The Casimir effect is analyzed. This effect consists of a polarization of the vacuum of quantized fields which arises as a result of a change in the spectrum of vacuum oscillations when the quantization volume is bounded or the topology of the space is non-Euclidean. Calculations of the effect for manifolds of various configurations and for fields with various spins are reported. Various definitions of the Casimir vacuum energy in the presence of walls are discussed. The quantum field theory of Casimir forces is generalized to incorporate the dispersion properties of the medium. Applications of the Casimir effect in various branches of physics are reviewed: from the theory of molecular forces to cosmology and elementary particle physics, including the bag model and supersymmetry.

http://iopscience.iop.org/article/10.1070/PU1988v031n11ABEH005641/pdf

The Casimir Effect and Its Applications

One of the most efficient methods to obtain the vacuum expectation values for the physical observables in the Casimir effect is based on using the Abel-Plana summation formula. This allows us to derive the regularized quantities in a manifestly cutoff independent way and present them in the form of strongly convergent integrals. However, the application of AbelPlana formula, in its usual form, is restricted by simple geometries when the eigenmodes have a simple dependence on quantum numbers. The author generalized the Abel-Plana formula which essentially enlarges its application range. Based on this generalization, formulae have been obtained for various types of series over the zeros of some combinations of Bessel functions and for integrals involving these functions. It has been shown that these results generalize the special cases existing in literature. Further, the derived summation formulae have been used to summarize series arising in the mode summation approach to the Casimir effect for spherically and cylindrically symmetric boundaries. This allows us to extract the divergent parts from the vacuum expectation values for the local physical observables in a manifestly cutoff independent way. The present paper reviews these results. Some new considerations are also added.

http://users.ictp.it/~pub_off/preprints-sources/2007/IC2007082P.pdf

The generalized Abel-Plana formula with applications to Bessel functions and casimir effect

Cosmic rays of the highest energy, above the Greisen-Zatsepin-Kuzmin (GZK) cutoff of the spectrum, may originate in decays of superheavy long-lived particles. We conjecture that these particles may be produced naturally in the early Universe from vacuum fluctuations during inflation and may constitute a considerable fraction of cold dark matter. We predict a new cutoff in the ultrahigh-energy cosmic ray spectrum E
cutoff<m
inflaton≈1013 GeV, the exact position of the cutoff and the shape of the cosmic ray spectrum beyond the GZK cutoff being determined by the QCD quark/gluon fragmentation. The Pierre Auger Project installation may in principle observe this phenomenon.

Ultrahigh-energy cosmic rays, superheavy long-lived particles, and matter creation after inflation

We examine critically the persistent belief that the physical interpretation of a quantum field theory in curved space-time is to be found by introducing a notion of particles at each time in terms of the exact eigenstates of some energylike operator at that time. It seems that all straightforward implementations of this principle suffer from either ambiguity or failure to produce finite particle densities and renormalizable energy-momentum tensors. Calculations are done for a cosmological model of Bianchi type I, containing a scalar fieldφ whose equation includes the term ξRφ. Diagonalization of thetotal energy suggests a certain particle definition, which, however, yields a finite particle creation rate only when ξ=1/6 and the universe is isotropic. Diagonalization of acanonical Hamiltonian is a prescription loose enough to be made consistent with almost any ansatz (including one that does yield physically acceptable expectation values), hence of no theoretical value. An apparently successful naive approach to the isotropic case (for any ξ) is explained as an unconscious first approximation to an adiabatic analysis of the notion of positive frequency.

Remarks on positive frequency and Hamiltonians in expanding universes.

The standard Friedmann-Robertson-Walker cosmological model which is widely accepted now provides a good description of the modern state of our Universe, but it encounters a fundamental difficulty, namely, the existence of a singularity at some finite moment of time t = 0 in the past. The Friedman solution cannot be analytically continued to the region t < 0 where, strictly speaking, there is no space and no time. So, in the scope of classical general relativity, it is in principle impossible to ask what was the state of our Universe before t = 0. Thus a bound to the human cognition is set here. The consideration of more general inhomogeneous and anisotropic cosmological models does not improve the situation because, as is well known, these models are also singular.1

S. G. Mamaev

Papers

Particle Creation from Vacuum Near an Homogeneous Isotropic Singularity

On field theory in the friedman space

Particle creation from the vacuum near a homogeneous isotropic singularity

Method of dimensional regularization for scalar and vector fields in homogeneous isotropic spaces

Quantum effects in intense external fields