ABSTRACT
This chapter continues the study of normal ordering in the Weyl algebra which we began in Chapter 5. In Section 6.1, normal ordering in the “abstract” Weyl algebra (generated by generators U and V satisfying UV − V U = h) is treated and many results are collected, some of which were already discussed in preceding chapters. Among the results we discuss are Viskov’s identity, the connection of normal ordering to rook numbers and the identity of Bender, Mead, and Pinsky. Turning to the extended Weyl algebra – allowing formal series in the generators –, several classical results (due to Crofton, McCoy, Scherk, Weyl, etc.) are presented. The normal ordered form of an arbitrary word in the generators can be described with the help of Wick’s theorem, which will be discussed from a combinatorial as well as a physical point of view in Section 6.2. A connection between normal ordering and umbral
calculus is drawn in Section 6.3, where the normal ordered form of eλ(q(aˆ †)aˆ+v(aˆ)) (where
aˆ† and aˆ are the usual creation and annihilation operators) is derived. If one normal orders words of a particular form, then connections to a variety of combinatorial problems appear, for instance, to that of counting trees with particular properties. Some of these connections are mentioned in Section 6.4. In Section 6.5, some different operator ordering schemes are presented and a few examples are treated, in particular in connection with operational relations. As far as the bosonic case is concerned, almost exclusively the single-mode case is treated since the multi-mode case does not introduce new interesting combinatorial aspects in the examples we consider. This is discussed in Section 6.6, where it is also shown that in the fermionic case the multi-mode case does introduce new aspects in comparison with the single-mode case.
