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

Chapter 3. Remarks on QCD in Coulomb Gauge 21
and setting all constraints of the second-class ensemble in the Hamiltonian to zero.
We will now apply this to Maxwell theory. The Hamiltonian density is given as 3
H Λ = 1 2 (Π2 + B 2 ) − (g 0 J 0 + Π · ∇)A 0 + ΛΠ 0 + g 0 A · J , (3.15)
where Λ is the unknown velocity Ȧ0. Because H is a Hamiltonian density, integrating by
part under the assumpion that the surface term vanishes does not change the Hamiltonian
and yields
H Λ = 1 2 (Π2 + B 2 ) − (g 0 J 0 − ∇ · Π)A 0 + ΛΠ 0 + g 0 A · J . (3.16)
By temporal derivation of (3.7) and demanding stability for the constraints we can derive
a chain of secondary constraints,
Π 0 = 0 ⇒ ∂ i Π i = g 0 J 0 ⇒ 0 = 0 . (3.17)
Since we are dealing with a gauge theory, this chain does not provide any condition for
Λ. For the purpose of eliminating the arbitrariness we have to fix the gauge. As we are
interested in Coulomb gauge, we impose transversality of the spatial components of the
photon field by the condition
∂ i A i = 0 . (3.18)
By demanding that the gauge condition is valid for all times we get the stability chain
∂ i A i = 0 ⇒ ∆A 0 = −∂ i Π i ⇒ ∆Λ = 0 . (3.19)
The constraints (3.17) and (3.19) are second-class, which means, that the dimension of the
matrix C of the Poisson brackets is maximum. If we choose φ 1 = Π 0 , φ 2 = ∂ i Π i − g 0 J 0 ,
φ 3 = ∂ i A i and φ 4 = ∆A 0 + ∂ i Π i , this matrix reads
⎛
⎞
0 0 0 −∆
0 0 ∆ 0
C = ⎜
⎟
⎝ 0 −∆ 0 −∆⎠ δ(3) (x − y) . (3.20)
∆ 0 ∆ 0
Thus Coulomb gauge belongs to the class I type of gauge conditions. The inverse of C is
⎛
⎞
0 −∆ −1 0 ∆ −1
C −1 ∆ −1 0 ∆ −1 0
= ⎜
⎟
⎝ 0 ∆ −1 0 0 ⎠ δ(3) (x − y) . (3.21)
−∆ −1 0 0 0
3 For the necessary conventions see appendix B .