The Casimir Effect

After a review of the standard calculation of the Casimir force between two metallic plates at zero and non-zero temperatures, we present the study of microscopic models to determine the large-distance asymptotic force in the high-temperature regime. Casimir's conducting plates are modelized by plasmas of interacting charges at temperature T. The charges are either classical, or quantum-mechanical and coupled to a (classical) radiation field. In these models, the force obtained is twice weaker than that arising from standard treatments neglecting the microscopic charge fluctutations inside the bodies. The enforcement of inert boundary conditions on the field in the usual calculations turns out to be inadequate in this regime. Other aspects of dispersion forces are also reviewed. The status of (non-retarded) van der Waals-London forces in a dilute medium of non-zero temperature and density is investigated. In a proper scaling regime called the atomic limit (high dilution and low temperature), one is able to give the exact large-distance atomic correlations up to exponentially small terms as T->0. Retarded van der Waals forces and forces between dielectric bodies are also reviewed. Finally, the Casimir effect in critical phenomena is addressed by considering the free Bose gas. It is shown that the grand-canonical potential of the gas in a slab at the critical value of the chemical potential has finite size corrections of the standard Casimir type. They can be attributed to the existence of long-range order generated by gapless excitations in the phase with broken continuous symmetry.Comment: Lecture notes prepared for the proceedings of the 1st Warsaw School of Statistical Physics, Kazimierz, Poland, June 2005. To appear in Acta Physica Polonica (2006). 52 pages, 0 figures. Available at http://th-www.if.uj.edu.pl/acta/vol37/pdf/v37p2503.pd

The Casimir force at high temperature

The standard expression of the high-temperature Casimir force between perfect conductors is obtained by imposing macroscopic boundary conditions on the electromagnetic field at metallic interfaces. This force is twice larger than that computed in microscopic classical models allowing for charge fluctuations inside the conductors. We present a direct computation of the force between two quantum plasma slabs in the framework of non relativistic quantum electrodynamics including quantum and thermal fluctuations of both matter and field. In the semi-classical regime, the asymptotic force at large slab separation is identical to that found in the above purely classical models, which is therefore the right result. We conclude that when calculating the Casimir force at non-zero temperature, fluctuations inside the conductors can not be ignored.Comment: 7 pages, 0 figure

Microscopic theory of the Casimir force at thermal equilibrium: large-separation asymptotics

We present an entirely microscopic calculation of the Casimir force $f(d)$ between two metallic plates in the limit of large separation $d$. The models of metals consist of mobile quantum charges in thermal equilibrium with the photon field at positive temperature $T$. Fluctuations of all degrees of freedom, matter and field, are treated according to the principles of quantum electrodynamics and statistical physics without recourse to approximations or intermediate assumptions. Our main result is the correctness of the asymptotic universal formula f(d) \sim -\frac{\zeta(3) \kB T}{8\pi d^3}, $d\to\infty$. This supports the fact that, in the framework of Lifshitz' theory of electromagnetic fluctuations, transverse electric modes do not contribute in this regime. Moreover the microscopic origin of universality is seen to rely on perfect screening sum rules that hold in great generality for conducting media.Comment: 34 pages, 0 figures. New version includes restructured intro and minor typos correcte

Thermal quantum electrodynamics of nonrelativistic charged fluids

The theory relevant to the study of matter in equilibrium with the radiation field is thermal quantum electrodynamics (TQED). We present a formulation of the theory, suitable for non relativistic fluids, based on a joint functional integral representation of matter and field variables. In this formalism cluster expansion techniques of classical statistical mechanics become operative. They provide an alternative to the usual Feynman diagrammatics in many-body problems which is not perturbative with respect to the coupling constant. As an application we show that the effective Coulomb interaction between quantum charges is partially screened by thermalized photons at large distances. More precisely one observes an exact cancellation of the dipolar electric part of the interaction, so that the asymptotic particle density correlation is now determined by relativistic effects. It has still the $r^{-6}$ decay typical for quantum charges, but with an amplitude strongly reduced by a relativistic factor.Comment: 32 pages, 0 figures. 2nd versio

Atom-wall dispersive forces: a microscopic approach

We present a study of atom-wall interactions in non-relativistic quantum electrodynamics by functional integral methods. The Feynman-Kac path integral representation is generalized to the case when the particle interacts with a radiation field, providing an additional effective potential that contains all the interactions induced by the field. We show how one can retrieve the standard van der Waals, Casimir-Polder and classical Lifshiftz forces in this formalism for an atom in its ground state. Moreover, when electrostatic interactions are screened in the medium, we find low temperature corrections that are not included in the Lifshitz theory of fluctuating forces and are opposite to them.Comment: 4 figure

Screening of classical Casimir forces by electrolytes in semi-infinite geometries

We study the electrostatic Casimir effect and related phenomena in equilibrium statistical mechanics of classical (non-quantum) charged fluids. The prototype model consists of two identical dielectric slabs in empty space (the pure Casimir effect) or in the presence of an electrolyte between the slabs. In the latter case, it is generally believed that the long-ranged Casimir force due to thermal fluctuations in the slabs is screened by the electrolyte into some residual short-ranged force. The screening mechanism is based on a "separation hypothesis": thermal fluctuations of the electrostatic field in the slabs can be treated separately from the pure image effects of the "inert" slabs on the electrolyte particles. In this paper, by using a phenomenological approach under certain conditions, the separation hypothesis is shown to be valid. The phenomenology is tested on a microscopic model in which the conducting slabs and the electrolyte are modelled by the symmetric Coulomb gases of point-like charges with different particle fugacities. The model is solved in the high-temperature Debye-H\"uckel limit (in two and three dimensions) and at the free fermion point of the Thirring representation of the two-dimensional Coulomb gas. The Debye-H\"uckel theory of a Coulomb gas between dielectric walls is also solved.Comment: 25 pages, 2 figure

Analytical and Numerical Demonstration of How the Drude Dispersive Model Satisfies Nernst's Theorem for the Casimir Entropy

In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersive model, assuming the relaxation is nonzero at zero temperature (which is the case when impurities are present), gives consistent results for the Casimir free energy at low temperatures. Specifically, we find that the free energy consists essentially of two terms, one leading term proportional to T^2, and a next term proportional to T^{5/2}. Both these terms give rise to zero Casimir entropy as T -> 0, thus in accordance with Nernst's theorem.Comment: 11 pages, 4 figures; minor changes in the discussion. Contribution to the QFEXT07 proceedings; matches version to be published in J. Phys.