Quantum Field Theory

Gordon W. Semenoff is Professor at The University of British Columbia in Vancouver, Canada. His research examines the nature at its most fundamental level. His recent interests have been in superstring theory and duality of string theories with strongly coupled gauge field theories and in quantum gravity.  He is also interested in quantum information theoretic questions in quantum field theory.

lt;p>1Many Particle Physics as a Quantum Field Theory

1.1   Introduction                           

1.2   Non-relativistic particles                       

1.2.1   Identical particles 

1.2.2   Spin 

1.3   Second Quantization 

1.4   The Heisenberg picture 

2 Degenerate Fermi and Bose Gases

2.1The limits of large volume and weakly interacting particles

2.2Degenerate Fermi gas and the Fermi surface 

 2.2.1  The ground state |O > 

2.2.2  Particle and holes 

2.2.3  The grand canonical ensemble

2.3 Bosons 

3 Classical field theory and the action principle

 3.1The Action Principle                                                 

3.1.1  The Action                     

3.1.2  The action principle and the Euler-Lagrange equations

3.1.3  Canonical momenta, Poisson brackets and Commutation relations                       

3.2Noether's theorem                                               

        3.2.1 Conservation laws and continuity equations

        3.2.3 Examples of symmetries

3.3Phase symmetry and the conservation of particle number

4 Non-relativistic space-time symmetries

4.2Galilean symmetry                                               

4.3Scale invariance                                                 

4.3.1  Improving the stress tensor                     

4.3.2  The consequences of scale invariance    

4.4Special Schrödinger symmetry

5 Space-time symmetry and relativistic field theory

5.1Quantum mechanics and special relativity         

5.2Coordinates                                 

5.3Scalars, vectors, tensors 

5.4The metric                              

5.5Symmetry of space-time          

 5.6The symmetries of Minkowski space            

6 Emergent relativistic symmetry

6.1Phonons                         

6.2The free scalar field theory           

 6.3The Debye theory of solids                 

 6.4Relativistic Fermions in Graphene         

7The Dirac Equation

 7.1The Dirac equation                        

 7.2Natural units                              

7.4Lorentz Invariance of the Dirac equation         

 7.5Spin of the Dirac field            

7.6 Phase symmetry and the conservation of particle number

7.6.1 Conserved number current

7.6.2 Relativistic Noether's Theorem for the Dirac equation

7.6.3 Alternative Proof of Noether's Theorem

7.7 Spacetime symmetries of the Dirac theory

7.7.1 Translation Invariance and the Stress Tensor

7.7.2 Lorentz Transformations

7.7.3 Stress Tensor and Killing Vectors

8 Photons                                                                                         

8.1 Relativistic Classical Electrodynamics

8.1.2 Massive photon (Optional reading)

8.2 Space-time symmetries of the photon

8.3 Quantum Electrodynamics

9 Functional methods                                                                      

9.1 Functional derivative

9.2 Functional integral

9.3 Real Scalar Field

9.3.1 Generating functional for free scalar fields

9.3.3 Generating functional as a functional integral

9.4 The self-interacting real scalar field

9.5 Analyticity

10 More Functional Integrals                                                          

10.1 Functional Integrals for the Photon Field

10.2 Functional Methods for Fermions

10.3 Generating functionals for non-relativistic Fermions

10.3.1 Interacting non-relativistic Fermions

10.4 The Dirac field

10.4.1 2-point function for the Dirac field

10.4.3 Functional integral for the Dirac field

10.5 Functional Quantum Electrodynamics

11 The Weakly Coupled Scalar Field Theory                                   

11.2 Counterterms

11.3 Computation of the 2-point function

11.4 Feynman diagrams

11.5 Simplifications of Feynman diagrams

11.6 The 4-point function

11.7 Computation of a one-loop Feynman Integral

11.7.1 Dimensional regularization

11.7.2 Wick Rotation

11.7.3 Feynman Parameters

11.7.4 Integration in 2w-dimensions

11.7.5 Asymptotic expansion at 2w - 4

11.7.6 Inverse Wick rotation

11.7.7 The mass tadpole

11.7.8 The 2- and 4-point functions

11.8 Subtraction Schemes

11.9 Elastic scattering amplitude

11.10 Connected Correlations and Goldstone's Theorem

11.10.1 Connected Correlation Functions

11.10.2 Goldstone's Theorem

11.10.3 Irreducible Correlation functions

11.11Integration formulae

11.11.1 Feynman Parameter Formula

11.11.2 Dimensional regularization integral

11.11.3 Euler's Gamma function

12 Perturbative Quantum Electrodynamics                                   

12.1 Counterterms

         12.2 The generating functional in perturbation theory

12.2.1 Wick's Theorem for Photons and Fermions

12.3 Feynman diagrams

12.3.1 Feynman rules

12.3.2 Fermion 2-point function

12.4 The photon 2-point function

12.5 Quantum corrections of the Coulomb potential

13 Generating Functionals and Quantum Electrodynamics

13.1 Connected Correlations and Goldstone's theorem

13.1.1 Connected Correlation Functions

13.1.2 Goldstone's Theorem

13.2 Furry's theorem

13.3 The Ward-Takahashi identities