MAT561: Physics for Mathematicians II (Spring 2008)

Instructor

Coordinates:

Outline

This is the second part of an ambitious two-part course in theoretical physics aimed at the graduate mathematics student. It is a fundamental part of the RTG Program in Geometry and Physics with the purpose of introducing many of the basic concepts, theories, and principles which form the basis of our current understanding of the Universe. The topics covered in the MAT 560 included the classical (non-)relativistic dynamics of particles and fields.

Prerequisite

In the second semester we will attempt to cover as much as possible from the following outline of topics:

• Classical field theory (continued):
  • Classical spinor fields
  • Sigma models
  • Supersymmetry
• Classical Mechanics:
  • Hamiltonian formalism
• Quantum Mechanics:
  • Heisenberg mechanics
  • Schrödinger mechanics
  • Dirac notation
  • Spinors
  • Feynmann Path Integral
• Quantum Fields I:
  • Canonical quantization
  • Path integral quantization
• Quantum Fields II:
  • Review of gauge systems
  • Gauge fixing and ghosts: Fadeev-Popov method
  • BRST method
  • BV method
• String theory I:
  • Bosonic string
  • Fermionic strings
  • Green-Schwarz string
• String theory II:
  • Covariant quantization
  • Backgrounds

As was the case in the first semester, the course will be taught from a physical perspective (mathematical rigor is not emphasized). The aim is to gain familiarity with and intuition for many of the concepts usually taught to a student of theoretical physics over the course of his/her undergraduate and graduate training.

Textbooks

The unorthodox nature of this program makes it impossible to follow a textbook. Nevertheless, I have assigned Theodore Frankel's "The Geometry of Physics: An Introduction" as a text. This book, as many others, is written to teach physicists mathematics and not the other way around. We will try to reverse-engineer and follow various sections of this book throughout the semester.

Due to the vast amount of material and the lack of centralized supporting literature, a lot of work will be required on the students' part to keep up. Besides the completion of regularly assigned homework sets, the student should read up on the topics being presented in the class.

Announcements

• The MAT 560 page has been updated to include all the old homework assignments.

Assignments

1. If you did not take MAT 560 you might want to look over the homework assignments from last semester.
2. Homework 1 due Wednesday March 5 in class: Do all the problems in this (PDF) set.
3. Homework 2 due Wednesday March 26 in class: Do all the problems in this (PDF) set.

Resources

• In class I recommended taking a look at V.I. Arnold's "Mathematical Methods of Classical Mechanics. I will try to remember to put it on reserve in the library.
• Matt Young reminded me of the notes on Classical Field Theory and Supersymmetry by Daniel S. Freed in the 1996-1997 IAS program in Quantum Field Theory. As Matt points out, there is quite a bit of overlap with what we are/will be doing in class from a pure math perspective.
• Thanks go out again to Matt Young for pointing out this article on how to derive the Euler-Lagrange equation in a coordinate invariant manner.
• Some of the material on special relativity I am presenting in class is based on Bernard Schutz' book "A first course in general relativity" [Spires]
• Matt Young has sent me a copy of the "Toronto Lectures on Physics" by Shlomo Sternberg on various topics covered in MAT 560. He recommends the first 5 chapters. The rest of the lectures also have some nice stuff which we will see in the coming weeks.