PHY406 Symmetries in Physics

S. G. Rajeev

Spring 2010 Schedule: B&L 407 10:00-12:45 on Wednesdays

Course web page

First Class Wed Jan 13

This course is about the applications of groups and Lie algebras to physics. I will assume some knowledge of linear algebra (e.g., hermitean and unitary matrices) but more important is the ability to grasp new ideas quickly. It will be accessible to advanced undergraduates as well as graduate students.

There will be no required textbook. I will post my notes below.

Jan 13 Lecture 1 Jan 20 Lecture 2 Jan 27 Lecture 3 & Problem Set 1 Feb 3 Lecture 4 & Problem Set 1 Due
Feb 10 Lecture 5 & Problem Set 2 & Solutions to Set 1 Feb 17 Lecture 6 & Problem Set 2 Due Feb 24 Lecture 7 & Problem Set 3 & Solutions to Set 2 Mar 3 Lecture 8

Mar 10 Spring Break

Lecture 9 & Problem Set 4 & Solutions to Set 3 Mar 24 Lecture 10 Mar 31 Lecture 11
Apr 7 Lecture 12 Apr 14 Lecture 13 & Problem Set 5 Last Class Apr 21 Lecture 14 .
The following are useful references.

  1. "Lie Groups in Physics" by G. 't Hooft available online at this link
  2. "Lie Algebras in Physics" by Howard Georgi. Benjamin-Cummings (1982)
  3. "Quarks, Leptons and Gauge Fields" 2nd edition by Kerson Huang World Scientific (1992)

The course will be graded Pass/Fail.

You should expect assignments of three problems roughly every couple of weeks.

I am teaching also a parallel course on Particle Physics, where the classification of elementary particles using groups such as SU(3), SU(2)xU(1) will be discussed.


  1. Symmetries in physics and geometry
  2. The Definition of a group; symmetries in quantum mechanics
  3. The Definition of a Lie algebra. The commutation relations of angular momentum.
  4. Representation of angular momentum by matrices. Spin.
  5. The rigid body. Classical equations and their solution.
  6. Euler equations of an ideal fluid; Lie algebra of incompressible vector fields.
  7. The spectrum of a quantum rigid body. Rotation spectrum of atoms and molecules.
  8. Isospin. Nuclear energy levels. Approximate symmetries.
  9. Static Quark Model. Color. Baryon Magnetic Moments.
  10. Bosonic and Fermionic spaces. Symmetric and exterior algebras.
  11. Invariant inner products.Ideals. Compact Simple Lie Algebras. Gell-Mann basis for SU(3).
  12. Random Matrices. Gaussian Unitary Ensemble (GUE) and GOE. Wigner surmise. Universality.
  13. Relativistic wave equations: Klein-Gordon,Dirac and Maxwell's equations.
  14. Yang-Mills Theory
  15. The symmetry group of the universe: the de Sitter group.

Jan 6, 2010.