Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

2.7 Free Spin-1 Particles• combine ⃗E and ⃗B into field strength tensor⎛⎞0 −E 1 −E 2 −E 3F µν = ∂ µ A ν − ∂ ν A µ = ⎜E 1 0 −B 3 B 2⎟⎝E 2 B 3 0 −B 1⎠ (72)E 3 −B 2 B 1 0• manifestly covariant Maxwell equations• current j µ necessarily conserved: ∂ µ j µ = 0• equation of motion for the vector potential A µ• gauge invariance: F µν does not change, if• with special gauge condition ∂ µ A µ = 0: ∂ 2 A µ = j µ• more, but not most, general case∂ µ F µν = j ν , ɛ µνρσ ∂ ν F ρσ = 0 (73)(g µν ∂ 2 − ∂ µ ∂ ν )A ν = j µ (73 ′ )A µ (x) → A µ (x) − ∂ µ ω(x) (74)(g µν ∂ 2 − (1 − ξ)∂ µ ∂ ν )A ν = j µ (75)• explicit mass termor∂ µ F µν + M 2 A ν = j ν (76)(g µν (∂ 2 + M 2 ) − ∂ µ ∂ ν )A ν = j µ (76 ′ )• contraction with ∂ µM 2 ∂ ν A ν = 0 (77)• massless vector bososns have two degrees of freedom, massive ones three• polarization vectors for massless vector bosons with momentum k = (k 0 ; 0, 0, k 0 )ɛ ± = ɛ ∗ ∓ = 1 √2(0; 1, ±i, 0) (78)• propertiesand with c = (1; 0, 0, −1)∑λ=−1,+1ɛ µ λ ɛ∗ λ ′ ,µ = −δ λλ ′ɛ µ λ k µ = 0ɛ µ λ ɛν,∗ λ= −g µν + c µk ν + c ν k µck(79a)(79b)(80)• polarization vectors for massive vector bosons with momentum14