersive objects, we find general results solely in terms of the scattering matrix. Specifically, we discus…
The Casimir force can be understood as resulting from the radiation pressure exerted by the vacuum fluctuations reflected by boundaries. We extend this local formulation to the case of partially transmitting boundaries by introducing reflectivity and transmittivity coefficients obeying conditions of unitarity, causality and high frequency transparency. We show that the divergences associated with the infiniteness of the vacuum energy do not appear in this approach. We give explicit expressions for the Casimir force which hold for any frequency dependent scattering and any temperature. The corr…
In this paper the Lifshitz formula for the Casimir energy between two dielectrics in zero temperature is derived using box renormalization. Although there are several derivations for the force in this case in the literature, including Lifshitz's own proof, so far there has been no unambiguous and rigorous derivation for energy that we are aware of. Since the energy becomes important in some cases, e.g. calculation of entropy or heat capacity, using the correct and precise definition of the Casimir energy, for the first time, we remove all of the infinities systematically without any ambiguity.…
We consider the Casimir energy due to a massless scalar field in a geometry of an infinite wedge closed by a Dirichlet circular cylinder, where the wedge is formed by $δ$-function potentials, so-called semitransparent boundaries. A finite expression for the Casimir energy corresponding to the arc and the presence of both semitransparent potentials is obtained, from which the torque on the sidewalls can be derived. The most interesting part of the calculation is the nontrivial nature of the angular mode functions. Numerical results are obtained which are closely analogous to those recently foun…
Casimir forces are a manifestation of the change in the zero-point energy of the vacuum caused by the insertion of boundaries. We show how the Casimir force can be efficiently computed by consideration of the vacuum fluctuations that are suppressed by the boundaries, and rederive the scalar Casimir effects for a series of the Dirichlet geometries. For the planar case a finite universal force is automatically found. Consistent with other calculations of the effect, for curved geometries divergent (non-universal) expressions are encountered. They are interpreted geometrically following Candelas …
One of the most efficient methods for the evaluation of the vacuum expectation values for physical observables in the Casimir effect is based on using the Abel-Plana summation formula. This enables to derive the renormalized quantities in a manifestly cutoff independent way and to present them in the form of strongly convergent integrals. However, applications of the Abel-Plana formula, in its usual form, are restricted by simple geometries when the eigenmodes have a simple dependence on quantum numbers. The author generalized the Abel-Plana formula which essentially enlarges its application r…
We evaluate the Hadamard function and the vacuum expectation value (VEV) of the current density for a charged scalar field, induced by flat boundaries in spacetimes with an arbitrary number of toroidally compactified spatial dimensions. The field operator obeys the Robin conditions on the boundaries and quasiperiodicity conditions with general phases along compact dimensions. In addition, the presence of a constant gauge field is assumed. The latter induces Aharonov-Bohm-type effect on the VEVs. There is a region in the space of the parameters in Robin boundary conditions where the vacuum stat…
We analyze the high temperature (or classical) limit of the Casimir effect. A useful quantity which arises naturally in our discussion is the ``relative Casimir energy", which we define for a configuration of disjoint conducting boundaries of arbitrary shapes, as the difference of Casimir energies between the given configuration and a configuration with the same boundaries infinitely far apart. Using path integration techniques, we show that the relative Casimir energy vanishes exponentially fast in temperature. This is consistent with a simple physical argument based on Kirchhoff's law. As a …
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 investigate quantum vacuum effects for a massive scalar field, induced by two planar boundaries in background of a linearly expanding spatially flat Friedmann-Robertson-Walker spacetime for an arbitrary number of spatial dimensions. For the Robin boundary conditions and for general curvature coupling parameter, a complete set of mode functions is presented and the related Hadamard function is evaluated. The results are specified for the most important special cases of the adiabatic and conformal vacuum states. The vacuum expectation values of the field squared and of the energy-momentum ten…
Quantities associated with Casimir forces are calculated in a model wave system of one spatial dimension with Dirichlet or Neumann boundary conditions. 1)Due to zero-point fluctuations, a partition is attracted to the walls of a box if the wave boundary conditions are alike for the partition and the walls, but is repelled if the conditions are different. 2)The use of Casimir energies in the presence of zero-point radiation introduces a natural maximum-entropy principle which is satisfied only by the Planck spectrum for both like and unlike boundary conditions between the box and partition. 3)T…
Several problems at the interface between the field-theoretical description of the Casimir effect and experiments on measuring the Casimir force are discussed. One of these problems is connected with the definition of the Casimir free energy in ideal metal rectangular boxes satisfying the general physical requirements. It is shown that the consideration of rectangular boxes with a partition (piston) does not negate the previously known results obtained for boxes without a piston. Both sets of results are found to be in mutual agreement. Another problem is related to the use of the proximity fo…
A path integral formulation is developed for the dynamic Casimir effect. It allows us to study arbitrary deformations in space and time of the perfectly reflecting (conducting) boundaries of a cavity. The mechanical response of the intervening vacuum is calculated to linear order in the frequency-wavevector plane, using which a plethora of interesting phenomena can be studied. For a single corrugated plate we find a correction to mass at low frequencies, and an effective shear viscosity at high frequencies that are both anisotropic. The anisotropy is set by the wavevector of the corrugation. F…
We discuss the quantization of sound waves in a fluid with a linear dispersion relation and calculate the quantum density fluctuations of the fluid in several cases. These include a fluid in its ground state. In this case, we discuss the scattering cross section of light by the density fluctuations, and find that in many situations it is small compared to the thermal fluctuations, but not negligibly small and might be observable at room temperature. We also consider a fluid in a squeezed state of phonons and fluids containing boundaries. We suggest that the latter may be a useful analog model …
We investigate combined effects of nontrivial topology, induced by a cosmic string, and boundaries on the fermionic condensate and the vacuum expectation value (VEV) of the energy-momentum tensor for a massive fermionic field. As geometry of boundaries we consider two plates perpendicular to the string axis on which the field is constrained by the MIT bag boundary condition. By using the Abel-Plana type summation formula, the VEVs in the region between the plates are decomposed into the boundary-free and boundary-induced contributions for general case of the planar angle deficit. The boundary-…
We consider and review the emergence of singular energy densities and field fluctuations at sharp boundaries or point-like field sources in the vacuum. The presence of singular energy densities of a field may be relevant from a conceptual point of view, because they contribute to the self-energy of the system. They should also generate significant gravitational effects. We first consider the case of the interface between a metallic boundary and the vacuum, and obtain the structure of the singular electric and magnetic energy densities at the interface through an appropriate limit from a dielec…
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.
When the vacuum is partitioned by material boundaries with arbitrary shape, one can define the zero-point energy and the free energy of the electromagnetic waves in it: this can be done, independently of the nature of the boundaries, in the limit that they become perfect conductors, provided their curvature is finite. The first examples we consider are Casimir's original configuration of parallel plates, and the experimental situation of a sphere in front of a plate. For arbitrary geometries, we give an explicit expression for the zero-point energy and the free energy in terms of an integral k…
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 the effects of light-cone fluctuations on the renormalized zero-point energy associated with a free massless scalar field in the presence of boundaries. In order to simulate light-cone fluctuations we introduce a space-time dependent random coefficient in the Klein-Gordon operator. We assume that the field is defined in a domain with one confined direction. For simplicity, we choose the symmetric case of two parallel plates separated by a distance $a$. The correction to the renormalized vacuum energy density between the plates goes as $1/a^{8}$ instead of the usual $1/a^{4}$ dependenc…
The review of vacuum and matter restructuring in space-time with boundaries is presented. We consider phase properties of confining gauge theories and strongly interacting fermion systems. In particular, the chiral and deconfinement phase transitions properties in the presence of Casimir plates. We also discuss mass scale shifts in such systems and their possible dynamical and geometrical nature.
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 have performed numerical simulations to evaluate the effect on the capacitance of finite size boundaries realistically present in the parallel plane, sphere-plane, and cylinder-plane geometries. The potential impact of edge effects in assessing the accuracy of the parameters obtained in the electrostatic calibrations of Casimir force experiments is then discussed.
We calculate the energy-momentum tensor due to electromagnetic vacuum fluctuations between two parallel hyperplanes in more than four dimensions, considering both metallic and MIT boundary conditions. Using the axial gauge, the problem can be mapped upon the corresponding problem with a massless, scalar field satisfying respectively Dirichlet or Neumann boundary conditions. The pressure between the plates is constant while the energy density is found to diverge at the boundaries when there are extra dimensions. This can be related to the fact that Maxwell theory is then no longer conformally i…
We calculate the Casimir interaction energy in $d=2$ spatial dimensions between two (zero-width) mirrors, one flat, and the other slightly curved, upon which {\em imperfect\/} conductor boundary conditions are imposed for an Electromagnetic (EM) field. Our main result is a second-order Derivative Expansion (DE) approximation for the Casimir energy, which is studied in different interesting limits. In particular, we focus on the emergence of a non-analyticity beyond the leading-order term in the DE, when approaching the limit of perfectly-conducting mirrors. We also show that the system conside…
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…
Using the Tridiagonal Representation Approach, we obtain solutions (energy spectrum and corresponding wavefunctions) for a new five-parameter potential box with inverse square singularity at the boundaries.
The static Casimir effect concerns quantum electrodynamic induced Lamb shifts in the mode frequencies and thermal free energies of condensed matter systems. Sometimes, the condensed matter constitutes the boundaries of a vacuum region. The static frequency shift effects have been calculated in the one photon loop perturbation theory approximation. The dynamic Casimir effect concerns two photon radiation processes arising from time dependent frequency modulations again computed in the one photon loop approximation. Under certain conditions the one photon loop computation may become unstable and…
We review recent developments in the Casimir effect which arises in quantization volumes restricted by material boundaries and in spaces with non-Euclidean topology. The starting point of our discussion is the novel exact solution for the electromagnetic Casimir force in the configuration of a cylinder above a plate. The related work for the scalar Casimir effect in sphere-plate configuration is also considered, and the application region of the proximity force theorem is discussed. Next we consider new experiments on the measurement of the Casimir force between metals and between metal and se…
Positive frequency Wightman function and vacuum expectation value of the energy-momentum tensor are computed for a massive scalar field with general curvature coupling parameter subject to Robin boundary conditions on two parallel plates located on $D+1$ - dimensional AdS background. The general case of different Robin coefficients on separate plates is considered. The mode summation method is used with a combination of a variant of the generalized Abel-Plana formula for the series over zeros of combinations of cylinder functions. This allows us to extract manifestly the parts due to the AdS s…
Recent experimental realizations of uniform confining potentials for ultracold atoms make it possible to create quantum acoustic resonators and explore nonequilibrium dynamics of quantum field theories. These systems offer a promising new platform for studying the dynamical Casimir effect, since they allow to achieve relativistic, i.e. near sonic, velocities of the boundaries. In comparison to previously studied optical and classical hydrodynamic systems ultracold atoms allow to realize a broader class of dynamical experiments combining both classical driving and vacuum squeezing. In this pape…
The wedge geometry closed by a circular-cylindrical arc is a nontrivial generalization of the cylinder, which may have various applications. If the radial boundaries are not perfect conductors, the angular eigenvalues are only implicitly determined. When the speed of light is the same on both sides of the wedge, the Casimir energy is finite, unlike the case of a perfect conductor, where there is a divergence associated with the corners where the radial planes meet the circular arc. We advance the study of this system by reporting results on the temperature dependence for the conducting situati…
