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Exact and Numerical Methods in Casimir Calculations K A Milton based on collaboration with K V Shajesh P Parashar J Wagner supported by NSF and DOE Oklahoma Center for High Energy Physics and H L Dodge Department of Physics and Astronomy University of Oklahoma TAMU August 6 2007 p 1 43 Introduction Recently there has been a flurry of papers concerning exact methods of calculating Casimir energies or forces between arbitrary distinct bodies Most notable is the recent paper by Emig Graham Jaffe and Kardar Casimir forces between arbitrary compact objects arXiv 0707 1862 cond mat stat mech TAMU August 6 2007 p 2 43 This paper has spawned responses noting that the methods are not so novel Duplantier stating that the idea was explicit in his famous papers with Balian 1977 Barton pointing out precursor in Sommerfeld 1909 and most explicitly the appearance of a earlier drafted paper by Kenneth and Klich Casimir forces in a T operator approach arXiv 0707 4017 quant ph TAMU August 6 2007 p 3 43 Green s Function Approach We agree with these critiques as to novelty of the formulation and note that indeed the derivation of the chief result of Emig et al is much more general than that given in their paper In fact it is a consequence of the general formula for Casimir energies for simplicity here we restrict attention to a massless scalar field is the infinite time that the configuration exists Schwinger 1975 i E Tr ln G 2 TAMU August 6 2007 p 4 43 where G is the Green s function satisfying matrix notation 2 V G 1 subject to some boundary conditions at infinity TAMU August 6 2007 p 5 43 Derivation Start from the vacuum amplitude in terms of sources h0 0 iK eiW K Z 1 dx dx K x G x x K x W K 2 From this the effective field is Z x dx G x x K x TAMU August 6 2007 p 6 43 If the geometry of the region is altered slightly as through moving one of the bounding surfaces the vacuum amplitude is altered Z 1 W K dx dx K x G x x K x 2 Z 1 dx dx x G 1 x x x 2 GG 1 1 TAMU August 6 2007 p 7 43



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