Quantum Field Theory

which is the claimed result. You can similarly check that the same substitution is legalin the diagramp/p∼/qe 2 [¯v(⃗q)γ µ u(⃗p)]D µν (k)[ū(⃗p ′ )γ ν v(⃗q ′ )] (6.90)qIn fact, although we won’t show it here, it’s a general fact that in every Feynman diagramwe may use the very nice, Lorentz invariant propagator D µν = −iη µν /p 2 . □Note: This is the propagator we found when quantizing in Lorentz gauge (using theFeynman gauge parameter). In general, quantizing the Lagrangian (6.37) in Lorentzgauge, we have the propagatorD µν = − i (ηp 2 µν + (α − 1) p )µp ν(6.91)p 2Using similar arguments to those given above, you can show that the p µ p ν /p 2 termcancels in all diagrams. For example, in the following diagrams the p µ p ν piece of thepropagator contributes as∼ ū(p ′ )γ µ u(p) k µ = ū(p ′ )( /p − /p ′ )u(p) = 0∼ ¯v(p)γ µ u(q) k µ = ū(p)( /p + /q ′ )u(q) = 0 (6.92)6.5 Feynman RulesFinally, we have the Feynman rules for QED. For vertices and internal lines, we write• Vertex: −ieγ µ• Photon Propagator: − iη µνp 2 + iǫi( /p + m)• Fermion Propagator:p 2 − m 2 + iǫFor external lines in the diagram, we attach• Photons: We add a polarization vector ǫ µ in /ǫµ out for incoming/outgoing photons.In Coulomb gauge, ǫ 0 = 0 and ⃗ǫ · ⃗p = 0.• Fermions: We add a spinor u r (⃗p)/ū r (⃗p) for incoming/outgoing fermions. We adda spinor ¯v r (⃗p)/v r (⃗p) for incoming/outgoing anti-fermions.– 143 –