Quantum probability and spectral analysis of graphs. With a foreword by Professor Luigi Accardi.

*(English)*Zbl 1141.81005
Theoretical and Mathematical Physics. Berlin: Springer (ISBN 978-3-540-48862-0/hbk). xviii, 371 p. (2007).

This very well written book by A. Hora and N. Obata has a double interest for the reader:

Firstly, it is a very accessible introduction for the non expert to a few rapidly evolving areas of mathematics such as spectral analysis of graphs, quantum probability theory and combinatorial representation theory of the symmetric group. On that side, this monograph seems to be the first publication providing a synthesis of a very vast mathematical literature in these areas by giving to the reader a concise and self contained panorama of existing results, and focusing efficiently on the main ideas and techniques of these areas.

Secondly, it uses non commutative probability theory as a common theory and technique to obtain results in a series of fields that are traditionally classified apart from non commutative probability theory (e.g. graph theory, representation theory, and even classical probability theory).

From that point of view, this book is important to the quantum probability community and emphasizes well many new applications of quantum probability to other areas of mathematics.

The unifying concepts in this book are those of moments of non commutative variables, adjacency matrix theory and Fock space theory, various notions of non commutative independence and their relations with operations on graphs and representations, quantum decomposition of a random variable, and limit theorems (in particular, the central limit theorem).

A list of errata of the book can be found at

http://www.math.nagoya-u.ac.jp/\(\sim\)hora/HObook-corrections.pdf .

Firstly, it is a very accessible introduction for the non expert to a few rapidly evolving areas of mathematics such as spectral analysis of graphs, quantum probability theory and combinatorial representation theory of the symmetric group. On that side, this monograph seems to be the first publication providing a synthesis of a very vast mathematical literature in these areas by giving to the reader a concise and self contained panorama of existing results, and focusing efficiently on the main ideas and techniques of these areas.

Secondly, it uses non commutative probability theory as a common theory and technique to obtain results in a series of fields that are traditionally classified apart from non commutative probability theory (e.g. graph theory, representation theory, and even classical probability theory).

From that point of view, this book is important to the quantum probability community and emphasizes well many new applications of quantum probability to other areas of mathematics.

The unifying concepts in this book are those of moments of non commutative variables, adjacency matrix theory and Fock space theory, various notions of non commutative independence and their relations with operations on graphs and representations, quantum decomposition of a random variable, and limit theorems (in particular, the central limit theorem).

A list of errata of the book can be found at

http://www.math.nagoya-u.ac.jp/\(\sim\)hora/HObook-corrections.pdf .

Reviewer: Benoit Collins (Ottawa)

##### MSC:

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

81S25 | Quantum stochastic calculus |

05Cxx | Graph theory |

20C30 | Representations of finite symmetric groups |