# Multiple Scattering Methods in Casimir Calculations

Kimball A. Milton
Jef Wagner
Oklahoma Center for High Energy Physics
and Homer L. Dodge Department of Physics and Astronomy,
University of Oklahoma, Norman, OK 73019, USA
http://www.nhn.ou.edu/

## I 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 Emig:2007cf . (Details, applied to a scalar field, are supplied in Ref.~Emig:2007me . See also Refs.~Emig:2007qw ; sidewalls .) Precursors include an early paper of Renne renne , rederiving the Lifshitz formula lifshitz in this way, the famous papers of Balian and Duplantier Balian:1977qr ; Balian:1976za ; Balian:2004jv , work of Kenneth and Klich Kenneth:2007jk based on the Lippmann-Schwinger formulation of scattering theory lippmann , the papers by Bulgac, Marierski, and Wirzba wirzba07 ; Bulgac:2005ku ; Wirzba:2005zn , who use the modified Krein formula krein , and by Bordag Bordag:2006vc ; Bordag:2006kx , who derives his results from a path integral formulation. Dalvit et al. Mazzitelli:2006ne ; Dalvit:2006wy use the argument principle to calculate the interaction between conducting cylinders with parallel axes. See also Reynaud et al.~reynaud and references therein. In fact, Emig and earlier collaborators buscher ; Emig:2006uh ; Emig:2002xz have published a series of papers, using closely related methods to calculate numerically forces between distinct bodies, starting from periodically deformed ones. Strong deviation from the proximity force approximation (PFA) is seen for when the distance between the bodies is large compared to their radii of curvature. Bordag Bordag:2006vc has precisely quantified the first correction to the PFA both for a cylinder and a sphere near a plane. As Gies and Klingmüller note Gies:2006bt , 1% deviations from the PFA occur when the ratio of the distance between the cylinder and the plate to the radius of the cylinder exceeds $0.01$. We will not discuss the worldline method of Gies and collaborators Gies:2006xe ; Gies:2006cq ; Gies:2003cv further, as that method lies rather outside our discussion here. Similar remarks apply to the work of Capasso et al.~capasso , who calculate forces from stress tensors using the familiar construction of the stress tensor in terms of Green’s dyadics Schwinger:1977pa ; Milton:1978sf . They use a numerical finite-difference engineering method. It is clear, then, with the exception of these last two methods, these approaches are fundamentally equivalent. We will refer to all of the former methods as multiple scattering techniques. We will now proceed to state the formulation in a simple, straightforward way, and apply it to various situations, all characterized by $δ$-function potentials. (A preliminary version of some of our results has already appeared Milton:2007gy .)## Ii Formalism

We begin by noting that the multiple-scattering formalism may be derived from the general formula for Casimir energies (for simplicity here we restrict attention to a massless scalar field) Schwinger75
E=i2τTrlnG,
(1)

where $τ$ is the “infinite” time that the
configuration exists, and $G$ is the Green’s function in the
presence of a potential $V$ satisfying
(matrix notation)
(−∂2+V)G=1,
(2)

subject to some boundary conditions at infinity. (For example,
we can use causal or Feynman boundary conditions, or, alternatively,
retarded Green’s functions.)
In Appendix A we give a heuristic derivation of this
fundamental formula.
The above formula for the Casimir energy is defined up to an infinite
constant, which can be at least partially compensated by inserting
a factor as do Kenneth and Klich Kenneth:2007jk :
E=i2τTrlnGG−10.
(3)

Here $G˙0$ satisfies, with the same boundary conditions as $G$, the
free equation
−∂2G0=1.
(4)

Now we define the $T$-matrix (note that our definition of $T$ differs by
a factor of 2 from that in Ref.~Emig:2007cf )
T=S−1=V(1+G0V)−1.
(5)

We then follow standard scattering theory lippmann , as reviewed
in Kenneth and Klich Kenneth:2007jk .
(Note that there seem to be some sign and ordering errors in that reference.)
The Green’s function can be alternatively written as
G=G0−G0TG0=11+G0VG0=V−1TG0,
(6)

which results in two formulæ for the Casimir energy
(7a)
(7b)

If the potential has two disjoint parts,
V=V1+V2,
(8)

it is easy to show that
T=(V1+V2)(1−G0T1)(1−G0T1G0T2)−1(1−G0T2),
(9)

where
Ti=Vi(1+G0Vi)−1,i=1,2.
(10)

Thus, we can write the general expression for the interaction between
the two bodies (potentials) in two alternative forms:
(11a)
(11b)

where
Gi=(1+G0Vi)−1G0,i=1,2.
(12)

The first form is exactly that given by Emig et al.~Emig:2007cf ,
and by Kenneth and Klich Kenneth:2007jk ,
while the latter is actually easily used if we know the individual
Green’s functions. (The effort involved in calculating with either is
identical.) In fact, the general form (11a) was recognized
earlier and applied to planar geometries by Maia Neto, Lambrecht, and
Reynaud maianeto ; lambrecht ; reynaud . In fact, Renne renne
essentially used Eq.~(11b) to derive the Lifshitz formula in 1971.
## Iii Casimir interaction between $δ$-plates

We now use the second formula above (11b) to calculate the Casimir energy between two parallel semitransparent plates, with potential
V=λ1δ(z−z1)+λ2δ(z−z2),
(13)

where the dimension of $λ˙i$ is $L^-1$.
The free reduced Green’s function is (where we have performed the
evident Fourier transforms in time and the transverse directions)
g0(z,z′)=12κe−κ|z−z′|,κ2=ζ2+k2.
(14)

Here $k=k˙⟂$ is the transverse momentum, and
$ζ=-iω$ is the Euclidean frequency.
The Green’s function associated with a single $δ$-function potential is
gi(z,z′)=12κ(e−κ|z−z′|−λiλi+2κe−κ|z−zi|e−κ|z′−zi|).
(15)

Then the energy/area is
E=116π3∫dζ∫d2k∫dzln(1−A)(z,z),
(16)

where, in virtue of the $δ$-function potentials ($a=—z˙2-z˙1—$)
(17)

We expand the logarithm according to
ln(1−A)=−∞∑s=1Ass.
(18)

For example, the leading term is easily seen to be
E(2)=−λ1λ216π3∫dζd2k4κ2e−2κa=−λ1λ232π2a,
(19)

which uses the change to polar coordinates,
dζd2k=dκκ2dΩ.
(20)

In general, it is easy to check that, because $A(z,z’)$ factorizes here,
$A(z,z’)=B(z)C(z’)$, $Tr A^n=(Tr A)^n$, or
Trln(1−A)=ln(1−TrA),
(21)

so the Casimir interaction between the two semitransparent plates is
E=14π2∫∞0dκκ2ln(1−λ1λ1+2κe−κaλ2λ2+2κe−κa),
(22)

which is exactly the well-known result Milton:2007ar .
## Iv Casimir self-energy for a single semitransparent sphere

Before we embark on new calculations, let us also confirm the known result for the self-stress on a single sphere of radius $a$ using this formalism. (This demonstrates, as did the rederivation of the Boyer result Boyer:1968uf by Balian and Duplantier Balian:1977qr , that the multiple scattering method is equally applicable to the calculation of self-energies.) We start from the general formula (7a), where
V(r,r′)=λδ(r−a)δ(r−r′).
(23)

We use the Fourier representation for the propagator in Euclidean space,
G0(r,r′)=e−|ζ||r−r′|4π|r−r′|=∫d3k(2π)3eik⋅(r−r′)k2+ζ2,
(24)

as well as the partial wave expansion of the plane wave
eik⋅r=∑lm4πiljl(kr)Ylm(^r)Y∗lm(^k).
(25)

Then, from the orthonormality of the spherical harmonics,
∫d^kY∗lm(^k)Yl′m′(^k)=δll′δmm′,
(26)

we obtain the representation
G0(r,r′)=2π∑lm∫∞0dkk2k2+ζ2jl(kr)jl(kr′)Ylm(^r)Y∗lm(^r′).
(27)

Now we combine the representation for the free Green’s function with the
spherical potential (23) to obtain
(G0V)(r,r′)=2λπδ(r′−a)∑lm∫∞0dkk2k2+ζ2jl(ka)jl(kr)Ylm(^r)Y∗lm(^r′).
(28)

When this, or powers of this, is traced (that is, $r$ and
$r’$
are set equal, and integrated over), we obtain a poorly defined expression;
to regulate this, we assume $r≠a$, for example, $r¡a$. (This is a type
of point-split regulation.) Then, because
jl(ka)=12(h(1)l(ka)+h(2)l(ka))=12(h(1)l(ka)+(−1)lh(1)l(−ka)),
(29)

while $j˙l(kr)=(-1)^lj˙l(-kr)$, we see that the $k$ integration in
Eq.~(28) can be evaluated as^{1}

∫∞0dkk2k2+ζ2jl(ka)jl(kr)=πaKl+1/2(|ζ|a)Il+1/2(|ζ|r),r<a.
(30)

Thus, it is easily seen that an arbitrary power of $G˙0V$ has trace
Tr(G0V)n=(λa)n∑lm(Kl+1/2(|ζ|a)Il+1/2(|ζ|a))n,
(31)

and that therefore the total self-energy of the semitransparent sphere is
given by the well-known expression barton04 ; Scandurra:1998xa
E=12πa∞∑l=0(2l+1)∫∞0dxln(1+λaIl+1/2(x)Kl+1/2(x)),x=|ζ|a.
(32)

Actually, a slightly different form involving integration by parts was
given in Refs.~Milton:2004vy ; Milton:2004ya ,
which results in the energy being finite though order $λ^2$.
In order $λ^3$ there is a divergence which is associated with
surface energy CaveroPelaez:2006kq .
## V $2+1$ Spatial Geometries

We now proceed to apply this method to the interaction between bodies, which leads, for example, as Emig et al. Emig:2007cf ; Emig:2007me point out, to a multipole expansion. In this section we illustrate this idea with a $2+1$ dimensional version, which allows us to describe, for example, cylinders with parallel axes. We seek an expansion of the free Green’s function for $R=R˙⟂$ entirely in the $x$-$y$ plane,
G0(R+r′−r)=ei|ω||r−R−r′|4π|r−R−r′|=∫dkz2πeikz(z−z′)g0(r⊥−R⊥−r′⊥),
(33)

where the reduced Green’s function is
g0(r⊥−R⊥−r′⊥)=∫(d2k⊥)(2π)2e−ik⊥⋅R⊥eik⊥⋅(r⊥−r′⊥)k2⊥+k2z+ζ2.
(34)

As long as the two potentials do not overlap, so that we have
$r˙⟂-R˙⟂-r’˙⟂≠0$, we can write an expansion in
terms of modified Bessel functions:
g0(r⊥−R⊥−r′⊥)=∑m,m′Im(κr)eimϕIm′(κr′)e−im′ϕ′~g0m,m′(κR),κ2=k2z+ζ2.
(35)

By Fourier transforming, and using the definition of the Bessel function
imJm(kr)=∫2π0dϕ2πe−imϕeikrcosϕ,
(36)

we easily find
~g0m,m′(κR)=12π∫∞0dkkk2+κ2Jm−m′(kR)Jm(kr)Jm′(kr′)Im(κr)Im′(κr′),
(37)

which is in fact independent of $r$, $r’$.
As in the previous section, the $k$ integral here can actually be
evaluated as a contour integral, as Bordag noted Bordag:2006vc .
No point-splitting is required here,
because the bodies are non-overlapping, so $r/R, r’/R¡ 1$. We write
the dominant Bessel function in terms of Hankel functions,
Jm−m′(x)=12[H(1)m−m′(x)+H(2)m−m′(x)]=12[H(1)m−m′(x)+(−1)m−m′+1H(1)m−m′(−x)],
(38)

and then we can carry out the integral over $k$ by closing the
contour in the upper half plane. We are left with
∫∞0dxxx2+y2Jm−m′(x)Jm(xr/R)Jm′(xr′/R)=(−1)m′Km−m′(y)Im(yr/R)Im′(yr′/R),
(39)

and therefore the reduced Green’s function has the simple form
~g0m,m′(κR)=(−1)m′2πKm−m′(κR).
(40)

Thus we can derive an expression for the interaction
energy per unit length between two bodies, in
terms of discrete matrices,
E≡EintL=18π2∫dζdkzlndet(1−~g0t1~g0⊤t2),
(41)

where $⊤$ denotes transpose, and
where the $T$ matrix elements are given by
tmm′=∫drrdϕ∫dr′r′dϕ′Im(κr)e−imϕIm′(κr′)eim′ϕ′T(r,ϕ;r′,ϕ′).
(42)

### v.1 Interaction between semitransparent cylinders

Consider, as an example, two parallel semitransparent cylinders, of radii $a$ and $b$, respectively, lying outside each other, described by the potentials
V1=λ1δ(r−a),V2=λ2δ(r′−b),
(43)

with the separation between the centers $R$ satisfying $R¿a+b$.
It is easy to work out the scattering matrix in this situation,
T1=V1−V1G0V1+V1G0V1G0V1−…,
(44)

so the matrix element is easily seen to be
(t1)mm′=2πλ1aδmm′I2m(κa)1+λ1aIm(κa)Km(κa).
(45)

Again, we used here the regularized integral^{2}

∫∞0dkkk2+κ2Jm(kr)Jm(kr′)=Km(κr′)Im(κr),r<r′.
(46)

Thus the Casimir energy per unit length is
E=14π∫∞0dκκtrln(1−A),
(47)

where
A=B(a)B(b),
(48)

in terms of the matrices
Bmm′(a)=Km+m′(κR)λ1aI2m′(κa)1+λ1aIm′(κa)Km′(κa).
(49)

### v.2 Interaction between cylinder and plane

As a check, let us rederive the result derived by Bordag Bordag:2006vc for a cylinder in front of a Dirichlet plane perpendicular to the $x$ axis. We start from the interaction (11a) written in terms of $¯G˙2$, the deviation from the free Green’s function induced by a single potential,
¯G2=G2−G0=−G0T2G0,
(50)

so the interaction energy has the form
E=−i2τTrln(1+T1¯G2).
(51)

When the second body is a Dirichlet plane, $¯G$ may be found by
the method of images, with the origin taken at the center of the cylinder,
¯G(r,r′)=−G0(r,¯r′),¯r′=(R−x′,y′,z′),
(52)

where $R$ is the distance between the center of the
cylinder and its image at $R˙⟂$, that is,
$R/2$ is the distance between the center of the cylinder and the plane.
(We keep $R$ here, rather than $R/2=D$, because of the close connection
to the two cylinder case.)
Now we encounter the 2-dimensional Green’s function
g(r⊥+r′⊥−R⊥)=∑mm′Im(κr)Im′(κr′)eimϕeim′ϕ′gmm′(κR),
(53)

(because the cylinder has $y→-y$ reflection symmetry)
where the argument given above yields
gmm′(κR)=12πKm+m′(κR).
(54)

Thus the interaction between the semitransparent cylinder and a Dirichlet
plane is
E=14π∫∞0κdκtrln(1−B(a)),
(55)

where $B(a)$ is given by Eq.~(49).
In the strong-coupling limit this result agrees with that given by Bordag,
because
trBs=tr~Bs,~Bmm′=1Km(κa)Km+m′(κR)Im′(κa).
(56)

### v.3 Weak-coupling

In weak coupling, the formula (47) for the interaction energy between two cylinders is
E=−λ1λ2ab4πR2∞∑m,m′=−∞∫∞0dxxK2m+m′(x)I2m(xa/R)I2m′(xb/R).
(57)

Similarly, the energy of interaction between a weakly-coupled cylinder
and a Dirichlet plane is from Eq.~(55)
E=−λa4πR2∞∑m=−∞∫∞0dxx K2m(x)I2m(xa/R).
(58)

### v.4 Power series expansion

It is straightforward to develop a power series expansion for the interaction between weakly-coupled semitransparent cylinders. One merely exploits the small argument expansion for the modified Bessel functions $I˙m(xa/R)$ and $I˙m’(xb/R)$:
I2m(x)=(x2)2|m|∞∑n=0Z|m|,n(x2)2n,
(59)

where the coefficients $Z˙m,n$ are
(60)

The Casimir energy per unit length (57) is now given as
E=−λ1aλ2b4πR2∫∞0dxx∞∑m=−∞∞∑m′=−∞∞∑n=0∞∑n′=0(xa2R)2|m|Z|m|,n(xa2R)2n(xb2R)2|m′|Z|m′|,n′(xb2R)2n′K2m+m′(x).
(61)

Reordering terms gives a more compact formula
E=−λ1aλ2b4πR2∞∑m=−∞∞∑m′=−∞∞∑n=0∞∑n′=0Z|m|,n(aR)2(|m|+n)Z|m′|,n′(bR)2(|m′|+n′)J|m|+|m′|+n+n′,m+m′,
(62)

where the two index symbol $J˙p,q$ represents
the integral over $x$, which evaluates to
Jp,q=2∫∞0dx(x2)2p+1K2q(x)=√πp!Γ(p+q+1)Γ(p−q+1)22p+2Γ(p+32).
(63)

In order to simplify the power series expansion in terms of
$aR$ and $bR$ we need to reorder the $m$ sums so that only
non-negative values of $m$ appear.
There are several ways to break up the $m$ sums;
one of them is to decompose the sum into the $m=m’=0$ term, the $m,m’$ same
sign terms, and the $m,m’$ different sign terms, giving
E=−λ1aλ2b4πR2[∞∑n=0∞∑n′=0Z0,n(aR)2nZ0,n′(bR)2n′Jn+n′,0+2∞∑m=1∞∑m′=0∞∑n=0∞∑n′=0Zm,n(aR)2(m+n)Zm′,n′(bR)2(m′+n′)Jm+m′+n+n′,m+m′+2∞∑m=0∞∑m′=1∞∑n=0∞∑n′=0Zm,n(aR)2(m+n)Zm′,n′(bR)2(m′+n′)Jm+m′+n+n′,m−m′].
(64)

It is now possible to combine the multiple infinite power series into a single
infinite power series, where each term is given by (possible multiple) finite
sum(s). In this case we get an amazingly simple result
E=−λ1aλ2b4πR212∞∑n=0(aR)2nPn(μ),
(65)

where $μ=b/a$, and
where by inspection we identify the binomial coefficients
Pn(μ)=n∑k=0(nk)2μ2k.
(66)

Remarkably, it is possible to perform the sums riordan ,
so we obtain the following
closed form for the interaction between two weakly-coupled cylinders:
E=−λ1aλ2b8πR2[(1−(a+bR)2)(1−(a−bR)2)]−1/2.
(67)

We note that in the limit $R-a-b=d→0$, $d$ being the distance between
the closest points on the two cylinders, we recover the proximity force
theorem in this case (124),
U(d)=−λ1λ232π√2abR1d1/2,d≪a,b.
(68)

In Figs.~1–2
we compare the exact energy (67) with the
proximity force approximation (68).
Evidently, the former approaches the latter
when the sum of the radii $a+b$ of the cylinders approaches the distance $R$
between their centers. The rate of approach is linear (with slope 3/2)
for the equal radius
case, but with slope $b^2/4a^2$ when $a≪b$.
More precisely, the ratio of the exact energy to the PFA is
EU≈1−1+μ+μ24μdR≈1−R2−aR+a24a(R−a)dR.
(69)

This correction to the PFA is derived by another method in Appendix C.
The reader should note that the the PFA is actually only defined in the
limit $d→0$, so the functional form away from that point is ambiguous.
Corrections to the PFA depend upon the specific form assumed for $U(d)$.
### v.5 Exact result for interaction between plane and cylinder

In exactly the same way, starting from Eq.~(58), we can obtain a closed-form result for the interaction energy between a Dirichlet plane and a weakly-coupled cylinder of radius $a$ separated by a distance $R/2$. The result is again quite simple:
E=−λa4πR2[1−(2aR)2]−3/2.
(70)

In the limit as $d→0$, this agrees with the PFA:
U(d)=−λ64π√2ad3/2.
(71)

Note again that this form is ambiguous: the proximity force theorem is
equally well satisfied if we replace $a$ by $R/2$, for example, in $U(d)$.
The comparison between this PFA and the exact result (70)
is given in Fig.~3.
### v.6 Strong coupling (Dirichlet) limit

The interaction between Dirichlet cylinders is given by Eq.~(47) in the limit $λ˙1=λ˙2→∞$, that is
E=14πR2∫∞0dxxtrln(1−A),
(72a)
where
Amm′=∑m′′Km+m′′(x)Km′′+m′(x)Im′′(xa/R)Km′′(xa/R)Im′(xb/R)Km′(xb/R).
(72b)

Here the trace of the logarithm can be interpreted as in Eq.~(18).
Because it no longer appears possible to obtain a closed-form solution,
we want to verify analytically that as the surfaces of the two cylinders
nearly touch, we recover the result of the proximity force theorem.
We use a variation of the scheme explained by Bordag for a cylinder
next to a plane Bordag:2006vc . [The
analysis is a bit simpler in the weak-coupling case,
which leads to Eq.~(68). See Appendix C.] First we
replace the products of Bessel functions in $A$ by their leading uniform
asymptotic approximants for all $m$’s large:
Bmm′′(a)Bm′′m′(b)∼12π1√m+m′′1√m′+m′′(1+(xm+m′′)2)−1/4(1+(xm′+m′′)2)−1/4e−χ,
(73)

where the exponent is
(74)

in terms of
η(z)=t−1+lnz1+t−1,η′(z)=1zt,η′′(z)=−tz2,
(75)

and
t=(1+z2)−1/2.
(76)

We write the trace of the $s$th power of $A$ as (summed on repeated indices)
(As)m1m1=Bm1m′1(a)Bm′1m2(b)Bm2m′2(a)Bm′2m3(b)⋯Bmsm′s(a)Bm′sm1(b).
(77)

We rescale variables in terms of a large variable $M$ and relatively small
variables:
m′i=Mαi,mi=Mβi,
(78)

where without loss of generality we take only $2s-1$ of the $α$’s and
$β$’s as independent:
s∑i=1(αi+βi)=s.
(79)

This normalization is chosen so at the critical point where $χ=0$
for $a+b=R$,
αi=aRβi=1−aR,∀i.
(80)

Away from this point, we consider fluctuations,
αi=aR+^αi,βi=1−aR+^βi,
(81)

with the constraint
s∑i=1(^αi+^βi)=0.
(82)

The Jacobian of this transformation is $sM^2s-1$.
Now, we expand the exponent in $tr A^s$, to first order in
$d=R-a-b$, and to second order in $^α˙i$, $^β˙i$. The result
is
χ=2MsdtR+Mt(Ra−1)s∑i=1[^αi−12aR−a(^βi+^βi+1)]2+Mt4aR−as∑i=1(^βi−^βi+1)2.
(83)

The $^α˙i$ terms lead to trivial Gaussian integrals. The difficulty
with the quadratic $^β˙i$ terms is that only $s-1$ of the differences
are independent. But, in view of the constraint (82) there are
only $s-1$ independent $β˙i$ variables. In fact, it is easy to check
that
s∑i=1(^βi−^βi+1)2=s−1∑i=1i+1i[^βi−^βi+1+1i+1s−1∑j=i+1(^βj−^βj+1)]2,
(84)

which now enables us to perform each successive $^β˙i-^β˙i+1$
integration. The Jacobian of the transformation to the
difference variables $u˙i=^β˙i-^β˙i+1$, $i=1,…,s-1$, is
$1/s$. Thus, we can immediately write down
(85)

which is exactly the result expected from the proximity force theorem,
according to Eq.~(125).
We will forego further discussion of strong coupling, and presentation of
numerical results, for these have been extensively discussed in several
recent papers, especially in Ref.~Emig:2007me .
## Vi 3-dimensional formalism

The three-dimensional formalism is very similar. In this case, the free Green’s function has the representation
G0(R+r′−r)=∑lm,l′m′jl(i|ζ|r)jl′(i|ζ|r′)Y∗lm(^r)Yl′m′(^r′)glm,l′m′(R).
(86)

The reduced Green’s function can be written in the form
g0lm,l′m′(R)=(4π)2il′−l∫(dk)(2π)3eik⋅Rk2+ζ2jl(kr)jl′(kr′)jl(i|ζ|r)jl′(i|ζ|r′)Ylm(^k)Y∗l′m′(^k).
(87)

Now we use the plane-wave expansion (25) once again, this time for
$e^ik⋅R$,
so now we encounter something new, an integral over three spherical
harmonics,
∫d^kYlm(^k)Y∗l′m′(^k)Y∗l′′m′′(^k)=Clm,l′m′,l′′m′′,
(88)

where
Clm,l′m′,l′′m′′=(−1)m′+m′′√(2l+1)(2l′+1)(2l′′+1)4π(ll′l′′000)(ll′l′′mm′m′′).
(89)

The three-$j$ symbols (Wigner coefficients)
here vanish unless $l+l’+l”$ is even. This fact
is crucial, since because of it we can follow the previous method of
writing $j˙l”(kR)$ in terms of Hankel functions of the first and second
kind, using the reflection property of the latter,
h(2)l′′(kR)=(−1)l′′h(1)l′′(−kR),
(90)

and then extending the $k$ integral over the entire real axis to a
contour integral closed in the upper half plane. The residue theorem
then supplies the result for the reduced Green’s function^{3}

g0lm,l′m′(R)=4πil′−l√2|ζ|πR∑l′′m′′Clm,l′m′,l′′m′′Kl′′+1/2(|ζ|R)Yl′′m′′(^R).
(91)

### vi.1 Casimir interaction between semitransparent spheres

For the case of two semitransparent spheres that are totally outside each other,
V1(r)=λ1δ(r−a),V2(r′)=λ2δ(r′−b),
(92)

in terms of spherical coordinates centered on each sphere, it is again
very easy to calculate the scattering matrices,
T1(r,r′)=λ1a2δ(r−a)δ(r′−a)∑lmYlm(^r)Y∗lm(^r′)1+λ1aKl+1/2(|ζ|a)Il+1/2(|ζ|a),
(93)

and then the harmonic transform is very similar to that seen in
Eq.~(45), ($k=i—ζ—$)
(94)

Let us suppose that the two spheres lie along the $z$-axis, that is,
$R=
R^z$. Then we can simplify the expression for the energy
somewhat by using $Y˙lm(θ=0)=δ˙m0(2l+1)/4π$.
The formula for the energy of interaction becomes
E=12π∫∞0dζtrln(1−A),
(95)

where the matrix
Alm,l′m′=δm,m′∑l′′Bll′′m(a)Bl′′l′m(b)
(96)

is given in terms of the quantities
(97)

Note that the phase always cancels in the trace in Eq.~(95).
For strong coupling, this result reduces to that found by Bulgac, Wirzba
et al.~Bulgac:2005ku ; wirzba07 for Dirichlet spheres, and recently
generalized by Emig et al.~Emig:2007me
for Robin boundary conditions. (See also Ref.~hensler .)
### vi.2 Weak coupling

For weak coupling, a major simplification results because of the orthogonality property,
l∑m=−l(ll′l′′m−m0)(ll′l′′′m−m0)=δl′′l′′′12l′′+1,l≤l′.
(98)

Then the formula for the energy of interaction between the two spheres is
E=−λ1aλ2b4R∫∞0dxx∑ll′l′′(2l+1)(2l′+1)(2l′′+1)(cccll′l′′000)2K2l′′+1/2(x)I2l+1/2(xa/R)I2l′+1/2(xb/R).
(99)

There is no infrared divergence because for small $x$ the product of Bessel
functions goes like $x^2(l+l’-l”)+1$, and $l”≤l+l’$.
As with the cylinders, we expand the modified Bessel functions
of the first kind in power series in $a/R,b/R¡1$. This expansion yields
the infinite series
E=−λ1aλ2b4πRabR2∞∑n=01n+1n∑m=0Dn,m(aR)2(n−m)(bR)2m,
(100)

where by inspection of the first several $D˙n,m$ coefficients
we can identify them as
Dn,m=12(2n+22m+1),
(101)

and now we can immediately sum the expression (100)
for the Casimir interaction energy to give
the closed form
E=λ1aλ2b16πRln⎛⎜
⎜⎝1−(a+bR)21−(a−bR)2⎞⎟
⎟⎠.
(102)

Again, when $d=R-a-b≪a,b$, the proximity force theorem (129)
is reproduced:
U(d)∼λ1λ2ab16πRln(d/R),d≪a,b.
(103)

However, as Figs.~4, 5 demonstrate, the
approach is not very smooth,
even for equal-sized spheres. The ratio of the energy to the PFA is
EU=1+ln[(1+μ)2/2μ]lnd/R,d≪a,b,
(104)

for $b/a=μ$. Truncating the power series (100) at $n=100$
would only begin to show the approach to the proximity force theorem limit.
The error in using the PFA between spheres can be very substantial.
Again we will forego discussion of the strong-coupling (Dirichlet) limit
here because of the extensive discussion already in the literature
Bulgac:2005ku ; wirzba07 ; Emig:2007me .
### vi.3 Exact result for interaction between plane and sphere

In just the way indicated above, we can obtain a closed-form result for the interaction energy between a weakly-coupled sphere and a Dirichlet plane. Using the simplification that
l∑m=−l(−1)m(lll′m−m0)(lll′000)=δl′0,
(105)

we can write the interaction energy
in the form
E=−λa2πR∫<