Rutwig campoamor stursberg, michel rausch de traubenberg group theory in physics a practitioners guide world scientific publishing co (2018)

Rather naturally, clas-sification of simple Lie algebras and their representations naturally occupy a central position, as well as representations of the Poincar´e group, withinduced rep

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Names: Rausch de Traubenberg, Michel, author | Campoamor-Stursberg, R., author.

Title: Group theory in physics : a practitioner’s guide / Michel Rausch de Traubenberg

(CNRS, France), Rutwig Campoamor-Stursberg (Universidad Complutense de Madrid, Spain) Description: New Jersey : World Scientific, 2018 | Includes bibliographical references and index Identifiers: LCCN 2018038941| ISBN 9789813273603 (hardcover : alk paper) |

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to be of direct importance for phenomena, but also more and more stractly, as is the case today in elementary particle physics Group theory

ab-is not only a tool to simplify computations, it provides elegant methods aswell as for classifications and for the determination of universal laws Evenmore important, it is a way to think and imagine the universe from theinfinitely small to the infinitely large, in other words to conceptualise theuniverse This aspect is corroborated by the successes obtained in particlephysics these last forty years, and illustrated by the theoretical attempts

in progress We note at this point the joint efforts of physicists and ematicians in the recent developments concerning quantum groups, supergroups, infinite-dimensional groups, etc Finally, let me add that the idea

math-of the unification math-of fundamental forces that the physicists math-of the infinitelysmall are looking for, could not be considered without the precious tool of

vii

group theory In this context, symmetry is acquiring a philosophical statusand one might imagine it playing a role in other domains of science, such

as the sciences of life, where some attempts have already been proposed.Therefore, it is highly desirable to offer to students, and also more advancedresearchers in theoretical physics, pedagogical guide books, such as the onepresented in the following pages Of course, good books on group theory forphysicists already exist But the subject is very broad, and only one man-ual cannot cover all the different aspects of this discipline The scientificexperience of the authors and their individual tastes lead them to selectwhat seems to them indispensable aspects of the subject One peculiarity

we could say power of group theory is that it stands at the crossroads ofalgebra, analysis and geometry This triple aspect is clearly considered inthis book It is nice noting that the main topics are developed in detail,with explicit computations which are easy to follow Rather naturally, clas-sification of simple Lie algebras and their representations naturally occupy

a central position, as well as representations of the Poincar´e group, withinduced representations `a la Wigner These subjects are preceded by gen-eral mathematical notions on algebra and differential geometry, and by apedagogical treatment of the most used Lie groups Discrete groups arenot neglected, two sections being devoted to this topic which finds appli-cation in modern neutrino mass models I appreciated that many otherimportant subjects are addressed, such as Jordan algebras and application

to gauge theories among others But it is not my purpose to exhaust thecontents of this series of lessons which are here delivered In this respect, Iconsider that the title of this book A Practitioner’s Guide to Group Theory

in Physics corresponds perfectly to what beginners as well as experiencedtheorists can expect I do hope that this study will convinced the readerthat Group Theory is the right way to describe, using the words of theFrench poet Ch Baudelaire, the order and beauty of our universe

Paul SorbaLAPTH, CNRS, FranceEmeritus Director of Research at CNRS

Why a new book of group theory, there being many excellent reviews on thesubject? Quite a natural question that arises and deserves an explanation.Both the authors have been involved for many years in research activitiesbelonging to the application of Group Theory in Physics, as well as havinggiven several lectures on the subject at various places, in particular at theDoctoral Schools of Madrid and Strasbourg

This book takes its origin from these lectures and from discussions withstudents and colleagues It turns out that, albeit various current standardtechniques used in group theoretical methods in physics being profuselycovered in the literature, there is no unified approach that is really of useand implementation for either the beginner and non-expert In this con-text, some of the special topics covered in this book result directly fromquestions from students and from the efforts of the authors to present anovel and more comprehensive explanation to aspects either usually notwell understood or to enlighten and motivate the use of a specific technique

in a general physical context In this sense, this book is by no means aformal mathematical introduction to Group Theory (for which there areexcellent monographs), a fact that justifies that many of the properties andtheorems will not be proved formally The motivation is to go beyond a

“Physical” description of Group Theory, and to emphasize on subtle andarduous points, where the difficulties will be clarified explicitly through ex-amples or mathematical recreations With this unusual approach in mind,this book is devoted to the study of symmetry groups in Physics from apractical perspective, that is, emphasising the explicit methods and algo-rithms useful for the practitioner, profusely illustrated by examples given

in general with many details

Even though this book has been written mainly with PhD students inmind (or even skilled Master Students for some parts of it), it is also ad-dressed to physicists interested in the practical application of symmetries

in physics, or to physicists that would like to have a better understanding

ix

of Group Theory Due to the professional background of the authors, most

of the applications studied are basically related to High Energy Physics

or spacetime symmetries, but the book is written in such a way that theaudience goes much beyond the High Energy Physics community All al-gorithms or explicit examples given to illustrate some delicate concepts arecertainly helpful for a better understanding of Group Theory, but moregenerally, they intend to illustrate why a certain tool the nature of whichmay appear as artificial turns out to be an effective procedure to extractphysically relevant information More formal readers can also find this book

of some interest, as more advanced topics that are commonly only found

in the technical literature are also covered, sometimes with some details

Of course, the reader not interested in these aspects can safely skip thecorresponding part These more difficult notions are explicitly indicatedthroughout

Finally, this book is meant as a self-contained introduction to grouptheory with applications in physics All concepts are gradually introducedand illustrated, through many examples, as already stated Few notionsare needed to read this book At the mathematical level some knowledge

on linear algebras (as e.g the notion of vector space) is needed However,the various algebraic structures relevant are smoothly introduced and illus-trated through examples For a better understanding of most of the physicalapplications, basic Quantum Mechanics (especially description of quantumstates) is supposed to be known In particular, the “bra” and “ket” nota-tion of Dirac is intensively used throughout the mathematical description

of what is called representations For the very last Chapters, basic cial Relativity is needed, and the Minkowski spacetime or the Lorentz andPoincar´e transformations are required for a better comprehension Further-more, the last Chapter is maybe the most knowledge demanding, as it isdevoted to symmetries in particle physics In order to benefit from thisChapter, the salient features of algebraic aspects of Quantum Field The-ory, needed for a comprehensive reading, are succinctly given We finallymention that some applications given in the very last Chapters are moreadvanced and technical, and can be studied in a second reading or as asupplement to more standard treatises on Group Theory

Spe-Madrid and Strasbourg

July, 2018

Rutwig Campoamor-StursbergMichel Rausch de Traubenberg

This book would certainly never have been written, had we not benefitedfrom the experience of many colleagues, who are too numerous to mention.The credit for the motivation of this book goes completely to students thatwith their curiosity and their thirst for knowledge constitute the guidingspirit for this book.

xi

2.1 Symmetry 9

2.1.1 Discrete symmetries 10

2.1.2 Continuous symmetries 10

2.2 Algebraic structures 11

2.2.1 Group 11

2.2.2 Basic structures 13

2.2.2.1 Rings 13

2.2.2.2 Field 14

2.2.2.3 Vector space 14

2.2.2.4 Algebra 16

2.3 Basic properties of tensors 21

2.4 Hilbert space 26

2.4.1 Finite-dimensional Hilbert spaces 27

2.4.2 Infinite-dimensional Hilbert spaces 28

2.5 Symmetries in Hilbert space 30

xiii

2.5.1 The Wigner theorem 30

2.5.2 Continuous transformations 32

2.6 Some matrix Lie groups and Lie algebras 37

2.7 Topological spaces 53

2.7.1 Topological spaces− definition 53

2.7.2 Continuous maps 55

2.7.3 Induced topology 56

2.7.4 Quotient topology and product spaces 57

2.7.5 Product spaces 59

2.7.6 Compacity Hausdorff and connected topological spaces 59

2.7.6.1 Hausdorff topological spaces 61

2.7.6.2 Connected topological spaces 62

2.8 Differentiable manifolds 64

2.8.1 Tangent spaces and vector fields 67

2.8.2 Differential forms 74

2.8.2.1 The exterior derivative 75

2.8.2.2 The Lie derivative 76

2.8.3 Lie groups Definition 78

2.8.4 Invariant forms Maurer-Cartan equations 84

2.9 Some definitions 86

2.9.1 Complexification and real forms 87

2.9.1.1 Complexification 87

2.9.1.2 Real forms 88

2.9.2 Linear representations 90

2.9.2.1 Reducible and irreducible representations 92 2.9.2.2 Complex conjugate and dual representations 92

2.9.2.3 Some important representations 93

2.9.2.4 Notations in Mathematics and in Physics 95 2.10 Differential and oscillator realisations of Lie algebras 95

2.10.1 Harmonic and fermionic oscillators 95

2.10.2 Differential operators 99

2.10.3 Oscillators and differential realisation of Lie algebras 101

2.11 Rough classification of Lie algebras 102

2.11.1 Elementary properties of Lie algebras 102

2.11.2 Solvable and nilpotent Lie algebras 104

2.11.3 Direct and semidirect sums of Lie algebras 107

2.11.4 Semisimple Lie algebras The Killing form 108

2.12 Enveloping algebras 111

2.12.1 Invariants of Lie algebras− Casimir operators 114

2.12.2 Rational invariants of Lie algebras 121

3 Finite groups: Basic structure theory 125 3.1 General properties of finite groups 125

3.1.1 Subgroups, factor groups 128

3.1.2 Homomorphims of groups 132

3.1.2.1 Direct and semidirect products 133

3.1.3 Conjugacy classes in groups 136

3.1.3.1 Class multiplication 137

3.1.4 Presentations of groups 139

3.1.4.1 Some important classes of groups 140

3.2 Permutations group 141

3.2.1 Characterisation of permutations 142

3.2.2 Conjugacy classes 147

3.2.3 The Cayley theorem 150

3.2.4 Action of groups on sets 151

3.3 Symmetry groups 154

3.3.1 The symmetry group of regular polygons 155

3.3.2 Symmetry groups of regular polyhedra 158

3.3.2.1 Symmetries of the tetrahedron 160

3.3.2.2 Symmetries of the hexahedron 162

3.3.2.3 Symmetries of the dodecahedron 164

3.4 Finite rotation groups 166

3.5 General symmetry groups 170

4 Finite groups: Linear representations 173 4.1 Linear representations 173

4.1.1 Some basic definitions and properties 174

4.1.2 Characters 178

4.1.3 Reducible and irreducible representations 180

4.1.4 Schur lemma 181

4.1.5 Irreducibility criteria for linear representations of finite groups 182

4.1.6 Complex real and pseudo-real matrix groups 184

4.1.7 Some properties of representations 185

4.1.8 General properties of character tables 190

4.1.9 The class coefficients 192

4.2 Construction of the character table 193

4.2.1 Diagonalisation algorithm 193

4.2.2 Algorithm implementation 195

4.3 Tensor product of representations 200

4.4 Representations of the symmetric group 202

4.4.1 Cycle classes and Young diagrams 206

4.4.2 Product of representations 213

5 Three-dimensional Lie groups 217 5.1 The group SU (2) and its Lie algebra su(2) 217

5.1.1 Defining representation 217

5.1.2 Representations 218

5.1.3 Some explicit realisations 222

5.1.4 The Lie group SU (2) 225

5.1.4.1 Fundamental representation 226

5.1.4.2 Matrix elements 226

5.2 The Lie group SO(3) 227

5.2.1 Properties of SO(3) 228

5.2.2 The universal covering group of SO(3) 230

5.3 The group USP(2) 231

5.4 Three-dimensional non-compact real Lie algebras and groups 231

5.4.1 Properties and definitions 231

5.4.2 Representations 235

5.4.3 Oscillators realisation of semi-infinite representations of SL(2, R) 238

5.4.4 Some explicit realisations 239

5.5 The complex Lie group SL(2, C) 246

6 The Lie group SU (3) 249 6.1 The su(3) Lie algebra 249

6.1.1 Definition 249

6.1.2 Casimir operators of su(3) 251

6.1.3 The Cartan-Weyl basis 253

6.1.4 Simple roots 256

6.1.5 The Chevalley basis 258

6.2 Some elements of representations 259

6.2.1 Differential realisation on C3 260

6.2.2 Harmonic functions on the five-sphere 265

7 Simple Lie algebras 269 7.1 Some preliminaries 269

7.1.1 Basic properties of linear operators 269

7.1.2 Semisimple and nilpotent elements 271

7.2 Some properties of simple complex Lie algebras 272

7.2.1 The Cartan subalgebra and the roots 272

7.2.2 Block structure of the Killing form 276

7.2.3 Commutation relations in the Cartan-Weyl basis 277 7.2.4 Fundamental properties of the roots 281

7.2.5 The Chevalley-Serre basis and the Cartan matrix 283 7.2.6 Dynkin diagrams− Classification 286

7.2.7 Classification of simple real Lie algebras 292

7.2.7.1 The compact Lie algebras 292

7.2.7.2 The split Lie algebras 293

7.2.7.3 General real Lie algebras 294

7.3 Reconstruction of the algebra 296

7.4 Subalgebras of simple Lie algebras 303

7.5 System of roots and Cartan matrices 306

7.6 The Weyl group 313

8 Representations of simple Lie algebras 319 8.1 Weights associated to a representation 319

8.1.1 The weight lattice and the fundamental weights 320 8.1.2 Highest weight representations 321

8.1.3 The multiplicity of the weight space and the Freudenthal formula 326

8.1.4 Characters and dimension of the representation space 335

8.1.5 Precise realisation of representations 341

8.2 Tensor product of representations: A first look 344 8.3 Complex conjugate, real and pseudo-real representations 349 8.4 Enveloping algebra and representations− Verma modules 350

9.1 The unitary algebra su(n) 362

9.1.1 Roots of su(n) 362

9.1.2 Young tableaux and representations of su(n) 363

9.1.3 Tensor product of representations 370

9.1.4 Differential realisation of su(n) 371

9.2 The orthogonal algebras so(2n) and so(2n + 1) 374

9.2.1 Roots of the orthogonal algebras 374

9.2.1.1 Roots of so(2n) 374

9.2.1.2 Roots of so(2n + 1) 376

9.2.2 Young tableaux and representations of O(p, q) and SO0(p, q) 377

9.2.2.1 Representations of O(p, q) 378

9.2.2.2 Representation of SO0(p, q) 383

9.2.2.3 Anti-symmetric tensors or k-forms 384

9.2.3 Spinor representations 385

9.2.3.1 The universal covering group of O(p, q) 385 9.2.3.2 Spinors 388

9.2.3.3 Real, pseudo-real and complex represen-tations of the Lie algebra so(1, d− 1) 395

9.2.3.4 Properties of (anti-)symmetry of the Γ-matrices 396

9.2.3.5 Product of spinors 398

9.2.3.6 Highest weights of the spinor representation(s) 400

9.2.4 Differential realisation of orthogonal algebras 403

9.2.4.1 Realisation of so(2n) 403

9.2.4.2 Realisation of so(2n + 1) 406

9.2.4.3 Note on the spinor representations 408

9.3 The symplectic algebra usp(2n) 408

9.3.1 Roots of usp(2n) 408

9.3.2 Young tableaux and representations of usp(2n) 409

9.3.3 Differential realisation of usp(2n) 412

9.3.4 Reality property of the representation of usp(2n) 416 9.3.5 The metaplectic representation of sp(2n, R) 416

9.4 Unitary representations of classical Lie algebras and differential realisations 419

9.5 Young tableaux via differential realisations 421

10.1 Some summary and reminder 426

10.2 Tensor product of representations 426

10.2.1 Practical methods for “partially” reducing the product of representations 430

10.2.1.1 The next-to-highest weight 430

10.2.1.2 The Dynkin method of reduction by parts 433 10.2.2 Conjugacy classes 434

10.2.3 Clebsch-Gordan coefficients 437

10.3 Subalgebras 442

10.3.1 Regular embeddings 444

10.3.2 Singular embeddings 454

11 Exceptional Lie algebras 461 11.1 Matrix Lie groups revisited 461

11.2 Division algebras and triality 462

11.2.1 Normed division algebras 462

11.2.2 Triality 464

11.2.3 Spinors and triality 465

11.2.4 Triality and Hurwitz algebras 469

11.2.5 The automorphism group of the division algebra and G2 471

11.3 The exceptional Jordan algebra and F4 475

11.3.1 Jordan algebras 475

11.3.2 The automorphism group of the exceptional Jordan algebra and F4 477

11.4 The magic square 479

11.5 Exceptional Lie algebras and spinors 485

12 Applications to the construction of orthonormal bases of states 489 12.1 Missing labels 490

12.1.1 The missing label problem 490

12.1.2 Missing label operators 491

12.1.3 Casimir operators 491

12.1.4 Labelling unambiguously irreducible representa-tions of semisimple algebras 492

12.1.5 Labelling states in the reduction process g⊂ g0 498

12.1.6 Special labelling operators: Decomposed Casimir operators 500

12.2 Berezin brackets of labelling operators 503

12.2.1 Properties of commutators of subgroup scalars 505

12.3 Algorithm for the determination of orthonormal bases of states 508

12.3.1 The algorithm 508

12.3.2 Orthonormal bases of eigenstates 510

12.4 Examples 514

12.4.1 The Wigner supermultiplet model 514

12.4.2 The chain so(7)⊃ su(2)3 517

12.4.3 The nuclear surfon model 522

13 Spacetime symmetries and their representations 533 13.1 Spacetime symmetries 534

13.1.1 Static symmetries 534

13.1.1.1 Representations of the rotation group revisited 534

13.1.1.2 The Euclidean group 539

13.1.2 Spacetime symmetries 540

13.1.2.1 The Minkowski spacetime, the Poincar´e group and conformal transformations 540

13.1.2.2 De Sitter and anti-de Sitter spaces 546

13.2 Representations of the symmetry group of spacetime 550

13.2.1 The Wigner method of induced representations 550

13.2.1.1 The little group or the little algebra 551

13.2.1.2 The method of induced representations 555 13.2.2 Unitary representation of the Euclidean group 558

13.2.3 Unitary representation of the Poincar´e group 558

13.2.4 Unitary representations of AdSd 563

13.2.5 Unitary representations of dSd or of the Lorentz group SO(1, d) 569

13.3 Relativistic wave equations 573

13.3.1 Relativistic wave equations and induced representations 574

13.3.2 An illustrative example: The Dirac equation 576

13.3.3 Infinite-dimensional representations 580

13.3.4 Majorana like equation for anyons 586

14.1 Algebras associated to the principle of equivalence 59114.2 Contractions of kinematical algebras 59714.2.1 In¨on¨u-Wigner contractions 59714.2.2 Kinematical algebras 60014.3 Kinematical groups 60614.3.1 Group multiplication 60714.3.2 Spacetime associated to kinematical groups 61014.4 Central extensions 61114.4.1 Lie algebra cohomology 61214.4.2 Central extensions of Lie algebras 61514.4.3 Central extensions of kinematical algebras 61614.5 Projective representations − Application to the Galileangroup 61714.5.1 Projective representations 61714.5.2 Relationship between projective representations

and central extensions 61914.5.3 Application to the Schr¨odinger equation 619

15.1 Symmetries in field theory 62515.1.1 Some basic elements of Quantum Field Theory 62615.1.2 Symmetries in Quantum Field Theory− The

Nœther Theorem 62715.1.3 Spin-statistics Theorem and Nœther Theorem 63015.1.4 Possible symmetries 63715.2 Spacetime symmetries 64115.2.1 Some reminder on the representations of the

Poincar´e group 64115.2.1.1 Massive particles 64215.2.1.2 Massless particles 64215.2.2 Possible particles 64315.2.2.1 Massless particles 64315.2.2.2 Massive particles 64915.2.3 Relativistic wave equations 65015.2.3.1 The scalar field 65015.2.3.2 The spinor field 65115.2.3.3 The vector field 65515.3 Internal symmetries 65815.3.1 Gauge interactions 659

15.3.1.1 Abelian transformations 66015.3.1.2 Non-Abelian transformations 66115.3.2 Possible spectra 66315.4 Fundamental interactions as a gauge theory 66615.4.1 The Standard Model of particles physics 66615.4.2 Possible gauge groups 67315.4.3 Unification of interactions: Grand-Unified Theories 67515.4.3.1 Unification with SU (5) 67815.4.3.2 Unification with SO(10) 68315.4.3.3 Unification with E6 69515.4.3.4 Can gauge groups be constrained? 70315.4.3.5 Discrete symmetries 704

List of Figures

2.1 The Fano plane 192.2 Cosets spaces 582.3 Differential map 662.4 Tangent vector 682.5 Left-invariant vector field 812.6 Complexification and real forms 903.1 Young diagram n = N1+· · · Nk 1493.2 The conjugacy classes of Σ3 with the number of elements perclass 1503.3 The conjugacy classes of Σ4 with the number of elements perclass 1503.4 Regular Polygon 1563.5 Symmetry axes of the triangle and the square 1573.6 Projection of the Dodecahedron from a face 1644.1 Standard Young tableaux of Σ4 2114.2 Hook structure of the diagram (12, 2) 2124.3 Hook structure of the diagram (22) 2134.4 Hook structure of the diagram (1, 3) 2135.1 Representations of SU (2) 2275.2 The parameter space of SO(3) 2295.3 The parameter space of SO(1, 2) 2356.1 Roots of su(3) 2566.2 Dynkin diagram of su(3) 257

xxiii

6.3 The fundamental representation of su(3) 2626.4 The anti-fundamental representation of su(3) 2626.5 The six-dimensional representation of su(3) 2636.6 The ten-dimensional representation of su(3) 2646.7 The ten-dimensional representation of su(3), cont’d 2656.8 The eight-dimensional representation of su(3) 2667.1 Roots of G2 3008.1 The representationD0,1 of G2 3238.2 The representationsD1,0,0,D0,1,0,D0,0,1, of su(4) 3258.3 The representationD2,0,0of su(4) 3268.4 The representationsD1,0,0,0 andD0,1,0,0 of su(5) 3278.5 The representationsD1,0,0,0,0andD0,0,0,1,0 of so(10) 3288.6 The representationD2,2 of su(3) 3318.7 The Verma moduleVn 3528.8 The Verma moduleV1,0 3548.9 The finite-dimensional representation Dk associated to Vs,k −case 1 3578.10 The representationD+k associated to Vs,k− case 2 3578.11 The representationD−k associated to Vs,k− case 3 3578.12 The unbounded representation associated to Vs,k − case 4 3579.1 Representation Λ2 of sp(6)− traceless two-forms 4149.2 Representation Λ4 of sp(6)− dual-traceless four-forms 41510.1 Roots connected to the weights µ(n)and µ(1) for Dn 43110.2 Roots connected to the weights µ(n)and µ(n −1) for Dn 43210.3 Dynkin diagram of D4 and triality 43610.4 Embedding of sl(3, C) into G2 44710.5 Weights of the D(1,0,0,0,0,0) representation of E6 45210.6 Dynkin diagrams of E6 and D5 45310.7 Embedding of su(3)× su(3) × su(3) into E6 − Dynkin diagrams

of ˆE6 and su(3)× su(3) × su(3) 45410.8 Extended weights of the D(1,0,0,0,0,0) representation of E6 45513.1 The cone of equation: (x0)2

− (x1)2

− (x2)2= 0 54513.2 The cone in the projective space RP2,d 54513.3 Two sheeted hyperboloid of equation: (x0)2

−(x1)2

−(x2)2= R2 547

13.4 One sheeted hyperboloid of equation: (x0)2

− (x1)2

− (x2)2 =

−R2 54713.5 The (n− 1)-sphere 55314.1 Contraction of SO(3) to E2 59914.2 Contraction of Kinematic algebras 60515.1 Emission of a spin s-particle 64515.2 Scattering of a massless spin-s particle by a graviton 64715.3 Chiral anomaly − Triangle diagram 66615.4 The Higgs potential 67215.5 Quantum corrections of the propagator and vertex 67615.6 Evolution of the coupling constant with the energy 677

List of Tables

2.1 Quaternions multiplication table 172.2 Octonions multiplication table 192.3 Casimir operators of complex simple Lie algebras 1193.1 Σ3Multiplication Table 1473.2 Conjugacy classes of Σ3 1483.3 Conjugacy classes of Σ4 1483.4 Platonic solids 1593.5 Symmetry objects and elements 1703.6 Finite rotation groups and axes of symmetry 1704.1 Elements of the matrix group G 1964.2 Multiplication table of G 1974.3 Class multiplication table of G 1994.4 Character table of G 2004.5 Character table of C2 2024.6 Character table of O× C2 2024.7 Number of partitions r(n) for low values of n 2034.8 Character table of Σ6 2056.1 Structure constants of su(3) 2526.2 The non-vanishing constants dabc of su(3) 2527.1 Relative length and angle between two roots 2827.2 Dynkin diagrams of simple complex Lie algebras 2917.3 Simple real Lie algebras and their maximal compact Lie subal-gebras 296

xxvii

7.4 Extended Dynkin diagrams 3057.5 Simple roots and positive roots of simple Lie algebras 3147.6 Weyl groups of simple complex Lie algebras 3179.1 Type of spinors 3979.2 Symmetry of Γ(k)C−1 39810.1 Correspondence roots/weights for the Classical Lie algebras 42710.2 Correspondence roots/weights for the Exceptional Lie algebras 42810.3 Highest roots expressed in terms of simple roots and fundamen-tal weights 42911.1 Identification for the quaternions multiplication 47011.2 Identification for the octonions multiplication 47111.3 Identification for the octonions multiplication cont’d 47211.4 Automorphism of the trialities and of the division algebras 47511.5 The n = 2, 3 magic squares 48211.6 The n = 2, 3 split magic squares 48311.7 The n = 2, 3 double split magic squares 48512.1 Polynomial realisation of the twenty-seven-dimensional repre-sentationD2,2 of su(3) 49512.2 Eigenvectors of J, h1, h2for the representationD2,2 of su(3) theindices indicate the eigenvalues 49612.3 so(3)3⊂ so(7)− commutators [Ta,b,c, Td,e,f] 51912.4 so(5) brackets in an so(3) ={L0, L±1} basis 52312.5 Indecomposable sixth order polynomial solutions to system(12.33) 52712.6 Indecomposable sixth order polynomial cont’d 52813.1 Unitary representations of the Euclidean group E(n) 55813.2 Unitary representations of the Poincar´e group ISO(1, d− 1) 56214.1 Kinematical algebras 59815.1 Degrees of freedom of the Dirac and the Weyl fields 65415.2 Degrees of freedom of Maxwell and Proca fields 65815.3 Particles content of the Standard Model 67015.4 Decomposition of the spinor representation in terms of su(3)c×su(2)L× u(1)Y × u(1)Q 687

Outline of the book

The notion of symmetry and its mathematical counterpart, given by GroupTheory, are central for the description of the laws of Physics and for a betterunderstanding of physical systems For instance, the principles of Galilean,Special and even General Relativity are based upon symmetries and theircorresponding groups These principles naturally lead to the structure ofspacetime, together with the corresponding laws for the description of phys-ical phenomena With Quantum Mechanics and Quantum Field Theory,where a physical state belongs to an infinite-dimensional Hilbert space, theconcept of symmetry is enlarged and takes another dimension In par-ticular, spectra in Quantum Mechanics are precisely classified in terms ofGroup Theory (or more precisely its representations) As an illustration,

we can now understand the Periodic Table of the Elements (Mendeleev)from rotational and conformal invariance Similarly, the Standard Model

of Particles Physics has its origin in the study of spacetime and internal (orgauge) symmetries which classify elementary particles (as the Mendeleevtable classifies the chemical elements) However, in the particle case, sym-metries go beyond a mere classification scheme, as they dictate the be-haviour of elementary particles, i.e., the way how they interact with eachother and the underlying section rules A group is an abstract notion and

in Physics, symmetries are not directly associated to groups, but moreprecisely to some representation A representation is a prescription thatdictates how the group elements act on the system For systems given by

by vector or tensor quantities, a representation of a symmetry group will

be associated to matrices acting on the system components, whereas in thecase of systems described in terms of functions, the relevant representationswill be codified by differential operators acting on functional spaces As anexample, everyone is familiar, at least unwittingly, with a realisation of the

rotation group in the daily life This is the three-dimensional representation

of the rotational group For completeness we mention that there are alsotwo important topics related to symmetries that will not be covered in fulldetail: symmetry breaking (spontaneous or explicit) and anomalies Theformer received a major attention with the discovery of the Higgs boson in2012

In order to facilitate the reading of this book, we now address succinctlyits contents chapter by chapter This can provide some help to the readerwho wants a precise answer to a specific question, giving some indications

on the way this book can be read Indeed, a comprehensive reading doesnot imply necessarily a linear reading of this text

Chapter 2 reviews the algebraic, topological and geometric notions derlying the theory of Lie groups and Lie algebras In particular, as apreamble to the study of Lie groups and Lie algebras, all matrix Lie groupsand matrix Lie algebras are studied in detail It should be observed thatmany of the concepts introduced in this chapter can be applied withoutusing a heavy formalism, merely knowing how to use the main basic prop-erties This chapter is relatively substantial as it is concerned with all thenotions we use implicitly or not We also mention that infinite-dimensionalLie algebras or Lie superalgebras are not studied in this book A linearreading of this chapter is not mandatory for understanding the book, andthe reader can simply refer to Chapter 2 (or to some part of it) when thecorresponding notion is used thereafter

un-This book is mainly concerned with continuous Lie groups, but Chapters

3 and 4 deal with finite groups and their linear representations Specialattention is given to the finite subgroups of the three-dimensional rotationalgroup and the relationship with regular solids or point groups (important

in condensed matter or crystallography, for instance) is exhibited We alsoillustrate several algorithms to compute the characters and study with somedetail the permutation group, that turns out to be important in the sequel.Chapters 5 and 6 serve as a link with the main purpose of this text, thestudy of Lie groups and their associated Lie algebras Indeed, many notions

of simple Lie groups and algebras are explicitly illustrated by consideringthe two simplest Lie algebras (and their associated Lie groups), namely thethree-dimensional Lie algebras and the Lie algebra related to the Hermitean

3× 3 matrices

Chapters 7 and 8 are important chapters, as they are concerned withthe study of simple real and simple complex Lie algebras together withtheir representations Chapter 7 deals with the structural theory of simple

complex Lie algebras In particular, the classification of simple (complexand real) Lie algebras is given Chapter 8 is devoted to the study of repre-sentations of Lie algebras (all unitary representations in the case of compactLie groups) Important notions as roots, weights (and their correspond-ing highest weight representations) and Dynkin diagrams are covered Westrongly emphasise the notion and distinction of real forms of complexLie algebras and the notion of complexification of real Lie algebras, no-tions that are often confused in the physical literature on the subject, andthat are fundamental tools to properly discuss the important real, complexand pseudo-real representations The main notions are highlighted throughmany examples and algorithms.

Chapter 9 is also important, as it is related to the so-called classicalalgebras This part is also highlighted with many examples, with a specificattention to oscillator and differential realisations of Lie algebras, of specialrelevance within the physical applications Various constructive algorithmsare given in detail, providing practical approaches that are generally notconsidered in the literature The spinor representation of the rotation group

is studied with a lot of detail The technique of Young tableaux is explicitlyused to have a complementary description of irreducible representations ofclassical Lie algebras in terms of tensors This method is alternative to thehighest weight representation of Chapter 8

Chapter 10 is dedicated to two important topics of representation ory: the tensor product of representations and the decomposition of anirreducible representation into irreducible summands when considered as

the-a representthe-ation of the-a subthe-algebrthe-a A lot of illustrthe-ative exthe-amples the-are given,

as well as further details and practical rules For instance, the polynomialrepresentations of all unitary representations of compact classical Lie alge-bras exhibited in Chapter 9 turns out to be of special importance for thecomputation of the Clebsch-Gordan coefficients, relevant for the coupling ofdifferent representations, in an algorithmic manner, as illustrated throughmany examples

Chapter 11 is somewhat out of the philosophy of this book and addressesthe relationship of exceptional Lie algebras with triality, octonions and theexceptional Jordan algebra This chapter can be omitted in a first reading,but may give a deeper insight to the applications presented in Chapter 15

A second part of the book is concerned with specific applications tophysics, mainly in the context of High Energy Physics, spacetime sym-metries and Grand-Unified Theories In these chapters we stress on anapproach not usually followed, especially in Chapter 15

In Chapter 12 we present a procedure based on analytical methods andthe Berezin bracket for the construction of orthonormal bases of statesfor a given chain of Lie group-subgroup This is an useful technique todescribe representations of a Lie algebra using a given subalgebra, thatusually corresponds to some internal symmetry of a system An algorithmicprocedure for the stepwise computation of bases of eigenstates is given,illustrated by various representative examples This chapter can be skipped

in a first reading

Chapter 13 is devoted to spacetime symmetries in arbitrary dimensions

In particular, the principles of symmetry, when applied to spacetime, arevery restrictive and the possible structures of spacetime are very limited(Minkowski or (anti-) de-Sitter) In this case the study of representations

is more involved as unitary representations are infinite-dimensional Themethod of induced representations of Wigner, applied to the Poincar´e group

in relation with relativistic wave equations, is investigated with some tail Moreover, all unitary representations of the Poincar´e algebra are ob-tained Some unitary representations of the (anti-) de-Sitter algebra arealso given, in particular in relation with higher spin algebras Since infinite-dimensional representations are more delicate, examples are developed with

de-a lot of detde-ails In pde-articulde-ar, two infinite-dimensionde-al relde-ativistic wde-aveequations (including the famous Majorana equation) are analysed with aspecial attention We also study the conformal group in relation with thecompactification of the Minkowski spacetime

Chapter 14 introduces the so-called kinematical algebras, used as a tification and a motivation to illustrate some important topics not studied

jus-so far The notion of contraction is introduced A contraction is simply

a mathematical way to obtain an algebra as a limit of another (e.g relativistic limit) It is shown that all kinematical algebras are related bycontractions Moreover, the spacetimes associated to kinematical algebrasare introduced by means of the homogeneous space technique Further,central extensions and projective representations (in relation with coho-mology) are investigated As an application, it is shown that the wavefunction for the Schr¨odinger equation can be interpreted within a projec-tive representation of the Galilean group

non-Chapter 15 is concerned with the application of symmetries in particlephysics We emphasise the fact that Quantum and Relativistic Physicsconsiderably restrict either the type of mathematical structures or the type

of particles (spin, mass, representation of Lie group and even the Lie groupitself) one is allowed to consistently consider This enables us to review the

possible symmetries in particles physics The Standard Model of particlesphysics is introduced, and in particular it is shown how all fundamentalinteractions can be unified in a simple Lie group in the context of Grand-Unified-Theories Three are the proposed unification models that will becovered, namely those basing on the Lie groups SU (5), SO(10) and E6.Explicit computations for these cases are provided.

This introductory chapter review the basic concepts that will be relevantthroughout this book Forthcoming chapters will often be based upon thenotions considered here, as well as on the many examples given

As the main purpose is to introduce notations and nomenclature, it ispossible to skip various parts of this chapter, and refer to them only whenneeded in the corresponding chapter In order to facilitate the reading ofthis book we indicate the correspondence between the introduced notionsand the later chapters

Section 2.1 is an introductory section that establishes a closed dence between symmetries and groups, meaning that the latter are centralfor the description of symmetries Next in Sec 2.2, the basic algebraicstructures are defined and examples are given In particular, importantexamples of algebras are introduced, such as the quaternions, octonionsand Clifford algebras The quaternions are useful in Chapter 7, where sim-ple real Lie algebras are studied The octonions will be strongly related

correspon-to exceptional Lie groups, and in particular will be useful in Chapter 11.Finally, Clifford algebras will become essential for the definition of spinors

in Chapter 9 Sections 2.3, 2.7 and 2.8 are perhaps a little bit beyond thephilosophy of this book, and can be safely skipped by the reader which isnot interested in formal aspects In Sec 2.3 tensors are introduced in anintrinsic way (this is useful for instance for the definition of the univer-sal enveloping algebra) Section 2.7 is devoted to a brief introduction totopology Topology is useful to define continuous functions on more generalspaces than the real lines or Euclidean spaces Finally Sec 2.8 is devoted

to a brief introduction of the concept of manifolds Basically manifolds arespaces that behave locally as Euclidean spaces (like the sphere S3 for in-stance) For instance the precise definition of Lie groups involves manifolds

Even if these sections are not necessary to the comprehension of this book,the notions presented throughout these three sections could be useful forthe reader interested by a more formal aspects Note also that these not soeasy notions are implicitly considered along this book.

Since one of the main aim of this book is to apply group theory in tum Mechanics, Sec 2.4 is devoted to Hilbert spaces (finite- and infinite-dimensional) Section 2.5 is a very important section, as it establishes thatunder some natural assumptions, Lie groups and Lie algebras turn out to becentral for the description of symmetries in Quantum Mechanics Modifyingslightly the assumptions that lead to Lie algebras, one is able to introduceLie superalgebras or coloured Lie (super)algebras The former are central

Quan-in the description of supersymmetry and supergravity These two examplesare given for completeness, in the exposition, but these structures will not

be studied in this book Matrix groups are very important, and Sec 2.6looks over the different matrix groups Matrix group over real, complexand quaternions will be introduced At the first glance, this section could

be seen as merely technical, in particular with the matrix groups associated

to the quaternions The reader can refer to quaternionic matrix Lie groupswhen studying the classical real Lie groups in Sec 7, and in particular theyare relevant for the description of the groups U SP (2n), SO∗(2n) or SU∗(n).Section 2.9 is related to the important topic of complexification and of theso-called real forms of Lie algebras Basically, complexification/real formsestablish how one can associate a real Lie algebra to a complex Lie algebraand conversely In this section some basic definitions are also given, as forinstance linear representations of Lie groups and Lie algebras Some em-phasis on the adjoint representation, which turns out to be very important,

is given The adjoint representation will be central for the classification ofsimple complex Lie algebras in Chapter 7 In physics, it is very convenient

to introduce explicit oscillator and differential realisations of Lie algebras.Section 2.10 is devoted to this subject The fermionic analogue of the har-monic oscillator will, be considered together with its associated Grassmannalgebra Oscillators and differential realisations will be used in Chapters 5,

6, 9, 13 and 15 In particular differential realisations are used in Chapter 9

to obtain easily explicit unitary representations of the classical (compact)Lie algebras Since Lie groups and Lie algebras are central in this book,the question of their classification is an important problem However, onlythe classification of simple real and simple complex Lie algebras is given inChapter 7

In Sec 2.11, a rough classification of Lie algebras is given, and various

important classes of Lie algebras such as nilpotent, solvable or ple algebras are introduced Semisimple Lie algebras will be studied, andclassified in great detail in Chapter 7 In this chapter we also introducealgebras with a semidirect sum structure that are essential for the descrip-tion of the spacetime symmetries, such as the Poincar´e algebra introduced

semisim-in Chapter 13 We end this section with the Levi decomposition of Liealgebras Casimir operators are operators that commute with all elements

of the Lie algebra and will constitute relevant tools in studying tations As Casimir operators are polynomials in the generators of the Liealgebra, strictly speaking there are not living in the Lie algebra itself, but

represen-in the so-called universal enveloprepresen-ing algebra Universal enveloprepresen-ing algebrasare defined in Sec 2.12 Universal enveloping algebras, and Verma mod-ules are considered in Chapter 8 (Sec 8.4) As these topics are relativelytechnical, both Secs 2.12 and 8.4 can be skipped since they will not beconsidered in the remaining part of this book

A symmetry is, essentially, a transformation that leaves invariant either asystem (classical or quantum) or a geometrical object In this section, itwill in turn be established that to the set of symmetries we can associate

a mathematical structure called group Symmetries and their associatedgroup structures play a crucial rˆole in physics: description of classical sys-tems, classification of spectra in Quantum Mechanics, classification of ele-mentary particles and description of their (gauge) interactions, etc Thus,for instance, the Galilean group is, in particular, extremely important inthe study of non-relativistic physics However, since there are a priori manypossible types of groups, there are a priori many possible types of symme-tries A first restriction on the possible types of groups allowed in physicscomes from Quantum Mechanics Indeed, the Wigner theorem stronglyconstraints the possible groups which act on Hilbert spaces In fact im-posing very reasonable hypothesis, only one type of group, the Lie groups(and their corresponding Lie algebras), is relevant for the descriptions ofsymmetries for quantum systems as shown in Sec 2.5 It is interesting topoint out the nice feature of this theorem: it selects naturally, among thelarge variety of groups, the type of groups which could be used in QuantumMechanics In other words, symmetries for a Quantum system arise in amore “restricted way” since only few types of groups are possible In factthis book will be mainly concerned with the study of Lie groups and Lie

algebras The concept of symmetry takes another dimension when the ciple of relativity is taken under consideration In particular, the Noethertheorem and the spin-statistics theorem, when applied in Quantum FieldTheory, restrict considerably the possible symmetries in spacetime that weare able to consider, as shown in Chapter 15.

A discrete symmetry is a set of transformations (finite or countable) thatleave a system invariant For instance, the following examples are standardrepresentatives of symmetries:

(1) A translation by a vector ~v in R3:

T : ~x→ ~x + ~v ,(2) The parity (or reflection) transformation in R3generated by

Id : ~x→ ~x ,

P : ~x→ −~x ,

is a symmetry of many classical and quantum systems

(3) Symmetries of a regular polygon (denoted Dn) with n−vertices aregenerated by the set of transformations

Rp=

cos2pπn − sin2pπn

sin2pπn cos2pπn

, p = 0,· · · , n − 1 ,

pRp = Id (with At the transpose of thematrix A) and det Rp = 1, although the matrices Sp = P Rp fulfil

(1) The rotations in R3 are completely determined by a direction (twoangles in spherical coordinates) and the angle of rotation around thisdirection or axis This corresponds to a three-parameter set of trans-formations Let us denote R(~α) a generic rotation of R3.

(2) The Galilei transformations of non-relativistic physics



~xt

(3) The Poincar´e transformations of special relativity

xµ→ x0µ= Λµνxν+ aµ ,with Λµ

νthe Lorentz transformations (the properties of Λ will be given

in Sec 2.6 and more into the details in Chapter 13) and aµ the time translations form a ten-parameter set of transformations

space-(4) The symmetry of the wave function in quantum mechanics

ψ(~x, t)→ eiαψ(~x, t) ,forms a one-parameter set of transformations

Besides its obvious geometrical or physical meaning, symmetries can bedescribed mathematically in terms of well defined algebraic structures, themost important of which will be briefly reviewed in this section