Group Theory and its Applications

Overview

An introduction to group theory and its applications to physics, developed from the notion of symmetry transformations through the abstract theory of groups, their representations, and applications to quantum mechanics.

Assessment

• Midterm exam
• Quiz
• Major (endterm) exam

Lecture outline

0. Course overview and logistics.
1. Transformations vs symmetries, examples, symmetry transformations of the isosceles triangle. (Zee I.1; Jones §1.3)
2. Active vs passive transformations, notation, defining symmetries, the set of allowed transformations, “more” symmetry of the equilateral triangle. (Zee I.1; Jones §1.2–1.3)
3. Symmetries and transformations in physics (examples), symmetry transformations of the equilateral triangle, rotations and reflections, non-commutativity of composition of transformations (order matters). (Jones §1.2–1.4)
4. Generators of symmetry transformations, symmetries of regular polygons with (un)directed sides. (Jones §1.4)
5. Definition of a group and physical motivation from symmetry transformations, Abelian groups, basic examples. (Jones §2.1–2.2)
6. Cyclic group of n elements (rotations of a regular polygon with n directed sides), group multiplication law of C_n. (Jones §2.3–2.4)
7. Group isomorphism (cyclic group example), multiplication table.
8. Homomorphisms, subgroups, dihedral group D_n (rotations of a regular polygon of n undirected sides), its non-Abelian nature. (Jones §2.4, 3.1, 3.3)
9. Permutation group of n elements S_n.
10. Permutation group (continued), Cayley’s theorem.
11. Transpositions, even/odd permutations, equivalence relations, conjugacy relations.
12. Conjugacy classes, partitioning of a group, examples. (Jones §3.3–3.5)
13. Subgroups, cosets, quotient set, Lagrange’s theorem.
14. Normal subgroups, quotient groups, examples. (Jones §3.5, 4.1)
15. Homomorphisms, image and kernel subgroups.
16. Isomorphism theorem. (Jones §4.1–4.3)
17. Group representations.
18. Group representations (continued): equivalence, character, reducibility.
19. Vector spaces, action on vector spaces, reducibility & complete reducibility revisited, linear transformations and matrix representations. (Jones §4.3–4.4)
20. Linear transformations (continued), connection to group representations, similarity of group representations revisited, scalar product. (Jones §4.4, 3.2, 4.5)
21. Group representation theory (continued).
22. Unitary representations, reducibility of unitary representations implies complete reducibility.
23. G-modules, submodules, Maschke’s theorem.
24. Schur’s lemmas.
25. Orthogonality of irreps. (Jones §5.3)
26. Number of inequivalent irreps, examples.
27. Orthogonality of characters, number of irreps. (Jones §5.3, 6.1)
28. Regular representation, its decomposition for Abelian groups, example of C_3, irreps of C_3.
29. Vector representation, its decomposition, example of C_3.
30. Invariant subspaces of vector representations of C_3, D_3, irreps of D_3.
31. Vector and regular representations of D_3, transformation of real-valued functions under the rotation group.
32. Transformation of real- and vector-valued functions under the rotation group.
33. Wavefunctions in QM and representation theory, direct products of representations.
34. Quiz.
35. Degeneracies in QM from representation theory.
36. Degeneracies in QM from representation theory (continued).
37. Breaking of degeneracies from representation theory.
38. SO(2), U(1) groups, irreps of SO(2), direct product of representations, character orthogonality for compact groups.

References

• A. Zee, Group Theory in a Nutshell for Physicists
• H. F. Jones, Groups, Representations and Physics