Commutation Relations, Normal Ordering, and Stirling Numbers: 1st Edition (Hardback) book cover

Commutation Relations, Normal Ordering, and Stirling Numbers

1st Edition

By Toufik Mansour, Matthias Schork

Chapman and Hall/CRC

504 pages | 20 B/W Illus.

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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.

About the Authors

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.

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Arithmetic
MATHEMATICS / Combinatorics
SCIENCE / Mathematical Physics