Quantum Field Theory Competitive Models

For more than 70 years, quantum field theory (QFT) can be seen as a driving force in the development of theoretical physics. The developed ideas and techniques of QFT have been successfully applied, in particular, within the phenomenological description of particle physics and solid state physics. Equally fascinating is the fruitful impact which QFT had in rather remote areas of mathematics, like Gromov-Witten and Donaldson-Witten invariants of low dimensional manifolds and for modular forms in relation to string theory. More recent developments in QFT also attack the problem to formulate a quantum version of gravity. However, there is no 'QFT as such', but instead there are only various mathematical approaches, aiming to make the basic ideas of QFT more rigorous. Such a rigorous understanding seems indispensable, in particular, to get a better understanding of how a physically reasonable quantum theory of gravity may look like. The present book features some of the different approaches, different physical viewpoints and techniques used to make the notion of quantum field theory more precise. This concerns algebraic, analytic, geometric, and stochastic aspects. For example, there will be discussed deformation theory, and the holographic AdS/CFT correspondence. The book also contains more recent developments like the use of category theory and topos theoretic methods to describe QFT. This volume emerged from the 3rd 'Blaubeuren Workshop: Recent Developments in Quantum Field Theory', held in July 2007 at the Max Planck Institute of Mathematics in the Sciences in Leipzig/Germany. All of the contributions to the volume are peer reviewed and committed to the idea of this workshop series: 'To bring together outstanding experts working in the field of mathematics and physics to discuss in an open atmosphere the fundamental questions at the frontier of theoretical physics'.