Noncommutative spacetimes. Symmetries in noncommutative geometry and field theory
In the first nine chapters, this book records approach to non-commutative differential geometry and its applications to physics developed over the past few years by the group led by Julius Wess. The final chapter summarises main themes of non-commutative geometry pursued by Julius Wess from 1989 until his death in 2007. The main theme of the book is physics and symmetries of spaces deformed by the twist or the \(*\)-product deformations (Moyal deformation of algebras of functions). The book is split into two parts. The first part is devoted to physical aspects of deformed field theory (chapters 1--5), the second part is devoted to foundations and applications of non-commutative geometries (chapters 6--9). The first three chapters, authored by Julius Wess, initially describe the \(*\)-product, differential calculus, differential operators and gauge transformations on deformed spaces. Subsequently gauge theory and the Einstein gravity are developed on deformed spaces. Chapters 4 and 5 are authored by Marija Dimitrijević and compare twisted gauge theory developed in Chapters 1--3 with the Seiberg-Witten approach to non-commutative field theory, and also describe gauge theory on the \(\kappa\)-deformed space. The second part starts with Chapter 6, written by Fedele Lizzi. This chapter is devoted to foundations of non-commutative geometry, and first describes the algebras of functions on classical spaces and then their deformation in terms of the Moyal \(*\)-product. Chapters 7 and 8 are written by Paolo Aschieri and start with an introduction to quantum groups (with particular attention paid to the deformations of Hopf algebras by twists) and culminate with the description of the twisted Poincaré algebra and the \(*\)-deformation of Einstein's gravity. Chapter 9, authored by Petr Kulish, provides one with a brief introduction to the Quantum Inverse Scattering Method (of solving integrable systems in quantum mechanics, in particular spin chains), and then analyses twists of integrable spin chains. Some versions of some of the material presented in the book have been published elsewhere. The authors write: ``Chapters 1--3 are, respectively, based on [\textit{J. Wess}, Gen. Relativ. Gravitation 39, No.~8, 1121--1134 (2007; Zbl 1181.83144), J.\ Phys.\ Conf.\ Ser.\ 53, 752 (2006), and SIGMA, Symmetry Integrability Geom. Methods Appl. 2, Paper 089, 9 pages, electronic only (2006; Zbl 1188.83069)]. A preliminary version of Chaps.\ 4 and 5, originally coauthored with F.\ Meyer, appeared, respectively, in the proceedings of the 4th and 3rd Summer School in Modern Mathematical Physics, Belgrade and Zlatibor, published in SFIN, Institute of Physics, Zemun (Serbia). Chapter 7 is an extended version of contribution for the proceedings of the Second Modave Summer School in Mathematical Physics, International Solvay Institutes. Chapter 8 is an extended version of J.\ Phys.\ Conf.\ Ser.\ 53, 799 (2006).''
scientific article; zbMATH DE number 5379961