Quantum Independent Increment Processes II: Structure of Quantum Levy Processes, Classical Probability, and Physics

353 Pages·2006·2.755 MB·English

by  Uwe Franz, Rolf Rolf (auth.), Michael Schüermann, Uwe Franz (eds.)

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
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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