From Classical Mechanics to Quantum Field Theory

210 From Classical Mechanics to Quantum Field Theory. A Tutorial

the plates modifies the vacuum energy in ad-dependent way. If the perturbation
increases the vacuum energy with the distance it will induce an attractive force
between the plates and this force will be repulsive if the energy decreases with the
distance.
The main modification introduced by the plates is the presence of boundary
conditions on the classical fields. Since the free electromagnetic field corresponds
to photons with two polarizations, the electromagnetic vacuum energy is twice the
vacuum energy of a massless scalar field with vanishing boundary conditions on
the plates. The physicalR^3 space is split into three disjoint domains:

ΩI={x∈R^3 ;−∞<x 3 ≤−d 2 },

ΩII={x∈R^3 ;−d 2 <x 3 ≤d 2 },

ΩIII={x∈R^3 ; d 2 <x 3 ≤∞}.

The physical vacuum is the product of the vacua of the different sectors

ψ 0 (φ)=ψI(φI)ψII(φII)ψIII(φIII)

and the vacuum energy the sum of the vacuum energies of the three domains

E 0 =EI+EII+EIII

The calculation of vacuum energy density in the domain between the plates
EII=EII/VIIby using the heat kernel regularization gives (see appendix A)

EII=

1

8 π^22

−

1

16

√

π^32

1

d

−

π^2

1440 d^4

+O(

(^12)
). (3.41)
In the other two domains, we only get
EI=

1

8 π^22

, EIII=

1

8 π^22

(3.42)

because of their infinite transversal size. The common divergent term corresponds
to the vacuum density in infinite volume. Thus, it disappears under vacuum
energy renormalization. The 32 divergent term between the plates correspond to
the selfenergy of the plates and has to be renormalized as well. The remaining
renormalized vacuum energydensity between the plates

ErenII =−

π^2

1440 d^4

(3.43)

is negative inducing an attractive force between the plates, which is the Casimir
effect of the vacuum energy on the plates.