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.

Classical field theories:

See Notes by D. Freed . For classical Chern-Simons theory see

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