'The book is primarily addressed to physicists. Nevertheless, as numerous examples are known in which exploration of the land where physics and mathematics overlap (and which quantum field theory definitely belongs to) resulted in important developments in mathematics, many mathematicians may also find this book interesting and even inspiring.'MathSciNetThis book is devoted to the subject of quantum field theory. It is divided into two volumes. The first can serve as a textbook on the main techniques and results of quantum field theory, while the second treats more recent developments, in particular the subject of quantum groups and noncommutative geometry, and their interrelation.The first volume is directed at graduate students who want to learn the basic facts about quantum field theory. It begins with a gentle introduction to classical field theory, including the standard model of particle physics, general relativity, and also supergravity. The transition to quantized fields is performed with path integral techniques, by means of which the one-loop renormalization of a self-interacting scalar quantum field, of quantum electrodynamics, and the asymptotic freedom of quantum chromodynamics is treated. In the last part of the first volume, the application of path integral methods to systems of quantum statistical mechanics is covered. The book ends with a rather detailed investigation of the fractional quantum Hall effect, and gives a stringent derivation of Laughlin's trial ground state wave function as an exact ground state.
The second volume covers more advanced themes. In particular Connes' noncommutative geometry is dealt with in some considerable detail; the presentation attempts to acquaint the physics community with the substantial achievements that have been reached by means of this approach towards the understanding of the elusive Higgs particle. The book also covers the subject of quantum groups and its application to the fractional quantum Hall effect, as it is for this paradigmatic physical system that noncommutative geometry and quantum groups can be brought together.

Quantum field theory certainly has to be counted as an essential part of our intel-lectual background. Many exceptional physicists have made decisive contributions to this subject. It has far reaching impact on disciplines such as elementary particle physics, quantum statistical mechanics, and also mathematics. The present text is a modest attempt to explain its main underlying concepts and major results.
Some comments on the kind of exposition should be made. According to my preferences, the book is mostly written in a rather condensed style, as I believe is appropriate for the present subject. I have abstained from writing a lengthy intro-duction to the topic of quantum field theory and what has been reached; the reason is that such an introduction would necessitate the whole vocabulary the newcomer wants to learn. Also I have refrained from further enlarging the text by exercises. Many calculations have only been indicated so that a lot of details remain to be filled in; hence there is room enough to test one's own status of comprehension.
Of course, quantum field theory as a highly developed discipline is also a 'vast field'. Therefore, the text is divided into two volumes; the first covers the more or less standard topics and can serve as a textbook, whereas the second is devoted to more recent developments. Both are organized in several parts; starred chapters and sections may be omitted on a first reading, but doubly starred ones may be entirely omitted since they are not necessary for the further understanding. Below a description of the contents of the book is given.
The first two parts are entirely devoted to the subject of classical relativis-tic field theory. In part I kinematical aspects are covered; we shall almost follow the historical development and introduce free fields, such as the Maxwell field de-scribing photons and the Dirac field for electrons. Their classical analogues are discussed by discretizing the fields on a cubical lattice. In the Dirac case it will be shown that already at the classical level Dirac wave functions must be anticommut-ing quantities. Furthermore the point is made that the famous doubling problem of fermions is void if the spinor field is put on a lattice of finite extent and antiperiodic boundary conditions are chosen in all spatial directions. For the discussion of the spin of the various fields some group theory is needed; in particular the importance of the Lorentz and the Poincaré group is emphasized in order to ensure the rela-tivistic invariance of the field equations; the relevant background is explained in an appendix on Lie groups and Lie algebras. We will end with a rather thorough treatment of Berezin's differential and integral calculus for classical anticommuting variables.
The second part II on classical relativistic field theory deals with dynamical prin-ciples. We shall rely on the Weyl principle to introduce local symmetries of both internal and external type. Here we will differ from common practice in that the Glashow-Salam-Weinberg theory of electroweak interactions, the standard model of particle physics, as well as general relativity and supergravity are all subsumed under the heading classical field theory. Indeed, what underlies all these theories are -more or less -geometric principles, for an understanding of which second quantization, i.e. quantum field theory, is not needed. In the last chapter a rather detailed account to cosmology is given, including a discussion of dark energy and quintessence.
Quantum field theory starts in the third part III, which is devoted to operator methods. Beginning with the canonical quantization of free fields, we shall then turn to a discourse on perturbation theory; it includes a discussion of the in-and out-picture, and the reduction formulas. Having available these techniques, we will derive the Wick theorem and can then compute cross sections of some elementary processes of quantum electrodynamics in lowest order.
In a more modern approach to quantum field theory though, operator methods are mostly avoided. Instead, one uses path integral techniques; the reason is that nonabelian gauge theories of the Yang-Mills type are extremely difficult to quantize directly since classically they are highly complicated constrained systems. Therefore we need path integral techniques, and in a first attempt we restrict ourselves to systems with a finite number of degrees of freedom. This is done in the fourth part IV, where the Feynman path integral for a nonrelativistic quantum mechanical particle is obtained from basic quantum mechanical principles. It comes in two versions, a Hamiltonian form and a Lagrangian form. The construction requires a latticization of the finite time interval considered; the continuum limit is to be performed at the very end. At least in all those cases we shall investigate, the continuum limit is well defined; this might prove to be useful in order to escape the measure theoretic intricacies connected with the continuum definition of the path integral.

Volume I

Preface V

I CLASSICAL RELATIVISTIC FIELD THEORY: KINEMATICAL ASPECTS 1

1. Relativistic Free Fields: Bosons 3

1.1 Maxwell's Equations in Relativistic Notation 3

1.2 Klein-Gordon Equation 9

1.3 Group Velocity and Special Relativity 11

1.4 Nonrelativistic Limit and the Schrödinger Equation 13

1.5 Classical Interpretation of Fields 14

1.6 Normal Coordinates 16

1.7 Quantized Harmonic Oscillator* 18

1.8 Quantization of the Klein-Gordon Field* 22

2. Lagrange Formalism for Fields 25

2.1 Functionals 25

2.2 Euler-Lagrange Equations for Fields 27

2.3 Variational Principle 29

      2.3.1 Nonrelativistic classical mechanics 30

      2.3.2 Relativistic classical mechanics 31

      2.3.3 Reparametrization invariance* 33

      2.3.4 Relativistic field theory 36

2.4 Appendix: Improper Bases* 37

3. Relativistic Invariance 45

3.1 Minkowski Space and its Symmetry Group 45

3.2 Transformation Law of Fields 54

3.3 Appendix: Lie Groups and Lie Algebras* 56

      3.3.1 Fundamentals of group theory 56

      3.3.2 Lie groups and their algebras 61

4. Special Relativity 75

4.1 Inertial Frames and Causality 75

4.2 Lengths and Time Intervals 77

4.3 Addition Theorem for Velocities 77

4.4 Rotating Frames 79

4.5 Accelerated Inertial Frames 81

4.6 Appendix: Product Integral 85

5. Relativistic Free Fields: Fermions 89

5.1 Dirac's Equation 89

5.2 Relativistic Invariance of Dirac's Equation 93

5.3 Variational Principle for the Dirac Equation 97

5.4 On the Origin of Gauge Invariance 99

5.5 Nonrelativistic Limit 100

5.6 'Classical' Interpretation of Fermions 102

      5.6.1 Fermions on a lattice 103

      5.6.2 Canonical quantization of fermions* 105

      5.6.3 The doubling problem 108

      5.6.4 Resolution of the doubling problem 109

5.7 Clifford Algebras and Spin Groups* 110

6. Relativistic Free Fields and Spin 119

6.1 Scalar Field 119

6.2 Dirac Field 120

6.3 Maxwell Field 122

6.4 Spin 124

      6.4.1 Representations of rotations 125

      6.4.2 Fields and their spin 128

6.5 Transformation Law of Fields and Induced Representations* 131

7. Neutral Fermions 133

7.1 Charge Conjugation 133

7.2 Majorana Spinors 135

7.3 Neutrinos 136

8. Symmetries and Conservation Laws 141

9. Differential and Integral Calculus for Anticommuting Variables 151

9.1 Real Grassmann Variables 151

9.2 Fourier Transformation 158

9.3 Complex Grassmann Variables 161

9.4 Appendix: Pfaffians* 164

II CLASSICAL RELATIVISTIC FIELD THEORY: DYNAMICAL ASPECTS 167

10. Dynamical Principles: Internal Symmetries 169

10.1 Internal Gauge Theories 169

      10.1.1 Gauge invariance in quantum mechanics 170

      10.1.2 Weyl's principle 172

10.2 Yang-Mills Theory 174

      10.2.1 Field equations 180

      10.2.2 Covariant current conservation 181

      10.2.3 Chern-Simons theory 185

10.3 Gauge Theories and Elementary Particle Physics 186

      10.3.1 Higgs-Kibble mechanism 187

      10.3.2 Glashow-Salam-Weinberg model 202

      10.3.3 Standard model 205

      10.3.4 Majorana neutrinos 210

      10.3.5 Appendix: SU(N) 214

11. Dynamical Principles: External Symmetries 223

11.1 Introduction 224

11.2 Dynamics of a Relativistic Point Particle in a Gravitational Field 226

      11.2.1 Geodesic equation 228

      11.2.2 Gravity and gauge invariance 231

      11.2.3 Newtonian limit 232

      11.2.4 Energy-momentum tensor of a relativistic point particle 234

11.3 Differential Geometry: A First Course 236

      11.3.1 Covariant differentiation 237

      11.3.2 Metric postulate 240

      11.3.3 Riemann's curvature tensor 242

      11.3.4 Bianchi identities and symmetry properties of the curvature tensor 243

      11.3.5 Parallel displacement 245

      11.3.6 Normal coordinates and the principle of equivalence 248

      11.3.7 Riemann normal coordinates 251

      11.3.8 Jacobi operator 253

      11.3.9 Vector analysis and integration 255

11.4 Einstein's Theory of Gravity 257

      11.4.1 Hilbert action 258

      11.4.2 Coupling of matter fields to gravity 263

      11.4.3 Gravity as a special relativistic field theory 267

      11.4.4 Linear approximation and gravitational waves 272

      11.4.5 Schwarzschild solution 274

11.5 Differential Geometry: A Second Course 279

      11.5.1 Coordinate transformations and the general linear group 279

      11.5.2 Geodetic function 283

      11.5.3 Geodetic parallel transport 286

      11.5.4 Coincidence limits* 287

      11.5.5 Riemann normal coordinates revisited 289

11.6 Differential Geometry: A Third Course 290

      11.6.1 Orthonormal frames 291

      11.6.2 Spin connexions 292

      11.6.3 Differential geometry as a gauge theory: Cartan connexions 298

11.7 Accelerated Observers and Inertial Systems** 301

11.8 Gravity as a Gauge Theory of the Poincaré Group 307

12. Supergravity 315

12.1 Super Poincaré Group 315

12.2 Supersymmetryand Differential Geometry: Cartan Connexions 319

12.3 Rarita-Schwinger Fermions 323

12.4 Supergravity as a Gauge Theory of the Super Poincaré Group 327

12.5 Summary 333

12.6 Appendix: Majorana Spinors in Higher Dimensions 333

      12.6.1 Properties of gamma matrices in four dimensions 335

      12.6.2 Fierz identities 335

      12.6.3 Properties of Majorana spinors in four dimensions 336

      12.6.4 Majorana spinors in higher dimensions 337

13. Cosmology 341

13.1 Gauss' Normal Coordinates 341

13.2 Symmetric Spaces 342

13.3 Realization of Maximally Symmetric Spaces 345

13.4 Robertson-Walker Metric 347

13.5 General Relativistic Hydrodynamics 348

13.6 Friedmann Equations 350

13.7 Models of the Universe 352

13.8 Present Status of the Universe 356

13.9 Observational Astronomy and Cosmological Parameters 359

13.10 Cosmological Constant Problem 360

13.11 Quintessence 362

13.12 Appendix: Geometric Optics in the Presence of Gravity 363

13.13 Appendix: Local Scale Invariance and Weyl Geometry* 366

      13.13.1 Weyl geometry 366

      13.13.2 Conformally invariant matter fields 371

III RELATIVISTIC QUANTUM FIELD THEORY: OPERATOR METHODS 377

14. Quantization of Free Fields 379

14.1 Scalar Field 379

      14.1.1 Canonical quantization 379

      14.1.2 Feynman propagator 381

14.2 Dirac Field 386

      14.2.1 'Canonical' quantization 386

      14.2.2 Feynman propagator 390

14.3 Maxwell Field 391

      14.3.1 'Canonical' quantization 391

      14.3.2 Feynman propagator 394

      14.3.3 Maxwell's theory as a constrained system* 395

15. Quantum Mechanical Perturbation Theory 397

15.1 Interactin Picture 397

15.2 Time Independent Perturbation Theory* 404

15.3 Formal Theory of Scattering* 412

15.4 In and Out Picture* 419

15.5 Gell-Mann & Low Formula* 421

15.6 Transcription to Quantum Field Theory* 423

15.7 Reduction Formulae* 425

      15.7.1 Scalar field 425

      15.7.2 Dirac field 429

      15.7.3 Maxwell field 430

16. Perturbative Quantum Electrodynamics 433

16.1 QED Hamiltonian in the Coulomb Gauge 433

16.2 QED Scattering Operator and States 436

16.3 Wick's Theorem 437

16.4 Scattering Matrix Elements 440

16.5 Feynman Rules for QED 445

16.6 Cross Sections 447

16.7 Elementary Processes 453

      16.7.1 Compton scattering 453

      16.7.2 Pair annihilation 457

      16.7.3 Moller scattering 458

      16.7.4 Electron-positron scattering 462

16.8 Appendix: Gamma 'Gymnastics' 462

IV NONRELATIVISTIC QUANTUM MECHANICS: FUNCTIONAL INTEGRAL METHODS 465

17. Path Integral Quantization 467

17.1 Feynman Path Integral 467

17.2 Gauge Invariance and the Midpoint Rule 477

17.3 Canonical Transformations and the Path Integral* 479

18. Path Integral Quantization of the Harmonic Oscillator 483

18.1 Harmonic Oscillator 483

18.2 Driven Harmonic Oscillator 487

19. Expectation Values of Operators 491

19.1 Expectation Values for a Finite Time Interval 491

19.2 Expectation Values for an Infinite Time Interval 494

19.3 Driven Harmonic Oscillator Revisited 497

20. Perturbative Methods 503

20.1 Perturbation Theory 503

20.2 Imaginary Time and Quantum Statistical Mechanics 504

20.3 Ground State Energy of the Quartic Anharmonic Oscillator 507

21. Nonperturbative Methods 511

21.1 Expansion in Terms of Planck's Constant 511

21.2 Small Deviations 513

21.3 Stationary Phase Approximation: Particle in an External Potential 516

21.4 Wentzel-Kramers-Brillouin and Stationary Phase Approximation: Compatibility 518

21.5 Stationary Phase Approximation: Charged Particle in an External Magnetic Field 521

21.6 Particle in an External Gravitational Field and Heat Kernel Expansion 526

21.7 Partition Functions and Functional Determinants 532

      21.7.1 Second order operators and the zeta function method 532

      21.7.2 First order operators and the zeta function method: Preliminary version 535

      21.7.3 First order operators and the zeta function method: Improved version 542

      21.7.4 First order operators and the zeta function method: Final version 545

22. Holomorphic Quantization 549

22.1 Coherent States: Bosons 549

22.2 Coherent State Path Integral: Bosons 553

22.3 Coherent States: Fermions 557

22.4 Path Integral for Fermions 562

22.5 Driven Harmonic Oscillator: Bosonic and Fermionic 565

23. Ghost Fermions 571

23.1 Schrödinger Representation 572

23.2 Vector Space Realization 575

23.3 Dirac States and Their Duals 576

23.4 Feynman Type Path Integral 580

23.5 Poisson Structures for Fermions 581

V RELATIVISTIC QUANTUM FIELD THEORY: FUNCTIONAL INTEGRAL METHODS 585

24. Quantum Fields on a Lattice 587

24.1 Lattice Bosons 587

24.2 Lattice Fermions 593

      24.2.1 Lattice fermions and the doubling problem 594

      24.2.2 Dirac-Kähler fermions 602

      24.2.3 Lattice fermions and the path integral 610

24.3 Lattice Gauge Fields 614

      24.3.1 Gauge theories on an infinite lattice 614

      24.3.2 Gauge theories on a finite lattice and the't Hooft algebra 618

25. Self Interacting Bosonic Quantum Field 621

25.1 Partition Function and Perturbation Theory 621

25.2 Effective Action 625

25.3 Effective Action and Perturbation Theory 629

25.4 Dimensional Regularization 632

25.5 Renormalization 644

25.6 'Cosmological' Constant 648

25.7 Renormalization Group Equations 652

25.8 Asymptotia 657

25.9 Coleman-Weinberg Effective Potential 659

      25.9.1 Stationary phase approximation 660

      25.9.2 Zeta-function evaluation 662

26. Quantum Electrodynamics 665

26.1 Path Integral for the Free Dirac Field 665

26.2 Path Integral for the Free Electromagnetic Field 666

26.3 Path Integral Representation of Quantum Electrodynamics 668

26.4 Ward's Identity 671

26.5 Regularization 674

      26.5.1 Regularization of the self-energy 676

      26.5.2 Regularization of the vacuum polarization 677

      26.5.3 Regularization of the vertex part 678

      26.5.4 Conclusion 679

26.6 Renormalization and the Callan-Symanzik Function 680

26.7 Application: Anomalous Magnetic Moment 682

26.8 Structure of the Physical Vacuum 684

      26.8.1 Ground state wave functional of the free electromagnetic field 684

      26.8.2 Casimir effect 688

      26.8.3 Euler-Heisenberg effective field theory 691

27. Quantum Chromodynamics 697

27.1 Faddeev-Popov Device 697

27.2 Becchi-Rouet-Stora Transformation 700

27.3 Zinn-Justin Equations 703

27.4 Feynman Rules 705

27.5 Regularization 708

27.6 Asymptotic Freedom 714

27.7 Conclusion 715

VI QUANTUM FIELD THEORY AT NONZERO TEMPERATURE 719

28. Nonrelativistic Second Quantization 721

28.1 Field Operators and the Fock Space Construction 723

28.2 Multilinear Algebra and the Fock Space Construction* 727

28.3 Second Quantized Form of the N-Particle Hamiltonian 729

29. Quantum Statistical Mechanics 733

29.1 Thermodynamics and the Partition Function 733

29.2Canonical Ensemble 736

29.3 Constant Mode Expansion of the Canonical Partition Function 738

30. Grand Canonical Ensemble 741

30.1 Path Integral Representation of Second Quantized Fields 741

30.2 Grand Canonical Partition Function as a Functional Integral 743

30.3 Perturbation Theory in Direct Space 745

30.4 Perturbation Theory in Fourier Space 747

30.5 Connect with Thermodynamic Quantities 749

30.6 Noninteracting Case 750

31. Bose-Einstein Condensation 753

31.1 Spontaneous Symmetry Breaking and Condensation 753

31.2 Condensation and Feynman Rules 756

31.3 Schwinger-Dyson-Beliaev equations 760

31.4 Hugenholtz-Pines Relation 761

31.5 Nonperturbative Approach 763

31.6 Superfluidity 767

32. Superconductivity 771

32.1 Introduction 771

32.2 Effective Action 772

33. Relativistic Quantum Field Theory at Nonzero Temperature 781

33.1 Relativistic Ideal Gas 781

      33.1.1 Bosons 781

      33.1.2 Fermions 782

      33.1.3 Gauge bosons 785

33.2 Symmetry Restoration 786

34. Fractional Quantum Hall Effect 789

34.1 Classical Hall Effect 790

34.2 Landau Problem 791

34.3 Second Quantization and the Integer Effect 794

34.4 Chern-Simons Theory and Ginzburg-Landau Effective Theory 797

34.5 Laughlin Theory 798

34.6 Excitations 804

34.7 Braid Statistics 808

      34.7.1 Path integral quantization in a non-simply connected space 808

      34.7.2 Artin's braid group 811

34.8 Chern-Simons Theory and Braid Statistics 814

34.9 Edge Excitations 816

34.10 Virasoro and Kac-Moody Algebras 817

34.11 Laughlin Ground State and Vertex Operators 827

34.12 Laughlin's Trial Wave Function as an Exact Ground State 832

      34.12.1 Canonical properties of Chern-Simons theory 834

      34.12.2 Quantization 838

      34.12.3 Regularization 845

      34.12.4 Excitations 847

Index I-1

Volume II

Preface V

VII SYMMETRIES AND CANONICAL FORMALISM 851

35. Hamiltonian Formalism and Symplectic Geometry 853

35.1 Introduction 853

35.2 Canonical Transformations 858

35.3 Generating Functions 864

35.4 Vector Fields as Generators of Diffeomorphisms 866

35.5 One Parameter Subgroups of Canonical Transformations 869

36. Conventional Symmetries 875

36.1 Symmetries and Conservation Laws: Lagrange Formalism 875

36.2 Symmetries and Conservation Laws: Hamilton Formalism 881

36.3 Gauge Invariance 887

37. Accidental Symmetries 891

37.1 Hydrogen Atom or Quantum Mechanical Kepler Problem 891

37.2 Three-Dimensional Harmonic Oscillator 895

38. Anomalous Symmetries 901

38.1 Generalized Noether Charges and Anomalies 901

38.2 Cochains and Boundaries 904

38.3 BRS Operator 907

38.4 Landau Problem: 1. Variation 909

38.5 Cohomology of Lie Groups and Algebras* 920

      38.5.1 Cohomology of Lie groups 920

      38.5.2 Cohomology of Lie algebras 925

      38.5.3 Heisenberg-Weyl group 929

VIII GAUGE SYMMETRIES AND CONSTRAINED SYSTEMS 933

39. Constrained Systems and Symplectic Reduction 935

39.1 Linear Reduction 935

39.2 Nonlinear Reduction 937

39.3 Constraints and Reduction 940

39.4 Symmetry and Marsden-Weinstein Reduction 942

39.5 Dirac Brackets 945

39.6 Dirac Brackets and Poisson Structures 947

40. Quantum Reduction of Constrained Systems 949

40.1 Gauge Theories as Constrained Systems 949

40.2 Finite Dimensional Analogue of Gauge Theories 953

40.3 Quantum Mechanical Time Evolution of Constrained Systems 957

40.4 Quantization of Constrained Systems 959

40.5 Geometry of Systems with First Class Constraints 961

40.6 Geometry of Yang-Mills Fields 965

40.7 Yang-Mills Theory and Poisson-Dirac Brackets 969

40.8 Faddeev's Path Integral Formula for Constrained Systems 975

41. BRS Quantization of Constrained Systems 981

41.1 BRS Invariance 981

41.2 Extended BRS Formalism 984

41.3 Fradkin-Vilkovisky Theorem 987

41.4 Zinn-Justin Equations 990

IX WEYL QUANTIZATION 993

42. Weyl Quantization of Bosons 995

42.1 Weyl Order: Real Representation 995

42.2 Weyl Order: Complex Representation 1000

42.3 Groenewold-Moyal Bracket 1003

42.4 Generalized Weyl Formalism 1006

42.5 Berezin's Path Integral 1008

42.6 Other Ordering Schemes and Symbols: Real Representation 1016

42.7 Other Ordering Schemes and Symbols: Complex Representation 1019

42.8 Generating Functions and Their Quantum Counterparts 1022

42.9 Weyl Ordering and the Path Integral 1025

42.10 Appendix: Pseudodifferential Operators and Weyl Quantization 1026

      42.10.1 Introduction 1027

      42.10.2 Symbol calculus 1028

      42.10.3 Symbol classes 1031

      42.10.4 Elliptic pseudodifferential operators 1036

      42.10.5 Elliptic pseudodifferential operators on manifolds 1037

43. Weyl Quantization of Bosons and Canonical Transformations 1039

43.1 Symplectic Vector Spaces and Symplectic Transformations 1039

43.2 Complex Structures and Complexifications 1041

43.3 Complex Realization of the Symplectic Group 1046

43.4 Heisenberg-Weyl Group and Quantization 1049

43.5 Metaplectic Operator 1051

43.6 Bargmann Transform 1057

43.7 Symplectic Transformations and Quantum Mechanics 1058

44. Geometric Quantization and Spin 1067

44.1 Generalized Coherent States: SU(2) 1067

44.2 Coherent States: Noncompact Picture 1075

44.3 Coherent State Path Integral: Noncompact Picture 1077

44.4 Coherent States: Compact Picture 1079

44.5 Coherent State Path Integral: Compact Picture 1080

44.6 Spin Models 1085

      44.6.1 Ferromagnets 1086

      44.6.2 Antiferromagnets: Ground state 1087

      44.6.3 Antiferromagnets: Quadratic approximation 1090

      44.6.4 Antiferromagnets and Chern-Simons term 1093

      44.6.5 Topological solitons 1096

      44.6.6 Topological solitons and Hopf fibration 1098

      44.6.7 Hopf invariant 1101

      44.6.8 Hopf invariant and Chern-Simons term 1102

45. Weyl Quantization of Fermions 1107

45.1 Canonical Symmetry: Weyl and Spinorial Operator 1107

45.2 Weyl Ordered Operators 1110

45.3 Fermionic Heisenberg-Weyl Transformation of Wave Functions 1111

45.4 Antiholomorphic Representation 1112

45.5 Complex Realization of Rotations 1114

45.6 Quantum Mechanical Representation of Canonical Transformations 1116

45.7 Fermionc Weyl Formalism 1118

45.8 Groenewold-Moyal Bracket for Fermions 1120

45.9 Generalized Weyl Formalism 1122

45.10 Berezin's Path Integral for Fermions 1124

45.11 Partition Function in the Weyl Approach 1130

X ANOMALIES IN QUANTUM FIELD THEORY 1137

46. Anomalies and Index Theorems 1139

46.1 Axial Anomaly 1139

      46.1.1 Chiral fermions 1139

      46.1.2 Quantization of chiral fermions 1141

      46.1.3 Computation of the axial anomaly: Heat kernel regularization 1143

      46.1.4 Computation of the axial anomaly: Zeta-function regularization 1147

      46.1.5 Physical origin of the axial anomaly 1151

      46.1.6 Axial anomaly and the supersymmetric proof of the index theorem 1160

46.2 Axial Gauge Anomaly 1169

46.3 Physical Consequences of Anomalies 1173

46.4 Anomalies and Geometry 1174

      46.4.1 Chern-Weil theory 1174

      46.4.2 Algebraic approach to anomalies 1184

      46.4.3 Anomalies and cohomology 1186

46.5 Gravitational Anomalies 1193

      46.5.1 Supersymmetric proof of the index theorem: External case 1194

      46.5.2 Index theorems for the classical complexes 1199

46.6 Supersymmetric Relativistic Point Particle with Spin 1210

46.7 Appendix: Spin and Spinc Structures 1214

      46.7.1 Orientations 1215

      46.7.2 Spin structures 1216

      46.7.3 Spinc structures 1217

46.8 Appendix: Geometric Gauge Fixing Conditions 1219

47. Integrated Anomalies 1227

47.1 Pure Non-Abelian Chern-Simons Theory 1227

      47.1.1 Classical properties 1228

      47.1.2 Quantization 1231

47.2 Nonabelian Schwinger Model 1236

47.3 Chiral Nonabelian Schwinger Model 1245

XI NONCOMMUTATIVE GEOMETRY 1251

48. Noncommutative Geometry: Algebraic Tools 1253

48.1 Basic Algebraic Tools 1254

      48.1.1 Modules 1254

      48.1.2 Algebras 1259

48.2 Noncommutative Differential Geometry 1268

      48.2.1 Universal differential graded algebras 1268

      48.2.2 Universal connexions 1273

      48.2.3 Hermitian connexions 1275

      48.2.4 Gauge transformations1276

48.3 Cyclic Cohomology 1277

      48.3.1 Elements of homological algebra 1277

      48.3.2 Noncommutative integral calculus 1281

48.4 Graded Cyclic Cohomology 1285

      48.4.1 Z/2-graded modules and algebras 1285

      48.4.2 Cyclic cohomology for Z/2-graded algebras 1290

48.5 Berezin Integration and Graded Cyclic Cohomology* 1296

49. Noncommutative Geometry: Analytic Tools 1321

49.1 Spectral Triples 1321

49.2 Spectral Triples and Universal Differential Calculus 1325

49.3 Dixmier Trace 1329

49.4 Wodzicki Residue 1335

49.5 Real Structures 1342

      49.5.1 Classification of Clifford algebras 1342

      49.5.2 Charge conjugation in four dimensions 1346

      49.5.3 Real structure in arbitrary dimension 1348

      49.5.4 Spectral triples with a real structure 1349

49.6 Order One and Orientation 1349

49.7 Regularity and Finiteness 1352

49.8 Axiomatic Foundation 1353

49.9 Internal Symmetries 1354

49.10 Appendix: Review of C*-Algebra Basics 1358

      49.10.1 Banach spaces 1358

      49.10.2 Continuous linear operators 1361

      49.10.3 Banach algebras 1362

      49.10.4 C*-algebras 1364

50. Noncommutative Geometry: Particle Physics 1367

50.1 Fermionic Action 1368

50.2 Bosonic Action 1369

      50.2.1 Spectral action 1370

      50.2.2 Spectral action for Yang-Mills fields coupled to gravity 1373

      50.2.3 Spectral action for a Higgs field coupled to gravity 1376

      50.2.4 Spectral action and cosmology 1382

      50.2.5 Spectral action and nonzero torsion 1386

50.3 Outlook 1389

51. A Glance at Noncommutative Quantum Field Theory 1391

51.1 Noncommutative Spaces 1391

      51.1.1 Noncommutative spacetime 1391

      51.1.2 Noncommutative 2-dimensional torus 1393

51.2 Landau Problem: 2. Variation 1397

51.3 Noncommutative Quantum Field Theory 1399

      51.3.1 Scalar field theory on noncommutative spacetime 1399

      51.3.2 Gauge theory on noncommutative spacetime 1402

      51.3.3 Scalar field theory on a noncommutative torus 1404

      51.3.4 Gauge theory on a noncommutative torus 1407

XII QUANTUM GROUPS 1413

52. Hopf Algebras 1415

52.1 Motivation 1415

52.2 Algebras 1417

52.3 Coalgebras 1422

52.4 Bialgebras 1427

52.5 Hopf Algebras 1430

52.6 Hopf*-Algebras 1441

53. Quasitriangular Hopf Algebras 1447

53.1 Almost Cocommutative Hopf Algebras 1447

53.2 Quasitriangular Hopf Algebras 1451

53.3 Ribbon Hopf Algebras 1456

53.4 Matrix Realizations of the Universal R-Operator and Artin's Braid Group 1462

53.5 Quasitriangular Hopf Algebras and*-Structures 1465

54. Quantum Groups: Basic Example 1467

54.1 Motivation 1467

54.2 Uq(sl2) as an Algebra

54.3 Uq(sl2) as a Hopf Algebra 14691470

54.4 Uq(sl2) as a Quasitriangular Hopf Algebra 1473

54.5 Uq(sl2) as a Quasitriangular Ribbon Hopf Algebra 1475

54.6 Elements of q-Analysis 1476

54.7 Real Forms of Uq(sl2) 1479

54.8 Representation Theory of Uq(sl2) 1483

      54.8.1 Deformation parameter not a root of unity 1483

      54.8.2 Deformation parameter a primitive root of unity 1485

      54.8.3 Unitarity 1488

      54.8.4 Quasitriangularity 1491

      54.8.5 Example: Deformed harmonic oscillator 1495

XIII NONCOMMUTATIVE GEOMETRY AND QUANTUM GROUPS 1497

55. Quantum Groups and the Noncommutative Torus 1499

55.1 Landau Problem: 3. Variation 1500

55.2 Weyl Quantization and Quantum Groups 1501

56. Quantum Hall Effect with Realistic Boundary Conditions 1505

Index I-1

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