Probability on Real Lie Algebras (Cambridge Tracts in Mathematics, Series Number 206)

This monograph is a progressive introduction to non-commutativity in probability theory, summarizing and synthesizing recent results about classical and quantum stochastic processes on Lie algebras. In the early chapters, focus is placed on concrete examples of the links between algebraic relations and the moments of probability distributions. The subsequent chapters are more advanced and deal with Wigner densities for non-commutative couples of random variables, non-commutative stochastic processes with independent increments (quantum Lévy processes), and the quantum Malliavin calculus. This book will appeal to advanced undergraduate and graduate students interested in the relations between algebra, probability, and quantum theory. It also addresses a more advanced audience by covering other topics related to non-commutativity in stochastic calculus, Lévy processes, and the Malliavin calculus. Cover 1 Half-title 3 Series information 4 Title page 5 Copyright information 6 Dedication 7 Table of contents 9 Notation 13 Preface 15 Introduction 17 1 Boson Fock space 23 1.1 Annihilation and creation operators 23 1.2 Lie algebras on the boson Fock space 26 1.3 Fock space over a Hilbert space 28 1.3.1 Creation and annihilation operators on Γ[sub(s)](eufrak h) 29 1.3.2 Weyl operators 30 Exercises 31 2 Real Lie algebras 32 2.1 Real Lie algebras 32 2.1.1 Adjoint action 33 2.2 Heisenberg–Weyl Lie algebra eufrak hw 34 2.2.1 Boson Fock space representation 34 2.2.2 Matrix representation 34 2.2.3 Representation on eufrak h= L[sup(2)] (mathbb R, dt ) 35 2.3 Oscillator Lie algebra eufrak osc 35 2.3.1 Matrix representation 35 2.3.2 Boson Fock space representation 35 2.3.3 The harmonic oscillator 36 2.4 Lie algebra eufrak sl[sub(2)] (mathbb R) 36 2.4.1 Boson Fock space representation 37 2.4.2 Matrix representation 37 2.4.3 Adjoint action 38 2.4.4 Representation of eufrak sl[sub(2)] (mathbb R) on L[sup(2)][sub(mathbb C)](mathbb R[sub(+)] , γ[sub(β)](τ) dτ) 38 2.4.5 Construction on the one-dimensional Gaussian space - β = 1/2 40 2.4.6 Construction on the two-dimensional Gaussian space - β = 1 41 2.5 Affine Lie algebra 42 2.6 Special orthogonal Lie algebras 43 2.6.1 Lie algebra so(2) 43 2.6.2 Lie algebra so(3) 44 2.6.3 Finite-dimensional representations of so(3) 45 2.6.4 Two-dimensional representation of so(3) 45 2.6.5 Adjoint action 46 Notes 48 Exercises 48 3 Basic probability distributions on Lie algebras 49 3.1 Gaussian distribution on eufrak hw 49 3.1.1 Gaussian Hilbert space representation 50 3.1.2 Hermite representation 52 3.2 Poisson distribution on eufrak osc 53 3.2.1 Poisson Hilbert space representation 54 3.2.2 Poisson-Charlier representation on the boson Fock space 55 3.2.3 Adjoint action 57 3.3 Gamma distribution on eufrak sl[sub(2)] (mathbb R) 58 3.3.1 Probability distributions 60 3.3.2 Laguerre polynomial representation 61 3.3.3 The case β = 1 65 3.3.4 Adjoint action 66 Exercises 66 4 Noncommutative random variables 69 4.1 Classical probability spaces 69 4.2 Noncommutative probability spaces 70 4.2.1 Noncommutative examples 72 4.3 Noncommutative random variables 76 4.3.1 Where are noncommutative random variables valued? 78 4.4 Functional calculus for Hermitian matrices 79 4.5 The Lie algebra so(3) 81 4.5.1 Two-dimensional representation of so(3) 81 4.5.2 Three-dimensional representation of so(3) on eufrak h = mathbb C[sup(3)] 84 4.5.3 Two-dimensional representation of so(3) 85 4.6 Trace and density matrix 87 4.7 Spin measurement and the Lie algebra so(3) 92 Notes 94 Exercises 94 5 Noncommutative stochastic integration 97 5.1 Construction of the Fock space 97 5.1.1 Construction from a positive definite function 97 5.1.2 Construction via tensor products 100 5.2 Creation, annihilation, and conservation operators 102 5.3 Quantum stochastic integrals 105 5.4 Quantum Itô table 108 Notes 110 Exercises 110 6 Random variables on real Lie algebras 112 6.1 Gaussian and Poisson random variables on osc 112 6.2 Meixner, gamma, and Pascal random variables on eufrak sl[sub(2)] (mathbb R) 116 6.3 Discrete distributions on so(2) and so(3) 118 6.3.1 The Bernoulli distribution on so(2) 118 6.3.2 The three-point distribution on so(3) 119 6.4 The Lie algebra e(2) 119 6.4.1 The case α = 0 120 6.4.2 The case α neq 0 121 Notes 121 Exercises 121 7 Weyl calculus on real Lie algebras 125 7.1 Joint moments of noncommuting random variables 125 7.2 Combinatorial Weyl calculus 128 7.2.1 Lie-theoretic Weyl calculus 129 7.3 Heisenberg–Weyl algebra 129 7.3.1 Functional calculus on the Heisenberg–Weyl algebra 129 7.3.2 Wigner functions on the Heisenberg–Weyl algebra 131 7.4 Functional calculus on real Lie algebras 136 7.5 Functional calculus on the affine algebra 139 7.6 Wigner functions on so(3) 144 7.6.1 Group-theoretical Wigner function 146 7.6.2 Probabilistic Wigner density 148 7.7 Some applications 150 7.7.1 Quantum optics 150 7.7.2 Time-frequency analysis 151 Notes 152 Exercises 152 8 Lévy processes on real Lie algebras 153 8.1 Definition 153 8.2 Schürmann triples 156 8.2.1 Quantum stochastic differentials 161 8.3 Lévy processes on eufrak hw and eufrak osc 162 8.4 Classical processes 164 Notes 169 Exercises 170 9 A guide to the Malliavin calculus 171 9.1 Creation and annihilation operators 171 9.1.1 Multiple stochastic integrals 172 9.1.2 Annihilation operator 173 9.1.3 Duality relation 175 9.2 Wiener space 177 9.2.1 Gradient and divergence operators 181 9.3 Poisson space 184 9.3.1 Finite difference gradient 187 9.3.2 Divergence operator 190 9.4 Sequence models 190 Notes 195 Exercises 195 10 Noncommutative Girsanov theorem 200 10.1 General method 200 10.2 Quasi-invariance on eufrak osc 202 10.3 Quasi-invariance on eufrak sl[sub(2)] (mathbb R) 205 10.4 Quasi-invariance on eufrak hw 206 10.5 Quasi-invariance for Lévy processes 207 10.5.1 Brownian motion 207 10.5.2 The Poisson process 208 10.5.3 The gamma process 209 10.5.4 The Meixner process 210 Notes 211 Exercises 211 11 Noncommutative integration by parts 212 11.1 Noncommutative gradient operators 212 11.2 Affine algebra 214 11.3 Noncommutative Wiener space 219 11.3.1 Noncommutative gradient 219 11.3.2 Integration by parts 221 11.3.3 Closability 224 11.3.4 Divergence operator 226 11.3.5 Relation to the commutative case 233 11.4 The white noise case 234 11.4.1 Iterated integrals 237 Notes 237 Exercises 238 12 Smoothness of densities on real Lie algebras 239 12.1 Noncommutative Wiener space 239 12.2 Affine algebra 244 12.3 Towards a Hörmander-type theorem 246 12.3.1 Derivative of a quantum stochastic integral 248 12.3.2 Derivative of the solution 249 12.3.3 The other flow 250 Exercises 252 Appendix 253 A.1 Polynomials 253 A.1.1 General idea 253 A.1.2 Finite support 253 A.1.3 Infinite support 254 A.1.3.1 Hermite polynomials 255 A.1.3.2 Poisson–Charlier polynomials 257 A.1.3.3 Meixner polynomials 259 A.2 Moments and cumulants 261 A.3 Fourier transform 263 A.4 Cauchy–Stieltjes transform 265 A.5 Adjoint action 266 A.6 Nets 267 A.7 Closability of linear operators 268 A.8 Tensor products 269 A.8.1 Tensor products of Hilbert spaces 269 A.8.2 Tensor products of L[sup(2)] spaces 270 Exercise solutions 271 Chapter 1 271 Chapter 2 272 Chapter 3 275 Chapter 4 278 Chapter 5 281 Chapter 6 282 Chapter 7 288 Chapter 8 288 Chapter 9 289 Chapter 10 291 Chapter 11 292 Chapter 12 292 References 293 Index 301 Machine Generated Contents Note: 1.boson Fock Space -- 1.1.annihilation And Creation Operators -- 1.2.lie Algebras On The Boson Fock Space -- 1.3.fock Space Over A Hilbert Space -- Exercises -- 2.real Lie Algebras -- 2.1.real Lie Algebras -- 2.2.heisenberg -- Weyl Lie Algebra -- 2.3.oscillator Lie Algebra Osc -- 2.4.lie Algebra Sl2(r) -- 2.5.affine Lie Algebra -- 2.6.special Orthogonal Lie Algebras -- Exercises -- 3.basic Probability Distributions On Lie Algebras -- 3.1.gaussian Distribution On -- 3.2.poisson Distribution On Osc -- 3.3.gamma Distribution On Sl2(r) -- Exercises -- 4.noncommutative Random Variables -- 4.1.classical Probability Spaces -- 4.2.noncommutative Probability Spaces -- 4.3.noncommutative Random Variables -- 4.4.functional Calculus For Hermitian Matrices -- 4.5.the Lie Algebra So(3) -- 4.6.trace And Density Matrix -- 4.7.spin Measurement And The Lie Algebra So(3) -- Exercises -- 5.noncommutative Stochastic Integration -- 5.1.construction Of The Fock Space --^ 5.2.creation, Annihilation, And Conservation Operators -- 5.3.quantum Stochastic Integrals -- 5.4.quantum Ito Table -- Exercises -- 6.random Variables On Real Lie Algebras -- 6.1.gaussian And Poisson Random Variables On Osc -- 6.2.meixner, Gamma, And Pascal Random Variables On Sl2(r) -- 6.3.discrete Distributions On So(2) And So(3) -- 6.4.the Lie Algebra E(2) -- Exercises -- 7.weyl Calculus On Real Lie Algebras -- 7.1.joint Moments Of Noncommuting Random Variables -- 7.2.combinatorial Weyl Calculus -- 7.3.heisenberg -- Weyl Algebra -- 7.4.functional Calculus On Real Lie Algebras -- 7.5.functional Calculus On The Affine Algebra -- 7.6.wigner Functions On So(3) -- 7.7.some Applications -- Exercises -- 8.levy Processes On Real Lie Algebras -- 8.1.definition -- 8.2.schurmann Triples -- 8.3.levy Processes On And Osc -- 8.4.classical Processes -- Exercises -- 9.a Guide To The Malliavin Calculus -- 9.1.creation And Annihilation Operators -- 9.2.wiener Space -- 9.3.poisson Space --^ 9.4.sequence Models -- Exercises -- 10.noncommutative Girsanov Theorem -- 10.1.general Method -- 10.2.quasi-invariance On Osc -- 10.3.quasi-invariance On Sl2(r) -- 10.4.quasi-invariance On -- 10.5.quasi-invariance For Levy Processes -- Exercises -- 11.noncommutative Integration By Parts -- 11.1.noncommutative Gradient Operators -- 11.2.affine Algebra -- 11.3.noncommutative Wiener Space -- 11.4.the White Noise Case -- Exercises -- 12.smoothness Of Densities On Real Lie Algebras -- 12.1.noncommutative Wiener Space -- 12.2.affine Algebra -- 12.3.towards A Hormander-type Theorem -- Exercises -- Appendix -- A.1.polynomials -- A.2.moments And Cumulants -- A.3.fourier Transform -- A.4.cauchy -- Stieltjes Transform -- A.5.adjoint Action -- A.6.nets -- A.7.closability Of Linear Operators -- A.8.tensor Products -- Exercise Solutions -- Chapter 1 -- Chapter 2 -- Chapter 3 -- Chapter 4 -- Chapter 5 -- Chapter 6 -- Chapter 7 -- Chapter 8 -- Chapter 9 -- Chapter 10 -- Chapter 11 -- Chapter 12. Uwe Franz, Université De Franche-comté, Nicolas Privault, Nanyang Technological University, Singapore. Includes Bibliographical References And Index.