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 connects unitary 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.
- Book Title Reflection Positivity
- Book Subtitle A Representation Theoretic Perspective
- Series Title SpringerBriefs in Mathematical Physics
- Series Abbreviated Title SpringerBriefs in Mathematical Physics
- DOI https://doi.org/10.1007/978-3-319-94755-6
- Copyright Information The Author(s) 2018
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
- Softcover ISBN 978-3-319-94754-9
- eBook ISBN 978-3-319-94755-6
- Series ISSN 2197-1757
- Series E-ISSN 2197-1765
- Edition Number 1
- Number of Pages VIII, 139
- Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
Topological Groups, Lie Groups
Quantum Field Theories, String Theory
Abstract Harmonic Analysis
Probability Theory and Stochastic Processes
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“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)