4. Casimir Energy for Fields with Spin

The mode energies for fields satisfying periodicity constraints are always the same as in the corresponding scalar (spinless) case. (7) Hence the total Casimir energy is given by the scalar field result multiplied by relevant spin multiplicity factors (negative for fermions).

Dirichlet and Neumann boundary conditions for scalar fields have no direct generalization to fields with spin.

The simplest general condition is that the action for a field should vanish outside a specified volume. For massless vector (spin 1) fields, this corresponds to bag boundary conditions, as discussed below. For a massless spinor (spin 1/2) field between two parallel planes this boundary condition yields a Casimir energy: In higher dimensional hyper-cuboids, the presence of corners prevents solutions to the massless Dirac equation with these boundary conditions. (8) Even with , no solutions exist for a massive spinor field [15].

Consistent boundary conditions may be formulated for a massless vector (``photon'') field in space dimensions. The field strength is represented by a totally antisymmetric rank-2 tensor . The dual of this tensor is defined as . The field satisfies the equations

We first consider a cavity with walls of infinite conductivity. The field then satisfies the boundary condition (9)

on each wall with spacelike normal vector . In determining modes of the field, it is convenient to introduce potentials such that

and to work in the ``radiation'' gauge

The modes of the field in a conducting hyper-cuboid with pairs of faces in -dimensional space are then

where gauge condition (4.3) implies (cf. [24])

The modes in (4.4) may have . Their energies are the same as in the scalar case (2.4). Condition (4.5) connects quantum numbers associated with different directions, and forbids modes for which two or more of the vanish. The total Casimir energy may then be written in terms of the scalar field result (3.4, 3.8) as

The Casimir energies obtained in the symmetrical configuration for several cases are given in the fourth column of Table I. (10) For low values of , the energy is positive for odd , and negative otherwise. Figure 4.1 shows the energy divided by volume with as a function of . For , the minimum energy is achieved with ; for larger gives smaller energy. Figure 4.2 gives a contour plot of the energy divided by volume in the case as a function of and . When , gives the maximum energy. The minimum energy is achieved when . In three dimensions, Casimir energy should thus tend to deform closed conducting boxes of fixed volume into long tubes. For , is the minimum energy configuration.

[ Figure 4.1 ] Casimir energy divided by volume for a massless vector field enclosed in a perfectly conducting rectangular ``tube'' in space dimensions.

[ Figure 4.2 ] Contour plot for the energy divided by volume of a massless vector field in a perfectly conducting box in three space dimensions. Regions shaded darker correspond to higher energies. The thick contour is at zero energy.

An alternative to the perfect conductor boundary condition (4.2) is obtained by requiring the action for the vector field to vanish outside a bounded region. Inserting the necessary step function in the action integral, and varying with respect to the potential , one obtains at the boundary the constraint (11)

where is a spacelike normal vector. This boundary condition is assumed in the bag model for hadrons in QCD. Constraint (4.7) is dual to constraint (4.2). For , this implies identical Casimir energies in the two cases. The modes of a field subject to (4.7) are obtained from (4.4) on interchanging sin and cos. The total energy is given in terms of the scalar field result ((3.4) and (3.8)) by

The last column of Table I lists results for this energy with in a variety of cases. The energy with is shown as a function of in Fig. 4.3. Maximum energy appears to be achieved when . The minimum energy for a given volume is attained when .

[ Figure 4.3 ] Casimir energy divided by volume for a massless vector field satisfying ``bag'' boundary conditions on the sides of an ``tube'' in space dimensions.

A massive vector field satisfying the Proca equation in space dimensions has degrees of freedom; the corresponding massless field would have degrees of freedom. One may formally impose boundary conditions (4.2) and (4.7) even on a massive vector field. The results are obtained by replacement of the factors and in Eqs. (4.6) and (4.8) with and , respectively, and with equal to the mass of the vector field. The zero mass limits of the corresponding energies do not coincide with the zero mass results obtained above (and given in Table I). However, as shown in [25], Eq. (4.2) no longer gives the boundary condition on a perfect conductor for massive vector fields. The additional (longitudinal) polarization state for massive vector fields is found to decouple in the limit , and thus presumably contributes no Casimir energy (12) (cf. [26]). The bag boundary condition (4.6), which allows no momentum flux outside a bag, nevertheless maintains its physical significance even for massive vector fields. For the small mass limit of the Casimir energy with this boundary condition is for and for . These results clearly differ from those for a genuinely massless vector field given in Table I.

For massless spin-2 fields, no physical boundary conditions appear to be consistent with the gauge invariance of the Lagrangian.

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