Table Of Content

This page intentionally left blank CAMBRIDGETRACTSINMATHEMATICS GeneralEditors B.BOLLOBAS,W.FULTON,A.KATOK,F.KIRWAN, P.SARNAK,B.SIMON 169QuantumStochasticProcessesandNoncommutative Geometry CAMBRIDGETRACTSINMATHEMATICS GENERALEDITORS B.BOLLOBAS,W.FULTON,A.KATOK,F.KIRWAN,P.SARNAK,B.SIMON Acompletelistofbooksintheseriescanbefoundat http://www.cambridge.org/series/sSeries.asp?code=CTM Recenttitlesincludethefollowing: 140. DerivationandIntegration.ByW.F.PFEFFER 141. FixedPointTheoryandApplications.ByR.P.AGARWAL,M.MEEHAN, andD.O’REGAN 142. HarmonicMapsbetweenRiemannianPolyhedra.ByJ.EELLSandB.FUGLEDE 143. AnalysisonFractals.ByJ.KIGAMI 144. TorsorsandRationalPoints.ByA.SKOROBOGATOV 145. IsoperimetricInequalities.ByI.CHAVEL 146. RestrictedOrbitEquivalenceforActionsofDiscreteAmenableGroups. ByJ.KAMMEYERandD.RUDOLPH 147. FloerHomologyGroupsinYang–MillsTheory.ByS.K.DONALDSON 148. GraphDirectedMarkovSystems.ByD.MAULDINandM.URBANSKI 149. CohomologyofVectorBundlesandSyzygies.ByJ.WEYMAN 150. HarmonicMaps,ConservationLawsandMovingFrames.ByF.HE´LEIN 151. FrobeniusManifoldsandModuliSpacesforSingularities.ByC.HERTLING 152. PermutationGroupAlgorithms.ByA.SERESS 153. AbelianVarieties,ThetaFunctionsandtheFourierTransform. ByA.POLISHCHUK 154. FinitePackingandCovering,K.BO¨RO¨CZKY,JR 155. TheDirectMethodinSolitonTheory.ByR.HIROTA.Editedandtranslated byA.NAGAI,J.NIMMO,andC.GILSON 156. HarmonicMappingsinthePlane.ByP.DUREN 157. AffineHeckeAlgebrasandOrthogonalPolynomials.ByI.G.MACDONALD 158. Quasi-FrobeniusRings.ByW.K.NICHOLSONandM.F.YOUSIF 159. TheGeometryofTotalCurvature.ByK.SHIOHAMA,T.SHIOYA, andM.TANAKA 160. ApproximationbyAlgebraicNumbers.ByY.BUGEAUD 161. EquivalenceandDualityforModuleCategories.ByR.R.COLBY, andK.R.FULLER 162. Le´vyProcessesinLieGroups.ByMINGLIAO 163. LinearandProjectiveRepresentationsofSymmetricGroups.ByA.KLESHCHEV 164. TheCoveringPropertyAxiom,CPA.K.CIESIELSKIandJ.PAWLIKOWSKI 165. ProjectiveDifferentialGeometryOldandNew.ByV.OVSIENKO andS.TABACHNIKOV 166. TheLe´vyLaplacian.ByM.N.FELLER 167. Poincare´DualityAlgebras,Macaulay’sDualSystems,andSteenrodOperations. ByD.M.MEYERandL.SMITH 168. TheCube:AWindowtoConvexandDiscreteGeometry.ByCHUANMINGZONG 169. QuantumStochasticProcessesandNoncommutativeGeometry. ByKALYANB.SINHAandDEBASHISHGOSWAMI QUANTUM STOCHASTIC PROCESSES AND NONCOMMUTATIVE GEOMETRY KalyanB.Sinha IndianStatisticalInstitute NewDelhi,India DebashishGoswami IndianStatisticalInstitute Kolkata,India CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridg e.org /9780521834506 © K. Sinha and D. Goswami This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2007 ISBN-13 978-0-511-26904-2 eBook(EBL) ISBN-10 0-511-26904-8 eBook(EBL) ISBN-13 978-0-521-83450-6 hardback ISBN-10 0-521-83450-3 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface vii Notation ix 1 Introduction 1 2 Preliminaries 7 ∗ 2.1 C andvonNeumannalgebras 7 2.2 Completelypositivemaps 20 2.3 Semigroupsoflinearmapsonlocallyconvexspaces 26 2.4 FockspacesandWeyloperators 30 3 Quantumdynamicalsemigroups 33 3.1 Generatorsofuniformlycontinuousquantumdynamical semigroups:thetheoremsofLindbladandChristensen–Evans 33 3.2 Thecaseofstronglycontinuousquantumdynamical semigroups 39 4 Hilbertmodules 79 ∗ 4.1 HilbertC -modules 79 4.2 HilbertvonNeumannmodules 88 4.3 GroupactionsonHilbertmodules 95 5 Quantumstochasticcalculuswithboundedcoefficients 103 5.1 Basicprocesses 103 5.2 StochasticintegralsandquantumItoˆ formulae 113 5.3 Hudson–Parthasarathy(H–P)typeequations 127 5.4 Map-valued,E–H-typequantumstochasticcalculus 133 6 Dilationofquantumdynamicalsemigroupswith boundedgenerator 147 6.1 Hudson–Parthasarathy(H–P)dilation 149 6.2 ExistenceofstructuremapsandE–HdilationofT 150 t 6.3 Adualityproperty 154 v vi Contents 6.4 AppearanceofPoissontermsinthedilation 156 6.5 ImplementationofE–Hflow 160 ∗ 6.6 DilationonaC -algebra 161 6.7 Covariantdilationtheory 164 7 Quantumstochasticcalculuswithunboundedcoefficients 169 7.1 Notationandpreliminaryresults 169 7.2 Q.S.D.E.withunboundedcoefficients 173 7.3 Application:quantumdampedharmonicoscillator 181 8 Dilationofquantumdynamicalsemigroupswithunbounded generator 185 8.1 Dilationofaclassofcovariantquantumdynamical semigroups 185 8.2 Dilationofquantumdynamicalsemigroups onU.H.F.algebra 218 9 Noncommutativegeometryandquantumstochasticprocesses 231 9.1 BasicsofdifferentialandRiemanniangeometry 231 9.2 HeatsemigroupandBrownianmotiononclassicalmanifolds 238 9.3 Noncommutativegeometry 248 9.4 Examples 253 9.5 AsymptoticanalysisofheatsemigroupsandLaplacians 264 9.6 QuantumBrownianmotiononnoncommutativemanifolds 273 References 281 Index 289 Preface On the one hand, in almost all the scientific areas, from physical to social sciences, biology to economics, from meteorology to pattern recognition in remote sensing, the theory of classical probability plays a major role and on the other much of our knowledge about the physical world at least is based on the quantum theory [12]. In a way, quantum theory itself is a new kind of theoryofprobability(inthelanguageofvonNeumannandBirkhoff)(seefor example[106])whichcontainstheclassicalmodel,andthereforeitisnatural toextendtheotherareasofclassicalprobabilitytheory,inparticularthetheory ofMarkovprocessesandstochasticcalculustothisquantummodel. Therearemorethanonepossibleways(seeforexample[127])toconstruct theabove-mentionedextensionandinthisbookwehavechosentheoneclos- est to the classical model in spirit, namely that which contains the classical theoryasasubmodel.Thisrequirementhasruledoutanydiscussionofareas such as freeand monotone-probability models. Once we accept this quantum probabilistic model, the ‘grand design’ that engages us is the ‘canonical con- structionofa∗-homomorphicflow(satisfyingasuitabledifferentialequation) onagivenalgebraofobservablessuchthattheexpectationsemigroupispre- ciselythegivencontractivesemigroupofcompletelypositivemapsonthesaid algebra’. This problem of ‘dilation’ is here solved completely for the case when the semigroup has a bounded generator, and also for the more general case (of anunboundedgenerator)withcertainadditionalconditionssuchassymmetry and/orcovariancewithrespecttoaLiegroupaction.However,acertainamount ofspacehastobedevotedtodeveloptheneededtechniquesandstructures,and thereaderisexpectedtobewellequippedwiththebasicsoffunctionalanalysis, theoryofHilbertspacesandofoperatorsinthemandofprobabilitytheoryin ordertomasterthese. Abeginnerwiththeabove-mentionedbackgroundmayreadChapters1to6 atfirstandmayleavetherestforasecondreading.Insomeplaces,mathemat- ical assertions have been made without proof wherever we felt that the proof is essentially similar to a detailed proof of an earlier statement or when the verificationofthesamecanbeleftasanexercise. vii viii Preface Duetolackofspace,notallequationshavebeendisplayedandlongexpres- sions had to be broken at the end of a line, any inconvenience due to this is regretted.Theopensquaresymboldenotestheendofaproof.Thereferencelist isfarfromcomplete,wehaveoftenincludedonlyarecentorarepresentative paper.Weapologizeforanyunintendedexclusionofareference. Itisapleasuretorememberherepeoplewhohavecontributedtothepreparation ofthisbook.ProfessorK.R.Parthasarathywasinstrumentalinintroducingus to the subject and one of us (K.B.S.) has collaborated with him extensively overnearlytwodecades;withouttheinsightsandmasterlyexpositionsofhim and of Professor P. A. Meyer, the subject may not have reached the stage it is in now. We thank all our friends, collaborators and members of the Q–P club who have helped us directly or indirectly in this endeavor. In particular, we must mention Professors Luigi Accardi, Robin Hudson, V. P. Belavkin, MartinLindsay,FrancoFagnola,StephaneAttal,Jean-LucSauvageot,Burkhard Ku¨mmerer, Hans Maassen, Rajarama Bhat and Dr Arup Pal and Dr Partha Sarathi Chakraborty. We are grateful to the Indian Statistical Institute (both DelhiandKolkatacampuses)forprovidingthenecessaryfacilities,Indo-French CentreforthePromotionofAdvancedResearchandDST-DAADagenciesfor making many collaborations possible. One of us (D.G.) would like to thank theAlexandervonHumboldtFoundationforapostdoctoralfellowshipduring 2000–01(andalsolatervisitsunderitsschemeof‘resumptionoffellowship’), when part of the work covered by this book was done. We must also thank Dr Lingraj Sahu, who as a graduate student at a critical stage of writing the monograph, helped with introduction of a part of the material and Mr Joydip Janaforhelpwithproofreading.Oneoftheauthors(D.G.)dedicatesthisbook to his wife, Gopa and the youngest addition to his family, expected possibly before this book sees the light of the day; and acknowledges with gratitude the constant encouragement and support from his parents, mother-in-law and Amit-daduringthewritingofthebook. Asisoftenthecaseinanysuchenterprise,someimportanttopics(e.g.stop times)havebeenleftout.Theresponsibilityforthechoiceoftopicsaswellas for any omissions and shortcomings of the text is entirely ours. We can only hopethatthismonographwillenthusesomeresearchersandstudentstosolve someoftheproblemsleftunsolved. K.B.Sinha DebashishGoswami