國立高雄大學圖資館 |

Language: English

Back

Reflection positivitya representatio...

Neeb, Karl-Hermann.

Reflection positivitya representation theoretic perspective /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Reflection positivityby Karl-Hermann Neeb, Gestur Olafsson.
Reminder of title:	a representation theoretic perspective /
Author:	Neeb, Karl-Hermann.
other author:	Olafsson, Gestur.
Published:	Cham :Springer International Publishing :2018.
Description:	viii, 139 p. :digital ;24 cm.
Contained By:	Springer eBooks
Subject:	Representations of groups.
Online resource:	http://dx.doi.org/10.1007/978-3-319-94755-6
ISBN:	9783319947556$q(electronic bk.)

Reflection positivitya representation theoretic perspective /
Neeb, Karl-Hermann.

Reflection positivitya representation theoretic perspective /[electronic resource] :by Karl-Hermann Neeb, Gestur Olafsson. - Cham :Springer International Publishing :2018. - viii, 139 p. :digital ;24 cm. - SpringerBriefs in mathematical physics,v.322197-1757 ;. - SpringerBriefs in mathematical physics ;v.1..

Preface -- Introduction -- Reflection positive Hilbert spaces -- 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 representations -- 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 -- Reflection positive 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.

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.

ISBN: 9783319947556$q(electronic bk.)

Standard No.: 10.1007/978-3-319-94755-6doiSubjects--Topical Terms:

191076
Representations of groups.

LC Class. No.: QA171 / .N443 2018

Dewey Class. No.: 512.22

Reflection positivitya representation theoretic perspective /
LDR:04469nmm a2200337 a 4500 001 540782
003 DE-He213
005 20190110102321.0
006 m d
007 cr nn 008maaau
008 190308s2018 gw s 0 eng d
020 $a 9783319947556$q(electronic bk.)
020 $a 9783319947549$q(paper)
024 7 $a 10.1007/978-3-319-94755-6 $2 doi
035 $a 978-3-319-94755-6
040 $a GP $c GP
041 0 $a eng
050 4 $a QA171 $b .N443 2018
072 7 $a PBG $2 bicssc
072 7 $a MAT014000 $2 bisacsh
072 7 $a MAT038000 $2 bisacsh
082 0 4 $a 512.22 $2 23
090 $a QA171 $b .N373 2018
100 1 $a Neeb, Karl-Hermann. $3 509377
245 1 0 $a Reflection positivity $h [electronic resource] : $b a representation theoretic perspective / $c by Karl-Hermann Neeb, Gestur Olafsson.
260 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2018.
300 $a viii, 139 p. : $b digital ; $c 24 cm.
490 1 $a SpringerBriefs in mathematical physics, $x 2197-1757 ; $v v.32
505 0 $a Preface -- Introduction -- Reflection positive Hilbert spaces -- 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 representations -- 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 -- Reflection positive 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.
520 $a 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.
650 0 $a Representations of groups. $3 191076
650 0 $a Mathematical physics. $3 190854
650 1 4 $a Mathematics. $3 184409
650 2 4 $a Topological Groups, Lie Groups. $3 273787
650 2 4 $a Quantum Field Theories, String Theory. $3 389186
650 2 4 $a Mathematical Physics. $3 522725
650 2 4 $a Abstract Harmonic Analysis. $3 274074
650 2 4 $a Probability Theory and Stochastic Processes. $3 274061
700 1 $a Olafsson, Gestur. $3 819234
710 2 $a SpringerLink (Online service) $3 273601
773 0 $t Springer eBooks
830 0 $a SpringerBriefs in mathematical physics ; $v v.1. $3 683312
856 4 0 $u http://dx.doi.org/10.1007/978-3-319-94755-6
950 $a Mathematics and Statistics (Springer-11649)