Commutation Relations, Normal Ordering, and Stirling Numbers

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 lettersU and V subject to the commutation relation UV - VU = I.

The book discusses the Stirling numbers, closely related generalizations, and their role as normal ordering coefficients in the Weyl algebra. It also considers several relatives of this algebra, all of which are special cases of the algebra in whichUV - qVU = hV^s holds true. The authors describe combinatorial aspects of these algebras and of 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 and explores the physical motivation for ordering words in the Weyl algebra arising from quantum theory, along with some applications. The text includes a comprehensive bibliography with many references to original publications.

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.