An outline of the class:


The class meets on MWF, 11:00-12:00, at 891 Evans


We will start with basics of classical field theory: the Lagrangian formulation, the Hamiltonian formulation. We will focus on the following examples: classical mechanics as classical field theory with one dimensional space time, non-diffeomorphism invariant models, scalar field, Yang-Mills, spinors. Effective classical field theories such as Euler equation and others. Classical field theories with discrete space time, discrete field theories.

Then we will focus on the general framework of quantum field theory. Will discuss the difference between statistical and unitary quantum field theories. Quantum mechanics is an example of quantum field theory with one dimensional space time. We will discuss the deformation quantization of Hamiltonian classical mechanics, geometric quantization. After this the notion of path integral quantization of classical mechanics we will be discussed. The precise definition of formal semiclassical path integral will be given and will be compared with the Schrodinger quantization. Another version of a mathematical framework for path integrals the Wiener integral, which will be discussed briefly.

Quantum field theory as a theory of infinitely many particles will be discussed for space times which are cylinders and tori. Two examples of such quantum field theories will be given: free theory and a one dimensional Bose gas with delta-function interaction. The semiclassical limit in this approach with be briefly discussed.

In the last part of the course we will focus on formal semiclassical quantization via path integral. In this approach the amplitudes are formal power series with coefficients given by Feynman diagrams. First we will discuss the quantization of classical theories with non-degenerate action functional. Then will see how to do it for classical gauge theories. Topological field theories are particularly "friendly" for such quantization because Feynman diagrams in such theories do not develop ultraviolet divergencies.

If time permit, we will discuss the problem of ultraviolet divergencies in Feynman diagrams and the renormalization in perturbation theory and will see how it applies to the Yang-Mills theory.

Lecture 1 General overview of the course.
Lecture 2 Classical scalar theory. Lagrangian 2nd order formulation.
Lecture 3 Pure Yang-Mills theory. Self-duality equations in 4d. Chern-Simons classical field theory started.
Lecture 4 Classical Lagrangian mechanics. Newtonian mechanics. Non-degenerate 2nd order Lagrangians and non-degenrte first order Lagrangians. Hamiltonian framework for first order Lagragians. Lecture 5 Recollection of general facts about Hamiltonian mechanics on symplectic manifolds. Isotropic, Lagrangian, coisotropic submanifolds in a symplectic manifold. Variational boundary conditions.
Lecture 6 Legendre transform. Hamiltonian framework for Newtonian mechanics on Riemannian manifolds. First order Lagrangian formulation of Newtonian classical mechanics (Hamilton-Jacobi action).
Lecture 7 Non-degenerate first order classical mechanics as a 1-dimensional classical field theory. Boundary conditions as Lagrangian fibrations.
Lecture 8 First order formulation of the scalar classical field theory.
Lecture 9 First order formulation of scalar classical field theory. The discussion of differential forms on the space of fields. The boundary structure: the image of the Euler-Lagrange space in the space of boundary fields in Lagrangian provided there is a unique solution to the Dirichlet problem (true in the liner case).
Lecture 10 Plan: First order formulation of the classical pure Yang-Mills theory. Gauge invariance. The action of the gauge group on the space of boundary fields is Hamiltonian. The Hamiltonian reduction of the space boundary fields. The reduced Hamiltonian Yang-Mills theory.
Lecture 11 Plan: The Chern-Simons theory. The gauge invariance. Moduli spaces of flat connections. The reduced classical Chern-Simons theory. Degenerate first order classical theories.
Lecture 12 Plan: Hamilton-Jacobi function in classical mechanics. Critical values of action functionals in classical field theories. Finite dimensional model.
Lecture 13 Plan: The functorial framework of quantum field theory. Quantum mechanics as an example.


Some references:

Classical field theories:

See Notes by D. Freed . For classical Chern-Simons theory see Classical Chern-Simons Theory also by D. Freed and Lie Groups and Chern-Simons Theory by Benjamin Himpel.

To compare with more physics oriented expositions see notes by E. Fradkin .

For Hamiltonian aspects of classical field theory on space time manifolds with boundaries see and references therein:
1) N. Reshetikhin, Lectures on quantization of gauge systems , arXiv:1008.1411.
2) A. S. Cattaneo, P. Mnev, N. Reshetikhin, Classical BV theories on manifolds with boundary , arXiv:1201.0290.
3) Alberto S. Cattaneo, Pavel Mnev, Nicolai Reshetikhin, Classical and quantum Lagrangian field theories with boundary , arXiv:1207.0239.

A year long program on QFT was organized in 1996-1997 at IAS. See the site of the program for Lecture Notes from this year.

Sklyanin