FYTN13, Symmetries and Group Theory

Running spring semesters odd years, first half.

VT 2019

Formal course specs (in Swedish)
Course pages at Live@Lund

Literature

Discrete part: Lecture notes by Ferdi Aryasetiawan

Continuous part: Lecture notes by Hugo Seriodio and Malin Sjödahl. These notes will be frequently updated during the course. Parts that we don't have time to go through will be gray shaded. Please report typos.

Course: Groups, representations and Physics, 2nd edition: H.F. Jones. From within Lund university, this book can be downloaded here.

People

Lecturers:
Ferdi Aryasetiawan, Mathematical Physics, Discrete part
Malin Sjödahl, Theoretical Particle Physics, Continuous part

Schedule

The schedule for 2019 can be found at the Live@Lund FYTN13 pages.

Preliminary course plan, continuous part, VT 2019

Lecture 1
Def of group and connection to physical transformations; Let's rotate, Lie Group ~ continuous group, SO(2) def, 1D rep and infinitesimal generator; SO(3) def and generators, generators in terms of the Levi-Cevita symbol, anti-commutator in terms of Levi-Cevita symbol, SO(3) algebra, the Axis-angle parametrization
Ch 2.1-2.1.3, 2.2-2.2.3 in lecture notes
Lecture 2
Lie groups: general concepts, topological properties, matrix groups, including general linear GL(n,F), (pseudo) orthogonal O(p,q), (pseudo) unitary U(p,q), and their special versions SL(n,F), SO(p,q), SU(p,q) Lie algebras: general concepts, field, linear vector space, algebra, definition of the Lie algebra, the Jacobi identity, Lie group as exp (Lie algebra); The so(3) Lie algebra with Hermitian matrices, the Casimir operator
Ch 1, 2.2.4-2.2.5 in lecture notes
Lecture 3
The 2l+1 dimensional representations of SO(2); The SU(2) group, definition, form from constraints, the su(2) algebra in terms of anti-symmetric matrices and Hermitian matrices; The relation between SU(2) and SO(3), (see that rotating we get the same transformation),
Ch 2.2.6, 3.1-3.3 in lecture notes
Lecture 4
The U(1) subgroup of SU(2); The 2j +1 dimensional irrep of SU(2); The direct product space, definition, the spin 1/2 x spin 1 example, Clebsch-Gordan coefficients, SU(2) weight diagrams
Ch 3.4-3.7 in lecture notes
Lecture 5
Real, pseudoreal, complex representations; SU(N), simple group (non-abelian with no continuous invariant subgroup), semi-simple (non-abelian no continouos abelian subgroup), simple Lie algebra, semi-simple Lie algebra, the Lie algebras su(N), u(N), the Killing form, the adjoint matrices, reality property of SU(2); SU(3) form of generators rank 2 group (2 simultaneously commuting generators, Cartan subalgebra) step operators in SU(3), multiplets in SU(3)
Ch 3.8, 4.1-4.3.1 in lecture notes
Lecture 6
SU(3) multiplets and weight diagrams; Classification of semi-simple Lie algebras
Ch 4.4 in the notes and Ch 9.2-9.5 in Jones
Lecture 7
The Lorentz group, defining properties, generators, algebra in terms of K' and L, the algebra split into N+, N-; Lorentz group representations, the (0,0) representation, the (1/2,0) and (0,1/2) representations, Van der Waerden notation/spinor helicity notation, the (1/2,1/2) representation, (the Poincare group)
Ch 6.1-6.2.5 in lecture notes
Lecture 8
Repetition

Oral exams:

Week 12, March 18-22
Oral exams, continuous and discrete part.

Hand-in exam:

May 2nd-9th
Written re-exam for the continuous part.

The hand-in reexam for the continuous part is now available here and should be handed in latest Thursday May 9th at 5 pm.

The tasks should be solved individually and handed in via email, in person, or in the mailbox of Malin Sjödahl. (The mailbox is located on the same floor as the HUB lecture room.)

Exercises

There will be three exercise sessions for the continuous part of the course. The exercises chosen for the problem sessions will be posted here.