From Classical Mechanics to Quantum Field Theory

208 From Classical Mechanics to Quantum Field Theory. A Tutorial

in the large Λ limit. Whereas in the heat kernel regularization

E 0 (Λ) =

1

8 π^22

−

m^2

16 π^2

+

m^4

64 π^2

(

2+γ+log

m^2

4

)

+O( )

for small values of =

√ 2
Λ^2. The leading quartic and logarithmic divergent terms
are the same in both regularizations whereas the quadratically divergent term is
different.
The source of divergence is of ultraviolet origin because it comes from the
integration ofω(k)=

√

k^2 +m^2 at large values of the momentum. The quantum
field theory of free scalar fields is an infinite set of harmonic oscillators, each one
labelled byk. Each of these oscillators contribute to the vacuum energy with
their zero-point energy^12 ω(k). This total contribution of zero-points energies to
the vacuum energy density is divergent, since there are modes with arbitrary high
momentum. This is why this divergence has ultraviolet origin. It appears in any
quantum field theory and not only in the free scalar quantum field. It is something
intrinsic to the theory of quantum fields.

3.4.2 Momentumoperator.....................

The generator of space translations is the momentum operatorPˆi.Inthefree
scalar field theory it is given by

Pˆi=

∫

d^3 x(ˆπ∂iφ),i=1, 2 , 3. (3.37)

Since the vacuum state is translation invariant,

PˆiΨ 0 =i

∫

d^3 xφ

√

∇^2 +m^2 ∂iφΨ 0

=

i

2

∫

d^3 x∂i

(

φ

√

∇^2 +m^2 φ

)

Ψ 0 =0, (3.38)

which apparently does not require renormalization as the vacuum energy. However,
the above analysis is too naive. If we write Eq. (3.38) in terms of the Fourier modes
of the field,

PˆiΨ 0 =^2 π

L

∑

n∈Z^3

(niωn)Ψ 0 , (3.39)

we realize that the sum is divergent. However, in the cut-off or heat kernel regu-
larizations, the regularized eigenvalues vanish

PˆiΨ 0 =^2 π

L

|ω∑n|<Λ

n∈Z^3

(niωn)Ψ 0 =^2 π

L

∑

n∈Z^3

(niωne−ω

(^2) n
)Ψ 0 =0, (3.40)