Commutation Relations, Normal Ordering, and Stirling Numbers
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Commutation Relations, Normal Ordering, and Stirling Numbers

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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.show more

Product details

  • Hardback | 528 pages
  • 178 x 254 x 33.02mm | 1,133g
  • Taylor & Francis Inc
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • 20 black & white illustrations, 5 black & white tables
  • 1466579889
  • 9781466579880

About Toufik Mansour

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.show more

Table of contents

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.show more