Heat Kernel Coefficients and Divergencies of the Casimir Energy for the Dispersive Sphere

The first heat kernel coefficients are calculated for a dispersive ball whose permittivity at high frequency differs from unity by inverse powers of the frequency. The corresponding divergent part of the vacuum energy of the electromagnetic field is given and ultraviolet divergencies are seen to be present. Also in a model where the number of atoms is fixed the pressure exhibits infinities. As a consequence, the ground-state energy for a dispersive dielectric ball cannot be interpreted easily. The ground-state energy for a dielectric ball shows ultraviolet divergencies still lacking physical understanding. This is an unsatisfactory situation, not only for general reasons but also in view of the rapid experimental progress. The canonical way to investigate the ultraviolet divergencies is to calculate the corresponding heat kernel coefficients. For the dielectric (nondispersive) ball this had been done in [1] and for the dielectric cylinder in [2], where it had been shown, for instance, that the coefficient a2 is zero in dilute order and nonzero beyond. In the present note we calculate the relevant heat kernel coefficients for a dielectric ball with dispersion. Dispersion means a frequency dependent permittivity, (ω). This is motivated by the expectation that for high frequency the permittivity tends to unity so ∗e-mail: [email protected] †e-mail: [email protected]