Casimir Force of Piston Systems with Arbitrary Cross Sections under Different Boundary Conditions

We study the Casimir force between two pistons under different boundary conditions inside an infinite cylinder with arbitrary cross section. It is found that the attractive or repulsive character of the Casimir force for a scalar field is determined only by the boundary condition along the longitudinal direction and is independent of the cross section, transverse boundary conditions and the mass of the field. Under symmetric Dirichlet-Dirichlet, Neumann-Neumann and periodic longitudinal boundary conditions the Casimir force is always attractive, but is repulsive under non-symmetric Dirichlet-Neumann and anti-periodic longitudinal boundary conditions. The Casimir force of the electromagnetic field in an ideal conductive piston is also investigated. This force is always attractive regardless of the shape of the cross section and the transverse boundary conditions.     

Casimir Effect at Finite Temperature in the Presence of One Fractal Extra Compactified Dimension

We discuss the Casimir effect for massless scalar fields subject to the Dirichlet boundary conditions on the parallel plates at finite temperature in the presence of one fractal extra compactified dimension. We obtain the Casimir energy density with the help of the regularization of multiple zeta function with one arbitrary exponent and further the renormalized Casimir energy density involving the thermal corrections. It is found that when the temperature is sufficiently high, the sign of the Casimir energy remains negative no matter how great the scale dimension δ is within its allowed region. We derive and calculate the Casimir force between the parallel plates affected by the fractal additional compactified dimension and surrounding temperature. The stronger thermal influence leads the force to be stronger. The nature of the Casimir force keeps attractive.     

The Casimir Effect for Parallel Plates in the Spacetime with a Fractal Extra Compactified Dimension

The Casimir effect for massless scalar fields satisfying Dirichlet boundary conditions on the parallel plates in the presence of one fractal extra compactified dimension is analyzed. We obtain the Casimir energy density by means of the regularization of multiple zeta function with one arbitrary exponent. We find a limit on the scale dimension like $\delta>\frac{1}{2}$ to keep the negative sign of the renormalized Casimir energy which is the difference between the regularized energy for two parallel plates and the one with no plates. We derive and calculate the Casimir force relating to the influence from the fractal additional compactified dimension between the parallel plates. The larger scale dimension leads to the greater revision on the original Casimir force. The two kinds of curves of Casimir force in the case of integer-numbered extra compactified dimension or fractal one are not superposition, which means that the Casimir force show whether the dimensionality of additional compactified space is integer or fraction.     

Nonequilibrium electromagnetic fluctuations: heat transfer and interactions

The Casimir force between arbitrary objects in equilibrium is related to scattering from individual bodies. We extend this approach to heat transfer and Casimir forces in nonequilibrium cases where each body, and the environment, is at a different temperature. The formalism tracks the radiation from each body and its scatterings by the other objects. We discuss the radiation from a cylinder, emphasizing its polarized nature, and obtain the heat transfer between a sphere and a plate, demonstrating the validity of proximity transfer approximation at close separations and arbitrary temperatures.     

Lateral Casimir force beyond the proximity-force approximation

We argue that the appropriate variable to study a nontrivial geometry dependence of the Casimir force is the lateral component of the Casimir force, which we evaluate between two corrugated metallic plates outside the validity of the proximity-force approximation. The metallic plates are described by the plasma model, with arbitrary values for the plasma wavelength, the plate separation, and the corrugation period, the corrugation amplitude remaining the smallest length scale. Our analysis shows that in realistic experimental situations the proximity-force approximation overestimates the force by up to 30%.     

Casimir interaction of dielectric gratings

We derive an exact solution for the Casimir force between two arbitrary periodic dielectric gratings and illustrate our method by applying it to two nanostructured silicon gratings. We also reproduce the Casimir force gradient measured recently [H. B. Chan, Y. Bao, J. Zou, R. A. Cirelli, F. Klemens, W. M. Mansfield, and C. S. Pai, Phys. Rev. Lett. 101, 030401 (2008)10.1103/PhysRevLett.101.030401] between a silicon grating and a gold sphere taking into account the material dependence of the force. We find good agreement between our theoretical results and the measured values both in absolute force values and the ratios between the exact force and proximity force approximation predictions.     

Surface plasmon modes and the Casimir energy

We show the influence of surface plasmons on the Casimir effect between two plane parallel metallic mirrors at arbitrary distances. Using the plasma model to describe the optical response of the metal, we express the Casimir energy as a sum of contributions associated with evanescent surface plasmon modes and propagative cavity modes. In contrast to naive expectations, the plasmonic mode contribution is essential at all distances in order to ensure the correct result for the Casimir energy. One of the two plasmonic modes gives rise to a repulsive contribution, balancing out the attractive contributions from propagating cavity modes, while both contributions taken separately are much larger than the actual value of the Casimir energy. This also suggests possibilities to tailor the sign of the Casimir force via surface plasmons.     

Finite temperature Casimir effect of massive fermionic fields in the presence of compact dimensions

We consider the finite temperature Casimir effect of a massive fermionic field confined between two parallel plates, with MIT bag boundary conditions on the plates. The background spacetime is Mp+1×Tq which has q dimensions compactified to a torus. On the compact dimensions, the field is assumed to satisfy periodicity boundary conditions with arbitrary phases. Both the high temperature and the low temperature expansions of the Casimir free energy and the force are derived explicitly. It is found that the Casimir force acting on the plates is always attractive at any temperature regardless of the boundary conditions assumed on the compact torus. The asymptotic limits of the Casimir force in the small plate separation limit are also obtained.     

The Casimir effect for fields with arbitrary spin

The Casimir force arises when a quantum field is confined between objects that apply boundary conditions to it. In a recent paper we used the two-spinor calculus to derive boundary conditions applicable to fields with arbitrary spin in the presence of perfectly reflecting surfaces. Here we use these general boundary conditions to investigate the Casimir force between two parallel perfectly reflecting plates for fields up to spin-2. We use the two-spinor calculus formalism to present a unified calculation of well-known results for spin-1/2 (Dirac) and spin-1 (Maxwell) fields. We then use our unified framework to derive new results for the spin-3/2 and spin-2 fields, which turn out to be the same as those for spin-1/2 and spin-1. This is part of a broader conclusion that there are only two different Casimir forces for perfectly reflecting plates—one associated with fermions and the other with bosons.     

THE CASIMIR FORCE BETWEEN POLARIZERS

The Casimir force between two parallel lincar polarizers oriented at an arbitrary angle is calculated. We compare two approaches, one by solving the field mode function in three-dimensional space with highly anisotropic dielectric slabs, the other by calculating the interaction between dipolcs induced by vacuum fluctuations in these slabs.     

卡西米尔力

量子电动力学中的卡西米尔力是真空零点能的体现.广义的卡西米尔力则依赖于涨落介质的类型广泛地出现于物理中,包括量子,临界,戈德斯通模,以及非平衡卡西米尔力.长程关联的涨落介质和约束是产生卡西米尔力的两个条件.本文通过回顾卡西米尔物理的发展,讨论了不同类型的卡西米尔力,几种正规化方法,并对卡西米尔物理的进一步发展做了展望.     

On a theory of the Casimir effect

We develop a mathematically precise framework for the Casimir effect. A major role is played by Dietz's idea of identifying the Casimir energy as the regularization-independent Ramanujan sum of an asymptotic series. As an illustration, we treat two cases: parallel plates and the sphere. We finally discuss the open problem of the Casimir force for the cube. We propose an Ansatz for the exterior force and argue why it may provide the exact solution, as well as an explanation of the repulsive sign of the force.     

Zeta functions and the Casimir energy

We use zeta function techniques to give a finite definition for the Casimir energy of an arbitrary ultrastatic spacetime with or without boundaries. We find that the Casimir energy is intimately related to, but not identical to, the one-loop effective energy. We show that in general the Casimir energy depends on a normalization scale. This phenomenon has relevance to applications of the Casimir energy in bag models of QCD.Within the framework of Kaluza-Klein theories we discuss the one-loop corrections to the induced cosmological and Newton constants in terms of a Casimir like effect. We can calculate the dependence of these constants on the radius of the compact dimensions, without having to resort to detailed calculations.     

Casimir force on amplifying bodies

Based on a unified approach to macroscopic QED that allows for the inclusion of amplification in a limited space and frequency range, we study the Casimir force as a Lorentz force on an arbitrary partially amplifying system of linearly locally responding (isotropic) magnetoelectric bodies. We demonstrate that the force on a weakly polarisable/magnetisable amplifying object in the presence of a purely absorbing environment can be expressed as a sum over the Casimir-Polder forces on the excited atoms inside the body. As an example, the resonant force between a plate consisting of a dilute gas of excited atoms and a perfect mirror is calculated.     

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.     

Experiment and theory in the Casimir effect

Casimir effect is the attractive force which acts between two plane parallel, closely spaced, uncharged, metallic plates in vacuum. This phenomenon was predicted theoretically in 1948 and reliably investigated experimentally only in recent years. In fact, the Casimir force is similar to the familiar van der Waals force in the case of relatively large separations when the relativistic effects come into play. We review the most important experiments on measuring the Casimir force by means of torsion pendulum, atomic force microscope and micromechanical torsional oscillator. Special attention is paid to the puzzle of the thermal Casimir force, i.e. to the apparent violation of the third law of thermodynamics when the Lifshitz theory of dispersion forces is applied to real metals. Thereafter we discuss the role of the Casimir force in nanosystems including the stiction phenomenon, actuators, and interaction of hydrogen atoms with carbon nanotubes. The applications of the Casimir effect for constraining predictions of extra-dimensional unification schemes and other physics beyond the standard model are also considered.     

Optical detection of the Casimir force between macroscopic objects

We report the optical detection of mechanical deformation of a macroscopic object induced by the Casimir force. An adaptive holographic interferometer based on a photorefractive BaTiO3:Co crystal was used to measure periodical nonlinear deformations of a thin pellicle caused by an oscillating Casimir force. A reasonable agreement between the experimental and calculated values of the first and second harmonics of the Casimir force oscillations has been obtained.     

Finite-temperature Casimir effect in piston geometry and its classical limit

We consider the Casimir force acting on a d-dimensional rectangular piston due to a massless scalar field with periodic, Dirichlet and Neumann boundary conditions and an electromagnetic field with perfect electric-conductor and perfect magnetic-conductor boundary conditions. The Casimir energy in a rectangular cavity is derived using the cut-off method. It is shown that the divergent part of the Casimir energy does not contribute to the Casimir force acting on the piston, thus renders an unambiguously defined Casimir force acting on the piston. At any temperature, it is found that the Casimir force acting on the piston increases from −∞ to 0 when the separation a between the piston and the opposite wall increases from 0 to ∞. This implies that the Casimir force is always an attractive force pulling the piston towards the closer wall, and the magnitude of the force gets larger as the separation a gets smaller. Explicit exact expressions for the Casimir force for small and large plate separations and for low and high temperatures are computed. The limits of the Casimir force acting on the piston when some pairs of transversal plates are large are also derived. An interesting result regarding the influence of temperature is that in contrast to the conventional result that the leading term of the Casimir force acting on a wall of a rectangular cavity at high temperature is the Stefan–Boltzmann (or black-body radiation) term which is of order T ^d+1, it is found that the contributions of this term from the two regions separating the piston cancel with each other in the case of piston. The high-temperature leading-order term of the Casimir force acting on the piston is of order T, which shows that the Casimir force has a nontrivial classical ℏ→0 limit. Explicit formulas for the classical limit are computed.     

Casimir Force at a Finite Cut-Off

We calculate the Casimir force at a finite cut-off Λ by summing the forces induced by the all fluctuation modes. We show that the Casimir force is independent of the cut-off function in the limit LΛ → ∞. There is a correction in the order of (LΛ)^-2, when LΛ is finite and large. This correction becomes remarkable when L is comparable with the microscopic length scale Λ^-1. It has been demonstrated that the Casimir force at a finite cut-off should be defined by summing forces of all fluctuation modes, instead of the derivative of Casimir energy with respect to L where an additional derivative of the cut-off function has been introduced.     

Casimir Force in a One-Dimensional Cavity with Quasimode

We calculate the Casimir force between a perfect reflective wall and a semitransparent wall in the laser cavity. Using the Fox-Li quasimode theory to describe the electromagnetic field in the laser cavity, the vacuum energy and the Casimir force are calculated. We compare our results to the force in the ideal situation and find it smaller in the dissipative cavity. We also find that the Casimir force decreases with the increase of the wall-wall distance and the decay rate of the quasimodes in the laser cavity.