IntroductionGreen's Function ApproachDerivationEffective two-particle sourceAmbiguity in formula$T$-matrixExpression for Green's functionInteraction between two potentialsInteraction in terms of $T_i$ or $G_i$Green's functions for $delta $-platesCasimir interaction between $delta $-platesCasimir EnergyGeneral $lambda $Multipole expansionReduced Green's functionExpression for $g^0_{m,m'}$Discrete matrix realizationNumerics of multipole expansionCapasso MethodEarlier Emig methodApplicationsBulgac et al. methodApplicationsBordag methodApplicationsDalvit methodMultiple-Scattering ApproachComments and Prognosis“Exact” and Numerical Methodsin Casimir CalculationsK. A. Miltonbased on collaboration with K. V. Shajesh, P. Parashar, J. Wagnersupported by NSF and DOEOklahoma Center for High Energy Physics, andH. L. Dodge Department of Physics and AstronomyUniversity of OklahomaTAMU, August 6, 2007 – p.1/43IntroductionRecently, there has been a flurry of pa-pers concerning “exact” methods of calculat-ing Casimir energies or forces between arbi-trary distinct bodies. Most notable is the re-cent paper byEmig, Graham, Jaffe, and Kardar,“Casimir forces between arbitrary compact ob-jects,” arXiv:0707.1862 [cond-mat.stat-mech].TAMU, August 6, 2007 – p.2/43This paper has spawned responses noting thatthe methods are not so novel:Duplantier, stat-ing that the idea was explicit in his famous pa-pers with Balian (1977);Barton, pointing out pre-cursor in Sommerfeld (1909); and most explicitly,the appearance of a earlier drafted paper byKen-neth and Klich, “Casimir forces in a T operator ap-proach.” arXiv:0707.4017 [quant-ph]TAMU, August 6, 2007– p.3/43Green’s Function ApproachWe agree with these critiques, as to novelty ofthe formulation, and note that indeed, thederivation of the chief result of Emig et al. ismuch more general than that given in their paper.In fact, it is a consequence of the general formulafor Casimir energies (for simplicity here werestrict attention to a massless scalar field)(τ isthe “infinite” time that the configurationexists)[Schwinger, 1975]E =i2τTrln G,TAMU, August 6, 2007 – p.4/43where G is the Green’s function satisfying (matrixnotation)(−∂2+ V )G = 1,subject to some boundary conditions at infinity.TAMU, August 6, 2007 – p.5/43DerivationStart from the vacuum amplitude in terms ofsources,h0+|0−iK= eiW [K],W [K] =12Z(dx)(dx′)K(x)G(x, x′)K(x′).From this the effective field isφ(x) =Z(dx′)G(x, x′)K(x′).TAMU, August 6, 2007 – p.6/43If the geometry of the region is altered slightly, asthrough moving one of the bounding surfaces,the vacuum amplitude is altered:δW [K] =12Z(dx)(dx′)K(x)δG(x, x′)K(x′)= −12Z(dx)(dx′)φ(x)δG−1(x, x′)φ(x′),GG−1= 1.TAMU, August 6, 2007– p.7/43Effective two-particle sourceUpon comparison with the two particle term ineiW [K]= eiR(dx)K(x)φ(x)+iR(dx)L= · · · +12iZ(dx)K(x)φ(x)2,we deduceiK(x)K(x′)eff= −δG−1(x, x′).TAMU, August 6, 2007 – p.8/43Thus the change in the generating functional isδW =i2Z(dx)(dx′)G(x, x′)δG−1(x′, x)= −i2Z(dx)(dx′)δG(x, x′)G−1(x′, x).From this, in matrix notationδW = −i2Trln G ⇒ E =i2τTrln G.TAMU, August 6, 2007 – p.9/43Ambiguity in formulaThe above formula for the Casimir energy isdefined up to an infinite constant, which can beat least partially compensated by inserting afactor as do Kenneth and Klich:E =i2τTrln GG−10.Here G0satisfies, with the same boundaryconditions as G, the free equation−∂2G0= 1.TAMU, August 6, 2007– p.10/43T -matrixNow we define the T -matrix,T = S − 1 = V (1 + G0V )−1.The following is just standard scatteringtheory a la Lippmann-Schwinger (1950).There seem to be some sign and orderingerrors in Kenneth & Klich.TAMU, August 6, 2007 – p.11/43Expression for Green’s functionThe Green’s function can be alternatively writtenasG = G0− G0T G0=11 + G0VG0= V−1T G0,which results in two formulæ for the CasimirenergyE =i2τTrln11 + G0V=i2τTrln V−1T.TAMU, August 6, 2007 – p.12/43Interaction between two potentialsIf the potential has two disjoint parts,V = V1+ V2it is easy to show thatT = (V1+V2)(1−G0T1)(1−G0T1G0T2)−1(1−G0T2),whereTi= Vi(1 + G0Vi)−1, i = 1, 2.TAMU, August 6, 2007 – p.13/43Interaction in terms of Tior GiThus, we can write the general expression for theinteraction between the two bodies (potentials) intwo alternative forms:E12= −i2τTrln(1 − G0T1G0T2)= −i2τTrln(1 − V1G1V2G2),whereGi= (1 + G0Vi)−1G0, i = 1, 2.TAMU, August 6, 2007– p.14/43The first form is exactly that given by Emig et al.,and by Kenneth and Klich, while the latter is actu-ally easily used if we know the individual Green’sfunctions.TAMU, August 6, 2007 – p.15/43Green’s functions for δ-platesWe now use the second formula above tocalculate the Casimir energy between twoparallel semitransparent plates, with potentialV = λδ(z − z1) + λδ(z − z2).The free reduced Green’s function isg(z, z′) =12κe−κ|z−z′|, κ2= ζ2+ k2.Here k = k⊥and ζ = −iω is the Euclidean fre-quency.TAMU, August 6, 2007 – p.16/43The Green’s function associated with a singlepotential isgi(z, z′) =12κe−κ|z−z′|−λλ + 2κe−κ|z−zi|e−κ|z′−zi|.TAMU, August 6, 2007 – p.17/43Casimir interaction between δ-platesThen the energy/area is (a = |z2− z1|)E1=116π3ZdζZd2kZdz ln(1 − A)(z, z),A(z, z′) =λ24κ2δ(z − z1)1 −λλ + 2κe−κ|z1−z2|×1 −λλ + 2κe−κ|z′−z2|=λλ + 2κ2e−κae−κ|z′−z2|δ(z − z1).TAMU, August 6, 2007 – p.18/43Casimir EnergyWe expand the logarithm according toln(1 − A) = −∞Xn=1Ann.For example the leading term is easily seen tobe (a = |z2− z1|)E(2)= −λ216π3Zdζ d2k4κ2e−2κa= −λ232π2a,which uses the change to polar coordinates,dζ d2k = dκ κ2dΩ.TAMU, August 6, 2007 – p.19/43General λIn general, it is easy to check that, becauseA(z, z′) factorizes here,A(z, z′) = B(z)C(z′),Tr ln(1 − A) = ln(1 − TrA),so the Casimir interaction between the twosemitransparent plates isE =14π2Z∞0dκ κ2ln 1 −λλ + 2κe−κa2!,which is exactly the well-known result.TAMU, August 6, 2007 – p.20/43Multipole expansionTo proceed to apply this method to generalbodies, Emig et al. revert to an even oldertechnique, the multipole expansion. Let’sillustrate this with a 2 + 1 dimensional version,which allows us to describe cylinders withparallel