Quantization, Geometry and Noncommutative Structures in Mathematics and Physics

Nonfiction, Science & Nature, Science, Physics, Mathematical Physics, Quantum Theory
Cover of the book Quantization, Geometry and Noncommutative Structures in Mathematics and Physics by , Springer International Publishing
View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart
Author: ISBN: 9783319654270
Publisher: Springer International Publishing Publication: October 26, 2017
Imprint: Springer Language: English
Author:
ISBN: 9783319654270
Publisher: Springer International Publishing
Publication: October 26, 2017
Imprint: Springer
Language: English

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.

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

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.

More books from Springer International Publishing

Cover of the book Advances in National Brand and Private Label Marketing by
Cover of the book Gender in Transnational Knowledge Work by
Cover of the book Fading and Shadowing in Wireless Systems by
Cover of the book Comprehensive Cardiovascular Medicine in the Primary Care Setting by
Cover of the book GeNeDis 2014 by
Cover of the book Ethnobotany of Tuberculosis in Laos by
Cover of the book Representation Theory, Number Theory, and Invariant Theory by
Cover of the book Distributed, Ambient and Pervasive Interactions: Technologies and Contexts by
Cover of the book Educational Research: Ethics, Social Justice, and Funding Dynamics by
Cover of the book Water Resources and Food Security in the Vietnam Mekong Delta by
Cover of the book The Bleak Political Implications of Socratic Religion by
Cover of the book Advanced Dynamics of Mechanical Systems by
Cover of the book Nonlinear Systems, Vol. 1 by
Cover of the book Global Networking, Communication and Culture: Conflict or Convergence? by
Cover of the book Biomass and Green Chemistry by
We use our own "cookies" and third party cookies to improve services and to see statistical information. By using this website, you agree to our Privacy Policy