Noncommutative Spacetimes

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

There are many approaches to noncommutative geometry and its use in physics, the ? operator algebra and C -algebra one, the deformation quantization one, the qu- tum group one, and the matrix algebra/fuzzy geometry one. This volume introduces and develops the subject by presenting in particular the ideas and methods recently pursued by Julius Wess and his group. These methods combine the deformation quantization approach based on the - tion of star product and the deformed (quantum) symmetries methods based on the theory of quantum groups. The merging of these two techniques has proven very fruitful in order to formulate ?eld theories on noncommutative spaces. The aim of the book is to give an introduction to these topics and to prepare the reader to enter the research ?eld himself/herself. This has developed from the constant interest of Prof. W. Beiglboeck, editor of LNP, in this project, and from the authors experience in conferences and schools on the subject, especially from their interaction with students and young researchers. In fact quite a few chapters in the book were written with a double purpose, on the one hand as contributions for school or conference proceedings and on the other handaschaptersforthepresentbook.Thesearenowharmonizedandcomplemented by a couple of contributions that have been written to provide a wider background, to widen the scope, and to underline the power of our methods.

Includes supplementary material: sn.pub/extras

Inhalt
Deformed Field Theory: Physical Aspects.- Differential Calculus and Gauge Transformations on a Deformed Space.- Deformed Gauge Theories.- Einstein Gravity on Deformed Spaces.- Deformed Gauge Theory: Twist Versus Seiberg#x2013;Witten Approach.- Another Example of Noncommutative Spaces: #x03BA;-Deformed Space.- Noncommutative Geometries: Foundations and Applications.- Noncommutative Spaces.- Quantum Groups, Quantum Lie Algebras, and Twists.- Noncommutative Symmetries and Gravity.- Twist Deformations of Quantum Integrable Spin Chains.- The Noncommutative Geometry of Julius Wess.