From Classical Mechanics to Quantum Field Theory

230 From Classical Mechanics to Quantum Field Theory. A Tutorial

grant FPA2015-65745-P and DGA-FSE grant 2015-E24/2 and the COST Action
MP1405 QSPACE, supported by COST (European Cooperation in Science and
Technology).

3.10 Appendix1.Casimireffect

In the domain ΩIIbetween two parallel plates, the normal modes with Dirichlet
boundary conditionsφ(x 1 ,x 2 ,−d 2 )=φ(x 1 ,x 2 ,d 2 ) = 0 of the scalar fieldφare wave
functions

φk 1 ,k 2 ,n(x)=

1

4 π^2 d

eik^1 x^1 +ik^2 x^2 sin

(nπx 3

d

+

nπ

2

)

. (3.57)

with arbitrary real values of the transverse momentak 1 andk 2 and discrete values
for the longitudinal modesk 3 =nπd withn=1, 2 ,···.
For the other two domains outside the plates ΩI,ΩIIIthe Fourier modes of the
classical fieldφare the same (3.57) but with continuous values of the transverse
modesn∈R+.
The vacuum energy density in the domain ΩIIbetween the plates is given by

EII=^1

2 d

∑∞

n=1

∫∞

−∞

∫∞

−∞

dk 1 dk 2

(2π)^2

√

k^21 +k^22 +

(nπ

d

) 2

.

In the heat kernel regularization, the vacuum energy density between the plates
is

EII =^1

2 d

∑∞

n=1

∫∞

−∞

∫∞

−∞

dk 1 dk 2

(2π)^2

e−(k

(^21) +k (^22) +(nπd)^2 )

√

k^21 +k^22 +

(nπ

d

) 2

.

To calculate this energy density, let us define the function

f(z)=

1

2 πd

∫∞

0

kdke−

√

k^2 +(zπd)^2

√

k^2 +

(zπ

d

) 2

=

1

4 πd

∫∞

(zπd)^2

dκ e−

√κ√

κ (3.58)

and to use the Euler-MacLaurin formula

∑∞

n=1

f(n)=

∫∞

0

dz f(z)+

1

2 [f(∞)−f(0)] +

1

12 [f

′(∞)−f′(0)]

−^1

720

[f′′′(∞)−f′′′(0)] +...