Quantum probability communications / managing editors, R.L. Hudson, J.M. Lindsay ; editorial board, Ph. Biane ... [et al.] ; advisory board, L. Accardi ... [et al.].

Format:

Book

Contributor:

Hudson, R. L. (Robin Lyth), 1960-

Lindsay, J. M.

Biane, P. (Philippe)

Accardi, L. (Luigi), 1947-

Series:

QP-PQ: QUANTUM PROBABILITY AND WHITE NOISE ANALYSIS

QP-PQ ; v. 10

Language:

English

Subjects (All):

Probabilities.

Quantum theory.

Stochastic processes.

Markov processes.

Physical Description:

1 online resource (372 p.)

Place of Publication:

Singapore ; London : World Scientific, 1998.

Language Note:

English

Summary:

Much has changed in the world of quantum probability since the publication of the last volume in this series. Giants in the field, such as P-A Meyer, K R Parthasarathy and W von Waldenfels, have reached the age of retirement. Readers will, however, be pleased to see evidence in the present volume that Partha remains as creatively active as ever. The field itself, regarded at one time as the esoteric province of a small group of devotees, has come of age. It has attracted the enthusiastic commitment of an ever-growing army of young mathematicians and physicists, many of whom are represented her

Contents:

CONTENTS; PREFACE; CLASSICAL AND QUANTUM STOCHASTIC CALCULUS; Introduction; Notations; I From classical to quantum stochastic calculus; I.1 Ito calculus approch to quantum stochastic calculus; I.1.1 Ito calculus on Fock space; I.1.2 Adaptedness revisited; I.1.3 Quantum stochastic integrals revisited; I.1.4 Complete solution of A-M's equations; I.2 Quantum semimartingales; I.2.1 An algebra of quantum semimartingales; I.2.2 Quantum square and angle brackets; I.2.3 Functional quantum Ito formulas; I.3 Chaotic expansion of operators; I.3.1 Maassen kernels as iterated integral representations

I.3.2 The case of Hilbert-Schmidt operatorsI.4 Hilbertian quasimartingales and quantum stop times; I.4.1 Enchev's Hilbertian quasimartingales; I.4.2 Quantum stop times and quasimartingales; I.4.3 Quantum strong Markov process; II From quantum to classical stochastic calculus; II.1 Extension and unification of classical stochastic calculus; II.1.1 Probabilistic interpretations of Fock space; II.1.2 Interpretation of the Ito calculus; II.1.3 Extension of classical stochastic calculus; II.1.4 Extension of the classical Ito formula; II .2 Representation of Wiener space endomorphisms

II.2.1 The martingale preserving endomorphismsII.2.2 The kernel mystery; II.2.3 Some more general endomorphisnas; II.2.4 Levy transform as a counter-example to Q.S.D.E.; Conclusion; REFERENCES; QUANTUM MARKOV PROCESSES AND GROUP REPRESENTATIONS; 0 INTRODUCTION; 1 DILATIONS AND GROUP REPRESENTATIONS; 1.1 Dilations of quantum Markov transition systems.; 1.2 Quantum Markov transition semigroups associated to locally compact groups.; 2 BROWNIAN MOTION, RAY-KNIGHT THEOREMS, AND HEISENBERG GROUPS; 2.1 Bessel processes.; 2.2 Ray-Knight theorems.

2.3 The approximation of local time and the Yule process.2.4 Heisenberg group and its representations.; 2.5 Gelfand pairs associated with the Heisenberg group.; 2.6 Non-commutative brownian motion.; 2.7 Restriction to Gelfand algebras.; 3 FREE BROWNIAN MOTION AND AUTOMORPHISM GROUPS OF TREES; 3.2 Harmonic analysis on automorphism groups of trees.; 3.3 Semigroup associated to the distance function.; REFERENCES; Elements of Quantum Probability; INTRODUCTION; 1. WHY CLASSICAL PROBABILITY DOES NOT SUFFICE; 1.1 An experiment with polarisers; 1.2 An improved experiment; 1.3 The decisive experiment

1.4 The Orsay experiment as a card game2. TOWARDS A MATHEMATICAL MODEL; 2.1 A mathematical description of polarization; 2.2 The full quantum mechanical truth about polarization; 2.3 Finite dimensional models; 2.4 The mathematical model of Aspect's experiment; 3. EXAMPLES OF QUANTUM COIN TOSSING; 3.1 Model of a quantum coin toss; 3.2 Classical coin tossing; 3.3 Polarizers lead to quantum coins; 3.4 From coin tossing to braids; 3.5 From coin tossing to nuclear magnetic resonance; 4. APPENDIX: AN OUTLOOK ONTO THE FULL MATHEMATICAL MODEL; 4.1 Real-valued random variables

4.2 Towards infinite dimensions

Notes:

Description based upon print version of record.

Includes bibliographical references.

ISBN:

9781283635844

1283635844

9789812816054

9812816054

OCLC:

813395998