Natural Heisenberg algebras

Normal ordered product

Let be a space carrying a representation of a natural Heisenberg algebra for or , and let denote the strictly positive and negative parts of respectively. The normal ordered product is a procedure for obtaining well-defined operators from infinite expressions.¹²

Definition

In general, if for we have the formal sum of operators

where is a ^homogenous operator of degree , we define lie

where we have used an Iverson bracket. Then the normal ordered product is defined recursively for by

which induces a map . In particular

and

For

Let be the inverse degree operator on , so

and

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §3.3, p. 73–76 ↩

2. 1988. Vertex operator algebras and the Monster, §4.2, p. 89–92 ↩