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

Chapter 3. Remarks on QCD in Coulomb Gauge 23
and therefore we get
H ⊥ = 1 [
Π
2
2 ⊥ + B 2] − 1 2 g2 0 J 0∆ −1 J 0 + g 0 A ⊥ · J . (3.34)
If we use the equality E ⊥ = Π ⊥ and integrate over the spatial coordinates, we get the
following expression for the Hamiltonian:
∫ [ ]
1
H = d 3 x
2 (E2 ⊥ + B2 ) + g 0 A ⊥ · J + H Coul , (3.35)
where we defined
H Coul := 1 2 g2 0
∫
d 3 xd 3 y ρ(x)V Coul (x,y)ρ(y) (3.36)
and used the notation
V Coul (x,y) = −∆ −1 | (x,y) . (3.37)
This is our final expression for the Hamiltonian of Maxwell theory and will be compared
later to the Hamiltonian of Yang-Mills theory in Coulomb gauge.
Some remarks are in order at this point. The fields A ⊥ and Π ⊥ possess only two
linearly independent components due to the transversality conditions
∂ i A i ⊥ = 0 , ∂ i Π i ⊥ = 0 . (3.38)
These components represent the two transversal photon polarisations. Using the Coulomb
gauge eliminates the unphysical degrees of freedom at the level of the Hamiltonian, which
means that timelike and longitudinal photons are no more present in the quantised theory.
Furthermore we notice, that the constraint φ 2 is Gauss’s law. It is therefore automatically
satisfied and does not have to be imposed after the quantisation as a constraint in terms
of operators on the physical states.
Our introduction of the first-class fields into the Hamiltonian made the quantity J 0
reappear in the expression
− 1 2 g2 0 J 0∆ −1 J 0 = 1 ∫
2 g2 0 d 3 y J 0(x)J 0 (y)
. (3.39)
4π|x − y|
This term is the well-known Coulomb energy density. The inverse of the Laplacian is
thereby known to be
∆ −1 | (x,y) = − 1 1
4π |x − y| . (3.40)
We already mentioned that the gauge condition (3.18) eliminates the unphysical degrees
of freedom. Therefore Coulomb gauge is called a physical gauge. The price we have to
pay for this feature is the violation of Lorentz invariance of Maxwell theory. However, the