1. One particle states: Wave Packets States

2. Heisenberg Picture

3. Combine the two eq. KG Equation

4. Dirac field and Lagrangian The Dirac wavefunction is actually a field, though unobservable! Dirac eq. can be derived from the following Lagrangian.

5. Negative energy!

6. Anti-commutator! A creation operator!

7. b annihilate an antiparticle!

8. Exclusion Principle

9. Now add interactions: For example, we can add to our Klein-Gordon or Dirac Lagrangian. Interaction Hamiltonian:

10. Schrodinger Picture

11. Heisenberg Picture We can move the time evolution t the operators: Heisenberg Equation

12. Interaction picture S States and Operators both evolve with time in interaction picture:

13. Evolution of Operators Operators evolve just like operators in the Heisenberg picture but with the full Hamiltonian replaced by the free Hamiltonian Field operators are free, as if there is no interaction!

14. Evolution of States S States evolve like in the Schrodinger picture but with Hamiltonian replaced by V(t). V(t) is just the interaction Hamiltonian H I in interaction picture! That means, the field operators in V(t) are free.

15. Operators evolve just like in the Heisenberg picture but with the full Hamiltonian replaced by the free Hamiltonian States evolve like in the Schrodinger picture but with the full Hamiltonian replaced by the interaction Hamiltonian. Interaction Picture

16. Define time evolution operator U All the problems can be answered if we are able to calculate this operator. It’s determined by the evolution of states. U operator

17. Solve it by a perturbation expansion in small parameters in H I. To leading order: Perturbation expansion

18. Define S matrix: It is Lorentz invariant if the interaction Lagrangian is invariant.

19. Vertex Add an interaction term in the Lagrangian: The transition amplitude for the decay of A: can be computed: To leading order: In ABC model, every particle corresponds to a field:

20. A B C ig Numerical factors remain Momentum Conservation

21. A B C Every field operator in the interaction corresponds to one leg in the vertex. Every field is a linear combination of a and a + interaction Lagrangianvertex Every leg of a vertex can either annihilate or create a particle! This diagram is actually the combination of 8 diagrams!

22. This is in momentum space. The integration yields a momentum conservation. A B C interaction Lagrangianvertex There is a spacetime integration. Interaction could happen anytime anywhere and their amplitudes are superposed.

23. Every field operator in the interaction corresponds to one leg in the vertex. interaction Lagrangianvertex Every leg of a vertex can either annihilate or create a particle! Every field operator in the interaction corresponds to one leg in the vertex. interaction Lagrangianvertex Every leg of a vertex can either annihilate or create a particle?

24. interaction Lagrangianvertex Every leg of a vertex can either annihilate or create a particle? can either annihilate a particle or create an antiparticle! can either annihilate an antiparticle or create a particle! The charge flow is consistent! So we can add an arrow for the charge flow.

25. Feynman Rules for an incoming particle External line When Dirac operators annihilate states, they leave behind a u or v ! Feynman Rules for an incoming antiparticle

27. Propagator The integration of two identical interaction Hamiltonian H I. The first H I is always later than the second H I This definition is Lorentz invariant!

28. Amplitude for scattering Propagator between x 1 and x 2 Fourier Transformation p 1 - p 3 pour into x 2 p 2 - p 4 pour into x 1

29. C x2x2 x1x1 A(p 1 )A(p 2 ) B(p 3 ) C(p 1 -p 3 ) B(p 4 ) A(p 1 )A(p 2 ) B(p 3 ) B(p 4 ) A particle is created at x 2 and later annihilated at x 1.

30. C x2x2 x1x1 C x2x2 x1x1 A(p 1 )A(p 2 ) B(p 3 ) C(p 1 -p 3 ) B(p 4 ) A(p 1 ) A(p 2 ) B(p 3 ) B(p 4 ) A particle is created at x 1 and later annihilated at x 2.

31. C x2x2 x1x1 C x2x2 x1x1 A(p 1 )A(p 2 ) B(p 3 ) C(p 1 -p 3 ) B(p 4 ) A(p 1 ) A(p 2 ) B(p 3 ) B(p 4 )

33. This doesn’t look explicitly Lorentz invariant. But it is!

36. Every field either couple with another field to form a propagator or annihilate (create) external particles! Otherwise it will vanish!

37. Antiparticles can be introduced easily by assuming that the field operator is a complex number field. Complex KG field can either annihilate a particle or create an antiparticle! Its conjugate either annihilate an antiparticle or create a particle! The charge flow is consistent! So we can add an arrow for the charge flow. Scalar Antiparticle

38. vertex Charge non-conserving

39. vertex Charge conserving

40. C x2x2 x1x1 A(p 1 )A(p 2 ) B(p 3 ) C(p 1 -p 3 ) B(p 4 ) A(p 1 )A(p 2 ) B(p 3 ) B(p 4 ) An antiparticle is created at x 2 and later annihilated at x 1. Propagator:

41. C x2x2 x1x1 C x2x2 x1x1 A(p 1 )A(p 2 ) B(p 3 ) C(p 1 -p 3 ) B(p 4 ) A(p 1 ) A(p 2 ) B(p 3 ) B(p 4 ) A particle is created at x 1 and later annihilated at x 2.

42. C x2x2 x1x1 C x2x2 x1x1 A(p 1 )A(p 2 ) B(p 3 ) C(p 1 -p 3 ) B(p 4 ) A(p 1 ) A(p 2 ) B(p 3 ) B(p 4 )

43. C x2x2 x1x1 C x2x2 x1x1 A(p 1 )A(p 2 ) B(p 3 ) C(p 1 -p 3 ) B(p 4 ) A(p 1 ) A(p 2 ) B(p 3 ) B(p 4 )

44. U(1) Abelian Symmetry The Lagrangian is invariant under the phase transformation of the field operator: invariant

45. A B C If A,B,C become complex, they carry charges! The interaction is invariant only if U(1) symmetry is related to charge conservation!

46. The Dirac Fermion Lagrangian is also invariant under U(1)

47. SU(N) Non-Abelian Symmetry Assume there are N kinds of fields If they are similar, we have a SU(N) symmetry! are invariant under SU(N)!

48. u-d 互換對稱 量子力學容許量子態的疊加 a+ b c+ d u u u d d dd u 量子力學下互換群卻變得更大！ 古典 量子

49. They are invariant under SU(N)!

50. Gauge symmetry

51. Gauge (Local) symmetry Kinetic energy is not invariant under gauge transformation! Global Symmetry

52. Could we find a new “derivative” that works as if the transformation is global? To get rid of the extra term, we introduce a new vector field:

53. Gauge (Local) symmetry is invariant under gauge transformation! Global Symmetry Replacing derivative with covariant derivative,

54. The scalar photon interaction vertices

55. To force it to be gauge invariant, you only need to replace derivative with coariant derivative. is gauge invariant!

56. This gauge invariant Lagrangian gives a definite interaction between fermions and photons

58. This form is forced upon us by gauge symmetry! It is really a Fearful Symmetry! Tony Zee Tyger! Tyger! burning bright In the forests of the night What immortal hand or eye Could frame thy fearful symmetry! William Blake

59. Let there be light! In the name of gauge symmetry!

60. Hermann Weyl, 1885-1955

61. Yang and Mills

62. SU(N) Non-Abelian Symmetry Assume there are N kinds of fields If they are similar, we have a SU(N) symmetry! are invariant under SU(N)!

63. Non-Abelian Gauge Symmetry We need one gauge field for each generator. Gauge fields transform as: is invariant under gauge transformation!

65. 2 × 2 matrices

68. Vacua happen at: Choose: For infinitesimal transformation: SU(2)χU(1) Y is broken into U(1) EM

69. W become massive Z become massive Photon is massless.