Kimball A. Milton

The Casimir Effect: Physical Manifestations of Zero-Point Energy ∗

Zero-point fluctuations in quantum fields give rise to observable forces between material bodies, the so-called Casimir forces. In these lectures I present the theory of the Casimir effect, primarily formulated in terms of Green’s functions. There is an intimate relation between the Casimir effect and van der Waals forces. Applications to conductors and dielectric bodies of various shapes will be given for the cases of scalar, electromagnetic, and fermionic fields. The dimensional dependence of the effect will be described. Finally, we ask the question: Is there a connection between the Casimir effect and the phenomenon of sonoluminescence?

A new perturbative approach to nonlinear problems

A recently proposed perturbative technique for quantum field theory consists of replacing nonlinear terms in the Lagrangian such as φ4 by (φ2)1+δ and then treating δ as a small parameter. It is shown here that the same approach gives excellent results when applied to difficult nonlinear differential equations such as the Lane–Emden, Thomas–Fermi, Blasius, and Duffing equations.

Casimir effect in dielectrics

We reconsider the Casimir (van der Waals) forces between dielectrics with plane, parallel surfaces for arbitrary temperature, using the methods of source theory. The general results of Lifshitz are confirmed, and are shown to imply the correct forces on metal surfaces. The same phenomena give rise to contributions to the surface tension and the latent heat of an idealized liquid, contributions which, unfortunately, are not well defined since they depend upon a momentum cutoff. However, with a reasonable value for this cutoff, qualitative agreement with the experimentally observed surface tension and latent heat of liquid helium at absolute zero is obtained.

Repulsive Casimir and Casimir–Polder forces

Casimir and Casimir?Polder repulsions have been known for more than 50 years. The general ?Lifshitz? configuration of parallel semi-infinite dielectric slabs permits repulsion if they are separated by a dielectric fluid that has a value of permittivity that is intermediate between those of the dielectric slabs. This was indirectly confirmed in the 1970s, and more directly by Capasso?s group recently. It has also been known for many years that electrically and magnetically polarizable bodies can experience a repulsive quantum vacuum force. More amenable to practical application are situations where repulsion could be achieved between ordinary conducting and dielectric bodies in vacuum. The status of the field of Casimir repulsion with emphasis on some recent developments will be surveyed. Here, stress will be placed on analytic developments, especially on Casimir?Polder (CP) interactions between anisotropically polarizable atoms, and CP interactions between anisotropic atoms and bodies that also exhibit anisotropy, either because of anisotropic constituents, or because of geometry. Repulsion occurs for wedge-shaped and cylindrical conductors, provided the geometry is sufficiently asymmetric, that is, either the wedge is sufficiently sharp or the atom is sufficiently far from the cylinder.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker?s 75th birthday devoted to ?Applications of zeta functions and other spectral functions in mathematics and physics?.

Casimir self-stress on a perfectly conducting spherical shell☆

Abstract The Casimir stress on a perfectly conducting uncharged sphere, due to occurrence of fluctuations in the electromagnetic field, is calculated using a source theory formulation. Two independent methods are employed: we compute (1) the total Casimir energy inside and outside the sphere, and (2) the radial component of the stress tensor on the surface. It is necessary to exercise care in allowing the field points to overlap; a correct limiting procedure supplies a “cutoff” in the frequency integration. In spite of numerous technical improvements, the result of Boyer, that the self-stress is repulsive (and not attractive as Casimir hoped), is confirmed unambiguously. The magnitude of the Casimir energy of a sphere of radius a is found, by numerical and analytic techniques, to be E = ( h c 2a )(0.09235) , also in agreement with the very recent result of Balian and Duplantier.

The Casimir effect: recent controversies and progress

The phenomena implied by the existence of quantum vacuum fluctuations, grouped under the title of the Casimir effect, are reviewed, with emphasis on new results discovered in the past four years. The Casimir force between parallel plates is rederived as the strong-coupling limit of δ-function potential planes. The role of surface divergences is clarified. A summary of effects relevant to measurements of the Casimir force between real materials is given, starting from a geometrical optics derivation of the Lifshitz formula, and including a rederivation of the Casimir–Polder forces. A great deal of attention is given to the recent controversy concerning temperature corrections to the Casimir force between real metal surfaces. A summary of new improvements to the proximity force approximation is given, followed by a synopsis of the current experimental situation. New results on Casimir self-stress are reported, again based on δ-function potentials. Progress in understanding divergences in the self-stress of dielectric bodies is described, in particular the status of a continuing calculation of the self-stress of a dielectric cylinder. Casimir effects for solitons, and the status of the so-called dynamical Casimir effect, are summarized. The possibilities of understanding dark energy, strongly constrained by both cosmological and terrestrial experiments, in terms of quantum fluctuations are discussed. Throughout, the centrality of quantum vacuum energy in fundamental physics is emphasized.

Theoretical and experimental status of magnetic monopoles

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 and 2004 results from an experiment (Fermilab E882) searching for real magnetically charged particles bound to elements from the CDF and D0 detectors were reported. The strongest direct experimental limits, from the CDF collaboration, have been reported in 2005. Less strong, but complementary, limits from the H1 collaboration at HERA were reported in the same year. Interpretation of these experiments also require new developments in theory. Earlier experimental and observational constraints on point-like (Dirac) and non-Abelian monopoles were given from the 1970s through the 1990s, with occasional short-lived positive evidence for such exotic particles reported. The status of the experimental limits on monopole masses will be reported, as well as the limitation of the theory of magnetic charge at present.

Scalar Casimir effect for a D-dimensional sphere.

The Casimir stress on a [ital D]-dimensional sphere (the stress on a sphere is equal to the Casimir force per unit area multiplied by the area of the sphere) due to the confinement of a massless scalar field is computed as a function of [ital D], where [ital D] is a continuous variable that ranges from [minus][infinity] to [infinity]. The dependence of the stress on the dimension is obtained using a simple and straightforward Greens function technique. We find that the Casimir stress vanishes as [ital D][r arrow]+[infinity] ([ital D] is a noneven integer) and also vanishes when [ital D] is a negative even integer. The stress has simple poles at positive even integer values of [ital D].

Equivalence of a Complex PT-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly

In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian PT -symmetric wrong-sign quartic Hamiltonian H = 1 p 2 − gx 4 has the same spectrum as the conventional Hermitian Hamiltonian ˜ H = 1 p 2 + 4gx 4 − √ 2g x. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian PT -symmetric Hamiltonian. This anomaly in the Hermitian form of a PT -symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into H. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to −φ 4 quantum field theory in higher-dimensional space-time are discussed.

Analytic perturbation theory in QCD and Schwinger{close_quote}s connection between the {beta} function and the spectral density

We argue that a technique called analytic perturbation theory leads to a well-defined method for analytically continuing the running coupling constant from the spacelike to the timelike region, which allows us to give a self-consistent definition of the running coupling constant for timelike momentum. The corresponding {beta} function is proportional to the spectral density, which confirms a hypothesis due to Schwinger. {copyright} {ital 1997} {ital The American Physical Society}