FYTN13, Symmetries and Group Theory
Running spring semesters odd years, first half.VT 2019
Links
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 LeviCevita symbol,
anticommutator in terms of LeviCevita symbol, SO(3) algebra,
the Axisangle parametrization
Ch 2.12.1.3, 2.22.2.3 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, 2.2.42.2.5 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
antisymmetric matrices
and Hermitian matrices;
The relation between SU(2) and SO(3),
(see that rotating we get the same transformation),
Ch 2.2.6, 3.13.3 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,
ClebschGordan coefficients,
SU(2) weight diagrams
Ch 3.43.7 in lecture notes  Lecture 5

Real, pseudoreal, complex representations;
SU(N),
simple group (nonabelian with no continuous invariant subgroup),
semisimple (nonabelian no continouos abelian subgroup),
simple Lie algebra,
semisimple 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, 4.14.3.1 in lecture notes  Lecture 6

SU(3) multiplets and weight diagrams;
Classification of semisimple Lie algebras
Ch 4.4 in the notes and Ch 9.29.5 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 6.16.2.5 in lecture notes  Lecture 8
 Repetition
Oral exams:
 Week 12, March 1822
 Oral exams, continuous and discrete part.
Handin exam:
 May 2nd9th
 Written reexam for the continuous part.
The handin 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.)