Quantum Probability And Infinite Dimensional Analysis: From Foundations To Applications

Cover......Page 1
QP-PQ: Quantum Probability and White Noise Analysis......Page 3
Quantum Probability And Infinite Dimensional Analysis......Page 4
ISBN 9789812561473......Page 5
CONTENTS......Page 8
Introduction......Page 6
1. Creation, annihilation, and neutral operators......Page 12
2. Polynomially symmetric measures......Page 14
3. Polynomially factorizable measures......Page 15
4. Probability measures by means of the CAN operators......Page 16
5.1. Probability measures on R with finite support......Page 17
5.2. Probability measums on R with infinite support......Page 18
6. Classical probability measures on the real line......Page 19
Acknowledgments......Page 21
References......Page 22
1. Introduction et prkliminaires......Page 23
1.2. Fonctions entieres a croissance exponentielle......Page 25
1.3. Transformation de Laplace......Page 27
1.4. Reprdsentation intdgmle de distributions positives......Page 28
2. Laplacien de LQvy agissant sur les distributions......Page 29
3. Semi groupe de LQvy agissant sur l’espace des fonctions......Page 32
References......Page 37
1. Introduction......Page 39
2. Hamiltonian of the model......Page 41
3. The \"Commutative\" case......Page 43
4.1. Quasi-commutation mles for bt and Ut.......Page 44
4.2. Normally ordered equation for Ut .......Page 46
4.3. Inner Langevin equation.......Page 47
4.4. Canonical form of the Jangevin equation.......Page 49
5. \"Anti-commutative\" case......Page 50
6. Conclusions......Page 51
References......Page 52
1. Accardi-Boiejko unitary isomorphism......Page 53
2. Pre-generating and generating functions......Page 55
3. Computation of orthogonal polynomials and Jacobi-Szegii parameters......Page 57
4. Orthogonal polynomials in terms of differential or difference operators......Page 60
5. Classical examples......Page 63
Acknowledgments......Page 65
References......Page 66
2. The physicals model and its stochastic limit......Page 67
3. Possible generalizations......Page 73
References......Page 77
1. Introduction......Page 78
2. The algebraic theory of finite dimensional multi-particle sys tems......Page 79
3. Transitional transformations......Page 85
4. Markovian multi-particle processes......Page 90
5. Interaction free branching processes......Page 97
6. Branching systems with interaction strength......Page 104
References......Page 115
1. Introduction......Page 116
2. Preliminaries......Page 117
3. Result......Page 121
4. Example......Page 123
References......Page 125
1. Introduction and preliminaries......Page 126
2.1. The Operator symbol......Page 128
2.2. Convergence of operators......Page 129
3. The Convolution product of operators......Page 130
3.1. Application to quantum stochastic diflerential equations......Page 132
References......Page 135
Regular Quantum Stochastic Cocycles have Exponential Product Systems B.V. Rajarama Bhat and J. Martin Lindsay......Page 137
1. Product systems of Hilbert W*-modules......Page 138
2. Quantum stochastic cocycles......Page 144
3. Product system of a stochastic cocycle......Page 147
4. Irreducible quantum stochastic dilations......Page 148
Acknowledgements......Page 149
References......Page 150
1. Introduction......Page 152
2. Definitions and Preliminaries......Page 153
3. Interaction of a single two-level atom with a single mode field......Page 155
4. Evolution of the atom-field system......Page 157
References......Page 163
Quantum Mechanics on the Circle through Hopf q-Deformations of the Kinematical Algebra with Possible Applications to LQvy Processes V.K. Dobrev, H.-D. Doebner and R. Twarock......Page 164
1. Introduction......Page 165
2. Short review of Borel quantisation with application to S1......Page 166
2.1. The kinematics......Page 167
2.2. The dynamics......Page 168
2.3. The kinematical algebra K(S1) and a family of evolution equations......Page 169
3. A discrete quantum kinematics based on a Hopf q-deformation of the kinematical algebra......Page 171
3.1.1. The position observables......Page 172
3.1.3. Coupling between the subalgebras: the Hopf q-deformation of the kinematical algebra......Page 173
3 . 2 . The quantum kinematics on S1......Page 174
4. The quantum dynamics with Hopf q-structure......Page 175
4.1.1. The right hand side of (37)......Page 176
4.1.2. The left hand side of (37)......Page 177
4.2. The q-deformed evolution equation......Page 178
4.3. The continuous limit......Page 180
4.4. Comparison with previous results......Page 181
Acknowledgments......Page 182
References......Page 183
1. Introduction......Page 185
2.1. The case of R......Page 187
2.2. The case of ZN......Page 192
2.3. The case of hw-Algebra......Page 196
3.1. The case of 2......Page 201
4. Relation of Algebraic and Quantum Random Walks......Page 204
5. Discussion......Page 209
References......Page 210
Dual Representations for the Schrodinger Algebra Philip Feinsilver and Rent! Schott......Page 212
2.1. Dual representations and r-matrices......Page 213
2.2. Commutation relations: Kirillov matriz, adjoint representation......Page 214
2.4. Dual representations and flags......Page 215
2.5. Dual representations for the Schrodinger algebra......Page 216
2.6. Double dual......Page 218
3.1. Splitting lemma......Page 219
3.2. Coodinates of the second kind found......Page 220
3.3. Basic polynomials and quantum observables......Page 223
References......Page 226
2. Coxeter groups......Page 227
3. Geometric Representation......Page 228
4. Adjoint Representation......Page 229
5. A Product of Trees......Page 230
6. Weak Amenability......Page 232
7. Haagerup Inequality......Page 233
8. Simplicity of the regular C*-algebra......Page 234
References......Page 236
1. Introduction......Page 238
2. Basics......Page 239
3. The Results......Page 241
4. Proofs......Page 242
References......Page 246
1. Introduction......Page 248
2. Definitions and preliminary results......Page 249
3. Construction of Gibbs states for the A-model......Page 252
4. Diagonal states and corresponding von Neumann algebras......Page 254
5.1. Potts model......Page 259
5.2. Markov random fields......Page 260
References......Page 261
1. Introduction......Page 263
2. The General Model......Page 265
3. The Boson Fock Space......Page 267
4. The Quantum Channel......Page 270
5. Control Gate......Page 273
6. Final Output AEter Second Beam Splitting......Page 275
References......Page 277
1. Introduction......Page 280
2. The Problem......Page 281
3. The Solutions......Page 283
4. Variant 1: Coincidences......Page 284
5. Variant 2: Demanding More......Page 285
Acknowledgments......Page 286
References......Page 287
1. Introduction: are quantum probabilities fixed by quantum determinism?......Page 288
2. Assumptions: degeneracy, functional invariance, unitary invariance......Page 291
3. The first part of the proof......Page 293
4. Completing the proof......Page 296
5. Discussion......Page 299
Acknowledgments......Page 302
References......Page 303
1. Introduction......Page 304
2. Stationary processes......Page 306
3. Volterra representations......Page 309
4. Concluding remark......Page 311
References......Page 312
1. Introduction.......Page 314
2. The It6 Hopf algebra.......Page 317
3. Calculus in the It6 Hopf algebra.......Page 319
4. Simple product integrals with system algebra.......Page 320
5. Double product integrals.......Page 323
6. The quantum Yang-Baxter equation.......Page 326
References......Page 329
1. Introduction......Page 331
2. Boolean Independence......Page 333
3. Monotone Independence......Page 335
4. F’reeness......Page 338
5. Mixed types of independence......Page 342
References......Page 346
1. Introduction......Page 348
2. Jacobi Matrix and its Spectral Measure......Page 350
3. Chaos Expansion for Gaussian and Poisson Process......Page 354
4. Jacobi field of a LQvy process......Page 358
5. Process of Meixner\'s type......Page 363
6. The usual Fock space representation of the Jacobi field of a LQvy process......Page 366
References......Page 370
1. Introduction......Page 373
2. Tomita-Takesaki scheme for transposition......Page 375
3. k-decomposability at the Hilbert-space level......Page 378
4. Application of local decomposability to low dimensional cases......Page 380
References......Page 384
1. Some general theory......Page 386
2. LQvy processes in compact Lie groups......Page 391
3. Invariant Markov processes in homogeneous spaces......Page 394
4. Limiting properties of LQvy processes......Page 397
5. Dynamical aspect......Page 401
References......Page 405
1. Introduction......Page 407
2. Correspondences and operator algebras......Page 409
3. Duality of W*-correspondences......Page 415
4. Applications of duality: Commutants and Canonical models......Page 419
References......Page 424
1. Introduction......Page 426
2. Definition and some examples......Page 427
3. Extendability of generalized Markov states on gauge-invariant parts......Page 430
References......Page 438
1. Introduction......Page 439
2. Test Particle Interacting with a Dilute Bose Gas......Page 443
3. The White Noise Approach......Page 446
3.1. Master Field......Page 447
3.2. Fock Representation of the Master Field......Page 448
3.3. Quantum White Noise, Langevin and Boltzmann Equations......Page 449
4. White Noise Approach without Fock-antiFock Representat ion......Page 451
4.1. Causal Time-Energy Quantum White Noise......Page 452
References......Page 457
1. Introduction......Page 459
2. The meaning of large deviation......Page 460
3. Main result......Page 462
References......Page 467
Quantum Optical Scenarios for Stochastic Resonance Vyacheslav Shatokhin, Thomas Wellens and Andreas Buchleitner......Page 468
References......Page 482
On a Classical Scheme in a Noncommutative Multiparameter Ergodic Theory Adam G. Skalski......Page 484
1. A classical scheme for proving individual ergodic theorems in the noncommutative context......Page 487
2. Unrestricted convergence of multiparameter averages......Page 492
3. Ergodic theorems for multiparameter Free Group Actions......Page 495
Acknowledgments......Page 501
References......Page 502
LBvy Processes and Tensor Product Systems of Hilbert Modules Michael Skeide......Page 503
1. The Arveson system of a LQvy process and the product system of its CP-semigroup......Page 504
2. The product system of a classical Markov semigroup......Page 507
3. The product system of the brownian semigroup and of the Ornstein-Uhlenbeck semigroup......Page 508
4. Hunt\'s formula for generators of CP-semigroup?......Page 511
References......Page 513
1. Introduction......Page 515
2. Respresentations of B(W)......Page 517
3. Generalizations of Bhat’s approach......Page 520
4. The generalization of Arveson\'s approach......Page 523
4.2 Observation.......Page 525
4.4 Applications.......Page 526
References......Page 527
The Hamiltonian of a Simple Pure Number Process Wilhelm von Waldenfels......Page 529
References......Page 534
1. Introduction......Page 536
2.1. Nuclear and Ezact C*-Algebms......Page 537
2.2. Noncommutative Entropy......Page 538
2.3. LijXngs......Page 539
2.4. Equivariant Lifting8......Page 542
3. Locally Equivariant LiRs and Entropy......Page 543
References......Page 546