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Phys 7212: Mathematical Methods in Theoreticall Physics II, Spring 2023

Symmetry, Topology & Emergence in Quantum Physics

J. A. Sauls  

Department of Physics & Astronomy, Louisiana State University 

Lectures: Monday, Wednesday, Friday from 10:30 - 11:30 in Nich 106

Symmetry and topology - the mathematics of group theory and homotopy theory - play special roles in atomic, nuclear, high-energy and condensed matter physics. The first half of this course will focus on Lie groups that play central roles in quantum theory of atomic and nuclear structure and processes, the standard model of particle physics and quantum field theories of condensed matter systems.

The second part of the course will explore the role of spontaneous symmetry breaking (SSB) in theories ranging from phase transitions to mass generation in quantum field theory, particularly as SSB relates to the emergence of new elementary particles and quanta, as well as novel excitations, "topological defects". We will develop homotopy theory related to the underlying topology of broken symmetry ground states and use it to classify and study the stable topological defects in a wide range of broken symmetry states of quantum field theories.


  • Format: Lecture and Discussion
  • Course Material: Textbooks, Reading & Lecture Notes
  • Evaluation: Attendance & written report (10 pages) on a topic chosen in consultation with the instructor, plus a 10+5 minute oral presentation of the report to the class.


  • Group Theory in a Nutshell, A. Zee, Princeton Press.
  • Symmetries in Quantum Physics, U. Fano & A. R. P. Rao, Academic Press.
  • Symmetry, Broken Symmetry & Topology, M. Guidry & Y. Sun, Cambridge.
  • Topological Quantum Numbers in Physics, David Thouless, World Scientific.
  • Cosmic Strings & Topological Defects, A. Velinkin & E. Shellard, Cambridge.


  • Finite, Discrete & Point Groups
  • Representations of Point Groups
  • Lie Groups, Lie Algebras & their Representations
  • Direct Product Spaces - Reducible representations
  • Symmetry, Symmetry Breaking & Phase transitions
  • Discrete Symmetries: Parity, Time-Reversal & Charge Conjugation
  • Landau Theory and Lagrangian Field Theories
  • Topology, Homotopy Groups and Topological Defects
  • Applications of Group Theory & Topology in QFT
    • CMT: Liquid Crystals, Ferromagnets & Superconductors
    • QFT: Mass Generation, Higgs & Gauge Bosons
    • CMT: Higgs Bosons & Topology of Helium-Three
    • QFT: Nonequilibrium Phase Transitions
    • COS: Nucleation of Topological Defects

Course Website Teaching HITP LSU Physics