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

Chapter 3. Remarks on QCD in Coulomb Gauge 27
Integrating over A a 0 leads to
∫
Z = DADΠexp
{ ∫ [
i Π a i Ȧi a − 1 ]}
2 (Π2 + B 2 ) − g 0 A a · J a ·
δ(∂ i A i a ) δ([D iΠ i ] a − g 0 J0 a )Det[−∇ · D] , (3.60)
which exhibits the validity of Gauss’s law in the integrand. This expression is the analogue
to (3.45). If we separate the transverse and longitudinal parts of the conjugated fields,
Π a = Π a ⊥ − ∇φ a , (3.61)
the integration measure factorises to DΠ ⊥ Dφ. Using ∇ · D = D · ∇, which is due to
the transversality of the gauge fields, and the definition of the colour-charge density of
the gluons ρ a gl = fabc A b i Ei c we can absorb the determinant in the delta function of (3.60),
which enforces Gauss’s law. Integrating over φ yields finally
∫
{ ∫
}
Z = DA ⊥ DΠ ⊥ exp i (Π a ⊥,iȦi,a − H ⊥ ) . (3.62)
The Hamiltonian density H ⊥ is derived from (3.54) by substituting A → A ⊥ and Π → Π ⊥
and setting all constraints to zero. Supressing colour indices it is
H ⊥ = 1 2 (Π2 ⊥ + B2 ) − 1 2 g2 0 (ρ gl + J 0 )(−∇ · D) −1 ∆(−∇ · D) −1 (ρ gl + J 0 ) + g 0 A ⊥ · J . (3.63)
Therefore the Yang-Mills analogues to (3.35) and (3.36) are
∫ [ ]
1
H = d 3 x
2 (E2 ⊥ + B2 ) + g 0 A a ⊥ · Ja + H Coul (3.64)
and
H Coul = 1 2 g2 0
∫
d 3 xd 3 yρ a (x)V ab
Coul (x,y)ρb (y) . (3.65)
Thereby we used the definitions
V ab
Coul(x,y) := M −1 (−∆)M −1 | ab
(x,y) , (3.66)
ρ := ρ gl + J 0 , (3.67)
M := −∇ · D . (3.68)
The operator M is called the Faddeev-Popov operator and plays an important role in the
Gribov-Zwanziger scenario of colour confinement. In the Abelian case of Maxwell theory
the Faddeev-Popov operator reduces simply to the Laplacian, M = −∆, and V ab
Coul (x,y),
which is the analogue of (3.37), becomes the Coulomb potential of electrodynamics.