Noncommutative Mathematics for Quantum Systems

199 Pages·2016·1.862 MB·English

by  Uwe Franz, Adam Skalski

Cambridge - IISc Series
NoncommutativeMathematicsforQuantumSystems
Noncommutativemathematicsisasignificantnewtrendofmathematics
in the twentieth century. Initially motivated by the formulation and
developmentofquantumphysicsduetoHeisenbergandvonNeumann,
the idea of ‘making a theory noncommutative’ has been extended to
many areas of pure and applied mathematics. An example is quantum
probability, describing the probabilistic aspects of quantum mechanics.
Thegeneralizationfromclassicaltoquantumhappenshereintwosteps:
first the theory is reformulated in terms of algebras of functions on
probabilityspaces,thenthecommutativityconditionisdropped.
Thisbookfocusesontwocurrentareasofnoncommutativemathematics:
quantumprobabilityandquantumdynamicalsystems.
The first part of the book provides an introduction to quantum
probability and quantum Le´vy processes. It provides introduction to the
notion of independence in quantum probability. The theory of quantum
stochastic processes with independent and stationary increments is also
highlighted. The second part provides an introduction to quantum
dynamical systems. It focuses on analogies with fundamental problems
studiedinclassicaldynamics.
The text underlines the balance between two crucial aspects of
noncommutative mathematics. On one hand the desire to build an
extension of the classical theory provides new, original ways to
understandwell-known’commutative’resultsandontheotherhandthe
richness of the quantum mathematical world presents completely novel
phenomena, never encountered in the classical setting. This text will be
useful to students and researchers in noncommutative probability,
mathematicalphysicsandoperatoralgebras.
Uwe Franz is Professor at the University of Franche-Comte´ in Besanc¸on
(France) since 2005. He authored the book ’Stochastic Processes and
Operator Calculus on Quantum Groups’, edited four books on quantum
probability and has written over fifty peer-reviewed research papers in
the area of noncommutative mathematics. His areas of interest include
noncommutative probability, quantum stochastic processes, quantum
stochasticcalculusandprobabilityonquantumgroups.
Adam Skalski is Associate Professor at the Institute of Mathematics,
PolishAcademyofSciencesandattheUniversityofWarsaw,Poland.He
has been awarded the Kuratowski Prize of the Polish Mathematical
Society in 2008 and the Sierpin´ski Prize of the Polish Academy of
Sciences in 2014. His areas of interest include topological quantum
groups,operatoralgebrasandquantumstochasticprocesses.
CAMBRIDGE-IIScSERIES
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The editorial board has been constituted with experts from a range of
disciplines in diverse fields of engineering, science and technology from
theIndianInstituteofScience,Bangalore.
IIScPressEditorialBoard:
G.K.Ananthasuresh,Professor,DepartmentofMechanicalEngineering
K.KesavaRao,Professor,DepartmentofChemicalEngineering
GadadharMisra,Professor,DepartmentofMathematics
T.A.Abinandanan,Professor,DepartmentofMaterialsEngineering
DiptimanSen,Professor,CentreforHighEnergyPhysics
Cambridge - IISc Series
Noncommutative Mathematics
for Quantum Systems
UweFranz
AdamSkalski
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Franz,Uwe.
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pagescm
Includesbibliographicalreferencesandindex.
Summary: ”Discusses two current areas of noncommutative mathematics, quantum
probabilityandquantumdynamicalsystems”–Providedbypublisher.
ISBN978-1-107-14805-5(hardback)
1. Probabilities. 2. Quantum theory. 3. Potential theory (Mathematics) I. Skalski, Adam,
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Contents
Preface xi
Conferencephoto xiv
Introduction xv
1 IndependenceandLe´vyProcessesinQuantumProbability 1
1.1 Introduction 1
1.2 WhatisQuantumProbability? 4
1.2.1 Distinguishingfeaturesofclassicalandquantum
probability 11
1.2.2 Dictionary‘Classical↔Quantum’ 13
1.3 WhydoweNeedQuantumProbability? 15
1.3.1 Mermin’sversionoftheEPRexperiment 15
1.3.2 Gleason’stheorem 20
1.3.3 TheKochen–Speckertheorem 21
1.4 InfiniteDivisibilityinClassicalProbability 23
1.4.1 Stochasticindependence 23
1.4.2 Convolution 23
1.4.3 Infinitedivisibility,continuousconvolution
semigroups,andLe´vyprocesses 23
1.4.4 TheDeFinetti–Le´vy–Khintchineformulaon
(R+,+) 25
1.4.5 Le´vy–Khintchineformulaeoncones 25
1.4.6 TheLe´vy–Khintchineformulaon(Rd,+) 26
1.4.7 TheMarkovsemigroupofaLe´vyprocess 27
1.4.8 Hunt’sformula 27
1.5 Le´vyProcessesonInvolutiveBialgebras 29
1.5.1 DefinitionofLe´vyprocessesoninvolutive
bialgebras 29
1.5.2 ThegeneratingfunctionalofaLe´vyprocess 34
1.5.3 TheSchu¨rmanntripleofaLe´vyprocess 36
1.5.4 Examples 41
1.6 Le´vyProcessesonCompactQuantumGroupsandtheir
MarkovSemigroups 42
viii Contents
1.6.1 Compactquantumgroups 43
1.6.2 TranslationinvariantMarkovsemigroups 48
1.7 IndependencesandConvolutionsinNoncommutative
Probability 55
1.7.1 NevanlinnatheoryandCauchy–Stieltjes
transforms 55
1.7.2 Freeconvolutions 57
1.7.3 AusefulLemma 60
1.7.4 Monotoneconvolutions 60
1.7.5 Booleanconvolutions 73
1.8 TheFiveUniversalIndependences 83
1.8.1 Algebraicprobabilityspaces 87
1.8.2 Classicalstochasticindependenceandtheproduct
ofprobabilityspaces 88
1.8.3 Productsofalgebraicprobabilityspaces 90
1.8.4 Classificationoftheuniversalindependences 94
1.9 Le´vyProcessesonDualGroups 98
1.9.1 Dualgroups 98
1.9.2 DefinitionofLe´vyprocessesondualgroups 101
1.9.3 Timereversal 103
1.10 OpenProblems 105
References 107
2 QuantumDynamicalSystemsfromthePointofViewof
NoncommutativeMathematics 121
2.1 NoncommutativeMathematicsandQuantum/
Noncommutativedynamicalsystems 121
2.1.1 NoncommutativeMathematics–Gelfand–Naimark
Theorem 122
2.1.2 Quantumtopologicaldynamicalsystemsandsome
propertiesoftransformationsofC∗-algebras 123
2.1.3 Cuntzalgebras 126
2.1.4 GraphC∗-algebras 128
2.1.5 C∗-algebrasassociatedwithdiscretegroups 130
2.2 NoncommutativeTopologicalEntropyofVoiculescu 131
2.2.1 EndomorphismsofCuntzalgebrasandaquantum
shift 132
Contents ix
2.2.2 TheshifttransformationonagraphC∗-algebra 134
2.2.3 ClassicaltopologicalentropyasdefinedbyRufus
Bowenandextensionstotopologicalpressure 134
2.2.4 Finite-dimensionalapproximationsinthetheory
ofC∗-algebras 136
2.2.5 Noncommutativetopologicalentropy(Voiculescu
topologicalentropy) 137
2.3 VoiculescuEntropyoftheShiftandotherExamples 141
2.3.1 EstimatingtheVoiculescuentropyofthe
permutativeendomorphismsofCuntzalgebras 141
2.3.2 TheVoiculescuentropyofthenoncommutative
shiftandrelatedquestions 145
2.3.3 GeneralizationstographC∗-algebrasandto
noncommutativetopologicalpressure 146
2.3.4 AutomorphismwhoseVoiculescuentropyis
genuinelynoncommutative 147
2.3.5 Automorphismthatleavesnonon-trivialabelian
subalgebrasinvariant 149
2.4 CrossedProductsandtheEntropy 152
2.4.1 Crossedproducts 152
2.4.2 TheVoiculescuentropycomputationsforthemaps
extendedtocrossedproducts 154
2.5 Quantum‘Measurable’DynamicalSystemsand
ClassicalErgodicTheorems 157
2.5.1 Measurabledynamicalsystemsandindividual
ergodictheorem 157
2.5.2 GNSconstructionandthepassagefromtopological
tomeasurablenoncommutativedynamical
systems 158
2.6 NoncommutativeErgodicTheoremofLance;
ClassicalandQuantumMultiRecurrence 163
2.6.1 Meanergodictheorem(s)invonNeumann
algebras 163
2.6.2 AlmostuniformconvergenceinvonNeumann
algebras 166
2.6.3 Lance’snoncommutativeindividualergodic
theoremandsomecommentsonitsproof 167
2.6.4 Classicalmultirecurrence 168