1. locality,Lorentz inv.gauge inv. spinor   vector A  interaction reprensatationperturbation S-matrix provability Wick's theorem

2. S-matrix provability

3. provability P ? k pk'k' p'p' electron quark scattering  provability amplitude ( p '≠ p, k '≠ k ) Let us consider electron quark scattering. We want to know the provability. For this purpose, we calculate the provability amplitude.

4. The higher contribution are the smller since g i is small. We examine it term by term. The first two terms vanish. The first non-trivial contribution comes from the n =2 term.  provability amplitude k pk'k' p'p' electron quark scattering provability P ? = bq†bq† be†be†    〉 0   p'≠p,k'≠k p'≠p,k'≠k 〈bq'be'〈bq'be' be†bq†〉be†bq†〉 0  0 bq'bq' be'be'    〈 = abbreviations 〈bq'be'〈bq'be' be†bq†〉be†bq†〉 0  0   e, q, … ∴∴ Let us see the n =2 term in detail.

5.  provability amplitude k pk'k' p'p' electron quark scattering provability P ? expand square q(x ')q(x ') A ( x ')   0  0  ' ' ' ' be†bq†〉be†bq†〉 〈 b' q b' e abbreviations change index  to change variable x to x' to avoid confusion with this  to avoid confusion with this x p'≠p p'≠p ∴ k'≠kk'≠k ∴ =

6. k pk'k' p'p' electron quark scattering provability P ? Wick's theorem because e & q are independent because b q † can be anihilated only by  q because b e † can be anihilated only by  e  provability amplitude

7. k pk'k' p'p' electron quark scattering provability P ?

8.  provability amplitude

9. T : T-matrix   1 k'  k   q q q q  k k' q collect all the x -dependent parts collect all the x '-dependent parts

10.  Feynman rules T : T-matrix 

11. 内線 photon fermion vertex 外線 incomingoutgoing 粒子 反粒子 loop fermion loop T matrix internal line external line particle ant- particle

14. loop diagram

16. Feynman rules

25. scattering cross section  target cross section A area N 2 # of targets N 1 # of projectiles N s # of scatterings scattering provability flux reaction a 1 +a 2 → a 3 +a 4 momentum mass energy volume of space total time

28. invariant variables 不変変数 center-of-mass frame 重心系 laboratory frame 実験室系 time s-channel t-channel s-channel t-channel u-channel reaction 反応

29. center-of-mass frame laboratory frame

31. decay width  = decay provability/total time = # of decays/ # of particles /total time # of decays # of particles total time

33. center-of-mass frame

34. k pk'k' p'p' fermion fermion scattering

36. 散乱断面積 scattering cross section 微分断面積 differential scattering cross section

37. differential scattering cross section 微分散乱断面積 scattering cross section 散乱断面積 n i :# of sipn states of particle a i unpolarized scattering cross section 非偏極散乱断面積 j i :spin quantum # (massive particle) (massless particle)