Normal ordered product

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

Definition

In general, if for $𝛼 \in 𝔥$ we have the formal sum of operators

𝛼 (𝑧) = \sum 𝑛 \in ℤ 𝛼 (𝑛) 𝑧 - 𝑛

where $𝛼 (𝑛)$ is a homogenous operator of degree $𝑛$ , we define #m/def/lie

𝛼 (𝑧) \pm = 12 𝛼 (0) [0 \in 𝑍] + \sum 𝑛 \in 𝑍 \pm 𝛼 (𝑛) 𝑧 - 𝑛 ⟹ 𝛼 (𝑧) = 𝛼 (𝑧) + + 𝛼 (𝑧) -

where we have used an Iverson bracket. Then the normal ordered product is defined recursively for ${𝛼 𝑖} 𝑘 𝑖 = 1 \subseteq 𝔥$ by

: 𝛼 1 (𝑧) : = 𝛼 1 (𝑧) : 𝛼 1 (𝑧) \dots 𝛼 𝑘 (𝑧) : = 𝛼 𝑘 (𝑧) - : 𝛼 1 (𝑧) \dots 𝛼 𝑘 - 1 (𝑧) : + : 𝛼 1 (𝑧) \dots 𝛼 𝑘 - 1 (𝑧) : 𝛼 𝑘 (𝑧) +

which induces a map $𝑆 ∙ 𝔥 \to (E n d 𝑉) {𝑧}$ . In particular

: 𝛼 1 (𝑛 1) 𝛼 2 (𝑛 2) : = {𝛼 1 (𝑛 1) 𝛼 2 (𝑛 2) 𝑛 1 \leq 𝑛 2 𝛼 2 (𝑛 2) 𝛼 1 (𝑛 1) 𝑛 2 \leq 𝑛 1

and

: 𝛼 (𝑧) 𝑘 : = 𝑘 \sum ℓ = 0 (𝑘 ℓ) (𝑎 (𝑧) -) ℓ (𝑎 (𝑧) +) 𝑘 - ℓ

For $𝐷 - 1$

Let $𝐷 - 1$ be the inverse degree operator on $𝐷 ((E n d 𝑉) {𝑧})$ , so

𝐷 - 1 𝛼 (𝑧) = 𝐷 - 1 𝛼 (𝑧) + + 𝐷 - 1 𝛼 (𝑧) - : 𝐷 - 1 𝛼 1 (𝑧) : = 𝐷 - 1 𝛼 1 (𝑧) : 𝐷 - 1 𝛼 1 (𝑧) \dots 𝐷 - 1 𝛼 𝑘 (𝑧) : = 𝐷 - 1 𝛼 𝑘 (𝑧) - : 𝐷 - 1 𝛼 1 (𝑧) \dots 𝐷 - 1 𝛼 𝑘 - 1 (𝑧) : = + : 𝐷 - 1 𝛼 1 (𝑧) \dots 𝐷 - 1 𝛼 𝑘 - 1 (𝑧) : 𝐷 - 1 𝛼 𝑘 (𝑧) +

and

: e x p 𝐷 - 1 𝛼 (𝑧) : = \sum 𝑘 \in ℕ 0 : (𝐷 - 1 𝛼 (𝑧)) 𝑘 : 𝑘!

#state/develop | #lang/en | #SemBr

1988. Vertex operator algebras and the Monster, §3.3, p. 73–76 ↩
1988. Vertex operator algebras and the Monster, §4.2, p. 89–92 ↩

Normal ordered product

Definition

For 𝐷−1

Footnotes

For $𝐷 - 1$