© 2018

Reflection Positivity

A Representation Theoretic Perspective


Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 32)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Karl-Hermann Neeb, Gestur Ólafsson
    Pages 1-8
  3. Karl-Hermann Neeb, Gestur Ólafsson
    Pages 9-20
  4. Karl-Hermann Neeb, Gestur Ólafsson
    Pages 21-34
  5. Karl-Hermann Neeb, Gestur Ólafsson
    Pages 35-50
  6. Karl-Hermann Neeb, Gestur Ólafsson
    Pages 51-67
  7. Karl-Hermann Neeb, Gestur Ólafsson
    Pages 69-78
  8. Karl-Hermann Neeb, Gestur Ólafsson
    Pages 79-102
  9. Karl-Hermann Neeb, Gestur Ólafsson
    Pages 103-111
  10. Karl-Hermann Neeb, Gestur Ólafsson
    Pages 113-122
  11. Back Matter
    Pages 123-139

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.


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

Authors and affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA

Bibliographic information


“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)