The finite-temperature Casimir effect for a dilute ball satisfying ϵµ = 1

Abstract

The finite temperature Casimir free energy is calculated for a dielectric ball of radius a embedded in an infinite medium. The condition ǫµ = 1 is assumed for the inside/outside regions. Both the Green function method and the mode summation method are considered, and found to be equivalent. For a dilute medium we find, assuming a simple "square" dispersion relation with an abrupt cutoff at imaginary frequencyω = ω0, the high temperature Casimir free energy to be negative and proportional to x0 ≡ ω0a. Also, a physically more realistic dispersion relation involving spatial dispersion is considered, and is shown to lead to comparable results.

Casimir effect for a dilute dielectric ball at finite temperature

“…Usually the transition to finite temperature in calculations of vacuum energy is accomplished by substituting the integration over imaginary frequencies by summation over the discrete Matsubara frequencies in the integral representation for the Casimir energy at zero temperature [16,21,22]. However, following this way one should control how many times the integration by parts in the integral expression at hand has been done [23,24].…”

“…(2.12) in [9]. Thus proceeding from the well justified equations for the free energy (2.2) and for the internal energy (2.6) at finite temperature, one can escape necessity to solve the problem: which energy (free or internal) is obtained on the substitution (3.10) in the initial integral representation for the Casimir energy at zero temperature [16,24].…”

“…It is sufficient to calculate the contribution of the W 2 l term alone and then to change the sign of this contribution to the opposite one. Hence 16) and only the term proportional to ∆n 2 should be preserved in this expression. By making use of the addition theorem for the Bessel functions [28] the sum over the angular momentum l in Eqs.…”

“…The calculation of the vacuum electromagnetic energy of a compact ball with the same velocity of light inside the ball and in the surrounding medium proves to be not more involved. In papers by Brevik and co-authors [3,[12][13][14][15][16] this problem has been studied at zero temperature and at finite temperature, as well as with allowance for dispersion (see also Refs. [17][18][19][20]).…”

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

“…This is of importance since we consider ǫ depending on k. We mention that these representations hold in the presence of arbitrary frequency dispersion, as has been noted in [5] (see also [6]). …”