One of the basic problems of quantum field theories, namely the ultraviolet infinities, is connected to the fact that we consider point particles and spaces which allow the definition of points. The infinities which arise in the gauge theories of the strong and electroweak interactions can be removed by a procedure called renormalization. However, the quantum field theory of gravity is not renormalizable. This is the reason why we say that quantum mechanics and general relativity are not compatible. Although supersymmetry improves its renormalization properties, the theory of supergravity is still not renormalizable.
ISBN 10: 3-86573-034-5ISBN 13: 978-3-86573-034-3 142 S. 18 EUR. 2004 (Diss.) |

**1 Introduction . . . . . 52 Noncommutative spaces and star products . . . . . 17**

2.1 Noncommutative Spaces

2.2 Star Product

2.3 Examples

2.3.1 Canonical case

2.3.2 Lie algebra case

2.3.3 Quantum space case

2.4 Noncommutative spaces from string theory

3.1 Algebraic description

3.2 Generalities on super Poisson brackets and star products

3.3 Deformed superspace and classical supertranslation symmetry

3.4 Non(anti)commutative superspaces from superstring theory

4.1 Basic relations and symmetries

4.2 Covariant derivatives and superfields

4.3 Star product

4.4 Gauge theory and restriction of the gauge group

4.5 Construction of the Seiberg-Witten map in terms of component fields

4.6 Construction of the Seiberg-Witten map in terms of superfields

4.7 Seiberg-Witten map on Minkowski space

4.7.1 No-go theorem

4.7.2 Nonchiral solution

4.7.3 Nonlocal solution

4.7.4 Nonsupersymmetric solution

4.7.5 Nonsupersymmetric noncommutative electrodynamics

4.8 Seiberg-Witten map on Euclidean space

5.1 Basic relations, symmetries and fields

5.2 Star product

5.3 Gauge theory and restriction of the gauge group

5.4 Construction of the Seiberg-Witten map in terms of component fields

5.5 Construction of the Seiberg-Witten map in terms of superfields

5.5.1 Seiberg-Witten maps for the gauge parameters

5.5.2 Seiberg-Witten maps for the matter and gauge fields

5.5.3 Seiberg-Witten maps for the field strengths and Yang-Mills action

6.1 Basic relations, symmetries and star product

6.2 Construction of the Seiberg-Witten maps in terms of superfields

6.2.1 Seiberg-Witten maps for gauge parameters

6.2.2 Seiberg-Witten maps for matter and gauge fields

A Conventions and superspace notation . . . . . 101

A.1 Metrics and the total antisymmetric tensor

A.2 Sigma matrices and SL(2,C) generators

A.3 Weyl spinors

A.4 N =1, d=4 Minkowski superspace

A.5 N =1, d=4 supersymmetry algebra

A.6 Representation of the supersymmetry generators

A.7 Covariant spinor derivatives

A.8 (Anti)chiral projectors

A.9 Euclidean superspace