A Calculus Based on a q-deformed Heisenberg Algebra

TL;DR

This paper develops a differential calculus framework over a q-deformed Heisenberg algebra, enabling the definition of derivatives, differential forms, and an exterior calculus in a non-commutative setting.

Contribution

It introduces a novel differential calculus on a q-deformed algebra, including derivatives and forms, extending classical calculus to a non-commutative quantum algebra setting.

Findings

Constructed a derivative that preserves the coordinate algebra

Developed a generalized Leibniz rule for the algebra

Built a differential and exterior calculus framework

Abstract

We show how one can construct a differential calculus over an algebra where position variables x and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by x and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based on this derivative differential forms and an exterior differential calculus can be constructed.