Casimir effect at nonzero temperature for wedges and cylinders Simen ˚ A. Ellingsen ∗ and Iver Brevik † Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Kimball A. Milton ‡ Oklahoma Center for High Energy Physics and Homer L. Dodge Department of Physics and Astronomy, The University of Oklahoma, Norman, OK 73019, USA (Dated: October 25, 2018) We consider the Casimir-Helmholtz free energy at nonzero temperature T for a circular cylinder and perfectly conducting wedge closed by a cylindrical arc, either perfectly conducting or isorefrac- tive. The energy expression at nonzero temperature may be regularized to obtain a finite value, except for a singular corner term in the case of the wedge which is present also at zero temperature. Assuming the medium in the interior of the cylinder or wedge be nondispersive with refractive index n , the temperature dependence enters only through the non-dimensional parameter 2 πnaT , a being the radius of the cylinder or cylindrical arc. We show explicitly that the known zero temperature result is regained in the limit aT → 0 and that previously derived high temperature asymptotics for the cylindrical shell are reproduced exactly. PACS numbers: 42.50.Lc, 11.10.Wx, 11.10.Gh, 42.50.Pq arXiv:1001.2630v1 [hep-th] 15 Jan 2010 I. INTRODUCTION The Casimir effect [1] is the name given to energies and forces due to field fluctuations in the presence of boundaries. Once a theoretical curiosity, the effect has gained enormous and still increasing attention since its first quantitative measurement a good decade ago [2]. Reviews of recent progress include Refs. [3–5]. The first geometry, considered in Casimir’s classic pa- per [1] was that of two perfectly conducting plates, gen- eralized to arbitrary dispersive materials by Lifshitz [6]. The force between parallel plates of any purely dielec- tric material is attractive, and it was therefore surprising when it was shown by Boyer that the Casimir stress on a perfectly conducting spherical shell is repulsive [7]. While it was clear from Boyer’s result that the Casimir effect has a strong geometry dependence, results for new geometries were slow in coming for a long time, and it was only in 1981 that DeRaad and Milton calculated the Casimir energy for a circularly cylindrical shell [8]. Since then a number of analytical efforts have added to the knowledge of Casimir effect in cylindrical cavities, both perfectly conducting [9–12] and (magneto)dielectric [13– 18]. Most treatments of the cylindrical geometry have dealt with the zero temperature situation, and only a few calculations have concerned finite temperature [19– 21], and in these references only the high-temperature asymptotics were derived. No analytical expression valid for all temperatures exists for the cylindrical geometry to our knowledge. A related geometry is the wedge. First considered with $^{∗}$Electronic address: simen.a.ellingsen@ntnu.no $^{†}$Electronic address: iver.h.brevik@ntnu.no $^{‡}$Electronic address: milton@nhn.ou.edu where the physical range is p ∈ [ 1 $^{$_{2}$,}$∞ ), but which we will in general allow to take any real positive value. Through- out our calculations we set c = ℏ = k$_{B}$ = 1. It was shown [34] that ˆ E could be rendered finite provided the arc be- come transparent at high frequencies. The calculations in [34] were extended to the first con- sideration of a wedge which is not perfectly conducting but instead assumed to be isorefractive (diaphanous), i.e., spatially uniform speed of light [35]. In that case where ˜ E 0 is a finite, regularizable energy closely analo- gous to that found for a cylinder [8, 10], whereas ˆ E is a divergent term associated with the corners where the arc meets the wedge. Here and in the following we will make frequent use of the symbol p = π/α, (1.2) E 0 = ˜ E $_{0}$( p ) + ˆ E , (1.1) We recently revisited the latter geometry to calculate the energy, at zero temperature, of a perfectly conducting wedge closed by a cylindrical boundary, either perfectly conducting or magnetodielectric [34]. We showed how that energy could be written on the form (subscript 0 indicates zero temperature) respect to Casimir effect in the 1970s [22, 23], it has been the subject of several treatments later [24–28]. The ge- ometry is inviting in that it is analytically solvable and contains the geometries of parallel plates and a single semi-infinite plate as limiting cases. The geometry of a wedge intercut by a cylindrical shell was considered by Nesterenko and co-workers [29, 30] and energy densities in the same geometry were calculated by Saharian and co-workers [31–33]. We are not aware of any previous efforts to tackle the Casimir energy problem for a wedge at non-zero temperature.