Operator Identities, Representations of Algebras and the Problem of Normal Ordering

TL;DR

This paper explores operator identities derived from algebra representations, demonstrating their reformulation in terms of creation and annihilation operators to simplify the normal ordering problem in second quantization.

Contribution

It introduces a new approach to rewriting operator identities from algebra representations, aiding the normal ordering process in quantum mechanics.

Findings

Operator identities linked to algebra representations are established.

Reformulation in terms of creation/annihilation operators simplifies normal ordering.

Application to quantum algebra and differential operator algebras demonstrated.

Abstract

Families of operator identities appeared as a consequence of an existence of finite-dimensional representation of (super) Lie algebras of first-order differential operators and $q$-deformed (quantum) algebras of first-order finite-difference operators are presented. It is shown that those identities can be rewritten in terms of creation/annihilation operators and it leads to a simplification of the problem of the normal ordering in the second quantization method.