On the Casimir Effect for N-dimensional Spheres

Advances in High Energy Physics, 2010

A global approach with cutoff exponential functions is used to obtain the Casimir energy of a massless scalar field in the presence of a spherical shell. The proposed method, mathematically well defined at the outset, makes use of two regulators, one of them to make the sum of the orders of Bessel functions finite and the other to regularize the integral involving the zeros of Bessel function. This procedure ensures a consistent mathematical handling in the calculations of the Casimir energy and allows a major comprehension on the regularization process when nontrivial symmetries are under consideration. In particular, we determine the Casimir energy of a scalar field, showing all kinds of divergences. We consider separately the contributions of the inner and outer regions of a spherical shell and show that the results obtained are in agreement with those known in the literature, and this gives a confirmation for the consistence of the proposed approach. The choice of the scalar field was due to its simplicity in terms of physical quantity spin.

Physics Letters B, 1984

We derive exact expressions for the Casimir interaction energy per unit area between two parallel, perfectly conducting, metal plates, in n (~ 2) dimensions.

1977

A n a l y t i c r e g u l a r i z a t i o n f o r t h e Casimir E f f e c t f o r rectangular systems i n one-, two-and three-dimensions, as w e l l as f o r p a r a l l e l conducting p l a t e s , i s discussed. We consider t h e a n a l y t i c r e g u l a r i z a t i o n by employing t h e Riemann zeta f u n c t i o n as w e l l as t h e zeta f u n c t i o n s introduced by Epstein. The forces, i n t h i s case, come o u t a u t o m a t i c a l l y f i n i t e , i. e., no s u b t r a c t i o n s a r e needed. We show t h a t t h e a n a l y t i c c o n t i n u a t i o n , * Work supported by FINEP, Rio de Janeiro, under c o n t r a c t 356/CT. ** W i t h a f e l l o w s h i p o f FAPESP, São

The original computations deriving the Casimir energy and force consists of first taking limits of the spectral zeta function and afterwards analytically extending the result. This process of computation presents a weakness in Hendrik Casimir's original argument since limit and analytic continuation do not commute. A case of the Laplacian on a parallelepiped box representing the space as the vacuum between two plates modelled with Dirichlet and periodic Neumann boundary conditions is constructed to address this anomaly. It involves the derivation of the regularised zeta function in terms of the Riemann zeta function on the parallelepiped. The values of the Casimir energy and Casimir force obtained from our derivation agree with those of Hendrik Casimir.

Journal of Physics A: Mathematical and General, 1980

ing t h e Riemann zeta f u n c t i o n as w e l l as t h e zeta f u n c t i o n s introduced by Epstein. The forces, i n t h i s case, come o u t a u t o m a t i c a l l y f i n i t e , i. * Work supported by FINEP, Rio de Janeiro, under c o n t r a c t 356/CT.

2015

The European Physical Journal C, 2002

We develop a mathematically precise framework for the Casimir effect. Our working hypothesis, verified in the case of parallel plates, is that only the regularization-independent Ramanujan sum of a given asymptotic series contributes to the Casimir pressure. As an illustration, we treat two cases: parallel plates, identifying a previous cutoff-free version (by G. Scharf and W.W.) as a special case, and the sphere. We finally discuss the open problem of the Casimir force for the cube. We propose an Ansatz for the exterior force and argue why it may provide the exact solution, as well as an explanation of the repulsive sign of the force.

arXiv.org of Cornell University, 2009

A global approach with cut-off exponential functions previously proposed is used to obtain the Casimir energy of a massless scalar field in the presence of a spherical shell. This method makes the use of two regulators, one of them to turn the sum of the orders of Bessel functions finite and a second, to turn the integral involving the zeros of Bessel function regularized. This proposed procedure ensures a consistent mathematical handling in the calculations of the Casimir energy for a scalar field as well it does show all types of divergences of interest. We separately consider the contributions of the inner and outer regions of a spherical shell and we show that the results obtained are in agreement with those known in the literature and this gives a confirmation for the consistence of the proposed approach.

Physical Review D, 1998

A simple method for calculating the Casimir energy for a sphere is developed which is based on a direct mode summation and counter integration in a complex plane of eigenfrequencies. The method uses only classical equations determining the eigenfrequencies of the quantum field under consideration. Efficiency of this approach is demonstrated by calculation of the Casimir energy for a perfectly conducting spherical shell and fora massless scalar field obeying the Dirichlet and Neumann boundary conditions on sphere. The possibility of rationalizing the removal of divergences in this problem as a renormalization of both the energy and the radius of the sphere is discussed.

2005

This survey summarizes briefly results obtained recently in the Casimir energy studies devoted to the following subjects: i) account of the material characteristics of the media in calculations of the vacuum energy (for example, Casimir energy of a dilute dielectric ball); ii) application of the spectral geometry methods for investigating the vacuum energy of quantized fields with the goal to