## Lecture course "Quantum Field Theory I", SS 2011

Quantum field theory is a universal tool which has applications ranging from condensed matter physics to elementary particle physics. A crucial role is played by symmetries which are at the heart of the universality shown by many physical systems. In this lecture the focus will be on foundational aspects of quantum field theory, with emphasis on condensed matter applications. Someone who is interested in going into high energy physics is recommended to attend a different lecture with focus on relativistic quantum field theory and quantum electrodynamics such as it is currently offered by Prof. A. Klemm in Bonn.

**Outline of topics:**

- Classical fields
- Functional calculus
- Variational principle and equations of motion
- Symmetries and conservation laws
- Quantum fields
- Canonical quantization
- Path integrals
- Coherent states
- Grassmann calculus
- Saddle point approximation
- Non-perturbative effects
- Correlation functions
- Perturbation theory
- Applications
- Lattice vibrations (phonons)
- Hubbard model and Mott insulator transition
- Spin waves
- The free and the weakly interacting electron gas

**Prerequisites:** Solid knowledge in quantum mechanics.

**Note:**
Hand-written lecture notes are available here.

**Schedule**

- Lecture 01: Motivation and overview, review of Lagrangian mechanics, small oscillations around equilibrium positions
- Lecture 02: The harmonic chain (discrete and continuum versions), Lagrangian formulation of field theories, Euler-Lagrange equations and Noether's Theorem
- Lecture 03: Examples of symmetries and conservation laws, Hamiltonian formulation of field theories, canonical quantization of the free scalar field
- Lecture 04: Continuation: canonical quantization of the free scalar field, relativistic formulation of the electromagnetic field
- Lecture 05: Canonical quantization of the electromagnetic field, Casimir effect
- Lecture 06: Continuation: Casimir effect, Particles & anti-particles, negative energy solutions and causality
- Lecture 07: Second quantization: Quantization of the Schrödinger field, occupation number representation of multiple particle states
- Lecture 08: Bosonic and fermionic Fock space, particle interactions
- Lecture 09: Models of electrons in solids: Free electron gas, jellium model
- Lecture 10: Jellium model, electrons in periodic potentials
- Lecture 11: Tight binding model, Hubbard model and its phenomenology
- Lecture 12: Spin chains
- Lecture 13: Continuation: Anti-ferromagnetic Heisenberg spin chain, Bogoliubov transformations, path integrals in single particle quantum mechanics
- Lecture 14: Construction of the path integral, Gaussian integrals, functional calculus
- Lecture 15: Path integral and statistical physics, saddle point approximation, semi-classical transition amplitudes
- Lecture 16: Propagator of a particle in a harmonic potential, non-perturbative effects: Instantons
- Lecture 17: Level splitting in a double well potential, path integrals in field theory (functional integrals): Bosonic coherent states
- Lecture 18: Grassmann calculus, fermionic coherent states
- Lecture 19: The coherent state path integral, Pauli magnetism
- Lecture 20: Matsubara sums, application to Pauli magnetism and the free energy of non-interacting particles
- Lecture 21: Effective actions, effective attractive electron interactions from electron-phonon interactions, correlation functions and generating functionals
- Lecture 22: Wick's theorem, the basics of perturbation theory
- Lecture 23: Asymptotic series, phi^4 theory, the propagators of the Klein-Gordon field in various dimensions
- Lecture 24: Feynman diagrams and their combinatorics
- Lecture 25: Linked cluster theorem, free energy, momentum space, outlook: divergencies and renormalization
- Lecture 26: Feynman diagrams for non-relativistic many-particle physics (Schrödinger field), Hartree-Fock approximation to the perturbed ground state energy
- Lecture 27: The weakly interacting electron gas: Hartree-Fock and Random Phase Approximation (RPA), ground state energy, vacuum polarization and screening of the Coulomb interaction

**Exercises**

- Exercise 01 (4.4.2011)
- Exercise 02 (11.4.2011)
- Exercise 03 (18.4.2011)
- Exercise 04 (26.4.2011)
- Exercise 05 (2.5.2011)
- Exercise 06 (9.5.2011)
- Exercise 07 (16.5.2011)
- Exercise 08 (23.5.2011)
- Exercise 09 (30.5.2011)
- Exercise 10 (6.6.2011)
- Exercise 11 (6.6.2011) [Bonus Sheet]
- Exercise 12 (20.6.2011)
- Exercise 13 (27.6.2011)
- Exercise 14 (4.7.2011)