Course Meeting Times
Lectures: 2 sessions / week, 80 minutes / session
Prerequisites
Familiarity with analysis in several variables, complex analysis, linear algebra and basic functional analysis, and basic differential geometry. For some parts: Riemann surfaces and fundamentals of Lie groups. Knowledge of advanced physics is not required.
Course Description
The development of quantum field theory (QFT) and string theory in the last four decades led to an unprecedented level of interaction between physics and mathematics, incorporating into physics such “pure” areas of mathematics as algebraic topology, algebraic geometry, representation theory, combinatorics, and even number theory. This interaction has been highly fruitful in both directions, and led to a necessity for physicists to know basic mathematics and for mathematicians to know basic physics. Physicists have been quick to learn, and nowadays good physicists know relevant areas of mathematics as deeply as professional mathematicians. On the other hand, many mathematicians have been slower, intimidated by the absence of rigor in physical texts, and, more importantly, by a different manner of presentation. In particular, even the basic setting of quantum field theory, necessary for understanding its more advanced (and mathematically exciting) parts, is already largely unfamiliar to mathematicians. Nevertheless, many of the basic ideas of quantum field theory can in fact be presented in a completely rigorous and mathematical way. Doing so will be the main goal of this course.
Note: This is a course primarily for mathematicians. Physicists will not learn much in it, except how to present the ideas of QFT (which they already know) in a mathematical way. It is important to note that the instructor knows less QFT than a graduate student specializing in QFT or string theory.
Topics
- Generalities on classical and quantum mechanics and field theory.
- 0-dimensional QFT: Stationary phase (steepest descent) formula. Calculus of Feynman diagrams with applications to combinatorics. Matrix models, large \(N\) limits. Applications to moduli space of curves and planar graphs.
- 1-dimensional QFT: Formalism of classical mechanics. Lagrangians, Hamiltonians, and the least action principle. The path integral approach to quantum mechanics. Correlation functions. Perturbative expansion using Feynman diagrams. The Hamiltonian approach and operator formalism. The Feynman-Kac formula.
- \(d\)-dimensional QFT for \(d > 1\): Formalism of classical field theory. Gauge theories. Path integral and Hamiltonian approaches to QFT. Wightman axioms. Free field theories. Perturbative expansion. Divergences. Renormalization theory.
- Supergeometry and field theories with fermions.
- Introduction to 2-dimensional conformal field theory.
Textbook and Lecture Notes
A recommended textbook for the course is Quantum Fields and Strings: A Course for Mathematicians, AMS, 1998 (but we won’t closely follow it). Instead, we will follow the lecture notes.
Assignments and Grading
The course grade is based on weekly homework, which is required for those taking the course for credit.
