Group Theory Lecture Notes

1 Group Theory Lecture NotesHugh Osbornlatest update: March 22, 2023 Based on part III lectures Symmetries and Groups, Michaelmas Term 2008, revised andextended at various times subsequentlyBooksBooks developing Group Theory by physicists from the perspective of particle physics areH. F. Jones,Groups, Representations and Physics,2nd ed., IOP Publishing (1998).A fairly easy going Georgi,Lie Algebras in Particle Physics,Perseus Books (1999).Describes the basics of Lie algebras for classical Fuchs and C. Schweigert,Symmetries, Lie Algebras and Representations,2nd ed., CUP(2003).This is more comprehensive and more mathematically sophisticated, and does not describephysical applications in any Ma, Group Theory for Physicists,World Scientific (2007).Quite Ramond, Group Theory , A Physicists Survey,CUP (2010).A relatively gentle physics motivated treatment, and includes discussion of finite Zee, Group Theory in a Nutshell for Physicists.

2 Princeton University Press (2016).Quite lengthy, comprehensive with many physics applications, some nice anecdotal Cvitanovi c, Group Theory : Birdtracks, Lie s and Exceptional Lie Groups, Princeton Uni-versity Press (2009), , but full of material not found elswhere. Great for doing following books contain useful discussions, in chapter 2 of Weinberg there is a proofof Wigner s theorem and a discussion of the Poincar e Group and its role in field Theory ,and chapter 1 of Buchbinder and Kuzenko has an extensive treatment of spinors in Weinberg,The Quantum Theory of Fields,(vol. 1), CUP (2005).J. Buchbinder and S. Kuzenko,Ideas and Methods of Supersymmetry and Supergravity, ora Walk Through Superspace,2nd ed., Institute of Physics Publishing (1998).They are many mathematical books with titles containing references to Groups, Represen-tations, Lie Groups and Lie Algebras. The motivations and language is often very different,and hard to follow, for those with a traditional theoretical physics background.

3 Particularbooks which may be useful Hall,Lie Groups, Lie Algebras, and Representations,Springer (2004), for an earlierversion see arXiv:math- focuses on matrix accessible than mostW. Fulton and J. Harris,Representation Theory ,Springer (1991).Historically the following book, first published in German in 1931, was influential in showingthe relevance of Group Theory to atomic physics in the early days of quantum mechanics. Itintroduces anti-unitary representations. For an English Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spec-tra,Academic Press (1959).iPrologueThe following excerpts are from Strange Beauty, by G. Johnson, a biography of MurrayGell-Mann1, the foremost particle physicist of the 1950 s and 1960 s who proposedSU(3)as a symmetry Group for hadrons and later quarks as the fundamental building blocks. Itreflects a time when most theoretical particle physicists were unfamiliar with groups beyondthe rotation Group , and perhaps also a propensity for some to invent mathematics as theywent it happened,SU(2)could also be used to describe the Isopspin symmetry- the groupof abstract ways in which a nucleon can be rotated in isospin space to get a neutron ora proton, or a pion to get negative, positive or neutral versions.

4 These rotations were whatGell-Mann had been calling currents. The groups were what he had been calling couldn t believe how much time he had wasted. He had been struggling in the darkwhile all these algebras, these groups- these possible classification schemes- had been studiedand tabulated decades ago. All he would have to do was to go to the library and look Paris, as Murray struggled to expand the algebra of the isospin doublet,SU(2), toembrace all hadrons, he had been playing with a hierarchy of more complex groups, withfour, five, six, seven rotations. He now realized that they had been simply combinationsof the simpler groupsU(1)andSU(2). No wonder they hadn t led to any interesting newrevelations. What he needed was a new, higher symmetry with novel properties. The nextone in Cartan s catalogue wasSU(3), a Group that can have eight of the cumbersome way he had been doing the calculations in Paris, Murrayhad lost the will to try an algebra so complex and inclusive.

5 He had gone all the way up toseven and Gell-Mann, 1929-2019, American, Nobel prize Notational Conventions01 Introduction, Definitions and Basic Definitions and Terminology .. and Cosets .. Subgroups, Simple Groups and Composition Series .. Product of Groups .. Cyclic, Dihedral and Permutation Groups .. Group .. Group .. Group .. Group .. Orbit Stabiliser Theorem .. Further Definitions .. and Semi-Direct Product .. Products and Central Products .. Classes .. , Normaliser, Centraliser and Commutator Subgroups .. Coset .. s Lemma .. Quaternion Groups .. , Octahedral and Icosahedral Groups .. Matrix Groups .. Group .. and Non Compact ..252 Schur s Lemmas .. Induced Representations .. Unitary Representations .. Dimensional Unitary Representations .. and Pseudo-Real Representations .. Orthogonality Relations .. Characters .. Constraints on Dimensions of Irreducible Representations.

6 Tensor Products .. and Antisymmetric Products .. Character Tables .. Molien Series .. Molien Series .. of Molien Series .. Series and Wreath Products .. Symmetries in Quantum Mechanics, Projective and Anti-Unitary Represen-tations .. Representations .. Representations ..513 Rotations and Angular Momentum,SO(3)andSU(2) Three Dimensional Rotations .. Isomorphism ofSO(3)andSU(2)~Z2.. Compact Isomorphisms .. Infinitesimal Rotations and Generators .. Representations of Angular Momentum Commutation Relations .. TheSj m basis .. of Time Reversal .. Matrices .. overSO(3)and orthogonality relations .. forSU(2).. Tensor Products and Angular Momentum Addition .. Examples of the calculation of Clebsch-Gordan coefficients .. of Singlet States .. of Highest Weight States .. Cases of Clebsch-Gordan Coefficients .. 3jSymbol .. 6jSymbol .. Relations .. Tensor Products and Characters.

7 (3)Tensors .. Spherical Harmonics .. Molien Series forSU(2).. Irreducible Tensor Operators .. Spinors .. Spinor Representation for Angular Momentum .. Calculation of Action of Derivatives .. Isospin .. 1064 Relativistic Symmetries, Lorentz and Poincar e Lorentz Group .. of Linearity .. of Lorentz Group .. Infinitesimal Lorentz Transformations and Commutation Relations .. Lorentz Group and Spinors .. (3,1) Sl(2,C)~Z2.. , Dotted and Undotted Indices .. Representations .. Poincar e Group .. Irreducible Representations of the Poincar e Group .. Representations .. States .. Representations .. Particle States and Angular Momentum Decomposition .. Treatment .. Casimir Operators .. Quantum Fields .. 1315 Lie Groups and Lie Fields, Differential Forms and Lie Brackets .. Lie Algebras .. Lie Algebra Definitions.

8 Matrix Lie Algebras and Matrix Lie Groups .. (2)Example .. Triangular Matrices .. and Lie Algebras .. Relation of Lie Algebras to Lie Groups .. Subgroups .. Cambell Hausdorff Formula .. Simply Connected Lie Groups and Covering Groups .. Group .. Representations .. Lie Algebra and Projective Representations .. Group .. Integration over a Lie Group , Compactness .. (2)Example .. CompactSl(2,R)Example .. Adjoint Representation and its Corollaries .. Form .. for Non Degenerate Killing Form .. of Semi-simple Lie Algebras .. Operators and Central Extensions .. 1606 Lie Algebras for Matrix Unitary Groups .. Symplectic Groups .. Orthogonal and Spin Groups .. Groups and Gamma Matrices .. and Traces of Gamma Matrices .. of Representations of the Clifford Algebra .. Matrix for Gamma Matrices .. Cases.

9 Identities .. 1727SU(3)and its Recap ofsu(2).. Asu(3)Lie algebra basis and its automorphisms .. Highest Weight Representations forsu(3).. of the Weight Diagram .. (3)Characters .. operator .. (3)Representations .. (3)Tensor Representations .. (3)Lie algebra again .. (3)and Physics .. (3)FSymmetry Breaking .. (3)and Colour .. Tensor Products forSU(3).. Discussion of Tensor Products .. 2018 Gauge Groups and Gauge Abelian Gauge Theories .. Non Abelian Gauge Theories .. Theory .. Gauge Invariants and Wilson Loops .. Loops .. 2159 Integrations over Spaces Modulo Group Integrals over Spheres .. Integrals over Symmetric and Hermitian Matrices .. Integrals over Compact Matrix Groups .. Integration over a Gauge Field and Gauge Fixing .. 239vi0 Notational ConventionsHopefully we use standard conventions.

10 For anyMij,ibelonging to an ordered set withmelements, not necessarily 1,..m, and similarlyjbelonging to an ordered set withnelements,M=[Mij]is the correspondingm nmatrix, with of courseilabelling the rows,jthe the unit matrix, on occasion1ndenotes then nunit any ( ),T[ ]denote the symmetric, antisymmetricparts, obtained by summing over all permutations of the order of the indices ,with an additional 1 for odd permutations in the antisymmetric case, and then dividingbyn!. Thus forn=2,T(ij)=12(Tij+Tji), T[ij]=12(Tij Tji).( )If some indices are to be omitted from symmetrisation or antisymmetrisation they aresurrounded , thusT[iSkSj]=12(Tikj Tjki).We use , , , as space-time indices,i,j,kare spatial indices while , , are a set of elementsxthen{x P}denotes the subset satisfying a vector spaceVmay be defined in terms of linear combinations of basis vectors{vr},r=1,..,dimVso that an arbitrary vector can be expressed as rarvr.