A Concise Introduction to Quantum Field Theory 215Quantum Field Theory is a natural way to describe such process in an accurate
manner.
Although we have identified the Fock structure of the space of quantum states
with the stratification given by the number of particles, the notion of particle is
secondary and it is just the quantum field that is fundamental. The explanation
of why two particles are identical come from the fact that they are generated by
the same field, which permeates the whole Universe. In this way we understand
why the particles coming in cosmic rays from far galaxies are identical to the
same particles on earth. The link is the quantum field. Moreover, there are field
theories where the particlecomposition is unclear. For example in gauge theories,
the fundamental fields are quark and gluon fields, but the physical particles are
mesons, baryons and glueballs which are bounded composites of quarks and gluons.
3.5.2 Wicktheorem
An important consequence of the Gaussian nature of the ground state of a free
field theory is the clustering property of the vacuum expectation values of the
product of field operators
〈 0 |φ(f 1 )φ(f 1 )···φ(fn)| 0 〉=〈f 1 f 1 ···fn〉=∫
δμφ(f 1 )φ(f 1 )···φ(fn).The cluster property is a fundamental characteristic of Gaussian measures
which gives rise to the Wick theorem which states that
〈f 1 f 2 ···fn〉=⎧
⎪⎪⎨
⎪⎪⎩
0forn=2m+1,
1
2!m!∑
σ∈Sn〈fσ(1)fσ(2)〉···〈fσ(2m−1)fσ(2m)〉 forn=2m.3.6 FieldsinInteraction
The free QFT considered in the previous section shows the basic properties of a
relativistic quantum field theory, but the challenge is to quantize field theories of
interacting fields. The procedure is basically the same. The main difference is that
the interacting Hamiltonian is not exactly solvable. For example, let us consider
the4!λφ^4 theory Hamiltonian
Hˆ=^1
2(
‖πˆ‖^2 +‖∇φ‖^2 +m^2 ‖φ‖^2 +λ
12‖φ^2 ‖^2)
.
Using the same quantization rules as in Eq. (3.25) we get a formal quantum Hamil-
tonianHˆwhich is defined in the space of functionals in the space of classical fields