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

Chapter 3. Remarks on QCD in Coulomb Gauge 25
If we insert this expression in the partition function, we find
∫ { ∫ [
Z = DADΠ exp i dx Π i ( A ˙i
− ∂ i A 0 ) − 1 2 (Π2 + B 2 ) + g 0 A µ J
]}δ(∂ µ i A i ) . (3.47)
Performing the Gaussian integral over Π we obtain the final result:
∫ { ∫ }
Z = DA exp i Ldx δ(∂ i A i ) , L = − 1 4 F µνF µν + g 0 A µ J µ . (3.48)
Our transformations end with the appearance of the Maxwell Lagrangian and the Coulomb
gauge condition as the only constraint in form of a delta function. It is possible to recast
this in a way, which emphasises the physical degrees of freedom A ⊥ and Π ⊥ . To this end
we start from (3.45) and separate the transverse and longitudinal parts of the Π field,
Π = Π ⊥ − ∇φ. The integration measure thus becomes DΠ ≃ DΠ ⊥ Dφ. Reintroducing
the irrelevant factor Det[−∆], we can use
Det[−∆]δ(∂ i Π i − g 0 J 0 ) = Det[−∆]δ(−∆ −1 φ − g 0 J 0 ) = δ(φ + g 0 ∆ −1 J 0 ) . (3.49)
and perform the φ-integration to obtain
∫
Z = DA ⊥ DΠ ⊥ exp
{ ∫
i
}
dx(Π ⊥,i Ȧ i − H ⊥ )
, (3.50)
where H ⊥ is the Hamiltonian density (3.34).
3.5 Quantisation of Yang-Mills theory
At this point we are ready to quantise a non-Abelian gauge theory of the gauge group
SU(N) in Coulomb gauge.
We start with the Lagrangian density
The conjugate momenta are
L = − 1 4 F a µνF µν
a + g 0 A a µJ µ a . (3.51)
Π 0 a = 0 , Πi a = F i0
a , (3.52)
and thus we obtain a primary constraint as we did in (3.7). Postulating the stability of
this condition gives us the chain of secondary constraints
Π 0 a = 0 ⇒ [D iΠ i ] a = g 0 J 0 a ⇒ 0 = 0 . (3.53)