Field Theory of Quantum Many-Body Systems

Lecture

Thomas Gasenzer

Tuesday, 11:15-13:00, Pw 12, kHS (first lecture on 16/10); even weeks, starting 19/10: Fri, 11:15 - 13:00: Pw 19, SR. [LSF]
Practice group: Odd weeks, Friday, 11:15-13:00 hrs, Pw 19, SR (Please register here.)

Attention! Exam on 05.02.13, 11:15-12:45 hrs, Pw 12, Room 106. Lecture on 08.02.13.

Content - Prerequisites - Literature - Exercises - Script WS 10/11 (different focus)

The lecture course provides an introduction to field theoretic methods for systems with many degrees of freedom. A focus will be set on applications to ultracold, mostly bosonic, atomic gases, superfluids and superconductors as they are the subject of many fore-front present-day experiments. Methodologically, the lecture will introduce the basics of the path-integral approach to quantum mechanics and field theory. In applying these techniques it will in particular concentrate on dynamical properties of the considered systems. Depending on time the topics marked by a star will lead into the field of present research on non-equilibrium critical dynamics. Knowledge of the basics of quantum mechanics and statistical mechanics is presumed while the course is designed to be self-contained on the quantum-field-theory side.

Content (preliminary):
  1. Introduction
    [ Script | Slides ]
  2. Equilibrium and nonequilibrium quantum field theory
    - Basics of quantum field theory - Non-linear Schrödinger model - Correlation functions - KMS boundary condition - Fluctuation-dissipation theorem - Physical information in the 2-point function
    [ Script | Slides | Lecture 01 | 02 | 03 | 04 | 05 ]
  3. Path-integral approach to quantum mechanics
    - Feynman path integral - Functional calculus - Saddle-point expansion - Perturbation theory - Some applications of the path-integral formulation
    [ Script | Lecture 06 | 07 ]
  4. Interacting Bose systems
    - Mean-field theory of a superfluid - Bogoliubov quasiparticles - Path-integral approach to Bose systems - Low-energy effective theory - Superfluid phase transition and spontaneous symmetry breaking - Nambu-Goldstone theorem - Superfluid phase in low dimensions - Finite-temperature superfluids - Superfluid to Mott insulator transition - Superfluidity and superconductivity - Anderson-Higgs mechanism
    [ Script | Lecture 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 ]
  5. Non-equilibrium quantum fields
    - Generating functional - Schwinger-Keldysh contour - Quantum vs classical path integral - The one-particle irreducible effective action - 2PI effective action - Dynamic equations - Kadanoff-Baym equations - Mean-field approximation - Conservation laws - Scattering effects and kinetic theory - Quantum Boltzmann equation
    [ Script | Lecture 16 | 17 | 18 | 19 | 20 ]
  6. *Non-equilibrium critical dynamics and wave turbulence
    - Four-wave kinetic and (Quantum) Boltzmann equations - Implications from conservation laws for nonequilibrium distributions - Stationary nonequilibrium distributions - Dimensional estimates and self-similarity - Stationary spectra of weak wave turbulence - Exact stationary solutions for the four-wave kinetic equation - Zakharov transformations - Constant fluxes of action and energy
    [ Script | Lecture 21 ]

Prerequisites:
  • Quantum Mechanics (Theoretical Physics III), Statistical Mechanics (Theor. Phys. IV), Quantum Field Theory I or Quantum Optics

Literature:

General texts
  • Brian Hatfield, Quantum Field Theory of Point Particles and Strings. Addison Wesley, Oxford, 2010. [ Google books | HEIDI ]
  • Michael E. Peskin, Daniel V. Schroeder An introduction to quantum field theory. Westview, Boulder, 2006. [ Google books | HEIDI ]
  • Xiao-Gang Wen, Quantum Field Theory of Many-Body Systems. OUP, Oxford, 2010. [ Google books | HEIDI ]
Non-equilibrium quantum field theory and quantum kinetic theory
  • L.P. Kadanoff and G. Baym, Quantum statistical mechanics. Addison-Wesley, Redwood City, 1989. [ HEIDI ]
  • Jørgen Rammer, Quantum field theory of non-equilibrium states. CUP, Cambridge, 2007. [ Online edition | HEIDI ]
  • J. Berges, Introduction to nonequilibrium quantum field theory, AIP Conf. Proc. 739, 3 (2005); arXiv.org: hep-ph/0409233 .
  • P. Danielewicz, Quantum Theory of Nonequilibrium Processes, Annals of Physics 152, 239 (1984) .
  • M. Bonitz, Quantum kinetic theory. Teubner, Stuttgart, 1998. [ Contents | HEIDI ]
  • E. Calzetta and B.-L. Hu, Nonequilibrium quantum field theory. CUP, Cambridge, 2008. [ Online fulltext | HEIDI ]
Ultracold atomic gases Wave turbulence
  • Sergey Nazarenko, Wave turbulence. Springer, Berlin, 2011. [ Google books | HEIDI | Full Text ]
  • V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I. Springer, Berlin, 1992. [ Google books ]
Transport kinetics and hydrodynamics
Exercises:

Exercises will be held, in general, on Fridays, 11:15-12:45 hrs, in SR, Pw19, during odd weeks, starting on 26/10/12. (Please register here.)

Problem sets:

    Sheet 01:
    Lagrangian Formalism - Field operators - Grand-canonical ensemble - Non-linear Schrödinger model.
    Sheet 02:
    Callan-Welton FDT - KMS relations - Correlation functions of damped harmonic oscillator - Pressure from Green's function.
    Sheet 03:
    Lippmann-Schwinger equation - Free propagator of harmonic oscillator - Goldstone theorem - Gelfand-Yaglom method.
    Sheet 04:
    Bogoliubov transformation and phonons - Gross-Pitaevskii equation - Glauber coherent states - Bose-Einstein condensates and coherent states.
    Sheet 05:
    Vortices - Analytic continuation - Stability of vortices - Abelian Higgs model and Meissner effect.
    Sheet 06:
    Cumulants - Generating functional of a quadratic action - Propagator and inverse propagtor - Classical Bose gas and critical exponents.