Exponential Formulas, Normal Ordering and the Weyl-Heisenberg Algebra

TL;DR

This paper explores exponential formulas within the Weyl-Heisenberg algebra, deriving normal ordered forms via differential equations, with applications to noncommutative geometry, star products, and quantum algebra structures.

Contribution

It introduces new differential equations for normal ordering of exponentials in the Weyl-Heisenberg algebra and provides propositions for 2D and higher dimensions, applicable to noncommutative spaces.

Findings

Derived differential equations for normal ordered exponentials.

Presented propositions for 2D Weyl-Heisenberg algebra and generalizations.

Connected results to star products, coproducts, and the BCH formula.

Abstract

We consider a class of exponentials in the Weyl-Heisenberg algebra with exponents of type at most linear in coordinates and arbitrary functions of momenta. They are expressed in terms of normal ordering where coordinates stand to the left from momenta. Exponents appearing in normal ordered form satisfy differential equations with boundary conditions that could be solved perturbatively order by order. Two propositions are presented for the Weyl-Heisenberg algebra in 2 dimensions and their generalizations in higher dimensions. These results can be applied to arbitrary noncommutative spaces for construction of star products, coproducts of momenta and twist operators. They can also be related to the BCH formula.