Quantum Independent Increment Processes II: Structure of Quantum Levy Processes, Classical Probability, and Physics PDF
Preview Quantum Independent Increment Processes II: Structure of Quantum Levy Processes, Classical Probability, and Physics
Lecture Notes in Mathematics 1866 Editors: J.-M.Morel,Cachan F.Takens,Groningen B.Teissier,Paris · · Ole E. Barndorff-Nielsen Uwe Franz Rolf Gohm · Burkhard Kümmerer Steen Thorbjørnsen Quantum Independent Increment Processes II Structure of Quantum Lévy Processes, Classical Probability, and Physics Editors: Michael Schüermann Uwe Franz ABC EditorsandAuthors OleE.Barndorff-Nielsen BurkhardKümmerer DepartmentofMathematicalSciences FachbereichMathematik UniversityofAarhus TechnischeUniversitätDarmstadt NyMunkegade,Bldg.350 Schlossgartenstr.7 8000Aarhus 64289Darmstadt Denmark Germany e-mail:[email protected] e-mail:kuemmerer@mathematik. tu-darmstadt.de MichaelSchuermann SteenThorbjørnsen RolfGohm DepartmentofMathematicsand UweFranz ComputerScience InstitutfürMathematikundInformatik UniversityofSouthernDenmark UniversitätGreifswald Campusvej55 Friedrich-Ludwig-Jahn-Str.15a 5230Odense 17487Greifswald Denmark Germany e-mail:[email protected] e-mail:[email protected] [email protected] [email protected] LibraryofCongressControlNumber:2005934035 MathematicsSubjectClassification(2000):60G51,81S25,46L60,58B32,47A20,16W30 ISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 ISBN-10 3-540-24407-7SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-24407-3SpringerBerlinHeidelbergNewYork DOI10.1007/11376637 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2006 PrintedinTheNetherlands Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorsandTechbooksusingaSpringerLATEXpackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11376637 41/TechBooks 543210 Preface Thisvolume isthesecondoftwo volumes containing thelecturesgiven atthe School “Quantum Independent Increment Processes: Structure and Applica- tions to Physics”. This school was held at the Alfried Krupp Wissenschafts- kolleginGreifswaldduringtheperiodMarch9–22,2003.Wethankthelectur- ersforallthehardworktheyaccomplished.Theirlecturesgiveanintroduction to current research in their domains that is accessible to Ph. D. students. We hopethatthetwovolumeswillhelptobringresearchersfromtheareasofclas- sical and quantum probability, operator algebras and mathematical physics together and contribute to developing the subject of quantum independent increment processes. We are greatly indebted to the Volkswagen Foundation for their finan- cial support, without which the school would not have been possible. We also acknowledge the support by the European Community for the Research TrainingNetwork“QP-Applications:QuantumProbabilitywithApplications toPhysics,InformationTheoryandBiology”undercontractHPRN-CT-2002- 00279. Special thanks go to Mrs. Zeidler who helped with the preparation and organisation of the school and who took care of all of the logistics. Finally, we would like to thank all the students for coming to Greifswald and helping to make the school a success. Neuherberg and Greifswald, Uwe Franz August 2005 Michael Schu¨rmann Contents Random Walks on Finite Quantum Groups Uwe Franz, Rolf Gohm........................................... 1 1 Markov Chains and Random Walks in Classical Probability........................................ 3 2 Quantum Markov Chains...................................... 5 3 Random Walks on Comodule Algebras .......................... 7 4 Random Walks on Finite Quantum Groups ...................... 11 5 Spatial Implementation ....................................... 12 6 Classical Versions ............................................ 18 7 Asymptotic Behavior ......................................... 22 A Finite Quantum Groups....................................... 24 B The Eight-Dimensional Kac-Paljutkin Quantum Group............ 26 References ...................................................... 30 Classical and Free Infinite Divisibility and L´evy Processes Ole E. Barndorff-Nielsen, Steen Thorbjørnsen ....................... 33 1 Introduction ................................................. 34 2 Classical Infinite Divisibility and L´evy Processes ................. 35 3 Upsilon Mappings ............................................ 48 4 Free Infinite Divisibility and L´evy Processes ..................... 92 5 Connections between Free and Classical Infinite Divisibility ...............................113 6 Free Stochastic Integration ....................................123 A Unbounded Operators Affiliated with a W∗-Probability Space ..................................150 References ......................................................155 L´evy Processes on Quantum Groups and Dual Groups Uwe Franz ......................................................161 VIII Contents 1 L´evy Processes on Quantum Groups ............................163 2 L´evy Processes and Dilations of Completely Positive Semigroups ...184 3 The Five Universal Independences..............................198 4 L´evy Processes on Dual Groups ................................229 References ......................................................254 Quantum Markov Processes and Applications in Physics Burkhard Ku¨mmerer .............................................259 1 Quantum Mechanics ..........................................262 2 Unified Description of Classical and Quantum Systems............265 3 Towards Markov Processes ....................................268 4 Scattering for Markov Processes................................281 5 Markov Processes in the Physics Literature ......................294 6 An Example on M ..........................................297 2 7 The Micro-Maser as a Quantum Markov Process .................302 8 Completely Positive Operators .................................308 9 Semigroups of Completely Positive Operators and Lindblad Generators ......................................312 10 Repeated Measurement and its Ergodic Theory ..................315 References ......................................................328 Index..........................................................331 Contents of Volume I L´evy Processes in Euclidean Spaces and Groups David Applebaum ................................................ 1 1 Introduction ................................................. 2 2 Lecture 1: Infinite Divisibility and L´evy Processes in Euclidean Space 5 3 L´evy Processes............................................... 15 4 Lecture 2: Semigroups Induced by L´evy Processes ................ 25 5 Analytic Diversions........................................... 29 6 Generators of L´evy Processes .................................. 33 7 Lp-Markov Semigroups and L´evy Processes ...................... 38 8 Lecture 3: Analysis of Jumps .................................. 42 9 Lecture 4: Stochastic Integration ............................... 55 10 Lecture 5: L´evy Processes in Groups ............................ 69 11 Lecture 6: Two L´evy Paths to Quantum Stochastics .............. 84 References ...................................................... 95 Locally compact quantum groups Johan Kustermans ............................................... 99 1 Elementary C*-algebra theory .................................102 2 Locally compact quantum groups in the C*-algebra setting ........112 3 Compact quantum groups .....................................115 4 Weight theory on von Neumann algebras ........................129 5 The definition of a locally compact quantum group ...............144 6 Examples of locally compact quantum groups ....................157 7 Appendix : several concepts ...................................172 References ......................................................176 Quantum Stochastic Analysis – an Introduction J.Martin Lindsay ................................................181 1 Spaces and Operators.........................................183 2 QS Processes ................................................214 3 QS Integrals .................................................221 X Contents 4 QS Differential Equations .....................................238 5 QS Cocycles .................................................243 6 QS Dilation .................................................253 References ......................................................264 Dilations, Cocycles and Product Systems B. V. Rajarama Bhat.............................................273 1 Dilation theory basics.........................................273 2 E -semigroups and product systems ............................277 0 3 Domination and minimality....................................282 4 Product systems: Recent developments..........................286 References ......................................................290 Index..........................................................293
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