Notes

(1) The ``antiperiodic'' or ``twisted'' condition is also consistent.

(2) One of the many formal derivations of this result is given in Sect. 7. So long as non-linear (self) interactions are absent [9], the equal weighting of modes would also follow from a classical field with random amplitude and phases constrained to have a Lorentz invariant spectrum [10]. The normal ordering used to remove disconnected vacuum diagrams in the usual formulation of quantum field theory applies only in an infinite volume: different normal ordering prescriptions must be used in finite volumes, allowing Casimir effects.

(3) If the boundary conditions are charge-conjugation invariant, any density of quantum numbers must cancel between particle and antiparticle modes. Energy is the unique quantity which is positive for both positive and negative frequency modes.

(4) This is analogous to the fact that supersymmetry is broken in finite temperature systems because boson fields satisfy periodic boundary conditions in imaginary time, while fermion fields satisfy antiperiodic ones [16]. Both boson and fermion fields may, however, satisfy periodic boundary conditions in space.

(5) The third form given is an example of the Walfisz formula discussed in [19].

(6) Accurate numerical evaluation of the necessary sums is easily achieved by a direct summation in which progressively larger sets of terms are averaged together as increases.

(7) Fields in equilibrium at finite temperature satisfy such constraints; the energy density of a (noninteracting) field at finite temperature is correspondingly given by the spin multiplicity factor times the result for a scalar field at the same temperature.

(8) Solutions nevertheless exist in a sphere [23].

(9) In the usual case of superconductors in four dimensions, this condition becomes , . In space dimensions, define a generalized electric field where is a space index . To avoid infinite currents in the conductor, all components of not along must vanish. In the conductor, so that ; for all directions orthogonal to . These constraints imply condition (4.2).

(10) The result for , agrees with that obtained in [7] with momentum cutoff regularization. The speculation of [13] based on the scalar field calculation that this result should be different in dimensional or zeta function regularization is unfounded: it was implicitly based on incorrect counting of electromagnetic field modes.

(11) In three dimensions this becomes , , as appropriate at the surface of a material containing infinitely mobile magnetic monopoles.

(12) The modes of a massive vector field in a cuboidal cavity are complicated when correct boundary conditions are imposed [25]: the sum necessary to obtain the Casimir energy cannot be performed analytically.

(13) Finite results require the regularization parameter to be taken independent of the size of the system: if a cutoff on mode number is used, divergences may remain [27].

(14) Interactions within a collection of positive and negative classical electric charges contributes a negative energy, but this negative energy is always overwhelmed by positive self energy of the charges. Similarly, the gravitational binding energy of a classical collection of masses is apparently always overwhelmed by kinetic energy, yielding a positive total energy. Similar compensation may occur for quantum mechanical Casimir energy.

(15) The lowest energy state in quantum gravity might nevertheless consist of an assembly of small (almost) closed universes with negative Casimir energies.

(16) As discussed in Section 6, however, the ``walls'' of such systems may nevertheless compensate their negative energy to yield a net positive energy which respects bound (8.1).

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