II. NONCOMMUTATIVE QED$_{4}$ Classically, the noncommutative QED$_{4}$ is described by the action where F$_{µν}$ = ∂$_{µ}$A$_{ν}$ − ∂$_{ν}$A$_{µ}$ − ie [ A$_{µ}$, A$_{ν}$ ]$_{⋆}$, with [ A$_{µ}$, A$_{ν}$ ]$_{⋆}$ = A$_{µ}$ ⋆ A$_{ν}$ − A$_{ν}$ ⋆ A$_{µ}$ , is the field strength, D$_{µ}$ψ = ∂ψ − ieA$_{µ}$ ⋆ ψ is a gauge covariant derivative and the star (Moyal) product is defined by where θ$_{µν}$ is a real antisymmetric matrix and ξ is a parameter which sets the strength of the noncommutativity. The above action is invariant under the gauge transformations To complete the quantum version of the model, we need to add to (1) a gauge fixing, S$_{GF}$ , and the corresponding Faddeev-Popov, S$_{F P}$ , actions. For the general class of Lorentz gauges in which we will work where α is the gauge fixing parameter. As it happens in the commutative gauge theories, the total action S = S$_{IN V}$ + S$_{GF}$ + S$_{F P}$ is not invariant under gauge transformations anymore but instead has a BRST symmetry such that where λ is a constant Grassmmanian parameter. At a formal level, the invariance of the action under these transformations imply in relations between the Green functions as it will be shortly verified. For an explicit calculation, we will need the Feynman rules for the model which are fixed S$_{IN V}$ = ∫ d $^{4}$x [ − 1 4 F$_{µν}$ ⋆ F µν + ¯ ψ ⋆ ( i / D − m ) ψ ] , (1) φ$_{1}$ ( x ) ⋆ φ$_{2}$ ( x ) ≡ e i $_{2}$ξθ µν ∂ ∂x µ ∂ ∂y µ φ$_{1}$ ( x ) φ$_{2}$ ( y ) |$_{y}$$_{=}$$_{x}$ , (2) δA$_{µ}$ = 1 e D$_{µ}$ Λ ≡ 1 e ( ∂$_{µ}$ Λ − ie [ A$_{µ}$, Λ]$_{⋆}$) , δψ = i Λ ⋆ ψ ( δ ¯ ψ = − i ¯ ψ ⋆ Λ) . (3) S$_{GF}$ + S$_{F P}$ = ∫ d $^{4}$x [ − 1 2 α ( ∂$_{µ}$A $^{µ}$) 2 ⋆ + ∂$_{µ}$ ¯ C ⋆ ( ∂ $^{µ}$C − ie [ A $^{µ}$, C ]$_{⋆}$) ] , (4) δA$_{µ}$ = − 1 e ( ∂$_{µ}$C − ie [ A$_{µ}$, C ]$_{⋆}$) λ, δψ = − iCλ ⋆ ψ ( δ ¯ ψ = i ¯ ψ ⋆ Cλ ) , δC = iC ⋆ Cλ, δ ¯ C = − 1 αe ( ∂$_{µ}$A $^{µ}$) λ, (5) 3