From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 209

and thus, the renormalized value of the vacuum eigenvalue ofPˆivanish. This
is due to the spherical symmetry of both regularizations. What is remarkable is
that in the case of the vacuum energy, any choice of the regularization provides
a non-vanishing value. In this sense there is a difference between the quantum
generators of space and time translations, which seems to be not in agreement with
the Lorentz symmetry. This is a genuine characteristic of canonical quantization
as will be emphasized later in the course.

3.4.3 Casimireffect

The existence of UV divergences in vacuum energy is not a special property of
scalar fields. Any quantum field theory faces the same problem, e.g. the electro-
magnetic field in quantum electrodynamics or the fields of the Standard Model
have UV divergent vacuum energies. We have renormalized the divergences by
removing the whole contribution of vacuum energy. However, this does not mean
that it is an unphysical quantity. The fact that the vacuum energy can have
observable consequences was first pointed out by Casimir[ 6 ].Heremarkedthat
although we can remove a fixed vacuum energy for the free fields, the variation of
the vacuum energy under external conditions could be detected and observed.
Consider a pair of infinite, perfectly conducting plates placed parallel to each
other at a distanced(see Figure 2). The conducting character of the plates implies
that the electromagnetic forces vanishes at both plate surfaces. The presence of

Fig. 3.2 Three different domains of vacuum fluctuations in the Casimir effect.