Quantum Field Theory I

1.3. RELATIVITY AND QUANTUM THEORY 45
1.3.2 The logic of the particle-focused approach to QFT
The relativistic quantum theory describes, above all, the physics of elementary
particles. Therefore the particle-focused approach looks like the most natural.
Nevertheless, it is by far not the most common, for reasons which are mainly
historical. Now we have to confess (embarrassed) that in these lectures we are
going to follow the less natural, but more wide-spread field-focused approach. 45
The particle-focused approach is only very briefly sketched in this paragraph.
Not everything should and could be understood here, it is sufficient just to catch
the flavor. If too dense and difficult (as it is) the paragraph should be skipped.
One starts with an irreducible representation of the Poincaré group on some
1-particle Hilbert space. The usual basis vectors in the Hilbert space are of the
form |p,σ〉, where p is the (overall) momentum of the state and all the other
characteristics are included in σ. For a multiparticle state, the σ should contain
a continuousspectrum ofmomentaofparticularparticles. This providesus with
a natural definition of 1-particle states as the ones with discrete σ. In this case
it turns out that values of σ correspond to spin (helicity) projections.
Irreducible representations are characterized by eigenvalues of two Casimir
operators (operators commuting with all generators), one of them being m 2 , the
eigenvalue of the Casimir operator P 2 , and the second one having to do with
the spin. The states in the Hilbert space are therefore characterized by eigenvalues
of 3-momentum, i.e. the notation |⃗p,σ〉 is more appropriate than |p,σ〉
(nevertheless, when dealing with Lorentz transformations, the |p,σ〉 notation is
very convenient). The |⃗p,σ〉 states are still eigenstates of the Hamiltonian, with
the eigenvalues E ⃗p = √ ⃗p 2 +m 2 .
Once a representation of the Poincaré group on a 1-particle Hilbert space
is known, one can systematically build up the corresponding Fock space from
direct products of the Hilbert ones. The motivation for such a construction is
thatthiswouldbeanaturalframeworkforprocesseswithnonconservednumbers
of particles, and such processes are witnessed in the nature. This Fock space
benefits from having a natural representation of Poincarégroup, namely the one
defined by the direct products of the representations of the original 1-particle
Hilbert space. The Hamiltonian constructed in this way, as well as all the other
generators, correspond to a system of noninteracting particles. In terms of
creation and annihilation operators, which are defined as very natural operators
in the Fock space the free Hamiltonian has a simple form H 0 = ∫ d 3 p
(2π) 3 E ⃗p a + ⃗p a ⃗p.
45 The explanation for this is a bit funny.
As to what we call here the particle-centered approach, the textbook is the Weinberg’s one.
We strongly recommend it to the reader, even if it would mean that he/she will quit these
notes. The present author feels that he has nothing to add to the Weinberg’s presentation.
But even if the approach of the Weinberg’s book is perhaps more natural than any other,
it is certainly not a good idea to ignore the traditional development, which we call here the
field-centered approach. If for nothing else, then simply because it is traditional and therefore
it became a part of the standard background of the majority of particle physicists.
Now as to the textbooks following the traditional approach, quite a few are available. But
perhaps in all of them there are points (and unfortunately not just one or two) which are not
explained clearly enough, and are therefore not easy to grasp. The aim of the present notes is
to provide the standard material with perhaps a bit more emphasis put on some points which
are often only glossed over. The hope is, that this would enable reader to penetrate into the
subject in combination of a reasonable depth with a relative painlessness.
Nevertheless, beyond any doubt, this hope is not to be fulfilled. The reader will surely find
a plenty of disappointing parts in the text.