Overview
- Inspires especially young researchers who will get a global picture and technical tools at the same time
- Bridges different active areas of research in mathematics and physics
- Written with a notable pedagogical effort
- Includes supplementary material: sn.pub/extras
Part of the book series: Mathematical Physics Studies (MPST)
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Table of contents (10 chapters)
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Front Matter
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Back Matter
Keywords
- deformation quantization
- noncommutative geometry
- Poisson manifold
- principal fibre boundles
- group actions
- Elloptic fibrations
- Poisson geometry
- Quantum groups
- Fedosov's star products
- Hopf algebras
- Hopf Galois extensions
- Spectral triples
- Toeplitz operators
- Nichols algebras
- Pure spinor superstrings
- Kodaira-Neron classification
- Miranda models
- Batalin-Vilkovisky formalism
- polyvectors
- Chern-Simons theory
About this book
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.
Editors and Affiliations
Bibliographic Information
Book Title: Quantization, Geometry and Noncommutative Structures in Mathematics and Physics
Editors: Alexander Cardona, Pedro Morales, Hernán Ocampo, Sylvie Paycha, Andrés F. Reyes Lega
Series Title: Mathematical Physics Studies
DOI: https://doi.org/10.1007/978-3-319-65427-0
Publisher: Springer Cham
eBook Packages: Physics and Astronomy, Physics and Astronomy (R0)
Copyright Information: Springer International Publishing AG 2017
Hardcover ISBN: 978-3-319-65426-3Published: 06 November 2017
Softcover ISBN: 978-3-319-88026-6Published: 18 May 2018
eBook ISBN: 978-3-319-65427-0Published: 26 October 2017
Series ISSN: 0921-3767
Series E-ISSN: 2352-3905
Edition Number: 1
Number of Pages: X, 341
Number of Illustrations: 6 b/w illustrations
Topics: Quantum Field Theories, String Theory, Mathematical Physics, Algebraic Geometry