M. Bordag

New developments in the Casimir effect

Abstract We provide a review of both new experimental and theoretical developments in the Casimir effect. The Casimir effect results from the alteration by the boundaries of the zero-point electromagnetic energy. Unique to the Casimir force is its strong dependence on shape, switching from attractive to repulsive as function of the size, geometry and topology of the boundary. Thus, the Casimir force is a direct manifestation of the boundary dependence of quantum vacuum. We discuss in depth the general structure of the infinities in the field theory which are removed by a combination of zeta-functional regularization and heat kernel expansion. Different representations for the regularized vacuum energy are given. The Casimir energies and forces in a number of configurations of interest to applications are calculated. We stress the development of the Casimir force for real media including effects of nonzero temperature, finite conductivity of the boundary metal and surface roughness. Also, the combined effect of these important factors is investigated in detail on the basis of condensed matter physics and quantum field theory at nonzero temperature. The experiments on measuring the Casimir force are also reviewed, starting first with the older measurements and finishing with a detailed presentation of modern precision experiments. The latter are accurately compared with the theoretical results for real media. At the end of the review we provide the most recent constraints on the corrections to Newtonian gravitational law and other hypothetical long-range interactions at submillimeter range obtained from the Casimir force measurements.

Advances in the Casimir effect

1. Introduction I: PHYSICAL AND MATHEMATICAL FOUNDATIONS OF THE CASIMIR EFFECT FOR IDEAL BOUNDARIES 2. Simple models of the Casimir effect 3. Field quantization and vacuum energy in the presence of boundaries 4. Regularization and renormalization of the vacuum energy 5. The Casimir effect at nonzero temperature 6. Approximate and numerical approaches to the Casimir effect 7. The Casimir effect for two ideal metal planes 8. The Casimir effect in rectangular boxes 9. Single spherical and cylindrical boundaries 10. The Casimir force between objects of arbitrary shape 11. Spaces with non-Euclidean topology II: THE CASIMIR FORCE BETWEEN REAL BODIES 12. The Lifshitz theory of van der Waals and Casimir forces between plane dielectrics 13. The Casimir interaction between plates made of real metals at zero temperature 14. The Casimir interaction between real metals at nonzero temperature 15. The Casimir interaction between metal and dielectric 16. The Lifshitz theory of atom-wall interaction 17. The Casimir force between rough and corrugated surfaces III: MEASUREMENTS OF THE CASIMIR FORCE AND THEIR APPLICATIONS IN BOTH FUNDAMENTAL PHYSICS AND NANOTECHNOLOGY 18. General requirements for Casimir force measurements 19. Measurements of the Casimir force between equals 20. Measurements of the Casimir force with semiconductors 21. Measurements of the Casimir force in configurations with corrugated surfaces 22. Measurement of the Casimir-Polder force 23. Applications of the Casimir force in nanotechnology 24. Constraints on hypothetical interactions from the Casimir effect 25. Conclusions and outlook

Heat kernel coefficients of the Laplace operator on the D-dimensional ball

We present a very quick and powerful method for the calculation of heat kernel coefficients. It makes use of rather common ideas, as integral representations of the spectral sum, Mellin transforms, non‐trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane. We apply our method to the case of the heat kernel expansion of the Laplace operator on a D‐dimensional ball with either Dirichlet, Neumann or, in general, Robin boundary conditions. The final formulas are quite simple. Using this case as an example, we illustrate in detail our scheme —which serves for the calculation of an (in principle) arbitrary number of heat kernel coefficients in any situation when the basis functions are known. We provide a complete list of new results for the coefficients B3,..., B10, corresponding to the D‐dimensional ball with all the mentioned boundary conditions and D=3,4,5.

Heat-kernels and functional determinants on the generalized cone

We consider zeta functions and heat-kernel expansions on the bounded, generalized cone in arbitrary dimensions using an improved calculational technique. The specific case of a global monopole is analysed in detail and some restrictions thereby placed on the

Casimir energies for massive scalar fields in a spherical geometry

A_{5/2}

Casimir effect for a sphere and a cylinder in front of a plane and corrections to the proximity force theorem

coefficient. The computation of functional determinants is also addressed. General formulas are given and known results are incidentally, and rapidly, reproduced.We consider zeta functions and heat-kernel expansions on the bounded, generalized cone in arbitrary dimensions using an improved calculational technique. The specific case of a global monopole is analysed in detail and some restrictions thereby placed on theA5/2 coefficient. The computation of functional determinants is also addressed. General formulas are given and known results are incidentally, and rapidly, reproduced.

Casimir energy for a massive fermionic quantum field with a spherical boundary

The Casimir energy corresponding to a massive scalar field with Dirichlet boundary conditions on a spherical bag is obtained. The field is considered, separately, inside and outside the bag. The renormalization procedure that is necessary to apply in each situation is studied in detail, in particular the differences occurring with respect to the case when the field occupies the whole space. The final result contains several constants that experience renormalization and can be determined only experimentally. The non-trivial finite parts that appear in the massive case are found exactly, providing a precise determination of the complete, renormalized zero-point energy for the first time. PACS: 02.30.+g, 02.40.+m hep-th/9608071 E-mail address: bordag@qft.physik.uni-leipzig.d400.de E-mail address: eli@zeta.ecm.ub.es E-mail address: kirsten@tph100.physik.uni-leipzig.de E-mail address: lese@zeta.ecm.ub.es 1

Quantum field theoretic treatment of the Casimir effect

Using a path integral approach we rederive a recently found representation of the Casimir energy for a sphere and a cylinder in front of a plane and derive the first correction to the proximity force theorem.

Heat kernels and zeta-function regularization for the mass of the supersymmetric kink

The vacuum energies corresponding to massive Dirac fields with the boundary conditions of the MIT bag model are obtained. The calculations are carried out with the fields occupying the regions inside and outside the bag, separately. The renormalization procedure for each of the situations is studied in detail, in particular the differences occurring with respect to the case when the field extends over the whole space. The final result contains several constants that undergo renormalization and can be determined experimentally only. The non-trivial finite parts which appear in the massive case are found exactly, providing a precise determination of the complete, renormalized zero-point energy in the fermionic case. The vacuum energy behaves as an inverse power of the mass, for large mass of the field.The vacuum energies corresponding to massive Dirac fields with the boundary conditions of the MIT bag model are obtained. The calculations are done with the fields occupying the regions inside and outside the bag, separately. The renormalization procedure for each of the situations is studied in detail, in particular the differences occurring with respect to the case when the field extends over the whole space. The final result contains several constants undergoing renormalization, which can be determined only experimentally. The non-trivial finite parts which appear in the massive case are found exactly, providing a precise determination of the complete, renormalized zero-point energy for the first time, in the fermionic case. The vacuum energy behaves like inverse powers of the mass for large masses. PACS: 11.10.Gh, 02.30.-f Running title: Casimir energy in the MIT bag E-mail address: eli@zeta.ecm.ub.es, elizalde@io.ieec.fcr.es E-mail address: michael.bordag@itp.uni-leipzig.de E-mail address: klaus.kirsten@itp.uni-leipzig.de 2 1

Zeta function determinant of the Laplace operator on the D-dimensional ball

A nontrivial quantum field theoretical treatment of the Casimir effect demands the quantization of spinor electrodynamics with boundary conditions. The boundary conditions are realized by two super conducting infinitely thin parallel plates. As a technical tool we use the path integral method. It is shown that in perturbation theoretical calculations the standard Feynman rules remain valid up to a modification of the photon propagator. One advantage of our procedure is the derivation of a closed expression for this modified photon propagator in a covariant gauge which allows the explicit calculation of loop diagrams. Up to the order e2 we determine the radiative correction for vacuum energy density expression and the Casimir force. It turns out that the distance dependent part of the energy density and thereby the Casimir force is ultraviolet finite. An explicit value is obtained in the limit of large distances (in comparison with the Compton wave length).