2. Casimir Energy in a Simple System

This section describes in some detail the calculation of the Casimir energy for the simple case of a non-interacting scalar field with mass in -dimensional space with one direction of finite length . The field satisfies the free Klein-Gordon equation

away from any boundaries. One form of constraint on the field is achieved by introduction of explicit boundaries consisting of -dimensional hyperplanes located at and . For points on these boundaries, the field obeys either

or

or a linear combination of these (Robin). The Neumann condition (2.2b) is analogous to ``bag'' boundary conditions, and implies that the momentum flux of the field through the boundaries vanishes. Instead of introducing physical boundaries, one may require the periodicity condition (1) (compactification of to )

We consider first the case of Dirichlet boundary conditions. The modes of the field are then

where is a positive integer. In the ground state (vacuum) each of these modes contributes an energy . (2) For normalization purposes, we take the transverse coordinates to be restricted by where . The total energy of the field between the planes is thus given by (3)

High modes render this sum formally divergent. The contributions of such modes should however be independent of . They should thus cancel in calculations of forces, or in comparisons of the energy density of the field in the presence and the absence of the planes. The sum in Eq. (2.5) may be regularized by a variety of techniques. The simplest is by analytic continuation in . Using the result (e.g. [11])

(2.5) becomes

The relation

then yields

We consider first the case . The necessary sum is then formally given by

where is the usual Riemann function. Using the reflection formula (e.g. [12])

together with the reduplication formula

the energy (2.9) may be written

This result is finite for all positive , and is always negative. The force per unit area on the boundary plates is attractive for any value of and of magnitude

The pressure of the vacuum between the plates is thus negative. In the limit , (2.13) gives the energy of the field in the absence of the plates. The regularization by analytic continuation in used here sets this energy to zero. Special cases of the result (2.13) are

For small ; the dimensionless coefficient in decreases for intermediate , but tends to as , taking on a minimum value of at .

An alternative scheme for regularizing the sum (2.9) is to impose an (exponential) cutoff in longitudinal momentum, as for electromagnetic modes in material media (e.g. [3]). Writing , the analogue of Eq. (2.10) is

where is the Lerch transcendent [12]. When this reduces to (2.10), except for a singular term whose dependence on is cancelled by the factor in (2.9). As mentioned above, terms independent of have no physical significance, and may differ between regularization schemes. Identical -dependent terms are always obtained in momentum-cutoff and dimensional continuation schemes. Each term in the sum (2.16) is positive, and the result is correspondingly positive. However, the physically relevant -dependent piece in the energy need not be positive.

The cutoff in longitudinal momentum should be distinguished from a cutoff at a fixed mode quantum number . The latter scheme cannot be implemented through properties of the boundaries. If used, it would yield a divergent -dependent energy.

Another method consists in writing the sum as . Analytic continuation of the resulting function then yields a form identical to (2.10) [13,14].

Equation (2.5) is for fields satisfying Bose-Einstein statistics. The sign is reversed for the case of Fermi-Dirac statistics. When equal numbers of fermion and boson states exist (as in supersymmetric models), the leading divergent contributions to (2.5) cancel. The total vacuum energy of an infinite volume vanishes when corresponding boson and fermion states have equal masses (for the divergent part alone to cancel, only the sums of the and need be equal [15]). The total Casimir energy in a finite volume will not vanish except with special boundary conditions (such as the periodic ones discussed below). (4) (With bag boundary conditions, or in a spherical Einstein universe [17], the total Casimir energy does not vanish.)

The sum (2.5) directly gives the energy for a field which exists in a cavity, and vanishes outside. The field could instead exist everywhere, but vanish on two thin plates. It is convenient for regularization to enclose such a system in a large cavity. The total energy of the system is then a sum of contributions from the resulting three spaces. With dimensional regularization, the energy of the field in the outer spaces goes to zero as their size goes to infinity. With other regularization schemes, a divergent contribution may remain.

We now treat the case of a massive scalar field. The analytic continuation of the sum (2.9) for non-zero is derived in the Appendix. The result for the energy is

where is a modified Bessel function. The first term in brackets gives a contribution to the total energy independent of and is therefore dropped. The first term in parentheses corresponds to a constant energy density and would occur even in the absence of the planes. It is cancelled by addition of a constant to the Hamiltonian density. The finite physically relevant energy is thus (cf. [18])

where is a Jacobi theta function (see Appendix), and is a regular Bessel function. (5) Numerical results for (2.18) (obtained using the second form given) are shown in Fig. 2.1. For small ,

Low modes may be considered responsible for the dependence of the Casimir energy on . When , the energies of these low modes are dominated by and approximately independent of , so that the physical Casimir energy decreases:

As increases, the contribution of transverse momenta become more important, and decreases less rapidly with , as seen in Fig. 2.1.

For the Neumann boundary condition (2.2b), the sin appearing in the modes (2.4) becomes cos, but the final results (2.13) and (2.18) for the physical energy remain the same.

[ Figure 1 ] Casimir energy density for a mass scalar field satisfying Dirichlet boundary conditions on two planes a distance apart in space dimensions.

With the periodicity condition (2.3) the modes of the field become

and the energy in this case is , where denotes the physical energy (2.18) for Dirichlet boundary conditions.

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