Mathematical Framework of Quantum Field Theory

Department of Mathematics, 917 Evans Hall, University of California, Berkeley,
email: reshetik at math.berkeley.edu
KdV Institute, C4.145, Science Park 904 1098 XH Amsterdam,
email: nresheti at science.uva.nl

Tentative syllabus

The syllabus will evolve as the seminars will go on. I will follow my Lecture Notes from the Quantum Field Theory course which was taught in Aarhus University in the Fall 2006, and in Berkeley in the Fall 2007. These Notes will appear on this site as seminars will go on. They are evolving. If you have comments, you are welcome to send them to me. Another set of notes which I will use is Lecture Notes for the summer school at Holbaeck, 2008. More references and reading material will appear on the left side of this page later. The homework list for this class will be updated regularly. Below is the short description of lectures and seminars as the semester will evolve.

This is an introductory meeting. The outline of classical Lagrangian mechanics and classical Lagrangian field theory. The outline of this lecture is here.

Classical Lagrangian and Hamiltonian mechanics. Legendre transform. Poisson brackets. Elements of geometry: smooth, Riemannian, and symplectic geometry. The outline of this lecture is here.

Lagrangian and Hamiltonian mechanics on Riemannian manifolds. Elements of symplectic geometry. The outline of this lecture is here.

Elements of quantum mechanics. The outline of this lecture is here.

Quantization in Hamiltonian picture and Schoedinger equation. The outline of this lecture is here.

The subject of this lecture is the semiclassical limit and the beginning of the discussion of path integrals in quantum mechanics. The outline of this lecture is here.

This lecture starts with the description of asymptotical expansion of oscillating integrals and on how to describe such expansions in terms of Feynman diagrams. Then we pass to the semiclassical expansion of the propagator in quantum mechanics is described in terms of Feynman integrals. This lecture follows the corresponding section from Holbaek lecture notes.

The framework of a $d$-dimensional local quantum field theory is outlined: (1) vector spaces (spaces of pure states) are assigned to $d-1$-dimensional manifolds, (2) a vector $Z(M)\in H(\pa M)$ is assigned to each $d$-manifold where $H(\pa M)$ is the vector space assigned to the $d-1$-dimensional boundary of $M$ according to (1). These assignments should satisfy certain axioms. Such structures were long known statistical mechanics on lattices with boundaries. In topological and conformal quantum field theories they were formulated by Atiyah and Segal. This lecture follows the corresponding section from Holbaek lecture notes.

Lagrangian classical field theory is the focus on this lecture. Examples are classical mechanics, classical scalar Bose field. Then the discussion was focused on principal $G$-bundles and connections. The lecture was close to corresponding sections of Holbaek lectures. Notes of connections are here.

In this lecture we will finish the description of the classical Yang-Mills theory . After this we will apply techniques of Feynman diagrams to the semiclassical quantization of classical field theories. The main example with be the scalar field with polynomial interaction. We will see that ultraviolet divergencies is one of the main problems in this direction.

In this lecture I will focus on the semiclassical quantization of gauge systems. We will discuss quantization of gauge theories, Faddeev-Popov ghost field, BRST quantization and corresponding algebraical tools from the theory of Lie groups and Lie algebras.

Continuation of the quantization of Yang-Mills in terms of Feynman diagrams.

A simple model: 2-dimensional Yang-Mills theory.

This is a take home exam. Please mail the pdf file with solutions to reshetik at math.berkeley.edu. It can be tex type file or scanned hand written text (in this case please make sure it is readable). Homework will add to the final grade up to 3 points (out of 10). The problems should be solved individually, the exam is here.

1. Glimm, James; Jaffe, Arthur Quantum physics. A functional integral point of view. Springer-Verlag, New York-Berlin, 1981. A textbook on constructive field theory. Somewhere half way it becomes somewhat technical, unless you want to know more detains on constructive field theory.

2. Weinberg, Steven The quantum theory of fields. Vol. I- III. Cambridge University Press, Cambridge, 2005. This is a classical textbook which is used at physics departments.

3. Faddeev, L. D.; Yakubovskii, O. A. Lectures on quantum mechanics for mathematics students. Translated from the 1980 Russian original by Harold McFaden. With an appendix by Leon Takhtajan. Student Mathematical Library, 47. American Mathematical Society, Providence, RI, 2009. This is the excellent mathematically minded introduction into quantum mechanics. It corresponds to a course given for undergraduate students at Leningrad University (St. Petersburg University now) in 1970's-1980's. A generation of mathematical physicists was rased on this text.

4. Takhtajan, Leon A. Quantum mechanics for mathematicians. Graduate Studies in Mathematics, 95. American Mathematical Society, Providence, RI, 2008.
This book is a graduate student version of Faddeev's book. It is great for mathematics graduate students interested in modern physics and for physics students who want to see more mathematically consistent treatment of quantum mechanics then in many physics textbooks.