Table of Contents

Preface to the Second Edition v

Preface to the First Edition vii

Introduction 1

1 Lorentz and Poincaré Group, SL(2, $$$), Dirac and Majorana Spinors 7

1.1 The Lorentz Group 7

1.2 The Poincaré Group 21

1.3 SL(2, $$$), Dotted and Undotted Indices 32

1.3.1 Spinor Algebra 32

1.3.2 Calculations with Spinors 48

1.3.3 Connection between SL(2, $$$) and L^↑₊ 53

1.3.4 The Fierz-Reordering Formula 64

1.3.5 Further Calculations with Spinors 65

1.3.6 Higher Order Weyl Spinors and their Representations 78

1.4 Dirac and Majorana Spinors 83

1.4.1 The Weyl Basis or Chiral Representations 85

1.4.2 The Canonical Basis or Dirac Representation 92

1.4.3 The Majorana Representation 95

1.4.4 Charge Conjugation, Dirac and Weyl Representations 101

1.4.5 Majorana Spinors 109

1.4.6 Calculations with Dirac Spinors 112

1.4.7 Calculations with Majorana Spinors 114

2 No-Go Theorems and Graded Lie Algebras 117

2.1 The Theorems of Coleman-Mandula and Haag, &Lstoke;opuszański, Sohnius 117

2.1.1 The Theorem of Coleman-Mandula 117

2.1.2 The Theorem of Haag, &Lstoke;opuszański and Sohnius 119

2.2 Graded Lie Algebras 121

2.2.1 Lie Algebras 121

2.2.2 Graded Algebras 123

2.2.3 Graded Lie Algebras 123

2.3 The Graded Lie Algebra of SU (2, $$$) 125

2.4 $$$₂ Graded Lie Algebras 131

2.5 Graded Matrices 139

3 The Supersymmetric Extension of the Poincaré Algebra 147

3.1 Four-Component Dirac Formulation 147

3.2 Two-Component Weyl Formulation 161

4 Representations of the Super-Poincaré Algebra 163

4.1 Casimir Operators 163

4.2 Classification of Irreducible Representations 172

4.2.1 N = 1 Supersymmetry 172

4.2.2 N >1 Supersymmetry 180

5 The Wess-Zumino Model 185

5.1 The Lagrangian and the Equations of Motion 185

5.2 Symmetries 187

5.3 Plane Wave Expansions 194

5.4 Projection Operators 205

5.5 Anticommutation Relations 209

5.6 The Energy-Momentum Operator of the Wess-Zumino Model 222

5.6.1 The Hamilton Operator 225

5.6.2 The Three-Momentum P_i 232

5.7 Infinitesimal Supersymmetry Transformations 235

6 Superspace Formalism and Superfields 243

6.1 Superspace 243

6.2 Grassmann Differentiation 246

6.3 Supersymmetry Transformations in the Weyl Formalism 250

6.3.1 Finite Supersymmetry Transformations 250

6.3.2 Infinitesimal Supersymmetry Transformations and Differential Operator Representations of the Generators 256

6.4 Consistency with the Majorana Formalism 262

6.5 Covariant Derivatives 264

6.6 Projection Operators 271

6.7 Constraints 278

6.8 Transformations of Component Fields 279

7 Constrained Superfields and Supermultiplets 287

7.1 Chiral Superfields 287

7.2 Vector Superfields, Generalized Gauge Transformations 300

7.3 The Supersymmetric Field Strength 306

8 Supersymmetric Lagrangians 317

8.1 Grassmann Integration 317

8.2 Lagrangians and Actions 323

8.2.1 Construction of Lagrangians from Scalar Superfields 323

8.2.2 Construction of Lagrangians from Vector Superfields 332

8.2.3 Remarks 340

9 Spontaneous Breaking of Supersymmetry 343

9.1 The Superpotential 343

9.2 Projection Technique 347

9.3 Spontaneous Symmetry Breaking 364

9.3.1 The Goldstone Theorem 367

9.3.2 Remarks on the Wess-Zumino Model 371

9.4 The O'Raifeartaigh Model 372

9.4.1 Spontaneous Breaking of Supersymmetry 372

9.4.2 The Mass Spectrum of the O'Raifeartaigh Model 377

10 Supersymmetric Gauge Theories 387

10.1 Minimal Coupling 387

10.2 Super Quantum Electrodynamics 395

10.3 The Fayet-Iliopoulos Model 400

10.4 Supersymmetric Non-Abelian Gauge Theory 412

Bibliography 423

Index 433