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Table of contents (9 chapters)
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Front Matter
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Back Matter
Keywords
- symmetric Lie groups
- reflection positive Hilbert space
- Kubo-Martin-Schwinger condition
- Euclidean group
- Poincare group
- stochastic processes
- Representation theory
- Lax-Phillips scattering theory
- Constructive Quantum Field Theory
- Hardy-Littlewood-Sobolev inequality
- Symmetric Lie groups
- Lorentz group
- Stochastic processes
- Lattice gauge theory
- Cartan dual group
- Wick rotation
- Riemannian geometry
About this book
Refection Positivity is a central theme at the crossroads of Lie group representations, euclidean and abstract harmonic analysis, constructive quantum field theory, and stochastic processes.
This book provides the first presentation of the representation theoretic aspects of Refection Positivity and discusses its connections to those different fields on a level suitable for doctoral students and researchers in related fields.
It starts with a general introduction to the ideas and methods involving refection positive Hilbert spaces and the Osterwalder--Schrader transform. It then turns to Reflection Positivity in Lie group representations. Already the case of one-dimensional groups is extremely rich.
For the real line it connects naturally with Lax--Phillips scattering theory and for the circle group it provides a new perspective on the Kubo--Martin--Schwinger (KMS) condition for states of operator algebras.
For Lie groups Reflection Positivity connectsunitary representations of a symmetric Lie group with unitary representations of its Cartan dual Lie group.
A typical example is the duality between the Euclidean group E(n) and the Poincare group P(n) of special relativity. It discusses in particular the curved context of the duality between spheres and hyperbolic spaces. Further it presents some new integration techniques for representations of Lie algebras by unbounded operators which are needed for the passage to the dual group. Positive definite functions, kernels and distributions and used throughout as a central tool.
Reviews
“This small monograph by Karl-Hermann Neeb and Gestur Ólafsson covers a wide range of problems concerning the concept of Reflection Positivity (RP). … The monograph will be useful for both professional mathematicians as well as doctoral students.” (Roman Urban, zbMATH 1403.22001, 2019)
Authors and Affiliations
Bibliographic Information
Book Title: Reflection Positivity
Book Subtitle: A Representation Theoretic Perspective
Authors: Karl-Hermann Neeb, Gestur Ólafsson
Series Title: SpringerBriefs in Mathematical Physics
DOI: https://doi.org/10.1007/978-3-319-94755-6
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Author(s) 2018
Softcover ISBN: 978-3-319-94754-9Published: 09 July 2018
eBook ISBN: 978-3-319-94755-6Published: 28 June 2018
Series ISSN: 2197-1757
Series E-ISSN: 2197-1765
Edition Number: 1
Number of Pages: VIII, 139
Topics: Topological Groups, Lie Groups, Quantum Field Theories, String Theory, Mathematical Physics, Abstract Harmonic Analysis, Probability Theory and Stochastic Processes