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Integrable quantum field theory
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page of Dr. Benjamin Doyon
This is a short series of lectures
on integrable quantum field theory in the Spring term of 2008 (Durham
University), see the list
of graduate lectures. Integrable quantum field theory is a wide
subject with applications, for instance, to condensed matter and
statistical physics, and with many connections to diverse areas of
mathematics. It 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. Knowing
some of integrable quantum field theory also provides many useful tools
for theoretical physicists, and gives a wider understanding of objects
of interest to mathematicians.
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 4 lectures, for a
total of 6 hours. Lectures are in OC218 (Ogden Centre), at the
following dates and times:
- 29
April, 9:00 - 11:00
- 1 May,
11:00 - 13:00
- 5 May,
15:15 - 16:15
- 6 May,
12:00 - 13:00
Lecture
notes (updated regularly)
There are many aspects to
integrable quantum field theory. The course 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.
Plan:
- Main
ideas behind integrability. What is quantum field theory, and its main
concepts: asymptotic states and the scattering matrix,
local fields and the operator product expansion, correlation functions.
- Short
distances and large distances: the renormalisation group flow and
correlation functions.
- Local
conserved charges. Factorisation of the scattering matrix, analytic
properties and the Yang-Baxter equation. Exact solutions.
- Properties
of form factors: a Riemann-Hilbert problem that can be solved exactly.
Evaluation of correlation functions. Twist fields.
- Other
topics (as time permits): Bethe ansatz; conformal perturbation theory;
integrable differential equations from quantum field theory.
Reading (reviews, theses):
- 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).
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