Quantum stochastic processes and noncommutative geometry.

*(English)*Zbl 1144.81002
Cambridge Tracts in Mathematics 169. Cambridge: Cambridge University Press (ISBN 0-521-83450-3/hbk). x, 290 p. (2007).

Quantum probability is a non-commutative version of probability theory that has applications to the physics of open and irreversible quantum systems. It comprises a solidly developed theory of non-commutative processes, stochastic calculus, stochastic differential equations, and stopping times.

Several reference books have already been published on the subject, including [K. R. Parthasarathy, An introduction to quantum stochastic calculus. Monographs in Mathematics. 85. Basel etc.: Birkhäuser (1992; Zbl 0751.60046)] and [P.-A. Meyer, Quantum probability for probabilists. Lecture Notes in Mathematics. 1538. Berlin: Springer-Verlag (1993; Zbl 0773.60098)], following the fundamental paper on quantum stochastic calculus by R. L. Hudson and K. R. Parthasarathy [Commun. Math. Phys. 93, 301–323 (1984; Zbl 0546.60058)].

The book under review deals with two important developments of the theory, namely the extension of quantum stochastic calculus to unbounded operator processes and its connection with non-commutative geometry. Indeed, the idea of unifying the geometric and probabilistic non-commutative frameworks has attracted a lot of interest and raised high expectations since the early developments of quantum probability. In addition to that, the book contains useful background material in algebra and differential geometry. The treatment of unbounded operator processes is partly based on the authors’ own work.

The text is divided into 8 chapters on non-commutative probability and stochastic processes, plus one chapter devoted to relations with non-commutative geometry. The approach adopted throughout the book relies on symmetric Fock spaces and modules, thus excluding other free or monotone probability models.

The first three chapters contain algebraic preliminaries. Basic material on von Neumann algebras, complete positivity, and symmetric Fock spaces is reviewed in Chapter 2, followed by Chapter 3 on the structure of quantum dynamical semigroups (i.e. contractive semigroups of completely positive maps) and of their generators. This includes the Linblad and Christensen-Evans theorems, and a thorough review of strongly continuous quantum dynamical semigroups.

In Chapter 4 the theory of Hilbert \(C^*\)- and von Neumann modules, and of their group actions, is presented. Hilbert modules will be useful later on for the treatment of Evans-Hudson type quantum stochastic calculus.

In Chapter 5, quantum stochastic integrals and the quantum Itô formula are introduced for bounded operator-valued integrand processes, and the existence and uniqueness of solutions to Hudson-Parthasarathy quantum stochastic differential equations on Fock spaces are discussed in this context. Evans-Hudson quantum stochastic differential equations on Fock modules are also considered, and an equation with bounded coefficients is solved in this framework.

In Chapter 6 the Hudson-Parthasarathy and Evans-Hudson dilations of uniformly continuous quantum dynamical semigroups of completely positive maps on a given algebra are realized as the solutions of quantum stochastic differential equations. These equations are used to describe the stochastic evolution of a quantum system coupled to a reservoir which is modeled according to a quantum stochastic process in Fock space.

Existence results for quantum stochastic differential equations with unbounded coefficients are stated in Chapter 7 by adapting the tools of Chapter 5, based on results by Mohari and Sinha. An application to a physical model (the quantum damped harmonic oscillator) is given, in which unbounded coefficients arise naturally.

In Chapter 8, the dilations of quantum dynamical semigroups with unbounded generators are constructed, in relation to the stochastic differential equations with unbounded coefficients of Chapter 7. This is done under specific conditions, for quantum dynamical semigroups that are symmetric covariant and covariant under a Lie group action, respectively in the Hudson-Parthasarathy and Evans-Hudson cases.

Chapter 9 provides an introduction to classical and non-commutative differential geometry which is of interest in its own right. Actually only a few pages of the last section are dealing with quantum Brownian motion on non-commutative manifolds. Nevertheless they provide a good starting point in this direction with a number of concrete examples.

This book represents a valuable and up to date introduction to quantum stochastic calculus that includes recent developments in the unbounded operator process case. It will also provide solid foundations for mathematicians and researchers wishing to further explore the interactions between the probabilistic and geometrical aspects of non-commutativity.

Several reference books have already been published on the subject, including [K. R. Parthasarathy, An introduction to quantum stochastic calculus. Monographs in Mathematics. 85. Basel etc.: Birkhäuser (1992; Zbl 0751.60046)] and [P.-A. Meyer, Quantum probability for probabilists. Lecture Notes in Mathematics. 1538. Berlin: Springer-Verlag (1993; Zbl 0773.60098)], following the fundamental paper on quantum stochastic calculus by R. L. Hudson and K. R. Parthasarathy [Commun. Math. Phys. 93, 301–323 (1984; Zbl 0546.60058)].

The book under review deals with two important developments of the theory, namely the extension of quantum stochastic calculus to unbounded operator processes and its connection with non-commutative geometry. Indeed, the idea of unifying the geometric and probabilistic non-commutative frameworks has attracted a lot of interest and raised high expectations since the early developments of quantum probability. In addition to that, the book contains useful background material in algebra and differential geometry. The treatment of unbounded operator processes is partly based on the authors’ own work.

The text is divided into 8 chapters on non-commutative probability and stochastic processes, plus one chapter devoted to relations with non-commutative geometry. The approach adopted throughout the book relies on symmetric Fock spaces and modules, thus excluding other free or monotone probability models.

The first three chapters contain algebraic preliminaries. Basic material on von Neumann algebras, complete positivity, and symmetric Fock spaces is reviewed in Chapter 2, followed by Chapter 3 on the structure of quantum dynamical semigroups (i.e. contractive semigroups of completely positive maps) and of their generators. This includes the Linblad and Christensen-Evans theorems, and a thorough review of strongly continuous quantum dynamical semigroups.

In Chapter 4 the theory of Hilbert \(C^*\)- and von Neumann modules, and of their group actions, is presented. Hilbert modules will be useful later on for the treatment of Evans-Hudson type quantum stochastic calculus.

In Chapter 5, quantum stochastic integrals and the quantum Itô formula are introduced for bounded operator-valued integrand processes, and the existence and uniqueness of solutions to Hudson-Parthasarathy quantum stochastic differential equations on Fock spaces are discussed in this context. Evans-Hudson quantum stochastic differential equations on Fock modules are also considered, and an equation with bounded coefficients is solved in this framework.

In Chapter 6 the Hudson-Parthasarathy and Evans-Hudson dilations of uniformly continuous quantum dynamical semigroups of completely positive maps on a given algebra are realized as the solutions of quantum stochastic differential equations. These equations are used to describe the stochastic evolution of a quantum system coupled to a reservoir which is modeled according to a quantum stochastic process in Fock space.

Existence results for quantum stochastic differential equations with unbounded coefficients are stated in Chapter 7 by adapting the tools of Chapter 5, based on results by Mohari and Sinha. An application to a physical model (the quantum damped harmonic oscillator) is given, in which unbounded coefficients arise naturally.

In Chapter 8, the dilations of quantum dynamical semigroups with unbounded generators are constructed, in relation to the stochastic differential equations with unbounded coefficients of Chapter 7. This is done under specific conditions, for quantum dynamical semigroups that are symmetric covariant and covariant under a Lie group action, respectively in the Hudson-Parthasarathy and Evans-Hudson cases.

Chapter 9 provides an introduction to classical and non-commutative differential geometry which is of interest in its own right. Actually only a few pages of the last section are dealing with quantum Brownian motion on non-commutative manifolds. Nevertheless they provide a good starting point in this direction with a number of concrete examples.

This book represents a valuable and up to date introduction to quantum stochastic calculus that includes recent developments in the unbounded operator process case. It will also provide solid foundations for mathematicians and researchers wishing to further explore the interactions between the probabilistic and geometrical aspects of non-commutativity.

Reviewer: Nicolas Privault (Hong Kong)

##### MSC:

81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |

81S25 | Quantum stochastic calculus |

46L53 | Noncommutative probability and statistics |

46L55 | Noncommutative dynamical systems |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81R15 | Operator algebra methods applied to problems in quantum theory |