_{1}

^{*}

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, , 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 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 , the zeta function, the Casimir energy, and the Casimir force are obtained. The obtained Casimir energy and force are exact up to order and 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.

The electromagnetic Casimir effect was first predicted theoretically by H. G. B. Casimir [

Since their discovery, Casimir forces have been found to have many applications from nanotechnology to string theory, and a large effort has gone into studying the generalization of the Casimir effect to quantum fields other than the electromagnetic field: fermions were first considered by Johnson [

It is well known that Casimir forces are very sensitive to the boundary conditions of the quantum fields at the plates. In the case of scalar fields, Dirichlet and Neuman boundary conditions are most commonly used, in the case of fermion fields or other fields with spin [

Massive or massless scalar fields appear in many areas of physics from the Higgs field in the Standard Model, to the dilaton field that breaks the conformal symmetry in string theory, to the Ginzburg-Landau scalar field in superconductivity, etc.

The Casimir effect due to a scalar field has been studied extensively in the parallel plate and spherical geometry. Different regularization techniques have been used to remove the singularities of the Casimir energy such as, for example, the zeta function technique and the Casimir piston technique. While in the context of this work I will use the zeta function technique, the Casimir piston technique [

The spherical Casimir effect for massless [

In Section 2, I describe the model and, for the case of a scalar field confined inside the spherical surface, obtain the zeta function

In 3-dimensional space the equation of motion of a scalar field,

where m is the scalar field mass. Using spherical coordinates, this equation becomes

where

the radial part of Equation (1) is found to be

A complete set of solutions of Equation (2), finite at the origin, is

where

we find

where

and, when the scalar field is confined inside the spherical surface, the zeta function is given by

where

Since

where the closed contour

where

logarithm does not change the result, since no additional pole is enclosed. A simple change of the integration variable allows me to rewrite Equation (4) as

and to exploit the Debye uniform asymptotic expansion of the modified Bessel functions [

where

and

I use Equation (6) and find

where the

and are polynomials of degree 3i

while the coefficients

with

and, for

Equation (8) displays the same feature as Equation (7): as N grows the sum on the right side becomes a more accurate approximation of the zeta function. Notice that the coefficients

In this section I evaluate the

and therefore, in the large mass limit, I find

Similarly

when

After I change the integration variable from z to

and integrate over the new variable y, I obtain

and, for

The integrals over

and, for

where

is the Hurwitz zeta function.

The Casimir energy for a massive scalar field confined inside a spherical surface of radius a, is given by

where

for

Mascheroni constant, and where I neglected all terms of order

where I used

the contribution of

where the

The appearance of divergencies in the calculation of

where

and h do not have names. The quantum part of the system under consideration is a scalar field satisfying Dirichlet boundary condition on the spherical surface. The ground state energy of this scalar field,

efficients

are F and h, since

If the scalar field is confined outside the spherical surface, the zeta function is [

and can be used to calculate

exact to order

with the

terms as

Finally, I discuss the situation where the scalar field is present in both the interior and exterior regions. In this case the Casimir energy is

and, using Equations (22) and (24), I find a finite value for the large mass limit of E

again different from what appears in the literature [

and I find a repulsive force

indicating an outward pressure on the spherical surface, that vanishes as

energy E and force F that I find in Equations (25) and (26), are exact to order

In this manuscript I used the zeta function regularization technique to study the spherical Casimir effect of a massive scalar field in

one valid outside, which are exact to order

gencies, as I expected, and can be renormalized following the renormalization procedure described in Ref. [

Finally, I studied the case of a scalar field present both inside and outside the spherical surface, and obtained the large mass limit of the Casimir energy (25) and force (26) in this case. Both quantities are finite and thus do

not need to be renormalized, and are exact to order

For a scalar field with mass

AndreaErdas, (2015) Spherical Casimir Effect for a Massive Scalar Field on the Three Dimensional Ball. Journal of Modern Physics,06,1104-1112. doi: 10.4236/jmp.2015.68115