LTCC Lectures
home page of Dr. Benjamin Doyon


Integrability is a wide subject that comprises many deep ideas and that can be applied to very diverse physical systems. This course will give an overview of some of the ideas behind integrability, and of the way they are developed in applications to four areas of theoretical physics: finite dynamical systems, classical field theory, quantum chains, and quantum field theory. In each case, slightly different techniques are involved, but one of the goal of this course is to emphasise the conceptual similarity between them. In particular, I will try to show how the basic concepts in integrable dynamical systems, such as action-angle variables, conserved quantities, and the simplification of the phase space trajectories that ensues, are realised in other contexts. I will discuss simple, standard examples in order to illustrate the concepts.
  1. Integrable classical dynamical systems: conserved quantities, Liouville theorem, action-angle variables, Lax pairs
  2. Integrable classical field theory: zero-curvature formulation and monodromy matrix, inverse scattering method, example: the sine-Gordon model
  3. Integrable quantum chains: Bethe ansatz, Yang-Baxter equations and quantum inverse scattering method, correlation functions, example: the Heisenberg spin chain
  4. Integrable quantum field theory: Factorised scattering theory, form factors, correlation functions, example: the quantum sine-Gordon model
(Optional- Other possible topics include: isomonodromic deformations, Riemann-Hilbert problems, tau functions, spectral curves, T-Q relations, integrable statistical models, integrable deformations of conformal field theory, classical integrable structure of quantum correlation functions, thermodynamic Bethe ansatz).

Introductory remarks

Notes for the course (written!)
Some solutions to proposed exercises (in writing...)

due 17 December 2012
some additional information to help with the assessment!

Section 2: Chapter 2 of [1]
Section 3: Chapter 13 of [1]
Section 4: Chapters 3 and 4 of [2], parts of [5], and some additional stuff
Section 5: from my old notes, but main source for the subject is [7]

Recommended reading:
  • [1] “Introduction to classical integrable systems”, O. Babelon, D. Bernard and M. Talon, Cambridge University Press, 2003 (mainly chapters 2, 3, 13)
  • [2] “How algebraic Bethe ansatz works for integrable models”, L.D. Faddeev, Les Houches School on Symétries Quantiques, North-Holland (1995), pp. 149-219, hep-th/9605187, 1996
  • [3] “Introduction to integrable quantum field theory”, B. Doyon, Lecture notes for a course given at Durham University (2008),
Additional Optional reading:
  • [4] “Algebraic analysis of solvable lattice models”, M. Jimbo and T. Miwa, Conference Board of the Mathematical Sciences 85, American Mathematical Society ,1993
  • [5] “On the quantum inverse scattering problem”, J. M. Maillet and V. Terras, Nucl. Phys. B575, 627-644, hep-th/9911030, 2000
  • [6] “Quantum inverse scattering method and correlation functions”, V.E. Korepin, N.M. Bogoliubov, A.G. Izergin, Cambridge University Press, 1993
  • [7] “Form factors in completely integrable models of quantum field theory”, F. A. Smirnov, World Scientific, 1992

Preliminary reading: Chapter 2 of O. Babelon, D. Bernard and M. Talon cited above.

Prerequisites: complex analysis, classical mechanics, special relativity, quantum mechanics, quantum field theory (introductory level).

Timetable and location:

There will be 5 two-hour lectures, on Mondays 13:10-15:10 at these dates:

29 October
5 November
12 November
26 November
3 December

Lectures will be held at the De Morgan House.

latest update: 2 December 2012