1st Edition

Commutation Relations, Normal Ordering, and Stirling Numbers

By Toufik Mansour, Matthias Schork Copyright 2016
    528 Pages 20 B/W Illustrations
    by Chapman & Hall

    Commutation Relations, Normal Ordering, and Stirling Numbers provides an introduction to the combinatorial aspects of normal ordering in the Weyl algebra and some of its close relatives. The Weyl algebra is the algebra generated by two letters U and V subject to the commutation relation UV − VU = I. It is a classical result that normal ordering powers of VU involve the Stirling numbers.

    The book is a one-stop reference on the research activities and known results of normal ordering and Stirling numbers. It discusses the Stirling numbers, closely related generalizations, and their role as normal ordering coefficients in the Weyl algebra. The book also considers several relatives of this algebra, all of which are special cases of the algebra in which UV − qVU = hVs holds true. The authors describe combinatorial aspects of these algebras and the normal ordering process in them. In particular, they define associated generalized Stirling numbers as normal ordering coefficients in analogy to the classical Stirling numbers. In addition to the combinatorial aspects, the book presents the relation to operational calculus, describes the physical motivation for ordering words in the Weyl algebra arising from quantum theory, and covers some physical applications.

    Introduction
    Set Partitions, Stirling, and Bell Numbers
    Commutation Relations and Operator Ordering
    Normal Ordering in the Weyl Algebra and Relatives
    Content of the Book

    Basic Tools
    Sequences
    Solving Recurrence Relations
    Generating Functions
    Combinatorial Structures
    Riordan Arrays and Sheffer Sequences

    Stirling and Bell Numbers
    Definition and Basic Properties of Stirling and Bell Numbers
    Further Properties of Bell Numbers
    q-Deformed Stirling and Bell Numbers
    (p, q)-Deformed Stirling and Bell Numbers

    Generalizations of Stirling Numbers
    Generalized Stirling Numbers as Expansion Coefficients in Operational Relations
    Stirling Numbers of Hsu and Shiue: A Grand Unification
    Deformations of Stirling Numbers of Hsu and Shiue
    Other Generalizations of Stirling Numbers

    The Weyl Algebra, Quantum Theory, and Normal Ordering
    The Weyl Algebra
    Short Introduction to Elementary Quantum Mechanics
    Physical Aspects of Normal Ordering

    Normal Ordering in the Weyl Algebra—Further Aspects
    Normal Ordering in the Weyl Algebra
    Wick’s Theorem
    The Monomiality Principle
    Further Connections to Combinatorial Structures
    A Collection of Operator Ordering Schemes
    The Multi-Mode Case

    The q-Deformed Weyl Algebra and the Meromorphic Weyl Algebra
    Remarks on q-Commuting Variables
    The q-Deformed Weyl Algebra
    The Meromorphic Weyl Algebra
    The q-Meromorphic Weyl Algebra

    A Generalization of the Weyl Algebra
    Definition and Literature
    Normal Ordering in Special Ore Extensions
    Basic Observations for the Generalized Weyl Algebra
    Aspects of Normal Ordering
    Associated Stirling and Bell Numbers

    The q-Deformed Generalized Weyl Algebra
    Definition and Literature
    Basic Observations
    Binomial Formula
    Associated Stirling and Bell Numbers

    A Generalization of Touchard Polynomials
    Touchard Polynomials of Arbitrary Integer Order
    Outlook: Touchard Functions of Real Order
    Outlook: ComtetTouchard Functions
    Outlook: q-Deformed Generalized Touchard Polynomials

    Appendices

    Bibliography

    Indices

    Exercises appear at the end of each chapter.

    Biography

    Toufik Mansour is a professor at the University of Haifa. His research interests include enumerative combinatorics and discrete mathematics and its applications. He has authored or co-authored numerous papers in these areas, many of them concerning the enumeration of normal ordering. He earned a PhD in mathematics from the University of Haifa.

    Matthias Schork is a member of the IT department at Deutsche Bahn, the largest German railway company. His research interests include mathematical physics as well as discrete mathematics and its applications to physics. He has authored or coauthored many papers focusing on Stirling numbers and normal ordering and its ramifications. He earned a PhD in mathematics from the Johann Wolfgang Goethe University of Frankfurt.