Noncommutative spacetimes : symmetries in noncommutative geometry and field theory / P. Aschieri ... [and others].

Format:

Book

Author/Creator:

Aschieri, P. (Paolo)

Series:

Lecture notes in physics ; 774.

Lecture notes in physics, 0075-8450 ; 774

Language:

English

Subjects (All):

Noncommutative differential geometry.

Field theory (Physics).

Physical Description:

xiv, 199 pages : illustrations ; 24 cm.

Place of Publication:

Berlin ; New York : Springer, [2009]

Summary:

There are many approaches to noncommutative geometry and to its use in physics. This volume addresses the subject by combining the deformation quantization approach, based on the notion of star-product, and the deformed quantum symmetries methods, based on the theory of quantum groups.

The aim of this work is to give an introduction to this topic and to prepare the reader to enter the research field quickly. The order of the chapters is "physics first" : the mathematics follows from the physical motivations (e.g. gauge field theories) in order to strengthen the physical intuition. The New mathematical tools, in turn, are used to explore further physical insights. A last chapter has been added to briefly trace Julius Wess' (1934-2007) seminal work in the field.

Contents:

Part I Deformed Field Theory: Physical Aspects

1 Differential Calculus and Gauge Transformations on a Deformed Space / Julius Wess 3

1.1 Introduction 3

1.2 The algebra 5

1.3 The star product 8

1.4 A deformed differential Calculus 10

1.5 A deformed algebra of differential operators 11

1.6 Gauge transformations 13

1.7 Diffeomorphism 16

1.8 Conclusion 17

1.9 Appendix 17

References 20

2 Deformed Gauge Theories / Julius Wess 23

2.1 Introduction 23

2.2 Gauge transformations 24

2.3 Hopf algebra techniques 26

2.4 Field equations 27

2.5 Matter fields 30

2.6 Examples 31

References 36

3 Einstein Gravity on Deformed Spaces / Julius Wess 39

3.1 Introduction 39

3.2 Differential operators 40

3.3 Tensor fields 43

3.4 Einstein-Hilbert gravity 46

References 51

4 Deformed Gauge Theory: Twist Versus Seiberg-Witten Approach / Marija Dimitrijević 53

4.1 Introduction 53

4.2 θ-deformed space 54

4.3 Twisted gauge theory 59

4.3.1 Gauge transformations 59

4.3.2 Field strength tensor 61

4.3.3 Equations of motion 62

4.4 Seiberg-Witten gauge theory 64

4.4.1 Enveloping algebra approach 65

4.4.2 Seiberg-Witten map 66

4.5 Comments 69

References 70

5 Another Example of Noncommutative Spaces: κ-Deformed Space / Marija Dimitrijević 73

5.1 Introduction 73

5.2 κ-deformed space 74

5.3 Star product approach 78

5.4 Gauge theory on the κ-deformed space 79

5.5 Gauge fields 81

5.6 Integral and the action 82

References 84

Part II Noncommutative Geometries: Foundations and Applications

6 Noncommutative Spaces / Fedele Lizzi 89

6.1 Commutative geometry (and topology) 89

6.1.1 Topology and algebras 90

6.1.2 Reconstructing the space from the algebra 92

6.1.3 Geometrical structures 94

6.2 Noncommutative spaces 95

6.2.1 The GNS construction 96

6.2.2 Commutative and noncommutative spaces 99

6.2.3 Deformations of spaces 100

6.3 The noncommuntative geometry of canonical Commutation relations 101

6.4 Final remarks 108

References 108

7 Quantum Groups, Quantum Lie Algebras and Twists / Paolo Aschieri 111

7.1 Introduction 111

7.2 Hopf algebras from groups 112

7.3 Quantum groups and SLq(2) 114

7.4 Universal enveloping algebras and Uq(sl(2) 117

7.5 Duality 119

7.6 Quantum Lie algebra 122

7.7 Deformation by twist and quantum Poincaré Lie algebra 124

Appendix 124

7.8 Algebras, coalgebras, and Hopf algebras 127

7.9 Hopf algebra twists 130

References 131

8 Noncommutative Symmetries and Gravity / Paolo Aschieri 133

8.1 Introduction 133

8.2 Deformation by twists 135

8.2.1 The twist F 136

8.2.2 *-Tensor algebra 139

8.2.3 *-Diffeomorphism symmetry 143

8.2.3.1 Relation between UΞ* and UΞ $$ 149

8.2.4 Twisted versus spontaneously broken symmetries 150

8.3 Poincaré symmetry 152

8.3.1 *-Poincaré algebra 152

8.3.2 Twisted Poincaré algebra 155

8.4 Covariant derivative, torsion, and curvature 156

8.5 Metric and Einstein equations 158

Appendix 159

8.6 Differential operators and vector fields 159

8.7 Proof that the coproduct δ* is coassociative 161

8.8 Proof that the bracket [u v]* is the adjoint action 162

References 162

9 Twist Deformations of Quantum Integrable Spin Chains / Petr Kulish 165

9.1 Introduction 165

9.2 Algebraic Bethe ansatz (QISM) 170

9.2.1 QISM for the XXX model 171

9.2.1.1 The Yangian Y (sl(2) 175

9.2.1.2 Higher spins and generalizations 175

9.2.2 Anisotropic XXZ spin chain 176

9.3 Twists and QISM 179

9.3.1 Jordanian twist 181

9.3.2 Abelian twist 182

9.3.3 Generalities on twist transformations 183

9.3.4 Coboundary twists and the jordanian deformation 185

9.4 Conclusions 187

References 187

10 The Noncommutative Geometry of Julius Wess / Paolo Aschieri 189

References 193.

Notes:

Includes bibliographical references and index.

ISBN:

9783540897927

3540897925

OCLC:

302080524