10 lectures

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

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

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Gauß curve 14

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Symmetry in Quantum Physics 16

A. external symmetries B. internal symmetries 17

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

external symmetries: exact in Minkowski space 19

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

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

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

symmetries are described mathematically by groups 23

symmetry groups n finite or infinite 24

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 = x = 1 25

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

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

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

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

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

e.g. rotations of space:

Casimir operator of Poincare group 39

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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discrete symmetries 42

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P: exact symmetry in the strong and electromagnetic interactions 45

P: maximal violation in the weak interactions 46

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

Lee Yang Wu 49

Experiment of Wu: beta decay of cobalt 50

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

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

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

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

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

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Goudsmit – Uhlenbeck 1924 a new discrete quantum number 69

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angular momentum: 73

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

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

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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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positron 84