Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

• this is a special case of the Gordon decomposition for arbitrary solutions of theDirac equation:ū k (p)γ µ u l (q) = ū k (p) /pγ µ + γ µ /q2mu l(q)= ū k (p) p ν(g νµ + 1[γ 2 ν, γ µ ] − ) + q ν (g µν + 1[γ 2 µ, γ ν ] − )u l (q)2m= p µ + q µ2m ūk(p)u l (q) + q ν − p ν2im ūk(p) i 2 [γ µ, γ ν ] − u l (q) (63)• there are alltogether 16 independent (anti-)hermitian 4 × 4-matrices:1 1 “scalar” (64a)γ µ 4 “vector” (64b)σ µν = i 2 [γ µ, γ ν ] −6 “tensor” (64c)γ 5 γ µ 4 “axial vector” (64d)γ 5 = iγ 0 γ 1 γ 2 γ 3 1 “pseudo scalar” (64e)• NB: the bare gamma matrices do not transform like vectors, tensors, axial vectors,or pseudo scalars!• there are additional nontrivial transformations L(Λ) that have to be applied onthe left and right, e. g.γ µ → Λµ ν L(Λ)γ ν L −1 (Λ) (65)• however sinceψ(x) → L(Λ)ψ(Λ −1 x) , ¯ψ(x) → ¯ψ(Λ −1 x)L −1 (Λ) (66)the L(Λ) compensate each other in matrix elements (a. k. a. “sandwiches”)¯ψ(x)γ µ ψ(y) → ¯ψ(Λ −1 x)L −1 (Λ)Λ νµ L(Λ)γ ν L −1 (Λ)L(Λ)ψ(Λ −1 x)= Λ νµ ¯ψ(Λ −1 x)γ ν ψ(Λ −1 x) (67)∴ and the L(Λ) can be ignored in the computation of matrix elements∴ the characterization as vector, tensor, axial vector, or pseudo scalars is meaningfulProblem 7. Compute [γ 5 , γ µ ] +Problem 8. Show the conservation of the vector current for two solutions ψ 1 (x) und ψ 2 (x) ofthe Dirac equation (32) and (51)∂ µ [ ¯ψ 1 (x)γ µ ψ 2 (x) ] = 0 . (68)12