Quantum Probability for Probabilists

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November 1995



In recent years, the classical theory of stochastic integration and stochastic differential equations has been extended to a non-commutative set-up to develop models for quantum noises. The author, a specialist of classical stochastic calculus and martingale theory, tries to provide an introduction to this rapidly expanding field in a way which should be accessible to probabilists familiar with the Ito integral. It can also, on the other hand, provide a means of access to the methods of stochastic calculus for physicists familiar with Fock space analysis. For this second edition, the author has added about 30 pages of new material, mostly on quantum stochastic integrals.


I: Non Commutative Probability.-
II: Spin.-
III: The Harmonic Oscillator.-
IV: Fock Space (1).- V. Multiple Fock Spaces.- VI. Stochastic Calculus on Fock Space.- VII. Independence.-
Appendix 1: Functional Analysis.- Hilbert-Schmidt operators (1).- Trace class operators (2).- Duality properties (3) Weak convergence properties (4).- Weak topologies for operators (5).- Tensor products of Hilbert spaces (6-7).-
Appendix 2: Conditioning and Kernels.- Conditioning: discrete case (1).- Conditioning: continuous case (2).- Example of the canonical pair (3).- Multiplicity theory (4).- Classical kernels (5).- Non commutative kernels, first form (6).- second form (7).- Completely positive maps (8).- Some difficulties (9).-
Appendix 3: Two Events.- 1. Elementary theory.- Application of spectral theory (2).- Some elementary properties (3).- Positive elements (4).- Symbolic calculus for s.a. elements (5).- Applications (6).- Characterization of positive elements (7).- A few inequalities (8).- Existence of many states (1).- Representations and the GNS theorem (2-3).- Examples from toy Fock space theory (4).- Quotient algebras and approximate units (5).- 3. Von Neumann algebras.- Weak topologies and normal states (1).- Von Neumann's bicommutant theorem (2-3).- Kaplanski's density theorem (4).- The predual (5).- Normality and order continuity (6).- About integration theory (7).- Measures with bounded density (8).- The linear Radon-Nikodym theorem (9).- The KMS condition (10).- Entire vectors (11).- 4. The Tomita-Takesaki theory.- Elementary geometric properties (1).- The main operators (2-3).- Interpretation of the adjoint (4).- The modular property (5).- Using the linear RN theorem (6).- The main computation (7).- The three main theorems (8).- Additional results (9).- Examples (10).-
Appendix 5: Local Times and Fock Space.- 1. Dynkin's formula.- Symmetric Markov semigroups and processes (1).- Dynkin's formula (2).- Sketch of the Marcus-Rosen approach to the continuity of local times (3).- 2. Le Jan's "supersymmetric" approach.- Notations of complex Brownian motion (1).- Computing the Wiener product (2).- Stratonovich integral and trace (4).- Expectation of the exponential of an element of the second chaos (5).- Exponential formula in the antisymmetric case (7).- Supersymmetric Fock space: the Wick and Wiener products (8).- Properties of the Wiener product (9).- Applications to local times (sketch) (10).- References.- Index of Notation.
EAN: 9783540602705
ISBN: 3540602704
Untertitel: 2nd ed. 1995. Book. Sprache: Englisch.
Verlag: Springer
Erscheinungsdatum: November 1995
Seitenanzahl: 328 Seiten
Format: kartoniert
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