1. 10 lectures

2. classical physics: a physical system is given by the functions of the coordinates and of the associated momenta – 2

3. quantum physics: coordinates and momenta are Hermitean operators in the Hilbert space of states 3

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5. 5

6. 6

7. 7

8. 8

9. 9

11. 11

12. 12

13. 13

14. Gauß curve 14

15. 15

16. Symmetry in Quantum Physics 16

17. A. external symmetries B. internal symmetries 17

18. external symmetries: Poincare group conservation laws: energy momentum angular momentum 18

19. external symmetries: exact in Minkowski space 19

20. General Relativity: no energy conservation no momentum conservation no conservation of angular momentum 20

21. internal symmetries: Isospin SU(3) Color symmetry Electroweak gauge symmetry Grand Unification: SO(10) Supersymmetry 21

22. internal symmetries broken by interaction: isospin broken by quark masses: SU(3) broken by SSB: electroweak symmetry unbroken: color symmetry 22

23. symmetries are described mathematically by groups 23

24. symmetry groups n finite or infinite 24

25. examples of groups: integer numbers: 3 + 5=8, 3 + 0=3, 5 + (-5) = 0 real numbers: 3.20 x 2.70=8.64, 3.20 x 1 = 3.20 3.20 x 0.3125 = 1 25

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27. 27

28. A symmetry is a transformation of the dynamical variables, which leave the action invariant. 28

29. Classical mechanics:  translations of space and time – ( energy, momentum ) rotations of space ( angular momentum ) 29

30. Special Relativity => Poincare group: translations of 4 space - time coordinates + Lorentz transformations 30

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32. Symmetry in quantum physics ( E. Wigner, 1930 … ) U: unitary operator 32

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35. 35

36. 36

37. Poincare group P: - time translations - - space translations - - rotations of space - - „rotation“ between time and space - 37

38. e.g. rotations of space: - - 38

39. Casimir operator of Poincare group 39

40. The operator U commutes with the Hamiltonoperator H: If U acts on a wave function with a specific energy, the new wave function must have the same energy ( degenerate energy levels ). 40

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42. discrete symmetries 42

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44. 44

45. P: exact symmetry in the strong and electromagnetic interactions 45

46. P: maximal violation in the weak interactions 46

47. 47

48. theory of parity violation: 1956: T. D. Lee and C.N.Yang experiment: Chien-Shiung Wu ( Columbia university ) 48

49. Lee Yang Wu 49

50. Experiment of Wu: beta decay of cobalt 50

51. 51

52. electrons emitted primarily against Cobalt spin (  violation of parity ) 52

53. 1958 Feynman, Gell-Mann Marshak, Sudarshan maximal parity violation lefhanded weak currents 53

54. CP – violation: weak interactions were CP invariant, until 1964: CP violation found at the level of 0.1% of the parity violation in decay of neutral K-mesons (James Cronin and Val Fitch, 1964 ) 54

55. present theory of CP-violation: phase in the mixing matrix of the quarks 55

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58. 58

59. V. Weisskopf – W. Pauli (~1933) the Klein-Gordon field is not a wave function, but describes a scalar field 59

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68. 68

69. Goudsmit – Uhlenbeck 1924 a new discrete quantum number 69

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72. 72

73. angular momentum: 73

74. 74

75. Spin of particles: pi-meson: 0 electron, proton: ½ photon: 1 delta resonance: 3/2 graviton: 2 75

76. 76

77. matter particles have spin ½ => fermions ( electron, proton, neutron ) force particles have spin 1 => bosons ( photon, gluons, weak bosons ) 77

78. 78

79. Klein-Gordon equation: no positive definite probability density exists Dirac 1927: search for a wave equation, in which the time derivative appears only in the first order ( Klein- Gordon equation: second time derivate is needed ) 79

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82. 82

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84. positron 84