Quantum Field Theory of Many-Body Systems

Lecture (MVSpec)

Thomas Gasenzer

Tuesday, 11:15-13:00; Thursday, 11:15-13:00; (starting on 19/10) INF 227 (KIP), SR 3.403+4. [LSF]  

Note (update: 29/11/21): Coronavirus: According to the presently active safety and prevention measures of the university (look here for up-to-date information by the university, as well as Cornelis Dullemond's FAQs to the examination committee) on-site teaching will be resumed in the forthcoming winter semester but will be subject to a number of rules. The main condition for taking part in the course on site will be adhering to the GG rule ("Alarmstufe II" applies from Monday, 29/11/21, on). As all participants' 2G status will be checked at the entrance to the building, please bring your documents (your RFID-card, from next week on) to each of the on-site lectures and exercises. For general and detailed information about the presently valid regulations we refer to the respective webpages by the university. Please let me know if you cannot come under these circumstances and need further help. We will find a way that everyone has the possibility to continue and finish the course.

Lecture format: In this lecture, I plan to make extensive use of online material (videos, quizzes, script), combining it with two on-site sessions per week of 1,5 hrs each (times as indicated above). Registered participants gain access to this material via the internal webpage of the lecture and are asked to prepare for each on-site lecture session, starting with that on Thu, 21/10. The on-site sessions will then serve to discuss further details, the solutions to the quiz questions as well as a further deepening of the subject matter.

Therefore, registration in the exercise system will be required in order to be able to take part in the course. I will send more details to registered participants by email.

Exercises  Tutor: Philipp Heinen

Register and view group list here. (Open 13/09-22/10)
Classes take place, exclusively on-site, on Fridays, 14:15-16:00 hrs, INF 227, SR 1.403+4 (starting on 29/10)

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 operator as well as the path-integral approach to quantum field theory. In applying these techniques I will in particular concentrate on thermal and dynamical properties of the considered systems. 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.


  1. Introduction
  2. Quantum field theory of matter
    From classical to quantum fields - Lagrangian and Hamiltonian field theory - *Constrained quantisation - Quantisation of the Bose field - Mode expansion - Harmonic oscillator - One- and multiparticle operators - Fock space - Identical particles - Bosons and fermions - Coherent states - Wigner function and phase space - Free systems and Wick's theorem - Cumulant expansion
  3. Mean-field theory of a weakly interacting Bose gas
    Non-linear Schrödinger model - Bogoliubov quasiparticles - Phase and Number fluctuations - Renormalisation of the ground-state energy - *Low-energy scattering theory - Ground state: Two-mode squeezing - *SU(1,1) coherent states - Thermal Bogoliubov quasiparticles
  4. Path-integral approach to quantum field theory
    A quick reminder of the Feynman path integral - Functional calculus - Saddle-point expansion and free propagator - Perturbation expansion, Dyson series, and resummation - Correlation functions - Connected functions and cumulants - Feynman diagrammatics - Low-energy effective theory - Linear-response theory - Retarded and advanced Greens functions - Spectral and statistical functions - Thermal path integral - *The quantum effective action - *Spontaneous Symmetry Breaking
  5. Low-temperature properties of dilute Bose systems
    Path-integral representation of the interacting Bose gas - Ginsburg-Landau theory of spontaneous symmetry breaking - The Luttinger-liquid description - Superfluid phase transition and spontaneous symmetry breaking - Nambu-Goldstone theorem - *The Lieb-Liniger model of a one-dimensional Bose gas - Superfluid phase in low dimensions - Superfluids at non-zero temperatures - Dimensionally reduced path integral - Hydrodynamic formulation and vortices - Thomas-Fermi approximation - The Berezinskii-Kosterlitz-Thouless transition - *Superfluid to Mott insulator transition - *Superfluidity and superconductivity - *Anderson-Higgs mechanism


  • Quantum Mechanics (PTP 3), Statistical Mechanics (PTP 4); Quantum Field Theory I or Quantum Optics (useful but not a precondition)

Skriptum :

  • The notes will available for download here.
  • The Script of the lecture on QFT of Many-Body Systems in WT 20/21 can be downloaded here.


General texts on quantum field theory

  • 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 ]
  • A. Zee, Quantum Field Theory in a Nutshell. Princeton UP, 2010. [ Google books | HEIDI ]

Greens functions

  • Gabriel Barton, Elements of Green's functions and propagation. Clarendon, Oxford, 2005. [ Google books | HEIDI ]

Quantum optics and phase-space methods

  • S.M. Barnett, P.M. Radmore, Methods in Theoretical Quantum Optics. Clarendon Press, Oxford, 1997. [ HEIDI ]
  • C.W. Gardiner, Quantum Noise. 2nd Ed. Springer Verlag, Berlin, 2000. [ HEIDI ]
  • L. Mandel, E. Wolf, Optical Coherence and Quantum Optics. CUP, Cambridge, 2008 (ISBN 0-521-41711-2). [ HEIDI ]
  • W.P. Schleich, Quantum Optics in Phase Space. Wiley-VCH, Weinheim, 2001. [ HEIDI ]
  • M.O. Scully, M.S. Zubairy, Quantum Optics. CUP, Cambridge, 2008. [ HEIDI | Google Books ]

Ultracold atomic gases: General texts and theory reviews

  • N. P. Proukakis, D. W. Snoke, P. B. Littlewood (Eds.), Universal Themes of Bose-Einstein Condensation. CUP, Cambridge, 2017. [ Google books | HEIDI ]
  • A. Griffin, D. W. Snoke, S. Stringari (Eds.), Bose-Einstein condensation. CUP, Cambridge, 2002. [ Google books | HEIDI ]
  • C.J. Pethick and H. Smith, Bose-Einstein condensation in Dilute Gases. CUP, Cambridge, 2002. [ Google books | HEIDI | Full Text ]
  • A. Leggett, Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-matter Systems. OUP Oxford, 2006. [ Google books | HEIDI ]
  • A. Leggett, Bose-Einstein condensation in the alkali gases: Some fundamental concepts. Review of Modern Physics 73, 307 (2001).
  • L.P. Pitaevskii and S. Stringari, Bose-Einstein condensation. Clarendon Press, Oxford, 2003. [ Google books | HEIDI ]
  • F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases. Review of Modern Physics 71, 463 (1999).
  • A. Fetter, Theory of a dilute low-temperature trapped Bose condensate. arXiv.org:cond-mat/9811366 (1998).

Ultracold atomic gases: A few original experimental perspectives

Few-body scattering theory

  • K. Burnett, P.S. Julienne, P.D. Lett, E. Tiesin, and C.J. Williams, Quantum encounters of the cold kind. Nature (London) 416, 225 (2002).
  • J. Dalibard, Collisional dynamics of ultra-cold atomic gases. Proc. Int. School Phys. Enrico Fermi, Course CXL: Bose-Einstein condensation in gases, Varenna 1998, M. Inguscio, S. Stringari, C. Wieman edts.
  • C.J. Joachain, Quantum Collision Theory. North-Holland, Amsterdam, 1983. [ HEIDI | Scribd Full Text ]
  • L.D. Landau and E. M. Lifshitz, Quantum Mechanics. Non-relativistic theory. (see Chapters XVII & XVIII.) Pergamon Press, Oxford, 1977. [ HEIDI | Online Full Text ]
  • R.G. Newton, Scattering Theory of Waves and Particles. Dover publications, 2002. [ HEIDI | Google Books ]
  • F. Schwabl, Quantum Mechanics. Springer, 2007. [ HEIDI | Google Books ]

Non-equilibrium quantum field theory and quantum kinetic theory

Additional material

  • D. Ebert and V. S. Yarunin, Functional-Integral Approach to the Quantum Dynamics of Nonrelativistic Bose and Fermi Systems in the Coherent-state Representation. Fortschr. Phys. 42, 7, 589 (1994).


Exercises will be held in general (exceptions posted above) on Fridays, 14:15-16:00 hrs, on-site, starting on 29/10/21. Tutor: Philipp Heinen (Please register here.)

The problem sets will available for download here.

Credit Points:

Passing the written exam, which will take place on 17/02/2022, 11:15-13:15 hrs, SR 3.403+4, will be the condition to obtain 8 CPs for the lecture.