3. Casimir Energy In More Complicated Regions

This section derives the Casimir energy for a scalar field in a general hypercuboidal region, with sides of finite length and sides with length .

The simplest results are obtained by taking the field periodic in the finite length directions (case ), corresponding to compactification of space dimensions to a hypertorus . The modes of the field then consist of a simple product of modes analogous to (2.21 ) for each direction. The total energy is obtained by summing separately over the for each set of modes:

The necessary multiple sum may be written in terms of the Epstein zeta function [20,21]

where the prime indicates that the term for which all is to be omitted. This function obeys the reflection formula [20]

analogous to (2.11) and derived in the Appendix. Using (3.3) the Casimir energy (3.1) becomes

This result is again finite for all and . It is always negative.

The asymptotic formulae given in the Appendix show that when one of the lengths becomes much larger than the others, the energy tends exponentially to that obtained with only finite .

When all the some simplification may occur. Figure 3.1 gives a plot of in this case for several values of . (6) The corresponding physical energies for a few values of are listed in the first column of Table I. Notice that if all are constrained to be equal, the Casimir energy is negative and leads to a force which tends to contract the system.

[ Figure 2 ] Values of the function defined in Eq. (3.3), giving the Casimir energy of a massless scalar field constrained to be periodic with unit period along orthogonal directions in -dimensional space.

[ Table 1 ] Casimir Energies Divided by Volume for Massless Scalar () and Vector () Fields in Dimensions Satisfying Periodic (), Dirichlet (), Neumann (), Perfect Conductor (), and ``Bag'' () Boundary Conditions on Each Surface of -Dimensional Unit Hypercubes

Some special values of are [22]:

where and ( is Catalan's constant), . apparently cannot be expressed as a product of one-dimensional sums.

Figure 3.2 shows the energy divided by volume for systems as a function of . In all cases, the minimum energy is achieved when . When the energy tends exponentially to the result obtained for infinite , as mentioned above.

[ Figure 3 ] Casimir energy divided by volume for a massless scalar field in space dimensions constrained to be periodic with periods and along two orthogonal directions.

Figure 3.3 gives contour plots of the energy divided by volume for systems with in the case as a function of and . Again, the minimum energy is achieved in the symmetric configuration . Similar behaviour appears to occur for higher values of . A system of fixed volume containing scalar fields with periodic constraints therefore attains its minimum energy when all are equal.

[ Figure 4 ] Contour plot for the energy divided by volume of a massless scalar field in three space dimensions constrained to have periods and along the three directions. The energy is always negative. Regions shaded lighter correspond to more negative energies.

The analogue of Eq. (3.4) for a massive scalar field is

Notice that as in Eq. (2.17) above, the first term here corresponds to a constant energy density throughout the volume, and may be cancelled by a constant term in the Hamiltonian density. The remaining physical energy is always negative.

Imposing only periodicity constraints, the quantum numbers specifying modes of a scalar field may each run over all integer values. The mode for which all carries zero energy when and is therefore irrelevant. However, with the Dirichlet boundary conditions (2.2a), each quantum number is restricted to the range . Similar, the Neumann condition (2.2b) implies . Using the results

valid for functions even in the , one may write the energies obtained with Dirichlet (D) and Neumann (N) boundary conditions in terms of the result (3.4) as

where the final sum is over distinct sets with all and . Since the are always negative, this result implies that the energy obtained with Neumann boundary conditions is also negative in all cases. With Dirichlet boundary conditions, however, the energy may be positive. In the limit so ( fixed), the sum is dominated by the term in which the smallest of the appears. The limiting energy is infinitely negative, as for the case .

The second two columns of Table I give the Casimir energies of massless scalar fields satisfying Dirichlet and Neumann boundary conditions on the surface of a hypercube (all equal).

As in the periodic case, the Casimir energy for a field satisfying Neumann boundary conditions leads to a force which tends to deform a fixed volume confining region into a cube.

The case of Dirichlet boundary conditions is more complicated. In the minimum energy shape for a particular value of d and a given volume it appears that some number of lengths are small and equal, and the rest are large. At least for when is even, and when is odd. When becomes greater than about 5.7, the energy for the symmetrical case ceases to be positive, and for , the energy for symmetrical becomes lower than that for . Figure 3.4 shows the energy divided by volume for the case as a function of : the development of a minimum at as increases is evident. Figure 3.5 gives a contour plot of the Casimir energies with for the case as a function of and . Again the lowest-energy configuration is the symmetrical one .

[ Figure 5 ] Casimir energy divided by volume for a massless scalar field satisfying Dirichlet boundary conditions on the sides of a rectangular ``tube'' in space dimensions.

[ Figure 6 ] Contour plot for the energy divided by volume of a massless scalar field satisfying Dirichlet boundary conditions on the sides of an , box in three space dimensions. Regions shaded darker correspond to higher energies. The thick contour is at zero energy.

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