Topic: Mathematics of Quantum Field Theory
Partial set of lecture notes:
Lecture 1: Overview. Presentation of the main problem regarding the definition and construction of a QFT model. Example of the Euclidean scalar field model with a quartic interaction.
Lecture 2: Continued. Basic example of a renormalization procedure.
Lecture 3: Multivariate formal power series. Explicit composition.
Lecture 4: The Inverse Function Theorem (existence and uniqueness) for multivariate formal power series. Introduction to tensors and diagrammatic algebra.
Lecture 5: The Cayley-Hamilton Theorem in pictures. Diagrammatic algebra for homogeneous polynomials. A glimpse of classical invariant theory. Graphical formulas for the discriminant of the binary quadratic, and for the cubic, in relation to Cardano's formula for the roots of a cubic.
Lecture 6: Explicit formulas for the reversion of formal power series, i.e., the Inverse Function Theorem using diagrammatic expansions in terms of tree graphs. Results about trees. Caylay's Theorem for counting trees.
Lecture 7: Continued. Simple one-dimensional example as a sanity check. Comparison with formula for root of a quadratic equation. Proof of convergence of the tree expansion.
Lecture 8: Detailed proofs of results about trees. Introduction to Gaussian integrals. Statement of the Isserlis-Wick Theorem for moments of multivariate Gaussians.
Lecture 9: Proof of the Isserlis-Wick Theorem. Quick introduction to category theory: categories and functors. Lots of examples.
Lecture 10: Introduction to QFT perturbation theory using category theory. Discussion of convergence issues. Example of zero-dimensional QFT, and Borel summability.
Lecture 11: Rigorous presentation of QFT perturbation theory in terms of Feynman diagrams, using Joyal's theory of combinatorial species. Symmetry factors.
Lecture 12: Continued. Natural transformations or morphisms of functors. Statement of the main "integration over the fiber" theorem.
Lecture 13: Proof of the main theorem.