Group Theory
PHYS 7550 Group Theory
Instructor: Mengxing Ye (JFB 305)
Time: MW 9:40-11:00AM
Location: JFB 210
Textbooks:
No textbooks are required to purchase for the course. Reading materials will be suggested for each lecture. Below are the references used for the readings unless otherwise specified.
Arovas: Lecture notes on Group Theory in Physics by Daniel Arovas from UC San Diego.
Dresselhaus: Group Theory -- Application to the Physics of Condensed Matter (lecture notes version) by M. S. Dresselhaus, G. Dresselhaus and A. Jorio.
Tinkham: Group Theory and Quantum Mechanics by M. Tinkham.
Zee: Group Theory in a Nutshell for Physicists by A. Zee [link to the pdf book]
Georgi: Lie Algebras in Particle Physics by Howard Georgi [link to the pdf book]
Course Syllabus:
Plan to cover: Elementary group theory and Applications in physics, including -- basic features of finite discrete groups; group representations; Lie group and Lie algebra; applications of discrete group and continuous group in (quantum) mechanical systems.
(Reading materials listed in parenthesis; A -- Arovas, D -- Dresselhaus, T -- Tinkham; Required unless marked with *)
Week 1:
08/19/24 -- Introduction: Symmetries in quantum mechanics (T Cha.1; A 1.2)
08/21/24 -- Introduction: Group definition and multiplication (T 2-1, 2-2; A 1.3.1, 1.3.2, 1.3.4)
Week 2:
08/26/24 -- Basic features of finite discrete groups: representation, rearrangement theorem, subgroup, cosets (T 2-3; A 1.3.7, 1.4.1)
08/28/24 -- Basic features of finite discrete groups: normal subgroup, quotient group (T 2-4, 2-5, 2-6, 2-8; A 1.4.1)
Week 3:
09/02/24 -- no lecture
09/04/24 -- Basic features of finite discrete groups: conjugacy, (semi)direct product (T 2-7; A 1.4.1)
HW 1 (due on Sep 18 -- submit through Canvas)
Link to the Mathematica note book for finding multiplication table and conjugacy classes automatically.
Week 4:
09/09/24 -- Class multiplications, spontaneous symmetry breaking; Brief introduction of Lie group and Lie algebra (Part 1: T 2-8; Part 2: A 1.5.1, 1.5.5)
09/11/24 -- Group representation: basic definitions (A 2.1.1 - 2.1.3)
Week 5:
09/16/24 -- Group representations: the Great Orthogonality Theorem (G.O.T) part 1 (A 2.1.4)
09/18/24 -- Group representations: the Great Orthogonality Theorem (G.O.T) part 2 (A 2.2.1 - 2.2.3; D 2.5 - 2.8)
Week 6:
09/23/24 -- Group characters (D 3.1 - 3.4 ; A* 2.3.1, 2.3.2, 2.4.1)
09/25/24 -- More on character table; Direct sum, direct product of representations (D 3.5-3.8 ; A 2.4.8)
HW 2 (due on Oct 11 -- submit through Canvas)
Week 7:
09/30/24 -- Application of group theory to (quantum) mechanical systems I: general discussions, coupled oscillators (A. Zee Group theory in a nutshell for physicists, Part III.1, III.2 [link to the pdf book])
10/02/24 -- Application of group theory to (quantum) mechanical systems II: coupled oscillators (A. Zee Group theory in a nutshell for physicists, III.2; more about group generators: A 1.4.6)
Week 8: Fall break
Week 9:
Week 10:
10/21/24 -- Introduction to Lie group, SO(2)
10/23/24 -- SO(3): finding irreps using the highest weight construction
Week 11:
10/28/24 -- SO(3): characters, orthogonality and Haar measure; SU(2); direct product of irreps (Haar measure: A 4.3.4; direct product of irreps: H Georgi Lie Algebras in Particle Physics 3.4-3.5 [see the link to the textbook pdf in the "Textbook" session] )
10/30/24 -- direct product of irreps cont'd; adjoint representation (direct product of irreps: Georgi 3.4-3.5; adjoint rep.: Georgi 2.4)
HW 3 (due on Nov 11 -- submit through Canvas)
Week 12:
Week 13:
Week 14:
Week 15:
11/25/24 -- Group Theory Applications in Kondo Anyon Physics (guest lecture given by Dr. Guangjie Li)
11/27/24 -- Application: coherent state path integral of quantum spins
Week 16:
12/2/24 -- Final presentations 1
12/4/24 -- Final presentations 2