Casimir Piston Research Articles

Casimir pistons with generalized boundary conditions: a step forward

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 piston. We provide explicit results for the Casimir force when the manifold $N$ is a $d$-dimensional sphere and a disk.

First-order correction to the Casimir force within an inhomogeneous medium

For the Casimir piston filled with an inhomogeneous medium, we regularized and expressed the Casimir energy with cylinder kernel coefficients by using the first-order perturbation theory. When the refractive index of the medium is smoothly inhomogeneous (i.e., derivatives of all orders exist), a logarithmically cutoff-dependent term and a quadratically cutoff-dependent term in the Casimir energy are found. We show that in the piston model these terms vanish in the force and thus the Casimir force is always cutoff independent, but these terms will remain in the force in the half-space model and must be removed by additional regularizations. We give explicit benchmark solutions to the first-order corrections of both Casimir energy and Casimir force for an exponentially decaying profile. The present method can be extended to other inhomogeneous profiles. Our results should be useful for future relevant calculations and experimental studies.

Casimir pistons with general boundary conditions

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×N, with I=[0,L]⊂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 and the parameters describing the general boundary conditions imposed. These results are then specialized to the case in which the manifold N is a d-dimensional sphere.

Some developments of the Casimir effect in p-cavity of (D + 1)-dimensional space–time

The Casimir effect for rectangular boxes has been studied for several decades. But there are still some unclear points. Recently, there are new developments related to this topic, including the demonstration of the equivalence of the regularization methods and the clarification of the ambiguity in the regularization of the temperature-dependent free energy. Also, the interesting quantum spring was raised stemming from the topological Casimir effect of the helix boundary conditions. We review these developments together with the general derivation of the Casimir energy of the p-dimensional cavity in (D + 1)-dimensional space–time, paying special attention to the sign of the Casimir force in a cavity with unequal edges. In addition, we also review the Casimir piston, which is a configuration related to rectangular cavity.

THERMAL FLUCTUATIONS IN CASIMIR PISTONS

We present analytical and simple expressions to determine the free energy, internal energy, entropy, as well as the pressure acting at the interface of a perfectly conducting rectangular Casimir piston. We show that infrared divergencies linear in temperature become cancelled within the piston configuration, and show a continuous behavior consistent with intuitive expectations.

Conical Casimir pistons with hybrid boundary conditions

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 modeled by a d-dimensional sphere. In this particular situation, explicit results are given for d = 2, 3, 4, 5.

Electromagnetic Casimir piston in higher-dimensional spacetimes

We consider the Casimir effect of the electromagnetic field in a higher-dimensional spacetime of the form $M\ifmmode\times\else\texttimes\fi{}\mathcal{N}$, where $M$ is the four-dimensional Minkowski spacetime and $\mathcal{N}$ is an $n$-dimensional compact manifold. The Casimir force acting on a planar piston that can move freely inside a closed cylinder is investigated. Different combinations of perfectly conducting boundary conditions and infinitely permeable boundary conditions are imposed on the cylinder and the piston. It is verified that if the piston and the cylinder have the same boundary conditions, the piston is always going to be pulled towards the closer end of the cylinder. However, if the piston and the cylinder have different boundary conditions, the piston is always going to be pushed to the middle of the cylinder. By taking the limit where one end of the cylinder tends to infinity, one obtains the Casimir force acting between two parallel plates inside an infinitely long cylinder. The asymptotic behavior of this Casimir force in the high temperature regime and the low temperature regime are investigated for the case where the cross section of the cylinder in $M$ is large. It is found that if the separation between the plates is much smaller than the size of $\mathcal{N}$, the leading term of the Casimir force is the same as the Casimir force on a pair of large parallel plates in the ($4+n$)-dimensional Minkowski spacetime. However, if the size of $\mathcal{N}$ is much smaller than the separation between the plates, the leading term of the Casimir force is $1+h/2$ times the Casimir force on a pair of large parallel plates in the four-dimensional Minkowski spacetime, where $h$ is the first Betti number of $\mathcal{N}$. In the limit the manifold $\mathcal{N}$ vanishes, one does not obtain the Casimir force in the four-dimensional Minkowski spacetime if $h$ is nonzero. Therefore the data obtained from Casimir experiments suggest that the first Betti number of the extra dimensions should be zero.

Casimir piston of real materials and its application to multilayer models

In this article, we derive the formula for the Casimir force acting on a piston made of real material moving inside a perfectly conducting rectangular box. It is shown that by taking suitable limits, one recovers the formula for the Casimir force acting on a perfectly conducting piston or an infinitely permeable piston. The Lifshitz formula for finite temperature Casimir force acting on parallel plates made of real materials is re-derived by considering the five-layer model in the context of the piston approach. It is observed that the divergences of the Casimir force will only cancel under certain conditions, for example, when the regions separated by the plates are filled with media of the same refractive index.

3Dscalar model as a4Dperfect conductor limit: Dimensional reduction and variational boundary conditions

Under dimensional reduction, a system in $D$ spacetime dimensions will not necessarily yield its $D\ensuremath{-}1$-dimensional analog version. Among other things, this result will depend on the boundary conditions and the dimension $D$ of the system. We investigate this question for scalar and Abelian gauge fields under boundary conditions that obey the symmetries of the action. We apply our findings to the Casimir piston, an ideal system for detecting boundary effects. Our investigation is not limited to extra dimensions and we show that the original piston scenario proposed in 2004, a toy model involving a scalar field in $3D$ ($2+1$) dimensions, can be obtained via dimensional reduction from a more realistic $4D$ electromagnetic (EM) system. We show that for perfect conductor conditions, a $D$-dimensional EM field reduces to a $D\ensuremath{-}1$ scalar field and not its lower-dimensional version. For Dirichlet boundary conditions, no theory is recovered under dimensional reduction and the Casimir pressure goes to zero in any dimension. This ``zero Dirichlet'' result is useful for understanding the EM case. We then identify two special systems where the lower-dimensional version is recovered in any dimension: systems with perfect magnetic conductor (PMC) and Neumann boundary conditions. We show that these two boundary conditions can be obtained from a variational procedure in which the action vanishes outside the bounded region. The fields are free to vary on the surface and have zero modes, which survive after dimensional reduction.

Three-dimensional Casimir piston for massive scalar fields

We consider Casimir force acting on a three-dimensional rectangular piston due to a massive scalar field subject to periodic, Dirichlet and Neumann boundary conditions. Exponential cut-off method is used to derive the Casimir energy. It is shown that the divergent terms do not contribute to the Casimir force acting on the piston, thus render a finite well-defined Casimir force acting on the piston. Explicit expressions for the total Casimir force acting on the piston is derived, which show that the Casimir force is always attractive for all the different boundary conditions considered. As a function of a – the distance from the piston to the opposite wall, it is found that the magnitude of the Casimir force behaves like 1 / a 4 when a → 0 + and decays exponentially when a → ∞ . Moreover, the magnitude of the Casimir force is always a decreasing function of a. On the other hand, passing from massless to massive, we find that the effect of the mass is insignificant when a is small, but the magnitude of the force is decreased for large a in the massive case.

Numerical and semiclassical analysis of some generalized Casimir pistons

The Casimir force due to a scalar field on a piston in a cylinder of radius $r$ with a spherical cap of radius $R>r$ is computed numerically in the world-line approach. A geometrical subtraction scheme gives the finite interaction energy that determines the Casimir force. The spectral function of convex domains is obtained from a probability measure on convex surfaces that is induced by the Wiener measure on Brownian bridges the convex surfaces are the hulls of. The vacuum force on the piston by a scalar field satisfying Dirichlet boundary conditions is attractive in these geometries, but the strength and short-distance behavior of the force depends crucially on the shape of the piston casing. For a cylindrical casing with a hemispherical head, the force for $a/R\sim 0$ does not depend on the dimension of the casing and numerically approaches $\sim - 0.00326(4)\hbar c/a^2$. Semiclassically this asymptotic force is due to short, closed and non-periodic trajectories that reflect once off the piston near its periphery. The semiclassical estimate $-\hbar c/(96\pi a^2)(1+2\sqrt{R^2-r^2}/a)$ for the force when $a/r\ll r/R\leq 1$ reproduces the numerical results within statistical errors.

Numerical calculation of the force on some Generalized Casimir Pistons

In this talk I presented numerical calculations of the Casimir force due to a scalar field on a piston in a cylinder of radius τ with a spherical cap of radius R > τ. Geometrical subtractions give a finite interaction energy. Due to reflection positivity, the vacuum force on the piston by a scalar field satisfying Dirichlet boundary conditions is attractive for these geometries, but the strength and short-distance behavior of the force depends strongly on the shape of the piston casing. For a cylindrical casing with a hemispherical head of large radius, the attractive force on the piston is inversely proportional to the square of the height of the piston.

Kaluza-Klein models as pistons

We consider the influence of extra dimensions on the force in Casimir pistons. Suitable analytical expressions are provided for the Casimir force in the range where the plate distance is small, and where it is large, compared to the size of the extra dimensions. We show that the Casimir force tends to move the center plate toward the closer wall; this result is true independently of the cross section of the piston and the geometry or topology of the additional Kaluza-Klein dimensions. The statement also remains true at finite temperature. In the limit where one wall of the piston is moved to infinity, the result for parallel plates is recovered. If only one chamber is considered, a criterion for the occurrence of Lukosz-type repulsion, as opposed to the occurrence of renormalization ambiguities, is given; we comment on why no repulsion has been noted in some previous cosmological calculations that consider only two plates.

Finite-temperature Casimir pistons for an electromagnetic field with mixed boundary conditions and its classical limit

In this paper, the finite-temperature Casimir force acting on a two-dimensional Casimir piston due to an 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 toward the equilibrium position, regardless of the boundary conditions assumed on the walls transverse to the piston. In contrast, if pure boundary conditions are assumed on the piston and the opposite wall, then the Casimir force always tends to pull the piston toward the closer wall and away from the equilibrium position. The nature of the force is not affected by temperature. However, in the high-temperature regime, the magnitude of the Casimir force grows linearly with respect to temperature. This shows that the Casimir effect has a classical limit as has been observed in other literature.

The Casimir force on a piston in the spacetime with extra compactified dimensions

A Casimir piston for massless scalar fields obeying Dirichlet boundary conditions in high-dimensional spacetimes within the frame of Kaluza–Klein theory is analyzed. We derive and calculate the exact expression for the Casimir force on the piston. We also compute the Casimir force in the limit that one outer plate is moved to the extremely distant place to show that the reduced force is associated with the properties of additional spatial dimensions. The more dimensionality the spacetime has, the stronger the extra-dimension influence is. The Casimir force for the piston in the model including a third plate under the background with extra compactified dimensions always keeps attractive. Further we find that when the limit is taken the Casimir force between one plate and the piston will change to be the same form as the corresponding force for the standard system consisting of two parallel plates in the four-dimensional spacetimes if the ratio of the plate-piston distance and extra dimensions size is large enough.

Perfect magnetic conductor Casimir piston ind+1dimensions

Perfect magnetic conductor (PMC) boundary conditions are dual to the more familiar perfect electric conductor (PEC) conditions and can be viewed as the electromagnetic analog of the boundary conditions in the bag model for hadrons in QCD. Recent advances and requirements in communication technologies have attracted great interest in PMC's, and Casimir experiments involving structures that approximate PMC's may be carried out in the not-too-distant future. In this paper, we make a study of the zero-temperature PMC Casimir piston in d+1 dimensions. The PMC Casimir energy is explicitly evaluated by summing over p+1-dimensional Dirichlet energies where p ranges from 2 to d inclusively. We derive two exact d-dimensional expressions for the Casimir force on the piston and find that the force is negative (attractive) in all dimensions. Both expressions are applied to the case of 2+1 and 3+1 dimensions. A spin-off from our work is a contribution to the PEC literature: we obtain a useful alternative expression for the PEC Casimir piston in 3+1 dimensions and also evaluate the Casimir force per unit area on an infinite strip, a geometry of experimental interest.

Cancellation of nonrenormalizable hypersurface divergences and thed-dimensional Casimir piston

Using a multidimensional cut-off technique, we obtain expressions for the cut-off dependent part of the vacuum energy for parallelepiped geometries in any spatial dimension d. The cut-off part yields nonrenormalizable hypersurface divergences and we show explicitly that they cancel in the Casimir piston scenario in all dimensions. We obtain two different expressions for the d-dimensional Casimir force on the piston where one expression is more convenient to use when the plate separation a is large and the other when a is small (a useful a→1/a duality). The Casimir force on the piston is found to be attractive (negative) for any dimension d. We apply the d-dimensional formulas (both expressions) to the two and three-dimensional Casimir piston with Neumann boundary conditions. The 3D Neumann results are in numerical agreement with those recently derived in 0705.0139 using an optical path technique providing an independent confirmation of our multidimensional approach. We limit our study to massless scalar fields.

Casimir pistons with hybrid boundary conditions

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.

Vacuum energy and repulsive Casimir forces in quantum star graphs

Casimir pistons are models in which finite Casimir forces can be calculated without any suspect renormalizations. It has been suggested that such forces are always attractive. We present three scenarios in which that is not true. Two of these depend on mixing two types of boundary conditions. The other, however, is a simple type of quantum graph in which the sign of the force depends upon the number of edges.

Casimir piston for massless scalar fields in three dimensions

We study the Casimir piston for massless scalar fields obeying Dirichlet boundary conditions in a three-dimensional cavity with sides of arbitrary lengths $a$, $b$, and $c$ where $a$ is the plate separation. We obtain an exact expression for the Casimir force on the piston valid for any values of the three lengths. As in the electromagnetic case with perfect-conductor conditions, we find that the Casimir force is negative (attractive) regardless of the values of $a$, $b$, and $c$. Though cases exist where the interior contributes a positive (repulsive) Casimir force, the total Casimir force on the piston is negative when the exterior contribution is included. We also obtain an alternative expression for the Casimir force that is useful computationally when the plate separation $a$ is large.