International Journal of Modern Physics A

The precision instrumental developments and the modern unification theories using compact dimensions have motivated a resurgence in the field of Casimir force measurements. Here, after a brief discussion of the history of Casimir force measurements, the experimental results of Casimir force measurements using an Atomic Force Microscope will be presented. Precision measurements of the normal Casimir force, demonstration of the shape dependent Casimir force and the lateral Casimir force will be presented.

We consider Yukawa-type forces inspired by extra dimensional physics and by exchange of light elementary particles. Constraints on these forces following from the recent measurements of the Casimir force are analysed. We show that the experiments with the Casimir force gave the possibility to strengthen the previously known constraints up to several thousand times in a wide interaction range. Further strengthening is expected in near future.

The Tevatron has inspired new interest in the subject of magnetic monopoles. First there was the 1998 D0 limit on the virtual production of monopoles, based on the theory of Ginzburg and collaborators. In 2000 the first results from an experiment (Fermilab E882) searching for real magnetically charged particles bound to elements from the CDF and D0 detectors were reported. This also required new developments in theory. The status of the experimental limits on monopole masses will be discussed, as well as the limitation of the theory of magnetic charge at present.

The situation with the temperature corrections to the Casimir force between real metals of finite conducitivity is reported. It is shown that the plasma dielectric function is well adapted to the Lifshitz formula and leads to reasonable results for real conductors. The Drude dielectric function which describes media with dissipation is found not to belong to the application range of the Lifshitz formula at nonzero temperature. For Drude metals the special modification of the zero-frequency term of this formula is suggested. The contradictory results on the subject in recent literature are analysed and explained.

When comparing experimental results with theoretical predictions of the Casimir force, the accuracy of the theory is as important as the precision of experiments. Here we evaluate the Casimir force when finite conductivity of the reflectors and finite temperature are simultaneously taken into account. We show that these two corrections are correlated, i.e. that they can not, in principle, be evaluated separately and simply multiplied. We estimate the correlation factor which measures the deviation from this common approximation. We focus our attention on the case of smooth and plane plates with a metallic optical response modeled by a plasma model.

The Casimir energies of single bodies (as opposed to the interaction between mutually disjoint bodies) have accumulated deceptive folklore which this talk will try to exorcise, by mean of calculations for atomic solids that, though optically dilute, are realistically dispersive. This is easy, because quantum electrodynamics then yields identically the same energy as one gets from the properly retarded interatomic potentials. The problem of regularizing (nominal) divergences turns out to be quite distinct from the appropriate process of renormalization: simply discarding all nominally divergent contributions would prevent one from understanding the physics. Contrary to legend, the pertinent Casimir energies for dielectric spheres are attractive.

The Casimir mutual free energy F for a system of two dielectric concentric nonmagnetic spherical bodies is calculated, at arbitrary temperatures. Whereas F has recently been evaluated for the special case of metals (refractive index n = ∞), here analogous results are presented for dielectrics, and shown graphically when n = 2.0. Our calculational method relies upon quantum statistical mechanics. The Debye expansions for the Riccati-Bessel functions when carried out to a high order are found to be very useful in practice (thereby overflow/underflow problems are easily avoided), and also to give accurate results even for the lowest values of l. Another virtue of the Debye expansions is that the limiting case of metals becomes quite amenable to an analytical treatment in spherical geometry. We first discuss the zero-frequency TE mode problem from a mathematical viewpoint and then, as physical input, invoke the actual dispersion relations. The result of our analysis, based upon adoption of the Drude dispersion relation as the most correct one at low frequencies, is that the zero-frequency TE mode does not contribute for a metal. Accordingly, F turns out in this case to be only one half of the conventional value.

The Casimir energy of a dilute homogeneous nonmagnetic dielectric ball at zero temperature is derived analytically within a microscopic realistic model of dielectrics for an arbitrary physically possible frequency dispersion of dielectric permittivity ε(iω). Divergences are absent in calculations, a minimum interatomic distance λ is a physical cut-off. Casimir surface force is proved to be a attractive. A physical definition of the Casimir energy is discussed.

The basic results in calculations of the thermodynamic functions of electromagnetic field in the background of a dilute dielectric ball at zero and finite temperature are presented. Summation over the angular momentum values is accomplished in a closed form by making use of the addition theorem for the relevant Bessel functions. The behavior of the thermodynamic characteristics in the low and high temperature limits is investigated. The T³-term in the low temperature expansion of the free energy is recovered (this term has been lost in our previous calculations).

We compute the non-retarded dispersive force between an electrically polarizable atom and a magnetically polarizable one. We show, in agreement with Feinberg and Sucher's result, that while the retarded force is proportional to 1/r⁸, the non-retarded one is proportional to 1/r⁵. We give a physical interpretation for this rather unexpected result.

The Casimir force between inhomogeneous slabs that exhibit a band-like structure is calculated. The slabs are made of basic unit cells each made of two layers of different materials. As the number of unit cells increases the Casimir force between the slabs changes, since the reflectivity develops a band-like structure characterized by frequency regions of high reflectivity. This is also evident in the difference of the local density of states between free and boundary distorted vacuum, that becomes maximum at frequencies corresponding to the band gaps. The calculations are restricted to vacuum modes with wave vectors perpendicular to the slabs.

Vacuum fluctuations produce a force acting on a rigid Casimir cavity in a weak gravitational field. Such a force is here evaluated and is found to have opposite direction with respect to the gravitational acceleration; the order of magnitude for a multi-layer cavity configuration is analyzed and experimental detection is discussed, bearing in mind the current technological resources.

We study the density of Casimir energy in dielectric and magnetic media. To do so we derive Born series for the Green's function of the electromagnetic field in a medium with arbitrary ∊ and μ. Within this framework the case of uniform velocity of light (∊μ = const) is studied. As an application we consider the Casimir energy of a thick dielectric-diamagnetic shell, as a function of the radii and the permeabilities. We show that there is a range of parameters in which the stress on the outer shell is inward, and a range where the stress on the outer shell is outward.

The first heat kernel coefficients are calculated for a dispersive ball whose permittivity at high frequency differs from unity by inverse powers of the frequency. The corresponding divergent part of the vacuum energy of the electromagnetic field is given and ultraviolet divergencies are seen to be present. Also in a model where the number of atoms is fixed the pressure exhibits infinities. As a consequence, the ground-state energy for a dispersive dielectric ball cannot be interpreted easily.

We use the substitution m² → M²(x) = m² + ξR(x) to construct a new form of the proper-time expansion of the Feynman propagator and the corresponding form of the renormalization counterterms < ϕ²>_DS and < T_μν >_DS for the scalar field.

In the present talk we briefly demonstrate an elegant and effective technique for calculation of the trace expansion in the derivatives of background fields. One of main advantages of the technique is manifestly (super)symmetrical and gauge invariant form of expressions on all stages of calculations. Other advantage is the universality of the calculation method. It shows itself to be good for calculation of one-loop effective action in several well-known models and new results in supersymmetrical models has already been found.

In the context of a model of space-time foam, made by N wormholes we discuss the possibility of having a foam formed by different configurations. An equivalence between Schwarzschild and Schwarzschild-Anti-de Sitter wormholes in terms of Casimir energy is shown. An argument to discriminate which configuration could represent a foamy vacuum coming from Schwarzschild black hole transition frequencies is used. The case of a positive cosmological constant is also discussed. Finally, a discussion involving charged wormholes leads to the conclusion that they cannot be used to represent a ground state of the foamy type.

We investigate the vacuum state of a charged scalar field coupled to two different electromagnetic (EM) backgrounds on two different spaces.

(i) is first defined on cylindrical space E² × S¹ with compact direction z ∈ S¹ having circumference L₃. This spatial compactification enables (but does not compel) an EM vacuum density j₃ to flow around the spatial cylinder, parallel to z. However, a nonzero gauge potential A₃ prefers one direction around z over the other, causing j₃ to become nonzero. We analyze this situation for A₃ = A₃ (y) an arbitrary function of coordinate y, with A_0,1,2 = 0.

(ii) is next defined on Minkowski spacetime in the presence of a background EM plane wave of arbitrary shape and amplitude. As the plane wave sweeps through the charged virtual particle sea at the speed of light it locally excites or churns the sea, making the vacuum nonstationary (without producing vacuum pairs). This nonstationary excitation of the vacuum is nondissipative. A summary of recent calculations is presented.

These two examples clearly display that, in gauge theory, potentials are more fundamental then fields.

We present a framework for the study of one–loop quantum corrections to extended field configurations in renormalizable quantum field theories. We work in the continuum, transforming the standard Casimir sum over modes into a sum over bound states and an integral over scattering states weighted by the density of states. We express the density of states in terms of phase shifts, allowing us to extract divergences by identifying Born approximations to the phase shifts with low order Feynman diagrams. Once isolated in Feynman diagrams, the divergences are canceled against standard counterterms. Thus regulated, the Casimir sum is highly convergent and amenable to numerical computation. Our methods have numerous applications to the theory of solitons, membranes, and quantum field theories in strong external fields or subject to boundary conditions.

We calculate the self-energy for an electrically charged particle at rest in the background of Gott-Hiscock cosmic string space-time. The self-energy continuously falls down out-ward from string's with maximum at the origin of the string. The plots of the numerical of the self-energy and the height of barrier are shown.

A role of the renormalization group invariance in calculations of the ground state energy for models with confined fermion fields is discussed. The case of the (1+1)D MIT bag model with the massive fermions is studied in detail.

In this paper we calculate the thermal Green functions associated with massless scalar and spin-1/2 fields in the pointlike global monopole spacetime. The thermal average values of their respective energy-momentum tensors are formally presented and explicit calculations are performed for specific values of the parameter α which codify the presence of the monopole. Moreover their high-temperature limit behaviors are explicitly exhibited by numerical analysis.

We solve for the spectrum and eigenfunctions of Dirac operator on the sphere. The igenvalues are nonzero whole numbers. The eigenfunctions are two-component spinors which may be classified by representations of the SU(2) group with half-integer angular momenta. They are special linear combinations of conventional spherical spinors.

The induced fermion number at zero temperature is topological (in the sense that it is only sensitive to global asymptotic properties of the background field), and is a sharp observable (in the sense that it has vanishing rms fluctuations). At finite temperature, it is shown to be generically nontopological, and it is not a sharp observable.

We study the spectrum of stable static fermion bags in the 1 + 1 dimensional Gross-Neveu model with N flavors of Dirac fermions, in the large N limit. In the process, we discover a new kink, heavier than the Callan-Coleman-Gross-Zee (CCGZ) kink, which is marginally stable (at least in the large N limit). The connection of this new kink and the conjectured S matrix of the Gross-Neveu model is obscured at this point. After identifying all stable static fermion bags, we arrange them into a periodic table, according to their O(2N) and topological quantum numbers.

The properties of plasmons, Nambu-Goldstone bosons and gapless Carlson-Goldman collective modes in color-flavor locked phase of color superconducting dense quark matter at finite temperature are reviewed. A possibility of a kaon condensation with an abnormal number of the NG bosons is also discussed.

Time-dependent solution of field equations for the 3-d gauge field model with a Chern-Simons (CS) topological mass is found. After Lorentz boost, this solution describes a massive nonlinear non-abelian plane wave. The CS-massive gauge fields interacting with Higgs fields are demonstrated to display stochastic character. Effective action for an SU(2) 4-d gauge model with a vortex is calculated in the 1-loop approximation. The minimum of the effective potential is found.

Casimir energy for solid conducting ball is considered on the base of some finite models. One model is physical and built of a battery of parallel metallic plates. Two finite models are based on the Higgs model of superconductivity. One of them is supersymmetric and based on the Witten field model for superconducting strings. Treatment shows that contribution of Casimir energy can be very essential for superdence state in the neutron stars and nuclear matter.

Yang–Mills theory in four dimensions is studied by using the Coulomb gauge. The Coulomb gauge Hamiltonian involves integration of matrix elements of an operator built from the Laplacian and from a first-order differential operator. The operator is studied from the point of view of spectral theory of pseudo-differential operators on compact Riemannian manifolds, both when self-adjointness holds and when it is not fulfilled. In both cases, well-defined matrix elements of are evaluated as a first step towards the more difficult problems of quantized Yang–Mills theory.

The current status of bounds on and limits of fermion determinants in two, three and four dimensions in QED and QCD is reviewed. A new lower bound on the two-dimensional QED determinant is derived. An outline of the demonstration of the continuity of this determinant at zero mass when the background magnetic field flux is zero is also given.

We consider the mechanism of obtaining compatible Poisson-Lie tensors, based on the existence of some special algebraic structures - families of Lie brackets defined on one and the same space.

Some remarks on determinant line bundles and topological invariants are given. The index of the twisted Dirac operator acting on real hyperbolic manifolds is computed. We briefly discuss physical applications of results.

We present a simple closed formula for the two-loop scalar QED Euler-Heisenberg Lagrangian in a Euclidean self-dual field. The use of Borel techniques for the construction of Euler-Heisenberg Lagrangians is shortly discussed.

The application of world-line techniques to calculation of the vacuum polarization and effective action in scalar and spinor QED with external arbitrary constant electromagnetic field configuration is presented.

A numerical technique for calculating effective actions of electromagnetic backgrounds is proposed, which is based on the string-inspired worldline formalism. As examples, we consider scalar electrodynamics in three and four dimensions to one-loop order. Beyond the constant-magnetic-field case, we analyze a step-function-like magnetic field exhibiting a nonlocal and nonperturbative phenomenon: "magnetic-field diffusion". Finally, generalizations to fermionic loops and systems at finite temperature are discussed.

We describe some specific quantum black hole model. It is pointed out that the origin of a black hole entropy is the very process of quantum gravitational collapse. The quantum black hole mass spectrum is extracted from the mass spectrum of the gravitating source. The classical analog of quantum black hole is constructed.

As shown recently 2d quantum gravity theories — including spherically reduced Einstein-gravity — after an exact path integral of its geometric part can be treated perturbatively in the loops of (scalar) matter. Obviously the classical mechanism of black hole formation should be contained in the tree approximation of the theory. This is shown to be the case for the scattering of two scalars through an intermediate state which by its effective black hole mass is identified as a "virtual black hole". We discuss the lowest order tree vertex for minimally and non-minimally coupled scalars and find a non-trivial finite S-matrix for gravitational s-wave scattering in the latter case.

We summarize recent evidence supporting the conjecture that four-dimensional Quantum Einstein Gravity (QEG) is nonperturbatively renormalizable along the lines of Weinberg's asymptotic safety scenario. This would mean that QEG is mathematically consistent and predictive even at arbitrarily small length scales below the Planck length. For a truncated version of the exact flow equation of the effective average action we establish the existence of a non-Gaussian renormalization group fixed point which is suitable for the construction of a nonperturbative infinite cutoff-limit. The cosmological implications of this fixed point are discussed, and it is argued that QEG might solve the horizon and flatness problem of standard cosmology without an inflationary period.

We study the decoherence process associated with the scattering of stochastic gravitational waves. We discuss the case of macroscopic systems, such as the planetary motion of the Moon around the Earth, for which gravitational scattering is found to dominate decoherence though it has a negligible influence on damping. This contrast is due to the very high effective temperature of the background of gravitational waves in our galactic environment.

Static solutions with a finite curvature at the horizon may exist in dilaton gravity at temperatures T ≠ T_H (including T = 0) where T_H is the Hawking temperature. In so doing, regular geometries are compatible with infinite quantum stresses. This phenomenon cannot be obtained within the perturbative scheme. It occurs in the weak-coupling regime between gravitation and dilaton and enables us to obtain well-defined entropy of extremal horizons S = 0 on the semiclassical level.

The question is examined of a mirror which starts from rest and either (i) accelerates for some time and eventually reverts to motion at constant velocity, (ii) continues accelerating forever. A sharp distinction in made between cases (i) and (ii) concerning the spectrum of the emitted radiation, and the qualitative difference between the two cases is pointed out. The Bogolubov coefficients are calculated for a trajectory of type (i). A type (ii) trajectory is entirely unphysical as far as any realistic mirror is concerned, however it is of interest in that it has been used as a simple analog of black hole collapse. The spectrum emitted for the type (ii) trajectory z = -ln (cosh t) is examined and it is shown that it is indeed that of a black body. Inconsistencies in previous derivations of the above result are pointed out.

We investigate the influence of the field-theoretical divergence on the thermal equilibrium property by employing the model of a uniformly accelerated monopole detector nonlinearly coupled to a scalar field in the flat space with the Dirichlet boundary condition. Introducing the Gaussian switching factor to the interaction between the detector and the field, we estimate the correction to the response function coming from this swithing factor. We find that the deviation from the thermal equilibrium of the response of this detector is mostly due to the interplay between the divergence and the existence of the boundary, when the detector is kept switched on during a large but finite amount of time.

The symmetry between the creation of pairs of massless bosons or fermions by accelerated mirror in 1+1 space and the emission of single photons or scalar quanta by electric or scalar charge in 3+1 space is deepened in this paper. The relation of Bogoliubov coefficients, describing the processes generated by the mirror, with Fourier's components of current or charge density leads to the coicidence of the spin of any disturbances bilinear in scalar or spinor field with the spin of quanta emitted by the electric or scalar charge. The mass and invariant momentum transfer of these disturbances are essential for the relation of Bogoliubov coefficients with invariant singular solutions and Green's functions of wave equations both for 1+1 and 3+1 spaces and especially for the integral relation (20) between these solutions. Namely the last relation leads to the coincidence of the self-action changes and vacuum-vacuum amplitudes for the accelerated mirror in two-dimentional space-time and charge in four-dimentional space-time. Thus, both invariants of the Lorentz group, spin and mass, perform intrinsic role in established symmetry.

The nonconformal scalar field is considered in N-dimensional space-time with metric which includes, in particular, the cases of nonhomogeneous spaces and anisotropic spaces of Bianchi type-I. The modified Hamiltonian is constructed. Under the diagonalization of it the energy of quasiparticles is equal to the oscillator frequency of the wave equation. The density of particles created by nonstationary metric is investigated. It is shown that the densities of conformal and nonconformal particles created in Friedmann radiative-dominant universe coincide.

Aharonov-Bohm solenoid changes the energy spectrum of charge particles in pure magnetic field. In particular, the degeneracy with respect to azimuthal quantum number l is partially lifted. In turn, this complicates the radiation spectrum of a charged particle in magnetic field in the presence of the solenoid (Aharonov-Bohm effect). In particular, the degeneracy of the radiation intensity with respect to the azimuthal quantum number is lifted completely. In the present work we study l-dependence (induced by Aharonov-Bohm solenoid) of synchrotron radiation intensity in semiclassical approximation.

We discuss the relativistic spin- particles in pseudo-classical models involving anticommuting Grassmann variables for the spin degrees of freedom. The constants of motion are expressed in terms of Killing vectors and Killing-Yano tensors. Passing from the spinning spaces to the Dirac equation in curved backgrounds we point out the role of the Killing-Yano tensors in the construction of the Dirac-type operators. The general results are applied to the case of the four-dimensional Euclidean Taub-NUT space.

Using the real time formalism of the finite temperature field theory we calculate the 1-loop polarization tensor in the presence of a background magnetic field in a medium. The expression is obtained to linear order in the background field strength. We discuss the Faraday rotation as well as the photon absorption probabilities in this context.

The inner crust of neutron stars consists of nuclei of various shapes immersed in a neutron gas and stabilized by the Coulomb interaction in the form of a crystal lattice. The scattering of neutrons on nuclear inhomegeneities leads to the quantum correction to the total energy of the system. This correction resemble the Casimir energy and turn out to have a large influence on the structure of the crust.