Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

k = (k 0 ; |⃗k| sin θ cos φ, |⃗k| sin θ sin φ, |⃗k| cos θ):ɛ ± = ɛ ∗ ∓ = e∓iφ√2(0; cos θ cos φ ∓ i sin φ, cos θ sin φ ± i cos φ, − sin θ)(81a)• properties (79) and3 Interactions3.1 Propagatorsɛ 0 = k 0M (| ⃗k|/k 0 ; sin θ cos φ, sin θ sin φ, cos θ) = ɛ ∗ 0∑λ=−1,0,+1ɛ µ λ ɛν,∗ λ• Photons far from all electrical charges satisfyand we have already seen the solutions.(81b)= −g µν + k µk νM 2 (82)∂ 2 A (0)µ (x) = 0 (83)• In the presence of electrical charges, the photons couple to the electromagneticcurrent∂ 2 A µ (x) = j µ (x) = −e ¯ψ(x)γ µ ψ(x) + . . . (84)and the solutions turn out to be more complicated.• Assumption: there is a “function” D, that solves• Then(∂ 2 + m 2 )D(x, m) = −δ 4 (x) (85)∫A µ (x) = A (0)µ (x) − d 4 yD(x − y, 0)j µ (y) (86)is a solution of the inhomogeneous equation (84) for each solution of the homogeneousequation (83), since∫ [∂ 2 A µ (x) = ∂ 2 A (0)µ (x) − d 4 y]∂ 2 D(x − y, 0) j µ (y)= 0 −∫ [ ]d 4 y −δ 4 (x − y) j µ (y) = j µ (x) (87)• interpretation: the current j µ (y) acts at the space time point y as a source ofphotons, that are “propagated” by the propagator D(x − y, 0) to the space timepoint xj µ (y)D(x − y, 0)A µ (x)(86 ′ )15