Abstract
This chapter provides the reader with a general overview of the various topics discussed in this volume, emphasizing the deep relations existing between them. Following a brief historical account of the emergence of the concept of “quantization” both in physics and mathematics, a description of the main concepts and tools appearing in subsequent chapters is presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
V.N. Gribov, Quantization of non-abelian gauge theories. Nuclear Phys. B 139(1), 1–19 (1978)
V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, 1989)
P.A.M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, 1982)
B. Kostant, Quantization and unitary representations, in Lectures in Modern Analysis and Applications III, vol. 170, Lecture Notes in Mathematics (Springer, 1970), pp. 87–208
J.-M. Souriau, Structure des Systèmes Dynamiques (Dunod, Paris, 1970)
N.M.J. Woodhouse, Geometric Quantization (Clarendon Press, 1991)
I. Vaisman, Lectures on the Geometry of Poisson Manifolds (Birkhauser, 1991)
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Deformation Theory and Quantization. I. Deformations of Symplectic Structures (Annals of Physics, 111, 1978)
B. Fedosov, A simple geometrical construction of deformation quantization. J. Differ. Geom. 40(2), 213–238 (1994)
A. Connes, Noncommutative Geometry (Academic press, 1995)
A. Cardona, C. Neira-Jiménez, H. Ocampo, S. Paycha, A.F. Reyes-Lega, (eds.) Geometric, Algebraic and Topological Methods for Quantum Field Theory (World Scientific, 2013)
A. Connes, Cyclic cohomology, quantum group symmetries and the local index formula for \(SU_q (2)\). J. Inst. Math. Jussieu 3, 203–225 (2004)
L. Dabrowski, G. Landi, A. Sitarz, W. van Suijlekom, J.C. Varilly, The Dirac operator on \(SU_q (2)\). Commun. Math. Phys. 259, 729–759 (2005)
G. Landi, W. van Suijlekom, Principal fibrations from noncommutative spheres. Commun. Math. Phys. 260, 729–759 (2005)
A. Cardona, H. Ocampo, S. Paycha (eds.), Geometric and topological methods for quantum field theory, in Proceedings of the Summer School held in Villa de Leyva, July 9–27, 2001 (World Scientific Publishing Co., Inc., River Edge, 2003)
F. Scheck, Quantum Physics (Springer, 2007)
J.M. Gracia-Bondía, J.C. Várilly, H. Figueroa, Elements of Noncommutative Geometry (Springer Science & Business Media, 2013)
D. Manchon. Hopf algebras, from basics to applications to renormalization (2004), arXiv:math/0408405
R. Brunetti, C. Dappiaggi, K. Fredenhagen, J. Yngvason, Advances in Algebraic Quantum Field Theory (Springer, 2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Cardona, A., Paycha, S., Reyes Lega, A.F. (2017). Prelude: A General Overview. In: Cardona, A., Morales, P., Ocampo, H., Paycha, S., Reyes Lega, A. (eds) Quantization, Geometry and Noncommutative Structures in Mathematics and Physics. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-65427-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-65427-0_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-65426-3
Online ISBN: 978-3-319-65427-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)