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Supergravity

Supergravity, together with string theory, is one of the most significant developments intheoretical physics. Although there are many books on string theory, this is the first-everauthoritative and systematic account of supergravity.

Written by two of the most respected workers in the field, it provides a solid introduc-tion to the fundamentals of supergravity. The book starts by reviewing aspects of relativisticfield theory in Minkowski spacetime. After introducing the relevant ingredients of differ-ential geometry and gravity, some basic supergravity theories (D

=4 and D

=11) and the

main gauge theory tools are explained. The second half of the book is more advanced: com-plex geometry and N= 1 andN= 2 supergravity theories are covered. Classical solutionsand a chapter on anti-de Sitter/conformal field theory (AdS/CFT) correspondence completethe text.

Numerous exercises and examples make it ideal for Ph.D. students, and with appli-cations to model building, cosmology, and solutions of supergravity theories, thistext is an invaluable resource for researchers. A website hosted by the authors, fea-turing solutions to some exercises and additional reading material, can be found atwww.cambridge.org/supergravity.

Daniel Z. Freedmanis Professor of Applied Mathematics and Physics at the MassachusettsInstitute of Technology. He has made many research contributions to supersymmetry andsupergravity: he was a co-discoverer of the first supergravity theory in 1976. This discoveryhas been recognized by the award of the Dirac Medal and Prize in 1993, and the DannieHeineman Prize of the American Physical Society in 2006.

Antoine Van Proeyenis Head of the Theoretical Physics Section at the KU Leuven, Belgium.Since 1979, he has been involved in the construction of various supergravity theories, theresulting special geometries, and their applications to phenomenology and cosmology.

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Our conventions

The metric is mostly plus, i.e. + . . . +. The curvature is

R= g (

+

)

= ea eb

ab ab+ ac c b ac c b

.

Ricci tensor and energymomentum tensors are defined by

R= R , R=g R,R 12 gR= 2T.

Covariant derivatives involving the spin connection are, for vectors and spinors,

D Va

= V

a

+

ab Vb, D

=

+ 14

ab ab .

We use (anti)symmetrization of indices with weight 1, i.e.

A[ab]= 12 (AabAba ) and A(ab)= 12 (Aab+Aba ) .The Levi-Civita tensor is

0123= 1 , 0123 = 1 .The dual, self-dual, and anti-self-dual of antisymmetric tensors are defined by

Hab 12 iabcdHcd, Hab= 12 (Hab Hab ) , Hab=

Hab

.

Structure constants are defined by

TA, TB

= fA B CTC.

The Clifford algebra is+ = 2g, = [] , . . .() = 00 ,= (i)(D/2)+101. . . D1 ;

in four dimensions:

= i0123, abcdd = iabc .The Majorana and Dirac conjugates are

= TC, = i0 .We mostly use the former. For Majorana fermions the two are equal.

The main SUSY commutator is (1),(2)

= 12 21.p-form components are defined by

p= 1p! 1p dx

1 dxp .

The differential acts from the left:

dA= Adx dx , A= Adx .

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Supergravity

D A NI E L Z . F R E E D MA NMassachusettsInstituteof Technology, USA

and

A N T O I N E V A N P R O E Y E NKU Leuven, Belgium

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C A M B R I D G E U N I V E R S I T Y P R E S S

Cambridge, New York, Melbourne, Madrid, Cape Town,Singapore, So Paulo, Delhi, Mexico City

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9780521194013

c D. Z. Freedman and A. Van Proeyen 2012

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2012

Printed in the United Kingdom at the University Press, Cambridge

A catalog record for this publication is available from the British Library

Library of Congress Cataloging in Publication data

Freedman, Daniel Z.Supergravity / Daniel Z. Freedman and Antoine Van Proeyen.

p. cm.ISBN 978-0-521-19401-3 (hardback)

1. Supergravity. I. Van Proeyen, Antoine. II. Title.QC174.17.S9F735 2012

530.1423dc23

2011053360

ISBN 978-0-521-19401-3 Hardback

Additional resources for this publication at www.cambridge.org/supergravity

Cambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party internet websites referred to

in this publication, and does not guarantee that any content on suchwebsites is, or will remain, accurate or appropriate.

http://www.cambridge.org/http://www.cambridge.org/9780521194013http://www.cambridge.org/9780521194013http://www.cambridge.org/9780521194013http://www.cambridge.org/

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Contents

Preface pagexvAcknowledgements xvii

Introduction 1

Part I Relativistic field theory in Minkowski spacetime 5

1 Scalar field theory and its symmetries 71.1 The scalar field system 71.2 Symmetries of the system 8

1.2.1 SO(n)internal symmetry 91.2.2 General internal symmetry 101.2.3 Spacetime symmetries the Lorentz and Poincar groups 12

1.3 Noether currents and charges 181.4 Symmetries in the canonical formalism 211.5 Quantum operators 221.6 The Lorentz group for D= 4 24

2 The Dirac field 252.1 The homomorphism of SL(2,C) SO(3, 1) 252.2 The Dirac equation 282.3 Dirac adjoint and bilinear form 312.4 Dirac action 322.5 The spinorsu( p, s)andv( p, s)for D= 4 332.6 Weyl spinor fields in even spacetime dimension 352.7 Conserved currents 36

2.7.1 Conserved U(1)current 362.7.2 Energymomentum tensors for the Dirac field 37

3 Clifford algebras and spinors 393.1 The Clifford algebra in general dimension 39

3.1.1 The generating-matrices 393.1.2 The complete Clifford algebra 403.1.3 Levi-Civita symbol 413.1.4 Practical-matrix manipulation 423.1.5 Basis of the algebra for even dimensionD= 2m 43

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vi Contents

3.1.6 The highest rank Clifford algebra element 443.1.7 Odd spacetime dimensionD= 2m+ 1 463.1.8 Symmetries of-matrices 47

3.2 Spinors in general dimensions 493.2.1 Spinors and spinor bilinears 49

3.2.2 Spinor indices 503.2.3 Fierz rearrangement 523.2.4 Reality 54

3.3 Majorana spinors 553.3.1 Definition and properties 563.3.2 Symplectic Majorana spinors 583.3.3 Dimensions of minimal spinors 58

3.4 Majorana spinors in physical theories 593.4.1 Variation of a Majorana Lagrangian 593.4.2 Relation of Majorana and Weyl spinor theories 60

3.4.3 U(1)symmetries of a Majorana field 61Appendix 3A Details of the Clifford algebras for D= 2m 623A.1 Traces and the basis of the Clifford algebra 623A.2 Uniqueness of the-matrix representation 633A.3 The Clifford algebra for odd spacetime dimensions 653A.4 Determination of symmetries of-matrices 653A.5 Friendly representations 66

4 The Maxwell and YangMills gauge fields 684.1 The abelian gauge field A(x) 69

4.1.1 Gauge invariance and fields with electric charge 69

4.1.2 The free gauge field 714.1.3 Sources and Greens function 734.1.4 Quantum electrodynamics 764.1.5 The stress tensor and gauge covariant translations 77

4.2 Electromagnetic duality 774.2.1 Dual tensors 784.2.2 Duality for one free electromagnetic field 784.2.3 Duality for gauge field and complex scalar 804.2.4 Electromagnetic duality for coupled Maxwell fields 83

4.3 Non-abelian gauge symmetry 86

4.3.1 Global internal symmetry 864.3.2 Gauging the symmetry 884.3.3 YangMills field strength and action 894.3.4 YangMills theory forG= SU(N) 90

4.4 Internal symmetry for Majorana spinors 93

5 The free RaritaSchwinger field 955.1 The initial value problem 97

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Contents vii

5.2 Sources and Greens function 995.3 Massive gravitinos from dimensional reduction 102

5.3.1 Dimensional reduction for scalar fields 1025.3.2 Dimensional reduction for spinor fields 1035.3.3 Dimensional reduction for the vector gauge field 104

5.3.4 Finally(x,y) 104

6 N= 1 global supersymmetry inD= 4 1076.1 Basic SUSY field theory 109

6.1.1 Conserved supercurrents 1096.1.2 SUSY YangMills theory 1106.1.3 SUSY transformation rules 111

6.2 SUSY field theories of the chiral multiplet 1126.2.1 U(1)Rsymmetry 1156.2.2 The SUSY algebra 116

6.2.3 More chiral multiplets 1196.3 SUSY gauge theories 1206.3.1 SUSY YangMills vector multiplet 1216.3.2 Chiral multiplets in SUSY gauge theories 122

6.4 Massless representations ofN-extended supersymmetry 1256.4.1 Particle representations ofN-extended supersymmetry 1256.4.2 Structure of massless representations 127

Appendix 6A Extended supersymmetry and Weyl spinors 129Appendix 6B On- and off-shell multiplets and degrees of freedom 130

Part II Differential geometry and gravity 133

7 Differential geometry 1357.1 Manifolds 1357.2 Scalars, vectors, tensors, etc. 1377.3 The algebra and calculus of differential forms 1407.4 The metric and frame field on a manifold 142

7.4.1 The metric 1427.4.2 The frame field 1437.4.3 Induced metrics 145

7.5 Volume forms and integration 146

7.6 Hodge duality of forms 1497.7 Stokes theorem and electromagnetic charges 1517.8 p-form gauge fields 1527.9 Connections and covariant derivatives 154

7.9.1 The first structure equation and the spin connectionab 1557.9.2 The affine connection 1587.9.3 Partial integration 160

7.10 The second structure equation and the curvature tensor 161

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viii Contents

7.11 The nonlinear-model 1637.12 Symmetries and Killing vectors 166

7.12.1 -model symmetries 1667.12.2 Symmetries of the Poincar plane 169

8 The first and second order formulations of general relativity 1718.1 Second order formalism for gravity and bosonic matter 1728.2 Gravitational fluctuations of flat spacetime 174

8.2.1 The graviton Greens function 1778.3 Second order formalism for gravity and fermions 1788.4 First order formalism for gravity and fermions 182

Part III Basic supergravity 185

9 N= 1 pure supergravity in four dimensions 1879.1 The universal part of supergravity 1889.2 Supergravity in the first order formalism 1919.3 The 1.5 order formalism 1939.4 Local supersymmetry ofN= 1, D= 4 supergravity 1949.5 The algebra of local supersymmetry 1979.6 Anti-de Sitter supergravity 199

10 D= 11 supergravity 20110.1 D 11 from dimensional reduction 20110.2 The field content ofD= 11 supergravity 20310.3 Construction of the action and transformation rules 203

10.4 The algebra ofD= 11 supergravity 210

11 General gauge theory 21211.1 Symmetries 212

11.1.1 Global symmetries 21311.1.2 Local symmetries and gauge fields 21711.1.3 Modified symmetry algebras 219

11.2 Covariant quantities 22111.2.1 Covariant derivatives 22211.2.2 Curvatures 223

11.3 Gauged spacetime translations 22511.3.1 Gauge transformations for the Poincar group 22511.3.2 Covariant derivatives and general coordinate transformations 22711.3.3 Covariant derivatives and curvatures in a gravity theory 23011.3.4 Calculating transformations of covariant quantities 231

Appendix 11A Manipulating covariant derivatives 23311A.1 Proof of the main lemma 23311A.2 Examples in supergravity 234

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Contents ix

12 Survey of supergravities 23612.1 The minimal superalgebras 236

12.1.1 Four dimensions 23612.1.2 Minimal superalgebras in higher dimensions 237

12.2 The R-symmetry group 238

12.3 Multiplets 24012.3.1 Multiplets in four dimensions 24012.3.2 Multiplets in more than four dimensions 242

12.4 Supergravity theories: towards a catalogue 24412.4.1 The basic theories and kinetic terms 24412.4.2 Deformations and gauged supergravities 246

12.5 Scalars and geometry 24712.6 Solutions and preserved supersymmetries 249

12.6.1 Anti-de Sitter superalgebras 25112.6.2 Central charges in four dimensions 252

12.6.3 Central charges in higher dimensions 253

Part IV Complex geometry and global SUSY 255

13 Complex manifolds 25713.1 The local description of complex and Khler manifolds 25713.2 Mathematical structure of Khler manifolds 26113.3 The Khler manifoldsCPn 26313.4 Symmetries of Khler metrics 266

13.4.1 Holomorphic Killing vectors and moment maps 26613.4.2 Algebra of holomorphic Killing vectors 268

13.4.3 The Killing vectors ofCP1 269

14 General actions withN= 1 supersymmetry 27114.1 Multiplets 271

14.1.1 Chiral multiplets 27214.1.2 Real multiplets 274

14.2 Generalized actions by multiplet calculus 27514.2.1 The superpotential 27514.2.2 Kinetic terms for chiral multiplets 27614.2.3 Kinetic terms for gauge multiplets 277

14.3 Khler geometry from chiral multiplets 27814.4 General couplings of chiral multiplets and gauge multiplets 280

14.4.1 Global symmetries of the SUSY-model 28114.4.2 Gauge and SUSY transformations for chiral multiplets 28214.4.3 Actions of chiral multiplets in a gauge theory 28314.4.4 General kinetic action of the gauge multiplet 28614.4.5 Requirements for anN= 1 SUSY gauge theory 286

14.5 The physical theory 288

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x Contents

14.5.1 Elimination of auxiliary fields 28814.5.2 The scalar potential 28914.5.3 The vacuum state and SUSY breaking 29114.5.4 Supersymmetry breaking and the Goldstone fermion 29314.5.5 Mass spectra and the supertrace sum rule 296

14.5.6 Coda 298Appendix 14A Superspace 298Appendix 14B Appendix: Covariant supersymmetry transformations 302

Part V Superconformal construction of supergravity theories 305

15 Gravity as a conformal gauge theory 30715.1 The strategy 30815.2 The conformal algebra 30915.3 Conformal transformations on fields 310

15.4 The gauge fields and constraints 31315.5 The action 31515.6 Recapitulation 31715.7 Homothetic Killing vectors 317

16 The conformal approach to pureN= 1 supergravity 32116.1 Ingredients 321

16.1.1 Superconformal algebra 32116.1.2 Gauge fields, transformations, and curvatures 32316.1.3 Constraints 32516.1.4 Superconformal transformation rules of a chiral multiplet 328

16.2 The action 33116.2.1 Superconformal action of the chiral multiplet 33116.2.2 Gauge fixing 33316.2.3 The result 334

17 Construction of the matter-coupledN= 1 supergravity 33717.1 Superconformal tensor calculus 338

17.1.1 The superconformal gauge multiplet 33817.1.2 The superconformal real multiplet 33917.1.3 Gauge transformations of superconformal chiral multiplets 340

17.1.4 Invariant actions 34217.2 Construction of the action 343

17.2.1 Conformal weights 34317.2.2 Superconformal invariant action (ungauged) 34317.2.3 Gauged superconformal supergravity 34517.2.4 Elimination of auxiliary fields 34717.2.5 Partial gauge fixing 351

17.3 Projective Khler manifolds 351

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Contents xi

17.3.1 The example ofCPn 35217.3.2 Dilatations and holomorphic homothetic Killing vectors 35317.3.3 The projective parametrization 35417.3.4 The Khler cone 35717.3.5 The projection 358

17.3.6 Khler transformations 35917.3.7 Physical fermions 36317.3.8 Symmetries of projective Khler manifolds 36417.3.9 T-gauge and decomposition laws 36517.3.10 An explicit example: SU(1, 1)/ U(1)model 368

17.4 From conformal to Poincar supergravity 36917.4.1 The superpotential 37017.4.2 The potential 37117.4.3 Fermion terms 371

17.5 Review and preview 373

17.5.1 Projective and KhlerHodge manifolds 37417.5.2 Compact manifolds 375Appendix 17A KhlerHodge manifolds 376

17A.1 Dirac quantization condition 37717A.2 KhlerHodge manifolds 378

Appendix 17B Steps in the derivation of (17.7) 380

Part VI N= 1 supergravity actions and applications 38318 The physicalN= 1 matter-coupled supergravity 385

18.1 The physical action 386

18.2 Transformation rules 38918.3 Further remarks 390

18.3.1 Engineering dimensions 39018.3.2 Rigid or global limit 39018.3.3 Quantum effects and global symmetries 391

19 Applications ofN= 1 supergravity 39219.1 Supersymmetry breaking and the super-BEH effect 392

19.1.1 Goldstino and the super-BEH effect 39219.1.2 Extension to cosmological solutions 395

19.1.3 Mass sum rules in supergravity 39619.2 The gravity mediation scenario 397

19.2.1 The Polnyi model of the hidden sector 39819.2.2 Soft SUSY breaking in the observable sector 399

19.3 No-scale models 40119.4 Supersymmetry and anti-de Sitter space 40319.5 R-symmetry and FayetIliopoulos terms 404

19.5.1 The R-gauge field and transformations 405

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xii Contents

19.5.2 FayetIliopoulos terms 40619.5.3 An example with non-minimal Khler potential 406

Part VII ExtendedN= 2 supergravity 409

20 Construction of the matter-coupledN= 2 supergravity 41120.1 Global supersymmetry 412

20.1.1 Gauge multiplets forD= 6 41220.1.2 Gauge multiplets forD= 5 41320.1.3 Gauge multiplets forD= 4 41520.1.4 Hypermultiplets 41820.1.5 Gauged hypermultiplets 422

20.2 N= 2 superconformal calculus 42520.2.1 The superconformal algebra 42520.2.2 Gauging of the superconformal algebra 427

20.2.3 Conformal matter multiplets 43020.2.4 Superconformal actions 43220.2.5 Partial gauge fixing 43420.2.6 Elimination of auxiliary fields 43620.2.7 Complete action 43920.2.8 D= 5 and D= 6, N= 2 supergravities 440

20.3 Special geometry 44020.3.1 The family of special manifolds 44020.3.2 Very special real geometry 44220.3.3 Special Khler geometry 44320.3.4 Hyper-Khler and quaternionic-Khler manifolds 452

20.4 From conformal to Poincar supergravity 45920.4.1 Kinetic terms of the bosons 45920.4.2 Identities of special Khler geometry 45920.4.3 The potential 46020.4.4 Physical fermions and other terms 46020.4.5 Supersymmetry and gauge transformations 461

Appendix 20A SU(2)conventions and triplets 463Appendix 20B Dimensional reduction 6 5 4 464

20B.1 Reducing from D= 6 D= 5 46420B.2 Reducing from D

=5

D

=4 464

Appendix 20C Definition of rigid special Khler geometry 465

21 The physicalN= 2 matter-coupled supergravity 46921.1 The bosonic sector 469

21.1.1 The basic (ungauged)N= 2, D= 4 matter-coupledsupergravity 469

21.1.2 The gauged supergravities 47121.2 The symplectic formulation 472

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Contents xiii

21.2.1 Symplectic definition 47221.2.2 Comparison of symplectic and prepotential formulation 47421.2.3 Gauge transformations and symplectic vectors 47421.2.4 Physical fermions and duality 475

21.3 Action and transformation laws 476

21.3.1 Final action 47621.3.2 Supersymmetry transformations 477

21.4 Applications 47921.4.1 Partial supersymmetry breaking 47921.4.2 Field strengths and central charges 48021.4.3 Moduli spaces of CalabiYau manifolds 480

21.5 Remarks 48221.5.1 FayetIliopoulos terms 48221.5.2 -model symmetries 48221.5.3 Engineering dimensions 482

Part VIII Classical solutions and the AdS/CFT correspondence 485

22 Classical solutions of gravity and supergravity 48722.1 Some solutions of the field equations 487

22.1.1 Prelude: frames and connections on spheres 48722.1.2 Anti-de Sitter space 48922.1.3 AdSDobtained from its embedding in RD+1 49022.1.4 Spacetime metrics with spherical symmetry 49622.1.5 AdSSchwarzschild spacetime 49822.1.6 The ReissnerNordstrm metric 49922.1.7 A more general ReissnerNordstrm solution 501

22.2 Killing spinors and BPS solutions 50322.2.1 The integrability condition for Killing spinors 50522.2.2 Commuting and anti-commuting Killing spinors 505

22.3 Killing spinors for anti-de Sitter space 50622.4 Extremal ReissnerNordstrm spacetimes as BPS solutions 50822.5 The black hole attractor mechanism 510

22.5.1 Example of a black hole attractor 511

22.5.2 The attractor mechanism real slow and simple 51322.6 Supersymmetry of the black holes 51722.6.1 Killing spinors 51722.6.2 The central charge 51922.6.3 The black hole potential 521

22.7 First order gradient flow equations 52222.8 The attractor mechanism fast and furious 523Appendix 22A Killing spinors for pp-waves 525

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xiv Contents

23 The AdS/CFT correspondence 52723.1 The N= 4 SYM theory 52923.2 Type IIB string theory andD3-branes 53223.3 The D3-brane solution of Type IIB supergravity 53323.4 KaluzaKlein analysis on AdS5 S5 53423.5 Euclidean AdS and its inversion symmetry 53623.6 Inversion and CFT correlation functions 53823.7 The free massive scalar field in Euclidean AdSd+1 53923.8 AdS/CFT correlators in a toy model 54123.9 Three-point correlation functions 54323.10 Two-point correlation functions 54523.11 Holographic renormalization 550

23.11.1 The scalar two-point function in a CFTd 55423.11.2 The holographic trace anomaly 555

23.12 Holographic RG flows 558

23.12.1 AAdS domain wall solutions 55923.12.2 The holographicc-theorem 56223.12.3 First order flow equations 563

23.13 AdS/CFT and hydrodynamics 564

AppendixA Comparisonof notation 573A.1 Spacetime and gravity 573A.2 Spinor conventions 575A.3 Components of differential forms 576A.4 Covariant derivatives 576

Appendix B Lie algebrasand superalgebras 577B.1 Groups and representations 577B.2 Lie algebras 578B.3 Superalgebras 581

References 583Index 602

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Preface

The main purpose of this book is to explore the structure of supergravity theories at theclassical level. Where appropriate we take a general D-dimensional viewpoint, usuallywith special emphasis on D= 4. Readers can consult the Contents for a detailed list ofthe topics treated, so we limit ourselves here to a few comments to guide readers. Wehave tried to organize the material so that readers of varying educational backgroundscan begin to read at a point appropriate to their background. Part I should be accessibleto readers who have studied relativistic field theory enough to appreciate the importanceof Lagrangians, actions, and their symmetries. Part II describes the differential geometricbackground and some basic physics of the general theory of relativity. The basic super-gravity theories are presented in Part III using techniques developed in earlier chapters. InPart IV we discuss complex geometry and apply it to matter couplings in global N

=1

supersymmetry. In Part V we begin a systematic derivation ofN= 1 matter-coupledsupergravity using the conformal compensator method. The going can get tough on thissubject. For this reason we present the final physical action and transformation rules andsome basic applications in two separate short chapters in Part VI. Part VII is devotedto a systematic discussion ofN = 2 supergravity, including a short chapter with theresults needed for applications. Two major applications of supergravity, classical solu-tions and the AdS/CFT correspondence, are discussed in Part VIII in considerable detail.It should be possible to understand these chapters without full study of earlier parts of thebook.

Many interesting aspects of supergravity, some of them subjects of current research,

could not be covered in this book. These include theories in spacetime dimensions D < 4,higher derivative actions, embedding tensors, infinite Lie algebra symmetries, and the pos-itive energy theorem.

Like many other subjects in theoretical physics, supersymmetry and supergravity arebest learned by readers who are willing to get their hands dirty. This means activelyworking out problems that reinforce the material under discussion. To facilitate this aspectof the learning process, many exercises for readers appear within each chapter. We give arough indication of the level of each exercise as follows:

xv

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xvi Preface

Level 1. The result of this exercise will be used later in the book.Level 2. This exercise is intended to illuminate the subject under discussion, but it is not

needed in the rest of the book.Level 3. This exercise is meant to challenge readers, but is not essential.

These levels are indicated respectively by single, double or triple gray bars in the outsidemargin.

A website featuring solutions to some exercises, errata and additional reading material,can be found at www.cambridge.org/supergravity.

Dan FreedmanToine Van Proeyen

October 2011

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Acknowledgements

We thank Eric Bergshoeff, Paul Chesler, Bernard de Wit, Eric DHoker, Henriette Elvang,John Estes, Gary Gibbons, Joaquim Gomis, Renata Kallosh, Hong Liu, Marin Lled,Samir Mathur, John McGreevy, Michael Peskin, Leonardo Rastelli, Kostas Skenderis,Stefan Vandoren, Bert Vercnocke and Giovanni Villadoro. We thank the students in var-ious courses (Leuven advanced field theory course, Doctoral schools in Paris, Barcelona,Hamburg), and also Frederik Coomans, Serge Dendas, Daniel Harlow, Andrew Larkoski,Jonathan Maltz, Thomas Rube, Walter Van Herck and Bert Van Pol for their input in the

preparation of this text and their critical remarks.Our home institutions have supported the writing of this book over a period of years, andwe are grateful. We also thank the Galileo Galilei Institute in Florence and the Departmentof Applied Mathematics and Theoretical Physics in Cambridge for support during extendedvisits, and the Stanford Institute for Theoretical Physics for support and hospitality, indeeda home away from home, during multiple visits when we worked closely together.

A.V.P. wil in het bijzonder zijn moeder bedanken voor de sterkte en voortdurende steundie hij van haar gekregen heeft. He also thanks Marleen and Laura for the strong supportduring the work on this book. D.Z.F. thanks his wife Miriam for her encouragement to startthis project and continuous support as it evolved.

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Introduction

Two developments in the late 1960s and early 1970s set the stage for supergravity. Firstthe standard model took shape and was decisively confirmed by experiments. The keytheoretical concept underlying this progress was gauge symmetry, the idea that symme-try transformations act independently at each point of spacetime. In the standard modelthese are internal symmetries, whose parameters are Lorentz scalarsA(x)that are arbi-trary functionsof the spacetime point x. These parameters are coordinates of the com-pact Lie group SU(3)

SU(2)

U(1). Scalar, spinor, and vector fields of the theory

are each classified in representations of this group, and the Lagrangian is invariant undergroup transformations. The special dynamics associated with the non-abelian gauge princi-ple allows different realizations of the symmetry in the particle spectrum and interactionsthat would be observed in experiments. For example, part of the gauge symmetry maybe spontaneously broken. In the standard model this produces the unification of weakand electromagnetic interactions. The observed strength and range of these forces are verydifferent, yet the gauge symmetry gives them a common origin.

The other development was global (also called rigid) supersymmetry [1,2,3].It is theunique framework that allows fields and particles of different spin to be unified in rep-resentations of an algebraic system called a superalgebra. The symmetry parameters are

spinorsthat areconstant, independent ofx. The simplest N= 1 superalgebra containsa spinor supercharge Qand the energymomentum operator Pa . The anti-commutator oftwo supercharges is a translation in spacetime. The N= 1 supersymmetry algebra hasrepresentations containing massless particles of spins (s, s 1/2)for s=1/2, 1, . . .andsomewhat larger representations containing particles with a common non-vanishing mass.Thus supersymmetry always unites bosons, integer spin, with fermions, half-integer spin.The focus of early work was interacting field theories of the (1/2, 0)and (1, 1/2)mul-tiplets. It was found that the ultraviolet divergences of supersymmetric theories are lesssevere than in the standard model due to the cancelation between bosons and fermions inloop diagrams.

Unbroken supersymmetry requires a spectrum of particles in equal-mass bosonfermionpairs. This is decidedly not what is observed in experiments. So if supersymmetry is real-ized in Nature, it must appear as a broken symmetry. Through the years much theoreticaleffort has been devoted to the construction of extensions of the standard model with brokensupersymmetry. It is hypothesized that the as yet unseen superpartners of the known par-ticles will be produced at the Large Hadron Collider (LHC) accelerator, thus confirming asupersymmetric version of the standard model. The advantages of supersymmetric modelsinclude the following:

1

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2 Introduction

Milder ultraviolet divergences permit an improved and more predictive treatment of elec-troweak symmetry breaking.

When extrapolated using the renormalization group, the three distinct gauge couplingsof the standard model approach a common value at high energy. The unification of cou-plings is a major success.

Supersymmetry provides natural candidates for the particles of cosmological cold darkmatter.

The role of gauge symmetry in the standard model suggested that a gauged form ofsupersymmetry would be interesting and perhaps more powerful than the global form. Sucha theory would contain gauge fields for both spacetime translations Paand SUSY transfor-mations generated by Q . Thus gauged supersymmetry was expected to be an extensionof general relativity in which the graviton acquires a fermionic partner called the gravitino.The name supergravity is certainly appropriate and was used even before the theory wasactually found. It was reasonable to think that the gauge fields of the theory would be the

vierbein,ea

(x), needed to describe gravity coupled to fermions, and a vectorspinor field,(x), for the gravitino. The graviton and gravitino belong to the (2, 3/2)representa-tion of the algebra. A Lagrangian field theory of supergravity was formulated in the springof 1976 in[4]. The approach taken was to modify the known free field Lagrangian forto agree with gravitational gauge symmetry and then find, by a systematic procedure,the additional terms necessary for invariance under supersymmetry transformations witharbitrary(x). Soon an alternative approach appeared[5] in which the most complicatedcalculation required in[4] is avoided.

Research in supergravity became a very intense activity in the years following its dis-covery. One early direction was the construction of Lagrangian field theories in which thespin-(2, 3/2) gravity multiplet is coupled to the (1/2, 0) and (1, 1/2) multiplets of global

supersymmetry. This is the framework of matter-coupled supergravity. It shares the posi-tive features of global symmetry listed above. In addition supergravity provides new sce-narios for the breaking of supersymmetry. In particular, the structure of the supergravityLagrangians allows SUSY breaking with vanishing vacuum energy and thus vanishingcosmological constant. This feature is not available without the coupling of matter fieldsto supergravity. Matter-coupled supergravity theories typically contain scalar fields, whichcan be useful in constructing phenomenological models of inflationary cosmology.

A spin-3/2 particle is the key prediction of supergravity. SUSY breaking gives it a masswhose magnitude depends on the breaking mechanism. Unfortunately it appears difficultto detect it at the LHC because it is coupled to matter with the feeble strength of quan-

tum gravity. However, gravitinos can be copiously produced in the ultra-high-temperatureenvironment at or near the big bang. Gravitino production leads to important constraintson early universe cosmology.

A second direction of research involves the construction of theories with several super-charges Qi ,i=1, 2, . . . ,N. Such extended supergravity theories can be constructed upto the limit N= 8 in spacetime dimension D= 4. Beyond that the superalgebra represen-tations necessarily contain particles of spin s 5/2, for which no consistent interactionsexist. Many of the ultraviolet divergences expected in a field theory containing gravity are

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Introduction 3

known to cancel in the maximal N= 8 theory, and some theorists speculate that it isultraviolet finite to all orders in perturbation theory.

Supergravity theories in spacetime dimensions D > 4 have been constructed up tothe bound D= 11 (which is again due to the higher spin consistency problem). Two10-dimensional supergravity theories, known as the Type IIA and Type IIB theories, are

related to the superstring theories that carry the same names. Roughly speaking, supergrav-ity appears as the low-energy limit of superstring theory. This means that the dynamics ofthe lowest-energy modes of the superstring are described by supergravity. But these state-ments do not do justice to the intimate and rich relation of these two theoretical frame-works.

The very important anti-de Sitter/conformal field theory (AdS/CFT) correspondenceprovides one example of this relation. It was based on the remarkable conjecture that TypeIIB string theory on the product manifold AdS5 S5 is equivalent to the maximal N= 4global supersymmetric gauge theory. However, concrete tests and predictions of AdS/CFTusually involve working in the limit in which classical supergravity is a valid approxima-

tion to string theory.The scope of supergravity is broad. There is a supergravity-inspired approach to posi-tive energy and stability in gravitational theories. Many classical solutions of supergravityhave the special BogomolnyiPrasadSommerfield (BPS) property and therefore satisfytractable first order field equations. The scalar sectors of supergravity theories involve non-linear-models on complex manifolds with new geometries of interest in both physics andmathematics.

To summarize: supergravity is based on the gauge principle of local supersymmetry andis thus connected to fundamental ideas in theoretical physics. Supergravity effects may turnout to be observable at the LHC. Further there is important input from cosmology. Thisreal side of the subject is far from confirmation, but it must be taken seriously. In addition

there are several more theoretical applications such as BPS solutions and AdS/CFT. Activeresearch continues on most branches of supergravity although 35 years have passed sinceit was first formulated.

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PART I

RELATIVISTIC FIELD THEORY INMINKOWSKI SPACETIME

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Scalar field theory and its symmetries 1

The major purpose of the early chapters of this book is to review the basic notions of rel-ativistic field theory that underlie our treatment of supergravity. In this chapter we discussthe implementation of internal and spacetime symmetries using the model of a system offree scalar fields as an example. The general Noether formalism for symmetries is also dis-cussed. Our book largely involves classical field theory. However, we adopt conventionsfor symmetries that are compatible with implementation at the quantum level.

Our treatment is not designed to teach the material to readers who are encountering

it for the first time. Rather we try to gather the ideas (and the formulas!) that are usefulbackground for later chapters. Supersymmetry and supergravity are based on symmetriessuch as the spacetime symmetry of the Poincar group and much more!

As in much of this book, we assume general spacetime dimension D, with specialemphasis on the case D= 4.

1.1 The scalar field system

We consider a system ofn real scalar fields i (x), i=1, . . . , n, that propagate in a flatspacetime whose metric tensor

= = diag(, +, . . . , +) (1.1)

describes one time and D 1 space dimensions. This is Minkowski spacetime, in whichwe use Cartesian coordinates x, = 0, 1, . . . ,D 1, with time coordinate x0 = t(withvelocity of lightc= 1).

Practicing physicists and mathematicians are largely concerned with fields that satisfynonlinear equations. However, linear wave equations, which describe free relativistic par-

ticles, have much to teach about the basic ideas. We therefore assume that our fields satisfythe KleinGordon equation

i (x) = m2i (x) , (1.2)

where = is the Lorentz invariant dAlembertian wave operator.7

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8 Scalar field theory and its symmetries

The equation has plane wave solutions ei( pxEt), which provide the wave functionsfor particles of spatial momentump, with spatial components pi , and energy E= p0 =

p2 + m2. The general solution of the equation is the sum of a positive frequency part,which can be expressed as the (D 1)-dimensional Fourier transform in the plane wavesei( pxEt), plus a negative frequency part, which is the Fourier transform in the ei( pxEt),

i (x) = i+(x)+ i(x) ,

i+(x) =

dD1 p(2 )(D1)2E

ei( pxEt)ai ( p) ,

i(x) =

dD1 p(2 )(D1)2E

ei( pxEt)ai( p). (1.3)

In the classical theory the quantitiesai ( p), ai( p)are simply complex valued functions ofthe spatial momentump. After quantization one arrives at the true quantum field theory1in whichai ( p), ai( p)are annihilation and creation operators2 for the particles describedby the field operator i (

x).

The KleinGordon equation(1.2) is the variational derivative S/ i (x)of the action

S=

dDxL(x) = 12

dDx

i

i + m2i i

. (1.4)

The repeated indexi is summed. The action is a functionalof the fields i (x). It is a realnumber that depends on the configuration of the fields throughout spacetime.

1.2 Symmetries of the system

Consider a set of fields such as the i (x)that satisfy equations of motion such as(1.2). Ageneral symmetry of the system is a mapping of the configuration space, i (x) i (x),with the property that if the original field configuration i (x)is a solution of the equationsof motion, then the transformed configuration i (x)is also a solution. For scalar fieldsand for most other systems of interest in this book, we can restrict attention to symmetrytransformations that leave the action invariant. Thus we require that the mapping has theproperty3,4

S[i ] = S[i ]. (1.5)

Here is an example.1 When desirable for clarity we use bold face to indicate the operator in the quantum theory that corresponds to

a given classical quantity.2 In the conventions above, creation and annihilation operators are normalized in the quantum theory by

[a( p), a( p)] = (2 )D12E( p p).3 Such mappings must also respect the boundary conditions. This requirement can be non-trivial, e.g. Neumann

and Dirichlet boundary conditions for the bosonic string lead to different spacetime symmetry groups. We willmostly assume that field configurations vanish at large spacetime distances.

4 One important exception is the electromagnetic duality symmetry, which is discussed in Sec.4.2.

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1.2 Symmetries of the system 9

Exercise 1.1 Verify that the map i (x) i (x)=i (x+ a)satisfies (1.5)if a is aconstant vector. This symmetry is called a global spacetime translation.

We consider both spacetime symmetries, which involve a motion in Minkowski space-time such as the global translation of the exercise, and internal symmetries, which do not.

Internal symmetries are simpler to describe, so we start with them.

1.2.1 SO(n) internal symmetry

Let Ri j be a matrix of the orthogonal group SO(n). This means that it is ann nmatrixthat satisfies

Ri ki jRj

= k , detR= 1. (1.6)It is quite obvious that the linear map,

i (x)

i (x)

= Ri

jj (x) , (1.7)

satisfies (1.5)and is an internal symmetry of the action (1.4). This symmetry is calleda continuous symmetry because a matrix of SO(n)depends continuously on 12 n(n1)independent group parameters. We will discuss one useful choice of parameters shortly.We also call the symmetry a global symmetry because the parameters are constants. In Ch.4we will consider local or gauged internal symmetries in which the group parameters arearbitrary functions ofx.

It is worth stating the intuitive picture of this symmetry. One may consider the field i

as ann-dimensional vector, that is an element ofRn . The transformation i Ri j j is arotation in this internal space. Such a rotation preserves the usual norm i i j j .

We now introduce the Lie algebra of the group SO(n). To first order in the smallparameter , we write the infinitesimal transformation

Ri j = ij ri j . (1.8)

This satisfies (1.6)ifri j= rj i . Any antisymmetric matrix ri j is called a generator ofSO(n). TheLie algebrais the linear space spanned by the 12 n(n 1)independent genera-tors, with the commutator product

[r, r] = r r rr. (1.9)Note that matrices are multiplied5 asri krkj .

A useful basis for the Lie algebra is to choose generators that act in each of the 12 n(n1)independent 2-planes ofRn . For the 2-plane in the directions ijthis generator is given by

r[i j]i

j ii

j j ij i j= r[ j i]i

j . (1.10)

5 Some mathematical readers may initially be perturbed by the indices used to express many equations in thisbook. We will follow the standard conventions used in physics. Unless ambiguity arises we use the Einsteinsummation convention for repeated indices, usually one downstairs and one upstairs. The summation conven-tion incorporates the standard rules of matrix multiplication.

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10 Scalar field theory and its symmetries

Note the distinction between the coordinate plane labels in brackets with hatted indices andthe row and column indices. The commutators of the generators defined in(1.10) are

[r[i j], r[kl]] = j kr[i l] i kr[ j l] j l r[i k] + i l r[ j k]. (1.11)

The row and column indices are suppressed in this equation, and this will be our practicewhen it causes no ambiguity. The equation implicitly specifies the structure constants ofthe Lie algebra in the basis of (1.10).

In this basis, a finite transformation of SO(n)is determined by a set of 12 n(n 1)realparametersij which specify the angles of rotation in the independent 2-planes. A generalelement of (the connected component) of the group can be written as an exponential

R= e 12 ij r[ij] . (1.12)

1.2.2 General internal symmetry

It will be useful to establish the notation for the general situation of linearly realized inter-nal symmetry under an arbitrary connected Lie group G, usually a compact group, ofdimension dim G. We will be interested in an n-dimensional representation ofG in whichthe generators of its Lie algebra are a set ofn nmatrices(tA)i j , A=1, 2, . . . , dim G.Their commutation relations are6

[tA, tB] = fAB CtC, (1.13)

and the fA B C are structure constants of the Lie algebra. The representative of a generalelement of the Lie algebra is a matrixthat is a superposition of the generators with realparametersA, i.e.

= AtA. (1.14)

An element of the group is represented by the matrix exponential

U() = e = eAtA . (1.15)

We consider a set of scalar fields i (x) which transforms in the representation just

described. The fields may be real or complex. If complex, the complex conjugate of everyelement is also included in the set. A group transformation acts by matrix multiplicationon the fields:

i (x) i (x) U()i j j (x). (1.16)6 Although it is common in the physics literature to insert the imaginary i in the commutation rule, we do not

do this in order to eliminate is in most of the formulas of the book. This means that compact generators tAin this book are anti-hermitian matrices.

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1.2 Symmetries of the system 11

If the system is complex, the representation ofG will typically be reducible, and the fullset{i } splits into equal numbers of fields and conjugates on which U() acts inde-pendently. It is not necessary to distinguish the real and complex cases explicitly in ournotation.

We assume that the equations of motion of the system are obtained from an action

S[i ] =

dDxL(i , i ) , (1.17)

which is invariant under (1.16).In this chapter we consider internal symmetry with theproperty that the Lagrangian density is invariant, that is

L(i , i ) = L(i , i ). (1.18)

This property is stronger than (1.5).

Exercise 1.2 Verify that Lagrangian density (1.4) is invariant under theSO(n) symmetry

of Sec.1.2.1,but not under the spacetime translation of Ex.1.1.

An infinitesimal transformation of the group is defined as the truncation of the expo-nential power series in (1.15)to first order in. This gives the field variation (matrix andvector indices suppressed)

= , (1.19)

which defines the action of a Lie algebra element on the fields.It is important to define iterated Lie algebra variations carefully. The definition we make

below may seem unfamiliar. However, we show that it does give a representation of the

algebra. Later, in Sec.1.4, we will see that the definition is compatible with implementationof symmetry transformations by Poisson brackets of their conserved Noether charges at theclassical level and (in Sec.1.5)by unitary transformation after quantization.

The action of a transformation2with Lie algebra element 2followed by a transfor-mation1with element1is defined by

12 21= 21. (1.20)

The second variation acts only on the dynamical variables of the system, the fields i , andis not affected by the matrix2that multiplies i . In detail,

12= A

1

B

2 tB tA. (1.21)The commutator of two symmetry variations is then

[1, 2]= [1, 2] 3 ,3 [1, 2] = fA B CA1 B2 tC. (1.22)

The commutator of two Lie algebra transformations is again an algebra transformation bythe element3= [1, 2].

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12 Scalar field theory and its symmetries

It follows that finite group transformations compose as

2 = U(2) 1 = U(2)U(1). (1.23)

We show in Sec.1.5that this agrees with the composition of the unitary transformationswhich implement the symmetry in the quantum theory.

1.2.3 Spacetime symmetries the Lorentz and Poincar groups

The Lorentz group is defined as the group of homogeneous linear transformations of coor-dinates in D-dimensional Minkowski spacetime that preserve the Minkowski norm of anyvector. We write

x = x or x = 1x , (1.24)and require that

xx = xx . (1.25)The Poincar group is defined by adjoining global translations and considering

x = 1x a . (1.26)

In this section we review the properties of these groups, their Lie algebras and the groupaction on fields such as i (x).

If(1.25) holds for any vector x, it follows that

= . (1.27)This is the defining property of the matrices. If the Minkowski metric were replaced bythe Kronecker delta, , these conditions would define the orthogonal group O(D), buthere we have the pseudo-orthogonal group O(D 1, 1). For most purposes we need onlythe connected component of this group, which we call the proper Lorentz group.

The metric tensor (and its inverse) are used to lower (or raise) vector indices. Thus onehas, for example, x= x and = . Upper or lower indices are calledcontravariant or covariant, respectively.

Exercise 1.3 Show that (1.24) and(1.27) imply

= (1) , = (1) ,x= (1) x= x . (1.28)

The first relation of the exercise resembles the standard matrix orthogonality property,but it holds for Lorentz when both indices are down (or both up), which is not the cor-rect position for their action as linear transformations. Indeed, Lorentz matrices must bemultiplied with indices in updown position, viz. .

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1.2 Symmetries of the system 13

We now introduce the Lie algebra of the Lorentz group, proceeding in parallel7 to thediscussion of the group SO(n) in Sec.1.2.1.For a small parameter , we write the infinites-imal transformation

= + m+ . (1.29)

It is straightforward to see that (1.29)satisfies (1.27)to first order in as long as thegenerator (mwith two lower indices) is antisymmetric, viz.

m m = m . (1.30)The Lie algebrais the real linear space spanned by the 12D(D 1)independent genera-tors, with the commutator product[m, m] = m m mm. These matrices must also bemultiplied asm m , but the forms with both indices down, as in (1.30),(or both up) areoften convenient.

A useful basis for the Lie algebra is to choose generators that act in each of the12D(D 1)coordinate 2-planes. For the 2-plane in the directions , this generator is

given bym[] = m[ ]. (1.31)

Note the distinction between the coordinate plane labels in brackets and the row andcolumn indices. In this basis, a finite proper Lorentz transformation is specified by a set of12D(D 1)real parameters = and takes exponential form

= e 12 m[] . (1.32)When matrix indices are restored, we have, with the representation (1.31),the series8

=

+

+12

+. . . . (1.33)

The commutators of the generators defined in (1.31)are

[m[], m[]] = m[] m[] m[] + m[]. (1.34)

These equations specify the structure constants of the Lie algebra, which may be written as

f[][][]= 8[[ []]]. (1.35)Note that antisymmetrization is always done with weight 1, see (A.8),such that the right-

hand side can be written as eight terms with coefficients 1.Exercise 1.4 Check that (1.34) leads to (1.35). To compare with (1.13), you have

to replace each of the indices A,B, C by antisymmetric combinations, e.g. A [].Moreover, you have to insert a factor 12 each time that you sum over such a combined index

to avoid double counting, as e.g. the factor 12 in (1.32).Therefore we rewrite (1.34)as

7 Specifically, it is1 which is the analogue ofR in (1.12) and ofU( )in (1.15).8 Thus, we now replace in (1.29) mby 12

m[] .

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14 Scalar field theory and its symmetries

[mA, mB] = fA B CmC [m[], m[]] = 12 f[][][]m[]. (1.36)

Under a symmetry, each group element is mapped to a transformation of the configura-tion space of the dynamical fields. This map must give a group homomorphism. For theLorentz matrix, the transformation of the scalar fields is defined as

i (x) i (x) i (x). (1.37)

Using(1.24), we find that i (x) = i (x).

Exercise 1.5 Show that the action(1.4) is invariant under the transformation(1.37).

We now define differential operators which implement the coordinate change due to aninfinitesimal transformation. A transformation in the [] 2-plane is generated by

L[]x x . (1.38)

The commutator algebra of these operators is isomorphic to (1.34), and we thus have arealization of the Lie algebra acting as differential operators on functions.

Exercise 1.6 Compute the commutators[L[],L []] and show that they agree withthat of(1.34) for matrix generators. Show that to first order in

i (x)

12

L[]i (x)

=i (x

+x ). (1.39)

We then define the differential operator

U() e12

L[] . (1.40)

Using this operator, the mapping(1.37) which defines the action of finite Lorentz transfor-mations on scalar fields can then be written as

i (x) i (x) = U()i (x) = i (x). (1.41)

Box 1.1 Symmetries

Symmetries are implemented by transformations that act on fields.

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1.2 Symmetries of the system 15

We see that both Lorentz and internal symmetries (see (1.16))are implemented by linearoperators acting on the classical fields, a differential operator for Lorentz and a matrixoperator for internal. Both operators depend on group parameters in the same way.

By expanding the exponential, this defines the infinitesimal Lorentz transformation ofthe scalars:

()i (x) = i (x) i (x) = i (x) i (x) = x i (x) = 12 L[]i (x).(1.42)

This definition does give a homomorphism of the group; namely the product of maps,first with 2, then with 1, produces the compound map associated with the product12. This can be seen from the sequence of steps

i (x) 2 i (x)= U(2)i (x) 1 i (x) = U(2)U(1)i (x)

= U(2)i (1x)

= i

(12x). (1.43)As mentioned above, a symmetry transformation actsdirectly on the fields. This conventiondetermines the order of operations in the first line. In the second and third lines, we usethe action of the differential operators to arrive at a transformation with matrix product12. This is exactly the same for internal symmetries in (1.23),where the action ofUisobtained by matrix multiplication.

Exercise 1.7 It is instructive to check (1.43) for Lorentz transformations which are close

to the identity. Specifically, use the definition (1.40)to show that the order1 2 terms in

the product U(2)U(1)i of differential operators acting on i agrees with terms of the

same order ini

(12x).Calculate the infinitesimal commutator[U(2), U(1)] to order12. Show that the

commutator is a Lorentz transformation with (matrix) parameters[1, 2]. Note that thesecond transformation acts on i (x), and not on the x-dependent factor in (1.42).

It is important to extend the Lorentz transformation rules to covariant and contravariantvector fields, which are, respectively, sections of the cotangent and tangent bundles ofMinkowski spacetime. The transformation of a general covariant vector field W(x)canbe modeled on that of the gradient of a scalar (x). From (1.37)we find

(x)

(x)=

x

(x)

=1 ( )(x) , (1.44)

where we have used the chain rule and (1.28)in the last step. Thus we define the transfor-mation of a general covariant field as

W(x) W(x) 1 W (x). (1.45)For contravariant vectors we assume a transformation of the form

V(x) BV(x). (1.46)

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16 Scalar field theory and its symmetries

The matrix can determined by requiring that the inner product V(x)W(x)transforms asa scalar. This fixes B= 1,and we have the transformation rule

V(x) V(x) 1 V (x). (1.47)

Next we define generators of Lorentz transformations appropriate to the covariantand contravariant vector representations. They are combined differential/matrix operatorsgiven by

J[]V(x) (L[]+ m[] )V (x) ,J[]W (x) (L[] + m[] )W(x). (1.48)

We avoid an overly decorative notation by suppressing row and column indices on J[].For each case a finite Lorentz transformation is implemented by the operator

U()

=e

12

J[] . (1.49)

Exercise 1.8 Verify that J[](VW) = L[](VW).

The Lorentz group has many representations, both higher rank tensor and spinor repre-sentations (to be discussed in Ch.2) and combinations thereof. Let i (x)denote a set offields where i is now an index of the components of a general representation. There is acorresponding Lie algebra representation with matricesm []which act on the indices anddifferential/matrix generators

J[]=L[] + m[] , (1.50)

in which i j= ij is the unit matrix. A finite Lorentz transformation is then the mapping(suppressing the indexi for simplicity of the notation)

(x) (x) = U()(x) = e12

m[](x). (1.51)

The precise forms of the operators m[], J[] and U()depend on the representationunder study.

Spacetime translations x x = x a are much simpler because they areimplemented in the same way in all representations of the Lorentz group, namely by themapping

i (x) i (x) = i (x+ a) = U(a) i (x) , (1.52)U(a)= ea P , (1.53)

P= = x

. (1.54)

In(1.52) we have defined the generator Pand the finite translation operatorU(a)whichare differential operators. Finite transformations of the Poincar group are implemented bythe operatorU(a, ) U()U(a), which acts as follows:

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1.2 Symmetries of the system 17

(x) (x) U(a,)(x) = U()U(a) (x)= e 12 m[] (x+ a). (1.55)

Exercise 1.9 Prove that U(()1a)U() = U()U(a). Verify for operators which areclose to the identity that

U(a)(x+ b) = (x+ a + b) ,U()(x+ b) = (x+ b). (1.56)

The Lie algebra of the Poincar group contains the D(D+ 1)/2 generators J[], P.The following commutation rules complete the specification of the Lie algebra:

[J

[

],J

[

]] =J

[

] J

[

] J

[

] +J

[

],

[J[],P] = P P ,[P,P] = 0. (1.57)

Exercise 1.10 Verify (1.57).

We now discuss the implementation of this Lie algebra on fields. The treatment is par-allel to the discussion of internal symmetry at the end of Sec. 1.2.2. The infinitesimalvariation of the fields i (x)is defined as the first order truncation of the exponential in(1.55):

= [a

P 1

2

J[]]. (1.58)The J[] operator appropriate to the representation is used, and representation indicesare suppressed for simplicity. Consider now the transformation 2with group parameters(a2, 2)followed by transformation1with parameters(a1, 1). The result is defined as

12= 1[a2 P 12 2 J[]]= [a2 P 12 2 J[]][a1P 12 1 J[]]. (1.59)

The second transformation acts only on the field variable. With some care one can calculatethe commutator of two variations, which yields a third transformation with parameters

(a3, 3), that is

[1, 2]= 3 ,

3 = [1, 2] = 14 f[][][]1 2 ,a

3= a1 2 a2 1 = 12 f[]

a1

2 a2 1

(1.60)

(remember Ex.1.4).Note that the parameters of infinitesimal transformations composewith the structure constants of the Lie algebra, exactly as in the case of internal symmetry.

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18 Scalar field theory and its symmetries

Exercise 1.11 Verify the rule of combination of parameters in(1.60).

As in the discussion of internal symmetry at the end of Sec.1.2.2, the product of twofinite transformations of the Poincar group, the first with parameters (a2, 2)followed bythe second with parameters(a1, 1), is given by

(x) (x) = U(a2, 2)U(a1, 1) (x). (1.61)The compound map is a representation, as we discuss at the end of Sec.1.5below.

1.3 Noether currents and charges

It is well known that, using the Noether method, one can construct a conserved currentfor every continuous global symmetry of the action of a classical field theory. Integrals ofthe time component of the currents are conserved charges, and an infinitesimal symmetrytransformation is implemented on fields by the Poisson bracket of charge and field. Poissonbrackets and the correspondence principle provide a useful bridge to the quantum theoryin which finite group transformations are implemented through unitary transformations. Inthis section and the next we review this formalism in order to show that the conventions forsymmetries used are compatible with readers previous study of symmetries in quantumfield theory. The Noether formalism has other important applications in supersymmetryand supergravity which enter in later chapters of this book.

We assume that we are dealing with a system of scalar fields i (x),i= 1, . . . , n, whoseLagrangian density is a function of the fields and their first derivatives, as described by the

action(1.17). The EulerLagrange equations of motion are

x

L

i (x)

L i (x)

= 0. (1.62)

We define a generic infinitesimal symmetry variation of the fields by

i (x) AAi (x) , (1.63)in which theA are constant parameters. This formula includes as special cases the variousinternal and spacetime symmetries discussed in Sec.1.2. In these cases each Ai (x)islinear in i and given by a matrix or differential operator applied to i . Specifically,

internal AAi A(tA)i j j , (1.64)

spacetime AAi

a 12 (x x )

i . (1.65)

If(1.63) is a symmetry of the theory, then the action is invariant, and the variation of theLagrangian density is an explicit total derivative, i.e. L= A KA . This must hold forallfield configurations, not merely those which satisfy the equations of motion(1.62). Indetail, the variation of the Lagrangian density is

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1.3 Noether currents and charges 19

L A

L

iA

i + L i

Ai

= A KA . (1.66)

Using (1.62)we can rearrange (1.66)to readJ

A= 0, where JA is the Noether current

JA= Li

Ai +KA . (1.67)

This is a conserved current, by which we mean that J

A = 0 for all solutions of theequations of motion of the system.

We temporarily assume that the symmetry parameters are arbitrary functions A(x). Inthis case the variation of the action is

S=

dDx

L

i(

AAi ) + L

iAA

i

= dDxA KA+ (A) LiAi=

dDx(

A)J

A . (1.68)

The use of varying parameters A(x)is usually an efficient way to obtain the conservedNoether current. Note that surface terms from partial integrations in the manipulationsabove have been neglected because the field configurations are assumed to vanish at largespacetime distances.

For each conserved current one can define an integrated Noether charge, which is aconstant of the motion, i.e. independent of time. Suppose that we have a foliation ofMinkowski spacetime by a family of space-like (D1)-dimensional surfaces (). Aspace-like surface has a time-like normal vectorn at every point.9 Then for each Noethercurrent there is an integrated charge

QA=

()

d J

A(x) , (1.69)

which is conserved, that is independent offor all solutions that are suitably damped atinfinity. The simplest foliation is given by the family of equal-time surfaces (t), whichare flat(D 1)-dimensional hyperplanes with fixed x0 = t. In this case

QA= dD1 x J0A(x, t). (1.70)We now discuss the specific Noether currents for the linear internal and spacetime trans-

formations of this chapter. To simplify the discussion we restrict attention to systems withconventional scalar kinetic term, so the Lagrangian density is

9 The Minkowski space norm of any vectorv isv v . A vector is called space-like if its norm is positive,time-like if the norm is negative, and null for vanishing norm. d is a vector proportional to n whichrepresents a surface element orthogonal to().

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20 Scalar field theory and its symmetries

L= 12 i i V(i ). (1.71)

For internal symmetry, substitution of the first line of(1.64) in(1.67) gives

JA

= tA, (1.72)

provided that the potential Vis invariant, that is, V= i V i = i V(tA)ij j =0. Forspacetime translations, the index Aof the generic current(1.67) is replaced by the vectorindex , and the Noether current obtained from(1.65) and(1.67) is the conventional stresstensor (or energymomentum tensor)

T= + L. (1.73)

For Lorentz transformations, the index Abecomes the antisymmetric pair , and(1.65)and(1.67) give the current

M[]= x T+ xT . (1.74)

The conserved charges for internal, translations, and Lorentz transformations aredenoted byTA, P,and M[], respectively. They are given by

TA=

dD1 x J0A,

P=

dD1 x T0 ,

M[]= dD1 x M0[]. (1.75)Note that one does not need the detailed form of the stress tensor (1.73) to show that the

current(1.74) is conserved. The situation is indeed simpler. A current of the form (1.74)is conserved ifT is both conserved and symmetric, T= T . For many systems offields, such as the Dirac field discussed in the next chapter, the stress tensor given by theNoether procedure is conserved but not symmetric. In all cases one can modify T torestore symmetry.

In general the symmetry currents obtained by the Noether procedure are not unique.They can be modified by adding terms of the form JA

S

A, where SA

=SA. The added term is identically conserved, and the Noether charges are not changedby the addition since J0Ainvolves total spatial derivatives. It is frequently the case thatthe Noether currents of spacetime symmetries need to be improved by adding such termsin order to satisfy all desiderata, such as symmetry ofT . Another example is the stresstensor of the electromagnetic field which we discuss in Ch.4. The Noether stress tensoris conserved but neither gauge invariant nor symmetric. It can be made gauge invariantand symmetric by improvement, and the improved stress tensor is naturally selected by thecoupling of the electromagnetic field to gravity.

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1.4 Symmetries in the canonical formalism 21

1.4 Symmetries in the canonical formalism

In this section we discuss the implementation of symmetries in the canonical formalism atthe classical and quantum level. In much modern work in quantum field theory the canon-

ical formalism has been superseded by the use of Feynman path integrals, but canonicalmethods provide a quick pedagogical treatment of the issues of immediate concern. Forscalar fieldsi , the canonical coordinates at fixed time t= 0 are the field variables (x, 0)at each pointxof space, and the canonical momenta are given by (x, 0) = S/t(x, 0).For the action (1.71),the canonical momentum is i= 0i = 0i .

We consider explicitly the special cases of internal symmetry, space translations androtations in which the vector KA of (1.66)has vanishing time component. In these casesthe formula (1.70) for the Noether charge simplifies to

QA= dD1 x L

0iA

i

= dD1 xi Ai . (1.76)We work in this generic notation and ask readers to verify the results using (1.64)forinternal symmetry and (1.65)for space translations and rotations. The following results arealso valid for time translations and Lorentz boosts, although the manipulations needed area little more complicated.

We remind readers that the basic (equal-time) Poisson bracket is{i (x), j (y)} =ij

D1(x y). The Poisson bracket of two observables A( ,)and B( ,)is

{A,B} dD1 x A i (x) Bi (x) Ai (x) B i (x) . (1.77)Poisson brackets {A, {B, C}} obey the Jacobi identity

{A, {B, C} }+ {B, {C,A} }+ {C, {A,B}} = 0. (1.78)

It is now easy to see that the infinitesimal symmetry transformation Ai is generatedby its Poisson bracket with the Noether charge QA. In detail

Ai (x)= {QA, i (x)} =

dD1 y {j (y)Aj (y), i (x)}. (1.79)

Further, Poisson brackets of the conserved charges obey the Lie algebra of the symmetrygroup,

{QA, QB} = fAB CQC. (1.80)

Exercise 1.12 Readers are invited to verify (1.80) for internal symmetry, spatial

translations and rotations using the Noether charges given in (1.75) and the structure

constants of the subalgebra of Poincar transformations that do not change the time

coordinate.

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22 Scalar field theory and its symmetries

In the Poisson bracket formalism an iterated symmetry variation, 2with parametersB2followed by1with parametersA1, is given by

12i = A1B2{QA, {QB , i }}. (1.81)

Using the Jacobi identity(1.78) one easily obtains the commutator

[1, 2] i = fA B CA1 B2{QC, i }. (1.82)

Note that the symmetry parameters compose exactly as in(1.22) and(1.60).

Exercise 1.13 Derive from(1.79) that[A, B ] = fAB CC.

When Poisson brackets are available, we define symmetry operators as in (1.79). How-ever, in practice we streamline the notation by omitting explicit Poisson brackets and sim-ply use the notationAi to indicate the transformation rules. In the three cases of interestwe replaceAby TA, Pand M and write the explicit transformation rules as

TAi = (tA)ij j ,

Pi = i ,

M[]i = J[]i . (1.83)

In the rest of the book, the symmetries described are produced by these operators. Theoperators satisfy the algebra(1.80) (with Qreplaced by T), and(1.57) (with J replacedby M). The minus signs are necessary due to the steps in(1.20)(1.22) and(1.58)(1.60),

which change the order of the operators when they act on fields.

1.5 Quantum operators

In the quantum theory each classical observable becomes an operator10 in Hilbert space,which we denote by bold-faced type, e.g. A( ,) A(,). The correspondence prin-ciple states that, if the Poisson bracket of two observables gives a third observable, i.e.

{A,B

} =C, then the commutator of the corresponding operators is

[A, B

]qu

=iC. Note

that we use = 1.After quantization the symmetry operators become the operator commutators

Ai = i

QA,

i

qu,

[QA, QB]qu= i fAB CQC. (1.84)10 Subtleties such as operator ordering in the definition ofA(,)are ignored because they are not relevant for

the questions of interest to us.

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1.5 Quantum operators 23

The first relation implies that a finite group transformation with parameters A is imple-mented by the unitary transformation

i (x) eiAQAi (x)eiAQA = U()i (x). (1.85)

Here U() is a generic notation for a finite group transformation. More specifically, for

internal symmetryU()U()of (1.15),for translationsU()U(a)of (1.53),andfor Lorentz U() U()of (1.49).For finite transformations of an internal symmetrygroupG or the Poincar group, (1.85)reads

eiATA i (x)ei

ATA = ei (x) ,ei[a

P+ 12 M[]]i (x)ei[aP+ 12 M[]]= i (x+ a). (1.86)

Exercise 1.14 Verify the corresponding quantum operator relation

[1, 2] i = ifAB CA1B2

QC,

i

qu. (1.87)

It is also useful to verify the composition of finite group transformations. A transforma-tion with parametersA2 followed by another one with parameters

A1 is found by applying

(1.85)twice. One obtains

eiA1 QA ei

B2 QB

i (x)eiB2 QB ei

A1 QA = U(2)eiA1 QAi (x)eiA1 QA

= U(2)U(1)i (x). (1.88)This agrees with (1.23)for internal symmetry and its analogue for spacetime transfor-mations. Furthermore the group composition law for the product e i

B2 QB ei

A1 QA of unitary

operators is the same as for the classical operatorsU(2)U(1), so we do get a consistentrepresentation of the symmetry group.

Exercise 1.15 For a free scalar field, use (1.75)and (1.73)to express the Hamiltonian

H= P0 in terms of the canonical momenta and coordinates

H= 12

dD1 x

2 + ()2

. (1.89)

Check that this leads, using (1.3),to the quantum commutation relation

[H, ]qu= i= i0

=

dD1 p(2 )(D1)2E

Eei( pxEt)a( p) + ei( pxEt)a( p)

. (1.90)

Express the Hamiltonian in terms of a( p)and a( p):H= 12

dD1 p E(2 )(D1)2E

a( p)a( p) + a( p)a( p) . (1.91)

Using a( p), a( p)

qu

= (2 )32E( p)3( p p). (1.92)

you can then reobtain (1.90).

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24 Scalar field theory and its symmetries

1.6 The Lorentz group forD= 4

In Sec.1.2.3we introduced fields transforming in a general finite-dimensional representa-tion of the D-dimensional Lorentz group SO(D

1, 1)without giving a detailed descrip-

tion of these representations. There are several special features of the case D= 4, whichreduce the description of the finite-dimensional representations of SO(3, 1)to those of thefamiliar representations of SU(2), as we now review.

First we note that the proper subgroup of O(3, 1)is characterized by the two conditionsdet()= 1 and 00 1. The latter means that the sign of the time coordinate of anypoint is preserved. There are three disconnected components, which contain the product ofthe discrete transformations P,T, and P T, describing inversion in space and/or time, witha proper transformation. Lorentz transformations in the disconnected components satisfyeither det() = 1 or 00 1 or both.

Let m [] denote the matrices of a representation of the Lie algebra (1.34) for D= 4.

The six independent matrices consist of three spatial rotations Ji= 1

2 i j km[j k] (wherei j kis the alternating symbol with123= 1) and three boostsKi= m[0i]. It is a straight-forward and important exercise to show, using(1.34), that the linear combinations

Ik= 12 (Jk iKk) , k= 1, 2, 3 ,Ik= 12 (Jk+ iKk) , (1.93)

satisfy the commutation relations of two independent copies of the Lie algebra su(2), viz.

[Ii ,Ij ] = i j kIk,[Ii ,Ij ] = i j kIk,

[Ii ,I

j

] =0. (1.94)

Note that the operators(1.93) are defined for the complexified algebra. The complexifiedLie algebra ofso(3, 1) is thus related to su(2)su(2). As such all finite-dimensionalirreducible representations ofso(3, 1)are obtained11 from products of two representationsofsu(2)and thus classified by the pair of non-negative integers or half-integers (j, j ).The (j, j ) representation has dimension (2j+1)(2j +1). The representations (j, j )and (j , j) are inequivalent representations when j = j . The four-dimensional vectorrepresentation of(1.31) for D= 4 is denoted by( 12 , 12 ).Exercise 1.16 Verify(1.94).

11 To be a representation of the group SO(3, 1)the sum j+ j must be an integer. But in fact we are interestedin the covering group SL(2,C), since this allows the representations for fermions.

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The Dirac field 2

The Dirac equation is based on special representations of the Lorentz group, called spinorrepresentations. They were discovered by lie Cartan in 1913 [6,7].These representationsare very different from the D-dimensional defining representation discussed in Ch.1(andfrom tensor products of the defining representation). It is remarkable and profound thatspinor representations are realized in Nature and are not just mathematical curiosities. Theydescribe fermionic particles such as the electron and quarks, and they are required forsupersymmetry.

Our treatment of the Dirac equation should be viewed as part of our review of the basicnotions of relativistic field theory needed to move ahead in this book. Many readers willalready be familiar with the Dirac field. We advise them to skim this chapter to learnour conventions and then move on to Ch. 3 in which the Clifford algebra and Majoranaspinors are discussed. It is this material that is really essential in applications to SUSY andsupergravity later in the book.

2.1 The homomorphism ofSL(2,C) SO(3, 1)

Spinor representations exist for all spacetime dimensions D . We introduce them for D= 4.In the notation of Sec.1.6,a general spinor representation is labeled (j, j )with ja half-integer and j an integer, or vice versa. The fundamental spinor representations are ( 12 , 0)and its complex conjugate(0, 12 ). We now discuss the important 2:1 homomorphism ofthe group SL(2,C)of unimodular 2 2 complex matrices onto the connected componentof O(3, 1). It will lead to an explicit description of the ( 12 , 0)and (0,

12 )representations,

and it is central to the treatment of fermions in quantum field theory.First we note that a general 2 2 hermitian matrix can be parametrized as

x

= x0 + x3 x1 ix2x1 + ix2 x0 x3 (2.1)

and that det x = xx , which is the negative of the Minkowski norm of the4-vector x. This suggests a close relation between the linear space of hermitian 22matrices and four-dimensional Minkowski space. Indeed, there is an isomorphism betweenthese spaces, which we now elucidate.

For this purpose we introduce two complete sets of 2 2 matrices= ( , i ), = = ( , i ), (2.2)

25

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26 The Dirac field

where is the unit matrix, and the three Pauli matrices are

1=

0 11 0

, 2=

0 ii 0

, 3=

1 00 1

. (2.3)

The indexon the matrices of (2.2) is a Lorentz vector index, suggesting that the matrices

are 4-vectors. We will make this precise shortly, and, in anticipation, we will raise andlower these indices using the Minkowski metric.

Exercise 2.1 Show that

+ = 2 , (2.4)tr( ) = 2. (2.5)

Using the matricesandand(2.5), we easily find

x = x , x = 12tr(x) , (2.6)

which gives the explicit form of the isomorphism. Given the 4-vector x, one can constructthe associated matrixxfrom the first equation of (2.6), and one can obtainx fromxusingthe second equation.

Let Abe a matrix of SL(2,C), and consider the linear map

x x AxA . (2.7)

The associated 4-vectors are also linearly related, i.e. x (A)x , and(2.6) can beused to obtain the explicit form of the matrix (A),

(A)

= 1

2

tr(A A

) . (2.8)

Since the transformation(2.7) preserves det x, the Minkowski norm xx is invariant.This means that the matrix (A)satisfies(1.27); it must be a Lorentz transformation, andwe can write, using(1.24),

1 = (A). (2.9)

Since the group SL(2,C)is connected[8], lies in the connected component of O(3, 1),i.e. the proper Lorentz group.

Here are some exercises to help familiarize readers with this important homomorphism.

Exercise 2.2 Verify that (2.8) is a group homomorphism by showing that (AB)=(A)(B).

Exercise 2.3 Show that the kernel of the homomorphism consists of the matrices

( , ).

Exercise 2.4 Show that AA = 1 and AA= . This gives precisemeaning to the statement that the matrices and are 4-vectors.

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2.1 The homomorphism ofSL(2,C) SO(3, 1) 27

Let us introduce two sets of matrices that will turn out to be generators of the Lie algebraso(3, 1)in the ( 12 , 0)and (0,

12 )representations. We define them in terms of the matrices

,as

= 14 ( ) ,=

14 ( ) . (2.10)

Note that = . The finite Lorentz transformation (1.32)is then represented as

L() = e 12 , (2.11)L() = e 12 . (2.12)

Exercise 2.5 Show that

L() =L() =L()1 . (2.13)

Exercise 2.6 Use (2.4) to show that the commutator algebras of[

]and[

]are iso-morphic to (1.34).According to (1.93)and (1.94),the commutators of the representatives

Ik= 12 ( 12 i j ki j+ i0k)and Ik= 12 ( 12 i j ki j i0k)should satisfy (1.94).Check thatin this case Ik=0 and Ikindeed satisfy these commutation relations. This means that thematrices are the generators of the(0,

12 )representation.

The representation matrices L and L of (2.11)and (2.12)are directly related to theSL(2,C)map (2.7).For a Lorentz transformation 1 related to SL(2,C)matrix Aby(2.9),we can identify A=L1 and A =L .

We now argue that one can reach any proper Lorentz transformation from SL(2,C)viathe homomorphism (2.7).We show explicitly that the SL(2,C)matrix A

= L()1 with

03= 30= and other = 0 maps to a Lorentz boost in the 3-direction under(2.7).The parameter is conventionally called the rapidity. We use AA= ,which was proven in Ex.2.4, to obtain the corresponding Lorentz transformation. Using

A= e 12 3 we computeA A= cosh 3 sinh ,

A3A= sinh + 3 cosh ,A1,2A= 1,2 . (2.14)

On the right-hand side we recognize the matrix of the Lorentz boost

1 =

cosh 0 0 sinh 0 1 0 00 0 1 0

sinh 0 0 cosh

. (2.15)Similarly, one can consider the SL(2,C)matrices A= L()1 with non-vanishing i jonly, and show that (2.7)maps them to rotations of the spatiali with any desired rotationangle.

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28 The Dirac field

Exercise 2.7 Show that(2.7) works as claimed for spatial rotations about the 1, 2, 3axes.

Since any proper Lorentz transformation can be expressed as a product of rotations anda boost in the 3-direction, our discussion shows that the homomorphism is onto.

In any even spacetime dimension D= 2m the situation is similar to what we havedescribed in detail in four dimensions. The Lie algebraso(2m 1, 1)has a pair of inequiv-alent fundamental spinor representations of dimension 2m1. The products of exponentialsof matrices of either representation satisfy the composition rules of a Lie group that iscalled Spin(2m 1, 1). The latter is the universal covering group of the connected compo-nent of O(2m 1, 1), related by a 2 : 1 homomorphism.

For odd D= 2m+ 1 the situation of spinor representations is somewhat different andwill be described in Ch.3.

2.2 The Dirac equation

We follow, at least roughly, the historical development, and introduce the Dirac equationas a relativistic wave equation describing a particle with internal structure. Such a particleis described by a multi-component field, e.g. M(x). The indices M label the compo-nents of a column vector that transforms under some finite-dimensional representation ofthe Lorentz group. For the particular case of the Dirac field the representation is closelyrelated to the fundamental spinor representations we have just discussed. We again workfor general D, with special emphasis on D= 4.

Dirac postulated that the electron is described by a complex valued multi-component

field (x)called a spinor field, which satisfies the first order wave equation

/(x) (x) = m(x) . (2.16)

The quantities , = 0, 1, . . . ,D1, are a set of square matrices, which act on theindices of the spinor field . Applying the Dirac operator again, one finds

/2

= m2 ,12 { + } = m2 . (2.17)

If we require that the second order differential operator on the left is equal to thedAlembertian, then we fulfill the physical requirement of plane-wave solutions discussedin Ch.1. This means that the matrices must satisfy

{, } + = 2 , (2.18)

where is the identity matrix in the spinor indices.

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2.2 The Dirac equation 29

The condition (2.18)is the defining relation of the Clifford algebra associated with theLorentz group. The D matrices are the generating elements of the Clifford algebra, anda basis of the algebra consists of the unit matrix and all independent products of the gener-ators. The structure of the Clifford algebra is important, and it is discussed systematicallyfor general D in Sec.3.1.For immediate purposes, we note that there is an irreducible

representation by square matrices of dimension 2[D/2], where [x] is the largest integer lessthan or equal to x. The representation is unique up to equivalence for even D= 2m,and there are two inequivalent representations for odd dimensions. It is always possibleto choose a representation in which the spatial -matrices i , i= 1, 2, . . . ,D1, arehermitian and 0 is anti-hermitian. We will always work in such a representation, whichwe call a hermitian representation.

For generic D, the matrices are necessarily complex, so the spinor field must havecomplex components. After Diracs work, Majorana discovered that there are real repre-sentations in D= 4 dimensions, and it is now known that such Majorana representationsexist in dimensions D= 2, 3, 4 mod 8. In these dimensions, one may impose the con-

straint that the field (x)is real. The special case of Majorana spinors is very importantfor supersymmetry and supergravity, and it will be discussed in Sec.3.3. In this chapter weassume that (x)is complex.

There are various levels of interpretation of the components of (x). In the firstquantized formalism and in many classical applications, they are simply complex num-bers. However, when second quantization is introduced through the fermionic path inte-gral, the components of (x) are anti-commuting Grassmann variables, which satisfy{(x), (y)} = 0. Finally in the second quantized operator formalism, they are oper-ators in Hilbert space. The equations we write are valid in both of the first two situ-ations. Although the quantized theory appears rarely in this book, our basic formulasare compatible with the canonical formalism, positive Hilbert space metric and positiveenergy.

Equivalent representations of the Clifford algebra describe equivalent physics. Thereforethe physical implications of the Dirac formalism should be independent of the choice ofrepresentation. Indeed much of the physics can be deduced in a representation independentfashion, but an explicit representation is convenient for some purposes. We display oneuseful representation for D= 4, namely a Weyl representation in which the 4 4 havethe 2 2 Weyl matrices of (2.2)in off-diagonal blocks:

=

0

0

. (2.19)

There are block off-diagonal representations of this type in all even dimensions. This isshown in Ex.3.11of Ch.3.

One important fact about the Clifford algebra in general dimension Dis that the com-mutators

14

,

(2.20)

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30 The Dirac field

are generators of a 2[D/2]-dimensional representation of the Lie algebra of SO(D 1, 1).It is a straightforward exercise to show, using only the Clifford property (2.18), that thecommutator algebra of the matrices is isomorphic to(1.34). An explicit representationis not needed.

Exercise 2.8 Do this straightforward exercise mentioned above.

Exercise 2.9 Show, using only(2.18), that[ , ] = 2[ ] = .

In the Weyl representation given above, one sees that the matrices are block diag-onal with the 2-component and of(2.11) and(2.12) as the diagonal entries. Thefour-dimen