Vorlesungsnotizen/Lecture Notes:

       Quantenfeldtheorie I/Quantum Field Theory I

       Prof. Michael G. Schmidt

Inhalt/Contents

1. Introduction
   1. Short repetition of quantum mechanics
   2. Observables
   3. Time evolution and Heisenberg equation
   4. Field equations
   5. Heisenberg uncertainty relation
   6. Interactions and Feynman graphs
   7. Perturbation theory
   
   
   
2. The free electromagnetic field, coherent states
   1. Coherent states for harmonic oscillator
   2. Coherent states and classicality
   3. Coherent states are overcomplete
   4. Operators in the coherent state representation
   
   
   
3. The free electromagnetic field (without sources)
   1. The hamiltonian for the electromagnetic (EM) field
   2. Naive quantization of the EM field
   3. Induced emission/absorbtion, spontaneous emission
   4. Electromagnetic waves in coherent state representation
   5. Exercise 1
   6. Exercises 2 and 3
   
   
   
4. Canonical formalism for the wave equation (Klein-Gordon equation)
   1. Field equation
   2. Klein-Gordon equation
   3. Klein-Gordon equation for massive particles
   4. Quantization
   5. The Hamiltonian in the N-representation
   6. Maxwell's equations in the Lagrange Formalism
   7. Maxwell's Lagrangian in the Coulomb gauge
   
   
   
5. Schrödinger equation in the language of field quantization ("2nd quantization")
   1. Lagrange formalism for the Schrödinger equation
   2. Heisenberg equation for the second quantized `wave function'
   3. The number and momentum operators
   4. The Hamiltonian in the second quantized formalism
   5. Multiparticle Schrödinger equation
   6. Construction of multiparticle states
   7. Interactions in the second quantized formalism
   8. Fermions and the Fermi-Dirac statistics
   9. The Fock space for fermions
   10. Multiparticle fermionic states
   
   
   
6. Relativistic covariance; the Lorentz and Poincaré group
   1. Properties of symmetry transformations
   2. Lie algebra of a symmetry group
   3. Relativistic invariance/covariance; Lorentz transformation
   4. Lorentz transformation of the metric tensor and vectors
   5. Lorentz group
   6. Lorentz algebra
   7. Lorentz boosts and the small group (rotations)
   8. Poincaré group
   9. Relativistic particles as unitary representations of the Poincaré group
   10. Representations of the small group
   11. Plane waves and Lorentz transformation
   
   
   
7. Covariant field equations and their quantization
   1. A general solution of the Klein-Gordon equation
   2. Canonical quantization of the Klein-Gordon (scalar) field
   3. Hamiltonian of the Klein-Gordon field
   4. Canonical quantization of a complex scalar field
   5. Charge conjugation
   6. Charge conjugation in scalar electrodynamics
   7. Microcausality
   8. Transition to the nonrelativistic theory
   9. Transition to the nonrelativistic theory (cont.)
   
   
   
8. The Dirac equation and its quantization: 0. Spinor representation of Lorentz group
   1. The Dirac gamma matrices
   2. Direct construction of the representation matrices in spinor space
   3. Lorentz boosts in spinor space
   4. Dirac spinors in terms of Lorentz representation matrices
   5. Transformation of the Lorentz spinor by the spinor representation matrices
   6. The Dirac field equation
   7. Representations of the Dirac algebra: Weyl or chiral representation
   8. The Dirac-Pauli representation
   9. Relativistic covariance of the Dirac field equation
   10. Charge conjugation C
   11. More on charge conjugation
   12. Projection operators onto positive and negative energy states
   13. The Lagrange and Hamilton functions for the spinor fields
   14. Quantization
   15. The creation and annihilation operators and the Fock space
   16. Charge conjugation revisited
   17. Clifford algebra and bilinear covariants
   
   
   
9. Quantum mechanical interpretation of the Dirac equation and its nonrelativistic limit; the Foldy-Wouthuysen transformation
   1. Positive and negative energy solutions
   2. Nonrelativistic limit
   3. The Pauli equation
   4. The Foldy-Wouthuysen transformation
   5. The Foldy-Wouthuysen transformation in presence of a constant electromagnetic field
   6. The Darwin term and the Thomas factor
   7. The hydrogen atom
   8. The hydrogen atom and the Foldy-Wouthuysen transformation
   
   
   
10. Symmetries and conservation laws
    1. Noether theorem
    2. Translation invariance and energy-momentum tensor
    3. Energy momentum tensors for the Klein-Gordon, Dirac and Maxwell fields
    4. Invariance under Lorentz transformations
    5. The orbit-angular momentum and spin density tensor
    6. Discrete symmetries: space inversion (parity)
    7. Discrete symmetries: time reversal
    8. Discrete symmetries: charge conjugation
    9. CPT theorem
    
    
    
11. Interacting fields, vacuum expectation values, S-matrix
    1. Interaction of particles in QFT and S-matrix
    2. The Yang-Feldman equation
    3. The Yang-Feldman equations and the in and out states
    4. The two-point functions (propagators)
    5. The retarded and advanced propagators
    6. The spectral representation of the commutator vev; the Kaellen-Lehmann bound on Z
    7. The LSZ reduction formalism
    8. The LSZ reduction formalism (continuation)
    9. The LSZ reduction formula for bosonic fields
    10. The LSZ reduction formula for fermionic fields
    11. Generating functional for the scattering operator
    
    
    
12. X. Invariant perturbation theory. 1) Dyson expansion
    1. 1)
    2. 1)
    3. 1)
    4. 1)
    5. 1)
    6. 1)
    7. 2) Wick Theorem
    8. 2)
    9. 2)
    10. 3) Feynman graphs
    11. 3)
    12. 3)
    13. 3)
    14. 3)
    15. 3)
    16. 3)
    17. 4) Back to S-matrix elements
    18. 4)
    19. 4)
    20. 5) Feynman rules for other interacting theories (a) Complex scalars
    21. 5) (b) Yukawa coupling
    22. 5) (c) Quantum electrodynamics
    23. 6) A simple example in Yukawa theory
    24. 6)
    25. 6)
    26. 7) From S-matrix to the Cross Section
    27. 7)
    28. 7)
    29. 7)
    30. 7)
    
    
    

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    Last modified: Thu Dec 13 2002 by T. Prokopec