18 Introductory Remarks on Quantized Fields
Quantum Field Theory (abbreviated QFT) deals with the quantization of fields. A familiar example
of a field is provided by the electromagnetic field. Classicalelectromagnetism describes the dynamics
of electric charges and currents, as well as electro-magnetic waves, such as radio waves and light,
in terms of Maxwell’s equations. At the atomic level, however, the quantum nature of atoms as
well as the quantum nature of electromagnetic radiation must be taken into account. Quantum
mechanically, electromagnetic waves turn out to be composed of quanta of light, whose individual
behavior is closer to that of a particle than to a wave. Remarkably, the quantization of the
electromagnetic field is in terms of the quanta of this field, which are particles, also calledphotons.
In QFT, this field particle correspondence is used both ways,and it is one of the key assumptions
of QFT that to every elementary particle, there correspondsa field. Thus, the electron will have
its own field, and so will every quark.
Quantum Field Theory provides an elaborate general formalism for the field–particle correspon-
dence. The advantage of QFT will be that it can naturally account for the creation and annihilation
of particles, which ordinary quantum mechanics of the Schr ̈odinger equation could not describe.
The fact that the number of particles in a system can change over time is a very important phe-
nomenon, which takes place continuously in everyone’s daily surroundings, but whose significance
may not have been previously noticed.
Inclassical mechanics, the number of particles in a closed system is conserved, i.e. the total
number of particles is unchanged in time. To each pointlike particle, one associates a set of position
and momentum coordinates, the time evolution of which is governed by the dynamics of the system.
Quantum mechanicsmay be formulated in two stages.
- The principles of quantum mechanics, such as the definitions of states, observables, are
general and do not make assumptions on whether the number of particles in the system is
conserved during time evolution. - Thespecific dynamics of the quantum system, described by the Hamiltonian, may or may
not assume particle number conservation. In introductory quantum mechanics, dynamics is
usually associated with non-relativistic mechanical systems (augmented with spin degrees of
freedom) and therefore assumes a fixed number of particles. In many important quantum
systems, however,the number of particles is not conserved.
A familiar and ubiquitous example is that ofelectromagnetic radiation. An excited atom may
decay into its ground state by emitting a single quantum of light or photon. The photon was not
“inside” the excited atom prior to the emission; it was “created” by the excited atom during its
transition to the grounds state. This is well illustrated asfollows. An atom in a state of sufficiently
high excitation may decay to its ground state in a single stepby emitting a single photon. However,
it may also emit a first photon to a lower excited state which inturn re-emits one or more photons
in order to decay to the ground state (see Figure??, (a) and (b)). Thus, given initial and final
states, the number of photons emitted may vary, lending further support to the fact that no photons
are “inside” the excited state to begin with.