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

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Library of Congress Cataloging-in-Publication Data
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) |
ISBN 9813273607 (hardcover : alk. paper)
Subjects: LCSH: Group theory. | Mathematical physics.
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` la m´emoire de la belle Sezen et du redoutable Moritz
A

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Foreword

It might be interesting to note that when Galileo Galilei asserted that the
book of nature is written in the language of mathematics the notion of
a group was not yet discovered. Indeed, the scientific community had to
wait for more than two centuries and, among others, the seminal work of
E. Galois, to read for the first time the word group in mathematics. A
few decades later, still in the nineteenth century, S. Lie expanded more
widely this path by creating the continuous symmetries. It was time for
physicists to step in: the twentieth century had arrived and the era of the
new scientific spirit, as expressed by the French philosopher G. Bachelard,
with Einsteinian relativity and quantum mechanics, had begun. In some
sense, we could say that the mathematicians had, once more, prepared the
terrain. Of course, all physics is not based on group theory, but one cannot
deny that it has become a corner stone in several domains of this science.
The notion of symmetry, and also of breaking of symmetry, is considered
to be of direct importance for phenomena, but also more and more abstractly, as is the case today in elementary particle physics. Group theory
is not only a tool to simplify computations, it provides elegant methods as
well as for classifications and for the determination of universal laws. Even
more important, it is a way to think and imagine the universe from the
infinitely small to the infinitely large, in other words to conceptualise the
universe. This aspect is corroborated by the successes obtained in particle
physics these last forty years, and illustrated by the theoretical attempts
in progress. We note at this point the joint efforts of physicists and mathematicians in the recent developments concerning quantum groups, super
groups, infinite-dimensional groups, etc. Finally, let me add that the idea
of the unification of fundamental forces that the physicists of the infinitely
small are looking for, could not be considered without the precious tool of
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Foreword

group theory. In this context, symmetry is acquiring a philosophical status
and 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 advanced
researchers in theoretical physics, pedagogical guide books, such as the one
presented in the following pages. Of course, good books on group theory for
physicists already exist. But the subject is very broad, and only one manual cannot cover all the different aspects of this discipline. The scientific
experience of the authors and their individual tastes lead them to select
what seems to them indispensable aspects of the subject. One peculiarity
we could say power of group theory is that it stands at the crossroads of
algebra, analysis and geometry. This triple aspect is clearly considered in
this book. It is nice noting that the main topics are developed in detail,
with explicit computations which are easy to follow. Rather naturally, classification of simple Lie algebras and their representations naturally occupy
a central position, as well as representations of the Poincar´e group, with
induced representations `
a la Wigner. These subjects are preceded by general mathematical notions on algebra and differential geometry, and by a
pedagogical treatment of the most used Lie groups. Discrete groups are
not neglected, two sections being devoted to this topic which finds application in modern neutrino mass models. I appreciated that many other
important subjects are addressed, such as Jordan algebras and application
to gauge theories among others. But it is not my purpose to exhaust the
contents of this series of lessons which are here delivered. In this respect, I
consider that the title of this book A Practitioner’s Guide to Group Theory

in Physics corresponds perfectly to what beginners as well as experienced
theorists can expect. I do hope that this study will convinced the reader
that Group Theory is the right way to describe, using the words of the
French poet Ch. Baudelaire, the order and beauty of our universe.
Paul Sorba
LAPTH, CNRS, France
Emeritus Director of Research at CNRS

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Preface
Why a new book of group theory, there being many excellent reviews on the
subject? Quite a natural question that arises and deserves an explanation.
Both the authors have been involved for many years in research activities
belonging to the application of Group Theory in Physics, as well as having
given several lectures on the subject at various places, in particular at the
Doctoral Schools of Madrid and Strasbourg.
This book takes its origin from these lectures and from discussions with
students and colleagues. It turns out that, albeit various current standard
techniques used in group theoretical methods in physics being profusely
covered in the literature, there is no unified approach that is really of use
and implementation for either the beginner and non-expert. In this context, some of the special topics covered in this book result directly from
questions from students and from the efforts of the authors to present a
novel and more comprehensive explanation to aspects either usually not
well 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 a
formal mathematical introduction to Group Theory (for which there are
excellent monographs), a fact that justifies that many of the properties and
theorems will not be proved formally. The motivation is to go beyond a

“Physical” description of Group Theory, and to emphasize on subtle and
arduous points, where the difficulties will be clarified explicitly through examples or mathematical recreations. With this unusual approach in mind,
this book is devoted to the study of symmetry groups in Physics from a
practical perspective, that is, emphasising the explicit methods and algorithms 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 in
mind (or even skilled Master Students for some parts of it), it is also addressed to physicists interested in the practical application of symmetries
in physics, or to physicists that would like to have a better understanding
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Preface

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 the
audience goes much beyond the High Energy Physics community. All algorithms or explicit examples given to illustrate some delicate concepts are
certainly helpful for a better understanding of Group Theory, but more
generally, they intend to illustrate why a certain tool the nature of which
may appear as artificial turns out to be an effective procedure to extract
physically 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 the
corresponding part. These more difficult notions are explicitly indicated
throughout.

Finally, this book is meant as a self-contained introduction to group
theory with applications in physics. All concepts are gradually introduced
and illustrated, through many examples, as already stated. Few notions
are 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 illustrated through examples. For a better understanding of most of the physical
applications, basic Quantum Mechanics (especially description of quantum
states) is supposed to be known. In particular, the “bra” and “ket” notation of Dirac is intensively used throughout the mathematical description
of what is called representations. For the very last Chapters, basic Special Relativity is needed, and the Minkowski spacetime or the Lorentz and
Poincar´e transformations are required for a better comprehension. Furthermore, the last Chapter is maybe the most knowledge demanding, as it is
devoted to symmetries in particle physics. In order to benefit from this
Chapter, the salient features of algebraic aspects of Quantum Field Theory, needed for a comprehensive reading, are succinctly given. We finally
mention that some applications given in the very last Chapters are more
advanced and technical, and can be studied in a second reading or as a
supplement to more standard treatises on Group Theory.
Madrid and Strasbourg
July, 2018

Rutwig Campoamor-Stursberg
Michel Rausch de Traubenberg

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Acknowledgments

We would like to express our gratitude to Marcus J. Slupinski, a long time
collaborator, for his helpful criticisms and suggestions. Many thanks also
to Luis J. Boya and Richard Kerner for their encouragements and their
wise advice. Alex Boeglin is kindly acknowledged for his careful reading of

the manuscript that helped to clarify some points in the presentation. We
are specially indebted to Paul Sorba for having accepted with enthusiasm
the laborious task of writing a foreword.
This book would certainly never have been written, had we not benefited
from the experience of many colleagues, who are too numerous to mention.
The credit for the motivation of this book goes completely to students that
with their curiosity and their thirst for knowledge constitute the guiding
spirit for this book.

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Contents

Foreword

vii

Preface

ix

Acknowledgments

xi

List of Figures

xxiii

List of Tables

xxvii

1.

Outline of the book

1

2.

Generalities

7

2.1

2.2

2.3
2.4

2.5

Symmetry . . . . . . . . . . . . . . . . . .
2.1.1 Discrete symmetries . . . . . . . .
2.1.2 Continuous symmetries . . . . . .
Algebraic structures . . . . . . . . . . . .
2.2.1 Group . . . . . . . . . . . . . . . .
2.2.2 Basic structures . . . . . . . . . .
2.2.2.1 Rings . . . . . . . . . . .
2.2.2.2 Field . . . . . . . . . . .
2.2.2.3 Vector space . . . . . . .
2.2.2.4 Algebra . . . . . . . . . .
Basic properties of tensors . . . . . . . . .
Hilbert space . . . . . . . . . . . . . . . .
2.4.1 Finite-dimensional Hilbert spaces
2.4.2 Infinite-dimensional Hilbert spaces
Symmetries in Hilbert space . . . . . . . .
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xiv

Contents

2.5.1 The Wigner theorem . . . . . . . . . . . . . . . .
2.5.2 Continuous transformations . . . . . . . . . . . . .
2.6 Some matrix Lie groups and Lie algebras . . . . . . . . . .
2.7 Topological spaces . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Topological spaces − definition . . . . . . . . . . .
2.7.2 Continuous maps . . . . . . . . . . . . . . . . . .
2.7.3 Induced topology . . . . . . . . . . . . . . . . . .
2.7.4 Quotient topology and product spaces . . . . . . .
2.7.5 Product spaces . . . . . . . . . . . . . . . . . . . .
2.7.6 Compacity. Hausdorff and connected topological
spaces . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.6.1 Hausdorff topological spaces . . . . . . .
2.7.6.2 Connected topological spaces . . . . . . .
2.8 Differentiable manifolds . . . . . . . . . . . . . . . . . . .
2.8.1 Tangent spaces and vector fields . . . . . . . . . .
2.8.2 Differential forms . . . . . . . . . . . . . . . . . .
2.8.2.1 The exterior derivative . . . . . . . . . .
2.8.2.2 The Lie derivative . . . . . . . . . . . . .
2.8.3 Lie groups. Definition . . . . . . . . . . . . . . . .
2.8.4 Invariant forms. Maurer-Cartan equations . . . .
2.9 Some definitions . . . . . . . . . . . . . . . . . . . . . . .
2.9.1 Complexification and real forms . . . . . . . . . .
2.9.1.1 Complexification . . . . . . . . . . . . . .
2.9.1.2 Real forms . . . . . . . . . . . . . . . . .

2.9.2 Linear representations . . . . . . . . . . . . . . . .
2.9.2.1 Reducible and irreducible representations
2.9.2.2 Complex conjugate and dual
representations . . . . . . . . . . . . . . .
2.9.2.3 Some important representations . . . . .
2.9.2.4 Notations in Mathematics and in Physics
2.10 Differential and oscillator realisations of Lie algebras . . .
2.10.1 Harmonic and fermionic oscillators . . . . . . . . .
2.10.2 Differential operators . . . . . . . . . . . . . . . .
2.10.3 Oscillators and differential realisation of Lie
algebras . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Rough classification of Lie algebras . . . . . . . . . . . . .
2.11.1 Elementary properties of Lie algebras . . . . . . .
2.11.2 Solvable and nilpotent Lie algebras . . . . . . . .
2.11.3 Direct and semidirect sums of Lie algebras . . . .

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Contents

xv

2.11.4 Semisimple Lie algebras. The Killing form . .
2.12 Enveloping algebras . . . . . . . . . . . . . . . . . . . .
2.12.1 Invariants of Lie algebras − Casimir operators
2.12.2 Rational invariants of Lie algebras . . . . . . .
3.

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Finite groups: Basic structure theory
3.1

3.2

3.3

3.4
3.5
4.

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General properties of finite groups . . . . . . . . .
3.1.1 Subgroups, factor groups . . . . . . . . . .
3.1.2 Homomorphims of groups . . . . . . . . . .

3.1.2.1 Direct and semidirect products . .
3.1.3 Conjugacy classes in groups . . . . . . . . .
3.1.3.1 Class multiplication . . . . . . . .
3.1.4 Presentations of groups . . . . . . . . . . .
3.1.4.1 Some important classes of groups
Permutations group . . . . . . . . . . . . . . . . . .
3.2.1 Characterisation of permutations . . . . . .
3.2.2 Conjugacy classes . . . . . . . . . . . . . .
3.2.3 The Cayley theorem . . . . . . . . . . . . .
3.2.4 Action of groups on sets . . . . . . . . . . .
Symmetry groups . . . . . . . . . . . . . . . . . . .
3.3.1 The symmetry group of regular polygons .
3.3.2 Symmetry groups of regular polyhedra . .
3.3.2.1 Symmetries of the tetrahedron . .
3.3.2.2 Symmetries of the hexahedron . .
3.3.2.3 Symmetries of the dodecahedron .
Finite rotation groups . . . . . . . . . . . . . . . .
General symmetry groups . . . . . . . . . . . . . .

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111
114
121
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166
170

Finite groups: Linear representations

173

4.1

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180
181

Linear
4.1.1
4.1.2

4.1.3
4.1.4
4.1.5
4.1.6
4.1.7

representations . . . . . . . . . . . . . . . . . . . .
Some basic definitions and properties . . . . . . .
Characters . . . . . . . . . . . . . . . . . . . . . .
Reducible and irreducible representations . . . . .
Schur lemma . . . . . . . . . . . . . . . . . . . . .
Irreducibility criteria for linear representations of
finite groups . . . . . . . . . . . . . . . . . . . . .
Complex real and pseudo-real matrix groups . . .
Some properties of representations . . . . . . . . .

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Contents

4.2

4.3

4.4

5.

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Three-dimensional Lie groups
5.1

5.2

5.3
5.4

5.5
6.

4.1.8 General properties of character tables
4.1.9 The class coefficients . . . . . . . . .
Construction of the character table . . . . . .
4.2.1 Diagonalisation algorithm . . . . . . .
4.2.2 Algorithm implementation . . . . . .

Tensor product of representations . . . . . . .
Representations of the symmetric group . . .
4.4.1 Cycle classes and Young diagrams . .
4.4.2 Product of representations . . . . . .

217

The group SU (2) and its Lie algebra su(2) . . . . . . .
5.1.1 Defining representation . . . . . . . . . . . . .
5.1.2 Representations . . . . . . . . . . . . . . . . .
5.1.3 Some explicit realisations . . . . . . . . . . . .
5.1.4 The Lie group SU (2) . . . . . . . . . . . . . .
5.1.4.1 Fundamental representation . . . . .
5.1.4.2 Matrix elements . . . . . . . . . . . .
The Lie group SO(3) . . . . . . . . . . . . . . . . . . .
5.2.1 Properties of SO(3) . . . . . . . . . . . . . . .
5.2.2 The universal covering group of SO(3) . . . .
The group USP(2) . . . . . . . . . . . . . . . . . . . .
Three-dimensional non-compact real Lie algebras and
groups . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Properties and definitions . . . . . . . . . . . .
5.4.2 Representations . . . . . . . . . . . . . . . . .
5.4.3 Oscillators realisation of semi-infinite
representations of SL(2, R) . . . . . . . . . . .
5.4.4 Some explicit realisations . . . . . . . . . . . .
The complex Lie group SL(2, C) . . . . . . . . . . . .

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. . 246

The Lie group SU (3)
6.1

190
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200
202
206
213

249

The su(3) Lie algebra . . . . . . . .
6.1.1 Definition . . . . . . . . . .
6.1.2 Casimir operators of su(3)
6.1.3 The Cartan-Weyl basis . .
6.1.4 Simple roots . . . . . . . .

6.1.5 The Chevalley basis . . . .

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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

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7.2

7.3
7.4
7.5
7.6
8.

Representations of simple Lie algebras

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8.1

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8.2
8.3
8.4
9.

Some preliminaries . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Basic properties of linear operators . . . . . . . .
7.1.2 Semisimple and nilpotent elements . . . . . . . . .
Some properties of simple complex Lie algebras . . . . . .
7.2.1 The Cartan subalgebra and the roots . . . . . . .
7.2.2 Block structure of the Killing form . . . . . . . . .
7.2.3 Commutation relations in the Cartan-Weyl basis .
7.2.4 Fundamental properties of the roots . . . . . . . .
7.2.5 The Chevalley-Serre basis and the Cartan matrix
7.2.6 Dynkin diagrams − Classification . . . . . . . . .
7.2.7 Classification of simple real Lie algebras . . . . . .
7.2.7.1 The compact Lie algebras . . . . . . . . .
7.2.7.2 The split Lie algebras . . . . . . . . . . .
7.2.7.3 General real Lie algebras . . . . . . . . .
Reconstruction of the algebra . . . . . . . . . . . . . . . .
Subalgebras of simple Lie algebras . . . . . . . . . . . . .
System of roots and Cartan matrices . . . . . . . . . . . .

The Weyl group . . . . . . . . . . . . . . . . . . . . . . . .

Weights associated to a representation . . . . . . . . . . .
8.1.1 The weight lattice and the fundamental weights .
8.1.2 Highest weight representations . . . . . . . . . . .
8.1.3 The multiplicity of the weight space and the
Freudenthal formula . . . . . . . . . . . . . . . . .
8.1.4 Characters and dimension of the representation
space . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.5 Precise realisation of representations . . . . . . . .
Tensor product of representations: A first look . . . . . .
Complex conjugate, real and pseudo-real representations .
Enveloping algebra and representations − Verma modules

Classical Lie algebras

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341
344
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350
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Contents

9.1

9.2

9.3

9.4
9.5

The unitary algebra su(n) . . . . . . . . . . . . . . . . . .
9.1.1 Roots of su(n) . . . . . . . . . . . . . . . . . . . .
9.1.2 Young tableaux and representations of su(n) . . .
9.1.3 Tensor product of representations . . . . . . . . .
9.1.4 Differential realisation of su(n) . . . . . . . . . . .
The orthogonal algebras so(2n) and so(2n + 1) . . . . . .
9.2.1 Roots of the orthogonal algebras . . . . . . . . . .
9.2.1.1 Roots of so(2n) . . . . . . . . . . . . . .
9.2.1.2 Roots of so(2n + 1) . . . . . . . . . . . .
9.2.2 Young tableaux and representations of O(p, q) and
SO0 (p, q) . . . . . . . . . . . . . . . . . . . . . . .
9.2.2.1 Representations of O(p, q) . . . . . . . .
9.2.2.2 Representation of SO0 (p, q) . . . . . . .
9.2.2.3 Anti-symmetric tensors or k-forms . . . .
9.2.3 Spinor representations . . . . . . . . . . . . . . . .
9.2.3.1 The universal covering group of O(p, q) .
9.2.3.2 Spinors . . . . . . . . . . . . . . . . . . .
9.2.3.3 Real, pseudo-real and complex representations of the Lie algebra so(1, d − 1) . .
9.2.3.4 Properties of (anti-)symmetry of the
Γ-matrices . . . . . . . . . . . . . . . . .

9.2.3.5 Product of spinors . . . . . . . . . . . . .
9.2.3.6 Highest weights of the spinor
representation(s) . . . . . . . . . . . . . .
9.2.4 Differential realisation of orthogonal algebras . . .
9.2.4.1 Realisation of so(2n) . . . . . . . . . . .
9.2.4.2 Realisation of so(2n + 1) . . . . . . . . .
9.2.4.3 Note on the spinor representations . . . .
The symplectic algebra usp(2n) . . . . . . . . . . . . . . .
9.3.1 Roots of usp(2n) . . . . . . . . . . . . . . . . . . .
9.3.2 Young tableaux and representations of usp(2n) . .
9.3.3 Differential realisation of usp(2n) . . . . . . . . . .
9.3.4 Reality property of the representation of usp(2n) .
9.3.5 The metaplectic representation of sp(2n, R) . . . .
Unitary representations of classical Lie algebras and
differential realisations . . . . . . . . . . . . . . . . . . . .
Young tableaux via differential realisations . . . . . . . . .

10. Tensor products and reduction of representations

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10.1 Some summary and reminder . . . . . . . . . . . . . . . .
10.2 Tensor product of representations . . . . . . . . . . . . . .
10.2.1 Practical methods for “partially” reducing the
product of representations . . . . . . . . . . . . .
10.2.1.1 The next-to-highest weight . . . . . . . .
10.2.1.2 The Dynkin method of reduction by parts
10.2.2 Conjugacy classes . . . . . . . . . . . . . . . . . .
10.2.3 Clebsch-Gordan coefficients . . . . . . . . . . . . .
10.3 Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1 Regular embeddings . . . . . . . . . . . . . . . . .
10.3.2 Singular embeddings . . . . . . . . . . . . . . . . .
11. Exceptional Lie algebras
11.1 Matrix Lie groups revisited . . . . . . . . . . . . . . . . .
11.2 Division algebras and triality . . . . . . . . . . . . . . . .
11.2.1 Normed division algebras . . . . . . . . . . . . . .
11.2.2 Triality . . . . . . . . . . . . . . . . . . . . . . . .
11.2.3 Spinors and triality . . . . . . . . . . . . . . . . .
11.2.4 Triality and Hurwitz algebras . . . . . . . . . . . .
11.2.5 The automorphism group of the division algebra
and G2 . . . . . . . . . . . . . . . . . . . . . . . .
11.3 The exceptional Jordan algebra and F4 . . . . . . . . . . .
11.3.1 Jordan algebras . . . . . . . . . . . . . . . . . . .
11.3.2 The automorphism group of the exceptional
Jordan algebra and F4 . . . . . . . . . . . . . . .
11.4 The magic square . . . . . . . . . . . . . . . . . . . . . . .
11.5 Exceptional Lie algebras and spinors . . . . . . . . . . . .
12. Applications to the construction of orthonormal bases of states
12.1 Missing labels . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.1 The missing label problem . . . . . . . . . . . . .
12.1.2 Missing label operators . . . . . . . . . . . . . . .

12.1.3 Casimir operators . . . . . . . . . . . . . . . . . .
12.1.4 Labelling unambiguously irreducible representations of semisimple algebras . . . . . . . . . . . . .
12.1.5 Labelling states in the reduction process g ⊂ g . .
12.1.6 Special labelling operators: Decomposed Casimir
operators . . . . . . . . . . . . . . . . . . . . . . .

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479

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12.2 Berezin brackets of labelling operators . . . . . . . . . . .
12.2.1 Properties of commutators of subgroup scalars . .
12.3 Algorithm for the determination of orthonormal bases of
states . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3.1 The algorithm . . . . . . . . . . . . . . . . . . . .
12.3.2 Orthonormal bases of eigenstates . . . . . . . . . .
12.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.1 The Wigner supermultiplet model . . . . . . . . .
12.4.2 The chain so(7) ⊃ su(2)3 . . . . . . . . . . . . . .
12.4.3 The nuclear surfon model . . . . . . . . . . . . . .
13. Spacetime symmetries and their representations
13.1 Spacetime symmetries . . . . . . . . . . . . . . . . . . . .
13.1.1 Static symmetries . . . . . . . . . . . . . . . . . .
13.1.1.1 Representations of the rotation group
revisited . . . . . . . . . . . . . . . . . .

13.1.1.2 The Euclidean group . . . . . . . . . . .
13.1.2 Spacetime symmetries . . . . . . . . . . . . . . . .
13.1.2.1 The Minkowski spacetime, the Poincar´e
group and conformal transformations . .
13.1.2.2 De Sitter and anti-de Sitter spaces . . . .
13.2 Representations of the symmetry group of spacetime . . .
13.2.1 The Wigner method of induced representations . .
13.2.1.1 The little group or the little algebra . . .
13.2.1.2 The method of induced representations .
13.2.2 Unitary representation of the Euclidean group . .
13.2.3 Unitary representation of the Poincar´e group . . .
13.2.4 Unitary representations of AdSd . . . . . . . . . .
13.2.5 Unitary representations of dSd or of the Lorentz
group SO(1, d) . . . . . . . . . . . . . . . . . . . .
13.3 Relativistic wave equations . . . . . . . . . . . . . . . . .
13.3.1 Relativistic wave equations and induced
representations . . . . . . . . . . . . . . . . . . . .
13.3.2 An illustrative example: The Dirac equation . . .
13.3.3 Infinite-dimensional representations . . . . . . . .
13.3.4 Majorana like equation for anyons . . . . . . . . .
14. Kinematical algebras

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14.1 Algebras associated to the principle of equivalence . . . .
14.2 Contractions of kinematical algebras . . . . . . . . . . . .
14.2.1 Ină
onă
u-Wigner contractions . . . . . . . . . . . . .
14.2.2 Kinematical algebras . . . . . . . . . . . . . . . .
14.3 Kinematical groups . . . . . . . . . . . . . . . . . . . . . .
14.3.1 Group multiplication . . . . . . . . . . . . . . . .
14.3.2 Spacetime associated to kinematical groups . . . .
14.4 Central extensions . . . . . . . . . . . . . . . . . . . . . .
14.4.1 Lie algebra cohomology . . . . . . . . . . . . . . .
14.4.2 Central extensions of Lie algebras . . . . . . . . .
14.4.3 Central extensions of kinematical algebras . . . .
14.5 Projective representations − Application to the Galilean
group . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.5.1 Projective representations . . . . . . . . . . . . . .
14.5.2 Relationship between projective representations
and central extensions . . . . . . . . . . . . . . . .
14.5.3 Application to the Schrăodinger equation . . . . . .
15. Symmetries in particles physics
15.1 Symmetries in field theory . . . . . . . . . . . . . . . .
15.1.1 Some basic elements of Quantum Field Theory
15.1.2 Symmetries in Quantum Field Theory − The
Nœther Theorem . . . . . . . . . . . . . . . . .
15.1.3 Spin-statistics Theorem and Nœther Theorem
15.1.4 Possible symmetries . . . . . . . . . . . . . . .
15.2 Spacetime symmetries . . . . . . . . . . . . . . . . . .
15.2.1 Some reminder on the representations of the
Poincar´e group . . . . . . . . . . . . . . . . . .

15.2.1.1 Massive particles . . . . . . . . . . . .
15.2.1.2 Massless particles . . . . . . . . . . .
15.2.2 Possible particles . . . . . . . . . . . . . . . . .
15.2.2.1 Massless particles . . . . . . . . . . .
15.2.2.2 Massive particles . . . . . . . . . . . .
15.2.3 Relativistic wave equations . . . . . . . . . . .
15.2.3.1 The scalar field . . . . . . . . . . . .
15.2.3.2 The spinor field . . . . . . . . . . . .
15.2.3.3 The vector field . . . . . . . . . . . .
15.3 Internal symmetries . . . . . . . . . . . . . . . . . . . .
15.3.1 Gauge interactions . . . . . . . . . . . . . . . .

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15.3.1.1 Abelian transformations . . . . . . . . .
15.3.1.2 Non-Abelian transformations . . . . . . .
15.3.2 Possible spectra . . . . . . . . . . . . . . . . . . .
15.4 Fundamental interactions as a gauge theory . . . . . . . .
15.4.1 The Standard Model of particles physics . . . . .
15.4.2 Possible gauge groups . . . . . . . . . . . . . . . .
15.4.3 Unification of interactions: Grand-Unified Theories
15.4.3.1 Unification with SU (5) . . . . . . . . . .
15.4.3.2 Unification with SO(10) . . . . . . . . .
15.4.3.3 Unification with E6 . . . . . . . . . . . .
15.4.3.4 Can gauge groups be constrained? . . . .
15.4.3.5 Discrete symmetries . . . . . . . . . . . .

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661
663
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666
673
675
678
683
695
703
704

Bibliography

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Index

723

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List of Figures

2.1
2.2
2.3
2.4
2.5
2.6

The Fano plane. . . . . . . . . .
Cosets spaces. . . . . . . . . . . .
Differential map. . . . . . . . . .
Tangent vector. . . . . . . . . . .
Left-invariant vector field. . . . .
Complexification and real forms.

3.1
3.2

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elements per
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elements per
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3.4
3.5
3.6

Young diagram n = N1 + · · · Nk . . . . . . . . . .
The conjugacy classes of Σ3 with the number of
class. . . . . . . . . . . . . . . . . . . . . . . . . .

The conjugacy classes of Σ4 with the number of
class. . . . . . . . . . . . . . . . . . . . . . . . . .
Regular Polygon. . . . . . . . . . . . . . . . . . .
Symmetry axes of the triangle and the square. .
Projection of the Dodecahedron from a face. . .

4.1
4.2
4.3
4.4

Standard Young tableaux of Σ4 . . . .
Hook structure of the diagram (12 , 2).
Hook structure of the diagram (22 ). .
Hook structure of the diagram (1, 3). .

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5.1
5.2
5.3

Representations of SU (2). . . . . . . . . . . . . . . . . . . . . . 227
The parameter space of SO(3). . . . . . . . . . . . . . . . . . . 229
The parameter space of SO(1, 2). . . . . . . . . . . . . . . . . . 235

6.1

6.2

Roots of su(3). . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Dynkin diagram of su(3). . . . . . . . . . . . . . . . . . . . . . 257

3.3

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156
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xxiv

List of Figures

6.3
6.4
6.5
6.6
6.7
6.8

The
The
The
The
The
The

fundamental representation of su(3). . . . . . .
anti-fundamental representation of su(3). . . . .
six-dimensional representation of su(3). . . . . .
ten-dimensional representation of su(3). . . . . .
ten-dimensional representation of su(3), cont’d.
eight-dimensional representation of su(3). . . . .

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7.1

Roots of G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

The representation D0,1 of G2 . . . . . . . . . . . . . . . . .
The representations D1,0,0 , D0,1,0 , D0,0,1 , of su(4). . . . . . .
The representation D2,0,0 of su(4). . . . . . . . . . . . . . .
The representations D1,0,0,0 and D0,1,0,0 of su(5). . . . . . .
The representations D1,0,0,0,0 and D0,0,0,1,0 of so(10). . . . .
The representation D2,2 of su(3). . . . . . . . . . . . . . . .
The Verma module Vn . . . . . . . . . . . . . . . . . . . . .
The Verma module V1,0 . . . . . . . . . . . . . . . . . . . . .
The finite-dimensional representation Dk associated to Vs,k
case 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.10 The representation Dk+ associated to Vs,k − case 2. . . . . .
8.11 The representation Dk− associated to Vs,k − case 3. . . . . .
8.12 The unbounded representation associated to Vs,k − case 4.

8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9

9.1
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323
325
326
327
328
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354
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357
357
357

Representation Λ2 of sp(6) − traceless two-forms. . . . . . . . . 414
Representation Λ4 of sp(6) − dual-traceless four-forms. . . . . 415

Roots connected to the weights µ(n) and µ(1) for Dn . . . . . . .
Roots connected to the weights µ(n) and µ(n−1) for Dn . . . . .
Dynkin diagram of D4 and triality. . . . . . . . . . . . . . . . .
Embedding of sl(3, C) into G2 . . . . . . . . . . . . . . . . . . .
Weights of the D(1,0,0,0,0,0) representation of E6 . . . . . . . . .
Dynkin diagrams of E6 and D5 . . . . . . . . . . . . . . . . . . .
Embedding of su(3) × su(3) × su(3) into E6 − Dynkin diagrams
ˆ6 and su(3) × su(3) × su(3). . . . . . . . . . . . . . . . . .
of E
10.8 Extended weights of the D(1,0,0,0,0,0) representation of E6 . . . .

10.1
10.2
10.3
10.4
10.5
10.6
10.7

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455

13.1 The cone of equation: (x0 )2 − (x1 )2 − (x2 )2 = 0 . . . . . . . . . 545
13.2 The cone in the projective space RP2,d . . . . . . . . . . . . . . 545
13.3 Two sheeted hyperboloid of equation: (x0 )2 −(x1 )2 −(x2 )2 = R2 547

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