5.23 The Abel-Plana Formula and the Casimir E ect 229

5.23 The Abel-Plana Formula and the Casimir E↵ect 229
and using f 00(w) = ⇢ ei and 2✓ + = ⇡ to show that (5.300)

⇢ e 2i✓ = ⇢ ei i⇡ = ⇢ ei = f 00 (w)

we get our formula for the saddle-point integral (5.294)

✓ 2⇡ ◆1/2
I(x) ⇡ xf 00 (w)
h(w) exf (w). (5.301)

If there are n saddle points wj for j = 1, . . . , n, then the integral I(x) is the

sum

XN ✓ 2⇡ ◆1/2
I(x) ⇡ xf 00 (wj
h(wj ) exf (wj ) . (5.302)
j=1
)

5.23 The Abel-Plana Formula and the Casimir E↵ect

This section is optional on a first reading.

Suppose the function f (z) is analytic and bounded for n1  Re z  n2.
Let C+ and C be two contours that respectively run counter-clockwise along

the rectangles with vertices n1, n2, n2 +i1, n1 +i1 and n1, n2, n2 i1, n1

i1 indented with tiny semi-circles and quarter-circles so as to avoid the

integers z = n1, n1 + 1, n1 + 2, . . . , n2 while keeping Imz > 0 in the upper

rectangle and Imz < 0 in the lower one (and n1 < Re z < n2). Then the

contour integrals Z

I± = f (z) 1 dz = 0 (5.303)
C± e⌥2⇡iz

vanish by Cauchy’s theorem (5.22) since the poles of the integrand lie outside

the indented rectangles.

The absolute value of the exponential exp( 2⇡iz) is arbitrarily large on

the top of the upper rectangle C+ where Imz = 1, and so that leg of
the contour integral I+ vanishes. Similarly, the bottom leg of the contour
integral I vanishes. Thus we can separate the di↵erence I+ I into a
term Tx due to the integrals near the x-axis between n1 and n2, a term
T1 involving integrals between n1 and n1 ± i1, and a term T2 involving
integrals between n2 and n2 ± i1, that is, 0 = I+ I = Tx + T1 + T2.

The term Tx = Ix + S consists of the integrals Ix along the segments of
the x-axis from n1 to n2 and a sum S over the tiny integrals along the semi-
circles and quarter circles that avoid the integers from n1 to n2. Elementary

230 Complex-Variable Theory

algebra simplifies the integral Ix to

Z n2  1 1 Z n2
e+2⇡ix f (x) dx.
Ix = f (x) e 2⇡ix 1 + 1 dx = (5.304)
n1
n1

The sum S is over the semi-circles that avoid n1 + 1, . . . , n2 1 and over

the quarter-circles that avoid n1 and n2. For any integer n1 < n < n2, the

integral along the semi-circle of Cn+ minus that along the semi-circle of Cn ,

both around n, contributes to S the quantity

ZZ f (z)
f (z) e2⇡iz
Sn = ZSCn+ e 2⇡iz 1 dz 1 dz
= 1
SCn+ e f (z) SCnZ f (z)

2⇡i(z n) dz SCn e2⇡i(z n) 1 dz (5.305)

since exp(±2⇡in) = 1. The first integral is clockwise in the upper half plane,

the second clockwise in the lower half plane. So if we make both integrals

counter-clockwise, inserting minus signs, we find as the radii of these semi-

circles shrink to zero I
Sn =
f (z) n) dz = f (n). (5.306)
2⇡i(z

One may show (exercise 5.39) that the quarter-circles around n1 and n2
contribute (f (n1) + f (n2))/2 to the sum S. Thus the term Tx is

Tx = 1 f (n1) + nX2 1 f (n) + 1 f (n2) Z n2 (5.307)
2 2 f (x) dx.
n=n1+1
n1

Since exp( 2⇡in1) = 1, the di↵erence between the integrals along the

imaginary axes above and below n1 is (exercise 5.40)

T1 = Z n1 e f (z) 1 dz Z n1 i1 f (z) 1 dz (5.308)
= f e2⇡iz (5.309)
n1Z+i11 2⇡iz iy) f n1
i e2⇡y iy) dy.
(n1 + (n1
0 1

Similarly, the di↵erence between the integrals along the imaginary axes

above and below n2 is (exercise 5.41)

Z n2+i1 f (z) Z n2 f (z)

T2 = e 2⇡iz 1 dz n2 i1 e2⇡iz 1 dz (5.310)
= (5.311)
i Zn2 1 f (n2 + iy) f (n2 iy) dy.
e2⇡y 1
0

5.23 The Abel-Plana Formula and the Casimir E↵ect 231

Since I+ I = Tx + T1 + T2 = 0, we can use (5.307) and (5.309–5.311)
to build the Abel-Plana formula (Whittaker and Watson, 1927, p. 145)

1 f (n1) + nX2 1 f (n) + 1 f (n2) Z n2 (5.312)
2 2 f (x) dx iy) dy
n=n1+1
Z 1 f (n1 iy) n1
=i f (n1 + iy) e2⇡y
f (n2 + iy) + f (n2
0 1

(Niels Abel 1802–1829 and Giovanni Plana 1781–1864).
In particular, if f (z) = z, the integral over y vanishes, and the Abel-Plana

formula (5.312) gives

1 n1 + nX2 1 n + 1 n2 Z n2 (5.313)
2 2 = x dx
n=n1+1
n1

which is an example of the trapezoidal rule.

Example 5.37 (The Casimir E↵ect) The Abel-Plana formula provides

one of the clearer formulations of the Casimir e↵ect. We will assume that

the hamiltonian for the electromagnetic field in empty space is a sum over

two polarizations and an integral over all momenta of a symmetric product

X2 Z ~ !(k) 1 h + i d3k (5.314)
H0 = 2 a†s(k)as(k) as(k)as†(k)

s=1

of the annihilation and creation operators as(k) and a†s(k) which satisfy the
commutation relations

[as(k), a†s0 (k0)] = ss0 (k k0) and [as(k), as0 (k0)] = 0 = [a†s(k), as†0 (k0)].
(5.315)

The vacuum state |0i has no photons, and so on it as(k)|0i = 0 (and
h0|a†s(k) = 0). But because the operators in H0 are symmetrically ordered,
the energy E0 of the vacuum as given by (5.314) is not zero; instead it is
quarticly divergent

X2 Z ~ !(k) 1 Z ~ !(k) d3k (5.316)
E0 = h0|H0|0i = 2 (0) d3k = V (2⇡)3

s=1

in which we used the delta-function formula
Z
(k k0) = e±i(k k0)·x d3x (5.317)
(2⇡)3

to identify (0) as the volume V of empty space divided by (2⇡)3. Since the

232 Complex-Variable Theory

photon has no mass, its (angular) frequency !(k) is c|k|, and so the energy

density E0/V is

E0 Z k3 dk = ~c K4 = ~c 1 (5.318)
V = ~c 2⇡2 8⇡2 8⇡2 d4

in which we cut o↵ the integral at some short distance d = K 1 below which
the hamiltonian (5.314) and the commutation relations (5.315) are no longer
valid. But the energy density of empty space is

⌦⇤⇢c = ⌦⇤ 3H02 ⇡ ~c 1 5 m)4 (5.319)
8⇡G 8⇡2 (2.8 ⇥ 10

which corresponds to a distance scale d of 28 micrometers. Since quantum

electrodynamics works well down to about 10 18 m, this distance scale is

too big by thirteen orders of magnitude.

If the universe were inside an enormous, perfectly conducting, metal cube

of side L, then the tangential electric and normal magnetic fields would

vanish on the surface of the cube Et(r, t) = 0 = Bn(r, t). The available

wave-numbers of the electromagnetic field inside the cube then would be

kn = 2⇡(n1, n2, n3)/L, and the energy density would be

E0 = 2⇡~c Xp (5.320)
V L4 n2.

n

The Casimir e↵ect exploits the di↵erence between the continuous (5.318)

and discrete (5.320) energy densities fpor the case of two metal plates of area
A separated by a short distance ` ⌧ A.

If the plates are good conductors, then at low frequencies the boundary

conditions Et(r, t) = 0 = Bn(r, t) hold, and the tangential electric and

normal magnetic field vanish on the surfaces of the metal plates. At high

frequencies, above the plasma frequency !p of the metal, these boundary

conditions fail because the relative electric permittivity of the metal

✏(!) ⇡ 1 !p2 ✓ ◆ (5.321)
!2 1 i
!⌧

has a positive real part. Here ⌧ is the mean time between electron collisions.

The modes that satisfy the low-frequency boundary conditions Et(r, t) =

0 = Bn(r, t) are (Bordag et al., 2009, p. 30)

r ⇣ ⇡n ⌘2
c k?2 `
!(k?, n) ⌘ + (5.322)

where n · k? = 0. The di↵erence between the zero-point energies of these

5.23 The Abel-Plana Formula and the Casimir E↵ect 233

modes and those of the continuous modes in the absence of the two plates

per unit area would be 2s s 3
1 `2k?2 `k? 5
E(`) ⇡~c Z1 k?dk? X1 `2k?2 Z 2⇡
A ` 2⇡ 4
= 0 n=0 ⇡2 + n2 0 ⇡2 + x2 dx

(5.323)

if the boundary conditions held at all frequencies. With p = `k?/⇡, we will

represent the failure of these boundary conditions at the plasma frequency
!p by means of a cuto↵ function like c(n) = (1+n/np) 4 where np = !p`/⇡c.

In terms of such a cuto↵ function, the energy di↵erence per unit area is
Z 1 "X1 #
⇡2~c p Z1 p
E(`) = 2`3 p dp c(n) p2 + n2 c(x) p2 + x2 dx p .
A 2
0 n=0 0

(5.324)

Since c(n) falls o↵ as (np/n)4 for n np, we may neglect terms in the sum

and integral beyond some integer M that is much larger than np

⇡2~c Z 1 "XM p ZM p #
2`3 c(x) p2 + x2 dx
E(`) = p dp c(n) p2 + n2 p .
A 0 2
0 n=0

(5.325)

The function p
p2 + z2
f (z) = p + z2 = (5.326)
c(z) p2 (1 + z/np)4

is analytic in the right half-plane Re z = x > 0 (exercise 5.42) and tends

to zero limx!1 |f (x + iy)| ! 0 as Re z = x ! 1. So we can apply the

Abel-Plana formula (5.312) with n1 = 0 and n2 = M to the term in the

square brackets in (5.325) and get

E(`) = ⇡2~c Z1 dp ⇢ ) p + M2 (5.327)
A 2`Z3 p c(M p2 iy)2

10h 2 p
p
+i c(iy) p2 + (✏ + iy)2 c( iy) p2 + (✏

0p
c(M + iy) p2 + (M + iy)2
i
+c(M p iy)2 dy 1
iy) p2 + (M e2⇡y

in which the infinitesimal ✏ reminds us that the contour lies inside the right
half-plane.

We now take advantage of the propertiespof the cuto↵ function c(z). Since
M np, we can neglect the term c(M ) p2 + M 2/2. The denominator

234 Complex-Variable Theory
p

exp(2⇡y) 1 also allows us to neglect the terms ⌥c(M ⌥iy) p2 + (M ⌥ iy)2.

We are left with
Z
E(`) = ⇡2~c h0 1 dp (5.328)
A 2Z`31 p
p

⇥ i c(iy) p2 + (✏ + iy)2 p i dy 1.
c( iy) p2 + (✏ iy)2 e2⇡y

0

Since the y integration involves the factor 1/(exp(2⇡y) 1), we can neglect

the detailed behavior of the cuto↵ functions c(iy) and c( iy) for y > np

where they di↵er appreciably from unity. The energy now is

E(`) ⇡2~c Z 1Z 1 p + (✏ + iy)2 p iy)2 dy.
A 2`3 p dp p2 e2⇡y p2 + (✏
= i 1 (5.329)

00

When y < p, the square-roots with the ✏’s cancel. But for y > p, they are
pp
p2 y2 ± 2i✏y = ±i y2 p2. (5.330)

p
Their di↵erence is 2i y2 p2, and so E(`) is

E(`) = ⇡2~c Z1 Z1 p p) dy (5.331)
A 2`3 p dp 2 y2 p2 ✓(y

00 e2⇡y 1

in which the Heaviside step function ✓(x) ⌘ (x + |x|)/(2|x|) keeps y > p

E(`) = ⇡2~c Zy dp Z 1py2 p2 dy. (5.332)
A `3 p 0 e2⇡y 1

0

The p-integration is elementary, and so the energy di↵erence is

E(`) = ⇡2~c Z 1 y3 dy = ⇡2~c B2 = ⇡2~c (5.333)
A 3`3 0 e2⇡y 1 3`3 8 720 `3

in which B2 is the second Bernoulli number (4.109). The pressure pushing
the plates together then is

p= 1 @E(`) = ⇡2~c (5.334)
A @` 240 `4

a result due to Casimir (Hendrik Brugt Gerhard Casimir, 1909–2000).
Although the Casimir e↵ect is very attractive because of its direct connec-

tion with the symmetric ordering of the creation and annihilation operators
in the hamiltonian (5.314), the reader should keep in mind that neutral
atoms are mutually attractive, which is why most gases are diatomic, and
that Lifshitz explained the e↵ect in terms of the mutual attraction of the
atoms in the metal plates (Lifshitz, 1956; Milonni and Shih, 1992) (Evgeny
Mikhailovich Lifshitz, 1915–1985).