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Chapter 8
Quantum Chromodynamics
8.1 Introduction: coloured quarks and gluons
In chapter 7 we have studied the behaviour of electrically charged particles and
the electromagnetic field embodied by photons. Notwithstanding the fact that
particles can have different charges, all these charges are of the same type in the
sense that they can be added. For instance, atoms are electrically neutral when
studied from the ‘outside’, since the positive charge of the nucleus is cancelled
out by the negative charge of the electron cloud. It is interesting to see what
happens if we enlarge our view to the possibility of ‘different types of charge’,
that cannot be meaningfully added in a simple way. In that case, a bound state
of particles with a different charge type might not look ‘neutral’ when seen from
the outside : the charges of the constituents would show through. To avoid
confusion with the electric charge we shall let the ‘new charges’ go by the name
of colours, and the dynamical theory of their interactions is called Quantum
Chromodynamics, or QCD.
We shall start our investigation with coloured fermions, called quarks 1 . The
number of colours is denoted by N, where of course N ≥ 2. The quarks are
described by Dirac spinors for given momentum and spin, and also by a colour
label which we shall denote by a, b, c, . . .. All these labels (or indices) run from
1 to N. A conjugate fermion (u or v) will carry an upper, a regular fermion (u
or v) a lower index.
In addition we expect vector particles to be present, that carry the colour
force. These we call gluons. In analogy to QED, we shall assume the gluons to
be massless, but since we have different colour types there must also be different
gluon types. The gluon type will be denoted by j, k, l, m, . . ., and it is up to us
to determine 2 how many gluon type occur for given N.
1 Historically, the notion of quark predates that of colour, and the colouring of quarks was
invented to explain the possibility of the existence of curious particles such as the ∆ ++ or
the Ω − . In this chapter, we are less interested in describing the world of hadrons than in
constructing an internally consistent theory, hence the unhistorical line of reasoning.
2 The usual approch is simply to postulate a local SU(N) gauge symmetry, from which the
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