Abstract
The zeta function regularization technique is used to study the Casimir effect for a scalar field of mass $m$ satisfying Dirichlet boundary conditions on a spherical surface of radius $a$. In the case of large scalar mass, $ma\gg1$, simple analytic expressions are obtained for the zeta function and Casimir energy of the scalar field when it is confined inside the spherical surface, and when it is confined outside the spherical surface. In both cases the Casimir energy is exact up to order $a^{-2}m^{-1}$ and contains the expected divergencies, which can be eliminated using the well established renormalization procedure for the spherical Casimir effect. The case of a scalar field present in both the interior and exterior region is also examined and, for $ma\gg 1$, the zeta function, the Casimir energy, and the Casimir force are obtained. The obtained Casimir energy and force are exact up to order $a^{-2}m^{-1}$ and $a^{-3}m^{-1}$ respectively. In this scenario both energy and force are finite and do not need to be renormalized, and the force is found to produce an outward pressure on the spherical surface.
Highlights
The electromagnetic Casimir effect was first predicted theoretically by H
While this paper investigates scalar fields within the context of a spherical geometry, the techniques that will be used in this paper can be extended to the case of other types of fields, such as fermions satisfying bag boundary conditions on the sphere
I obtain, using ζ int (s) from Section 3, the large mass limit of the zeta function and Casimir energy in the case of a scalar field confined outside the spherical surface
Summary
The electromagnetic Casimir effect was first predicted theoretically by H. In this manuscript I use the zeta function technique to study the spherical Casimir effect for a scalar field of mass m and, without using the heat kernel expansion, obtain simple analytic forms for the zeta function and Casimir energy when the scalar field is confined inside or outside the spherical surface, in the case of large scalar mass (ma 1) In both cases the Casimir energy is found to be divergent, as expected [15] [16]. I obtain, using ζ int (s) from Section 3, the large mass limit of the zeta function and Casimir energy in the case of a scalar field confined outside the spherical surface.
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