Reflection positivity: a representation theoretic perspective

145 Pages·2018·1.058 MB·English

by  Neeb, Karl-Hermann, Ólafsson, Gestur

SPRINGER BRIEFS IN MATHEMATICAL PHYSICS 32
Karl-Hermann Neeb · Gestur Ólafsson
Reflection
Positivity
A Representation
Theoretic
Perspective
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Ó
Karl-Hermann Neeb Gestur lafsson
(cid:129)
fl
Re ection Positivity
A Representation Theoretic Perspective
123
Karl-Hermann Neeb GesturÓlafsson
Department Mathematik Department ofMathematics
UniversitätErlangen-Nürnberg LouisianaState University
Erlangen,Germany BatonRouge,LA, USA
ISSN 2197-1757 ISSN 2197-1765 (electronic)
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Preface
The concept of reflection positivity (RP) occurs as an important theme in various
areas of mathematics and physics:
(cid:129) In the representation theory of Lie groups, it establishes a passage of a unitary
representationofasymmetricLiegroup(suchastheeuclideanmotiongroup)to
a unitary representation of the Cartan dual group (such as the Poincaré group)
[FOS83, JOl00, JOl98, NÓ14, JPT15].
(cid:129) In constructive Quantum Field Theorem (QFT), it arises as the condition of
Osterwalder–Schrader(OS)positivityforaeuclideanfieldtheorytocorrespond
to a relativistic one [GJ81, Ja08, Ja18, Os95a, Os95b, OS73, OS75].
(cid:129) For stochastic processes, it is weaker than the Markov property and specifies
processes arising in lattice gauge theory. It plays a central role in the mathe-
matical study of phase transitions and symmetry breaking [FILS78, JJ16, JJ17,
Nel73].
(cid:129) Inanalysis,itisacrucialconditionthatleadstoinequalitiessuchastheHardy–
Littlewood–Sobolev inequality [FL10].
Only recently, it became apparent that there are many hidden and still not
sufficiently well-understood structures underlying the duality between unitary
representations of a symmetric Lie group and its dual. Establishing reflection
positivityin this context requires new analytic methods and new geometric insight
into constructions and realizations of representations in analytic contexts. New
developments concern analytic issues such as criteria for integrating Lie algebra
representations to Lie group representations, reflection positive functions, distri-
butions and kernels, new dilation techniques for representations and unexpected
connections between Kubo–Martin–Schwinger (KMS) states of C(cid:1)-algebras and
reflection positive unitary representations.
This was our motivation to write this “light” introduction to the representation
theoretic aspects of reflection positivity to present this perspective on a level
suitable for doctoral students.
v
vi Preface
Acknowledgment: We are most grateful to Jan Frahm for reading earlier ver-
sions of this manuscript and for helping us to eliminate typos and inaccuracies.
Erlangen, Germany Karl-Hermann Neeb
Baton Rouge, USA Gestur Ólafsson
References
[FL10] Frank,R.L.,Lieb,E.H.:InversionpositivityandthesharpHardy-Littlewood-Sobolev
inequality.Calc.Var.Partial.Differ.Equ.39,85–99(2010)
[FILS78] Fröhlich, J., Israel, R., Lieb, E.H., Simon, B.: Phase transitions and reflection posi-
tivity.I.Generaltheoryandlongrangelatticemodels.Commun.Math.Phys.62(1),
1–34(1978)
[FOS83] Fröhlich,J.,Osterwalder,K.,Seiler,E.:Onvirtualrepresentationsofsymmetricspaces
andtheiranalyticcontinuation.Ann.Math.118,461–489(1983)
[GJ81] Glimm,J.,Jaffe,A.:QuantumPhysics-AFunctionalIntegralPointofView.Springer,
NewYork(1981)
[Ja08] Jaffe, A.: Quantum theory and relativity. In: Doran, R.S., Moore, C.C., Zimmer,
R.J. (eds.) Group Representations, Ergodic Theory, and Mathematical Physics:
ATributetoGeorgeW.Mackey.ContemporaryinMathematics,vol.449.American
MathematicalSociety,Providence(2008)
[Ja18] Jaffe, A.: Reflection positivity then and now, Oberwolfach Reports 55/2017,
“ReflectionPositivity”.doi:10.4171/OWR/2017/55
[JJ16] Jaffe,A.,Janssens,B.:Characterizationofreflectionpositivity:Majoranasandspins.
Commun.Math.Phys.346(3),1021–1050(2016)
[JJ17] Jaffe,A.,Janssens,B.:Reflectionpositivedoubles.J.Funct.Anal.272(8),3506–3557
(2017)
[JOl00] Jorgensen, P.E.T., Ólafsson, G.: Unitary representations and Osterwalder–Schrader
duality.In:DoranR.S.,VaradarajanV.S.(eds.)TheMathematicalLegacyofHarish–
Chandra. Proceedings of Symposia in Pure Mathematics, vol. 68. American
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[JOl98] Jorgensen,P.E.T.,Ólafsson,G.:UnitaryrepresentationsofLiegroupswithreflection
symmetry.J.Funct.Anal.158,26–88(1998)
[JPT15] Jorgensen, P.E.T., Pedersen, S., Tian, F.: Extensions of Positive Definite Functions:
ApplicationsandTheirHarmonicAnalysis.LectureNotesinMathematics,vol.2160.
Springer,Berlin(2016)
[Nel73] Nelson, E.: Construction of quantum fields from Markofffields. J. Funct. Anal. 12,
97–112(1973)
[NÓ14] Neeb, K.-H., Ólafsson, G.: Reflection positivity and conformal symmetry. J. Funct.
Anal.266,2174–2224(2014)
[Os95a] Osterwalder, K.: Supersymmetric quantum field theory. In: “Constructive Physics,”
(Palaiseau, 1994), Lecture Notes in Physics 446, V. Rivasseau, Ed., pp 117–130,
Springer,Berlin,(1995)
[Os95b] Osterwalder, K., Constructing supersymmetric quantum field theory models. In:
“AdvancesinDynamicalSystemsandQuantumPhysics,”(Capri,1993),pp209–224.
WorldScientificPublication,RiverEdge,NJ(1995)
[OS73] Osterwalder, K., Schrader, R.: Axioms for Euclidean green’s functions. I. Commun.
Math.Phys.31,83–112(1973)
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Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Reflection Positive Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Reflection Positive Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Reflection Positive Subspaces as Graphs. . . . . . . . . . . . . . . . . . 10
2.3 The Markov Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Reflection Positive Kernels and Distributions . . . . . . . . . . . . . . 12
2.5 Reflection Positivity in Riemannian Geometry. . . . . . . . . . . . . . 16
2.6 Selfadjoint Extensions and Reflection Positivity. . . . . . . . . . . . . 17
3 Reflection Positive Representations . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 The OS Transform of Linear Operators. . . . . . . . . . . . . . . . . . . 21
3.2 Symmetric Lie Groups and Semigroups . . . . . . . . . . . . . . . . . . 24
3.3 Reflection Positive Representations . . . . . . . . . . . . . . . . . . . . . 26
3.4 Reflection Positive Functions . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Reflection Positivity on the Real Line . . . . . . . . . . . . . . . . . . . . . . 35
4.1 Reflection Positive Functions on Intervals. . . . . . . . . . . . . . . . . 36
4.2 Reflection Positive One-Parameter Groups . . . . . . . . . . . . . . . . 38
4.3 Reflection Positive Operator-Valued Functions . . . . . . . . . . . . . 41
4.4 A Connection to Lax–Phillips Scattering Theory . . . . . . . . . . . . 47
5 Reflection Positivity on the Circle . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.1 Positive Definite Functions Satisfying KMS Conditions . . . . . . . 52
5.2 Reflection Positive Functions and KMS Conditions . . . . . . . . . . 57
5.3 Realization by Resolvents of the Laplacian. . . . . . . . . . . . . . . . 62
6 Integration of Lie Algebra Representations . . . . . . . . . . . . . . . . . . 69
6.1 A Geometric Version of Fröhlich’s Selfadjointness Theorem. . . . 70
6.2 Integrability for Reproducing Kernel Spaces . . . . . . . . . . . . . . . 71
6.3 Representations on Spaces of Distributions . . . . . . . . . . . . . . . . 74
6.4 Reflection Positive Distributions and Representations. . . . . . . . . 76
vii
viii Contents
7 Reflection Positive Distribution Vectors. . . . . . . . . . . . . . . . . . . . . 79
7.1 Distribution Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.1.1 Distributions on Lie Groups and Homogeneous
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.1.2 Smooth Vectors and Distribution Vectors. . . . . . . . . . . . 81
7.2 Reflection Positive Distribution Vectors . . . . . . . . . . . . . . . . . . 85
7.3 Spherical Representation of the Lorentz Group . . . . . . . . . . . . . 86
7.3.1 The Principal Series. . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.3.2 The Complementary Series. . . . . . . . . . . . . . . . . . . . . . 88
7.3.3 H-Invariant Distribution Vectors . . . . . . . . . . . . . . . . . . 92
7.4 Reflection Positivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.4.1 Reflection Positivity for the Conformal Group. . . . . . . . . 93
7.4.2 Resolvents of the Laplacian on the Sphere . . . . . . . . . . . 95
7.4.3 The Distribution Kernel of ðm2(cid:3)DÞ(cid:3)1. . . . . . . . . . . . . . 97
8 Generalized Free Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.1 Lorentz Invariant Measures on the Light Cone and Their
Relatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.2 From the Poincaré Group to the Euclidean Group . . . . . . . . . . . 105
8.3 The Conformally Invariant Case . . . . . . . . . . . . . . . . . . . . . . . 110
9 Reflection Positivity and Stochastic Processes . . . . . . . . . . . . . . . . 113
9.1 Reflection Positive Group Actions on Measure Spaces. . . . . . . . 113
9.2 Stochastic Processes Indexed by Lie Groups. . . . . . . . . . . . . . . 117
9.3 Associated Positive Semigroup Structures and Reconstruction. . . 118
Appendix: Background Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Chapter 1
Introduction
Inthecontextofquantumphysicsreflectionpositivityisoftenrela√tedtoWickrota-
tion,whichroughlymeansmultiplyingthetimecoordinatebyi = −1.Thiscanbe
madepreciseandusedinanalyticconstructionsifthisprocessisrelatedtoanalytic
continuationofthetimevariabletoadomaininthecomplexplanewhichprovides
ameanstogobackandforthbetweenrealandimaginarytime.
AdualityofasimilarflavoralsoexistsinthecontextofLiegroups,whereitarose
almost a century ago in the work of É. Cartan on the classification of symmetric
spaces.HereoneconsidersasymmetricLiegroup(G,τ),i.e.,aLiegroupG,endowed
withaninvolutiveautomorphismτ.ThentheLiealgebragof G decomposes into
τ-eigenspaces
h={x ∈g: τx = x} and q={x ∈g: τx =−x}.
Fromthebracketrelations[h,h]⊆h,[h,q]⊆q,and[q,q]⊆hitthenfollowsthat
theCartandual
gc :=h+iq
alsoisaLiesubalgebraofthecomplexifiedLiealgebragC =g⊕ig.Wethusobtain
a duality relation between symmetric Lie groups (G,τ) and (Gc,τc), where Gc
denotesaLiegroupwithLiealgebraGc andτc aninvolutiveautomorphismacting
by x +iy (cid:6)→ x −iy on the Lie algebra gc =h+iq. The classical examples from
quantumphysicsaretheeuclideanmotiongroupG = E(d)∼=Rd (cid:2)O (R)andthe
d
automorphismτ of E(d)inducesbytimereflection.Thisestablishesadualitywith
thePoincarégroupGc = P(d)∼=R1,d−1(cid:2)O1,d−1(R).
InmanycasesbothgroupsGandGcarecontainedinonecomplexLiegroupGC
andH =G∩GcisaLiesubgroupwithLiealgebrah,containedinboth.Therefore
anypassagefromG toGc shouldberelatedtoanalyticcontinuationtodomainsin
GC whose closure intersects both groups G and Gc. On the Lie algebra level the
passagefromg=h⊕qtogc =h⊕iqverymuchresemblesWickrotationbecause
theelementsofqaremultipliedbyi (cf.[HH17],wherethiscontextisdiscussedfor
pseudo-Riemannianmanifolds).Simpleexamplesare
©TheAuthor(s)2018 1
K.-H.NeebandG.Ólafsson,ReflectionPositivity,SpringerBriefsinMathematical
Physics,https://doi.org/10.1007/978-3-319-94755-6_1