ABSTRACT
This chapter focuses on the Weyl algebra and the process of normal ordering words in its generators. We consider an “abstract” version of the Weyl algebra which is characterized by two generators U and V satisfying the commutation relation UV − V U = h for some h ∈ C. More precisely, on the right-hand side of the commutation relation one has hI where I denotes the identity commuting with U and V (hence, with all words in U and V ). However, we identify cI with c, and since we are interested in the combinatorial consequences of the commutation relation this makes no difference. In Chapter 1, we considered several concrete representations of the Weyl algebra, for instance, by the operators X and D. The Weyl algebra is one of the simplest noncommutative algebras and arises from other wellknown algebras, for instance, the polynomials in two indeterminates, through various kinds of “deformations”. As such it aroused the interest of many mathematicians for different reasons and has been considered in depth. We mention some of these relations and give references to the literature, but the algebraic properties of the Weyl algebra are not in the scope of the present book. Another representation of the Weyl algebra is given by the annihilation and creation operators used in elementary quantum theory. This is the reason why physicists are also interested in the algebraic or combinatoric structure of the Weyl algebra. This combinatorial structure is transferred directly to the structure of physical expectation values.
