physics751: Group Theory (for Physicists)

1 Physics751: Group Theory (for Physicists) Christoph L AVZ 124, Phone 73 3163 16, 2010 Contents1 About These Notes .. Time and Place .. Tutorials .. Exam and Requirements .. Literature ..62 Groups: General Motivation: Why Group Theory ? .. Definition, Examples .. Examples .. Subgroups and Cosets .. Conjugates, Normal Subgroups .. Quotient Groups .. Group Homomorphisms .. Product Groups .. Summary ..213 General Group Actions .. Representations .. Unitarity .. Reducibility .. Direct Sums and Tensor Products.

2 Schur s Lemma .. Eigenstates in Quantum Mechanics .. Tensor Operators, Wigner Eckart Theorem .. Tensor Operators .. Matrix elements of Tensor Operators .. Summary ..364 Discrete The Symmetric Group .. Cycles, Order of a Permutation .. Cayley s Theorem .. Conjugacy Classes .. Representations .. The Regular Representation .. Orthogonality Relations .. Characters .. Finding Components of Representations .. Conjugacy Classes and Representations .. Representations of the Symmetric Group .. Reducibility and Projectors.

3 Young Tableaux and Young Operators .. Characters, the Hook Rule .. Summary ..565 Lie Groups and Lie Definition, Examples .. Examples, Topology .. Some Differential Geometry .. Tangent Vectors .. Vector Fields .. The Lie Algebra .. Matrix Groups .. The Lie Algebra of Matrix Groups .. Lie Algebras ofGL(n,K) and Subgroups .. From the Algebra to the Group .. Lie s Third Theorem .. The Exponential Map .. Summary ..736 Representations of Lie Generalities .. Structure Constants .. Representations.

4 The Adjoint Representation .. Killing Form .. Subalgebras .. Real and Complex Lie Algebras .. Representations ofsu(2) .. Diagonalising the Adjoint Representation .. Constructing an Irreducible Representation .. Decomposing a General Representation .. The Cartan Weyl Basis .. The Cartan Subalgebra .. Roots .. Roots and Weights .. The Master Formula .. Geometry of Roots .. Example:su(3) .. Positive and Simple Roots .. Example:su(3) .. Constructing the Algebra .. Representations and Fundamental Weights.

5 Highest Weight Construction .. Fundamental Weights .. Cartan Matrix, Dynkin Diagrams .. The Cartan Matrix .. Dynkin Diagrams .. Classification of Simple Lie Algebras .. Systems .. Constraints .. The Dynkin Diagrams of the Classical Groups .. (n) .. (n) .. (n) .. A Note on Complex Representations .. Young Tableaux forSU(n) .. Products of Representations .. Subalgebras .. Spinors and Clifford Algebra .. Casimir Operators .. Summary ..1234 Chapter About These NotesThese are the full lecture notes, including up to Chapter 6, which deals with repre-sentations of Lie algebras.

6 Most likely there are still errors please report them to meso I can fix them in later versions!When reading the notes, you should also do the exercises which testyour under-standing and flesh out the concepts with explicit examples and calculations. Doing theexercises is what makes you learn this (and any other) topic! Time and PlaceThis is a three-hour course. Lectures will be on Wednesday from14 to 16 and on Fridayfrom 9 to 10. Since lecture period is from April 12 to July 23, with a break from May25 to 29, 14 weeks, and Dies Academicus on Wednesday May 19, there will be13 two-hour and 14 one-hour problem sheets, lecture notes (chapter by chapter, delayedwith respect to the lecture) and further information will be posted TutorialsThere are three tutorial groups.

7 Monday 10 12, Raum 5 AVZ, Thomas Wotschke Tuesday 10 12, Raum 5 AVZ, Daniel Lopes Wednesday 10 12, Raum 118 AVZ (H orsaal), Marco will be handed out in weekn, solved during the week, collected and discussedin weekn+ 1, corrected during the next week and returned in weekn+ 2. Problemscan be solved in groups of two Exam and RequirementsTo be allowed in the exam, a student needs to have obtained 50 % of the problem sheetpoints AND presented two solution on the will be written the on the first Tuesday after the term, July 27. If required, aresit will be offered at the end of the term break, late September or early LiteratureThere are many books on this subject, ranging from formal to applied.

8 Here I givea selection. Furthermore, many lecture notes are available on the web. In particular,I have partially followed the lecture notes of Michael Ratz (TU Munich), which areunfortunately not freely available on the web. H. Georgi, Lie Algebras In Particle Physics. From Isospin To Unified Theories, Westview Press (Front. )(1982) classic, very accessible. A second edition has come out in 1999, containing alsoa nice chapter on discrete groups. M. Hamermesh, Group Theory and Its Application to Physical Problems, Addison Wesley Publishing (1962)A classical reference, in particular for discrete groups and applications inquantum mechanics.

9 H. Weyl, Quantum mechanics and Group Theory , Z. (1927) of the original foundations of the use of symmetry in quantum mechanics R. N. Cahn, Semisimple Lie Algebras And Their Representations, Menlo Park,USA: Benjamin/Cummings ( 1984) 158 P. ( Frontiers In Physics,59)(Availableonline ~rncahn/www/ )Short book, available for free H. F. Jones, Groups, representations and physics, Bristol, UK: Hilger (1990)287 pAlso discusses finite groups and quantum mechanics, mathematicallysimple R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, NewYork, USA: Wiley Interscience (1974)Covers mainly mathematical aspects of Lie groups, supplies someproofs omittedin the lecture W.

10 Fulton and R. Harris, Representation Theory : A First Course ,SpringerGraduate Text in Mathematics 1991 Modern Treatment, mathematical but very accessible6 J. Fuchs and C. Schweigert, Symmetries, Lie Algebras And Representations: AGraduate Course For Physicists, Cambridge, UK: Univ. Pr. (1997) 438 pRather formal and advanced treatment, for the mathematically interested R. Slansky, Group Theory For Unified Model Building, Phys. , 1(1981).Invaluable reference: Contains a large appendix with loadsof tables ofrepresentations and branching rules.