The QCD Quark Propagator in Coulomb Gauge and - Institut für Physik

Chapter 3. Remarks on QCD in Coulomb Gauge 19
whereas approximate equality was found with a different method for the SU(2) group
[CZ03]. In a second period of lattice calculations the eigenvalue spectrum of the Faddeev-
Popov operator became the main topic of investigation. There confinement is related to
the near-zero eigenmodes of the Faddeev-Popov operator [GOZ05] and relations to the
vortex confinement scenario are found.
3.4 Quantisation of Maxwell theory
In the following we will show how to formulate Yang-Mills theory in Coulomb gauge. It
will make apparent how this approach provides access to the Coulomb potential of colour
charges in an elegant way. For the sake of comparison with the Abelian case we begin by
quantising Maxwell theory in this gauge 1 .
For the quantisation of gauge theories standard procedures are not applicable. Looking
at the theory of electromagnetism this becomes clear. The Lagrangian density of this field
theory is given by
L = − 1 4 F µνF µν + g 0 A µ j µ . (3.5)
Here the four-current j µ = (ρ,j) is a function of the matter fields. Computing the conjugated
momentum fields gives
Explicitly they read
Π µ =
Imposing usual rules of quantisation 2 , we set
∂L
∂(∂ 0 A µ ) . (3.6)
Π 0 = 0 (3.7)
Π i = F i0 . (3.8)
[µ(x, t), ˆΠ ν (y, t)] = iδ ν µδ (3) (x − y) . (3.9)
Comparing this to (3.7) we have clearly a contradiction. Modified rules for quantisation
with constraints are required. The origin of their formulation is due to Dirac [Dir].
From the viewpoint of Dirac quantisation, the choice of a gauge consists of the replacement
of an arbitrary function by a well-defined one in the Hamiltonian. The gauge
conditions can be classified into three classes.
• Class I: Gauge conditions involving only A a k
canonically conjugate momenta πk a.
and their
1 This chapter employs the conventions A.1 and B .
2 A hat “ ˆ ” over a quantity is supposed to denote the associated operator.