Quantum differentiability on noncommutative Euclidean spaces

TL;DR

This paper investigates quantum differentiability on Moyal spaces, establishing precise conditions for the decay of singular values of the quantized differential, which is more complex than similar problems in classical and other quantum spaces.

Contribution

It provides necessary and sufficient conditions for singular value decay in quantum Euclidean spaces, advancing understanding of quantum differentiability in noncommutative geometry.

Findings

Identifies decay conditions for singular values of the quantized differential.

Shows the problem's complexity exceeds that of classical Euclidean and quantum tori cases.

Establishes a framework for analyzing quantum differentiability in noncommutative spaces.

Abstract

We study the topic of quantum differentiability on quantum Euclidean $d$-dimensional spaces (otherwise known as Moyal $d$-spaces), and we find conditions that are necessary and sufficient for the singular values of the quantised differential to have decay $O(n^{-\alpha})$ for $0 < \alpha \leq \frac{1}{d}$. This result is substantially more difficult than the analogous problems for Euclidean space and for quantum $d$-tori.