Reflection Positivity : A Representation Theoretic Perspective / by Karl-Hermann Neeb, Gestur Ólafsson.

Format:

Book

Author/Creator:

Neeb, Karl-Hermann., Author.

Ólafsson, Gestur., Author.

Series:

SpringerBriefs in Mathematical Physics, 2197-1757 ; 32

Language:

English

Subjects (All):

Topological groups.

Lie groups.

Quantum field theory.

String models.

Mathematical physics.

Harmonic analysis.

Probabilities.

Topological Groups, Lie Groups.

Quantum Field Theories, String Theory.

Mathematical Physics.

Abstract Harmonic Analysis.

Probability Theory and Stochastic Processes.

Local Subjects:

Topological Groups, Lie Groups.

Quantum Field Theories, String Theory.

Mathematical Physics.

Abstract Harmonic Analysis.

Probability Theory and Stochastic Processes.

Physical Description:

1 online resource (135 pages)

Edition:

1st ed. 2018.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2018.

Summary:

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.

Contents:

Preface

Introduction

Reflection positive Hilbert spaces

Reflection positive subspaces as graphs

The Markov condition

Reflection positive kernels and distributions

Reflection positivity in Riemannian geometry

Selfadjoint extensions and reflection positivity

Reflection positive representations

The OS transform of linear operators

Symmetric Lie groups and semigroups

Reflection positive functions

Reflection positivity on the real line

Reflection positive functions on intervals

Reflection positive one-parameter groups

Reflection positive operator-valued functions

A connection to Lax–Phillips scattering theory

Reflection positivity on the circle

Positive definite functions satisfying KMS conditions

Reflection positive functions and KMS conditions

Realization by resolvents of the Laplacian

Integration of Lie algebra representations

A geometric version of Fr¨ohlich’s Selfadjointness Theorem

Integrability for reproducing kernel spaces

Representations on spaces of distributions

Reflection positive distributions and representations

Reflection positive distribution vectors

Distribution vectors

Spherical representation of the Lorentz group

Generalized free fields

Lorentz invariant measures on the light cone and their relatives

From the Poincar´e group to the euclidean group

The conformally invariant case

Reflection positivity and stochastic processes

Reflection positive group actions on measure spaces

Stochastic processes indexed by Lie groups

Associated positive semigroup structures and reconstruction

A Background material

A.1 Positive definite kernels

A.2 Integral representations

Index.

ISBN:

3-319-94755-9