Quantum stochastic processes and noncommutative geometry

Kalyan B. Sinha, Debashish Goswami0521834503, 9780521834506, 9780511269974

The classical theory of stochastic processes has important applications arising from the need to describe irreversible evolutions in classical mechanics; analogously quantum stochastic processes can be used to model the dynamics of irreversible quantum systems. Noncommutative, i.e. quantum, geometry provides a framework in which quantum stochastic structures can be explored. This book is the first to describe how these two mathematical constructions are related. In particular, key ideas of semigroups and complete positivity are combined to yield quantum dynamical semigroups (QDS). Sinha and Goswami also develop a general theory of Evans-Hudson dilation for both bounded and unbounded coefficients. The unique features of the book, including the interaction of QDS and quantum stochastic calculus with noncommutative geometry and a thorough discussion of this calculus with unbounded coefficients, will make it of interest to graduate students and researchers in functional analysis, probability and mathematical physics.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Contents……Page 7
Preface……Page 9
Notation……Page 11
1 Introduction……Page 13
2.1.1 C-algebras……Page 19
2.1.2 von Neumann algebras……Page 22
2.1.3 Group actions on C and von Neumann algebras……Page 28
2.2 Completely positive maps……Page 32
2.3.1 Some general theory……Page 38
2.3.2 Some results on perturbation of semigroups……Page 41
2.4 Fock spaces and Weyl operators……Page 42
3.1 Generators of uniformly continuous quantum dynamical semigroups: the theorems of Lindblad and Christensen–Evans……Page 45
3.2.1 Construction of quantum dynamical semigroups from form-generators……Page 51
3.2.2 Structure theorem for a class of strongly continuous quantum dynamical semigroups on B(h)……Page 67
3.2.3 Structure of generator of symmetric quantum dynamical semigroups……Page 73
3.2.4 A class of quantum dynamical semigroups on uniformly hyperfinite (U.H.F.) C-algebra……Page 80
4.1 Hilbert C-modules……Page 91
4.2 Hilbert von Neumann modules……Page 100
4.3.1 The case of Hilbert C-modules……Page 107
4.3.2 The case of Hilbert von Neumann modules……Page 111
5.1 Basic processes……Page 115
5.2 Stochastic integrals and quantum Itô formulae……Page 125
5.3 Hudson–Parthasarathy (H–P) type equations……Page 139
5.4.1 The formalism of map-valued quantum stochastic integration……Page 145
5.4.2 Solution of a class of map-valued quantum stochastic differential equations (Q.S.D.E.) with bounded coefficients……Page 149
6 Dilation of quantum dynamical semigroups with bounded generator……Page 159
6.1 Hudson–Parthasarathy (H–P) dilation……Page 161
6.2 Existence of structure maps and Evans–Hudson (E–H) dilation of Tt……Page 162
6.3 A duality property……Page 166
6.4 Appearance of Poisson terms in the dilation……Page 168
6.5 Implementation of E–H flow……Page 172
6.6 Dilation on a C-algebra……Page 173
6.7 Covariant dilation theory……Page 176
6.7.1 Covariant E.H theory for a C-algebra……Page 177
6.7.2 The von Neumann algebra case……Page 179
7.1 Notation and preliminary results……Page 181
7.2 Q.S.D.E. with unbounded coefficients……Page 185
7.3 Application: quantum damped harmonic oscillator……Page 193
8.1 Dilation of a class of covariant Q.D.S…….Page 197
8.1.1 Notations and preliminaries……Page 198
8.1.2 H–P dilation of a class of symmetric covariant quantum dynamical semigroups……Page 209
8.1.3 E–H dilation of covariant quantum dynamical semigroups……Page 219
8.2.1 E–H dilation……Page 230
8.2.2 Covariance of the E–H flow……Page 240
9.1 Basics of differential and Riemannian geometry……Page 243
9.2 Heat semigroup and Brownian motion on classical manifolds……Page 250
9.3 Noncommutative geometry……Page 260
9.4.1 Spectral triples on noncommutative torus……Page 265
9.4.2 Noncommutative 2d-dimensional plane……Page 269
9.4.3 Spectral triples on quantum Heisenberg manifold……Page 270
9.5.1 Volume form and scalar curvature on noncommutative torus……Page 276
9.5.2 Volume form for noncommutative 2d-plane……Page 283
9.6 Quantum Brownian motion on noncommutative manifolds……Page 285
9.6.1 Brownian motion on noncommutative torus……Page 287
9.6.2 Noncommutative 2d-dimensional plane……Page 288
9.6.3 Quantum Heisenberg manifolds……Page 289
References……Page 293
Index……Page 301