Physics 262: Group Theory for Physicists (Winter 2019, UCR)
Syllabus
Course calendar
Webpage
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.
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.)
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.
A sketch of the solution to the twin paradox on a compact space.
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.
Georgi, Lie Algebras in Particle Physics. This is our official textbook, it has almost everything we’ll need (and then some) but not necessarily in the order that makes the most sense for us. It is convenient because it contains derivations of the main results.
Cahn, Semi-Simple Lie Algebras and their Representations. This is a bit more terse and goes beyond the scope of the course. It is available free from the author.
Zee, Group Theory in a Nutshell for Physicists. This is one of my favorite books for its conversational and pedagogical tone. We’ll basically be jumping into the second half of the book, which is a bit hefty.
Gutowski, lecture notes from Part III: Symmetry and Particle Physics. Introduces the geometry of Lie groups in careful detail. We will use this as a primary reference for tensor products.
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.
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.
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