Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

2.1 Klein-Gordon Equation(i∂ 0 ) 2 φ(x) =[(−i⃗∂) 2 + m 2 ]φ(x) (22)• is obviously a covariant wave equation, because(□ + m2 ) φ(x) = 0 (23)• fourier transformφ(x) =∫ d 4 p(2π) 4 e−ipx ˜φ(p) , i∂ µ φ(x) =∫ d 4 p(2π) 4 e−ipx p µ ˜φ(p) , etc (24)∴ algebraic equation (p 2 − m 2) ˜φ(p) = 0 (23 ′ )p 0“mass shell”:p 0 = + √ ⃗p 2 + m 2 ,p 2 = m 2 , p 0 0|⃗p|p 0 = − √ ⃗p 2 + m 2• correct relativistic dispersion relation E = + √ ⃗p 2 + m 2• but what about the other solution E = − √ ⃗p 2 + m 2 ?2.2 Free Spin-0 Particles• general solution of the Klein-Gordon equation∫ d 4 pφ(x) =(2π) 2πΘ(p 0)δ(p 2 − m 2 ) ( φ (+) (⃗p)e −ipx + φ (−) (⃗p)e ipx) (25)4∫ ∣d 3 ⃗p ∣∣∣∣p0 (=φ(2π) 3 2p 0=√(+) (⃗p)e −ipx + φ (−) (⃗p)e ipx) (26)⃗p 2 +m∫2= ˜dp ( φ (+) (⃗p)e −ipx + φ (−) (⃗p)e ipx) (27)• conserved current ∂ 0 j 0 (x) − ⃗∇⃗j(x) = ∂ µ j µ (x) = 0 out of two solutions φ 1 and φ 2of the Klein-Gordon equation with the same mass:j µ (x) = φ ∗ 1(x)i ←→ ∂ µ φ 2 (x) = φ ∗ 1(x)[i∂ µ φ 2 (x)] − [i∂ µ φ ∗ 1(x)]φ 2 (x) (28)5