##
**Quantum probability for probabilists.**
*(English)*
Zbl 0773.60098

Lecture Notes in Mathematics. 1538. Berlin: Springer-Verlag. x, 287 p. (1993).

These notes accumulate the material presented at the Strasbourg Probability Seminar to teach and learn the noncommutative probability. The text, a first version of which appeared in successive volumes of the Séminaire de Probabilités, has been unified, augmented and translated into English. The main topic of the notes is quantum stochastic calculus, a new quickly developing field on borders of stochastic processes and operator theory in Fock spaces and taking some inspirations from quantum theory of open systems [see the review of K. R. Parthasarathy’s book: ”An introduction to quantum stochastic calculus.” (1992), Zbl 0751.60046.]

The volume comprises 7 chapters and 5 appendices. Chapter I gives an introduction into the algebraic formulation of the noncommutative probability, while Chapter II deals in detail with the first nontrivial example of two-level systems. Formally this case plays the role of Bernoulli scheme in standard presentations of probability theory, however in the noncommutative case it has much reacher algebraic structure. The case of commuting spins leads to a generalization of the de Moivre- Laplace theorem, preparing for Gaussian states on canonical commutation relations to be introduced in Chapter III, while the case of anticommuting spins discloses a remarkable description of quantum Bernoulli laws as Gaussian states on Clifford algebras.

Chapter III is titled “Harmonic oscillator” and gives a brief account of canonical commutation relations, Weyl operators and Gaussian (quasi- free) states for one degree of freedom. Fock spaces for infinitely many degrees of freedom are described in Chapters IV and V, stressing the relation with the Wiener-Itô multiple stochastic integrals. Considerable attention is paid to Maassen’s kernel calculus, generalizing Berezin’s formalism of second quantization and describing products of operators in terms of their kernels.

The main topic “Stochastic calculus in Fock space” is reached in Chapter VI, which has 70 pages, i.e. 1/4 of the volume. It contains the definition of quantum stochastic integrals of adapted operator-valued processes in Fock space with respect to basic operator martingales, quantum Itô’s formula and the existence and uniqueness theorem for quantum stochastic differential equations and flows. Along with the basic results of Hudson-Parthasarathy-Evans on the case of bounded operator coefficients, a variety of more recent achievements is surveyed, such as Azéma martingales and their noncommutative interpretation, representation of solutions of quantum stochastic differential equations as time-ordered exponentials, approaches to the “quantum Stone’s theorem” and to equations with unbounded coefficients. Finally, Chapter VII gives an introduction to processes with independent increments on coalgebras, developed by the Heidelberg school.

Appendices 1-4 contain supplementary background material on operator theory, noncommutative conditioning, two-events and \(C^*\)-algebras. Of special interest is Appendix 5 devoted to applications of Fock space to the theory of local times, derivation of Dynkin’s formula for expectations relative to stochastic processes of local times of a symmetric Markov process, its “supersymmetric” version and applications to the self-intersections for two-dimensional Brownian motion.

The central concept of the volume is the Fock space, and within this frame the lectures are remarkable for their scope. Written with encyclopaedic erudition and designed as a pilot survey rather than a pedantic course, they will be interesting to a broad audience including probabilists eager to penetrate the noncommutative domain and to specialists in this field who will find a variety of new stimulating observations and connections with other branches of mathematics.

The volume comprises 7 chapters and 5 appendices. Chapter I gives an introduction into the algebraic formulation of the noncommutative probability, while Chapter II deals in detail with the first nontrivial example of two-level systems. Formally this case plays the role of Bernoulli scheme in standard presentations of probability theory, however in the noncommutative case it has much reacher algebraic structure. The case of commuting spins leads to a generalization of the de Moivre- Laplace theorem, preparing for Gaussian states on canonical commutation relations to be introduced in Chapter III, while the case of anticommuting spins discloses a remarkable description of quantum Bernoulli laws as Gaussian states on Clifford algebras.

Chapter III is titled “Harmonic oscillator” and gives a brief account of canonical commutation relations, Weyl operators and Gaussian (quasi- free) states for one degree of freedom. Fock spaces for infinitely many degrees of freedom are described in Chapters IV and V, stressing the relation with the Wiener-Itô multiple stochastic integrals. Considerable attention is paid to Maassen’s kernel calculus, generalizing Berezin’s formalism of second quantization and describing products of operators in terms of their kernels.

The main topic “Stochastic calculus in Fock space” is reached in Chapter VI, which has 70 pages, i.e. 1/4 of the volume. It contains the definition of quantum stochastic integrals of adapted operator-valued processes in Fock space with respect to basic operator martingales, quantum Itô’s formula and the existence and uniqueness theorem for quantum stochastic differential equations and flows. Along with the basic results of Hudson-Parthasarathy-Evans on the case of bounded operator coefficients, a variety of more recent achievements is surveyed, such as Azéma martingales and their noncommutative interpretation, representation of solutions of quantum stochastic differential equations as time-ordered exponentials, approaches to the “quantum Stone’s theorem” and to equations with unbounded coefficients. Finally, Chapter VII gives an introduction to processes with independent increments on coalgebras, developed by the Heidelberg school.

Appendices 1-4 contain supplementary background material on operator theory, noncommutative conditioning, two-events and \(C^*\)-algebras. Of special interest is Appendix 5 devoted to applications of Fock space to the theory of local times, derivation of Dynkin’s formula for expectations relative to stochastic processes of local times of a symmetric Markov process, its “supersymmetric” version and applications to the self-intersections for two-dimensional Brownian motion.

The central concept of the volume is the Fock space, and within this frame the lectures are remarkable for their scope. Written with encyclopaedic erudition and designed as a pilot survey rather than a pedantic course, they will be interesting to a broad audience including probabilists eager to penetrate the noncommutative domain and to specialists in this field who will find a variety of new stimulating observations and connections with other branches of mathematics.

Reviewer: A.S.Holevo (Moskva)

### MSC:

60K40 | Other physical applications of random processes |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60H99 | Stochastic analysis |

81S25 | Quantum stochastic calculus |

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