Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

• from which we can construct solutions for arbitrary on shell momenta with p 2 =m 2 u k (p) = √ /p + mp0 + m u k( ⃗ 0)(57a)v k (p) =/p − m √p0 + m v k( ⃗ 0)(57b)• since (/p+m)(/p−m) = p 2 −m 2 , these are obviously solutions of the Dirac equationfor on-shell momenta• the motivation for the not obviously covariant normalization will be apparentafter problem 4mathematical reminder:• compare the inner product of a row vector with a column vector⎛ ⎞b 1( )b 2a1 a 2 · · · a n ⎜ ⎟⎝ . ⎠ = ∑ na i b i (58)i=1b nto the outer product of a column vector with a row vector⎛ ⎞⎛⎞a 1a 1 b 1 a 1 b 2 . . . a 1 b ma 2⎜ ⎟⎝ . ⎠ ⊗ ( ) a 2 b 1 a 2 b 2 . . . a 2 b mb 1 b 2 · · · b m = ⎜⎟⎝ . . . ⎠a n a n b 1 a n b 2 . . . a n b m(59)• these are two very different operations– the inner product produces a number– the outer product produces a matrixProblem 4. Determine ū k (p) und ¯v k (p) from the definitions and show that for p 2 = m 22∑u k (p)ū k (p) = /p + m ,k=12∑v k (p)¯v k (p) = /p − m (60)k=1Problem 5. Compute (always assuming p 2 = m 2 )ū k (p)u l (p) , ¯v k (p)v l (p) , ū k (p)v l (p) , ¯v k (p)u l (p) . (61)Problem 6. Compute (always assuming p 2 = m 2 )ū k (p)γ µ u l (p) , ¯v k (p)γ µ v l (p) , ū k (⃗p)γ 0 v l (−⃗p) , ¯v k (−⃗p)γ 0 u l (⃗p) . (62)11