Notice that, because of the charge conjugation properties [9, 11], terms of the type are not allowed in the decomposition (29). Thus the longitudinal part of the propagator is the same as in the free approximation. Notice also that which will be useful in the next section when we will analyze the ST identity for the vectorial vertex function. The ST identity for the vectorial fermion-photon vertex, the proper part of 〈 0 | T ( ψ ¯ ψA$_{µ}$ ) | 0 〉 , can be derived by turning off all the sources after differentiating the functional equation (15) with respect the sources η ( y ), ¯ η ( x ), and ζ ( z ). The result is or, equivalently, i.e., where ∂$_{x}$ ∧ ∂$_{ˆ}$ x = 1 $_{2}$ξθ µν ∂ ∂x µ ∂ ∂ ˆ x $_{ν}$. Notice that as consequence of this identity We may translate the above equations into identities for the proper, one-particle irreducible, vertex functions. These functions are given by 7 ˜ q µ q ν + ˜ q ν q µ ˜ q 2 (30) q $^{µ}$D$_{µν}$ ( q ) = − i αq$_{ν}$ q 2 , (31) δ 〈 0 | T ( ψ ( x ) ¯ ψ ( y ) ¯ C ( z )) | 0 〉 = 0 . B. The vectorial vertex function 1 αe ∂ µ z δ $^{3}$W δ ¯ η ( x ) δη ( y ) δJ $^{µ}$( z ) ∣ = δ $^{3}$W δζ ( z ) δ ¯ η ( x ) δω ( y ) ∣ − δ $^{3}$W δζ ( z ) δη ( y ) δ ¯ ω ( x ) ∣ , (32) 1 αe ∂ µ z 〈 0 | T ( ψ ( x ) ¯ ψ ( y ) A$_{µ}$ ( z )) | 0 〉 = i 〈 0 | T ( ¯ C ( z ) ψ ( x ) ¯ ψ ( y ) ⋆ C ( y )) | 0 〉 − i 〈 0 | T ( ¯ C ( z ) ¯ ψ ( y ) C ( x ) ⋆ ψ ( x )) | 0 〉 , (33) 1 αe ∂ µ z 〈 0 | T ( ψ ( x ) ¯ ψ ( y ) A$_{µ}$ ( z )) | 0 〉 = ie i∂$_{x}$ ∧ ∂$_{ˆ}$ x 〈 0 | T ( ψ (ˆ x ) ¯ ψ ( y ) C ( x ) ¯ C ( z )) | 0 〉|$_{ˆ}$ x = x − ie i∂$_{y}$ ∧ ∂$_{ˆ}$ y 〈 0 | T ( ψ ( x ) ¯ ψ ( y ) C (ˆ y ) ¯ C ( z )) | 0 〉|$_{ˆ}$ y = $_{y}$, (34) 〈 0 | T ( ψ ( x ) ¯ ψ ( y ) A$_{µ}$ ( z )) | 0 〉 = − ∫ d $^{4}$x $^{′}$d $^{4}$y $^{′}$d $^{4}$z $^{′}$S$_{F}$ ( x − x $^{′}$)Γ $^{ν}$( x $^{′}$, y $^{′}$, z $^{′}$) S$_{F}$ ( y ′ − y ) D$_{µν}$ ( z − z $^{′}$) , (35)