From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 231

Now, from the last three terms only two,f(0) andf′′′(0) do not vanish because
the function Eq. (3.58) satisfies thatf(∞)=f′(∞)=f′′′(∞)=0andf′(0) = 0.
Thus, we get that

EII =^1

2

∫

d^3 k

(2π)^3

e−k

2 √

k^2

−

1

2 d

∫∞

−∞

∫∞

−∞

dk 1 dk 2

(2π)^2

e−(k

(^21) +k (^22) )√
k^21 +k^22

+

1

1440

f′′′(0) +O( ). (3.59)

The first term in Eq. (3.59) is the vacuum energy of the free field and gives a
divergent contribution

E(1)=

1

8 π^22. (3.60)

The second term corresponds to the selfenergy of the plates and gives another
divergent contribution

E(2)=

1

16

√

π^32

1

d

. (3.61)

On the contrary the contribution of the third term is finite

EII(3)=−

π^2

1440 d^4

. (3.62)

The calculation for the other two domains outside the plates ΩIand ΩIIIcan
be performed in a similar way, but the results can be derived from (3.60), (3.61)
and (3.62) just by taking the limitd→∞. The results are

EI=EIII =

1

8 π^22. (3.63)

3.11 Appendix2.Gaussianmeasures...................

Let us consider the following Gaussian probability measure of zero mean

dμc=

e−x

2

2 c

√

2 πc

dx ,

in the real lineR. The average of anyL^1 (R) functionf

〈f〉=

∫

R

dμc(x)f(x),