physics751: Group Theory (for Physicists) Christoph L¨udeling [email protected] Office AVZ 124, Phone 73 3163 http://www.th.physik.uni-bonn.de/nilles/people/luedeling/grouptheory/ August 16, 2010 Contents 1 Preliminaries 5 1.1 AboutTheseNotes .............................. 5 1.2 TimeandPlace ................................ 5 1.3 Tutorials.................................... 5 1.4 ExamandRequirements . 6 1.5 Literature ................................... 6 2 Groups: General Properties 8 2.1 Motivation: WhyGroupTheory? . 8 2.2 Definition,Examples ............................. 10 2.2.1 Examples ............................... 12 2.3 SubgroupsandCosets ............................ 13 2.4 Conjugates,NormalSubgroups . 14 2.5 QuotientGroups ............................... 16 2.6 GroupHomomorphisms ........................... 16 2.7 ProductGroups................................ 20 2.8 Summary ................................... 21 3 Representations 22 3.1 GeneralGroupActions. 22 3.2 Representations ................................ 24 3.3 Unitarity.................................... 25 3.4 Reducibility .................................. 26 3.5 DirectSumsandTensorProducts . 28 3.6 Schur’sLemma ................................ 29 3.7 EigenstatesinQuantumMechanics . 31 3.8 Tensor Operators, Wigner–Eckart Theorem . 32 3.8.1 TensorOperators ........................... 33 3.8.2 Matrix elements of Tensor Operators . 34 3.9 Summary ................................... 36 4 Discrete Groups 37 4.1 TheSymmetricGroup ............................ 38 4.1.1 Cycles, Order of a Permutation . 38 2 4.1.2 Cayley’s Theorem . 40 4.1.3 Conjugacy Classes . 40 4.2 Representations ................................ 42 4.2.1 The Regular Representation . 42 4.2.2 Orthogonality Relations . 43 4.3 Characters................................... 46 4.3.1 Finding Components of Representations . 47 4.3.2 Conjugacy Classes and Representations . 48 4.4 RepresentationsoftheSymmetricGroup . 51 4.4.1 Reducibility and Projectors . 51 4.4.2 Young Tableaux and Young Operators . 52 4.4.3 Characters,theHookRule . 55 4.5 Summary ................................... 56 5 Lie Groups and Lie Algebras 58 5.1 Definition,Examples ............................. 58 5.1.1 Examples,Topology . 60 5.2 SomeDifferentialGeometry . 62 5.2.1 TangentVectors. 62 5.2.2 VectorFields ............................. 63 5.3 TheLieAlgebra................................ 65 5.4 MatrixGroups ................................ 67 5.4.1 TheLieAlgebraofMatrixGroups . 67 5.4.2 Lie Algebras of GL(n, Ã)andSubgroups . 68 5.5 FromtheAlgebratotheGroup . 69 5.5.1 Lie’sThirdTheorem . 69 5.5.2 TheExponentialMap . 70 5.6 Summary ................................... 73 6 Representations of Lie Algebras 74 6.1 Generalities .................................. 74 6.1.1 StructureConstants . 74 6.1.2 Representations. 75 6.1.3 The Adjoint Representation . 75 6.1.4 KillingForm.............................. 76 6.1.5 Subalgebras .............................. 77 6.1.6 Real and Complex Lie Algebras . 78 6.2 Representations of su(2) ........................... 78 6.2.1 Diagonalising the Adjoint Representation . 79 6.2.2 Constructing an Irreducible Representation . .... 79 6.2.3 Decomposing a General Representation . 81 6.3 TheCartan–WeylBasis ........................... 82 6.3.1 TheCartanSubalgebra. 82 6.3.2 Roots ................................. 83 3 6.4 RootsandWeights .............................. 86 6.4.1 TheMasterFormula . 86 6.4.2 GeometryofRoots .......................... 87 6.4.3 Example: su(3)............................ 88 6.5 PositiveandSimpleRoots . 91 6.5.1 Example: su(3)............................ 94 6.5.2 ConstructingtheAlgebra . 94 6.6 Representations and Fundamental Weights . .... 95 6.6.1 Highest Weight Construction . 95 6.6.2 FundamentalWeights. 96 6.7 Cartan Matrix, Dynkin Diagrams . 98 6.7.1 TheCartanMatrix .......................... 98 6.7.2 DynkinDiagrams ........................... 99 6.8 Classification of Simple Lie Algebras . 99 6.8.1 ΠSystems............................... 99 6.8.2 Constraints .............................. 100 6.9 The Dynkin Diagrams of the Classical Groups . 104 6.9.1 su(n).................................. 104 6.9.2 so(n).................................. 107 6.9.3 sp(n).................................. 111 6.10 A Note on Complex Representations . 113 6.11 Young Tableaux for SU(n).......................... 115 6.11.1 ProductsofRepresentations . 116 6.12Subalgebras .................................. 117 6.13 SpinorsandCliffordAlgebra. 120 6.14CasimirOperators .............................. 121 6.15Summary ................................... 123 4 Chapter 1 Preliminaries 1.1 About These Notes These are the full lecture notes, i.e. including up to Chapter 6, which deals with repre- sentations of Lie algebras. Most likely there are still errors – please report them to me so I can fix them in later versions! When reading the notes, you should also do the exercises which test your under- standing and flesh out the concepts with explicit examples and calculations. Doing the exercises is what makes you learn this (and any other) topic! 1.2 Time and Place This is a three-hour course. Lectures will be on Wednesday from 14 to 16 and on Friday from 9 to 10. Since lecture period is from April 12 to July 23, with a break from May 25 to 29, i.e. 14 weeks, and Dies Academicus on Wednesday May 19, there will be 13 two-hour and 14 one-hour lectures. The problem sheets, lecture notes (chapter by chapter, delayed with respect to the lecture) and further information will be posted on http://www.th.physik.uni-bonn.de/nilles/people/luedeling/grouptheory/ 1.3 Tutorials There are three tutorial groups: 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 Rauch. • Problems will be handed out in week n, solved during the week, collected and discussed in week n + 1, corrected during the next week and returned in week n + 2. Problems can be solved in groups of two students. 5 1.4 Exam and Requirements To be allowed in the exam, a student needs to have obtained 50 % of the problem sheet points AND presented two solution on the blackboard. Exam will be written the on the first Tuesday after the term, July 27. If required, a resit will be offered at the end of the term break, i.e. late September or early October. 1.5 Literature There are many books on this subject, ranging from formal to applied. Here I give a 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 are unfortunately not freely available on the web. H. Georgi, “Lie Algebras In Particle Physics. From Isospin To Unified Theories,” • Westview Press (Front. Phys. 54) (1982) 1. A classic, very accessible. A second edition has come out in 1999, containing also a 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 in quantum mechanics. H. Weyl,“Quantum mechanics and group theory,” Z. Phys. 46 (1927) 1. • One 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) (Available online at http://phyweb.lbl.gov/~rncahn/www/liealgebras/book.html) Short book, available for free H. F. Jones, “Groups, representations and physics,” Bristol, UK: Hilger (1990) • 287 p Also discusses finite groups and quantum mechanics, mathematically simple R. Gilmore, “Lie Groups, Lie Algebras, and Some of Their Applications,” New • York, USA: Wiley Interscience (1974) Covers mainly mathematical aspects of Lie groups, supplies some proofs omitted in the lecture W. Fulton and R. Harris, “Representation Theory: A First Course”, Springer • Graduate Text in Mathematics 1991 Modern Treatment, mathematical but very accessible 6 J. Fuchs and C. Schweigert, “Symmetries, Lie Algebras And Representations: A • Graduate Course For Physicists,” Cambridge, UK: Univ. Pr. (1997) 438 p Rather formal and advanced treatment, for the mathematically interested R. Slansky, “Group Theory For Unified Model Building,” Phys. Rept. 79, 1 • (1981). Invaluable reference: Contains a large appendix with loads of tables of representations and branching rules. M. Nakahara, “Geometry, topology and physics,” Bristol, UK: Institute of • Physics Publishing (2003) 596 p. (Graduate student series in physics) Very useful book, covers in particular the differential geometry aspects of Lie groups. Generally recommended for particle theorists. 7 Chapter 2 Groups: General Properties 2.1 Motivation: Why Group Theory? Why are there lectures called “Group Theory for Physicists”? In the end, this is a math- ematical subject, so why don’t students interested in the topic attend a mathematics lecture? After all, there are very few lectures like “Number Theory for Physicists”. This is captured in a statement made by James Jeans in 1910 while discussing a syllabus1: “We may as well cut out the group theory. That is a subject that will never be of any use in physics.” Only few decades later, however, Heisenberg said2 “We will have to abandon the philosophy of Democritus and the concept of elementary particles. We should accept instead the concept of elementary symmetries.” This explains why group theory is im- portant in almost any area