Metric On Quantum Spaes

TL;DR

This paper introduces a quantum-deformed metric tensor in $q$-calculus, explores its implications on theory parameters, and examines examples like the Manin plane and $q$-deformed $SU(2)$, linking to Connes' framework.

Contribution

It presents the first formulation of a metric tensor in $q$-deformed differential calculus and analyzes its restrictive effects on the theory's parameters.

Findings

Restrictions on parameters due to the metric existence

Explicit analysis of Manin plane and $q$-deformed $SU(2)

Discussion of connections with Connes' approach

Abstract

We introduce the analogue of the metric tensor in case of $q$-deformed differential calculus. We analyse the consequences of the existence of such metric, showing that this enforces severe restrictions on the parameters of the theory. We discuss in detail the examples of the Manin plane and the $q$-deformation of $SU(2)$. Finally we touch the topic of relations with the Connes' approach.