Physics 262: Group Theory for Physicists (Winter 2019, UCR)

View the Project on GitHub Tanedo/Physics262-2019

Group Theory for Physicists

Course calendar
Triangular Graph Paper via Jan Gutowski

We meet Mondays, Wednesdays and every other Friday at 4:15 - 5:30pm (see schedule) in the Physics Conference Room.

Course Description

Graduate-level introduction to the representation of continuous (Lie) groups and how they appear in quantum mechanical systems. How are continuous symmetries manifested mathematically in our physical theories? What patterns of spontaneous symmetry breaking are allowed? How do representations combine? Examples will draw from particle and nuclear physics, but the course will focus on the mathematical formalism relevant to many theoretical disciplines. (Strongly recommended to hep-ex/ph students and cond-mat theory students.)



What we didn’t get to

There are a few topics that I regret that we did not cover. I list them here for future iterations of the course, but also so that students of this course know that these topics are “just within reach” of what they now already understand.

  1. Squeezing out more representation theory. Casimir operators, using Dynkin diagrams, Young tableaux for SU(N).
  2. More on spinors: Weyl, Majorana, Dirac; the Clifford algebra; spinors in condensed matter.
  3. Spontaneous symmetry breaking (the handbook is Slansky) and Goldstone Bosons (e.g. Burgess review)
  4. Conservation laws
  5. Topology and physics
  6. Anomalies

A sketch of the solution to the twin paradox on a compact space.

Course References

For our goals, I am not aware of any single textbook that introduces the topic at a pace and level appropriate for our class. I suggest using a combination of the following references. The lectures are meant to be connective tissue that gives an overarching narrative for what we are doing and why.

Other books I have used in the past are those by: Lipkin, Wu-Ki Tung, Jones.


3 problem sets (60%)

1 in-class presentation and essay on an application of group theory in your field (20% + 20%)


Familiarity with linear algebra at the level of Physics 221 (Quantum Mechanics). No background (or primary interest) in particle physics, field theory, or abstract algebra necessary.

About PHYS 262

Physics 262 is a graduate-level special topics course in high-energy physics. Topics change by year and instructor. The course may be taken multiple times for credit. Please register for this class, by registering, you are encouraging the department and the college to offer more of these special topics courses in the future.

Tentative course plan

We’ll see how things actually pan out as the course goes along.

Week 1: (iso)spin in quantum mechanics, groups and algebras (chapter 3)
Week 2: fundamental, anti-fundamental, adjoint: what is a representation (chapter 1-2)
Week 3: roots and weights (chapter 6)
Week 4: generalization to SU(3) (chapter 7)
Week 5: simple roots (chapter 8 - 9)
Week 6: tensor and irreducible representations, Clebsch-Gordan coefficients (chapter 10)
Week 7: Dynkin diagrams (chapter 6)
Week 8: spontaneous symmetry breaking in physics
Week 9: presentations
Week 10: advanced topics: the Poincare group and fermions