Analysis on q-deformed quantum spaces

TL;DR

This paper develops a comprehensive q-deformed analysis framework for quantum spaces like Manin plane, Euclidean, and Minkowski spaces, enabling the formulation of physical theories on these noncommutative geometries.

Contribution

It introduces and explains all key notions of q-deformed analysis, demonstrating their consistency and applicability to physical models on quantum spaces.

Findings

Consistent framework for q-deformed analysis established

Detailed discussion of star products and q-deformed derivatives

Foundations for formulating physical theories on quantum spaces

Abstract

A q-deformed version of classical analysis is given to quantum spaces of physical importance, i.e. Manin plane, q-deformed Euclidean space in three or four dimensions, and q-deformed Minkowski space. The subject is presented in a rather complete and selfcontained way. All relevant notions are introduced and explained in detail. The different possibilities to realize the objects of q-deformed analysis are discussed and their elementary properties are studied. In this manner attention is focused on star products, q-deformed tensor products, q-deformed translations, q-deformed partial derivatives, dual pairings, q-deformed exponentials, and q-deformed integration. The main concern of this work is to show that these objects fit together in a consistent framework, which is suitable to formulate physical theories on quantum spaces.