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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.
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).
- Integrable classical dynamical systems: conserved quantities, Liouville theorem, action-angle variables, Lax pairs
- Integrable classical field theory:
zero-curvature formulation and monodromy matrix, inverse scattering
method, example: the sine-Gordon model
- Integrable quantum chains: Bethe
ansatz, Yang-Baxter equations and quantum inverse scattering method,
correlation functions, example: the Heisenberg spin chain
- Integrable quantum field theory:
Factorised scattering theory, form factors, correlation functions,
example: the quantum sine-Gordon model
Additional Optional reading:
-  “Introduction to
classical integrable systems”, O. Babelon, D. Bernard and M. Talon,
Cambridge University Press, 2003 (mainly chapters 2, 3, 13)
-  “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
-  “Introduction to integrable quantum field theory”, B. Doyon, Lecture notes for a course given at Durham University (2008), http://www.mth.kcl.ac.uk/~bdoyon/notesIQFT08.pdf
Preliminary reading: Chapter 2 of O. Babelon, D. Bernard and M. Talon cited above.
-  “Algebraic analysis
of solvable lattice models”, M. Jimbo and T. Miwa, Conference Board of
the Mathematical Sciences 85, American Mathematical Society ,1993
-  “On the quantum inverse scattering problem”, J. M. Maillet and V. Terras, Nucl. Phys. B575, 627-644, hep-th/9911030, 2000
-  “Quantum inverse
scattering method and correlation functions”, V.E. Korepin, N.M.
Bogoliubov, A.G. Izergin, Cambridge University Press, 1993
-  “Form factors in completely integrable models of quantum field theory”, F. A. Smirnov, World Scientific, 1992
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:
Lectures will be held at the De Morgan House.
latest update: 2 December 2012