Group Theory for Physicists

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.

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.)

Homework

• HW1: Reps of SU(2) and SU(3). Updated 1/24.
• HW2: Tensor stuff

Lectures

• Lec 1: Introduction: rotations.
• Lec 2: Review of SU(2) (corrected SU(2) notes)
• Lec 3: SU(2) representations, the adjoint representation, introduction to SU(3). See: Cahn chapter 1-2, Gutowski, Georgi chapter 3.
• Lec 4: the adjoint representation, SU(3) representations. Checking commutation relations in Mathematica nb/pdf. Additional course references placed in iLearn.
• Lec 5: SU(3) representations, once again, slightly more abstractly. Fundamental and the anti-fundamental.
• Lec 6: SU(3) weight diagrams, multiplicity of states.
• Lec 7: Hexagonal weight diagrams
• Lec 7b On the adjoint
• Lec 8: Tensor representations
• Lec 9: Indices for SU(N)
• Lec 10: Invariant tensors, roots
• Lec 11: HW1 review, roots for SO(4) and SO(5)
• Lec 12: Root systems
• Lec 13: Towards a general classification
• Lec 14: The Lorentz Group, I (notes, mostly correct)
• Lec 15: The Lorentz Group, II
• Lec 16: Induced Representations
• Lec 17: A hint of holography, spinors (omitted in the lecture), the Mobius strip. Useful reference: Topology of Fibre bundles and Global Aspects of Gauge Theories (hep-th/0611201).
• Lec 18: What physicists should know about fiber bundles, how it shows up in electromagnetism. There are many good references for fiber bundles. I suggest 1607.03089 and any review articles on electrodynamics from a geometric picture. For more systematic treatments grounded in physics, Mathematical Methods in Classical Mechanics by Arnold is fantastic. Other introductions inclue Vol. II of Polchinski’s string theory books or the text by Nakahara. Mathematics for Physicists by Stone and Goldbart is also a good introduction at this level. I’ve also heard good things about Differential Geometry and Lie Groups for Physicists by Fecko.
• Lec 19: Final lecture: the Lie derivative on manifolds (and why it’s the same thing as the commutator of the Lie algebra). Hand-wavey comments towards directions where your work may touch on aspects of this construction. The lecture draws from Geometrical methods of mathemtical physics by Bernard Schutz.
• Lec 20: canceled due to an un-movable, last minute, required university-level meeting (I’m shaking my fist in the air as I type this). If we were to have had class, we may have talked about the solution to the twin paradox in a closed universe, the gauge theory of falling cats, parallel parking and holonomies, how to write Maxwell’s equations succinctly, differential forms and vector analysis. For those also taking GR, you may like John Baez’s tutorial on GR which gives a physical motivation for the ideas of the last 3 lectures and the accompanying explainer on the meaning of Einstein’s equation.

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.

• 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.

Evaluation

3 problem sets (60%)

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

Pre-requisites

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