Quantization, Geometry and Noncommutative Structures in Mathematics and Physics

Description

This monograph presents various ongoing approaches to the vast topic of quantization, which is the process of forming a quantum mechanical system starting from a classical one, and discusses their numerous fruitful interactions with mathematics.The opening chapter introduces the various forms of quantization and their interactions with each other and with mathematics.A first approach to quantization, called deformation quantization, consists of viewing the Planck constant as a small parameter. This approach provides a deformation of the structure of the algebra of classical observables rather than a radical change in the nature of the observables. When symmetries come into play, deformation quantization needs to be merged with group actions, which is presented in chapter 2, by Simone Gutt.The noncommutativity arising from quantization is the main concern of noncommutative geometry. Allowing for the presence of symmetries requires working with principal fiber bundles in a non-commutative setup, where Hopf algebras appear naturally. This is the topic of chapter 3, by Christian Kassel. Nichols algebras, a special type of Hopf algebras, are the subject of chapter 4, by Nicolás Andruskiewitsch. The purely algebraic approaches given in the previous chapters do not take the geometry of space-time into account. For this purpose a special treatment using a more geometric point of view is required. An approach to field quantization on curved space-time, with applications to cosmology, is presented in chapter 5 in an account of the lectures of Abhay Ashtekar that brings a complementary point of view to non-commutativity.An alternative quantization procedure is known under the name of string theory. In chapter 6 its supersymmetric version is presented. Superstrings have drawn the attention of many mathematicians, due to its various fruitful interactions with algebraic geometry, some of which are described here. The remaining chapters discuss further topics, as the Batalin-Vilkovisky formalism and direct products of spectral triples.This volume addresses both physicists and mathematicians and serves as an introduction to ongoing research in very active areas of mathematics and physics at the border line between geometry, topology, algebra and quantum field theory. Front Matter ....Pages i-x Prelude: A General Overview (Alexander Cardona, Sylvie Paycha, Andrés F. Reyes Lega)....Pages 1-16 Deformation Quantization and Group Actions (Simone Gutt)....Pages 17-73 Principal Fiber Bundles in Non-commutative Geometry (Christian Kassel)....Pages 75-133 An Introduction to Nichols Algebras (Nicolás Andruskiewitsch)....Pages 135-195 Quantum Field Theory in Curved Space-Time (Andrés F. Reyes Lega)....Pages 197-220 An Introduction to Pure Spinor Superstring Theory (Nathan Berkovits, Humberto Gomez)....Pages 221-246 Introduction to Elliptic Fibrations (Mboyo Esole)....Pages 247-276 Batalin–Vilkovisky Formalism as a Theory of Integration for Polyvectors (Pierre J. Clavier, Viet Dang Nguyen)....Pages 277-292 Split Chern–Simons Theory in the BV-BFV Formalism (Alberto S. Cattaneo, Pavel Mnev, Konstantin Wernli)....Pages 293-324 Weighted Direct Product of Spectral Triples (Kevin Falk)....Pages 325-335 Back Matter ....Pages 337-341