1st Edition

Commutation Relations, Normal Ordering, and Stirling Numbers

ISBN 9781466579880
Published September 21, 2015 by Chapman and Hall/CRC
504 Pages 20 B/W Illustrations

USD $140.00

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Book Description

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.

Table of Contents

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
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




Exercises appear at the end of each chapter.

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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.