and The identity (20) can be simplified by redefining Γ: so that we obtain and Let us now consider some specific applications of the above identities. As a first application of the identities derived in the previous section, we will now prove that the longitudinal part of the photon propagator is not modified by radiative corrections. To this end, we twice differentiate the generating functional of the connected Green functions (15) with respect to ζ ( y ) and J $^{ν}$( z ) and set all sources equal to zero, which gives where we have introduced the notation O| to imply that the object O at the left of the vertical bar has to be calculated with all sources equal to zero. But, from Eq. (16) it follows that so that the photon propagator D$_{µν}$ ( z − y ) = − i δ $^{2}$W δJ $^{ν}$( z ) δJ $^{µ}$( y ) | must satisfy which in momentum space becomes Now, compatibility with this constraint requires the propagator to have the general form i δ Γ δ ¯ C$_{cl}$ = − e∂$_{µ}$ δ Γ δK$_{µ}$ . (21) Γ → Γ − 1 2 α ∫ d $^{4}$x ( ∂$_{µ}$A µ $_{cl}$) 2 (22) ∫ d $^{4}$x [ δ Γ δA µ cl ⋆ δ Γ δK$_{µ}$ + δ Γ δC$_{cl}$ ⋆ δ Γ δv + δ Γ δψ ⋆ δ Γ δ ¯ ω + δ Γ δω ⋆ δ Γ δ ¯ ψ ] = 0 (23) i δ Γ δ ¯ C$_{cl}$ + e∂$_{µ}$ δ Γ δK$_{µ}$ = 0 . (24) − 1 αe ∂ µ y δ $^{2}$W δJ $^{ν}$( z ) J $^{µ}$( y ) ∣ + δ $^{2}$W δζ ( y ) δK $^{ν}$( z ) ∣ = 0 , (25) ∂ ν z δ $^{2}$W δζ ( y ) δK $^{ν}$( z ) ∣ = 1 e δ ( z − y ) (26) ∂ µ z ∂ ν $_{y}$D$_{µν}$ ( z − y ) = − iαδ ( z − y ) , (27) q $^{µ}$q$^{ν}$D$_{µν}$ ( q ) = − iα. (28) D$_{µν}$ ( q ) = ( g µν − q $^{µ}$qν q 2 ) D$_{T}$ ( q $^{2}$) + ˜ q $^{µ}$˜ q ν ˜ q 2 D$_{θ}$ ( q $^{2}$) − iα q 2 q $^{µ}$qν q 2 . (29) 6 A. The photon propagator