Two-dimensional quantum field theory:
integrability and other concepts
Home page of Dr. Benjamin Doyon

This is a series of lectures on two-dimensional quantum field theory in the Michaelmas term of 2009 (Durham University), see the list of graduate lectures. Quantum field theory is one of the most successful theories of modern theoretical physics. Its roots lie in two seemingly very different subjects: the quantum mechanics of relativistic particles, and the statistical physics of critical many-body systems. It has nowadays applications to the study of many-body problems in general and of the fundamental particles of physics, and it is an essential tool in string theory. But quantum field theory is much more than these applications. The unifying concept behind it is that of collective phenomena: quantum field theory describes what happens when "local" objects in interaction with each other start acting "together", giving rise to new behaviours. These behaviours are the source of the most interesting phenomena of modern physics, including, for instance, the fundamental particles of high-energy physics.

In these lectures I will introduce, in the context of two-dimensional quantum field theory, a range of concepts at the basis of its mathematical structure. I will start with many fundamental concepts that are valid generally in two dimensions, and some also in higher dimensions. The restriction to two dimensions is useful as it avoids the sometimes heavy higher-dimensional notations and complexities. I will then introduce the notion of integrability. There are many aspects to integrable quantum field theory. I will focus mainly on what is called factorised scattering theory, which is applicable when there is a scale in the model, more precisely for quantum models of massive relativistic particles. Integrability offers the unique opportunity to probe the properties of quantum field theory, both as a theory of relativistic particles and as a scaling limit of quantum chains and classical lattice systems, in a much deeper way than what can be done from standard methods.

I will aim my lectures at graduate students and advanced undergraduates. The course is intended to be an introduction to both quantum field theory, and integrability in this context. As an introduction to quantum field theory, it complements and does not replace standard courses, since the emphasis will be on the general structure and non-perturbative aspects.

I will assume knowledge of:
  • Quantum mechanics
  • Special relativity
  • Complex analysis
There will be 8 one-hour lectures. Lectures are in OC218 (Ogden Centre), Tuesdays at 14:15 and Thursdays at 16:15:
  • 24 February, 14:15 - 15:15
  • 26 February, 16:15 - 17:15
  • 3 March, 14:15 - 15:15
  • 5 March, 16:15 - 17:15
  • 10 March, 14:15 - 15:15
  • 12 March, 16:15 - 17:15
  • 17 March, 14:15 - 15:15
  • 19 March, 16:15 - 17:15

Approximate plan (unfortunately not all topics have been covered, and but some topics have been discussed more extensively than previously considered):
  • Introduction: quantum relativistic particles, scaling limits, collective phenomena, and QFT.
  • Hilbert space: asymptotic states and scattering matrix, local operators and operator product expansion.
  • Quantisation schemes: from fluctuating fields to Hilbert space.
  • Renormalisation group flow: short and large distance limits of correlation functions.
  • Concepts of locality: symmetries and twist fields.
  • Integrability: local conserved charges, factorisation of the scattering matrix, analytic properties and the Yang-Baxter equation. Exact solutions.
  • Form factors: Riemann-Hilbert problem, evaluation of correlation functions.
  • Other topics on integrability (if time permits).


Lecture notes  (in progress - written: introduction and local-density formulation)

Related lecture notes  (from previous integrability course - has discussion of asymptotic states)


Here is some reading mainly on integrability in two-dimensional quantum field theory:
  • P. Dorey, Exact S-matrices, preprint hep-th/9810026.
  • F. H. L. Essler, R. M. Konik, Applications of Massive Integrable Quantum
    Field Theories to Problems in Condensed Matter Physics. I. Kogan Memorial
    Volume, World Scientific, preprint cond-mat/0412421
  • G. Mussardo, Off-critical statistical models: Factorized scattering theories
    and bootstrap program. Phys. Rep. 218 (1992) 215-379
  • O. A. Castro Alvaredo, Bootstrap methods in 1+1 dimensional quantum field
    theory: the homogeneous sine-Gordon models. Universidade de Santiago de
    Compostela, Spain (2001) hep-th/0109212
  • B. D., Correlation functions in integrable quantum field theory. Rutgers
    University, USA (2004) .ps, .pdf
More reading (fundamental works):
  • A. B. Zamolodchikov and Al. B. Zamolodchikov, Factorized S-matrices in
    two dimensions as the exact solutions of certain relativistic quantum field
    theory models, Ann. Phys. 120 (1979) 253-291.
  • M. Karowski and P. Weisz, Exact form factors in (1+1)-dimensional field
    theoretic models with soliton behaviour. Nucl. Phys. B139 (1978) 455-476.
  • F. A. Smirnov, Form factors in completely integrable models of quantum
    field theory, World Scientific, Singapore (1992).