Prof. Dirk Kreimer
ph: 358 2393 dkreimer@bu.edu off: 265/Math.Dept.

Office Hours: Mo 11-1, Thu 11-12.

MA557 is a self-contained  introduction to the mathematical techniques in
quantum field
theory. There has been much progress in the understanding of the
mathematical structure of perturbative techniques in quantum field theory in
recent years.
This progress allows one to present sophisticated mathematical concepts of
quantum physics and field theory in a much more elementary way. The
mathematics involved is a combination of topics from analysis, algebra,
combinatorics, and even a little number theory.  In particular, the
algebraic structure of the Hopf algebra of rooted trees plays a central
role in the systematic renormalization of divergent integrals that plague
QFT.

Prerequisites: MA226, MA242 or consent of instructor.

Syllabus:

Week 1: Introduction to the Hopf algebra of rooted trees.

Week 2: Continuation of elementary prperties of Hopf algebras.

Week 3: The problem of renormalization. Application of a Hopf algebra to it.

Week 4: Basics from quantum field theory:
Canonical quantization, commutators, Fock space, Wick's
theorem, time- and normal ordering.

Week 5: Feynman rules for scalar field theories, generalization to
spin, momentum space Feynman integrals, coordinate space Feynman
integrals.

Week 6: Generating functions for graphs, graph counting, basic
notions of graph theory as needed.

Week 7: Power counting, existence of short-distance singularities,
one-loop examples.

Week 8: The concept of renormalization, diffeomorphisms of physical
observables, scale transformations.

Week 9: Basic elements from Hopf algebras and their dual Lie
algebras of graph insertions. Counit, coproduct, coinverse. Feynman rules
as characters of a Hopf algebra.

Week 10: The principle of multiplicative subtraction. Simple
examples from one-dimensional field theories.  Scale transformations and
renormalization group flow as an infinitesimal character.

Week 11: Zimmermann's forest formula. Overlapping divergences.
Two-loop examples worked out in scalar field theory.

Week 12: The treatment of form-factors and theories with spin in
general. Running couplings and anomalous dimensions.

Week 13: Exact renormalization group versus perturbative
renrmalization. Rooted trees versus graphs. The fundamental role of the
residue of a Feynman graph.

Week 14: Effective actions. Recovering the functional integral from
a perturbative expansion. Dyson-Schwinger equations in comparison with
Knizhnik-Zamolodchikov equations.